journal of computational analysis and applications€¦ · journal of computational analysis and...
TRANSCRIPT
Volume 21, Number 5 November 2016 ISSN:1521-1398 PRINT,1572-9206 ONLINE
Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC
803
Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (fourteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC 28601, USA. Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http://www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $700, Electronic OPEN ACCESS. Individual:Print $350. For any other part of the world add $130 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2016 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.
804
Editorial Board
Associate Editors of Journal of Computational Analysis and Applications
Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators. Ravi P. Agarwal Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 tel: 361-593-2600 [email protected] Differential Equations, Difference Equations, Inequalities George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,U.S.A Tel.901-678-3144 e-mail: [email protected] Approximation Theory, Real Analysis, Wavelets, Neural Networks, Probability, Inequalities. J. Marshall Ash Department of Mathematics De Paul University 2219 North Kenmore Ave. Chicago, IL 60614-3504 773-325-4216 e-mail: [email protected] Real and Harmonic Analysis Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara, Turkey, [email protected]
Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics Carlo Bardaro Dipartimento di Matematica e Informatica Universita di Perugia Via Vanvitelli 1 06123 Perugia, ITALY TEL+390755853822 +390755855034 FAX+390755855024 E-mail [email protected] Web site: http://www.unipg.it/~bardaro/ Functional Analysis and Approximation Theory, Signal Analysis, Measure Theory, Real Analysis.
Martin Bohner Department of Mathematics and Statistics, Missouri S&T Rolla, MO 65409-0020, USA [email protected] web.mst.edu/~bohner Difference equations, differential equations, dynamic equations on time scale, applications in economics, finance, biology. Jerry L. Bona Department of Mathematics The University of Illinois at Chicago 851 S. Morgan St. CS 249 Chicago, IL 60601 e-mail:[email protected] Partial Differential Equations, Fluid Dynamics Luis A. Caffarelli Department of Mathematics The University of Texas at Austin Austin, Texas 78712-1082 512-471-3160 e-mail: [email protected] Partial Differential Equations George Cybenko Thayer School of Engineering
805
Dartmouth College 8000 Cummings Hall, Hanover, NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail:[email protected] Approximation Theory and Neural Networks Sever S. Dragomir School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, MC 8001, AUSTRALIA Tel. +61 3 9688 4437 Fax +61 3 9688 4050 [email protected] Inequalities, Functional Analysis, Numerical Analysis, Approximations, Information Theory, Stochastics. Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR-06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications Saber N. Elaydi Department Of Mathematics Trinity University 715 Stadium Dr. San Antonio, TX 78212-7200 210-736-8246 e-mail: [email protected] Ordinary Differential Equations, Difference Equations Christodoulos A. Floudas Department of Chemical Engineering Princeton University Princeton,NJ 08544-5263 609-258-4595(x4619 assistant) e-mail: [email protected] Optimization Theory&Applications, Global Optimization J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 [email protected]
Partial Differential Equations, Semigroups of Operators H. H. Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany 011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design John R. Graef Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37304 USA [email protected] Ordinary and functional differential equations, difference equations, impulsive systems, differential inclusions, dynamic equations on time scales, control theory and their applications Weimin Han Department of Mathematics University of Iowa Iowa City, IA 52242-1419 319-335-0770 e-mail: [email protected] Numerical analysis, Finite element method, Numerical PDE, Variational inequalities, Computational mechanics Tian-Xiao He Department of Mathematics and Computer Science P.O. Box 2900, Illinois Wesleyan University Bloomington, IL 61702-2900, USA Tel (309)556-3089 Fax (309)556-3864 [email protected] Approximations, Wavelet, Integration Theory, Numerical Analysis, Analytic Combinatorics Margareta Heilmann Faculty of Mathematics and Natural Sciences, University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected]
806
Approximation Theory (Positive Linear Operators) Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected] Computational Mathematics Jong Kyu Kim Department of Mathematics Kyungnam University Masan Kyungnam,631-701,Korea Tel 82-(55)-249-2211 Fax 82-(55)-243-8609 [email protected] Nonlinear Functional Analysis, Variational Inequalities, Nonlinear Ergodic Theory, ODE, PDE, Functional Equations. Robert Kozma Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA [email protected] Neural Networks, Reproducing Kernel Hilbert Spaces, Neural Percolation Theory Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston, RI 02881,USA [email protected] Differential and Difference Equations Irena Lasiecka Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional Analysis, [email protected]
Burkhard Lenze Fachbereich Informatik Fachhochschule Dortmund University of Applied Sciences Postfach 105018 D-44047 Dortmund, Germany e-mail: [email protected] Real Networks, Fourier Analysis, Approximation Theory
Hrushikesh N. Mhaskar Department Of Mathematics California State University Los Angeles, CA 90032 626-914-7002 e-mail: [email protected] Orthogonal Polynomials, Approximation Theory, Splines, Wavelets, Neural Networks Ram N. Mohapatra Department of Mathematics University of Central Florida Orlando, FL 32816-1364 tel.407-823-5080 [email protected] Real and Complex Analysis, Approximation Th., Fourier Analysis, Fuzzy Sets and Systems Gaston M. N'Guerekata Department of Mathematics Morgan State University Baltimore, MD 21251, USA tel: 1-443-885-4373 Fax 1-443-885-8216 Gaston.N'[email protected] [email protected] Nonlinear Evolution Equations, Abstract Harmonic Analysis, Fractional Differential Equations, Almost Periodicity & Almost Automorphy M.Zuhair Nashed Department Of Mathematics University of Central Florida PO Box 161364 Orlando, FL 32816-1364 e-mail: [email protected] Inverse and Ill-Posed problems, Numerical Functional Analysis, Integral Equations, Optimization, Signal Analysis Mubenga N. Nkashama Department OF Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170 205-934-2154 e-mail: [email protected] Ordinary Differential Equations, Partial Differential Equations Vassilis Papanicolaou Department of Mathematics
807
National Technical University of Athens Zografou campus, 157 80 Athens, Greece tel:: +30(210) 772 1722 Fax +30(210) 772 1775 [email protected] Partial Differential Equations, Probability Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S. Korea, [email protected] Functional Equations Svetlozar (Zari) Rachev, Professor of Finance, College of Business, and Director of Quantitative Finance Program, Department of Applied Mathematics & Statistics Stonybrook University 312 Harriman Hall, Stony Brook, NY 11794-3775 tel: +1-631-632-1998, [email protected] Alexander G. Ramm Mathematics Department Kansas State University Manhattan, KS 66506-2602 e-mail: [email protected] Inverse and Ill-posed Problems, Scattering Theory, Operator Theory, Theoretical Numerical Analysis, Wave Propagation, Signal Processing and Tomography Tomasz Rychlik Polish Academy of Sciences Instytut Matematyczny PAN 00-956 Warszawa, skr. poczt. 21 ul. Śniadeckich 8 Poland [email protected] Mathematical Statistics, Probabilistic Inequalities Boris Shekhtman Department of Mathematics University of South Florida Tampa, FL 33620, USA Tel 813-974-9710 [email protected]
Approximation Theory, Banach spaces, Classical Analysis T. E. Simos Department of Computer Science and Technology Faculty of Sciences and Technology University of Peloponnese GR-221 00 Tripolis, Greece Postal Address: 26 Menelaou St. Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis H. M. Srivastava Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3R4 Canada tel.250-472-5313; office,250-477-6960 home, fax 250-721-8962 [email protected] Real and Complex Analysis, Fractional Calculus and Appl., Integral Equations and Transforms, Higher Transcendental Functions and Appl.,q-Series and q-Polynomials, Analytic Number Th. I. P. Stavroulakis Department of Mathematics University of Ioannina 451-10 Ioannina, Greece [email protected] Differential Equations Phone +3-065-109-8283 Manfred Tasche Department of Mathematics University of Rostock D-18051 Rostock, Germany [email protected] rostock.de Numerical Fourier Analysis, Fourier Analysis, Harmonic Analysis, Signal Analysis, Spectral Methods, Wavelets, Splines, Approximation Theory Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional
808
Analysis, [email protected]
Juan J. Trujillo University of La Laguna Departamento de Analisis Matematico C/Astr.Fco.Sanchez s/n 38271. LaLaguna. Tenerife. SPAIN Tel/Fax 34-922-318209 [email protected] Fractional: Differential Equations-Operators-Fourier Transforms, Special functions, Approximations, and Applications Ram Verma International Publications 1200 Dallas Drive #824 Denton, TX 76205, USA [email protected] Applied Nonlinear Analysis, Numerical Analysis, Variational Inequalities, Optimization Theory, Computational Mathematics, Operator Theory
Xiang Ming Yu Department of Mathematical Sciences Southwest Missouri State University Springfield, MO 65804-0094 417-836-5931 [email protected] Classical Approximation Theory, Wavelets Lotfi A. Zadeh Professor in the Graduate School and Director, Computer Initiative, Soft Computing (BISC) Computer Science Division University of California at Berkeley Berkeley, CA 94720 Office: 510-642-4959 Sec: 510-642-8271 Home: 510-526-2569 FAX: 510-642-1712 [email protected] Fuzzyness, Artificial Intelligence, Natural language processing, Fuzzy logic Richard A. Zalik Department of Mathematics Auburn University Auburn University, AL 36849-5310
USA. Tel 334-844-6557 office 678-642-8703 home Fax 334-844-6555 [email protected] Approximation Theory, Chebychev Systems, Wavelet Theory Ahmed I. Zayed Department of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Chicago, IL 60614-3250 773-325-7808 e-mail: [email protected] Shannon sampling theory, Harmonic analysis and wavelets, Special functions and orthogonal polynomials, Integral transforms Ding-Xuan Zhou Department Of Mathematics City University of Hong Kong 83 Tat Chee Avenue Kowloon, Hong Kong 852-2788 9708,Fax:852-2788 8561 e-mail: [email protected] Approximation Theory, Spline functions, Wavelets Xin-long Zhou Fachbereich Mathematik, Fachgebiet Informatik Gerhard-Mercator-Universitat Duisburg Lotharstr.65, D-47048 Duisburg, Germany e-mail:[email protected] duisburg.de Fourier Analysis, Computer-Aided Geometric Design, Computational Complexity, Multivariate Approximation Theory, Approximation and Interpolation Theory
809
Instructions to Contributors
Journal of Computational Analysis and Applications An international publication of Eudoxus Press, LLC, of TN.
Editor in Chief: George Anastassiou
Department of Mathematical Sciences University of Memphis
Memphis, TN 38152-3240, U.S.A.
1. Manuscripts files in Latex and PDF and in English, should be submitted via email to the Editor-in-Chief: Prof.George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis,TN 38152, USA. Tel. 901.678.3144 e-mail: [email protected] Authors may want to recommend an associate editor the most related to the submission to possibly handle it. Also authors may want to submit a list of six possible referees, to be used in case we cannot find related referees by ourselves. 2. Manuscripts should be typed using any of TEX,LaTEX,AMS-TEX,or AMS-LaTEX and according to EUDOXUS PRESS, LLC. LATEX STYLE FILE. (Click HERE to save a copy of the style file.)They should be carefully prepared in all respects. Submitted articles should be brightly typed (not dot-matrix), double spaced, in ten point type size and in 8(1/2)x11 inch area per page. Manuscripts should have generous margins on all sides and should not exceed 24 pages. 3. Submission is a representation that the manuscript has not been published previously in this or any other similar form and is not currently under consideration for publication elsewhere. A statement transferring from the authors(or their employers,if they hold the copyright) to Eudoxus Press, LLC, will be required before the manuscript can be accepted for publication.The Editor-in-Chief will supply the necessary forms for this transfer.Such a written transfer of copyright,which previously was assumed to be implicit in the act of submitting a manuscript,is necessary under the U.S.Copyright Law in order for the publisher to carry through the dissemination of research results and reviews as widely and effective as possible.
810
4. The paper starts with the title of the article, author's name(s) (no titles or degrees), author's affiliation(s) and e-mail addresses. The affiliation should comprise the department, institution (usually university or company), city, state (and/or nation) and mail code. The following items, 5 and 6, should be on page no. 1 of the paper. 5. An abstract is to be provided, preferably no longer than 150 words. 6. A list of 5 key words is to be provided directly below the abstract. Key words should express the precise content of the manuscript, as they are used for indexing purposes. The main body of the paper should begin on page no. 1, if possible. 7. All sections should be numbered with Arabic numerals (such as: 1. INTRODUCTION) . Subsections should be identified with section and subsection numbers (such as 6.1. Second-Value Subheading). If applicable, an independent single-number system (one for each category) should be used to label all theorems, lemmas, propositions, corollaries, definitions, remarks, examples, etc. The label (such as Lemma 7) should be typed with paragraph indentation, followed by a period and the lemma itself. 8. Mathematical notation must be typeset. Equations should be numbered consecutively with Arabic numerals in parentheses placed flush right, and should be thusly referred to in the text [such as Eqs.(2) and (5)]. The running title must be placed at the top of even numbered pages and the first author's name, et al., must be placed at the top of the odd numbed pages. 9. Illustrations (photographs, drawings, diagrams, and charts) are to be numbered in one consecutive series of Arabic numerals. The captions for illustrations should be typed double space. All illustrations, charts, tables, etc., must be embedded in the body of the manuscript in proper, final, print position. In particular, manuscript, source, and PDF file version must be at camera ready stage for publication or they cannot be considered. Tables are to be numbered (with Roman numerals) and referred to by number in the text. Center the title above the table, and type explanatory footnotes (indicated by superscript lowercase letters) below the table. 10. List references alphabetically at the end of the paper and number them consecutively. Each must be cited in the text by the appropriate Arabic numeral in square brackets on the baseline. References should include (in the following order): initials of first and middle name, last name of author(s) title of article,
811
name of publication, volume number, inclusive pages, and year of publication. Authors should follow these examples: Journal Article 1. H.H.Gonska,Degree of simultaneous approximation of bivariate functions by Gordon operators, (journal name in italics) J. Approx. Theory, 62,170-191(1990). Book 2. G.G.Lorentz, (title of book in italics) Bernstein Polynomials (2nd ed.), Chelsea,New York,1986. Contribution to a Book 3. M.K.Khan, Approximation properties of beta operators,in(title of book in italics) Progress in Approximation Theory (P.Nevai and A.Pinkus,eds.), Academic Press, New York,1991,pp.483-495. 11. All acknowledgements (including those for a grant and financial support) should occur in one paragraph that directly precedes the References section. 12. Footnotes should be avoided. When their use is absolutely necessary, footnotes should be numbered consecutively using Arabic numerals and should be typed at the bottom of the page to which they refer. Place a line above the footnote, so that it is set off from the text. Use the appropriate superscript numeral for citation in the text. 13. After each revision is made please again submit via email Latex and PDF files of the revised manuscript, including the final one. 14. Effective 1 Nov. 2009 for current journal page charges, contact the Editor in Chief. Upon acceptance of the paper an invoice will be sent to the contact author. The fee payment will be due one month from the invoice date. The article will proceed to publication only after the fee is paid. The charges are to be sent, by money order or certified check, in US dollars, payable to Eudoxus Press, LLC, to the address shown on the Eudoxus homepage. No galleys will be sent and the contact author will receive one (1) electronic copy of the journal issue in which the article appears. 15. This journal will consider for publication only papers that contain proofs for their listed results.
812
Mixed problems of fractional coupled systems of
Riemann-Liouville differential equations and Hadamard integral
conditions
S.K. Ntouyas1,2 Jessada Tariboon3 and Phollakrit Thiramanus3
1Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greecee-mail: [email protected] Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics,Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia3Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science,King Mongkut’s University of Technology North Bangkok, Bangkok, 10800 Thailande-mail: [email protected], [email protected]
Abstract
IIn this paper we study existence and uniqueness of solutions for mixed problems consisting non-local Hadamard fractional integrals for coupled systems of Riemann-Liouville fractional differentialequations. The existence and uniqueness of solutions is established by using the Banach’s contrac-tion principle, while the existence of solutions is derived by applying Leray-Schauder’s alternative.Examples illustrating our results are also presented.
Key words and phrases: Riemann-Liouville fractional derivative; Hadamard fractional integral;coupled system; existence; uniqueness; fixed point theorems.AMS (MOS) Subject Classifications: 34A08; 34A12; 34B15.
1 Introduction
The aim of this paper is to investigate the existence and uniqueness of solutions for nonlocal Hadamardfractional integrals for a coupled system of Riemann-Liouville fractional differential equations of theform:
RLDpx(t) = f(t, x(t), y(t)), t ∈ [0, T ], 1 < p ≤ 2,
RLDqy(t) = g(t, x(t), y(t)), t ∈ [0, T ], 1 < q ≤ 2,
x(0) = 0,
m1∑i=1
µiHIαix(ηi) =n1∑
j=1
δjHIβj y(ξj) + λ1,
y(0) = 0,
m2∑k=1
τkHIσkx(γk) =n2∑l=1
ωlHIνly(θl) + λ2,
(1)
where RLDq, RLDp are the standard Riemann-Liouville fractional derivative of orders q, p, two contin-uous functions f, g : [0, T ]×R2 → R, HIαi , HIβj , HIσk and HIνl are the Hadamard fractional integralof orders αi, βj , σk, νl > 0, λ1, λ2 ∈ R are given constants, ηi, ξj , γk, θl ∈ (0, T ), and µi, δj , τk, ωl ∈ R, form1,m2, n1, n2 ∈ N, i = 1, 2, . . . ,m1, j = 1, 2, . . . , n1, k = 1, 2, . . . ,m2, l = 1, 2, . . . , n2 are real constantssuch that (
m1∑i=1
µiηp−1i
(p − 1)αi
)(n2∑l=1
ωlθq−1l
(q − 1)νl
)6=
n1∑j=1
δjξq−1j
(q − 1)βj
(m2∑k=1
τkγp−1k
(p − 1)σk
).
Fractional calculus has a long history with more than three hundred years. Up to now, it has beenproved that fractional calculus is very useful. Many mathematical models of real problems arising
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
813 Ntouyas et al 813-828
S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS
in various fields of science and engineering were established with the help of fractional calculus, suchas viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, and electromagneticwaves. For examples and recent development of the topic, see ([1, 2, 3, 4, 5, 6, 7, 14, 16, 17, 18, 19,20, 21]). However, it has been observed that most of the work on the topic involves either Riemann-Liouville or Caputo type fractional derivative. Besides these derivatives, Hadamard fractional derivativeis another kind of fractional derivatives that was introduced by Hadamard in 1892 [12]. This fractionalderivative differs from the other ones in the sense that the kernel of the integral (in the definition ofHadamard derivative) contains logarithmic function of arbitrary exponent. For background material ofHadamard fractional derivative and integral, we refer to the papers [8, 9, 10, 13, 14, 15].
The paper is organized as follows: In Section 2 we will present some useful preliminaries andlemmas. The main results are given in Section 3, where existence and uniqueness results are obtained byusing Banach’s contraction principle and Leray-Schauder’s alternative. Finally the uncoupled integralboundary conditions case is studied in Section 4. Examples illustrating our results are also presented.
2 Preliminaries
In this section, we introduce some notations and definitions of fractional calculus and present preliminaryresults needed in our proofs later [18, 14].
Definition 2.1 The Riemann-Liouville fractional derivative of order q > 0 of a continuous functionf : (0,∞) → R is defined by
RLDqf(t) =1
Γ(n − q)
(d
dt
)n ∫ t
0
(t − s)n−q−1f(s)ds, n − 1 < q < n,
where n = [q]+1, [q] denotes the integer part of a real number q, provided the right-hand side is point-wisedefined on (0,∞), where Γ is the gamma function defined by Γ(q) =
∫ ∞0
e−ssq−1ds.
Definition 2.2 The Riemann-Liouville fractional integral of order q > 0 of a continuous functionf : (0,∞) → R is defined by
RLIqf(t) =1
Γ(q)
∫ t
0
(t − s)q−1f(s)ds,
provided the right-hand side is point-wise defined on (0,∞).
Definition 2.3 The Hadamard derivative of fractional order q for a function f : (0,∞) → R is definedas
HDqf(t) =1
Γ(n − q)
(td
dt
)n ∫ t
0
(log
t
s
)n−q−1f(s)
sds, n − 1 < q < n, n = [q] + 1,
where log(·) = loge(·).
Definition 2.4 The Hadamard fractional integral of order q ∈ R+ of a function f(t), for all t > 0, isdefined as
HIqf(t) =1
Γ(q)
∫ t
0
(log
t
s
)q−1
f(s)ds
s,
provided the integral exists.
Lemma 2.5 ([14], page 113) Let q > 0 and β > 0. Then the following formulas
HIqtβ = β−qtβ and HDqtβ = βqtβ
hold.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
814 Ntouyas et al 813-828
MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS
Lemma 2.6 Let q > 0 and x ∈ C(0, T )∩L(0, T ). Then the fractional differential equation RLDqx(t) =0 has a unique solution x(t) = c1t
q−1 + c2tq−2 + . . . + cntq−n, where ci ∈ R, i = 1, 2, . . . , n, and
n − 1 < q < n.
Lemma 2.7 Let q > 0. Then for x ∈ C(0, T ) ∩ L(0, T ) it holds
RLIqRLDqx(t) = x(t) + c1t
q−1 + c2tq−2 + . . . + cntq−n,
where ci ∈ R, i = 1, 2, . . . , n, and n − 1 < q < n.
Lemma 2.8 Given φ, ψ ∈ C([0, T ], R), the unique solution of the problem
RLDpx(t) = φ(t), t ∈ [0, T ], 1 < p ≤ 2,
RLDqy(t) = ψ(t), t ∈ [0, T ], 1 < q ≤ 2,
x(0) = 0,
m1∑i=1
µiHIαix(ηi) =n1∑
j=1
δjHIβj y(ξj) + λ1,
y(0) = 0,
m2∑k=1
τkHIσkx(γk) =n2∑l=1
ωlHIνly(θl) + λ2,
(2)
is
x(t) = RLIpφ(t) +tp−1
Ω
n2∑l=1
ωlθq−1l
(q − 1)νl
(n1∑
j=1
δjHIβjRLIqψ(ξj) −
m1∑i=1
µiHIαiRLIpφ(ηi) + λ1
)
−n1∑
j=1
δjξq−1j
(q − 1)βj
(n2∑l=1
ωlHIνlRLIqψ(θl) −
m2∑k=1
τkHIσkRLIpφ(γk) + λ2
),
(3)
and
y(t) = RLIqψ(t) +tq−1
Ω
m2∑k=1
τkγp−1k
(p − 1)σk
(n1∑
j=1
δjHIβjRLIqψ(ξj) −
m1∑i=1
µiHIαiRLIpφ(ηi) + λ1
)
−m1∑i=1
µiηp−1i
(p − 1)αi
(n2∑l=1
ωlHIνlRLIqψ(θl) −
m2∑k=1
τkHIσkRLIpφ(γk) + λ2
),
(4)
where
Ω =m1∑i=1
µiηp−1i
(p − 1)αi
n2∑l=1
ωlθq−1l
(q − 1)νl−
n1∑j=1
δjξq−1j
(q − 1)βj
m2∑k=1
τkγp−1k
(p − 1)σk6= 0. (5)
Proof. Using Lemmas 2.6-2.7, the equations in (2) can be expressed as equivalent integral equations
x(t) = RLIpφ(t) + c1tp−1 + c2t
p−2, (6)
y(t) = RLIqψ(t) + d1tq−1 + d2t
q−2, (7)
for c1, c2, d1, d2 ∈ R. The conditions x(0) = 0, y(0) = 0 imply that c2 = 0, d2 = 0. Taking the Hadamardfractional integral of order αi > 0, σk > 0 for (6) and βj > 0, νl > 0 for (7) and using the property ofthe Hadamard fractional integral given in Lemma 2.5 we get the system
m1∑i=1
µiHIαiRLIpφ(ηi) + c1
m1∑i=1
µiηp−1i
(p − 1)αi=
n1∑j=1
δjHIβjRLIqψ(ξj) + d1
n1∑j=1
δjξq−1j
(q − 1)βj+ λ1,
m2∑k=1
τkHIσkRLIpφ(γk) + c1
m2∑k=1
τkγp−1k
(p − 1)σk=
n2∑l=1
ωlHIνlRLIqψ(θl) + d1
n2∑l=1
ωlθq−1l
(q − 1)νl+ λ2,
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
815 Ntouyas et al 813-828
S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS
from which we have
c1 =1Ω
n2∑l=1
ωlθq−1l
(q − 1)νl
(n1∑
j=1
δjHIβjRLIqψ(ξj) −
m1∑i=1
µiHIαiRLIpφ(ηi) + λ1
)
−n1∑
j=1
δjξq−1j
(q − 1)βj
(n2∑l=1
ωlHIνlRLIqψ(θl) −
m2∑k=1
τkHIσkRLIpφ(γk) + λ2
)
and
d1 =1Ω
m2∑k=1
τkγp−1k
(p − 1)σk
(n1∑
j=1
δjHIβjRLIqψ(ξj) −
m1∑i=1
µiHIαiRLIpφ(ηi) + λ1
)
−m1∑i=1
µiηp−1i
(p − 1)αi
(n2∑l=1
ωlHIνlRLIqψ(θl) −
m2∑k=1
τkHIσkRLIpφ(γk) + λ2
).
Substituting the values of c1, c2, d1 and d2 in (6) and (7), we obtain the solutions (3) and (4). ¤
3 Main Results
Throughout this paper, for convenience, we use the following expressions
RLIwh(s, x(s), y(s))(v) =1
Γ(w)
∫ v
0
(v − s)w−1h(s, x(s), y(s))ds,
and
HIuRLIwh(s, x(s), y(s))(v) =
1Γ(u)Γ(w)
∫ v
0
∫ t
0
(log
v
t
)u−1
(t − s)w−1 h(s, x(s), y(s))t
dsdt,
where u ∈ ρi, γj, v ∈ t, T, ηi, θj, w = p, q and h = f, g, i = 1, 2, . . . , n, j = 1, 2, . . . ,m.Let C = C([0, T ], R) denotes the Banach space of all continuous functions from [0, T ] to R. Let us
introduce the space X = x(t)|x(t) ∈ C([0, T ]) endowed with the norm ‖x‖ = max|x(t)|, t ∈ [0, T ].Obviously (X, ‖ · ‖) is a Banach space. Also let Y = y(t)|y(t) ∈ C([0, T ]) be endowed with the norm‖y‖ = max|y(t)|, t ∈ [0, T ]. Obviously the product space (X × Y, ‖(x, y)‖) is a Banach space withnorm ‖(x, y)‖ = ‖x‖ + ‖y‖.
In view of Lemma 2.8, we define an operator T : X × Y → X × Y by T (x, y)(t) =(
T1(x, y)(t)T2(x, y)(t)
),
where
T1(x, y)(t) = RLIpf(s, x(s), y(s))(t) +tp−1
Ω
n2∑l=1
ωlθq−1l
(q − 1)νl
(n1∑
j=1
δjHIβjRLIqg(s, x(s), y(s))(ξj)
−m1∑i=1
µiHIαiRLIpf(s, x(s), y(s))(ηi) + λ1
)−
n1∑j=1
δjξq−1j
(q − 1)βj
(n2∑l=1
ωlHIνlRLIqg(s, x(s), y(s))(θl)
−m2∑k=1
τkHIσkRLIpf(s, x(s), y(s))(γk) + λ2
),
and
T2(x, y)(t) = RLIqg(s, x(s), y(s))(t) +tq−1
Ω
m2∑k=1
τkγp−1k
(p − 1)σk
(n1∑
j=1
δjHIβjRLIqg(s, x(s), y(s))(ξj)
−m1∑i=1
µiHIαiRLIpf(s, x(s), y(s))(ηi) + λ1
)−
m1∑i=1
µiηp−1i
(p − 1)αi
(n2∑l=1
ωlHIνlRLIqg(s, x(s), y(s))(θl)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
816 Ntouyas et al 813-828
MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS
−m2∑k=1
τkHIσkRLIpf(s, x(s), y(s))(γk) + λ2
).
Let us introduce the following assumptions which are used hereafter.
(H1) Assume that f, g : [0, T ] × R2 → R are continuous functions and there exist constants mi, ni, i =1, 2 such that for all t ∈ [0, T ] and ui, vi ∈ R, i = 1, 2,
|f(t, u1, u2) − f(t, v1, v2)| ≤ K1|u1 − v1| + K2|u2 − v2|
and|g(t, u1, u2) − g(t, v1, v2)| ≤ L1|u1 − v1| + L2|u2 − v2|.
(H2) Assume that there exist real constants ki, li ≥ 0 (i = 1, 2) and k0 > 0, l0 > 0 such that ∀xi ∈R, (i = 1, 2) we have
|f(t, x1, x2)| ≤ k0 + k1|x1| + k2|x2|, |g(t, x1, x2)| ≤ l0 + l1|x1| + l2|x2|.
For the sake of convenience, we set
M1 =1
Γ(p + 1)
(T p +
T p−1
|Ω|
n2∑l=1
|ωl|θq−1l
(q − 1)νl
m1∑i=1
|µi|ηpi
pαi+
T p−1
|Ω|
n1∑j=1
|δj |ξq−1j
(q − 1)βj
m2∑k=1
|τk|γpk
pσk
), (8)
M2 =T p−1
|Ω|Γ(q + 1)
(n2∑l=1
|ωl|θq−1l
(q − 1)νl
n1∑j=1
|δj |ξqj
qβj+
n1∑j=1
|δj |ξq−1j
(q − 1)βj
n2∑l=1
|ωl|θql
qνl
), (9)
M3 =T p−1
|Ω|
(|λ1|
n2∑l=1
|ωl|θq−1l
(q − 1)νl+ |λ2|
n1∑j=1
|δj |ξq−1j
(q − 1)βj
), (10)
M4 =1
Γ(q + 1)
(T q +
T q−1
|Ω|
m2∑k=1
|τk|γp−1k
(p − 1)σk
n1∑j=1
|δj |ξqj
qβj+
T q−1
|Ω|
m1∑i=1
|µi|ηp−1i
(p − 1)αi
n2∑l=1
|ωl|θql
qνl
), (11)
M5 =T q−1
|Ω|Γ(p + 1)
(m2∑k=1
|τk|γp−1k
(p − 1)σk
m1∑i=1
|µi|ηpi
pαi+
m1∑i=1
|µi|ηp−1i
(p − 1)αi
m2∑k=1
|τk|γpk
pσk
), (12)
M6 =T q−1
|Ω|
(|λ1|
m2∑k=1
|τk|γp−1k
(p − 1)σk+ |λ2|
m1∑i=1
|µi|ηp−1i
(p − 1)αi
), (13)
and
M0 = min1 − (M1 + M5)k1 − (M2 + M4)l1, 1 − (M1 + M5)k2 − (M2 + M4)l2, (14)
ki, li ≥ 0 (i = 1, 2).
The first result is concerned with the existence and uniqueness of solutions for the problem (1) andis based on Banach’s contraction mapping principle.
Theorem 3.1 Assume that (H1) holds. In addition, suppose that
(M1 + M5)(K1 + K2) + (M2 + M4)(L1 + L2) < 1,
where Mi, i = 1, 2, 4, 5 are given by (3.1)-(3.2) and (3.4)-(3.5). Then the boundary value problem (1)has a unique solution.
Proof. Define supt∈[0,T ] f(t, 0, 0) = N1 < ∞ and supt∈[0,T ] g(t, 0, 0) = N2 < ∞ such that
r ≥ max
M1N1 + M2N2 + M3
1 − (M1K1 + M2L1 + M1K2 + M2L2),
M4N2 + M5N1 + M6
1 − (M4L1 + M5K1 + M4L2 + M5K2)
,
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
817 Ntouyas et al 813-828
S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS
where M3 and M6 are defined by (3.3) and (3.6), respectively.
We show that T Br ⊂ Br, where Br = (x, y) ∈ X × Y : ‖(x, y)‖ ≤ r.
For (x, y) ∈ Br, we have
|T1(x, y)(t)| = maxt∈[0,T ]
RLIpf(s, x(s), y(s))(t) +
tp−1
Ω
[n2∑l=1
ωlθq−1l
(q − 1)νl
×
(n1∑
j=1
δjHIβjRLIqg(s, x(s), y(s))(ξj) −
m1∑i=1
µiHIαiRLIpf(s, x(s), y(s))(ηi) + λ1
)
−n1∑
j=1
δjξq−1j
(q − 1)βj
(n2∑l=1
ωlHIνlRLIqg(s, x(s), y(s))(θl)
−m2∑k=1
τkHIσkRLIpf(s, x(s), y(s))(γk) + λ2
)]
≤ RLIp(|f(s, x(s), y(s)) − f(s, 0, 0)| + |f(s, 0, 0)|)(T ) +T p−1
|Ω|
[n2∑l=1
|ωl|θq−1l
(q − 1)νl
×
(n1∑
j=1
|δj |HIβjRLIq(|g(s, x(s), y(s)) − g(s, 0, 0)| + |g(s, 0, 0)|)(ξj)
+m1∑i=1
|µi|HIαiRLIp(|f(s, x(s), y(s)) − f(s, 0, 0)| + |f(s, 0, 0)|)(ηi) + |λ1|
)
+n1∑
j=1
|δj |ξq−1j
(q − 1)βj
(n2∑l=1
|ωl|HIνlRLIq(|g(s, x(s), y(s)) − g(s, 0, 0)| + |g(s, 0, 0)|)(θl)
+m2∑k=1
|τk|HIσkRLIp(|f(s, x(s), y(s)) − f(s, 0, 0)| + |f(s, 0, 0)|)(γk) + |λ2|
)]
≤ RLIp(K1‖x‖ + K2‖y‖ + N1)(T ) +T p−1
|Ω|
[n2∑l=1
|ωl|θq−1l
(q − 1)νl
×
(n1∑
j=1
|δj |HIβjRLIq(L1‖x‖ + L2‖y‖ + N2)(ξj)
+m1∑i=1
|µi|HIαiRLIp(K1‖x‖ + K2‖y‖ + N1)(ηi) + |λ1|
)
+n1∑
j=1
|δj |ξq−1j
(q − 1)βj
(n2∑l=1
|ωl|HIνlRLIq(L1‖x‖ + L2‖y‖ + N2)(θl)
+m2∑k=1
|τk|HIσkRLIp(K1‖x‖ + K2‖y‖ + N1)(γk) + |λ2|
)]
= (K1‖x‖ + K2‖y‖ + N1)
RLIp(1)(T ) +
T p−1
|Ω|
n2∑l=1
|ωl|θq−1l
(q − 1)νl
m1∑i=1
|µi|HIαiRLIp(1)(ηi)
+T p−1
|Ω|
n1∑j=1
|δj |ξq−1j
(q − 1)βj
m2∑k=1
|τk|HIσkRLIp(1)(γk)
+ (L1‖x‖ + L2‖y‖ + N2)
T p−1
|Ω|
×n2∑l=1
|ωl|θq−1l
(q − 1)νl
n1∑j=1
|δj |HIβjRLIq(1)(ξj) +
T p−1
|Ω|
n1∑j=1
|δj |ξq−1j
(q − 1)βj
n2∑l=1
|ωl|HIνlRLIq(1)(θl)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
818 Ntouyas et al 813-828
MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS
+|λ1|T p−1
|Ω|
n2∑l=1
|ωl|θq−1l
(q − 1)νl+ |λ2|
T p−1
|Ω|
n1∑j=1
|δj |ξq−1j
(q − 1)βj
= (K1‖x‖ + K2‖y‖ + N1)
T p
Γ(p + 1)+
T p−1
|Ω|Γ(p + 1)
n2∑l=1
|ωl|θq−1l
(q − 1)νl
m1∑i=1
|µi|ηpi
pαi
+T p−1
|Ω|Γ(p + 1)
n1∑j=1
|δj |ξq−1j
(q − 1)βj
m2∑k=1
|τk|γpk
pσk
+ (L1‖x‖ + L2‖y‖ + N2)
T p−1
|Ω|Γ(q + 1)
×n2∑l=1
|ωl|θq−1l
(q − 1)νl
n1∑j=1
|δj |ξqj
qβj+
T p−1
|Ω|Γ(q + 1)
n1∑j=1
|δj |ξq−1j
(q − 1)βj
n2∑l=1
|ωl|θql
qνl
+|λ1|T p−1
|Ω|
n2∑l=1
|ωl|θq−1l
(q − 1)νl+ |λ2|
T p−1
|Ω|
n1∑j=1
|δj |ξq−1j
(q − 1)βj
= (K1‖x‖ + K2‖y‖ + N1)M1 + (L1‖x‖ + L2‖y‖ + N2)M2 + M3
= (M1K1 + M2L1)‖x‖ + (M1K2 + M2L2)‖y‖ + M1N1 + M2N2 + M3
≤ (M1K1 + M2L1 + M1K2 + M2L2)r + M1N1 + M2N2 + M3 ≤ r.
In the same way, we can obtain that
|T2(x, y)(t)| ≤ (L1‖x‖ + L2‖y‖ + N2)
T q
Γ(q + 1)+
T q−1
|Ω|Γ(q + 1)
m2∑k=1
|τk|γp−1k
(p − 1)σk
n1∑j=1
|δj |ξqj
qβj
+T q−1
|Ω|Γ(q + 1)
m1∑i=1
|µi|ηp−1i
(p − 1)αi
n2∑l=1
|ωl|θql
qνl
+ (K1‖x‖ + K2‖y‖ + N1)
T q−1
|Ω|Γ(p + 1)
×m2∑k=1
|τk|γp−1k
(p − 1)σk
m1∑i=1
|µi|ηpi
pαi+
T q−1
|Ω|Γ(p + 1)
m1∑i=1
|µi|ηp−1i
(p − 1)αi
m2∑k=1
|τk|γpk
pσk
+|λ1|T q−1
|Ω|
m2∑k=1
|τk|γp−1k
(p − 1)σk+ |λ2|
T q−1
|Ω|
m1∑i=1
|µi|ηp−1i
(p − 1)αi
= (L1‖x‖ + L2‖y‖ + N2)M4 + (K1‖x‖ + K2‖y‖ + N1)M5 + M6
= (M4L1 + M5K1)‖x‖ + (M4L2 + M5K2)‖y‖ + M4N2 + M5N1 + M6
≤ (M4L1 + M5K1 + M4L2 + M5K2)r + M4N2 + M5N1 + M6 ≤ r.
Consequently, ‖T (x, y)(t)‖ ≤ r.
Now for (x2, y2), (x1, y1) ∈ X × Y, and for any t ∈ [0, T ], we get
|T1(x2, y2)(t) − T1(x1, y1)(t)|
≤ RLIp|f(s, x2(s), y2(s)) − f(s, x1(s), y1(s))|(T ) +T p−1
|Ω|
[n2∑l=1
|ωl|θq−1l
(q − 1)νl
×
(n1∑
j=1
|δj |HIβjRLIq(|g(s, x2(s), y2(s)) − g(s, x1(s), y1(s))|)(ξj)
+m1∑i=1
|µi|HIαiRLIp(|f(s, x2(s), y2(s)) − f(s, x1(s), y1(s))|)(ηi)
)
+n1∑
j=1
|δj |ξq−1j
(q − 1)βj
(n2∑l=1
|ωl|HIνlRLIq(|g(s, x2(s), y2(s)) − g(s, x1(s), y1(s))|)(θl)
+m2∑k=1
|τk|HIσkRLIp(|f(s, x2(s), y2(s)) − f(s, x1(s), y1(s))|)(γk)
)]
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
819 Ntouyas et al 813-828
S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS
≤ (K1‖x2 − x1‖ + K2‖y2 − y1‖)
T p
Γ(p + 1)+
T p−1
|Ω|Γ(p + 1)
n2∑l=1
|ωl|θq−1l
(q − 1)νl
m1∑i=1
|µi|ηpi
pαi
+T p−1
|Ω|Γ(p + 1)
n1∑j=1
|δj |ξq−1j
(q − 1)βj
m2∑k=1
|τk|γpk
pσk
+ (L1‖x2 − x1‖ + L2‖y2 − y1‖)
×
T p−1
|Ω|Γ(q + 1)
n2∑l=1
|ωl|θq−1l
(q − 1)νl
n1∑j=1
|δj |ξqj
qβj+
T p−1
|Ω|Γ(q + 1)
n1∑j=1
|δj |ξq−1j
(q − 1)βj
n2∑l=1
|ωl|θql
qνl
= (K1‖x2 − x1‖ + K2‖y2 − y1‖)M1 + (L1‖x2 − x1‖ + L2‖y2 − y1‖)M2
= (M1K1 + M2L1)‖x2 − x1‖ + (M1K2 + M2L2)‖y2 − y1‖,
and consequently we obtain
‖T1(x2, y2)(t) − T1(x1, y1)‖ ≤ (M1K1 + M2L1 + M1K2 + M2L2)[‖x2 − x1‖ + ‖y2 − y1‖]. (15)
Similarly,
‖T2(x2, y2)(t) − T2(x1, y1)‖ ≤ (M4L1 + M5K1 + M4L2 + M5K2)[‖x2 − x1‖ + ‖y2 − y1‖]. (16)
It follows from (15) and (16) that
‖T (x2, y2)(t) − T (x1, y1)(t)‖ ≤ [(M1 + M5)(K1 + K2) + (M2 + M4)(L1 + L2)](‖x2 − x1‖ + ‖y2 − y1‖).
Since (M1 + M5)(K1 + K2) + (M2 + M4)(L1 + L2) < 1, therefore, T is a contraction operator. So, ByBanach’s fixed point theorem, the operator T has a unique fixed point, which is the unique solution ofproblem (1). This completes the proof. ¤
In the next result, we prove the existence of solutions for the problem (1) by applying Leray-Schauderalternative.
Lemma 3.2 (Leray-Schauder alternative) ([11], page.4.) Let F : E → E be a completely continuousoperator (i.e., a map that restricted to any bounded set in E is compact). Let
E(F ) = x ∈ E : x = λF (x) for some 0 < λ < 1.
Then either the set E(F ) is unbounded, or F has at least one fixed point.
Theorem 3.3 Assume that (H2) holds. In addition it is assumed that
(M1 + M5)k1 + (M2 + M4)l1 < 1 and (M1 + M5)k2 + (M2 + M4)l2 < 1,
where M1, M2, M4, M5 are given by (3.1)-(3.2) and (3.4)-(3.5). Then there exists at least one solutionfor the boundary value problem (1).
Proof. First we show that the operator T : X × Y → X × Y is completely continuous. By continuityof functions f and g, the operator T is continuous.
Let Θ ⊂ X × Y be bounded. Then there exist positive constants P1 and P2 such that
|f(t, x(t), y(t))| ≤ P1, |g(t, x(t), y(t))| ≤ P2, ∀(x, y) ∈ Θ.
Then for any (x, y) ∈ Θ, we have
‖T1(x, y)‖ ≤ RLIp|f(s, x(s), y(s))|(T ) +T p−1
|Ω|
[n2∑l=1
|ωl|θq−1l
(q − 1)νl
(n1∑
j=1
|δj |HIβjRLIq|g(s, x(s), y(s))|(ξj)
+m1∑i=1
|µi|HIαiRLIp|f(s, x(s), y(s))|(ηi) + |λ1|
)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
820 Ntouyas et al 813-828
MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS
+n1∑
j=1
|δj |ξq−1j
(q − 1)βj
(n2∑l=1
|ωl|HIνlRLIq|g(s, x(s), y(s))|(θl)
+m2∑k=1
|τk|HIσkRLIp|f(s, x(s), y(s))|(γk) + |λ2|
)]
≤
(T p
Γ(p + 1)+
T p−1
|Ω|Γ(p + 1)
n2∑l=1
|ωl|θq−1l
(q − 1)νl
m1∑i=1
|µi|ηpi
pαi
+T p−1
|Ω|Γ(p + 1)
n1∑j=1
|δj |ξq−1j
(q − 1)βj
m2∑k=1
|τk|γpk
pσk
)P1 +
(T p−1
|Ω|Γ(q + 1)
×n2∑l=1
|ωl|θq−1l
(q − 1)νl
n1∑j=1
|δj |ξqj
qβj+
T p−1
|Ω|Γ(q + 1)
n1∑j=1
|δj |ξq−1j
(q − 1)βj
n2∑l=1
|ωl|θql
qνl
)P2
+|λ1|T p−1
|Ω|
n2∑l=1
|ωl|θq−1l
(q − 1)νl+ |λ2|
T p−1
|Ω|
n1∑j=1
|δj |ξq−1j
(q − 1)βj
= M1P1 + M2P2 + M3.
Similarly, we get
‖T2(x, y)‖ ≤
(T q
Γ(q + 1)+
T q−1
|Ω|Γ(q + 1)
m2∑k=1
|τk|γp−1k
(p − 1)σk
n1∑j=1
|δj |ξqj
qβj
+T q−1
|Ω|Γ(q + 1)
m1∑i=1
|µi|ηp−1i
(p − 1)αi
n2∑l=1
|ωl|θql
qνl
)P2 +
(T q−1
|Ω|Γ(p + 1)
×m2∑k=1
|τk|γp−1k
(p − 1)σk
m1∑i=1
|µi|ηpi
pαi+
T q−1
|Ω|Γ(p + 1)
m1∑i=1
|µi|ηp−1i
(p − 1)αi
m2∑k=1
|τk|γpk
pσk
)P1
+|λ1|T q−1
|Ω|
m2∑k=1
|τk|γp−1k
(p − 1)σk+ |λ2|
T q−1
|Ω|
m1∑i=1
|µi|ηp−1i
(p − 1)αi
= M4P2 + M5P1 + M6.
Thus, it follows from the above inequalities that the operator T is uniformly bounded.
Next, we show that T is equicontinuous. Let t1, t2 ∈ [0, T ] with t1 < t2. Then we have
|T1(x(t2), y(t2)) − T1(x(t1), y(t1))|
≤ 1Γ(p)
∫ t1
0
[(t2 − s)p−1 − (t1 − s)p−1]|f(s, x(s), y(s))|ds
+1
Γ(p)
∫ t2
t1
(t2 − s)p−1|f(s, x(s), y(s))|ds +tp−12 − tp−1
1
|Ω|
[n2∑l=1
|ωl|θq−1l
(q − 1)νl
×
(n1∑
j=1
|δj |HIβjRLIq|g(s, x(s), y(s))|(ξj) +
m1∑i=1
|µi|HIαiRLIp|f(s, x(s), y(s))|(ηi) + |λ1|
)
+n1∑
j=1
|δj |ξq−1j
(q − 1)βj
(n2∑l=1
|ωl|HIνlRLIq|g(s, x(s), y(s))|(θl)
+m2∑k=1
|τk|HIσkRLIp|f(s, x(s), y(s))|(γk) + |λ2|
)]
≤ P1
Γ(p)
∫ t1
0
[(t2 − s)p−1 − (t1 − s)p−1]ds +P1
Γ(p)
∫ t2
t1
(t2 − s)p−1ds
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
821 Ntouyas et al 813-828
S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS
+tp−12 − tp−1
1
|Ω|
[n2∑l=1
|ωl|θq−1l
(q − 1)νl
(P2
n1∑j=1
|δj |ξqj
qβj Γ(q + 1)) + P1
m1∑i=1
|µi|ηpi
pαiΓ(p + 1)+ |λ1|
)
+n1∑
j=1
|δj |ξq−1j
(q − 1)βj
(P2
n2∑l=1
|ωl|θql
qνlΓ(q + 1)+ P1
m2∑k=1
|τk|γpk
pσkΓ(p + 1)+ |λ2|
)].
Analogously, we can obtain
|T2(x(t2), y(t2)) − T2(x(t1), y(t1))|
≤ P2
Γ(q)
∫ t1
0
[(t2 − s)q−1 − (t1 − s)q−1]ds +P2
Γ(q)
∫ t2
t1
(t2 − s)q−1ds
+tq−12 − tq−1
1
|Ω|
[m2∑k=1
|τk|γp−1k
(p − 1)σk
(P2
n1∑j=1
|δj |ξqj
qβj Γ(q + 1)) + P1
m1∑i=1
|µi|ηpi
pαiΓ(p + 1)+ |λ1|
)
+m1∑i=1
|µi|ηp−1i
(p − 1)αi
(P2
n2∑l=1
|ωl|θql
qνlΓ(q + 1)+ P1
m2∑k=1
|τk|γpk
pσkΓ(p + 1)+ |λ2|
)].
Therefore, the operator T (x, y) is equicontinuous, and thus the operator T (x, y) is completely contin-uous.
Finally, it will be verified that the set E = (x, y) ∈ X ×Y |(x, y) = λT (x, y), 0 ≤ λ ≤ 1 is bounded.Let (x, y) ∈ E , then (x, y) = λT (x, y). For any t ∈ [0, T ], we have
x(t) = λT1(x, y)(t), y(t) = λT2(x, y)(t).
Then
|x(t)| ≤ (k0 + k1‖x‖ + k2‖y‖)
(T p
Γ(p + 1)+
T p−1
|Ω|Γ(p + 1)
n2∑l=1
|ωl|θq−1l
(q − 1)νl
m1∑i=1
|µi|ηpi
pαi
+T p−1
|Ω|Γ(p + 1)
n1∑j=1
|δj |ξq−1j
(q − 1)βj
m2∑k=1
|τk|γpk
pσk
)+ (l0 + l1‖x‖ + l2‖y‖)
×
(T p−1
|Ω|Γ(q + 1)
n2∑l=1
|ωl|θq−1l
(q − 1)νl
n1∑j=1
|δj |ξqj
qβj+
T p−1
|Ω|Γ(q + 1)
n1∑j=1
|δj |ξq−1j
(q − 1)βj
n2∑l=1
|ωl|θql
qνl
)
+|λ1|T p−1
|Ω|
n2∑l=1
|ωl|θq−1l
(q − 1)νl+ |λ2|
T p−1
|Ω|
n1∑j=1
|δj |ξq−1j
(q − 1)βj
and
|y(t)| ≤ (l0 + l1‖x‖ + l2‖y‖)
(T q
Γ(q + 1)+
T q−1
|Ω|Γ(q + 1)
m2∑k=1
|τk|γp−1k
(p − 1)σk
n1∑j=1
|δj |ξqj
qβj
+T q−1
|Ω|Γ(q + 1)
m1∑i=1
|µi|ηp−1i
(p − 1)αi
n2∑l=1
|ωl|θql
qνl
)+ (k0 + k1‖x‖ + k2‖y‖)
×
(T q−1
|Ω|Γ(p + 1)
m2∑k=1
|τk|γp−1k
(p − 1)σk
m1∑i=1
|µi|ηpi
pαi+
T q−1
|Ω|Γ(p + 1)
m1∑i=1
|µi|ηp−1i
(p − 1)αi
m2∑k=1
|τk|γpk
pσk
)
+|λ1|T q−1
|Ω|
m2∑k=1
|τk|γp−1k
(p − 1)σk+ |λ2|
T q−1
|Ω|
m1∑i=1
|µi|ηp−1i
(p − 1)αi.
Hence we have
‖x‖ ≤ (k0 + k1‖x‖ + k2‖y‖)M1 + (l0 + l1‖x‖ + l2‖y‖)M2 + M3
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
822 Ntouyas et al 813-828
MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS
and
‖y‖ ≤ (l0 + l1‖x‖ + l2‖y‖)M4 + (k0 + k1‖x‖ + k2‖y‖)M5 + M6,
which imply that
‖x‖ + ‖y‖ ≤ (M1 + M5)k0 + (M2 + M4)l0 + [(M1 + M5)k1 + (M2 + M4)l1]‖x‖+[(M1 + M5)k2 + (M2 + M4)l2]‖y‖ + M3 + M6.
Consequently,
‖(x, y)‖ ≤ (M1 + M5)k0 + (M2 + M4)l0 + M3 + M6
M0,
for any t ∈ [0, T ], where M0 is defined by (14), which proves that E is bounded. Thus, by Lemma 3.2,the operator T has at least one fixed point. Hence the boundary value problem (1) has at least onesolution. The proof is complete. ¤
3.1 Examples
Example 3.4 Consider the following system of coupled Riemann-Liouville fractional differential equa-tions with Hadamard type fractional integral boundary conditions
RLD4/3x(t) =t
(t + 6)2|x(t)|
(1 + |x(t)|)+
e−t
(t2 + 3)3|y(t)|
(1 + |y(t)|)+
34, t ∈ [0, 2],
RLD3/2y(t) =118
sinx(t) +1
22t + 19cos y(t) +
54, t ∈ [0, 2],
x(0) = 0, 2HI2/3x(3/5) + πHI7/5x(1) =√
2HI3/2y(1/3) + e2HI5/4y(
√3) + 4,
y(0) = 0, −3HI9/5x(2/3) + 4HI7/4x(9/7) +25HI1/3x(
√2)
=e
2HI11/6y(8/5) − 2HI12/11y(1/4) − 10.
(17)
Here p = 4/3, q = 3/2, T = 2, λ1 = 4, λ2 = −10, m1 = 2, n1 = 2, m2 = 3, n2 = 2, µ1 = 2,µ2 = π, α1 = 2/3, α2 = 7/5, η1 = 3/5, η2 = 1, δ1 =
√2, δ2 = e2, β1 = 3/2, β2 = 5/4, ξ1 = 1/3,
ξ2 =√
3, τ1 = −3, τ2 = 4, τ3 = 2/5, σ1 = 9/5, σ2 = 7/4, σ3 = 1/3, γ1 = 2/3, γ2 = 9/7, γ3 =√
2,ω1 = e/2, ω2 = −2, ν1 = 11/6, ν2 = 12/11, θ1 = 8/5, θ2 = 1/4 and f(t, x, y) = (t|x|)/(((t + 6)2)(1 +|x|)) + (e−t|y|)/(((t2 + 3)3)(1 + |y|)) + (3/4) and g(t, x, y) = (sinx/18) + (cos y)/(22t + 19) + (5/4).Since |f(t, x1, y1) − f(t, x2, y2)| ≤ ((1/18)|x1 − x2| + (1/27)|y1 − y2|) and |g(t, x1, y1) − g(t, x2, y2)| ≤((1/18)|x1 − x2| + (1/20)|y1 − y2|). By using the Maple program, we can find
Ω =m1∑i=1
µiηp−1i
(p − 1)αi
n2∑l=1
ωlθq−1l
(q − 1)νl−
n1∑j=1
δjξq−1j
(q − 1)βj
m2∑k=1
τkγp−1k
(p − 1)σk≈ −218.9954766 6= 0.
With the given values, it is found that K1 = 1/18, K2 = 1/27, L1 = 1/18, L2 = 1/20, M1 '2.847852451, M2 ' 0.5295490231, M4 ' 4.723846069, M5 ' 1.276954854, and
(M1 + M5)(K1 + K2) + (M2 + M4)(L1 + L2) ' 0.9364516398 < 1.
Thus all the conditions of Theorem 3.1 are satisfied. Therefore, by the conclusion of Theorem 3.1, theproblem (17) has a unique solution on [0, 2].
Example 3.5 Consider the following system of coupled Riemann-Liouville fractional differential equa-
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
823 Ntouyas et al 813-828
S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS
tions with Hadamard type fractional integral boundary conditions
RLDπ/2x(t) =25
+1
(t + 6)2tan−1 x(t) +
120e
y(t), t ∈ [0, 3],
RLD7/4y(t) =√
π
2+
142
sinx(t) +1
t + 20y(t) cos x(t), t ∈ [0, 3],
x(0) = 0, 3HI1/4x(5/2) +√
5HI√
2x(7/8) + tan(4)HI√
3x(9/4)
=√
8π
3 HI5/3y(5/4) − 2HI6/11y(π/3) + 2,
y(0) = 0, −23HI2/3x(π/2) + 3HI6/5x(5/3) +
√2
πHI1/3x(
√2)
+79HI11/9x(
√5) = eHI7/6y(π/6) − log(9)HI3/4y(7/4) − 1.
(18)
Here p = π/2, q = 7/4, T = 3, λ1 = 2, λ2 = −1, m1 = 3, n1 = 2, m2 = 4, n2 = 2, µ1 = 3, µ2 =√
5,µ3 = tan(4), α1 = 1/4, α2 =
√2, α3 =
√3, η1 = 5/2, η2 = 7/8, η3 = 9/4, δ1 =
√8π/3, δ2 = −2,
β1 = 5/3, β2 = 6/11, ξ1 = 5/4, ξ2 = π/3, τ1 = −2/3, τ2 = 3, τ3 =√
2/π, τ4 = 7/9, σ1 = 2/3,σ2 = 6/5, σ3 = 1/3, σ4 = 11/9, γ1 = π/2, γ2 = 5/3, γ3 =
√2, γ4 =
√5, ω1 = e, ω2 = − log(9),
ν1 = 7/6, ν2 = 3/4, θ1 = π/6, θ2 = 7/4, f(t, x, y) = (2/5) + (tan−1 x)/((t + 6)2) + (y)/(20e) andg(t, x, y) = (
√π/2) + (sin x)/(42) + (y cos x)/(t + 20). By using the Maple program, we get
Ω =m1∑i=1
µiηp−1i
(p − 1)αi
n2∑l=1
ωlθq−1l
(q − 1)νl−
n1∑j=1
δjξq−1j
(q − 1)βj
m2∑k=1
τkγp−1k
(p − 1)σk≈ −59.01857601 6= 0.
Since |f(t, x, y)| ≤ k0 + k1|x| + k2|y| and |g(t, x, y)| ≤ l0 + l1|x| + l2|y|, where k0 = 2/5, k1 = 1/36,k2 = 1/(20e), l0 =
√π/2, l1 = 1/42, l2 = 1/20, it is found that M1 ' 7.406711671, M2 ' 1.110132269,
M4 ' 6.802999724, M5 ' 7.790182643. Furthermore, we can find that
(M1 + M5)k1 + (M2 + M4)l1 ≈ 0.6105438577 < 1,
and(M1 + M5)k2 + (M2 + M4)l2 ≈ 0.6751878489 < 1.
Thus all the conditions of Theorem 3.3 holds true and consequently the conclusion of Theorem 3.3, theproblem (18) has at least one solution on [0, 3].
4 Uncoupled integral boundary conditions case
In this section we consider the following system
RLDpx(t) = f(t, x(t), y(t)), t ∈ [0, T ], 1 < p ≤ 2,
RLDqy(t) = g(t, x(t), y(t)), t ∈ [0, T ], 1 < q ≤ 2,
x(0) = 0,
m1∑i=1
µiHIαix(ηi) =n1∑
j=1
δjHIβj x(ξj) + λ1,
y(0) = 0,
m2∑k=1
τkHIσky(γk) =n2∑l=1
ωlHIνly(θl) + λ2.
(19)
Lemma 4.1 (Auxiliary Lemma) For h ∈ C([0, T ], R), the unique solution of the problemRLDpx(t) = h(t), 1 < p ≤ 2, t ∈ [0, T ]
x(0) = 0,
m1∑i=1
µiHIαix(ηi) =n1∑
j=1
δjHIβj x(ξj) + λ1,(20)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
824 Ntouyas et al 813-828
MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS
is given by
x(t) = RLIph(t) +tp−1
Λ
n1∑j=1
δjHIβjRLIph(ξj) −
m1∑i=1
µiHIαiRLIph(ηi) + λ1
, (21)
where
Λ :=m1∑i=1
µiηp−1i
(p − 1)αi−
n1∑j=1
δjξp−1j
(p − 1)βj6= 0. (22)
4.1 Existence results for uncoupled case
In view of Lemma 4.1, we define an operator T : X × Y → X × Y by T(u, v)(t) =(
T1(u, v)(t)T2(u, v)(t)
)where
T1(u, v)(t) = RLIpf(s, u(s), v(s))(t) +tp−1
Λ
(n1∑
j=1
δjHIβjRLIpf(s, u(s), v(s))(ξj)
−m1∑i=1
µiHIαiRLIpf(s, u(s), v(s))(ηi) + λ1
),
and
T2(u, v)(t) = RLIqg(s, u(s), v(s))(t) +tq−1
Φ
(n2∑l=1
ωlHIνlRLIqg(s, u(s), v(s))(θl)
−m2∑k=1
τkHIσkRLIqg(s, u(s), v(s))(γk) + λ2
),
where
Φ =m2∑k=1
τkγq−1k
(q − 1)σk−
n2∑l=1
ωlθq−1l
(q − 1)νl6= 0.
In the sequel, we set
M1 =1
Γ(p + 1)
T p +T p−1
|Λ|
n1∑j=1
|δj |ξpj
pβj+
T p−1
|Λ|
m1∑i=1
|µi|ηpi
pαi
, (23)
M2 =T p−1λ1
|Λ|, (24)
M3 =1
Γ(q + 1)
(T q +
T q−1
|Φ|
n2∑l=1
|ωl|θql
qνl+
T q−1
|Φ|
m2∑k=1
|τk|γqk
qσk
), (25)
M4 =T q−1λ2
|Φ|. (26)
Now we present the existence and uniqueness result for the problem (19). We do not provide theproof of this result as it is similar to the one for Theorem 3.1.
Theorem 4.2 Assume that f, g : [0, T ] × R2 → R are continuous functions and there exist constantsKi, Li, i = 1, 2 such that for all t ∈ [0, T ] and ui, vi ∈ R, i = 1, 2,
|f(t, u1, u2) − f(t, v1, v2)| ≤ K1|u1 − v1| + K2|u2 − v2|
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
825 Ntouyas et al 813-828
S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS
and|g(t, u1, u2) − g(t, v1, v2)| ≤ L1|u1 − v1| + L2|u2 − v2|.
In addition, assume thatM1(K1 + K2) + M3(L1 + L2) < 1,
where M1 and M3 are given by (23) and (25) respectively. Then the boundary value problem (19) hasa unique solution.
Example 4.3 Consider the following system of coupled Riemann-Liouville fractional differential equa-tions with uncoupled Hadamard type fractional integral boundary conditions
RLDe/2x(t) =cos(πt)
(πt + 4)2|x(t)|
|x(t)| + 2+
3et/2
(t + 5)3|y(t)|
|y(t)| + 3+
2e, t ∈ [0, 4],
RLD√
3y(t) =sinx(t)
15(et + 3)+
2√|y(t)| + 1
7π(t + 3)+ 5, t ∈ [0, 4],
x(0) = 0,√
11HI5/2x(2/3) +tan2(5)
20 HI10/3x(π) =5e
HI3/7x(e)
− 76HI
√5x(
√2) +
π
2 HI2/5x(12/7) + 11,
y(0) = 0,log(15)
9 HI7/4y(1/4) + 2HI5/6y(√
7)
=π2
15 HI4/3y(1/e) +√
5HI9/7y(7/2) +√
8/3.
(27)
Here p = e/2, q =√
3, T = 4, λ1 = 11, λ2 =√
8/3, m1 = 2, n1 = 3, m2 = 2, n2 = 2, µ1 =√
11,µ2 = tan2(5)/20, α1 = 5/2, α2 = 10/3, η1 = 2/3, η2 = π, δ1 = 5/e, δ2 = −7/6, δ3 = π/2, β1 = 3/7,β2 =
√5, β3 = 2/5, ξ1 = e, ξ2 =
√2, ξ3 = 12/7, τ1 = log(15)/9, τ2 = 2, σ1 = 7/4, σ2 = 5/6,
γ1 = 1/4, γ2 =√
7, ω1 = π2/15, ω2 =√
5, ν1 = 4/3, ν2 = 9/7, θ1 = 1/e, θ2 = 7/2, f(t, x, y) =(cos(πt)|x|)/(((πt+4)2)(|x|+2))+(3et/2|y|)/(((t+5)3)(|y|+3))+(2/e) and g(t, x, y) = (sin x(t))/(15(et+3))+(2
√|y| + 1)/(7π(t+3))+5. Since |f(t, x1, y1)−f(t, x2, y2)| ≤ ((1/50)|x1 −x2|+(e2/125)|y1 −y2|)
and |g(t, x1, y1)− g(t, x2, y2)| ≤ ((1/60)|x1 −x2|+(1/(21π))|y1 − y2|). By using the Maple program, wecan find
Λ :=m1∑i=1
µiηp−1i
(p − 1)αi−
n1∑j=1
δjξp−1j
(p − 1)βj≈ 69.35947949 6= 0
and
Φ =m2∑k=1
τkγq−1k
(q − 1)σk−
n2∑l=1
ωlθq−1l
(q − 1)νl≈ −3.358717154 6= 0.
With the given values, it is found that K1 = 1/50, K2 = e2/125, L1 = 1/60, L2 = 1/(21π), M1 '5.673444294, M3 ' 15.54186374. In consequence,
M1(K1 + K2) + M3(L1 + L2) ≈ 0.9434486991 < 1.
Thus all the conditions of Theorem 4.2 are satisfied. Therefore, there exists a unique solution for theproblem (27) on [0, 4].
The second result dealing with the existence of solutions for the problem (19) is analogous toTheorem 3.3 and is given below.
Theorem 4.4 Assume that there exist real constants ki, li ≥ 0 (i = 1, 2) and k0 > 0, l0 > 0 such that∀xi ∈ R, (i = 1, 2) we have
|f(t, x1, x2)| ≤ k0 + k1|x1| + k2|x2|,
|g(t, x1, x2)| ≤ l0 + l1|x1| + l2|x2|.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
826 Ntouyas et al 813-828
MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS
In addition it is assumed that
k1M1 + l1M3 < 1 and k2M1 + l2M3 < 1,
where M1 and M3 are given by (23) and (25) respectively. Then the boundary value problem (19) hasat least one solution.
Proof. Setting
M0 = min1 − k1M1 − l1M3, 1 − k2M1 − l2M3, ki, li ≥ 0 (i = 1, 2),
the proof is similar to that of Theorem 3.3. So we omit it. ¤
References
[1] R.P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations,Comput. Math. Appl. 59 (2010), 1095-1100.
[2] B. Ahmad, J.J. Nieto, Boundary value problems for a class of sequential integrodifferential equa-tions of fractional order, J. Funct. Spaces Appl. 2013, Art. ID 149659, 8 pp.
[3] B. Ahmad, J.J. Nieto, Riemann-Liouville fractional integro-differential equations with fractionalnonlocal integral boundary conditions, Bound. Value Probl. 2011, 2011:36, 9 pp.
[4] B. Ahmad, S.K. Ntouyas, A. Alsaedi, A study of nonlinear fractional differential equations ofarbitrary order with Riemann-Liouville type multistrip boundary conditions, Math. Probl. Eng.(2013), Art. ID 320415, 9 pp.
[5] B. Ahmad, S.K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differentialequations with three-point integral boundary conditions, Adv. Differ. Equ. (2011) Art. ID 107384,11pp.
[6] D. Baleanu, K. Diethelm, E. Scalas, J.J.Trujillo, Fractional Calculus Models and Numerical Meth-ods. Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, 2012.
[7] D. Baleanu, O.G. Mustafa, R.P. Agarwal, On Lp-solutions for a class of sequential fractionaldifferential equations, Appl. Math. Comput. 218 (2011), 2074-2081.
[8] P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Compositions of Hadamard-type fractional integrationoperators and the semigroup property, J. Math. Anal. Appl. 269 (2002), 387-400.
[9] P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-typefractional integrals, J. Math. Anal. Appl. 269 (2002), 1-27.
[10] P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Mellin transform analysis and integration by parts forHadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002), 1-15.
[11] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
[12] J. Hadamard, Essai sur l’etude des fonctions donnees par leur developpment de Taylor, J. Mat.Pure Appl. Ser. 8 (1892) 101-186.
[13] A.A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001), 1191-1204.
[14] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional DifferentialEquations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
[15] A.A. Kilbas, J.J. Trujillo, Hadamard-type integrals as G-transforms, Integral Transform. Spec.Funct.14 (2003), 413-427.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
827 Ntouyas et al 813-828
S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS
[16] X. Liu, M. Jia, W. Ge, Multiple solutions of a p-Laplacian model involving a fractional derivative,Adv. Differ. Equ. 2013, 2013:126.
[17] D. O’Regan, S. Stanek, Fractional boundary value problems with singularities in space variables,Nonlinear Dynam. 71 (2013), 641-652.
[18] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[19] J. Tariboon, T. Sitthiwirattham, S. K. Ntouyas, Existence results for fractional differential inclu-sions with multi–point and fractional integral boundary conditions, J. Comput. Anal. Appl. 17(2014), 343-360.
[20] P. Thiramanus, J. Tariboon, S. K. Ntouyas, Average value problems for nonlinear second-orderimpulsive q-difference equations, J. Comput. Appl. Anal. 18 (2015), 590-611.
[21] L. Zhang, B. Ahmad, G. Wang, R.P. Agarwal, Nonlinear fractional integro-differential equationson unbounded domains in a Banach space, J. Comput. Appl. Math. 249 (2013), 51–56.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
828 Ntouyas et al 813-828
Ternary Jordan ring derivations on Banach ternary algebras: A fixed pointapproach
Madjid Eshaghi Gordji1, Shayan Bazeghi1, Choonkil Park2∗ and Sun Young Jang3∗
1Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran2Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea
3Department of Mathematics, University of Ulsan, Ulsan 680-749, Korea
e-mail: [email protected], [email protected], [email protected],
Abstract. Let A be a Banach ternary algebra. An additive mapping D : (A, [ ])→ (A, [ ]) is called a ternary Jordan ringderivation if D([xxx]) = [D(x)xx] + [xD(x)x] + [xxD(x)] for all x ∈ A.
In this paper, we prove the Hyers-Ulam stability of ternary Jordan ring derivations on Banach ternary algebras.
1. Introduction
We say that a functional equation (Q) is stable if any function g satisfying the equation (Q) approximately is near totrue solution of (Q). Also, we say that a functional equation is superstable if every approximately solution is an exactsolution of it.
Recently, Bavand Savadkouhi et al. [4] investigate the stability of ternary Jordan derivations on Banach ternary algebrasby direct methods.
Ternary algebraic operations were considered in the 19th century by several mathematicians. Cayley [7] introduced thenotion of cubic matrix, which in turn was generalized by Kapranov, Gelfand and Zelevinskii [17].
The comments on physical applications of ternary structures can be found in [3, 12, 13, 14, 22, 23, 26, 28, 31, 32].Let A be a Banach ternary algebra. An additive mapping D : (A, [ ])→ (A, [ ]) is called a ternary ring derivation if
D([xyz]) = [D(x)yz] + [xD(y)z] + [xyD(z)]
for all x, y, z ∈ A.An additive mapping D : (A, [ ])→ (A, [ ]) is called a ternary Jordan ring derivation if
D([xxx]) = [D(x)xx] + [xD(x)x] + [xxD(x)]
for all x ∈ A.Theorem 1.1. ([11]) Suppose that (Ω, d) is a complete generalized metric space and T : Ω → Ω is a strictly contractivemapping with the Lipschitz constant L. Then, for any x ∈ Ω, either
d(Tnx, Tn+1x) =∞, ∀n ≥ 0,
or there exists a positive integer n0 such that(1) d(Tnx, Tn+1x) <∞ for all n ≥ n0;(2) the sequence Tnx is convergent to a fixed point y∗ of T ;(3) y∗ is the unique fixed point of T in Λ = y ∈ Ω : d(Tn0x, y) <∞;(4) d(y, y∗) ≤ 1
1−Ld(y, Ty) for all y ∈ Λ.
The study of stability problems originated from a famous talk given by Ulam [30] in 1940: “Under what conditiondoes there exist a homomorphism near an approximate homomorphism?” In the next year 1941, Hyers [15] answeredaffirmatively the question of Ulam for additive mappings between Banach spaces. A generalized version of the theorem ofHyers for approximately additive mappings was given by Rassias [24] in 1978. The stability problems of several functionalequations have been extensively investigated by a number of authors and there are many interesting results concerningthis problem (see [1, 5, 8, 10, 18, 19, 20, 21, 25, 27, 29, 33, 34]).
In this paper, we prove the Hyers-Ulam stability and superstability of ternary Jordan ring derivations on Banachternary algebras by the fixed point method.
02010 Mathematics Subject Classification. Primary 39B52; 39B82; 47H10; 46B99; 17A40.0Keywords: Hyers-Ulam stability; ternary ring derivation; Banach ternary algebra; fixed point method; ternary Jordan
ring derivation.0∗Corresponding author.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
829 Gordji et al 829-834
M. Eshaghi Gordji, Sh. Bazeghi, C. Park, S. Y. Jang
2. Hyers-Ulam stability of ternary Jordan ring derivations
In this section, we prove the Hyers-Ulam stability of ternary Jordan ring derivations on Banach ternary algebras.Throughout this section, assume that A is a Banach ternary algebra.
Lemma 2.1. Let f : A→ A be an additive mapping. Then the following assertions are equivalent.
f([a, a, a]) = [f(a), a, a] + [a, f(a), a] + [a, a, f(a)] (2.1)
for all a ∈ A, and
f([a, b, c] + [b, c, a] + [c, a, b]) =[f(a), b, c] + [a, f(b), c] + [a, b, f(c)] + [f(b), c, a] + [b, f(c), a]
+ [b, c, f(a)] + [f(c), a, b] + [c, f(a), b] + [c, a, f(b)](2.2)
for all a, b, c ∈ A.
Proof. Replacing a by a+ b+ c in (2.1), we have
f([(a+ b+ c), (a+ b+ c), (a+ b+ c)]) =[f(a+ b+ c), (a+ b+ c), (a+ b+ c)] + [(a+ b+ c), f(a+ b+ c), (a+ b+ c)]
+ [(a+ b+ c), (a+ b+ c), f(a+ b+ c)]
and so
f([(a+ b+ c), (a+ b+ c), (a+ b+ c)])
= f([a, a, a] + [a, b, a] + [a, c, a] + [b, a, a] + [b, b, a] + [b, c, a] + [c, a, a] + [c, b, a] + [c, c, a]
+ [a, a, b] + [a, b, b] + [a, c, b] + [b, a, b] + [b, b, b] + [b, c, b] + [c, a, b] + [c, b, b] + [c, c, b]
+ [a, a, c] + [a, b, c] + [a, c, c] + [b, a, c] + [b, b, c] + [b, c, c] + [c, a, c] + [c, b, c] + [c, c, c])
= f([a, a, a]) + f([a, b, a]) + f([a, c, a]) + f([b, a, a]) + f([b, b, a]) + f([b, c, a]) + f([c, a, a]) + f([c, b, a]) + f([c, c, a])
+ f([a, a, b]) + f([a, b, b]) + f([a, c, b]) + f([b, a, b]) + f([b, b, b]) + f([b, c, b]) + f([c, a, b]) + f([c, b, b]) + f([c, c, b])
+ f([a, a, c]) + f([a, b, c]) + f([a, c, c]) + f([b, a, c]) + f([b, b, c]) + f([b, c, c]) + f([c, a, c]) + f([c, b, c]) + f([c, c, c])
= [f(a), a, a] + [a, f(a), a] + [a, a, f(a)] + [f(a), b, a] + [a, f(b), a] + [a, b, f(a)] + [f(a), c, a] + [a, f(c), a] + [a, c, f(a)]
+ [f(b), a, a] + [b, f(a), a] + [b, a, f(a)] + [f(b), b, a] + [b, f(b), a] + [b, b, f(a)] + [f(b), c, a] + [b, f(c), a] + [b, c, f(a)]
+ [f(c), a, a] + [c, f(a), a] + [c, a, f(a)] + [f(c), b, a] + [c, f(b), a] + [c, b, f(a)] + [f(c), c, a] + [c, f(c), a] + [c, c, f(a)]
+ [f(a), a, b] + [a, f(a), b] + [a, a, f(b)] + [f(a), b, b] + [a, f(b), b] + [a, b, f(b)] + [f(a), c, b] + [a, f(c), b] + [a, c, f(b)]
+ [f(b), a, b] + [b, f(a), b] + [b, a, f(b)] + [f(b), b, b] + [b, f(b), b] + [b, b, f(b)] + [f(b), c, b] + [b, f(c), b] + [b, c, f(b)]
+ [f(c), a, b] + [c, f(a), b] + [c, a, f(b)] + [f(c), b, b] + [c, f(b), b] + [c, b, f(b)] + [f(c), c, b] + [c, f(c), b] + [c, c, f(b)]
+ [f(a), a, c] + [a, f(a), c] + [a, a, f(c)] + [f(a), b, c] + [a, f(b), c] + [a, b, f(c)] + [f(a), c, c] + [a, f(c), c] + [a, c, f(c)]
+ [f(b), a, c] + [b, f(a), c] + [b, a, f(c)] + [f(b), b, c] + [b, f(b), c] + [b, b, f(c)] + [f(b), c, c] + [b, f(c), c] + [b, c, f(c)]
+ [f(c), a, c] + [c, f(a), c] + [c, a, f(c)] + [f(c), b, c] + [c, f(b), c] + [c, b, f(c)] + [f(c), c, c] + [c, f(c), c] + [c, c, f(c)]
for all a, b, c ∈ A.On the other hand, we have
f([(a+ b+ c), (a+ b+ c), (a+ b+ c)])
= [f(a), a, a] + [f(a), a, b] + [f(a), a, c] + [f(a), b, a] + [f(a), b, b] + [f(a), b, c] + [f(a), c, a] + [f(a), c, b] + [f(a), c, c]
+ [f(b), a, a] + [f(b), a, b] + [f(b), a, c] + [f(b), b, a] + [f(b), b, b] + [f(b), b, c] + [f(b), c, a] + [f(b), c, b] + [f(b), c, c]
+ [f(c), a, a] + [f(c), a, b] + [f(c), a, c] + [f(c), b, a] + [f(c), b, b] + [f(c), b, c] + [f(c), c, a] + [f(c), c, b] + [f(c), c, c]
+ [a, f(a), a] + [a, f(a), b] + [a, f(a), c] + [b, f(a), a] + [b, f(a), b] + [b, f(a), c] + [c, f(a), a] + [c, f(a), b] + [c, f(a), c]
+ [a, f(b), a] + [a, f(b), b] + [a, f(b), c] + [b, f(b), a] + [b, f(b), b] + [b, f(b), c] + [c, f(b), a] + [c, f(b), b] + [c, f(b), c]
+ [a, f(c), a] + [a, f(c), b] + [a, f(c), c] + [b, f(c), a] + [b, f(c), b] + [b, f(c), c] + [c, f(c), a] + [c, f(c), b] + [c, f(c), c]
+ [a, a, f(a)] + [a, b, f(a)] + [a, c, f(a)] + [b, a, f(a)] + [b, b, f(a)] + [b, c, f(a)] + [c, a, f(a)] + [c, b, f(a)] + [c, c, f(a)]
+ [a, a, f(b)] + [a, b, f(b)] + [a, c, f(b)] + [b, a, f(b)] + [b, b, f(b)] + [b, c, f(b)] + [c, a, f(b)] + [c, b, f(b)] + [c, c, f(b)]
+ [a, a, f(c)] + [a, b, f(c)] + [a, c, f(c)] + [b, a, f(c)] + [b, b, f(c)] + [b, c, f(c)] + [c, a, f(c)] + [c, b, f(c)] + [c, c, f(c)]
for all a, b, c ∈ A.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
830 Gordji et al 829-834
Ternary Jordan ring derivations on Banach ternary algebras
It follows that
([f(b), c, a] + [b, f(c), a] + [b, c, f(a)]) + ([f(c), a, b] + [c, f(a), b] + [c, a, f(b)]) + ([f(a), b, c] + [a, f(b), c] + [a, b, f(c)])
= f([b, c, a]) + f([c, a, b]) + f([a, b, c]) = f([b, c, a] + [c, a, b] + [a, b, c]) = f([a, b, c] + [b, c, a] + [c, a, b])
and
[f(a), b, c] + [f(b), c, a] + [f(c), a, b] + [c, f(a), b] + [a, f(b), c] + [b, f(c), a] + [b, c, f(a)] + [c, a, f(b)] + [a, b, f(c)]
= ([f(a), b, c] + [a, f(b), c] + [a, b, f(c)]) + ([f(b), c, a] + [b, f(c), a] + [b, c, f(a)]) + ([f(c), a, b] + [c, f(a), b] + [c, a, f(b)])
for all a, b, c ∈ A. Then
f([a, b, c] + [b, c, a] + [c, a, b]) =([f(a), b, c] + [a, f(b), c] + [a, b, f(c)]) + ([f(b), c, a] + [b, f(c), a]
+ [b, c, f(a)]) + ([f(c), a, b] + [c, f(a), b] + [c, a, f(b)])
for all a, b, c ∈ A. Hence (2.2) holds true.For the converse, replacing b and c by a in (2.2), we have
f([a, a, a] + [a, a, a] + [a, a, a]) =[f(a), a, a] + [a, f(a), a] + [a, a, f(a)] + [f(a), a, a] + [a, f(a), a]
+ [a, a, f(a)] + [f(a), a, a] + [a, f(a), a] + [a, a, f(a)]
and so
f(3[a, a, a]) = 3([f(a), a, a] + [a, f(a), a] + [a, a, f(a)])
for all a ∈ A. Thus
f([a, a, a]) = [f(a), a, a] + [a, f(a), a] + [a, a, f(a)]
for all a ∈ A. This completes the proof.
Theorem 2.2. Let f : A→ A be a mapping for which there exists function ϕ : A×A×A→ [0,∞) such that
‖f(x+ y)− f(x)− f(y)‖ ≤ ϕ(x, y, 0), (2.3)
‖f([x, y, z] + [y, z, x] + [z, x, y])− [f(x), y, z]− [x, f(y), z]− [x, y, f(z)]− [f(y), z, x]
−[y, f(z), x]− [y, z, f(x)]− [f(z), x, y]− [z, f(x), y]− [z, x, f(y)]‖ ≤ ϕ(x, y, z)(2.4)
for all x, y, z ∈ A. If there exists a constant 0 < L < 1 such that
ϕ(x
2,y
2,z
2
)≤ L
8ϕ(x, y, z) (2.5)
for all x, y, z ∈ A, then there exists a unique ternary Jordan ring derivation D : A→ A such that
‖f(x)−D(x)‖ ≤ L
8− 2Lϕ(x, x, 0) (2.6)
for all x ∈ A.
Proof. It follows from (2.5) that
limn→∞
23nϕ( x
2n,y
2n,z
2n
)= 0 (2.7)
for all x, y, z ∈ A. By (2.5), ϕ(0, 0, 0) = 0. Letting x = y = 0 in (2.3), we get ‖f(0)‖ ≤ ϕ(0, 0, 0) = 0 and so f(0) = 0. LetΩ = g : A→ X, g(0) = 0. We introduce a generalized metric on Ω as follows:
d(g, h) = dϕ(g, h) = infC ∈ (0,∞) : ‖g(x)− h(x)‖ ≤ Cϕ(x, x, 0), ∀x ∈ A .
It is easy to show that (Ω, d) is a generalized complete metric space [16]. Now, we consider the mapping T : Ω → Ωdefined by Tg(x) = 2g(x
2) for all x ∈ A and g ∈ Ω. Note that, for all g, h ∈ Ω and x ∈ A,
d(g, h) < C ⇒ ‖g(x)− h(x)‖ ≤ Cϕ(x, x, 0)
⇒ ‖2g(x
2)− 2h(
x
2)‖ ≤ 2 C ϕ(
x
2,x
2, 0)
⇒ ‖2g(x
2)− 2h(
x
2)‖ ≤ L
4C ϕ(x, x, 0)
⇒ d(Tg, Th) ≤ L
4C.
Hence we obtain that
d(Tg, Th) ≤ L
4d(g, h)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
831 Gordji et al 829-834
M. Eshaghi Gordji, Sh. Bazeghi, C. Park, S. Y. Jang
for all g, h ∈ Ω, that is, T is a strictly contractive mapping of Ω with the Lipschitz constant L. Putting y = x in (2.3), wehave
‖f(2x)− 2f(x)‖ ≤ ϕ(x, x, 0), (2.8)
and so ∥∥∥f(x)− 2f(x
2
)∥∥∥ ≤ ϕ(x2,x
2, 0)≤ L
8ϕ(x, x, 0)
for all x ∈ A. Let us denote
D(x) = limn→∞
2nf( x
2n
)(2.9)
for all x ∈ A. By the result in ([2, 6]), D is an additive mapping and so it follows from the definition of D, (2.4) and (2.7)that
‖D([x, y, z] + [y, z, x] + [z, x, y])− [D(x), y, z]− [x,D(y), z]− [x, y,D(z)]− [D(y), z, x]− [y,D(z), x]
− [y, z,D(x)]− [D(z), x, y]− [z,D(x), y]− [z, x,D(y)]‖
= limn→∞
8n‖f([x, y, z
23n] + [
y, z, x
23n] + [
z, x, y
23n])− [f(
x
2n),y
2n,z
2n]− [
x
2n, f(
y
2n),z
2n]− [
x
2n,y
2n, f(
z
2n)]
− [f(y
2n),z
2n,x
2n]− [
y
2n, f(
z
2n),x
2n]− [
y
2n,z
2n, f(
x
2n)]− [f(
z
2n),x
2n,y
2n]− [
z
2n, f(
x
2n),y
2n]− [
z
2n,x
2n, f(
y
2n)]‖
≤ limn→∞
8nϕ( x
2n,y
2n,z
2n
)= 0
for all x, y, z ∈ A and so D([x, y, z]+[y, z, x]+[z, x, y]) = [D(x), y, z]+[x,D(y), z]+[x, y,D(z)]+[D(y), z, x]+[y,D(z), x]+[y, z,D(x)] + [D(z), x, y] + [z,D(x), y] + [z, x,D(y)], which implies that D is a ternary Jordan ring derivation, by Lemma2.1. According to Theorem 1.1, since D is the unique fixed point of T in the set Λ = g ∈ Ω : d(f, g) < ∞, D is theunique mapping such that
‖f(x)−D(x)‖ ≤ C ϕ(x, x, 0)
for all x ∈ A and C > 0. By Theorem 1.1, we have
d(f,D) ≤ 1
1− L4
d(f, Tf) ≤ 4L
8(4− L)
and so
‖f(x)−D(x)‖ ≤ L
8− 2Lϕ(x, x, 0)
for all x ∈ A. This completes the proof.
Corollary 2.3. Let θ, r be nonnegative real numbers with r > 1. Suppose that f : A→ A is a mapping such that
‖f(x+ y)− f(x)− f(y)‖ ≤ θ(‖x‖r + ‖y‖r), (2.10)
‖f([x, y, z] + [y, z, x] + [z, x, y])− [f(x), y, z]− [x, f(y), z]− [x, y, f(z)]− [f(y), z, x]
−[y, f(z), x]− [y, z, f(x)]− [f(z), x, y]− [z, f(x), y]− [z, x, f(y)]‖ ≤ θ(‖x‖r + ‖y‖r + ‖z‖r)(2.11)
for all x, y, z ∈ A. Then there exists a unique ternary Jordan ring derivation D : A→ A satisfying
‖f(x)−D(x)‖ ≤ θ
2r+1 − 1‖x‖r
for all x ∈ A.
Proof. The proof follows from Theorem 2.2 by taking
ϕ(x, y, z) := θ(‖x‖r + ‖y‖r + ‖z‖r)
for all x, y, z ∈ A. Then we can choose L = 21−r and so we obtain the desired conclusion.
Remark 2.4. Let f : A → A be a mapping with f(0) = 0 such that there exists a function ϕ : A × A × A → [0,∞)satisfying (2.3) and (2.4). Let 0 < L < 1 be a constant such that
ϕ(2x, 2y, 2z) ≤ 2Lϕ(x, y, z)
for all x, y, z ∈ A. By a similar method as in the proof of Theorem 2.2, one can show that there exists a unique ternaryJordan ring derivation D : A→ A satisfying
‖f(x)−D(x)‖ ≤ 2
4− Lϕ(x, x, 0)
for all x ∈ A. For the caseϕ(x, y, z) := δ + θ(‖x‖r + ‖y‖r + ‖z‖r),
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
832 Gordji et al 829-834
Ternary Jordan ring derivations on Banach ternary algebras
(where θ, δ are nonnegative real numbers and 0 < r < 1, there exists a unique ternary Jordan ring derivation D : A→ Xsatisfying
‖f(x)−D(x)‖ ≤ 4δ
8− 2r+
8θ
8− 2r‖x‖r
for all x ∈ A.
Now, we formulate a theorem for the superstability of ternary Jordan ring derivations.
Theorem 2.5. Suppose that there exist a function ϕ : A×A×A→ [0,∞) and a constant 0 < L < 1 such that
ψ(x
2,y
2,z
2
)≤ L
8ϕ(x, y, z)
for all x, y, z ∈ A. Moreover, if f : A→ A is an additive mapping such that
‖f([x, y, z] + [y, z, x] + [z, x, y])− [f(x), y, z]− [x, f(y), z]− [x, y, f(z)]− [f(y), z, x]
−[y, f(z), x]− [y, z, f(x)]− [f(z), x, y]− [z, f(x), y]− [z, x, f(y)]‖ ≤ ϕ(x, y, z)
for all x, y, z ∈ A, then f is a ternary Jordan ring derivation.
Proof. The proof is similar to the proof of Theorem 2.2. We will omit it.
Corollary 2.6. Let θ, s be nonnegative real numbers and s > 3. If f : A→ A is an additive mapping such that
‖f([x, y, z] + [y, z, x] + [z, x, y])− [f(x), y, z]− [x, f(y), z]− [x, y, f(z)]− [f(y), z, x]
−[y, f(z), x]− [y, z, f(x)]− [f(z), x, y]− [z, f(x), y]− [z, x, f(y)]‖ ≤ θ(‖x‖s + ‖y‖s + ‖z‖s)
for all x, y, z ∈ A, then f is a ternary Jordan ring derivation.
Remark 2.7. Suppose that there exist a function ψ : A×A×A→ [0,∞) and a constant 0 < L < 1 such that
ϕ(2x, 2y, 2z) ≤ 2Lϕ(x, y, z)
for all x, y, z ∈ A. Moreover, if f : A→ A is an additive mapping such that
‖f([x, y, z] + [y, z, x] + [z, x, y])− [f(x), y, z]− [x, f(y), z]− [x, y, f(z)]− [f(y), z, x]
−[y, f(z), x]− [y, z, f(x)]− [f(z), x, y]− [z, f(x), y]− [z, x, f(y)]‖ ≤ ϕ(x, y, z)
for all x, y, z ∈ A, then f is a ternary Jordan ring derivation.
Acknowledgments
S. Y. Jang was supported by the Research Fund, University of Ulsan, 2014.
References
[1] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50-59.[2] J. Bae, W. Park, A functional equation having monomials as solutions, Appl. Math. Comput. 216 (2010), 87-94.[3] F. Bagarello, G. Morchio, Dynamics of mean-field spin models from basic results in abstract differential equations, J.
Stat. Phys. 66 (1992), 849-866.[4] M. Bavand Savadkouhi, M. Eshaghi Gordji, J. M. Rassias and N. Ghobadipour, Approximate ternary Jordan deriva-
tions on Banach ternary algebras, J. Math. Phys. 50, Art. ID 042303 (2009).[5] L. Cadariu, L. Gavruta, P. Gavruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013),
60-67.[6] L. Cadariu, V. Radu, On the stability of the Cauchy functional equation: A fixed point approach, Grazer Math. Ber.
346 (2004), 43-52.[7] A. Cayley, On the 34concomitants of the ternary cubic. Amer. J. Math. 4 (1881), 1-15.[8] A. Chahbi, N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6
(2013), 198-204.[9] Y. Cho, C. Park, M. Eshaghi Gordji, Approximate additive and quadratic mappings in 2-Banach spaces and related
topics, Int. J. Nonlinear Anal. Appl. 3 (2012), No. 1, 75-81.[10] Y. Cho, C. Park, M. Eshaghi Gordji, Approximate additive and quadratic mappings in 2-Banach spaces and related
topics, Int. J. Nonlinear Anal. Appl. 3 (2012), 75-81.[11] J. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric
space, Bull. Amer. Math. Soc. 74 (1968), 305-309.[12] M. Eshaghi Gordji, A. Ebadian, N. Ghobadipour, J. M. Rassias, M. B. Savadkouhi, Approximately ternary homo-
morphisms and derivations on C∗-ternary algebras, Abs. Appl. Anal. 2012, Art. ID 984160 (2012).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
833 Gordji et al 829-834
M. Eshaghi Gordji, Sh. Bazeghi, C. Park, S. Y. Jang
[13] P. Gavruta, L. Gavruta, A new method for the generalized Hyers-Ulam-Rassias stability, Int. J. Nonlinear Anal.Appl. 1 (2010), 11-18.
[14] N. Ghobadipour, C. Park, Cubic-quartic functional equations in fuzzy normed spaces, Int. J. Nonlinear Anal. Appl.1 (2010), 12-21.
[15] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl Acad. Sci. USA 27 (1941), 222-224.[16] K. Jun, H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl.
274 (2002), 867-878.[17] M. Kapranov, I. M. Gelfand, A. Zelevinskii, Discriminants, Resultants and Multidimensional Determinants,
Birkhauser, Berlin, 1994.[18] M. Kim, Y. Kim, G. A. Anastassiou, C. Park, An additive functional inequality in matrix normed modules over a
C∗-algebra, J. Comput. Anal. Appl. 17 (2014), 329-335.[19] M. Kim, S. Lee, G. A. Anastassiou, C. Park, Functional equations in matrix normed modules, J. Comput. Anal.
Appl. 17 (2014), 336-342.[20] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C∗-algebras, J. Comput.
Anal. Appl. 15 (2013), 365-368.[21] C. Park, A. Najati, S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput.
Anal. Appl. 15 (2013), 452-462.[22] C. Park, A. Najati, Generalized additive functional inequalities in Banach algebras, Int. J. Nonlinear Anal. Appl. 1
(2010), 54-62.[23] C. Park, Th. M. Rassias, Isomorphisms in unital C∗-algebras, Int. J. Nonlinear Anal. Appl. 1 (2010), 1-10.[24] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.[25] S. Schin, D. Ki, J. Chang, M. Kim, Random stability of quadratic functional equations: a fixed point approach, J.
Nonlinear Sci. Appl. 4 (2011), 37-49.[26] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach
algebras, J. Comput. Anal. Appl. 13 (2011), 1097-1105.[27] S. Shagholi, M. Bavand Savadkouhi, M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Frechet
algebras, J. Comput. Anal. Appl. 13 (2011), 1106-1114.[28] D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C∗-homomorphisms, J. Comput. Anal.
Appl. 16 (2014), 964-973.[29] D. Shin, C. Park, Sh. Farhadabadi, Stability and superstability of J∗-homomorphisms and J∗-derivations for a
generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125-134.[30] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed. Wiley, New York, 1940.[31] L. Vainerman, R. Kerner, On special classes of n-algebras, J. Math. Phys. 37, Art. ID 2553 (1996).[32] S. Zolfaghari, Stability of generalized QCA-functional equation in p-Banach spaces, Int. J. Nonlinear Anal. Appl. 1,
(2010) , 84-99.[33] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013),
51-59.[34] S. Zolfaghari, Approximation of mixed type functional equations in p-Banach spaces, J. Nonlinear Sci. Appl. 3
(2010), 110-122.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
834 Gordji et al 829-834
Initial value problems for a nonlinear
integro-differential equation of mixed type in
Banach spaces∗
Xiong-Jun Zheng†, Jin-Ming WangCollege of Mathematics and Information Science, Jiangxi Normal University
Nanchang, Jiangxi 330022, People’s Republic of China
Abstract
In this paper, we discuss the following initial value problem for first order nonlinearintegro-differential equations of mixed type in a Banach space:
u′ = f(t, u, Tu, Su)u(t0) = u0.
In the case of the integral kernel k(t, s) of the operator (Tu)(t) =∫ t
t0k(t, s)u(s)ds being
unbounded, we obtain the existence of maximal and minimal solutions for the aboveproblem by establishing a new comparison theorem.
Keywords: noncompactness measure, unbounded integral kernel, maximal andminimal solutions, integro-differential equations.
1 Introduction and Preliminaries
Suppose that E is a Banach space. In this paper, We consider the following initial valueproblem for first order nonlinear integro-differential equations of mixed type in E:
u = f(t, u, Tu, Su)u(t0) = u0,
(1.1)
where f ∈ C[J × E × E × E, E], J = [t0, t0 + a](a > 0), u0 ∈ E, and
(Tu)(t) =
∫ t
t0
k(t, s)u(s)ds, (Su)(t) =
∫ t0+a
t0
h(t, s)u(s)ds. (1.2)
∗The work was supported by the Natural Science Foundation of Jiangxi Province (No. 20122BAB201008,20143ACB21001) and Science and Technology Plan of Education Department of Jiangxi Province (No.GJJ08169).†Corresponding author. E-mail address: [email protected], [email protected].
1
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
835 Xiong-Jun Zheng et al 835-847
In (1.2), k(t, s) = ρ(t,s)(t−s)α (0 < α < 1), ρ(t, s) ∈ C[D,R+], and h(t, s) ∈ C[D0, R
+], where
R+ = [0,+∞), D =
(t, s) ∈ R2∣∣t0 ≤ s ≤ t ≤ t0 + a
, D0 =
(t, s) ∈ R2
∣∣(t, s) ∈ J × J.Here, k(t, s) is unbounded on D, ρ(t, s) is bounded on D, and h(t, s) is bounded on D0.Set R0 = max
ρ(t, s)
∣∣(t, s) ∈ D, h0 = maxh(t, s)
∣∣(t, s) ∈ D0
.
The study of initial value problems for nonlinear integro-differential equations has beenof great interest for many researchers for its physical backgrounds and applications inmathematical models. We refer the reader to [1, 5–12] and references therein for somerecent results on equation (1.1). However, in many earlier results, the kernel k(t, s) ofthe operator T is bounded. In this paper, we will make further study on the initial valueproblem (1.1) in the case of k(t, s) being unbounded. By establishing a comparison theorem,we achieve an existence theorem about minimal and maximal solutions for equation (1.1).
Throughout the rest of this paper, let (E, ‖ · ‖) be a real Banach space and P be a conein E which defines a partial ordering in E denoted by ”≤”.
Suppose that E∗ is the dual space of E, the dual cone of the cone P is P ∗ = ϕ ∈E∗|ϕ(x) ≥ 0,∀x ∈ P. A cone P ⊂ E is said to be normal there exists a constant γ > 0such that
θ ≤ x ≤ y =⇒ ‖x‖ ≤ γ‖y‖,∀x, y ∈ E.
The cone P is normal if and only if any order interval [x, y] = z ∈ E|x ≤ z ≤ y isbounded in norm(see [3]). Set
C[J,E] =u(t) : J → E
∣∣∣u(t) is continuous on J
,
C1[J,E] =u(t) : J → E
∣∣∣u(t) and u′(t) are continuous on J
.
Let ‖u‖c = maxt∈J‖u(t)‖ be a norm for u ∈ C[J,E], then C[J,E] will be a Banach space
with norm ‖·‖c. It is easy to know Pc =u ∈ C[J,E]
∣∣u(t) ≥ θ,∀t ∈ J
is a cone in C[J,E].The cone Pc defines an ordering in C[J,E] which also denoted by ”≤” here. Obviously,when the cone P is normal, Pc is a normal cone in C[J,E].
Assume that V is a bounded set in E. The Kuratowski measure of noncompactnessα(V ) and the Hausdorff measure of noncompactness β(V ) are defined respectively as follow:
α(V ) = infδ > 0|V can be expressed as the union S =m⋃i=1
Vi of a finite number of sets Vi
with diameter diam(Vi) ≤ δ,
β(V ) = infδ > 0
∣∣∣V can be covered by a finite number of closed balls Vi with diameter
diam(Vi) ≤ δ
.
The relationship of the two noncompactness measures is
β(V ) ≤ α(V ) ≤ 2β(V ). (1.3)
2
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
836 Xiong-Jun Zheng et al 835-847
For the basic properties of cones and noncompactness measures, we refer the reader to[2–4]. For convenience, the Kuratowski measure of noncompactness for bounded sets inE and C[J,E] are all denoted by α(·). In the sequel, we denote B(t) = u(t)|u ∈ B,(TB)(t) = (Tu)(t)|u ∈ B, (SB)(t) = (Su)(t)|u ∈ B for all B ⊂ C[J,E] with t ∈ J .
Lemma 1.1. Let m ∈ C1[J,R1] be such that
m′(t) ≥ −Mm(t)−N∫ t
t0
k(t, s)m(s)ds, m(t0) ≥ 0, t ∈ J, (1.4)
where M ≥ 0 and N ≥ 0 are two constants satisfying one of the following conditions:(i)
NR0eMa a
2−α
1− α≤ 1; (1.5)
(ii)
aM +NR0a
2−α
1− α≤ 1. (1.6)
Then m(t) ≥ 0 for all t ∈ J .
Proof. Case 1. If the condition (i) is established, let v(t) = m(t)eMt. From (1.4), we have
v′(t) ≥ −N∫ t
t0
k∗(t, s)v(s)ds, ∀t ∈ J, v(t0) ≥ 0, (1.7)
where k∗(t, s) = k(t, s)eM(t−s). Now, we prove that
v(t) ≥ 0, ∀t ∈ J. (1.8)
In fact, if there exists t0 ≤ t1 ≤ t0 + a such that v(t1) < 0 and let maxv(t) : t0 ≤ t ≤t1 = b, then b ≥ 0. If b = 0, then v(t) ≤ 0 for all t0 ≤ t ≤ t1 and so (1.7) implies that
v′(t) ≥ 0, ∀t0 ≤ t ≤ t1.
Hence we have v(t1) ≥ v(t0) = m(t0)eMt0 ≥ 0, which contradicts v(t1) < 0.
If b > 0, then there exists t0 ≤ t2 < t1 such that v(t2) = b > 0 and so there existst2 < t3 < t1 such that v(t3) = 0. Then, by the mean value theorem, there exists t2 < t4 < t3such that
v′(t4) =v(t3)− v(t2)
t3 − t2=−v(t2)
t3 − t2=−b
t3 − t2< − b
a. (1.9)
On the other hand, from (1.7), we have
v′(t4) ≥ −N∫ t4
t0
k∗(t4, s)v(s)ds
3
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
837 Xiong-Jun Zheng et al 835-847
≥ −Nb∫ t4
t0
k∗(t4, s)ds
= −Nb∫ t4
t0
ρ(t4, s)
(t4 − s)αeM(t4−s)ds
≥ −NbR0
∫ t4
t0
(t4 − s)−αeM(t4−s)ds
≥ −NbR0eMa
∫ t4
t0
(t4 − s)−αds
= −NbR0eMa (t4 − t0)1−α
1− α
≥ −NbR0eMa a
1−α
1− α.
Then from (1.9), we have NR0eMa a2−α
1−α > 1 which contradicts (1.5). Therefore, (1.8)is true and so m(t) ≥ 0 for all t ∈ J .
Case 2. If the assumption (ii) holds, but the conclusion does not hold, then thereexists t1 ∈ (t0, t0 + a] such that
m(t1) = mint∈J
m(t) < 0,
and so m′(t1) ≤ 0. If maxt0≤t≤t1
m(t) ≤ 0, from (1.4), we have
0 ≥ m′(t1) ≥ −Mm(t1)−N∫ t1
t0
k(t1, s)m(s)ds ≥ −Mm(t1) > 0,
which is a contradictory statement. Therefore, there exists t2 ∈ [t0, t1) such that m(t2) =maxt0≤t≤t1
m(t) = µ > 0. Then, by the mean value theorem, there exists t3 ∈ (t2, t1) such that
m′(t3) =m(t1)−m(t2)
t1 − t2< −µ
a.
It follows from (1.4) that
−µa> m′(t3) ≥ −Mm(t3)−N
∫ t3
t0
ρ(t3, s)
(t3 − s)αm(s)ds
≥ −Mµ−NR0µ
∫ t3
t0
1
(t3 − s)αds
= −Mµ−NR0µ(t3 − t0)1−α
1− α
≥ −Mµ−NR0µa1−α
1− α,
i.e. aM +NR0a2−α
1−α > 1 which contradicts (1.6). The Lemma is proved.
4
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
838 Xiong-Jun Zheng et al 835-847
Lemma 1.2. Let m ∈ C[J,R+] be such that
m(t) ≤M1
∫ t
t0
m(s)ds+M2(t− t0)∫ t0+a
t0
m(s)ds, t ∈ J (1.10)
where M1 > 0, M2 ≥ 0, are constants for satisfying one of the following conditions:(i)aM2(e
aM1 − 1) < M1, (ii)a(2M1 + aM2) < 2. Then m(t) ≡ 0, t ∈ J .
Proof. Case 1. If the condition (i) holds, letting v(t) = m(t)eMt, then m1(t0) = 0,m′1(t) = m(t), t ∈ J . If m1(t0 + a) 6= 0, it follows from (1.10) that
m′1(t) ≤M1m1(t) + aM2m1(t0 + a), t ∈ J
and from e−M1(t−t0) > 0 we have(m1(t)e
−M1(t−t0))′≤ aM2m1(t0 + a)e−M1(t−t0), t ∈ J.
Now, we integrate the above inequality between t0 and t with noticing m1(t0) = 0, we canobtain
m1(t)e−M1(t−t0) ≤ aM2m1(t0 + a)
∫ t
t0
e−M1(s−t0)ds
≤ aM2
M1m1(t0 + a)
(1− e−M1(t−t0)
), t ∈ J.
By choosing t = t0 + a, we can get
aM2
(eaM1 − 1
)≥M1
which contradicts (i). Consequently, m1(t0 + a) =∫ t0+at0
m(s)ds = 0 which implies m(t) ≡0, t ∈ J .
Case 2. If the condition (ii) is established, it follows from (1.10) that
m(t) ≤ [M1 +M2(t− t0)]∫ t0+a
t0
m(s)ds.
Integrating the above inequality between t0 and t0 + a, we get∫ t0+a
t0
m(t)dt ≤[aM1 +
a2M2
2
] ∫ t0+a
t0
m(s)ds.
From the above inequality and conditio (ii), it follows that∫ t0+at0
m(t)dt = 0, so m(t) ≡0, t ∈ J . This completes the proof.
Lemma 1.3. If B is a equicontinuous bounded set in ⊂ C[J,E], then α(B) = maxt∈J
α(B(t)).
5
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
839 Xiong-Jun Zheng et al 835-847
Lemma 1.4. If B is a equicontinuous bounded set in ⊂ C[J,E] with J = [a, b], thenα(u(t)|u ∈ B) is continuous with respect to t ∈ J and
α
(∫ b
au(t)dt
∣∣∣∣u ∈ B) ≤ ∫ b
aαu(t)
∣∣u ∈ B dt.Lemma 1.5. (see [2]) Let E be a separable Banach space, J = [a, b] and un : J → Ebe continuous abstract function sequences. If there exists a function φ ∈ L[a, b] such that‖un(t)‖ ≤ φ(t), t ∈ J, n = 1, 2, 3, · · · , then β
(un(t)
∣∣n = 1, 2, 3, · · ·)
is integrable on Jand
β
(∫ b
aun(t)dt
∣∣∣∣n = 1, 2, 3, · · ·)≤∫ b
aβ(un(t)
∣∣n = 1, 2, 3, · · ·)dt.
Now, we give our assumptions:(H1) There exist v0, ω0 ∈ C1[J,E] such that v0(t) ≤ ω0(t)(t ∈ J) and v0, ω0 are a lower
solution and an upper solution respectively for the initial value problem (1.1), that is
v′0 ≤ f(t, v0, T v0, Sv0), ∀t ∈ J ; v0(t0) ≤ u0,
ω′0 ≥ f(t, ω0, Tω0, Sω0), ∀t ∈ J ; ω0(t0) ≥ u0.
(H2) For any t ∈ J , any u, v ∈ [v0, ω0] = u ∈ C[J,E]|v0 ≤ u ≤ ω0 and u ≤ v, wehave
f(t, v, Tv, Sv)− f(t, u, Tu, Su) ≥ −M(v − u)−NT (v − u),
where M > 0, N ≥ 0 are constants satisfying the condition (i) or (ii) in Lemma 1.1.(H3) For any t ∈ J and equicontinuous bounded monotone sequences B = un ⊂
[v0, ω0], we have
α(f(t, B(t), (TB)(t), (SB)(t)) ≤ c1α(B(t)) + c2α((TB)(t)) + c3α((SB)(t)),
where ci(i = 1, 2, 3) are constants satisfying one of the following two conditions:
(i)ah0c3
(e2a(c1+M+
c2R0a1−α
1−α +2NR0a
1−α1−α ) − 1
)< c1 +M + c2R0a1−α
1−α + 2NR0a1−α
1−α ;
(ii)a(
2c1 + 2M + 2c2R0a1−α
1−α + 4NR0a1−α
1−α + ah0c3
)< 1.
2 Main results
Theorem 2.1. Let E be a real Banach space, P ⊂ E be a normal cone and the conditions(H1), (H2), (H3) be satisfied. Then the initial value problem (1.1) has a minimal solu-tion and a maximal solution u, u∗ ∈ C1[J,E] in [v0, ω0], and for the initial value v0 andω0, the iterative sequences vn(t) and ωn(t) defined by the following formulas convergeuniformly to u(t), u∗(t) on J according to the norm in E respectively:
vn(t) = u0e−M(t−t0) +
∫ t
t0
eM(s−t)[f(s, vn−1(s), (Tvn−1)(s), (Svn−1)(s))6
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
840 Xiong-Jun Zheng et al 835-847
+Mvn−1(s)−NT (vn − vn−1)(s)]ds, ∀t ∈ J, (2.1)
ωn(t) = u0e−M(t−t0) +
∫ t
t0
eM(s−t)[f(s, ωn−1(s), (Tωn−1)(s), (Sωn−1)(s))+Mωn−1(s)−NT (ωn − ωn−1)(s)
]ds, ∀t ∈ J (n = 1, 2, 3, · · · ). (2.2)
Moreover, there holds
v0 ≤ v1 ≤ · · · ≤ vn ≤ · · · ≤ u ≤ u∗ ≤ · · · ≤ ω1 ≤ ω0. (2.3)
Proof. For any η ∈ [v0, ω0], we consider the initial value problem of linear integro-differentialequation in Banach space E:
u′ = g(t)−Mu−NTu, u(t0) = u0, (2.4)
where g(t) = f(t, η(t), (Tη)(t), (Sη)(t)
)+Mη(t) +N(Tη)(t). It is easy to show that u is a
solution of the linear initial value problem (2.4) if and only if u is the fixed point in C[J,E]of the following operator
(Au)(t) = u0e−M(t−t0) +
∫ t
t0
eM(s−t)[g(s)−N(Tu)(s)]ds. (2.5)
In the following, we will prove there exists n0 such that An0 is a contraction operator.For any u, v ∈ C[J,E], t ∈ J , it follows from (2.5) that
‖(Au)(t)− (Av)(t)‖ ≤ N
∫ t
t0
‖T (u− v)(s)‖ds
≤ N
∫ t
t0
[∫ s
t0
k(s, τ)‖u(τ)− v(τ)‖dτ]ds
= N
∫ t
t0
[∫ s
t0
ρ(s, τ)
(s− τ)α‖u(τ)− v(τ)‖dτ
]ds
≤ NR0‖u− v‖c∫ t
t0
∫ s
t0
1
(s− τ)αdτds
=NR0(t− t0)2−α
(1− α)(2− α)‖u− v‖c. (2.6)
In the same way, by (2.5) and (2.6), we have
∥∥(A2u)(t)− (A2v)(t)∥∥ ≤ N
∫ t
t0
‖T (Au−Av)(s)‖ds
≤ N
∫ t
t0
[∫ s
t0
k(s, τ)‖(Au)(τ)− (Av)(τ)‖dτ]ds
7
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
841 Xiong-Jun Zheng et al 835-847
≤ NR0
∫ t
t0
[∫ s
t0
1
(s− τ)αNR0(τ − t0)2−α
(1− α)(2− α)‖u− v‖cdτ
]ds
=(NR0)
2
(1− α)(2− α)‖u− v‖c
∫ t
t0
[∫ s
t0
(τ − t0)2−α
(s− τ)αdτ
]ds
≤ (NR0)2‖u− v‖c
(1− α)(2− α)
∫ t
t0
∫ s
t0
(s− τ)−α(s− t0)2−αdτds
=(NR0)
2‖u− v‖c(1− α)(2− α)
∫ t
t0
(s− t0)3−2α
1− αds
=(NR0)
2
(1− α)22(2− α)2‖u− v‖c(t− t0)4−2α.
It is easy to prove that by mathematical induction
∥∥(Anu)(t)− (Anv)(t)∥∥ ≤ (NR0)
n
n![(1− α)(2− α)]n(t− t0)n(2−α)‖u− v‖c, t ∈ J, n = 1, 2, 3, · · · .
Thus
‖Anu−Anv‖c ≤(NR0a
2−α)n
n![(1− α)(2− α)]n‖u− v‖c, n = 1, 2, 3, · · · .
We can choose n0 ∈ 1, 2, 3, · · · such that (NR0a2−α)n
n![(1−α)(2−α)]n < 1, and so An0 a contraction
operator in C[J,E]. Therefore, it follows from the principle of contraction mapping thatAn0 , that is, A has a unique fixed point uη in C[J,E] which implies the linear initial valueproblem (2.4) has a unique solution uη in C[J,E]. Now, we define a operator
Bη = uη (2.7)
where uη is a unique solution for η of the linear initial value problem (2.4), and satisfies
u′η = f(t, η(t), (Tη)(t), (Sη)(t))−M(uη(t)− η(t))−NT (uη − η)(t), uη(t0) = u0.
Then B : [v0, ω0] −→ C[J,E], and the iterative sequences (2.1)(2.2) can be written
vn = Bvn−1, ωn = Bωn−1, n = 1, 2, 3, · · · . (2.8)
Moreover, we claim that the operator B defined by (2.7) satisfiesi)
v0 ≤ Bv0, Bω0 ≤ ω0; (2.9)
ii)Bη1 ≤ Bη2, ∀η1, η2 ∈ [v0, ω0], η1 ≤ η2. (2.10)
Next, we will prove i) and ii). Firstly, we prove the result i). Set v1 = Bv0, it followsfrom the definition of B that
8
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
842 Xiong-Jun Zheng et al 835-847
v′1 = f(t, v0, T v0, Sv0)−M(v1 − v0)−NT (v1 − v0), v1(t0) = u0. (2.11)
For any ϕ ∈ P ∗, let m(t) = ϕ(v1(t)−v0(t)), it follows from (2.11) and the assumption (H1)that
m′(t) ≥ −Mm(t)−N∫ t
t0
k(t, s)m(s)ds, m(t0) ≥ 0.
Thus, by lemma 1.1, it follows that m(t) ≥ 0 for all t ∈ J , which implies v1(t)− v0(t) ≥ 0for all t ∈ J . It follows theorem 2.4.3 in [3] that v0 ≤ Bv0. Similarly, we can prove thatBω0 ≤ ω0. Consequently, the result i) is proved.
Next, we prove ii). Let uη1 = Bη1, uη2 = Bη2, it follows from the hypothesis (H2) andthe definition of B that
u′η1 − u′η2 = f(t, η2, Tη2, Sη2)−M(uη2 − η2)−NT (uη2 − η2)
−f(t, η1, Tη1, Sη1) +M(uη1 − η1) +NT (uη1 − η1)≥ −M(uη2 − uη1)−NT (uη2 − uη1) (2.12)
anduη2(t0)− uη1(t0) = u0 − u0 = θ. (2.13)
For any ϕ ∈ P ∗, let m(t) = ϕ(uη2(t)− uη1(t)). From (2.12) and (2.13), it follows that
m′(t) ≥ −Mm(t)−N∫ t
t0
k(t, s)m(s)ds, m(t0) = 0
Thus, by lemma 1.1, it follows that m(t) ≥ 0 for all t ∈ J , which implies uη2(t)−uη1(t) ≥ θ,t ∈ J , that is, Bη1 ≤ Bη2. So the result ii) is proved.
Form (2.8)-(2.10) and observing that v0 ≤ ω0, it follows that
v0 ≤ v1 ≤ · · · ≤ vn ≤ · · · ≤ ωn ≤ · · · ≤ ω1 ≤ ω0. (2.14)
and B is a mapping with [v0, ω0] into [v0, ω0].In the following, we prove that vn(t) converges uniformly to some element u ∈ C[J,E]
in J . By the normality of P , the cone Pc is normal in C[J,E] which implies the orderinterval [v0, ω0] is a bounded set in C[J,E]. Then, it follows from (2.14) that vn is abounded set in C[J,E]. On the one hand, for any η ∈ [v0, ω0], by the conditions (H1) and(H2), we have
v′0 +Mv0 +NTv0 ≤ f(t, v0, T v0, Sv0) +Mv0 +NTv0
≤ f(t, η, Tη, Sη) +Mη +NTη
≤ f(t, ω0, Tω0, Sω0) +Mω0 +NTω0
≤ ω′0 +Mω0 +NTω0.
9
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
843 Xiong-Jun Zheng et al 835-847
Then, by the normality of Pc, the setf(t, η, Tη, Sη) + Mη + NTη
∣∣η ∈ [v0, ω0]
is abounded set in C[J,E]. On the other hand, the set
Tη∣∣η ∈ [v0, ω0]
is also a bounded set
in C[J,E], because it follows from the boundedness of [v0, ω0] that for any η ∈ [v0, ω0],
‖Tη(t)‖ ≤∫ t
t0
k(t, s)‖η(s)‖ds
≤ ‖η‖c∫ t
t0
ρ(t, s)
(t− s)αds
≤ R0‖η‖c∫ t
t0
1
(t− s)αds
= R0‖η‖c(t− t0)1−α
1− α.
Therefore, f(t, η, Tη, Sη)|η ∈ [v0, ω0] is a bounded set in C[J,E]. Thus, from
v′n = f(t, vn−1, T vn−1, Svn−1)−M(vn−vn−1)−NT (vn−vn−1), t ∈ J, n = 1, 2, 3, · · · , (2.15)
it follows that v′n|n = 1, 2, 3, · · · is a bounded set in C[J,E]. Applying the mean valuetheorem, we see that all the functions vn(t)|n = 1, 2, 3, · · · is equicontinuous on J . FromLemma 1.3, we have
α(vn|n = 1, 2, 3, · · · ) = maxt∈J
α(vn(t)|n = 1, 2, 3, · · · ). (2.16)
Now, we prove α(vn|n = 1, 2, 3, · · · ) = 0. From (2.4), (2.5), (2.7) and (2.8), it followsthat
vn(t) = u0e−M(t−t0) +
∫ t
t0
eM(s−t)[f (s, vn−1(s), (Tvn−1), (Svn−1)(s))
+Mvn−1(s)−NT (vn − vn−1)(s)]ds. (2.17)
Let m(t) = αvn(t)|n = 1, 2, 3, · · · , then m(t0) = α(u0) = 0, m ∈ C[J,R+]. Forevery n, by the continuity of vn(t), vn(t)|t ∈ J is a separable set in E, so vn(t)|t ∈J, n = 1, 2, 3, · · · is a separable set in E. Thus, we can assume that E is a separableBanach space without loss of generality
(otherwise, the closed subspace in E is spanned
by vn(t)|t ∈ J, n = 1, 2, 3, · · · can be used in place of E). By (2.17), (1.3) and Lemma
1.5 and observing 0 < eM(s−t) ≤ 1, (t, s) ∈ D, we can obtain
m(t) ≤ α
(∫ t
t0
eM(s−t)[f(s,B(s), (TB)(s), (SB)(s)) +MB(s)−NT (B1 −B)(s)]ds
)≤ 2β
(∫ t
t0
eM(s−t)[f(s,B(s), (TB)(s), (SB)(s)) +MB(s)−NT (B1 −B)(s)]ds
)≤ 2
∫ t
t0
β[f(s,B(s), (TB)(s), (SB)(s)) +MB(s)−NT (B1 −B)(s)
]ds
10
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
844 Xiong-Jun Zheng et al 835-847
≤ 2
∫ t
t0
[β (f (s,B(s), (TB)(s), (SB)(s)))
+Mβ(B(s)) +Nβ(T (B1 −B)(s))]ds. (2.18)
where B(s) = vn(s)|n = 0, 1, 2, · · · , B1(s) = vn(s)|n = 1, 2, 3, · · · . By the condition(H3) and (1.3), we have
β (f(s,B(s), (TB)(s), (SB)(s)))
≤ α (f(s,B(s), (TB)(s), (SB)(s)))
≤ c1α(B(s)) + c2α(((TB))(s)) + c3α((SB)(s)). (2.19)
From the uniform boundedness of B(s) and uniform continuity of h(t, s), it easy to prove(SB)(s) is a equicontinuous bounded set, so it follows from Lemma 1.4 that
α((SB)(s)) = α
(∫ t0+a
t0
h(s, τ)B(τ)dτ
)≤ h0
∫ t0+a
t0
m(τ)dτ. (2.20)
Now, we consider dealing with α((TB)(s)
). Firstly,∫ s
t0
k(s, τ)dτ =
∫ s
t0
ρ(s, τ)
(s− τ)αdτ ≤ R0
∫ s
t0
1
(s− t)αdτ ≤ R0a
1−α
1− α.
Since B(s) is equicontinuous bounded sequences and α(B(s)) = m(s), there exists a par-
tition B(s) =l⋃
i=1Bi such that the partition (TB)(s) =
l⋃i=1
TBi exists, where TBi =∫ st0k(s, τ)vi(τ)dτ
∣∣vi ∈ Bi, so we have
diam(TBi) = sup∀v1i ,v2i ∈Bi
∥∥∥∥∫ s
t0
k(s, τ)[v1i (τ)− v2i (τ)
]dτ
∥∥∥∥≤ R0a
1−α
1− αsup
∀v1i ,v2i ∈Bi
∥∥v1i (τ)− v2i (τ)∥∥
=R0a
1−α
1− αdiam(Bi)
<R0a
1−α
1− αα(B(s)) +
R0a1−α
1− α· ε.
By using the arbitrariness of ε, we have
α(TB(s) ≤ R0a1−α
1− αα(B(s)) =
R0a1−α
1− αm(s), (2.21)
and by (1.3), we have
β (T (B1 −B)(s)) ≤ α (T (B1 −B)(s)) ≤ 2R0a1−α
1− αm(s). (2.22)
11
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
845 Xiong-Jun Zheng et al 835-847
Thus, it follows from (2.18)-(2.22) that
m(t) ≤ 2
∫ t
t0
[c1m(s) +
c2R0a1−α
1− αm(s) + c3h0
∫ t0+a
t0
m(τ)dτ
+Mm(s) +2NR0a
1−α
1− αm(s)
]ds
= 2
(c1 +M +
c2R0a1−α
1− α+
2NR0a1−α
1− α
)∫ t
t0
m(s)ds
+2h0c3(t− t0)∫ t0+a
t0
m(s)ds.
Therefore, from Lemma 1.2 and the conditions (i)(ii) of the assumption (H3), we havem(t) ≡ 0, t ∈ J which implies αvn|n = 1, 2, 3, · · · = 0 from (2.16), that is, vn ⊂ [v0, ω0]is a relatively compact set in C[J,E]. Thus there exists a subsequence vnk ⊂ vn andsome u ∈ [v0, ω0] such that vnk converges to u in norm ‖ · ‖c. Further, from (2.14) andthe normality of Pc, it is easy to prove that vn converges to u in norm ‖ · ‖c, that is,vn(t) converges uniformly to u(t) on J according to the norm in E. Similarly, we canprove that ωn(t) converges uniformly to some u∗ ∈ [v0, ω0] on J according to the normin E. Clearly, the result (2.3) is true.
Finally, we prove that u and u∗ are a minimal solution and a maximal solution respec-tively of the initial value problem (1.1). Let
un(t) = −M (vn(t)− vn−1(t))−NT (vn − vn−1) (t), t ∈ J, n = 1, 2, 3, · · · .
Since vn(t) converges uniformly to u(t) on J , it is easy to prove ‖un‖c → 0(n → ∞).Setting εn = ‖un‖c, from (2.15), we get
v′n = f (t, vn−1, T vn−1, Svn−1) + un(t), vn(t0) = u0, ‖un(t)‖ ≤ εn, t ∈ J.
Applying Corollary 2.1.1 in [4], we know u is a solution of the initial value problem (1.1).Similarly, we can prove that ω∗ is also a solution of the initial value problem (1.1). If u isa solution in [v0, ω0] of the initial value problem (1.1), then Bu = u, so by v0 ≤ u ≤ ω0
and (2.8)-(2.10), it is easy to obtain
vn ≤ u ≤ ωn, n = 1, 2, 3, · · · .
Letting n → ∞ in above formula, we get u ≤ u ≤ u∗. Consequently, u, u∗ are theminimal solution and maximal solution of the initial value problem (1.1) respectively. Thiscompletes the proof.
References
[1] B. Ahmad, et al., Some existence theorems for fractional integrodifferential equation-s and inclusions with initial and non-separated boundary conditions, Bound. ValueProbl. 2014, 2014:249.
12
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
846 Xiong-Jun Zheng et al 835-847
[2] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
[3] D. Guo, Partial ordering methods in nonlinear problems, Shandong Science and Tech-nology Press, Jinan, China, 1997 (in Chinese).
[4] D. Guo, J. Sun, Ordinary differential equations in abstract spaces, Shandong Scienceand Technology Press, Jinan, China, 1989 (in Chinese).
[5] A. Jawahdou, Initial value problem of fractional integro-differential equations in Ba-nach space, Fract. Calc. Appl. Anal. 18 (2015), no. 1, 20–37.
[6] H. Lan, Y. Cui, Perturbation technique for a class of nonlinear implicit semilinearimpulsive integrodifferential equations of mixed type with noncompactness measure,Adv. Difference Equ. 2015, no. 1, 2015:11.
[7] L. Liu, C. Wu, F. Guo, A unique solution of initial value problems for first orderimpulsive integro-differential equations of mixed type in Banach spaces, J. Math. Anal.Appl., 275 (2002), 369–385.
[8] J. Machado et al., Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions, Fixed Point Theory Appl.2013, 2013:66.
[9] N. I. Mahmudov, S. Zorlu, Approximate controllability of fractional integrodifferentialequations involving nonlocal initial conditions, Bound. Value Probl. 2013, 2013:118.
[10] Z. Smarda, Y. Khan, Singular initial value problem for a system of integro-differentialequations unsolved with respect to the derivative, Appl. Math. Comput. 222 (2013),290–296.
[11] Y. Sun, Positive solutions for nonlinear integrodifferential equations of mixed type inBanach spaces, Abstr. Appl. Anal. 2013, Art. ID 787038.
[12] S. Xie, Existence of solutions for nonlinear mixed type integro-differential functionalevolution equations with nonlocal conditions, Bound. Value Probl. 2012, 2012:100.
13
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
847 Xiong-Jun Zheng et al 835-847
Solving the multicriteria transportation equilibrium system
problem with nonlinear path costs
Chaofeng Shi, Yingrui Wang
Abstract. In this paper, we present an self-adaptive algorithm for solving the multicriteriatransportation equilibrium system problem with variable demand and nonlinear path costs. Thepath cost function considered is comprised of three attributes, travel time, toll and travel fares,that are combined into a nonlinear generalized cost. Travel demand is determined endogenouslyaccording to a travel disutility function. Travelers choose routes with the minimum overall gener-alized costs. Numerical experiments are conducted to demonstrate the feasibility of the algorithmto this class of transportation equilibrium system problems.
Key Words and Phrases: multicriteria, general networks, nonlinear path costs, transportationequilibrium system
2000 Mathematics Subject Classification: 49J40, 90C33.
1 Introduction
Usually, there are more than one kind of goods transported through the traffic network, in reality. Aswe know, the transportation cost of one kind of goods can be affected by other kinds of goods under thesame traffic network. In detail, the flows of different kinds of goods are not independent. Generally, in2010, He et. al . [1] called this problem as dynamic traffic network equilibrium system. Several authors(see, for instance, [2-5]) study the model with elastic demands and develop some results in this contexttheoretical features and numerical procedures. For example, in the general economic case, the equilibriumcost will affect to the market demand of goods, so the O-D pair demand of these goods depends on theequilibrium cost and the equilibrium distribution. Therefore, it is reasonable to consider the trafficequilibrium problem with elastic demand when there are many kinds of goods transported through thesame traffic network. At the same time, the travel cost function is considered widely and deeply. It isgenerally accepted that travelers consider a number of criteria (e.g., time, money, distance, safety, routecomplexity, etc.) when selecting routes. Presumably, these criteria are then combined in some manner toform a generalized cost for each particular route or path under consideration, and a route selected basedon minimization of the generalized cost of the trip. Most commonly, it is assumed that travelers selectthe ’best’ route based on either a single criterion, such as travel time, or several criteria using a linear(or additive) path cost function. However, as pointed out by Gabriel and Bernstein [6], there are manysituations in which the linear path cost function is inadequate for addressing factors affecting a varietyof transportation policies. Such factors include: (i) Nonlinear valuation of travel time–small amountsof time are valued proportionately less than larger amounts of time. (ii) Emissions fees–emissions ofhydrocarbons and carbon monoxide are a nonlinear function of travel times. (iii) Path-specific tolls andfares–most existing fare and toll pricing structures are not directly proportional to either travel timeor distance. These, and other such factors, are generally difficult to accommodate without explicitlyusing path flows in the formulation and solution, particularly for traffic equilibrium problems involvingmulti-dimensional nonlinear path costs. Despite the obvious usefulness of incorporating multiple criteriaand relaxing the assumption of linear path costs for an important class of traffic equilibrium problems,there have been relatively few attempts to incorporate multiple criteria within route choice modeling.Under the assumption that the nonlinear path cost function is known a priori, Scott and Bernstein [7]solved a constrained shortest path problem (CSPP) to generate a set of Pareto optimal paths and thenidentify the best path by evaluating the cost values of the alternative paths. In a later study, Scottand Bernstein [8] embedded the CSPP into the gradient projection method to solve the non-additive
1
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
848 Chaofeng Shi et al 848-855
traffic equilibrium problem. Using a new gap function recently proposed by Facchinei and Soares [9], Loand Chen [10] reformulated the nonadditive traffic equilibrium problem as an equivalent unconstrainedoptimization and solved a special case involving fixed demand and route-specific costs. Chen et al. [11]provided a projection and contraction algorithm for solving the elastic traffic equilibrium problem withroute-specific costs. Recently, some formulations and properties of the non-additive traffic equilibriummodels were also explored, such as the nonlinear time/money relation [12], the uniqueness and convexityof the bicriteria traffic equilibrium problem [13]. Furthermore, Altman and Wynter [14] discussed the non-additive cost structures in both transportation and telecommunication networks. Howerver, there are fewresults to discuss the problem related to the transportation network system for the nonlinear multiciteriatransportation cost functions. On the other hand, Verma [15] investigated the approximation solvabilityof a new system of nonlinear variational inequalities involving strongly monotone mappings. In 2005,Bnouhachem [16] presented a new self-adaptive method for solving general mixed variational inequalities.In 2007, Shi [17] proposed a new self-adaptive iterative method for solving nonlinear variational ineualitysystem (SNVI) and proved the convergence of the proposed method. The numerical examples were givento illustrate the efficiency of the proposed method. In this paper, we consider the traffic equilibriumproblem with variable demand, fixed tolls, and a nonlinear path cost function. We first discuss themulticriteria traffic equilibrium problem and its equivalent nonlinear variational inequality formulation,and present the associated multicriteria shortest path problem and solution algorithm. We then explore anew self-adaptive iterative method (SI) developed by Shi [17] to solve SNVI that characterizes this class oftraffic equilibrium system problem. The SI method is simple and can handle a general monotone mapping.Unlike the non-smooth equations/sequential quadratic programming (NE/SQP) method proposed byGabriel and Bernstein [6] to solve the non-additive traffic equilibrium problem, the SI method does notrequire the mapping to be differentiable.
2 Preliminaries
Without loss of generality, we consider the case that there are only two kinds of goods transportedthrough the network. Suppose that a traffic network consists of a set N of nodes, a set Ω of origin-destination (O/D) pairs, and a set R of routes. Each route r ∈ R links one given origin-destination pairω ∈ Ω. The set of all r ∈ R which links the same origin-destination pair ω ∈ Ω is denoted by R(ω). Assumethat n is the number of the route in R and m is the number of origin-destination (O/D) pairs in Ω. Letvector Hi = (Hi
1,Hi2, · · · ,Hi
r, · · · ,Hin)T ∈ Rn i = 1, 2 denote the flow vector for the two kinds of goods,
where Hir, r ∈ R, denotes the flow in route r ∈ R. A feasible flow has to satisfy the capacity restriction
principle: λir ≤ Hi
r ≤ µir, for all r ∈ R , and a traffic conservation law:
∑r∈R(ω) Hi
r = ρiω(H1,H2) , for
all ω ∈ Ω, where λ andµ are given in Rn, is the travel demand related to the given pair ω ∈ Ω , andρi
ω(H1,H2) ≥ 0 denotes the travel demand vector, which generally depends on equilibrium cost and,essentially, on the equilibrium distribution H1andH2. Thus the set of all feasible flows is given by
Ki(H1,H2) := H ∈ Rn‖λi ≤ H ≤ µi,ΦH = ρi(H1,H2), (2.1)
where Φ = (δωr)m × n is defined as
δωr =
1 if r ∈ R(ω)0 Otherwise
Thus the set of feasible flows is given by K1(H1,H2)×K2(H1,H2) . We call that is a flow of the trafficnetwork system with elastic demands. As pointed out by Gabriel and Bernstein [6], the linear assumptionis rather restrictive and cannot adequately model certain important applications.
Let mappingCi : K → Rn be the cost function of the ith kinds of goods for i = 1, 2. Cir(H
1,H2)gives the marginal cost of transporting one additional unit of the ith kind of goods under the rth route.For the multicriteria traffic equilibrium problem with nonlinear path costs based on travel time, toll andtransportation fares, a possible nonlinear path cost function can be the following form:
Cir(H
1,H2) = gr(∑
a∈A
δrspatia(H1,H2)) +
∑
a∈A
δrspaτa +
∑
a∈A
δrspaf i
a(H1,H2), (2.2)
2
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
849 Chaofeng Shi et al 848-855
Where gr is a nonlinear function describing the value-of-time for path r, τa is the toll on link a , and fa
is the transportation fares function on link a .
Definition 2.1. (H1,H2) ∈ K1(H1,H2) ×K2(H1,H2) is an equilibrium flow if and only if for allω ∈ Ω and q, s, p, r ∈ R(ω) there holds
C1q (H1,H2) < C1
s (H1,H2) ⇒ H1q = µ1
q or H1s = λ1
s, (2.3)
C2p(H1,H2) < C2
r (H1,H2) ⇒ H2p = µ2
p or H2r = λ2
r,
3 Existence and Uniqueness of the solution for the multicriteriatransportation equilibrium system problem
The following result establishes relationship between the system of dynamic traffic equilibrium problemand a system of variational inequalities.
Theorem 3.1. (H1,H2) ∈ K1(H1,H2)×K2(H1,H2) is an equilibrium flow if and only if ,
< C1(H1,H2), F 1 −H1 >≥ 0 ∀F 1 ∈ K1(H1,H2), (3.1)< C2(H1,H2), F 2 −H2 >≥ 0 ∀F 2 ∈ K2(H1,H2),
Proof. First assume that (3.1) holds and (2.3) does not hold. Then there exist ω ∈ Ω and q, s ∈ R(ω)such that
Ciq(H
1,H2) < Cis(H
1,H2), Hiq < µi
s, Hiq > λi
s, i = 1, 2. (3.2)
Let δi = minµiq −Hi
q, his − λi
s, i = 1, 2.Then δi > 0, i = 1, 2.We define a vector Fi ∈ Ki(H1,H2), i = 1, 2, whose components are
F iq(t) = Hi
q + δi, F is(t) = Hi
s − δi, F ir = Hi
r, (3.3)
when r 6= q, s.Thus,
< Ci(H1,H2), F i −Hi >=n∑
j=1
Cij(H
1,H2)(F ij −Hi
j) = δi(Ciq(H
1,H2)− Cis(H
1,H2)) < 0, (3.4)
and so (3.1) is not satisfied. Therefore, it is proved that (3.1) implies (2.4).Next, assume that (2.4) holds. That is
Ciq(H
1,H2) < Cis(H
1,H2) ⇒ Hiq = µi
q, or His = µi
s, i = 1, 2. (3.5)
Let F i ∈ Ki(H1,H2) for i = 1, 2. Then (3.1) holds from Definition 2.1. The proof is completed.Furthermore, we discuss the existence and uniqueness of the solution for the dynamic traffic equilibriumsystem (3.1). In order to get our main results, the following definitions will be employed.
Definition 3.2. Ci(x, y)(i = 1, 2) is said to be θ-strictly monotone with respect to x on K1(H1,H2)×K2(H1,H2) if there exists θ > 0such that
< Ci(x1, y)− Ci(x2, y), x1 − x2 >≥ θ‖x1 − x2‖22, (3.6)
∀x1, x2 ∈ K1(H1,H2),∀y ∈ K2(H1,H2).
3
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
850 Chaofeng Shi et al 848-855
Definition 3.3. Ci(x, y)(i = 1, 2) is said to be L-Lipschitz continuous with respect to x onK1(H1,H2)×K2(H1,H2) if there exists θ > 0such that
‖Ci(x1, y)− Ci(x2, y)‖2 ≤ L‖x1 − x2‖2, (3.7)
∀x1, x2 ∈ K1(H1,H2),∀y ∈ K2(H1,H2).
Remark 3.4. Based on Definitions 3.2 and 3.3, we can similarly define the θ -strict monotonicityand L-Lipschitz continuity of Ci(x, y) with respect to y on K1(H1,H2)×K2(H1,H2), for i = 1, 2.
Theorem 3.5. (H1,H2) ∈ K1(H1,H2)×K2(H1,H2) is an equilibrium flow if and only if there existα > 0 and β > 0 such that
H1 = PK1(H2 − αC1(H1,H2)), (3.8)
H2 = PK2(H2 − βC1(H1,H2)),
where Pki: Rn → Ki(H1,H2) is a projection operator for i = 1, 2.
Proof. The proof is analogous to that of Theorem 5.2.4 of [18].Let ‖(x, y)1‖ be the norm on space K1(H1,H2)×K2(H1,H2) defined as follows:
‖(x, y)‖1 = ‖x‖2 + ‖y‖2,∀x ∈ K1(H1,H2), y ∈ K2(H1,H2). (3.9)
It is easy to see that (K1(H1,H2)×K2(H1,H2), ‖.‖1)is a Banach space. Similar to Theorem 3.9 in Heet. al. [1], one can easily obtain the following theorem, the proof is omitted.
Theorem 3.6. Suppose that C1(H1,H2) is θ1-strictly monotone and L11-Lipschitz continuous withrespect to H1 , and L12-Lipschitz continuous with respect to H2 on K1(H1,H2)×K2(H1,H2). Supposethat C2(H1,H2) is L21-Lipschitz continuous with respect to H1, θ2-strictly monotone, and L22-Lipschitzcontinuous with respect to H2 on K1(H1,H2)×K2(H1,H2). If there exist α > 0 and β > 0 such that
√1− 2γθ1 + α2L2
11 + βL21 < 1, (3.10)√
1− 2ηθ2 + β2L222 + αL12 < 1,
then problem (3.1) admits unique solution.
Remark 3.7. If f1j (H1,H2) is θ1
j -strictly monotone with respect to H1and g1j
∑nj=1 δrs
pj t1j is θ
1
j -strictly monotone with respect to H1 , then
θ1 =n∑
j=1
(θ1
j + δrspj θ
1j ).
In fact,
< C1j (H1,H2)− C2
j (H1,H2),H1 − H1 >
=< gr(n∑
j=1
δrspj t
1j (H
1,H2)) +n∑
j=1
δrspjτj +
n∑
j=1
δrspjf
1j (H1,H2)− gr(
n∑
j=1
δrspj t
1j (H
1,H2)) (3.11)
+n∑
j=1
δrspjτj +
n∑
j=1
δrspjf
1j (H1,H2),H1 − H1 >
≥n∑
j=1
(θ1
j + δrspj θ
1j )‖H1 − H1‖22
So,< Cj(H1,H2)− Cj(H1,H2),H1 − H1 >≥ θ1‖H1 − H1‖.
4
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
851 Chaofeng Shi et al 848-855
4 Algorithms for solving the multicriteria transportation equi-librium system problem
Here, we describe an iterative algorithm with fixed step-sizes, and also describe a self-adaptive algo-rithm, which uses a self-adaptive stratedgy of step-size choice.
Algorithm 4.1 Iterative Method with fixed step-sizesStep 1. Given ε > 0, α, β ∈ [0, 1), and (H0
1 ,H02 ) ∈ K1(H1,H2)×K2(H1,H2), set k = 0.
Step 2. Get the next iterate:
H1,k+1 = PK1(H2,k − αC1(H1,k,H2,k),
H2,k+1 = PK2(H2,k − βC1(H1,k,H2,k),
Step 3. Compute r1 = ‖H(k+1)1 −H
(k)1 ‖, r2 = ‖H(k+1)
2 −H(k)2 ‖ , if r1, r2 < ε, then stop; Otherwise,k = k+1,
go to step 2.
Algorithm 4.2 SI methodStep 1. Given ε > 0, γ ∈ [1, 2),µ ∈ (0, 1),ρ > 0 ,δ ∈ (0, 1) ,δ0 ∈ (0, 1) , and µ0 ∈ H, set k = 0.Step 2. Setρk = ρ , if ‖r1(H1K , ρ)‖ < ε and ‖r1(H1K , ρ)‖ < ε, then stop; otherwise, find the smallestnonnegative integer mk, such that ρk = ρµmk satisfying
‖ρk(C1(H1k,H2k)− C1(wk,H2k)‖ ≤ δ‖r(xk, ρk)‖, (4.1)
where wk = PK [H1k − ρkC1(H1k,H2k)].Step 3. Compute
d(H1k, ρk) = r(H1k, ρk)− ρkC2(H1k,H2k) + ρkC2(PK [H1k − ρkC(H1k,H2k)],H2k), (4.2)
where r(H1k, ρ) = H1k − PK [H1k − ρC2(H1k,H2k)] .Step 4. Get the next iterate:
H2k = PK [H1k − γd(H1k, ρk)− γC2(H1k,H2k)]; (4.3)H1,k+1 = PK [H1k − ρC1(H1k,H2k)]
Step 5. If ‖ρk(C(H1k,H2k) − C(wk,H2k)‖ ≤ δ0‖r(xk, ρk)‖ , then set ρ = ρk/µ, else set ρ = ρk. Setk = k + 1, and go to Step 2.
Remark 4.2. Note that Algorithm 4.2 is obviously a modification of the standard procedure. InAlgorithm 4.2, the searching direction is taken as H1k − γd(H1k, ρk) − γC(H1k,H2k) , which is closelyrelated to the projection residue, and differs from the standard procedure. In addition, the self-adaptivestrategy of step-size choice is used. The numerical results show that these modifications can introducecomputational efficiency substantially.
Theorem 4.3. Suppose that C1(H1,H2) is θ1 -strictly monotone and L11-Lipschitz continuouswith respect to H1, and L12-Lipschitz continuous with respect to H2 on K1(H1,H2) × K2(H1,H2).Suppose that C2(H1,H2) is L21-Lipschitz continuous with respect to H1 ,θ2 -strictly monotone, andL22-Lipschitz continuous with respect to H2 on K1(H1,H2) ×K2(H1,H2). Let H1∗,H2∗ ∈ K form asolution set for the SNVI (2.1) and let the sequences H1k and H2k be generated by Algorithm 4.2.If 0 < θ <
√1− 2ρθ1 + 2ρ2L2
12(1 + γL21)/(1− γL22) +√
2ρL211 + 2ρL11 < 1 , then the sequence H1k
converges to H1∗ and the sequence H2k converges to H2∗, for 0 < ρ < 2r/s2.Proof. Since (H1∗,H2∗) is a solution of transportation equilibrium system (3.2), it follows from Theorem3.5 that
H1∗ = PK1 [H2∗ − ρC1(H1∗,H2∗)], (4.4)
H2∗ = PK2 [H1∗ − γC2(H1∗,H2∗)]
5
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
852 Chaofeng Shi et al 848-855
Applying Algorithm 4.2, we know
‖H1,k+1 −H1∗‖ = ‖PK1 [H2k − ρC1(H1k,H2k)]− PK1 [H
2∗ − ρC1(H1∗,H2∗)]
≤ ‖H2k −H2∗ − ρC1(H1k,H2k) + ρC1(H1∗,H2∗)‖Since T is r-strongly monotone and s-Lipschitz continuous, we know
‖H2k −H2∗ − ρC1(H1∗,H2∗)‖2
≤ ‖H2k−H2∗‖2−2ρ < C1(H1k−H2k−C1(H1∗,H2∗,H2k−H2∗ > +ρ2‖C1(H1k,H2k)−C1(H1∗,H2∗)‖≤ ‖H2k −H2∗‖2 − 2ρθ1‖H2k −H2∗‖2 + 2ρL11‖H1k −H1∗‖2 + ρ2‖C1(H1k,H2k)− C1(H1∗,H2∗)‖
≤ ‖H2k−H2∗‖2−2ρθ1‖H2k−H2∗‖2 +2ρL11‖H1k−H1∗‖2 +2ρ2L211‖H1k−H1∗‖2 +2ρ2L2
12‖H2k−H2∗‖2
≤ (1− 2ρθ1 + 2ρ2L212)‖H2k −H2∗‖2 + (2ρ2L2
11 + 2ρL11)‖H1k −H1∗‖2
It follows that
‖H1,k+1 −H1∗‖ ≤√
1− 2ρθ1 + 2ρ2L212‖H2k −H2∗‖+
√2ρ2L2
11 + 2ρL11‖H1k −H1∗‖. (4.5)
Next, we consider
‖H2k −H2∗‖ = ‖PK2 [H1k − γd(H1k, ρk)− γC2(H1k,H2k)]− PK2 [H
1∗ − γC2(H1∗,H2∗)‖ (4.6)≤ ‖H1k − γd(H1k, ρk)− γC2(H1k,H2k)−H1∗ + γC2(H1∗,H2∗)‖≤ ‖H1k − γd(H1k, ρk)−H1∗‖+ γ‖C2(H1k,H2k)− C2(H1∗,H2∗)‖
where we use the property of the operator PK . Now, we consider
‖H1k −H1∗ − γd(H1k, ρk)‖2 (4.7)≤ ‖H1k −H1∗‖2 − 2γ < H1k −H1∗, d(H1k, ρk) > +γ2‖d(H1k, ρk)‖2
≤ ‖H1k −H1∗‖2,
where we use the definition of d(H2k, ρk).It follows that
‖H2,k −H2∗‖ ≤ (1 + γL21)‖H1k −H1∗‖+ γL22‖H2k −H2∗‖. (4.8)
From (4.5) to (4.8), we know
‖H1,k+1 −H1∗‖ ≤ (√
1− 2ρθ1 + 2ρ2L212(1 + γL21)/(1− γL22) +
√2ρL2
11 + 2ρL11)‖H1k −H1∗‖. (4.9)
Since 0 < θ < 1 , from (4.9), we know H1k → H1∗. Thus from (4.8), we know H2k → H2∗.
5 Numerical results
In this section, we presented some numerical results for the proposed method. we consider a simple trafficnetwork consisting of two nodes, only a origin-destination (O/D) pair, and a set R of routes. Each router ∈ R links the origin-destination pair in parallel. Assume that n is the number of the route in R.Let C1(H1,H2) = DH1(t) + cT
1 H2(t), C2(H1,H2) = DH1(t) + cT2 H2(t) , where
D =
4 −2 · · · · · ·1 4 · · · · · ·· · · · · · 4 −2· · · · · · 1 4
,
c1 = (−1,−1, · · · ,−1)T , c2 = (1, 1, · · · , 1)T ,H1(t) = H1 ∈ Rn,H1(t) = H2 ∈ Rn. let
K1(H1,H2) = H1|H1 ∈ [l, u],Hi1 + Hi
2 ≤ 2000, i = 1, 2, · · · , n,
6
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
853 Chaofeng Shi et al 848-855
K2(H1,H2) = H1|H1 ∈ [l, u],Hi1 + Hi
2 ≤ 2000, i = 1, 2, · · · , n.where l = (0, 0, · · · , 0)T , u = (1000, 1000, · · · , 1000)T . The calculations are started with vectors H1 =(0, 0, · · · , 0)T ,H2 = (5, 5, · · · , 5)T and stopped whenever r1, r2 < 10−5. Table 1 gives the numerical resultsof Algorithms 4. 1. Table 2 gives the numerical results of Algorithms 4. 2.Comparing Table 2 and Table 1, it show that Algorithm 4.2 is very effective for the problem tested. Inaddition, it seems that the computational time and the iteration numbers are not very sensitive to theproblem size.
Table 1: Computation performance with different scales by Algorithm 4.1
n Iteration CPU(s)50 366 29.5469100 183 28.5469200 93 27.2813300 63 30.4375
Table 2: Computation performance with different scales by Algorithm 4.2
n Iteration CPU(s)50 220 17.7282100 110 16.1281200 56 15.3687300 38 18.2625
References
[1] Y. P. He, J. P. Xu, N. J. Huang, and M. Wu, Dynamic traffic network equilibrium system, FixedPoint Theory and Application, vol. 2010, Art. 873025.
[2] S. C. Dafermos, The general multinodal network equilibrium problem with elastic demand, Networks,vol. 12, pp. 57-72, 1982.
[3] M. Fukushima, On the dual approach to the traffic assignment problem, Transportation Res. PartB, vol. 18, pp. 235-245, 1984.
[4] M. Fukushima and T. Itoh, A dual approach to asymmetric traffic equilibrium problems, Math.Japon. vol. 32 , pp. 701-721, 1987.
[5] A. Maugeri, Stability results for variational inequalities and applications to traffic equilibrium prob-lem, Supplemento Rendiconti Circolo Matematico di Palermo vol. 8, pp. 269-280, 1985.
[6] S. Gabriel, D. Bernstein, The traffic equilibrium problem with nonadditive path costs, TransportationScience, Vol 31, pp. 337-348, 1997.
[7] K. Scott, D. Bernstein, Solving a best path problem when the value of time function is nonlinear,in: The 77th Annual Meeting of Transportation Research Board, 1998.
[8] K. Scott, D. Bernstein, Solving a traffic equilibrium problem when paths are not additive, in: The78th Annual Meeting of Transportation Research Board, 1999.
[9] F. Facchinei, J. Soares, A new merit function for nonlinear complementarity problems and a relatedalgorithm, SIAM Journal of Optimization, Vol. 7, pp. 225-247 , 1997.
[10] H.K. Lo, A. Chen, Traffic equilibrium problem with route-specific costs: formulation and algorithms,Transportation Research Part B, Vol. 34 , pp. 493-513 , 2000.
7
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
854 Chaofeng Shi et al 848-855
[11] A. Chen, H.K. Lo, H. Yang, A self-adaptive projection and contraction algorithm for the trafficassignment problem with path-specific costs, European Journal of Operational Research, Vol. 135,pp. 27-41 , 2001.
[12] T. Larsson, P.O. Lindberg, J. Lundgren, M. Patriksson, C. Rydergren, On traffic equilibrium modelswith a nonlinear time/money relation, in: M. Patriksson, M. Labbe (Eds.), Transportation Planning:State of the Art, Kluwer Academic Publishers, 2002.
[13] D.H. Bernstein, L. Wynter, Issues of uniqueness and convexity in non-additive bi-criteria equilibriummodels, in: Proceedings of the EURO Working Group on Transportation, Rome, Italy, 2000.
[14] E. Altman, L. Wynter, Equilibrium, games, and pricing in transportation and telecommunicationnetworks, Networks and Spatial Economics , Vol. 4, pp. 7-21, 2004.
[15] R. U. Verma, Projection methods, algorithms, and a new system of nonlinear variational inequalities,Computers and Mathematics with Applications, vol. 41, no. 7-8, pp. 1025-1031, 2001.
[16] A. Bnouhachem, A self-adaptive method for solving general mixed variational inequalities, Journalof Mathematical Analysis and Applications, vol. 309, no. 1, pp. 136-150, 2005.
[17] C. F. Shi, A self-adaptive method for solving a system of nonlinear variational inequalities, Mathe-matical Problems in Engineering, Art. 23795, 2007.
[18] P. Daniele, Dynamic Networks and Evolutionary Variational Inequalities, New Dimensions in Net-works, Edward Elgar, Cheltenham, UK, 2006.
Chaofeng ShiDepartment of Financial and EconomicsChongqing Jiaotong UniversityChongqing, 400074, P. R. ChinaandDepartment of EconomicsUniversity of CaliforniaRiverside, CA 92507, USA.Correspondence should be addressed to Chaofeng Shi, E-mail: [email protected]
Yingrui WangDepartment of Financial and EconomicsChongqing Jiaotong UniversityChongqing, 400074, P. R. China
8
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
855 Chaofeng Shi et al 848-855
BARNES’ MULTIPLE FROBENIUS-EULER AND HERMITE
MIXED-TYPE POLYNOMIALS
DAE SAN KIM, DMITRY V. DOLGY, AND TAEKYUN KIM
Abstract. In this paper, we consider the Barnes’ multiple Frobenius-Eulerand Hermite mixed-type polynomials. Using the umbral calculus, we derive
several explicit formulas and recurrence relations for these polynomials. Also,we establish connections between our polynomials and several known familiesof polynomials.
1. Introduction
For λ = 1, s ∈ N, the Frobenius-Euler polynomials of order s are defined by thegenerating function
(1.1)
(1− λet − λ
)s
ext =∞∑
n=0
H(s)n (x | λ) t
n
n!, (see [7, 12, 19]) .
Let a1, a2, . . . , ar, λ1, λ2, . . . , λr ∈ C with a1, . . . , ar = 0, λ1, . . . , λr = 1. Thenthe Barnes’ multiple Frobenius-Euler polynomials Hn (x | a1, . . . , ar;λ1, . . . , λr) aregiven by the generating funciton(1.2)
r∏j=1
(1− λjeajt − λj
)ext =
∞∑n=0
Hn (x | a1, . . . , ar;λ1, . . . , λr)tn
n!, (see [13, 15]) .
When x = 0,Hn (a1, . . . , ar;λ1, . . . , λr) = Hn (0 | a1, . . . , ar;λ1, . . . , λr) are calledthe Barnes’ multiple Frobenius-Euler numbers (see [13]).
For a1 = a2 = · · · = ar = 1, λ1 = λ2 = · · · = λr = λ, we have Hn(x |1, 1, . . . , 1︸ ︷︷ ︸r−times
;λ, λ, . . . , λ︸ ︷︷ ︸)r−times
= H(r)n (x | λ). When x = 0, H(r)
n (λ) = H(r)n (0 | λ) are called
the Frobenius-Euler numbers of order r.The Hermite polynomials H
(ν)n (x) of variance ν (0 = ν ∈ R) are given by the
generating function
(1.3) e−νt2/2ext =∞∑
n=0
H(ν)n (x)
tn
n!, (see [24]) .
When x = 0, H(ν)n = H
(ν)n (0) are called the Hermite numbers of variance ν. It
is well known that the Bernoulli polynomials of order r (∈ N) are defined by thegenerating function
2010 Mathematics Subject Classification. 05A19, 05A40, 11B75, 11B83.Key words and phrases. Barnes’ multiple Frobenius-Euler and Hermite mixed-type polynomi-
als, umbral calculus.
1
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
856 DAE SAN KIM et al 856-870
2 DAE SAN KIM, DMITRY V. DOLGY, AND TAEKYUN KIM
(1.4)
(t
et − 1
)r
ext =
∞∑n=0
B(r)n (x)
tn
n!, (see [1–24, 26]) .
When x = 0, B(r)n = B
(r)n (0) are called the Bernoulli numbers of order r. For
n ≥ 0, the Stirling numbers of the first kind are given by
(1.5) (x)n = x (x− 1) · · · (x− n+ 1) =
n∑l=0
S1 (n, l)xl, (see [24]) .
The Stirling numbers of the second kind are defined by the generating function
(1.6)(et − 1
)n= n!
∞∑l=n
S2 (l, n)xl
l!, (see [24]) .
Let C be the complex field and let F be the set of all formal power series in thevariable t:
(1.7) F =
f (t) =
∞∑k=0
aktk
k!
∣∣∣∣∣ ak ∈ C
.
Let P = C [x] and let P∗ be the vector space of all linear functionals on P. Weuse the notation ⟨L | p (x)⟩ to denote the action of the linear functional L on thepolynomial p (x), and we recall that the vector space operations on P∗ are definedby ⟨L+M | p (x)⟩ = ⟨L | p (x)⟩+ ⟨M | p (x)⟩, and ⟨cL | p (x)⟩ = c ⟨L| p (x)⟩, wherec is a complex constant in C. The linear functional ⟨f (t)| ·⟩ on P is defined by
(1.8) ⟨f (t)|xn⟩ = an, (n ≥ 0) , where f (t) ∈ F , (see [17, 21, 24]) .
By (1.8), we easily get
(1.9)⟨tk∣∣xn⟩ = n!δn,k, (n, k ≥ 0) , (see [8, 21, 24]) ,
where δn,k is the Kronecker’s symbol.
Let fL (t) =∑∞
k=0⟨L|xk⟩
k! tk. Then, by (1.9), we get ⟨fL (t)|xn⟩ = ⟨L | xn⟩ . So,the map L 7→ fL (t) is a vector space isomorphism from P∗ onto F . Henceforth,F denotes both the algebra of formal power series in t and the vector space of alllinear functionals on P, and so an element f (t) of F will be thought of as botha formal power series and a linear functional. We call F the umbral algebra andthe umbral calculus is the study of umbral algebra. The order o (f (t)) of a powerseries f (t) = 0 is the smallest integer k for which the coefficient of tk does notvanish. If the order of f (t) is 1, then f (t) is called a delta series; if the order g (t)is 0, then g (t) is called an invertible series. Let f (t) , g (t) ∈ F with o (f (t)) = 1and o (g (t)) = 0. Then there exists a unique sequence sn (x) (deg sn (x) = n) such
that⟨g (t) f (t)
k | sn (x)⟩
= n!δn,k for n, k ≥ 0. Such a sequence sn (x) is called
the Sheffer sequence for (g (t) , f (t)) which is denoted by sn (x) ∼ (g (t) , f (t)) (see[21, 24]). In particular, if sn (x) ∼ (g (t) , t), then sn (x) is called an Appell sequencefor g (t). For f (t) , g (t) ∈ F , we have(1.10)⟨f (t) g (t)| p (x)⟩ = ⟨f (t)| g (t) p (x)⟩ = ⟨g (t)| f (t) p (x)⟩ = ⟨1| f (t) g (t) p (x)⟩ ,
(1.11) f (t) =∞∑k=0
⟨f (t)|xk
⟩ tkk!, p (x) =
∞∑k=0
⟨tk∣∣ p (x)⟩ xk
k!, (see [24]) .
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
857 DAE SAN KIM et al 856-870
BARNES’ MULTIPLE FROBENIUS-EULER AND HERMITE MIXED-TYPE POLYNOMIALS 3
Thus, by (1.11), we get(1.12)
tkp (x) = p(k) (x) =dkp (x)
dxk, eytp (x) = p (x+ y) , and
⟨eyt∣∣ p (x)⟩ = p (y) .
The sequence sn (x) is Sheffer for (g (t) , f (t)) if and only if
(1.13)1
g(f (t)
)eyf(t) = ∞∑k=0
sk (x)
k!tk, (y ∈ C) , (see [17, 21, 24]) ,
where f (t) is the compositional inverse of f (t) with f (f (t)) = f(f (t)
)= t. It is
well known that the Sheffer identity is given by(1.14)
sn (x+ y) =∞∑j=0
(n
j
)sj (x) pn−j (y) , where pn (x) = g (t) sn (x) , (see [17, 24]) .
For sn (x) ∼ (g (t) , f (t)), we have
(1.15) sn+1 (x) =
(x− g′ (x)
g (x)
)1
f ′ (x)sn (x) , (n ≥ 0) ,
(1.16) sn (x) =n∑
j=0
1
j!
⟨g(f (t)
)−1f (t)
j∣∣∣xn⟩xj ,
and
(1.17) ⟨f (t)|xp (x)⟩ = ⟨∂tf (t)| p (x)⟩ , f (t) sn (x) = nsn−1 (x) , (n ≥ 1) .
Let sn (x) ∼ (g (t) , f (t)) and rn (x) ∼ (h (t) , l (t)), (n ≥ 0). Then we have
(1.18) sn (x) =n∑
m=0
Cn,mrm (x) , (n ≥ 0) ,
where
(1.19) Cn,m =1
m!
⟨h(f (t)
)g(f (t)
) l (f (t))m∣∣∣∣∣xn⟩, (see [17, 21, 24]) .
In this paper, we consider the polynomials FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr) whose
generating function is given by
r∏j=1
(1− λjeajt − λj
)e−νt2/2ext =
r∏j=1
(1− λjeajt − λj
)ext−νt2/2(1.20)
=∞∑
n=0
FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr)
tn
n!,
where r ∈ Z>0, a1, . . . , ar, λ1, . . . , λr ∈ C with a1, . . . , ar = 0, λ1, . . . , λr = 1, and
ν ∈ R with ν = 0. FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr) are called Barnes’ multiple
Frobenius-Euler and Hermite mixed-type polynomials.
When x = 0, FH(ν)n (a1, . . . , ar;λ1, . . . , λr) = FH
(ν)n (0 | a1, . . . , ar;λ1, . . . , λr)
are called the Barnes’ multi[ple Frobenius-Euler and Hermite mixed-type numbers.
We observe here that FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr),Hn (x | a1, . . . , ar;λ1, . . . , λr),
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
858 DAE SAN KIM et al 856-870
4 DAE SAN KIM, DMITRY V. DOLGY, AND TAEKYUN KIM
andH(ν)n (x) are respectively Appell sequences for
∏rj=1
(eajt−λj
1−λj
)eνt
2/2,∏r
j=1
(eajt
n−λj
1−λj
),
and eνt2/2. That is,
(1.21) FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr) ∼
r∏j=1
(eajt − λj1− λj
)eνt
2/2, t
,
(1.22) Hn (x | a1, . . . , ar;λ1, . . . , λr) ∼
r∏j=1
(eajt − λj1− λj
), t
,
and
(1.23) H(ν)n (x) ∼
(eνt
2/2, t).
From the Barnes’ multiple Frobenius-Euler and Hermite mixed-type polynomi-als, we investigate some properties of those polynomials. Finally, we give some newand interesting identities which are derived from umbral calculus.
2. Barnes’ multiple Frobenius-Euler and Hermite mixed-typepolynomials
From (1.21), (1.22) and (1.23), we note that
tFH(ν)n (x | a1, . . . , ar;λ1, . . . , λr) =
d
dxFH(ν)
n (x | a1, . . . , ar;λ1, . . . , λr)(2.1)
= nFH(ν)n−1 (x | a1, . . . , ar;λ1, . . . , λr) ,
tHn (x | a1, . . . , ar;λ1, . . . , λr) =d
dxHn (x | a1, . . . , ar;λ1, . . . , λr)(2.2)
= nHn−1 (x | a1, . . . , ar;λ1, . . . , λr) ,and
(2.3) tH(ν)n (x) =
d
dxH(ν)
n (x) = nH(ν)n−1 (x) .
Now, we give explicit expressions related to the Barnes’ multiple Frobenius-Eulerand Hermite mixed-type polynomials.
From (1.13), we note that
FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr) = e−νt2/2
r∏j=1
(1− λjeajt − λj
)xn(2.4)
=e−νt2/2Hn (x | a1, . . . , ar;λ1, . . . , λr)
=∞∑
m=0
1
m!
(−ν2
)mt2mHn (x | a1, . . . , ar;λ1, . . . , λr)
=
[n2 ]∑m=0
1
m!
(−ν2
)m(n)2mHn−2m (x | a1, . . . , ar;λ1, . . . , λr)
=
[n2 ]∑m=0
(n
2m
)(2m)!
m!
(−ν2
)mHn−2m (x | a1, . . . , ar;λ1, . . . , λr) .
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
859 DAE SAN KIM et al 856-870
BARNES’ MULTIPLE FROBENIUS-EULER AND HERMITE MIXED-TYPE POLYNOMIALS 5
By (1.9), we get
FH(ν)n (y | a1, . . . , ar;λ1, . . . , λr)(2.5)
=
⟨ ∞∑i=0
FH(ν)i (y | a1, . . . , ar;λ1, . . . , λr)
ti
i!
∣∣∣∣∣xn⟩
=
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2eyt
∣∣∣∣∣∣xn⟩
=
⟨r∏
j=1
(1− λjeajt − λj
)∣∣∣∣∣∣ e−νt2/2eytxn
⟩
=
⟨r∏
j=1
(1− λjeajt − λj
)∣∣∣∣∣∣∞∑l=0
H(ν)l (y)
tl
l!xn
⟩
=n∑
l=0
(n
l
)H
(ν)l (y)
⟨r∏
j=1
(1− λjeajt − λj
)∣∣∣∣∣∣xn−l
⟩
=n∑
l=0
(n
l
)H
(ν)l (y)
⟨ ∞∑i=0
Hi (a1, . . . , ar;λ1, . . . λr)ti
i!
∣∣∣∣∣xn−l
⟩
=n∑
l=0
(n
l
)Hn−l (a1, . . . , ar;λ1, . . . , λr)H
(ν)l (y) .
Thus, by (2.5), we get(2.6)
FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr
) =n∑
l=0
(n
l
)Hn−l (a1, . . . , ar;λ1, . . . , λr)H
(ν)l (x) .
Therefore, by (2.4) and (2.6), we obtain the following theorem.
Theorem 2.1. For n ≥ 0, we have
FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr)
=
[n2 ]∑m=0
(n
2m
)(2m)!
m!
(−ν2
)mHn−2m (x | a1, . . . , ar;λ1, . . . , λr)
=n∑
l=0
(n
l
)Hn−l (a1, . . . , ar;λ1, . . . , λr)H
(ν)l (x) .
From (1.9), we have
FH(ν)n (y | a1, . . . , ar;λ1, . . . , λr)(2.7)
=
⟨ ∞∑i=0
FH(ν)i (y | a1, . . . , ar;λ1, . . . , λr)
ti
i!
∣∣∣∣∣xn⟩
=
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2eyt
∣∣∣∣∣∣xn⟩
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
860 DAE SAN KIM et al 856-870
6 DAE SAN KIM, DMITRY V. DOLGY, AND TAEKYUN KIM
=
⟨e−νt2/2
∣∣∣∣∣∣r∏
j=1
(1− λjeajt − λj
)eytxn
⟩
=
⟨e−νt2/2
∣∣∣ ∞∑l=0
Hl (y | a1, . . . , ar;λ1, . . . , λr)tl
l!xn
⟩
=n∑
l=0
(n
l
)Hl (y | a1, . . . , ar;λ1, . . . , λr)
⟨ ∞∑i=0
H(ν)i
ti
i!
∣∣∣∣∣xn−l
⟩
=n∑
l=0
(n
l
)Hl (y | a1, . . . , ar;λ1, . . . , λr)H(ν)
n−l.
Thus, by (2.7), we get(2.8)
FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr) =
n∑l=0
(n
l
)Hl (x | a1, . . . , ar;λ1, . . . , λr)H(ν)
n−l.
Now, we will use the conjugation representation in (1.16). For FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr) ∼(
g (t) =∏r
j=1
(eajt−λj
1−λj
)eνt
2/2, f (t) = t), we observe that⟨
g(f (t)
)−1f (t)
j∣∣∣xn⟩(2.9)
=
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2tj
∣∣∣∣∣∣xn⟩
=
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣ tjxn⟩
=(n)j
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣xn−j
⟩
=(n)j
⟨e−νt2/2
∣∣∣Hn−j (x | a1, . . . , ar;λ1, . . . , λr)⟩
=(n)j
⟨ ∞∑m=0
1
m!
(−ν2
)mt2m
∣∣∣∣∣Hn−j (x | a1, . . . , ar;λ1, . . . , λr)
⟩
=(n)j
[n−j2 ]∑
m=0
1
m!
(−ν2
)m(n− j)2mHn−j−2m (a1, . . . , ar;λ1, . . . , λr) .
From (1.16) and (2.9), we can derive the following equation:
FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr)(2.10)
=n∑
j=0
(n
j
) [n−j2 ]∑
m=0
1
m!
(−ν2
)m(n− j)2mHn−j−2m (a1, . . . , ar;λ1, . . . , λr)x
j .
Therefore, by (2.8) and (2.10), we obtain the following theorem.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
861 DAE SAN KIM et al 856-870
BARNES’ MULTIPLE FROBENIUS-EULER AND HERMITE MIXED-TYPE POLYNOMIALS 7
Theorem 2.2. For n ≥ 0, we have
FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr)
=n∑
l=0
(n
l
)H
(ν)n−lHl (x | a1, . . . , ar;λ1, . . . , λr)
=n∑
j=0
(n
j
) [n−j2 ]∑
m=0
1
m!
(−ν2
)m(n− j)2mHn−j−2m (a1, . . . , ar;λ1, . . . , λr)x
j .
Remark. From (1.14), we have
FH(ν)n (x+ y | a1, . . . , ar;λ1, . . . , λr)(2.11)
=n∑
j=0
(n
j
)FH
(ν)j (x | a1, . . . , ar;λ1, . . . , λr) yn−j .
By (1.15) and (1.21), we get
FH(ν)n+1 (x | a1, . . . , ar;λ1, . . . , λr)(2.12)
=
(x− g′ (t)
g (t)
)FH(ν)
n (x | a1, . . . , ar;λ1, . . . , λr) ,
where g (t) =∏r
j=1
(eajt−λj
1−λj
)eνt
2/2.
Now, we compute that
g′ (t)
g (t)= (log g (t))
′(2.13)
=
r∑j=1
log(eajt − λj
)−
r∑j=1
log (1− λj) +1
2νt2
′
=
r∑j=1
ajeajt
eajt − λj+ νt.
So
g′ (t)
g (t)FH(ν)
n (x | a1, . . . , ar;λ1, . . . , λr)(2.14)
=r∑
j=1
ajeajt
1− λj· 1− λjeajt − λj
r∏i=1
(1− λieait − λi
)e−νt2/2xn
+ νtFH(ν)n (x | a1, . . . , ar;λ1, . . . , λr)
=
r∑j=1
aj1− λj
FH(ν)n (x+ aj | a1, . . . , ar, aj ;λ1, . . . , λr, λj)
+ nνFH(ν)n−1 (x | a1, . . . , ar;λ1, . . . , λr)
=
r∑j=1
aj1− λj
FH(ν)n (x+ aj | a1, . . . , ar, aj ;λ1, . . . , λr, λj)
+ nνFH(ν)n−1 (x | a1, . . . , ar, aj ;λ1, . . . , λr) .
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
862 DAE SAN KIM et al 856-870
8 DAE SAN KIM, DMITRY V. DOLGY, AND TAEKYUN KIM
By (2.12) and (2.14), we get
FH(ν)n+1 (x | a1, . . . , ar;λ1, . . . , λr)(2.15)
=xFH(ν)n (x | a1, . . . , ar;λ1, . . . , λr)
−r∑
j=1
aj1− λj
FH(ν)n (x+ aj | a1, . . . , ar, aj ;λ1, . . . , λr, λj)
− nνFH(ν)n−1 (x | a1, . . . , ar;λ1, . . . , λr) .
For n ≥ 2, by (1.9), we get
FH(ν)n (y | a1, . . . , ar;λ1, . . . , λr)(2.16)
=
⟨ ∞∑i=0
FH(ν)i (y | a1, . . . , ar;λ1, . . . , λr)
ti
i!
∣∣∣∣∣xn⟩
=
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2eyt
∣∣∣∣∣∣xn⟩
=
⟨∂t
r∏j=1
(1− λjeajt − λj
)e−νt2/2eyt
∣∣∣∣∣∣xn−1
⟩
=
⟨∂t r∏j=1
(1− λjeajt − λj
) e−νt2/2eyt
∣∣∣∣∣∣xn−1
⟩
+
⟨r∏
j=1
(1− λjeajt − λj
)(∂te
−νt2/2)eyt
∣∣∣∣∣∣xn−1
⟩
+
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
(∂te
yt)∣∣∣∣∣∣xn−1
⟩.
The third term is
y
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2eyt
∣∣∣∣∣∣xn−1
⟩(2.17)
=yFH(ν)n−1 (y | a1, . . . , ar;λ1, . . . , λr) .
The second term is
− ν
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2eyt
∣∣∣∣∣∣ txn−1
⟩(2.18)
=− ν (n− 1)
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2eyt
∣∣∣∣∣∣xn−2
⟩
=− ν (n− 1)FH(ν)n−2 (y | a1, . . . , ar;λ1, . . . , λr) .
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
863 DAE SAN KIM et al 856-870
BARNES’ MULTIPLE FROBENIUS-EULER AND HERMITE MIXED-TYPE POLYNOMIALS 9
We observe that
∂t
r∏j=1
(1− λjeajt − λj
)(2.19)
=
r∑i=1
(1− λieait − λi
)′∏j =i
(1− λjeajt − λj
)
=−r∏
j=1
(1− λjeajt − λj
) r∑i=1
aieait
eait − λi,
where(2.20)
r∑i=1
aieait
eait − λi=
r∑i=1
ai1− λi
eait1− λieait − λi
=r∑
i=1
ai1− λi
eait∞∑
m=0
Hm (λi)amim!
tm.
So, by (2.19) and (2.20), we get(2.21)
∂t
r∏j=1
1− λjeajt − λj
= −r∏
j=1
(1− λjeajt − λj
) r∑i=1
ai1− λi
eait∞∑
m=0
Hm (λi)amim!
tm.
Now, the first term is
−r∑
i=1
ai1− λi
⟨e(y+ai)t
r∏j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣n−1∑m=0
Hm (λi)amim!
tmxn−1
⟩(2.22)
=−r∑
i=1
ai1− λi
n−1∑m=0
(n− 1
m
)Hm (λi) a
mi
×
⟨e(y+ai)t
r∏j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣xn−1−m
⟩
=−r∑
i=1
ai1− λi
n−1∑m=0
(n− 1
m
)Hm (λi) a
mi
×
⟨ ∞∑l=0
FH(ν)l (y + ai | a1, . . . , ar;λ1, . . . , λr)
tl
l!
∣∣∣∣∣xn−1−m
⟩
=−r∑
i=1
n−1∑m=0
(n− 1
m
)am+1i
1− λiHm (λi)FH
(ν)n−1−m (y + ai | a1, . . . , ar;λ1, . . . , λr) .
Therefore, by (2.16), (2.17), (2.18) and (2.22), we obtain the following theorem.
Theorem 2.3. For n ≥ 2, we have
FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr)
=xFH(ν)n−1 (x | a1, . . . , ar;λ1, . . . , λr)− ν (n− 1)FH
(ν)n−2 (x | a1, . . . , ar;λ1, . . . , λr)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
864 DAE SAN KIM et al 856-870
10 DAE SAN KIM, DMITRY V. DOLGY, AND TAEKYUN KIM
−r∑
i=1
n−1∑m=0
(n− 1
m
)am+1i
1− λiHm (λi)FH
(ν)n−1−m (x+ ai | a1, . . . , ar;λ1, . . . , λr) .
Remark. We compute the following in two different ways in order to derive anidentity: ⟨
r∏j=1
(1− λjeajt − λj
)e−νt2/2tm
∣∣∣∣∣∣xn⟩, (m,n ≥ 0) .
On one hand, it is ⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2tm
∣∣∣∣∣∣xn⟩
(2.23)
= (n)m
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣xn−m
⟩
=(n)m FH(ν)n−m (a1, . . . , ar;λ1, . . . , λr) .
On the other hand, it is⟨∂t
r∏j=1
(1− λjeajt − λj
)e−νt2/2tm
∣∣∣∣∣∣xn−1
⟩(2.24)
=
⟨∂t r∏j=1
(1− λjeajt − λj
) e−νt2/2tm
∣∣∣∣∣∣xn−1
⟩
+
⟨r∏
j=1
(1− λjeajt − λj
)(∂te
−νt2/2)tm
∣∣∣∣∣∣xn−1
⟩
+
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2 (∂tt
m)
∣∣∣∣∣∣xn−1
⟩.
From (2.23) and (2.24), we can derive the following equation: for n ≥ m+ 2,
FH(ν)n−m (a1, . . . , ar;λ1, . . . , λr)
(2.25)
=− ν (n−m− 1)FH(ν)n−m−2 (a1, . . . , ar;λ1, . . . , λr)
−r∑
i=1
n−m−1∑l=0
(n−m− 1
l
)al+1i
1− λiHl (λi)FH
(ν)n−1−l−m (ai; a1, . . . , ar;λ1, . . . , λr) .
For FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr) ∼
(∏rj=1
(eajt−λj
1−λj
)eνt
2/2, t), (x)n ∼ (1, et − 1),
we have
(2.26) FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr) =
n∑m=0
Cn,m (x)m ,
Cn,m(2.27)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
865 DAE SAN KIM et al 856-870
BARNES’ MULTIPLE FROBENIUS-EULER AND HERMITE MIXED-TYPE POLYNOMIALS11
=1
m!
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
(et − 1
)m∣∣∣∣∣∣xn⟩
=
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣ 1
m!
(et − 1
)mxn
⟩
=
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣∞∑
l=m
S2 (l,m)tl
l!xn
⟩
=
n∑l=m
(n
l
)S2 (l,m)
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣xn−l
⟩
=
n∑l=m
(n
l
)S2 (l,m)FH
(ν)n−l (a1, . . . , ar;λ1, . . . , λr) .
Therefore, by (2.26) and (2.27), we obtain the following theorem.
Theorem 2.4. For n ≥ 0, we have
FH(ν)n (x | a1, . . . , ar;λ1 . . . , λr) =
n∑m=0
n∑l=m
(n
l
)S2 (l,m)FH
(ν)n−l (a1, . . . , ar;λ1, . . . , λr) (x)m .
It is easy to show that
x(n) = x (x+ 1) · · · (x+ n− 1) ∼(1, 1− e−t
).
From (1.18) and (1.19), we have
(2.28) FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr) =
n∑m=0
Cn,mx(m),
where
Cn,m(2.29)
=1
m!
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
(1− e−t
)m∣∣∣∣∣∣xn⟩
=1
m!
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2e−mt
(et − 1
)m∣∣∣∣∣∣xn⟩
=
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2e−mt
∣∣∣∣∣∣ 1
m!
(et − 1
)mxn
⟩
=
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2e−mt
∣∣∣∣∣∣∞∑
l=m
S2 (l,m)tl
l!xn
⟩
=n∑
l=m
(n
l
)S2 (l,m)
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2e−mt
∣∣∣∣∣∣xn−l
⟩
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
866 DAE SAN KIM et al 856-870
12 DAE SAN KIM, DMITRY V. DOLGY, AND TAEKYUN KIM
=n∑
l=m
(n
l
)S2 (l,m)
⟨ ∞∑i=0
FH(ν)i (−m | a1, . . . , ar;λ1, . . . , λr)
ti
i!
∣∣∣∣∣xn−l
⟩
=n∑
l=m
(n
l
)S2 (l,m)FH
(ν)n−l (−m | a1, . . . , ar;λ1, . . . , λr) .
Therefore, by (2.28) and (2.29), we obtain the following theorem.
Theorem 2.5. For n ≥ 0, we have
FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr)
=
n∑m=0
n∑l=m
(n
l
)S2 (l,m)FH
(ν)n−l (−m | a1, . . . , ar;λ1, . . . , λr)x
(m).
From (1.4), (1.13), (1.18), (1.19) and (1.21), we have
(2.30) FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr) =
n∑m=0
Cn,mB(s)m (x) , (s ∈ N) ,
where
Cn,m
(2.31)
=1
m!
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
(et − 1
t
)s
tm
∣∣∣∣∣∣xn⟩
=
(n
m
)⟨ r∏j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣(et − 1
t
)s
xn−m
⟩
=
(n
m
)⟨ r∏j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣∞∑l=0
s!
(l + s)!S2 (l + s, s) tlxn−m
⟩
=
(n
m
) n−m∑l=0
s!
(l + s)!S2 (l + s, s) (n−m)l
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣xn−m−l
⟩
=
(n
m
) n−m∑l=0
(n−m
l
)(l+ss
) S2 (l + s, s)FH(ν)n−m−l (a1, . . . , ar;λ1, . . . , λr) .
Therefore, by (2.30) and (2.31), we obtain the following theorem.
Theorem 2.6. For n ≥ 0, and s ∈ N, we have
FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr)
=n∑
m=0
(n
m
) n−m∑l=0
(n−m
l
)(l+ss
) S2 (l + s, s)FH(ν)n−m−l (a1, . . . , ar;λ1, . . . , λr)B
(s)m (x) .
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
867 DAE SAN KIM et al 856-870
BARNES’ MULTIPLE FROBENIUS-EULER AND HERMITE MIXED-TYPE POLYNOMIALS13
From (1.1), (1.18), (1.19) and (1.21), we have
(2.32) FH(ν)n (x | a1, . . . , ar;λ1, , . . . , λr) =
n∑m=0
Cn,mH(s)m (x | λ) , (s ∈ N) ,
where
Cn,m(2.33)
=1
m!
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
(et − λ1− λ
)s
tm
∣∣∣∣∣∣xn⟩
=1
m! (1− λ)s
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
(et − λ
)s∣∣∣∣∣∣ tmxn⟩
=
(nm
)(1− λ)s
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2
∣∣∣∣∣∣s∑
j=0
(s
j
)(−λ)s−j
ejtxn−m
⟩
=
(nm
)(1− λ)s
s∑j=0
(s
j
)(−λ)s−j
⟨r∏
j=1
(1− λjeajt − λj
)e−νt2/2ejt
∣∣∣∣∣∣xn−m
⟩
=
(nm
)(1− λ)s
s∑j=0
(s
j
)(−λ)s−j
FH(ν)n−m (j | a1, . . . , ar;λ1, . . . , λr) .
Therefore, by (2.32) and (2.33), we obtain the following theorem.
Theorem 2.7. For n ≥ 0, we have
FH(ν)n (x | a1, . . . , ar;λ1, . . . , λr)
=1
(1− λ)sn∑
m=0
(n
m
) s∑j=0
(s
j
)(−λ)s−j
FH(ν)n−m (j | a1, . . . , ar;λ1, . . . , λr)H(s)
m (x|λ) .
References
1. T. Agoh and M. Yamanaka, A study of Frobenius-Euler numbers and polynomi-als, Ann. Sci. Math. Quebec 34 (2010), no. 1, 1–14. MR 2744192 (2012a:11024)
2. A. Bayad and T. Kim, Results on values of Barnes polynomials, Rocky Moun-tain J. Math. 43 (2013), no. 6, 1857–1869. MR 3178446
3. A. Bayad, T. Kim, W.J. Kim, S. H. Lee, Arithmetic properties of q-Barnespolynomials, J. Comput. Anal. Appl. 15 (2013), no. 1, 111–117. MR 3076723
4. M. Can, M. Cenkci, V. Kurt, and Y. Simsek, Twisted Dedekind type sumsassociated with Barnes’ type multiple Frobenius-Euler l-functions, Adv. Stud.Contemp. Math. (Kyungshang) 18 (2009), no. 2, 135–160. MR 2508979(2010a:11072)
5. I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, higher-order w-q-Genocchinumbers, Adv. Stud. Contemp. Math. (Kyungshang) 19 (2009), no. 1, 39–57.MR 2542124 (2011b:05010)
6. L. Carlitz, Some polynomials related to the Bernoulli and Euler polynomials,Utilitas Math. 19 (1981), 81–127. MR 624049 (82j:10023)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
868 DAE SAN KIM et al 856-870
14 DAE SAN KIM, DMITRY V. DOLGY, AND TAEKYUN KIM
7. , Some remarks on the multiplication theorems for the Bernoulli andEuler polynomials, Glas. Mat. Ser. III 16(36) (1981), no. 1, 3–23. MR 634291(83b:10009)
8. R. Dere and Y. Simsek, Applications of umbral algebra to some special polyno-mials, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 3, 433–438.MR 2976601
9. S. Gaboury, R. Tremblay, and B.-J. Fugere, Some explicit formulas for cer-tain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. JangjeonMath. Soc. 17 (2014), no. 1, 115–123. MR 3184467
10. H. W. Gould, Explicit formulas for Bernoulli numbers, Amer. Math. Monthly79 (1972), 44–51. MR 0306102 (46 #5229)
11. D. S. Kim and T. Kim, Higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials, Adv. Difference Equ. (2013), 2013:251, 13. MR 3108262
12. D. S. Kim, T. Kim, S.-H. Lee, and S.-H. Rim, A note on the higher-order Frobenius-Euler polynomials and Sheffer sequences, Adv. Difference Equ.(2013), 2013:41, 12. MR 3032702
13. D. S. Kim, T. Kim, q-Bernoulli polynomials and q-umbral calculus, Sci. ChinaMath. no. 9, 57 (2014), no. 9, 1867–1874. MR 3249396
14. D.S. Kim, T. Kim, T. Komatsu, S.-H. Lee Barnes-type Daehee of the first kindand poly-Cauchy of the first kind mixed-type polynomials, Adv. Difference Equ.2014 (2014), 2014:140, 22 pp. MR 3259855
15. T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10(2003), no. 3, 261–267. MR 2012900 (2004j:11106)
16. , An identity of the symmetry for the Frobenius-Euler polynomials asso-ciated with the fermionic p-adic invariant q-integrals on Zp, Rocky MountainJ. Math. 41 (2011), no. 1, 239–247. MR 2845943 (2012k:11027)
17. , Identities involving Laguerre polynomials derived from umbral calculus,Russ. J. Math. Phys. 21 (2014), no. 1, 36–45. MR 3182545
18. T. Kim and J. Choi, A note on the product of Frobenius-Euler polynomials aris-ing from the p-adic integral on Zp, Adv. Stud. Contemp. Math. (Kyungshang)22 (2012), no. 2, 215–223. MR 2961614
19. T. Kim and B. Lee, Some identities of the Frobenius-Euler polynomials, Abstr.Appl. Anal. (2009), Art. ID 639439, 7. MR 2487361 (2010a:11231)
20. T. Kim, A note on poly-Bernoulli and higher-order poly-Bernoulli polynomials,Russ. J. Math. Phys. 22 (2015), no. 1, 26–33.
21. T. Kim and T. Mansour, Umbral calculus associated with Frobenius-type Euler-ian polynomials, Russ. J. Math. Phys. 21 (2014), no. 4, 484–493. MR 3284958
22. Q.-M. Luo and F. Qi, Relationships between generalized Bernoulli numbers andpolynomials and generalized Euler numbers and polynomials, Adv. Stud. Con-temp. Math. (Kyungshang) 7 (2003), no. 1, 11–18. MR 1981601
23. H. Ozden, I. N. Cangul, and Y. Simsek, Remarks on q-Bernoulli numbers as-sociated with Daehee numbers, Adv. Stud. Contemp. Math. (Kyungshang) 18(2009), no. 1, 41–48. MR 2479746 (2009k:11037)
24. S. Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Aca-demic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.MR 741185 (87c:05015)
25. K. Shiratani, On Euler numbers, Mem. Fac. Sci. Kyushu Univ. Ser. A 27 (1973),1–5. MR 0314755 (47 #3307)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
869 DAE SAN KIM et al 856-870
BARNES’ MULTIPLE FROBENIUS-EULER AND HERMITE MIXED-TYPE POLYNOMIALS15
26. Y. Simsek, O. Yurekli, and V. Kurt, On interpolation functions of the twistedgeneralized Frobenius-Euler numbers, Adv. Stud. Contemp. Math. (Kyung-shang) 15 (2007), no. 2, 187–194. MR 2356176 (2008g:11193)
Department of Mathematics, Sogang University, Seoul 121-742, Republic of KoreaE-mail address: [email protected]
Institute of Mathematics and Computer Science, Far Eastern Federal University,690950 Vladivostok, Russia
E-mail address: d−[email protected]
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Ko-rea
E-mail address: [email protected]
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
870 DAE SAN KIM et al 856-870
Robust stability and stabilization of linear uncertain
stochastic systems with Markovian switching
Yifan Wu1
Department of Basic Courses Jiangsu Food & Pharmaceutical Science College,
HuaiAn, Jiangsu, 223003, China
Abstract. This paper is concerned with robust stability and stabilization problem for a class of
linear uncertain stochastic systems with Markovian switching. The uncertain system under con-
sideration involves parameter uncertainties both in the drift part and in the diffusion part. New
criteria for testing the robust stability of such systems are established in terms of bi-linear matrix
inequalities (BLMIs), and sufficient conditions are proposed for the design of robust state-feedback
controllers. An example illustrates the proposed techniques.
Keywords: Bi-linear matrix inequalities (BLMIs); Robust stabilization; Stochastic system with
Markovian switching; Uncertainty
1 Introduction
Stochastic systems with Markovian switching have been used to model many practical systems
where they may experience abrupt changes in their structure and parameters. Such systems have
played a crucial role in many applications, such as hierarchical control of manufacturing systems([4,
5, 16]), financial engineering ([19]) and wireless communications ( [6]).
In the past decades, the stability and control of Markovian jump systems have recently received
a lot of attention. For example, [3] and [15] systematically studied stochastic stability properties of
jump linear systems. [1] discussed the stability of a semi-linear stochastic differential equation with
Markovian switching. [7, 9, 10, 12] discussed the exponential stability of general nonlinear stochastic
systems with Markovian switching of the form
dx(t) = f(x(t), t, r(t))dt+ g(x(t), t, r(t))dB(t). (1.1)
1E-mail address: [email protected]
1
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
871 Yifan Wu 871-880
Over the last decade, stochastic control problems governed by stochastic differential equation with
Markovian switching have attracted considerable research interest, and we here mention[2, 11, 20,
23, 24]. It is well known that uncertainty occurs in many dynamic systems and is frequently a cause
of instability and performance degradation. In the past few years, considerable attention has been
given to the problem of designing robust controllers for linear systems with parameter uncertainty,
such as[8, 13, 17, 21, 22]. However, a literature search reveals that the issue of stabilization of
uncertain system under consideration involves parameter uncertainties both in the drift part and
in the diffusion part has not been fully investigated and remains important and challenging. This
situation motivates the present study on the robust stabilization of linear uncertain stochastic systems
with Markovian switching. We aim at designing a robust state-feedback controller such that, for all
admissible uncertainties , the closed-loop system is exponentially stable in mean square.
The structure of this paper is as follows. In Section 2, we introduce notations, definitions and
results required from the literature. In Section 3, we shall discuss the problem of mean square
exponential stabilization for a linear jump stochastic system. In Section 4, sufficient conditions are
proposed for the design of robust state-feedback controllers. An example is discussed for illustrating
our main results in Section 5.
2 Preliminaries
In this paper, we will employ the following notation. Let |.| be the Euclidean norm in Rn. The
interval [0,∞) be denoted by R+. If A is a vector or matrix, its transpose is denoted by AT .
In denotes the n × n identity matrix. If A is a symmetric matrix λmin(A) and λmax(A) mean
the smallest and largest eigenvalue, respectively. If A and B are symmetric matrices, by A > B
and A ≥ B we mean that A − B is positive definite and nonnegative definite, respectively. And
C2,1(Rn × R+ × S;R+) denotes the family of all R+-valued functions on Rn × R+ × S which are
continuously twice differentiable in x and once differentiable in t. We write diag(a1, ..., an) for a
diagonal matrix whose diagonal entries starting in the upper left corner are a1, ..., an.
Let (Ω,F , (F)t, P ) be a complete probability space with a filtration (F)t satisfying the usual
conditions. Let r(t), t ≥ 0, be a right-continuous Markov chain on the probability space taking
values in a finite state space S = 1, 2, ..., N with generator Q = (qij)N×N given by
P (r(t+ ∆) = j | r(t) = i) =
qij∆ + o(∆), if i 6= j
1 + qii∆ + o(∆), if i = j
where ∆ > 0, and qij ≥ 0 denotes the switching rate from i to j if i 6= j while qii = −∑i6=j
qij .
2
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
872 Yifan Wu 871-880
Definition 1 ([9]) The trivial solution of system (1), or simply system (1) is said to be exponentially
stable in mean square if
lim supt→∞
1
tlog(E|x(t; t0, x0, r0)|2) < 0,
for all (t0, x0, r0) ∈ R+ ×Rn × S.
If V ∈ C2,1(Rn ×R+ × S;R+) , define operator LV (x, t, i) associated with system (1) by
LV (x, t, i) =∂V (x, t, i)
∂t+∂V (x, t, i)
∂xf(x, t, i)
+1
2tr[gT (x, t, i)
∂2V (x, t, i)
∂x2g(x, t, i)] +
N∑j=1
qijV (x, t, i).
We have the following lemma.
Lemma 2.1([9]) Let λ, c1, c2 be positive numbers. Assume that there exists a function V (x, t, i) ∈
C2,1(Rn ×R+ × S;R+) such that
c1|x(t)|2 ≤ V (x, t, i) ≤ c2|x(t)|2
and
LV (x, t, i) ≤ −λ|x(t)|2
for all (x, t, i) ∈ Rn ×R+ × S, then system (1) is exponentially stable in mean square.
In this note, we consider the following linear uncertain stochastic systems with Markovian switch-
ing:
dx(t) = A(r(t))x(t)dt+d∑k=1
Bk(r(t))x(t)dwk(t),
x(t0) = x0 ∈ Rn, t ≥ t0,
(2.2)
where w(t) = (w1(t), w2(t), · · · , wd(t))T denotes a d-dimensional Brownian motion or Wiener process,
x(t) ∈ Rn is the system state, we assume that w(t) and r(t) are independent. For any i ∈ S, 1 ≤
k ≤ d, Ai = A(r(t) = i) and Bki = Bk(r(t) = i) are not precisely known a priori, but belong to the
following admissible uncertainty domains:
Da = Ai +D0iF0i(t)E0i : F0i(t)TF0i(t) ≤ I, i ∈ S,
Dbk = Bki +DkiFki(t)Eki : Fki(t)TFki(t) ≤ I, i ∈ S,
where Ai, Bki, D0i, E0i, Dki, Eki are known constant real matrices with appropriate dimensions, while
F0i(t) and Fki(t) denotes the uncertainties in the system matrices, for all i ∈ S.
3
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
873 Yifan Wu 871-880
Lemma 2.2 ([14, 18]) Let A,D,E,W and F (t) be real matrices of appropriate dimensions such
that FT (t)F (t) ≤ I and W > 0, then ,
1) For scalar ε > 0, DF (t)E + (DF (t)E)T ≤ εDDT + 1εE
TE
2) For any scalar ε > 0 such that W − εDDT > 0,
(A+DF (t)E)TW−1(A+DF (t)E) ≤ AT (W − εDDT )−1A+ 1εE
TE.
3 Robust stability analysis
This section, we discuss the robust stability for system (2). For convenience, we will let the initial
values x0 and r0 be non-random, namely x0 ∈ Rn and r0 ∈ S, but the theory developed in this
paper can be generalized without any difficulty to cope with the case of random initial values, and
we write x(t; t0, x0, r0) = x(t) simply.
Theorem 3.1 Suppose that there exist N symmetric positive-definite matrices Pi and positive
scalars εi, γi, and λi, such that ∀ i ∈ S, the following BLMIs hold:
Π11 ∗ ∗ ∗ ∗
E0iPi −γiI ∗ ∗ ∗
Π31 0 Π33 ∗ ∗
Π41 0 0 −εiI ∗
Π51 0 0 0 Π55
< 0, (3.3)
where the symbol ‘∗’ denotes the transposed element at the symmetric position, and
Π11 = AiPi + PiATi + qiiPi + λiPi + γiD0iD
T0i,
Π31 = [PiBT1i, PiB
T2i, . . . , PiB
Tdi]T ,
Π41 = [PiET1i, PiE
T2i, . . . , PiE
Tdi]T ,
Π33 = diag[εiD1iDT1i − Pi, . . . , εiDdiD
Tdi − Pi],
Π51 = [Pi, Pi, . . . , Pi︸ ︷︷ ︸N−1
]T ,
Π55 = diag[−1
qi1P1, . . . ,
−1
qi(i−1)Pi−1,
−1
qi(i+1)Pi+1, . . . ,
−1
qiNPN ],
then system (2) is exponentially stable in mean square.
Proof Let Xi = P−1i and define V (x, i) = xTXix for all i ∈ S. And let c1 = minλmin(Xi) : i ∈
S, c2 = maxλmax(Xi) : i ∈ S, it is clear that
c1|x(t)|2 ≤ V (x, i) ≤ c2|x(t)|2. (3.4)
4
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
874 Yifan Wu 871-880
On the other hand, a calculation shows that
LV (x, i) = x(t)T[Xi(Ai +D0iF0iE0i) + (Ai +D0iF0iE0i)
TXi +N∑j=1
qijXj
+d∑k=1
(Bki +DkiFkiEki)TXi(Bki +DkiFkiEki)
]x(t),
by Lemma 2.2, for all i ∈ S, if there exist positive scalars εi and γi such that εiDkiDTki − Pi < 0,
1 ≤ k ≤ d, then we have
LV (x, i) ≤ x(t)T[XiAi +ATi Xi + γiXiD0iD
T0iXi + 1
γiET0iE0i
+d∑k=1
BTki(Pi − εiDkiDTki)−1Bki +
d∑k=1
1
εiETkiEki +
N∑j=1
qijXj
]x(t).
Thus, there exists a λ > 0 such that
LV (x, i) ≤ −λ|x(t)|2
will hold if for any i ∈ S there exists a λi > 0 such that
XiAi +ATi Xi + γiXiD0iDT0iXi + 1
γiET0iE0i
+d∑k=1
BTki(Pi − εiDkiDTki)−1Bki +
d∑k=1
1
εiETkiEki +
N∑j=1
qijXj + λiXi
< 0.
(3.5)
Pre- and post-multiplying (5) by Pi yields
AiPi + PiATi + γiD0iD
T0i + 1
γiPiE
T0iE0iPi
+d∑k=1
PiBTki(Pi − εiDkiD
Tki)−1BkiPi
+d∑k=1
1
εiPiE
TkiEkiPi +
∑j 6=i
qijPiP−1j Pi + qiiPi + λiPi
< 0,
which is equivalent to inequality (3) in view of Schur complement equivalence. The assertion of this
theorem follows from Lemma 2.1 immediately.
Remark 1 Theorem 3.1 provides the analysis results for the exponential stability of the system
(2). It can be seen from (3) that we need to check whether there exist N symmetric positive-definite
matrices Pi and positive scalars εi, γi, and λi meeting the N coupled matrix inequalities. It is clear
that inequality (3) is BLMIs, and it is LMIs for a prescribed λi, then we are able to determine
exponential stability of the system (3) readily by checking the solvability of the LMIs.
5
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
875 Yifan Wu 871-880
4 Robust stabilization synthesis
This section deals with the robust stabilization problem for linear uncertain stochastic systems with
Markovian switching. Let us consider the uncertain stochastic control system of the form
dx(t) =[A(r(t))x(t) + C(r(t))u(t)
]dt+
d∑k=1
[Bk(r(t))x(t) + Ck(r(t))u(t)
]dwk(t),
x(t0) = x0 ∈ Rn, t ≥ t0.
(4.6)
We aim to design a state-feedback controller u(t) = K(r(t))x(t) such that the resulting closed-loop
system
dx(t) =[A(r(t)) + C(r(t))K(r(t))
]x(t)dt+
d∑k=1
[Bk(r(t)) + Ck(r(t))K(r(t))
]x(t)dwk(t),
x(t0) = x0 ∈ Rn, t ≥ t0.
(4.7)
is exponentially stable in mean square over all admissible uncertainty domains Da and Dbk, where
Ki = K(r(t) = i) (i ∈ S) is the controller to be determined.
The following results solve the robust stabilization problem for system (6).
Theorem 4.1 The closed-loop system (7) is exponentially stable in mean square with respect to
state-feedback gain Ki = YiP−1i , if there exist N symmetric positive-definite matrices Pi, N matrices
Yi and positive scalars εi, γi, and λi, such that∀ i ∈ S, the following BLMIs hold:
Π11 ∗ ∗ ∗ ∗
E0iPi −γiI ∗ ∗ ∗
Π31 0 Π33 ∗ ∗
Π41 0 0 −εiI ∗
Π51 0 0 0 Π55
< 0, (4.8)
where
Π11 = (AiPi + CiYi) + (AiPi + CiYi)T + qiiPi
+ λiPi + γiD0iDT0i,
Π31 = [(B1iPi + C1iYi)T , . . . , (BdiPi + CdiYi)
T ]T ,
Π41 = [PiET1i, PiE
T2i, . . . , PiE
Tdi]T ,
Π33 = diag[εiD1iDT1i − Pi, . . . , εiDdiD
Tdi − Pi],
Π51 = [Pi, Pi, . . . , Pi︸ ︷︷ ︸N−1
]T ,
Π55 = diag[−1
qi1P1, . . . ,
−1
qi(i−1)Pi−1,
−1
qi(i+1)Pi+1, . . . ,
−1
qiNPN ].
6
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
876 Yifan Wu 871-880
Proof The proof is similar to that of Theorem 3.1, so we only give an outlined one. Let Xi = P−1i
and define V (x, i) = xTXix . There exists a λ > 0 such that LV (x, i) ≤ −λ|x(t)|2 will hold if for
any i ∈ S there exist positive scalars εi , γi and λi, where εiDkiDTki − Pi < 0, 1 ≤ k ≤ d, such that
Xi(Ai + CiKi) + (Ai + CiKi)TXi + γiXiD0iD
T0iXi + 1
γiET0iE0i
+d∑k=1
(Bki + CkiKi)T (Pi − εiDkiD
Tki)−1(Bki + CkiKi)
+d∑k=1
1
εiETkiEki +
N∑j=1
qijXj + λiXi < 0.
(4.9)
Noting that Yi = KiPi, and Pre- and post-multiplying (9) by Pi yields
(AiPi + CiYi) + (AiPi + CiYi)T + γiD0iD
T0i + 1
γiPiE
T0iE0iPi
+d∑k=1
(BkiPi + CkiYi)T (Pi − εiDkiD
Tki)−1(BkiPi + CkiYi)
+d∑k=1
1
εiPiE
TkiEkiPi +
∑j 6=i
qijPiP−1j Pi + qiiPi + λiPi < 0,
which is equivalent to (8) in view of Schur complement equivalence. The assertion of this theorem
follows from Lemma 2.1 immediately.
Remark 2 It is shown in Theorem 4.1 that the robust exponentially stabilization of system (6)-
(7) is guaranteed if the inequalities (8) are valid. And the inequality (8) is linear in Yi and Pi for a
prescribed λi, thus the standard LMI techniques can be exploited to check the exponential stability
of the closed-loop system (7).
5 Example
Let w(t) be a one-dimensional Brownian motion, let r(t) be a right-continuous Markov chain
taking values in S = 1, 2 with generator Q =
−1 1
1 −1
, consider a two-dimensional stochastic
systems with Markovian switching of the form
dx(t) =
[(A(r(t)) +D0(r(t))F0(r(t), t)E0(r(t))
)x(t) + C(r(t))u(t)
]dt
+
[(B(r(t)) +D1(r(t))F1(r(t), t)E1(r(t))
)x(t) + C1(r(t))u(t)
]dw(t),
(5.10)
where
A1 =
0.5 0.2
0.3 0.8
, A2 =
1 0.1
0.2 2
, B1 =
−1 0.5
0.5 −1
, B2 =
−2 0.1
0.1 1
,7
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
877 Yifan Wu 871-880
D01 = diag(−1,−2), D02 = diag(0.2, 0.3), D11 = diag(−1,−1), D12 = diag(5,−0.5),
E01 = diag(0.2, 0.2), E02 = diag(−3,−5), E11 = diag(−0.9,−0.9), E12 = diag(0.5, 1),
C1 =
−8 0.1
0.05 −10
, C2 =
−20 0
0 −30
, C11 =
−1 0.5
2 3
, C12 =
−2 1
0.5 −4
,for i = 1, 2, F0i(t) and F1i(t) denote the uncertainties of system (10). Let λ1 = 1, λ2 = 2, by solving
LMIs (8), we find the feasible solution:
P1 =
98.708 4.383
4.383 85.385
, P2 =
233.108 −0.786
−0.786 180.327
, Y1 =
93.468 −16.376
−20.698 70.862
,
Y2 =
171.947 −64.056
75.520 82.513
, γ1 = 0.082, γ2 = 1.170, ε1 = 0.034, ε2 = 0.004,
therefore, by Theorem 4.1, closed-loop system (10) is exponentially stable in mean square with
respect to state-feedback gain Ki = YiP−1i .
6 Conclusions
Based on the exponential stability theory, we have investigated the robust stochastic stability
of the uncertain stochastic system with Markovian switching, sufficient stability conditions were
developed. The robust stability of such systems can be tested based on the feasibility of bi-linear
matrix inequalities An example has been presented to illustrate the effectiveness of the main results.
It is believed that this approach is one step further toward the descriptions of the uncertain stochastic
systems.
References
[1] G.K. Basak, A. Bisi, M.K. Ghosh, Stability of a random diffusion with linear drift. J. Math.
Anal. Appl. 202, 604-622 (1996)
[2] Y. Dong, J. Sun, On hybrid control of a class of stochastic non-linear Markovian switching
systems. Automatica. 44, 990-995 (2008)
8
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
878 Yifan Wu 871-880
[3] X. Feng, K.A. Loparo, Y. Ji, H.J. Chizeck, Stochastic stability properties of jump linear systems.
IEEE Trans. Automat. Contr. 37, 38-52 (1992)
[4] M.K. Ghosh, A. Arapostathis, S.I. Marcus, Ergodic control of switching diffusions. SIAM J.
Control Optim. 35, 1952-1988 (1997)
[5] M.K. Ghosh, A. Arapostathis, S.I. Marcus, Optimal control of switching diffusions with appli-
cation to flexible manufacturing systems. SIAM J. Contr. Optim. 35, 1183-204 (1993)
[6] V. Krishnamurthy, X. Wang, G. Yin,Spreading code optimization and adaptation in CDMA via
discrete stochastic approximation. IEEE Trans. Inform. Theory. 50, 1927-1949 (2004)
[7] R. Khasminskii, C. Zhu, G. Yin, Stability of regime-switching diffusions. Stoch. Proc. Appl.
117, 1037-1051 (2007)
[8] J. Lian, F. Zhang, P. Shi, Sliding mode control of uncertain stochastic hybrid delay systems
with average dwell time. Circuits Syst. Signal Process. 31, 539-553 (2012)
[9] X. Mao, Stability of stochastic differential equations with Markovian switching. Stoch. Proc.
Appl. 79, 45-67(1999)
[10] X. Mao, G. Yin, C. Yuan, Stabilization and destabilization of hybrid systems of stochastic
differential equations. Automatica. 43, 264-273(2007)
[11] X. Mao, J. Lam, L. Huang, Stabilisation of hybrid stochas- tic differential equations by delay
feedback control. Syst. Control Lett. 57, 927-935 (2008)
[12] S. Pang, F. Deng, X. Mao, Almost sure and moment exponential stability of eulercmaruyama
discretizations for hybrid stochastic differential equations. J. Comput. Appl. Math. 213, 127-141
(2008)
[13] P.V. Pakshin, Robust stability and stabilization of family of jumping stochastic systems. Non-
linear Analysis. 30, 2855-2866 (1997)
[14] I.R. Petersen, A stabilization algorithm for a class of uncertain linear systems. Syst. Control
Lett., 8:351-357,1987.
[15] M.A. Rami, L.E. Ghaoui, LMI optimization for nonstandard Riccati equations arising in s-
tochastic control. IEEE Trans. Automat. Contr. 41, 1666-1671(1996)
[16] S.P. Sethi, Q. Zhang. Hierarchical decision making in stochastic manufacturing systems(
Birkhauser, Boston, 1994)
9
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
879 Yifan Wu 871-880
[17] Z. Wang, H. Qiao, K.J. Burnham, On stabilization of bilinear uncertain time-delay stochastic
systems with Markovian jumping parameters. IEEE Trans. Automat. Contr. 47, 640-646 (2002)
[18] S. Xu, T. Chen. Robust H∞ control for uncertain stochastic systems with state delay, IEEE
Trans. Automat. Control. 47, 2089-2094 (2002)
[19] G. Yin, R.H. Liu, Q. Zhang, Recursive algorithms for stock liquidation: A stochastic optimiza-
tion approach. SIAM J. Contr. Optim. 13, 240-263(2002)
[20] C. Yuan, J. Lygeros, On the exponential stability of switching diffusion processes, IEEE Trans.
Automat. Control. 50, 1422-1426 (2005)
[21] C. Yuan, J. Lygeros, Stabilization of a class of stochastic differential equations with Markovian
switching. Syst. Control Lett. 54, 819-833 (2005)
[22] C. Yuan, X. Mao, Robust stability and controllability of stochastic differential delay equations
with Markovian switching. Automatica. 40, 343-354(2004)
[23] C. Zhu, Optimal control of the risk process in a regime-switching environment. Automatica. 47,
1570-1579 (2011)
[24] F.Zhu, Z.Han, J.Zhang, Stability analysis of stochastic differential equations with Markovian
switching. Syst. Control Lett. 61, 1209-1214(2012)
10
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
880 Yifan Wu 871-880
On interval valued functions and Mangasarian type duality involvingHukuhara derivative
Izhar Ahmad1,∗ Deepak Singh2, Bilal Ahmad Dar3, S. Al-Homidan4
1,4 Department of Mathematics and Statistics, King Fahd University of Petroleumand Minerals, Dhahran 31261, Saudi Arabia.2Department of Applied Sciences, NITTTR (under Ministry of HRD, Govt. ofIndia), Bhopal, M.P., India.3Department of Applied Mathematics, Rajiv Gandhi Proudyogiki Vishwavidyalaya(State Technological University of M.P.), Bhopal, M.P., India.
Abstract
In this paper, we introduce twice weakly differentiable and twice H-differentiable interval valued functions. The existence of twice H-differentiableinterval-valued function and its relation with twice weakly differentiable func-tions are presented. Interval valued bonvex and generalized bonvex func-tions involving twice H-differentiability are proposed. Under the proposedsettings, necessary conditions are elicited naturally in order to achieve LU -efficient solution. Mangasarian type dual is discussed for a nondifferentiablemultiobjective programming problem and appropriate duality results are alsoderived. The theoretical developments are illustrated through non-trivial nu-merical examples.
Keyword: Interval valued functions; twice weak differentiability; twice H- differ-entiability, LU -efficient solution; generalized bonvexity; duality.
Mathematics Subject Classification: 90C25, 90C29, 90C30.
1 Introduction
The study of uncertain programming is always challenging in its modern face. Sev-eral attempts to achieve optimal in the same have been made in several directions.However optimality conditions still needs to be optimized. In this direction intervalvalued programming is one of the several techniques which has got attention ofresearchers in the recent past. Existing literature [2, 4, 5, 7, 8, 9, 10, 11, 12, 13,17, 19, 20, 21, 22] contains many interesting results on the study of interval val-ued programming involving different types of differentiability concepts and varioustypes of convexity concepts of interval valued functions.
Second order duality gives tighter bounds for the value of the objective functionwhen approximations are used. For more information, authors may see ([11], pp
∗Corresponding author.
E-mail addresses: [email protected] (Izhar Ahmad), [email protected] (DeepakSingh), [email protected] (Bilal Ahmad Dar), [email protected] (S.Al-Homidan)
1
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
881 Izhar Ahmad et al 881-896
93). One more advantage is that if a feasible point in the primal problem is givenand first order duality does not use, then we can apply second order duality toprovide a lower bound of the value of the primal problem.
Note that the study of nondifferentiable interval valued programming problemshas not been studied extensively as quoted in Sun and Wang [18] therefore to studythe second order duals of the aforesaid problem is an interesting move, we considerthe following nondifferentiable vector programming problem with interval valuedobjective functions and constraint conditions and study its second order dual ofMangasarian type.(IP )
min f(x) + (xTBx)12 =
(f1(x) + (xTBx)
12 , ..., fk(x) + (xTBx)
12
)subject to gj(x) LU [0, 0], j ∈ Λm
where fi = [fLi , f
Ui ], i ∈ Λk and gj = [gLj , g
Uj ], j ∈ Λm are interval valued functions
with fLi , f
Ui , g
Lj , g
Uj : Rn → R, i ∈ Λk, j ∈ Λm be twice differentiable functions.
The remaining paper is designed as: section 2 is devoted to preliminaries. Sec-tion 3 represents the differentiation of interval valued functions with the introduc-tion of twice weakly differentiable and twice H-differentiable interval valued func-tions. Some properties of these functions are also presented. Section 4 highlightsthe concept of so-called bonvexity and its quasi and pseudo forms of interval valuedfunctions and their properties. In section 5, the necessary conditions for proposedsolution concept are elicited naturally by considering above settings. Finally withthe proposed settings the section 6 is devoted to study the Mangasarian type dualof primal problem (IP ). Lastly we conclude in section 7.
2 Preliminaries
Let Ic denote the class of all closed and bounded intervals in R. i.e.,
Ic = [a, b] : a, b ∈ R and a ≤ b.
And b − a is the width of the interval [a, b] ∈ Ic. Then for A ∈ Ic we adoptthe notation A = [aL, aU ], where aL and aU are respectively the lower and upperbounds of A. Let A = [aL, aU ], B = [bL, bU ] ∈ Ic and λ ∈ R, we have the followingoperations.
(i) A+B = a+ b : a ∈ A and b ∈ B = [aL + bL, aU + bU ](ii)
λA = λ[aL, aU ] =
[λaL, λaU ] if λ ≥ 0[λaU , λaL] if λ < 0;
(iii)A×B = [min
ab,max
ab],
whereminab
= minaLbL, aLbU , aUbL, aUbU
2
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
882 Izhar Ahmad et al 881-896
andmaxab
= maxaLbL, aLbU , aUbL, aUbU
In view of (i) and (ii) we see that
−B = −[bL, bU ] = [−bU ,−bL] and A−B = A+ (−B) = [aL − bL, aU − bL].
Also the real number a ∈ R can be regarded as a closed interval Aa = [a, a], thenwe have for B ∈ Ic
a+B = Aa +B = [a+ bL, a+ bU ].
Note that the space Ic is not a linear space with respect to the operations (i) and(ii), since it does not contain inverse elements.
3 Differentiation of interval valued functions
Definition 1. [20] Let X be open set in R. An interval-valued function f : X →Ic is called weakly differentiable at x∗ if the real-valued functions fL and fU aredifferentiable at x∗ (in the usual sense).
Given A,B ∈ Ic, if there exists C ∈ Ic such that A = B+C, then C is called theHukuhara difference of A and B. We also write C = AH B when the Hukuharadifference C exists, which means that aL− bL ≤ aU − bU and C = [aL− bL, aU − bU ].
Proposition 1. [20] Let A = [aL, aU ] and B = [bL, bU ] be two closed intervals in R.If aL−bL ≤ aU−bU , then the Hukuhara difference C exists and C = [aL−bL, aU−bU ].
Definition 2. [20] Let X be an open set in R. An interval-valued function f :X → Ic is called H-differentiable at x∗ if there exists a closed interval A(x∗) ∈ Icsuch that
limh→0+
f(x∗ + h)H f(x∗)
hand lim
h→0+
f(x∗)H f(x∗ + h)
h
both exist and are equal to A(x∗). In this case, A(x∗) is called the H-derivative off at x∗.
Proposition 2. [20] Let f be an interval-valued function defined on X ⊆ Rn. If fis H-differentiable at x∗ ∈ X, then f is weakly differentiable at x∗.
Next we introduce twice differentiable interval valued functions and study someproperties.
Definition 3. Let X be an open set in Rn, and let x∗ = (x∗1, ..., x∗n) ∈ X be fixed.
Then we say that f is twice weakly differentiable interval valued function at x∗ if
3
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
883 Izhar Ahmad et al 881-896
fL and fU are twice differentiable functions at x∗ (in usual sense). We denote by∇2f the second differential of f , then we have
∇2f(x∗) = ∇(∇f(x))x=x∗
= ∇(∇[fL(x), fU(x)])x=x∗
= ∇([∇fL(x),∇fU(x)])x=x∗
= [∇2fL(x),∇2fU(x)]x=x∗
=
[(∂2fL
∂xi∂xj
)i,j
(x),
(∂2fU
∂xi∂xj
)i,j
(x)
]x=x∗
.
Definition 3 is illustrated by the following example.
Example 1. Consider the interval valued function
f(x1, x2) = [fL = 2x1 + x22, fU = x21 + x22 + 1]. (1)
Therefore we have
∇f(x) = [(2, 2x2), (2x1, 2x2)]T
and
∇2f(x) =
[(0 00 2
),
(2 00 2
)].
Definition 4. Let X be an open set in Rn, and let x∗ = (x∗1, ..., x∗n) ∈ X be fixed.
Then we say that f is twice H-differentiable interval valued function if f ′ is H-differentiable at x∗, where f ′ is H-derivative of f . We denote by ∇2
Hf the secondorder H-differential of f , then we have
∇2Hf(x∗) = ∇H(∇Hf(x))x=x∗
= ∇H
(∂f
∂x1(x), ...,
∂f
∂xn(x)
)T
x=x∗
=
(∇H
[∂fL
∂x1(x),
∂fU
∂x1(x)
], ...,∇H
[∂fL
∂xn(x),
∂fU
∂xn(x)
])T
x=x∗
=
[∂2fL
∂2x1(x), ∂
2fU
∂2x1(x)]...[
∂2fL
∂x1∂xn(x), ∂2fU
∂x1∂xn(x)]
.... . .
...[∂2fL
∂xn∂x1(x), ∂2fU
∂xn∂x1(x)]...[∂2fL
∂2xn(x), ∂
2fU
∂2xn(x)]
n×n,x=x∗
.
(2)
Following example justifies the existence of twiceH-differentiable interval valuedfunction.
Example 2. Consider the interval valued function (1), then by definition we have
∇Hf(x) = ([2, 2x1], [2x2, 2x2])T
4
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
884 Izhar Ahmad et al 881-896
which exist for x1 ≥ 1. Therefore we have
∇2Hf(x) = ∇H([2, 2x1], [2x2, 2x2])
T
=
([0, 2] [0, 0][0, 0] [2, 2]
).
The relation between twice weakly differentiable and twice H-differentiable in-terval valued functions is furnished as follows.
Proposition 3. Let f be an interval-valued function defined on X ⊆ Rn. If f istwice H-differentiable at x∗ ∈ X, then f is twice weakly differentiable at x∗.
Proof. From (2) we have
∇2Hf(x∗) =
∂2fL
∂2x1(x) ... ∂2fL
∂x1∂xn(x)
.... . .
...∂2fL
∂xn∂x1(x) ... ∂2fL
∂2xn(x)
n×n
,
∂2fU
∂2x1(x) ... ∂2fU
∂x1∂xn(x)
.... . .
...∂2fU
∂xn∂x1(x) ... ∂2fU
∂2xn(x)
n×n
x=x∗
= [∇2fL(x),∇2fU(x)]x=x∗
= ∇2f(x∗).
We authenticate Proposition 3 by following example.
Example 3. From Example 2 we have
∇2Hf(x) =
([0, 2] [0, 0][0, 0] [2, 2]
)=
[(0 00 2
),
(2 00 2
)]=[∇2fL(x),∇2fU(x)
]= ∇2f(x). (see Example 1).
The converse of Proposition 3 is not true in general, however we have the fol-lowing result.
Proposition 4. Let f ∈ T , be twice weakly differentiable function at x∗, with(fL)′′(x∗) = aL
′(x∗) and (fU)′′(x∗) = aU
′(x∗).
1. if (fL)′(x∗+h)−(fL)′(x∗) ≤ (fU)′(x∗+h)−(fU)′(x∗) and (fL)′(x∗)−(fL)′(x∗−h) ≤ (fU)′(x∗)−(fU)′(x∗−h) for every h > 0, then f is twice H-differentiablewith second H-derivative [aL
′(x∗), aU
′(x∗)].
2. if aL′(x∗) > aU
′(x∗), then f is not twice H-differentiable at x∗.
Proof. The proof is similar as that of Proposition 4.3 of [20].
The existence of twice weakly differentiable interval valued functions which arenot twice H-differentiable is proved by following example.
Example 4. Consider f : [0, 2] → [x3 + x2 + 1, x3 + 2x + 2] be an interval valuedfunction defined on [0, 2]. Then f is twice weakly differentiable on (0, 2) but f isnot twice H-differentiable on (0, 2) as aL
′(x∗) > aU
′(x∗).
5
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
885 Izhar Ahmad et al 881-896
4 Interval valued bonvex functions
Convexity is an important concept in studying the theory and methods of mathe-matical programming, which has been generalized in several ways. For differentiablefunctions numerous generalizations of convexity exist in the literature. An impor-tant concept so-called second order convexity for twice differentiable real valuedfunctions was introduced in Mond [14], however Bector and Chandra [6] namedthem as bonvex functions. Now consider f to be real valued twice differentiablefunction, then for the definitions of (strictly) bonvexity, (strictly) pseudobonvexityand (strictly) quasibonvexity, one is refered to [3].
In this section, we introduce LU -bonvex, LU -pseudobonvex and LU -quasibonvexinterval valued functions and their strict conditions. We consider T to be the setof all interval valued functions defined on X ⊆ Rn.
Definition 5. Let f ∈ T be twice H-differentiable function at x∗ ∈ X. If we havefor every x ∈ X and P = (P1, ..., Pn) with Pi ∈ Ic such that PL
i ≥ 0, i ∈ Λk.
1.
f(x)H f(x∗) +1
2P T∇2
Hf(x∗)P LU
∇Hf(x∗) +∇2
Hf(x∗)P
(x− x∗)
then we say that f is LU-bonvex at x∗. We also say that f is strictly LU-bonvex at x∗(6= x) if the inequality is strict.
2. If
f(x)H f(x∗) +1
2P T∇2
Hf(x∗)P LU [0, 0],
⇒∇Hf(x∗) +∇2
Hf(x∗)P
(x− x∗) LU [0, 0]
then we say that f is LU-quasibonvex at x∗. We also say that f is strictlyLU-quasibonvex x∗(6= x) if the inequality is strict.
3. If ∇Hf(x∗) +∇2
Hf(x∗)P
(x− x∗) LU [0, 0],
⇒ f(x)H f(x∗) +1
2P T∇2
Hf(x∗)P LU [0, 0]
then we say that f is LU-pseudobonvex at x∗. We also say that f is strictly LU-pseudobonvex at x∗(6= x) if the inequality is strict.
Now we present some non-trivial examples which authenticates that the classof interval valued functions introduced in this section is non-empty.
Example 5. Consider an interval valued function f(x) = [x2 + 3x + 2, x2 + 3x +5], x ≥ 0. Then we have
∇Hf(x) = ([2x+ 3, 2x+ 3])
6
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
886 Izhar Ahmad et al 881-896
= ([3, 3])x=0
and∇2
Hf(x) = ([2, 2])
we have
[x2 + 3x+ 2, x2 + 3x+ 5]H [2, 5] +1
2([0, 1])T [2, 2][0, 1] = [x2 + 3x+ 2, x2 + 3x+ 2]
LU ([3, 3] + [2, 2][0, 1])(x)
= [3x, 5x]
therefore f is LU-bonvex at x = 0.
Next consider another interval valued functions defined as
f(x1, x2) = [x21 + x22 + 3, x21 + x22 + 5], x ≥ 0.
Then we have
∇Hf(x1, x2) = ([2x1, 2x1], [2x2, 2x2])T
=([4, 4], [4, 4]T
)T(x1,x2)=(2,2)
and
∇2Hf(x) =
([2, 2] [0, 0][0, 0] [2, 2]
)Now let
[x21 + x22 + 3, x21 + x22 + 5]H [11, 13] +1
2
([1, 1][1, 1]
)T ([2, 2] [0, 0][0, 0] [2, 2]
)([1, 1][1, 1]
)LU [0, 0]
then (([4, 4][4, 4]
)+
([2, 2] [0, 0][0, 0] [2, 2]
)([1, 1][1, 1]
))(x1 − 2x2 − 2
)LU [0, 0].
this shows that f is LU-quasibonvex at (2, 2).However if((
[4, 4][4, 4]
)+
([2, 2] [0, 0][0, 0] [2, 2]
)([1, 1][1, 1]
))(x1 − 2x2 − 2
)LU [0, 0].
then
[x21 + x22 + 3, x21 + x22 + 5]H [11, 13] +1
2
([1, 1][1, 1]
)T ([2, 2] [0, 0][0, 0] [2, 2]
)([1, 1][1, 1]
)LU [0, 0]
this shows that f is LU-pseudobonvex at (2, 2).
7
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
887 Izhar Ahmad et al 881-896
Proposition 5. Let f ∈ T be twice H-differentiable function at x∗ and P =(P1, ..., Pn) with Pi ∈ Ic such that PL
i ≥ 0, i ∈ Λn.
1. if f is LU-bonvex at x∗ then fL and fU are bonvex functions at x∗.
2. if f is LU-quasibonvex at x∗ then fL and fU are quasibonvex functions at x∗.
3. if f is LU-pseudobonvex at x∗ then fL and fU are pseudobonvex functions atx∗.
Proof. (i) Let f is LU -bonvex at x∗, then by definition we have
f(x)H f(x∗) +1
2P T∇2
Hf(x∗)P LU
∇Hf(x∗) +∇2
Hf(x∗)P
(x− x∗)
Since f is twice H-differentiable at x∗, then by Proposition 3 and Definition 3 fL
and fU are twice differentiable at x∗. Also since PLi ≥ 0, therefore we have
fL(x)− fL(x∗) +1
2PLT∇2fL(x∗)PL ≥
∇fL(x∗) +∇2fL(x∗)PL
(x− x∗),
and
fU(x)− fU(x∗) +1
2PUT∇2fU(x∗)PU ≥
∇fU(x∗) +∇2fU(x∗)PU
(x− x∗).
Therefore fL and fU are bonvex functions at x∗.(ii) and (iii) follows by similar treatment.
Note that the converse of Proposition 5 follows in the light of Proposition 4.
Proposition 6. Let f ∈ T be twice H-differentiable function at x∗ and P =(P1, ..., Pn) with Pi ∈ Ic such that PL
i ≥ 0, i ∈ Λn.
1. if f is strictly LU-bonvex at x∗ then either fL or fU or both are strictly bonvexfunctions at x∗.
2. if f is strictly LU-quasibonvex at x∗ then either fL or fU or both are strictlyquasibonvex functions at x∗.
3. if f is strictly LU-pseudobonvex at x∗ then either fL or fU or both are strictlypseudobonvex functions at x∗.
Proof. Proof is same as that of Proposition 5.
Remark 1. If we assume that fL = fU , then bonvexity comes as a sub-case ofLU-bonvexity, and similarly for quasi and pseudobonvexity.
8
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
888 Izhar Ahmad et al 881-896
5 Solution concept and necessary conditions
In this section we shall propose solution concept and derive some necessary condi-tions for problem (IP ). We define by SIP the set of feasible solutions of (IP ).
Definition 6. Let x∗ ∈ SIP . We say that x∗ is an efficient solution of (IP ) if thereexist no x ∈ SIP , such that
fi(x) LU fi(x∗), i ∈ Λk and fh(x) ≺LU fh(x∗), for at least one index h.
An efficient solution x∗ is said to be properly efficient solution of (IP ) if there existscalar M > 0, such that for all i ∈ Λk, fi(x) ≺LU fi(x
∗) and x ∈ SIP imply that
fi(x∗)H fi(x) LU Mfh(x)H fh(x∗)
for atleast one index h ∈ Λk − i such that fh(x∗) ≺LU fh(x).
Theorem 1. (Mond et al. [16]) Let x∗ be a properly efficient solution of (P ) (see,[3]) at which constraint qualification [15] is satisfied. Then there exist λ∗ ∈ Rk, u∗ ∈Rm and v∗i ∈ Rn, i ∈ ΛK such that
k∑i=1
λ∗i (fi(x∗) +Biv
∗i ) +∇u∗Tg(x∗) = 0,
u∗Tg(x∗) = 0,
(x∗TBix∗)
12 = x∗TBiv
∗i , i ∈ Λk,
v∗iTBiv
∗i ≤ 1, i ∈ Λk,
λ∗ > 0,k∑
i=1
λ∗i = 1, u∗ ≥ 0.
Now we present the necessary conditions for problem(IP ). Consider the follow-ing constraint qualification CQ1
dT∇Hgj(x∗) LU [0, 0], j ∈ J0(x∗)
dT∇Hfi(x∗) + dTBix
∗/(x∗TBix∗)
12 LU [0, 0], if x∗TBix
∗ > 0
dT∇Hfi(x∗) + (dTBid)
12 LU [0, 0], if x∗TBix
∗ = 0
Theorem 2. Let x∗ be a properly efficient solution of (IP ) at which a constraintqualification CQ1 is satisfied. Then there exist λ∗ ∈ Rk, u∗ ∈ Rm and v∗i ∈ Rn, i ∈ΛK such that
k∑i=1
λ∗i (∇Hfi(x∗) +Biv
∗i ) +∇Hu
∗Tg(x∗) = [0, 0],
u∗Tg(x∗) = [0, 0],
(x∗TBix∗)
12 = x∗TBiv
∗i , i ∈ Λk,
v∗itBiv
∗i ≤ 1, i ∈ Λk,
λ∗ > 0,k∑
i=1
λ∗i = 1, u∗ ≥ 0.
9
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
889 Izhar Ahmad et al 881-896
Proof. Since x∗ is properly efficient solution of (IP ) at which a constraint qualifica-tion CQ1 is satisfied. Then using the property of intervals and twice H-derivative,for 0 < ξLi , ξ
Ui ∈ R, i ∈ Λk with ξLi + ξUi = 1, i ∈ Λk, we have
CQ2dT∇gLj (x∗) > 0, j ∈ J0(x∗)
dT∇gUj (x∗) > 0, j ∈ J0(x∗)
dT (ξLi ∇fLi (x∗) + ξUi ∇fU
i (x∗)) + dTBix∗/(x∗TBix
∗)12 < 0, if x∗TBix
∗ > 0
dT (ξLi ∇fLi (x∗) + ξLi ∇fU
i (x∗)) + (dTBid)12 < 0, if x∗TBix
∗ = 0
Further using the property of intervals and twice H-derivative, for 0 < ξLi , ξUi ∈
R, i ∈ Λk with ξLi + ξUi = 1, i ∈ Λk we have new conditions as
k∑i=1
λ∗i ((ξLi ∇fL
i (x∗) + ξLi ∇fUi (x∗)) +Biv
∗i ) +∇u∗T (gL(x∗) + gU(x∗)) = 0,
u∗TgL(x∗) = 0,
u∗TgU(x∗) = 0,
(x∗TBix∗)
12 = x∗TBiv
∗i , i ∈ Λk,
v∗iTBiv
∗i ≤ 1, i ∈ Λk,
λ∗ > 0,k∑
i=1
λ∗i = 1, u∗ ≥ 0.
Now using constraint qualification CQ2 the above conditions are justified by Theo-rem 1 for the problem (say (IP1)) heaving objective function (ξL1 f
L1 (x) + ξL1 f
U1 (x),
..., ξLk fLk (x)+ξLk f
Uk (x)) and constraint functions gLj (x), gUj (x) ≤ 0, j ∈ Λm. Now it is
easy to see that the optimal solutions of (IP ) and (IP1) are same. This completesthe proof.
6 Mangasarian type duality
In this section, we propose the following Mangasarian type dual of primal problem(IP ).(MSD) V-maximize(
f1(y) + uTg(y) + yTB1v1 H1
2P T∇2
Hf1(y) + uTg(y)P, ...,
fk(y) + uTg(y) + yTBkvk H1
2P T∇2
Hfk(y) + uTg(y)P)
subject to
k∑i=1
λi(∇Hfi(y) +∇2Hfi(y)P +Bivi) +∇Hu
Tg(y) +∇2Hu
Tg(y)P = [0, 0] (3)
10
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
890 Izhar Ahmad et al 881-896
vTi Bivi ≤ 1, i ∈ Λk (4)
λ > 0,r∑
i=1
λi = 1 (5)
u = (u1, ..., um)T ≥ 0, g = (g1, ..., gm) such that gj = [gLj , gUj ], j = 1, ...,m, P =
(P1, ..., Pn) with Pi ∈ Ic such that PLi ≥ 0, i ∈ Λk. and y, vi ∈ Rn.
We define by SMSD the set of all feasible solutions of (MSD), therefore ifz ∈ SMSD then z = (y, u, v, λ, P ), such that v ∈ Rk with vi ∈ Rn, and Pi ∈ Ic suchthat PL
i ≥ 0, i ∈ Λk. We shall use the following generalized Schwartz inequality:
xTAz ≤ (xTAx)1/2(zTAz)1/2,
where x, z ∈ Rn and A is positive semidefinite symmetric matrix of order n.
Theorem 3. (weak duality) Let x ∈ SIP and z ∈ SMSD. Assume that fi(.) +(.)TBivi, i ∈ Λk and gj(.), j ∈ Λm are LU-bonvex at y, then the following can nothold.
fi(x)+(xTBix)12 LU fi(y)+uTg(y)+yTBiviH
1
2P T∇2
Hfi(y) + uTg(y)P, i ∈ Λk.
(6)and
fh(x)+(xTBhx)12 ≺LU fh(y)+uTg(y)+yTBhvhH
1
2P T∇2
Hfh(y) + uTg(y)P, (7)
for at least one index h.
Proof. From (3) we have
k∑i=1
λi(∇fL
i (y) +∇2fLi (y)PL +Bivi
)+∇uTgL(y) +∇2uTgL(y)PL = 0.
k∑i=1
λi(∇fU
i (y) +∇2fUi (y)PU +Bivi
)+∇uTgU(y) +∇2uTgU(y)PU = 0.
Adding we get,
k∑i=1
λi(∇fL
i (y) +∇fUi (y) +∇2fL
i (y)PL +∇2fUi (y)PU + 2Bivi
)+∇uTgL(y)
+∇uTgU(y) +∇2uTgL(y)PL +∇2uTgU(y)PU = 0. (8)
If possible let (6) and (7) holds then by definition we havefLi (x) + (xTBix)
12 ≤ fL
i (y) + uTgL(y) + yTBivi − 12PLT∇2fL
i (y) + uTgL(y)PL.
fUi (x) + (xTBix)
12 ≤ fU
i (y) + uTgU(y) + yTBivi − 12PUT∇2fU
i (y) + uTgU(y)PU .
for i ∈ Λk, andfLh (x) + (xTBhx)
12 < fL
h (y) + uTgL(y) + yTBhvh − 12PLT∇2fL
h (y) + uTgL(y)PL.
fUh (x) + (xTBhx)
12 ≤ fU
h (y) + uTgU(y) + yTBhvh − 12PUT∇2fU
h (y) + uTgU(y)PU .
11
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
891 Izhar Ahmad et al 881-896
orfLh (x) + (xTBhx)
12 ≤ fL
h (y) + uTgL(y) + yTBhvh − 12PLT∇2fL
h (y) + uTgL(y)PL.
fUh (x) + (xTBhx)
12 < fU
h (y) + uTgU(y) + yTBhvh − 12PUT∇2fU
h (y) + uTgU(y)PU .
orfLh (x) + (xTBhx)
12 < fL
h (y) + uTgL(y) + yTBhvh − 12PLT∇2fL
h (y) + uTgL(y)PL.
fUh (x) + (xTBhx)
12 < fU
h (y) + uTgU(y) + yTBhvh − 12PUT∇2fU
h (y) + uTgU(y)PU .
for atleast one index h.This yields for λ = (λ1, ..., λr);λi > 0
k∑i=1
λi
(fLi (x) + (xTBix)
12
)+(fUi (x) + (xTBix)
12
)<
k∑i=1
λi
fLi (y) + yTBivi −
1
2PLT∇2fL
i (y)PL
+ uTgL(y)− 1
2PLT∇2uTgL(y)PL+
k∑i=1
λi
fUi (y) + yTBivi −
1
2PUT∇2fU
i (y)PU
+ uTgU(y)− 1
2PUT∇2uTgU(y)PU .
(9)
From the hypothesis that fi(.) + (.)TBix, i ∈ Λk and gj, j ∈ Λm are LU -bonvex aty, we have
fi(x) + xTBivi H (fi(y) + yTBivi) +1
2P T∇2
Hfi(y)P LU(∇Hfi(y) +∇2
Hfi(y)P +Bivi)
(x− y), i ∈ Λk (10)
and
gj(x)H gj(y)+1
2P T∇2
Hgj(y)P LU
(∇Hgj(y) +∇2
Hgj(y)P)
(x−y), j ∈ Λm. (11)
After multiplying (10) by λi, i ∈ Λk and (11) by uj, j ∈ Λm and adding, yields
k∑i=1
λi
fLi (x) + xTBivi − fL
i (y)− yTBivi +1
2PLT∇2fL
i (y)PL
+uTgL(x)−uTgL(y)+
1
2PLT∇2uTgL(y)PL+
k∑i=1
λi
fUi (x)+xTBivi−fU
i (y)−yTBivi+1
2PUT∇2fU
i (y)PU
+
uTgU(x)− uTgU(y) +1
2PUT∇2uTgU(y)PU ≥
k∑i=1
λi(∇fLi (y) +∇2fL
i (y)PL +Bivi) +∇uTgL(y) +∇2uTgL(y)PL
(x− y)+
k∑
i=1
λi(∇fUi (y) +∇2fU
i (y)PU +Bivi) +∇uTgU(y) +∇2uTgU(y)PU
(x− y).
12
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
892 Izhar Ahmad et al 881-896
Now by (4), (9), Schewartz inequality and uTg(x) LU [0, 0], we get
k∑i=1
λi
∇fL
i (y) +∇2fLi (y)PL +∇fU
i (y) +∇2fUi (y)PU + 2Bvi
+∇uTgL(y)+
∇uTgU(y) +∇2uTgL(y)PL +∇2uTgU(y)PU < 0.
which is a contradiction to (8). This completes the proof.
Theorem 4. (Strong duality theorem) Assume that x∗ is properly efficient solutionof problem (IP ) at which constraint qualification CQ1 is satisfied. Then thereexist λ∗ ∈ Rk, u∗ ∈ Rm and v∗i ∈ Rn, i ∈ Λk, such that (x∗, u∗, v∗i , λ
∗, P ∗T =([0, 0], ..., [0, 0])) is feasible for (MSD) and the corresponding objective values of(IP ) and (MSD) are equal. Moreover assume that the weak duality between (IP )and (MSD) in Theorem are satisfied, then (x∗, u∗, v∗i , λ
∗, P ∗T = ([0, 0], ..., [0, 0])) isan efficient solution of (MSD).
Proof. Since x∗ is efficient solution of problem (IP ) at which constraint qualificationCQ1 is satisfied. Then by Theorem 2 there exist λ∗ ∈ Rk, u∗ ∈ Rm and v∗i ∈ Rn, i ∈Λk, such that
k∑i=1
λ∗i (∇Hfi(x∗) +Biv
∗i ) +∇Hu
∗Tg(x∗) = [0, 0],
u∗Tg(x∗) = [0, 0],
(x∗TBix∗)
12 = x∗TBiv
∗i , i ∈ Λk,
v∗iTBiv
∗i ≤ 1, i ∈ Λk,
λ∗ > 0,k∑
i=1
λ∗i = 1, u∗ ≥ 0.
Which yields that (x∗, u∗, v∗i , λ∗, P ∗T = ([0, 0], ..., [0, 0])) ∈ SMSD and the corre-
sponding objective values of (IP ) and (MSD) are equal.
Now let (x∗, u∗, v∗i , λ∗, P ∗T = ([0, 0], ..., [0, 0])) is not efficient solution of dual prob-
lem (MSD), then by Definition there exist (y∗, u∗, v∗i , λ∗, P ∗) ∈ SMSD, such that
fi(x∗) + x∗TBiv
∗i + u∗Tg(x∗) LU fi(y
∗) + u∗Tg(y∗) + y∗TBiv∗i
H1
2P ∗T∇2
Hfi(y∗) + u∗Tg(y∗)P ∗, i ∈ Λk
andfi(x
∗) + x∗TBiv∗i + u∗Tg(x∗) ≺LU fi(y
∗) + u∗Tg(y∗) + y∗TBiv∗i
H1
2P ∗T∇2
Hfi(y∗) + u∗Tg(y∗)P ∗,
for atleast one index h.
Now using (x∗TBix∗)
12 = x∗TBiv
∗i , i ∈ Λk and u∗Tg(y∗) = [0, 0], we get a contra-
diction to weak duality theorem. Therefore (x∗, u∗, v∗i , λ∗, P ∗T = ([0, 0], ..., [0, 0]))
is an efficient solution of dual problem (MSD).
13
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
893 Izhar Ahmad et al 881-896
Theorem 5. (Strict converse duality) Let x∗ ∈ SIP and z∗ ∈ SMSP such that
k∑i=1
λ∗i fi(x∗) + x∗TBiv∗i LU
k∑i=1
λ∗i
fi(y
∗) + y∗TBiv∗i H
1
2P ∗T∇2
Hfi(y∗)P ∗
+u∗Tg(y∗)H1
2P ∗T∇2
Hu∗Tg(y∗)P ∗. (12)
Assume that fi(.) + (.)TBiv∗i , i ∈ Λk are strictly LU-bonvex at y∗ and gj(.), j ∈ Λm
is LU-bonvex at y∗ then x∗ = y∗.
Proof. If possible let x∗ 6= y∗. Now since fi(.) + (.)TBiv∗i , i ∈ Λk are strictly LU -
bonvex at y∗ , we have
fi(x∗) + x∗TBiv
∗i H (fi(y
∗) + y∗TBiv∗i ) +
1
2P ∗T∇2
Hfi(y∗)P ∗ LU(
∇Hfi(y∗) +∇2
Hfi(y∗)P ∗ +Bivi
∗) (x∗ − y∗), i ∈ Λk. (13)
and
gj(x∗)Hgj(y
∗)+1
2P ∗T∇2
Hgj(y∗)P ∗ LU (∇Hgj(y
∗)+∇2Hgj(y
∗)P ∗)(x∗−y∗), j ∈ Λm.
(14)Now multiplying (13) by λ∗i , i ∈ Λk and (14) by u∗j , j ∈ Λm and then summing upwe get
k∑i=1
λ∗ifi(x
∗) + x∗TBiv∗i
+ u∗Tg(x∗)H
k∑i=1
λ∗i
fi(y
∗) + y∗TBiviH
1
2P ∗T∇2
Hfi(y∗)P ∗
H u∗Tg(y∗) +
1
2P ∗T∇2
Hu∗Tgj(y
∗)P ∗ LUk∑
i=1
λ∗j(∇Hfi(y∗)+∇2
Hfi(y∗)P ∗+Biv
∗i )+∇Hu
∗Tg(y∗)+∇2Hu∗Tg(y∗)P ∗
(x∗−y∗).
The above inequality on using (3) and u∗Tg(x∗) LU [0, 0] gives
k∑i=1
λ∗i fi(x∗) + x∗TBiv∗i LU
k∑i=1
λ∗i
fi(y
∗) + y∗TBiv∗iH
1
2P ∗T∇2
Hfi(y∗)P ∗
+ u∗Tg(y∗)H
1
2P ∗T∇2
Hu∗Tg(y∗)P ∗.
which is a contradiction to (12). Hence x∗ = y∗
7 Conclusions
This paper represents the study of nondifferentiable vector problem in which objec-tive functions and constraints are interval valued. Firstly the twice H- differentiableinterval valued functions are introduced, secondly the concepts of LU -bonvexity,LU -quasibonvexity and LU -pseudobonvexity are introduced, thirdly the necessaryconditions for proposed solution concept are obtained. And lastly the Mangasarian
14
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
894 Izhar Ahmad et al 881-896
type dual is proposed and the corresponding duality results are obtained. Althoughthe interval valued equality constraints are not considered in this paper, the similarmethodology proposed in this paper can also be used to handle the interval valuedequality constraints. However it will be interesting to study the Mond-Weir typeduality results [1] for the problem (IP ). Future research is oriented to consider theuncertain environment in order to study the optimality conditions involving Fuzzyparameters.
Acknowledgements
The research of the first and fourth author is financially supported by King FahdUniversity of Petroleum and Minerals, Saudi Arabia under the Internal ResearchProject No. IN131026.
References
[1] I. Ahmad, Z. Husain, Second order (F, α, ρ, d)-convexity and duality in multi-objective programming, Inform Sci 176 (2006) 3094-3103.
[2] I. Ahmad, A. Jayswal, J. Banerjee: On interval-valued optimization problemswith generalized invex functions, J Inequal Appl (2013) 1-14.
[3] I. Ahmad, S. Sharma, Second order duality for nondifferentiable multiobjectiveprogramming problems, Num Func Anal Optim 28 (9-10) (2007) 975-988.
[4] I. Ahmad, D. Singh, B. A. Dar, Optimality conditions for invex interval-valuednonlinear programming problems involving generalized H-derivative, Filomat(2015) (Accepted).
[5] A. Jayswal, I. Stancu-Minasian, J. Banerjee, A.M. Stancu, Sufficieny and dualityfor optimization problems involving interval-valued invex functions in paramet-ric form, Oper Res Int J 15(2015) 137-161.
[6] C. R. Bector, S. Chandra, Generalized-bonvexity and higher order duality forfractional programming, Opsearch 24 (1987)143-154.
[7] A. Bhurjee, G. Panda, Efficient solution of interval optimization problem, MathMeth Oper Res 76 (2012) 273-288.
[8] A. Bhurjee, G. Panda, Multiobjective optimization problem with bounded pa-rameters, Rairo-Oper. Res. 48(2014), 545-558.
[9] Y. Chalco-Cano, H. Roman-Flores, MD. Jimenez-Gamero, Generalized deriva-tive and π-derivative for set valued functions, Inform Sci 181 (2011) 2177-2188.
[10] Y. Chalco-Cano, W.A. Lodwick, A. Rufian-Lizana, Optimality conditions oftype KKT for optimization problem with interval-valued objective function viageneralized derivative, Fuzzy Optim Dec Making 12 (2013) 305-322.
15
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
895 Izhar Ahmad et al 881-896
[11] Y. Chalco-Cano, A. Rufian-Lizana, H. Roman-Flores, M.D. Jimenez-Gamero,Calculus for interval-valued functions using generalized Hukuhara derivativeand applications, Fuzzy Sets and Systems 219 (2013) 49-67.
[12] H. Ishibuchi, H. Tanaka, Multiobjective programming in optimization of in-terval valued objective functions, Eur J Oper Res 48 (1990) 219-225.
[13] L. Li, S. Liu, J. Zhang, Univex interval-valued mapping with differentiabilityand its application in nonlinear programming, J Appl Math Art. ID 383692,(2013). http://dx.doi.org/10.1155/2013/383692.
[14] B. Mond, Second order duality for nonlinear programs, Opsearch 11 (1974)90-99.
[15] B. Mond, A class of nondifferentiable mathematical programming problems.J. Math Anal Appl 46 (1974) 169-174.
[16] B. Mond, I. Husain, M.V. Durgaprasad, Duality for a class of nondifferentiablemultiple objective programming problems, J Inform Optim Sci 9 (1988) 331-341.
[17] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valuedfunctions and interval differential equations, Nonlinear Anal 71 (2009) 1311-1328.
[18] Y. Sun, L. Wang, Optimality conditions and duality in nondifferentiable inter-val valued progra-mming, J Indust Manag Optim 9 no. 1 (2013) 131-142.
[19] D. Singh, B. A. Dar, A. Goyal, KKT optimality conditions for interval valuedoptimization problems, J Nonl Anal Optim 5 no. 2 (2014) 91-103.
[20] H. C. Wu, The karush Kuhn tuker optimality conditions in an optimizationproblem with interval valued objective functions, Eur J Oper Res 176 (2007)46-59.
[21] H. C. Wu, The karush Kuhn tuker optimality conditions in multiobjectiveprogramming problems with interval valued objective functions, Eur J OperRes 196 (2009) 49-60.
[22] J Zhang, S. Liu, L. Li, Q. Feng, The KKT optimality conditions in a classof generalized convex optimization problems with an interval-valued objectivefunction, Optim Lett 8 (2014) 607-631.
16
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
896 Izhar Ahmad et al 881-896
ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN
β-HOMOGENEOUS NORMED SPACES
SUNGSIK YUN, GEORGE A. ANASTASSIOU AND CHOONKIL PARK∗
Abstract. In this paper, we solve the following additive-quadratic ρ-functional inequalities
‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖ (0.1)
≤∥∥∥ρ(2f
(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
)∥∥∥ ,where ρ is a fixed complex number with |ρ| < 1, and∥∥∥2f
(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
∥∥∥≤ ‖ρ(f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y))‖, (0.2)
where ρ is a fixed complex number with |ρ| < 12, and prove the Hyers-Ulam stability of the additive-
quadratic ρ-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banach spaces andprove the Hyers-Ulam stability of additive-quadratic ρ-functional equations associated with theadditive-quadratic ρ-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banachspaces.
1. Introduction and preliminaries
The stability problem of functional equations originated from a question of Ulam [24] concerningthe stability of group homomorphisms. The functional equation f(x + y) = f(x) + f(y) is calledthe Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additivemapping. Hyers [11] gave a first affirmative partial answer to the question of Ulam for Banachspaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [15]for linear mappings by considering an unbounded Cauchy difference. A generalization of theRassias theorem was obtained by Gavruta [8] by replacing the unbounded Cauchy difference by
a general control function in the spirit of Rassias’ approach. The functional equation f(x+y2
)=
12f(x) + 1
2f(y) is called the Jensen equation.The functional equation f(x+ y) + f(x− y) = 2f(x) + 2f(y) is called the quadratic functional
equation. In particular, every solution of the quadratic functional equation is said to be a quadraticmapping. The stability of quadratic functional equation was proved by Skof [23] for mappingsf : E1 → E2, where E1 is a normed space and E2 is a Banach space. Cholewa [6] noticed thatthe theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. The
functional equation 2f(x+y2
)+ 2
(x−y2
)= f(x) + f(y) is called a Jensen type quadratic equation.
The stability problems of various functional equations have been extensively investigated by anumber of authors (see [1, 4, 5, 13, 14, 18, 19, 20, 21, 22]).
2010 Mathematics Subject Classification. Primary 39B62, 39B72, 39B52, 39B82.Key words and phrases. Hyers-Ulam stability; β-homogeneous space; additive-quadratic ρ-functional equation;
additive-quadratic ρ-functional inequality.∗Corresponding author: Choonkil Park (email: [email protected]).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
897 SUNGSIK YUN et al 897-909
S. YUN, G. A. ANASTASSIOU, C. PARK
In [9], Gilanyi showed that if f satisfies the functional inequality
‖2f(x) + 2f(y)− f(xy−1)‖ ≤ ‖f(xy)‖ (1.1)
then f satisfies the Jordan-von Neumann functional equation
2f(x) + 2f(y) = f(xy) + f(xy−1).
See also [16]. Gilanyi [10] and Fechner [7] proved the Hyers-Ulam stability of the functionalinequality (1.1). Park, Cho and Han [12] proved the Hyers-Ulam stability of additive functionalinequalities.
Definition 1.1. Let X be a linear space. A nonnegative valued function ‖ · ‖ is an F -norm if itsatisfies the following conditions:
(FN1) ‖x‖ = 0 if and only if x = 0;(FN2) ‖λx‖ = ‖x‖ for all x ∈ X and all λ with |λ| = 1;(FN3) ‖x+ y‖ ≤ ‖x‖+ ‖y‖ for all x, y ∈ X;(FN4) ‖λnx‖ → 0 provided λn → 0;(FN5) ‖λxn‖ → 0 provided xn → 0.Then (X, ‖ · ‖) is called an F ∗-space. An F -space is a complete F ∗-space.
An F -norm is called β-homogeneous (β > 0) if ‖tx‖ = |t|β‖x‖ for all x ∈ X and all t ∈ C (see[17]).
In Section 2, we solve the additive-quadratic ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.1) in β-homogeneous complexBanach spaces. We moreover prove the Hyers-Ulam stability of an additive-quadratic ρ-functionalequation associated with the additive-quadratic ρ-functional inequality (0.1) in β-homogeneouscomplex Banach spaces.
In Section 3, we solve the additive-quadratic ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.2) in β-homogeneous complexBanach spaces. We moreover prove the Hyers-Ulam stability of an additive-quadratic ρ-functionalequation associated with the additive-quadratic ρ-functional inequality (0.2) in β-homogeneouscomplex Banach spaces.
Throughout this paper, let β1, β2 be positive real numbers with β1 ≤ 1 and β2 ≤ 1. Assumethat X is a β1-homogeneous real or complex normed space with norm ‖ · ‖ and that Y is a β2-homogeneous complex Banach space with norm ‖ · ‖.
2. Additive-quadratic ρ-functional inequality (0.1)
Throughout this section, assume that ρ is a fixed complex number with |ρ| < 1.In this section, we investigate the additive-quadratic ρ-functional inequality (0.1) in β-homogeneous
complex Banach spaces.
Lemma 2.1. An even mapping f : X → Y satisfies
‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖ (2.1)
≤∥∥∥∥ρ(2f
(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
)∥∥∥∥for all x, y ∈ X if and only if f : X → Y is quadratic.
Proof. Assume that f : X → Y satisfies (2.1).Letting x = y = 0 in (2.1), we get ‖2f(0)‖ ≤ |ρ|β2‖2f(0)‖. So f(0) = 0.Letting y = x in (2.1), we get ‖f(2x)− 4f(x)‖ ≤ 0 and so f(2x) = 4f(x) for all x ∈ X. Thus
f
(x
2
)=
1
4f(x) (2.2)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
898 SUNGSIK YUN et al 897-909
ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES
for all x ∈ X.It follows from (2.1) and (2.2) that
‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖
≤∥∥∥∥ρ(2f
(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
)∥∥∥∥=|ρ|β22β2‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖
and sof(x+ y) + f(x− y) = 2f(x) + 2f(y)
for all x, y ∈ X.The converse is obviously true.
Corollary 2.2. An even mapping f : X → Y satisfies
f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y) (2.3)
= ρ
(2f
(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
)for all x, y ∈ X if and only if f : X → Y is quadratic.
The functional equation (2.3) is called an additive-quadratic ρ-functional equation.We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (2.1) in
β-homogeneous complex Banach spaces for an even mapping case.
Theorem 2.3. Let r > 2β2β1
and θ be nonnegative real numbers, and let f : X → Y be an even
mapping such that
‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖ (2.4)
≤∥∥∥∥ρ(2f
(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
)∥∥∥∥+ θ(‖x‖r + ‖y‖r)
for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that
‖f(x)−Q(x)‖ ≤ 2θ
2β1r − 4β2‖x‖r (2.5)
for all x ∈ X.
Proof. Letting x = y = 0 in (2.4), we get ‖2f(0)‖ ≤ |ρ|β2‖2f(0)‖. So f(0) = 0.Letting y = x in (2.4), we get
‖f(2x)− 4f(x)‖ ≤ 2θ‖x‖r (2.6)
for all x ∈ X. So∥∥f(x)− 4f
(x2
)∥∥ ≤ 22β1r
θ‖x‖r for all x ∈ X. Hence∥∥∥∥4lf ( x2l)− 4mf
(x
2m
)∥∥∥∥ ≤ m−1∑j=l
∥∥∥∥4jf ( x2j)− 4j+1f
(x
2j+1
)∥∥∥∥ ≤ 2
2β1r
m−1∑j=l
4β2j
2β1rjθ‖x‖r (2.7)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.7) that thesequence 4nf( x
2n ) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence4nf( x
2n ) converges. So one can define the mapping Q : X → Y by
Q(x) := limn→∞
4nf(x
2n)
for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (2.7), we get (2.5).Since f : X → Y is even, the mapping Q : X → Y is even.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
899 SUNGSIK YUN et al 897-909
S. YUN, G. A. ANASTASSIOU, C. PARK
It follows from (2.4) that
‖Q(x+ y) +Q(x− y)− 2Q(x)−Q(y)−Q(−y)‖
= limn→∞
4β2n∥∥∥∥f (x+ y
2n
)+ f
(x− y
2n
)− 2f
(x
2n
)− f
(y
2n
)− f
(−y2n
)∥∥∥∥≤ lim
n→∞4β2n|ρ|β2
(∥∥∥∥2f (x+ y
2n+1
)+ 2f
(x− y2n+1
)− 3
2f
(x
2n
)+
1
2f
(−x2n
)−1
2f
(y
2n
)− 1
2f
(−y2n
)∥∥∥∥)+ limn→∞
4β2nθ
2β1nr(‖x‖r + ‖y‖r)
= |ρ|β2∥∥∥∥2Q(x+ y
2
)+ 2Q
(x− y
2
)− 3
2Q(x) +
1
2Q(−x)− 1
2Q(y)− 1
2Q(−y)
∥∥∥∥for all x, y ∈ X. So
‖Q(x+ y) +Q(x− y)− 2Q(x)−Q(y)−Q(−y)‖
≤∥∥∥∥ρ(2Q
(x+ y
2
)+ 2Q
(x− y
2
)− 3
2Q(x) +
1
2Q(−x)− 1
2Q(y)− 1
2Q(−y)
)∥∥∥∥for all x, y ∈ X. By Lemma 2.1, the mapping Q : X → Y is quadratic.
Now, let T : X → Y be another quadratic mapping satisfying (2.5). Then we have
‖Q(x)− T (x)‖ = 4β2n∥∥∥∥Q( x
2n
)− T
(x
2n
)∥∥∥∥≤ 4β2n
(∥∥∥∥Q( x
2n
)− f
(x
2n
)∥∥∥∥+
∥∥∥∥T ( x
2n
)− f
(x
2n
)∥∥∥∥)≤ 4 · 4β2n
(2β1r − 4β2)2β1nrθ‖x‖r,
which tends to zero as n→∞ for all x ∈ X. So we can conclude that Q(x) = T (x) for all x ∈ X.This proves the uniqueness of Q. Thus the mapping Q : X → Y is a unique quadratic mappingsatisfying (2.5).
Theorem 2.4. Let r < 2β2β1
and θ be nonnegative real numbers, and let f : X → Y be an even
mapping satisfying (2.4). Then there exists a unique quadratic mapping Q : X → Y such that
‖f(x)−Q(x)‖ ≤ 2θ
4β2 − 2β1r‖x‖r (2.8)
for all x ∈ X.
Proof. It follows from (2.6) that∥∥∥f(x)− 1
4f(2x)∥∥∥ ≤ 2θ
4β2‖x‖r for all x ∈ X. Hence∥∥∥∥ 1
4lf(2lx)− 1
4mf(2mx)
∥∥∥∥ ≤ m−1∑j=l
∥∥∥∥ 1
4jf(2jx)− 1
4j+1f(2j+1x)
∥∥∥∥ ≤ m−1∑j=l
2β1rj
4β2j2θ
4β2‖x‖r (2.9)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.9) that thesequence 1
4n f(2nx) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence
14n f(2nx) converges. So one can define the mapping Q : X → Y by
Q(x) := limn→∞
1
4nf(2nx)
for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (2.9), we get (2.8).The rest of the proof is similar to the proof of Theorem 2.3.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
900 SUNGSIK YUN et al 897-909
ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES
Lemma 2.5. An odd mapping f : X → Y satisfies (2.1) if and only if f : X → Y is additive.
Proof. Since f : X → Y is an odd mapping, f(0) = 0.Assume that f : X → Y satisfies (2.1).Letting y = x in (2.1), we get ‖f(2x)− 2f(x)‖ ≤ 0 and so f(2x) = 2f(x) for all x ∈ X. Thus
f
(x
2
)=
1
2f(x) (2.10)
for all x ∈ X.It follows from (2.1) and (2.10) that
‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖
≤∥∥∥∥ρ(2f
(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
)∥∥∥∥= |ρ|β2‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖
and so
f(x+ y) + f(x− y) = 2f(x) (2.11)
for all x, y ∈ X. Letting z = x+ y and w = z − y in (2.11), we get
f(z) + f(w) = 2f
(z + w
2
)= f(z + w)
for all z, w ∈ X.The converse is obviously true.
Corollary 2.6. An odd mapping f : X → Y satisfies (2.3) if and only if f : X → Y is additive.
We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (2.1) inβ-homogeneous complex Banach spaces for an odd mapping case.
Theorem 2.7. Let r > β2β1
and θ be nonnegative real numbers, and let f : X → Y be an odd
mapping satisfying (2.4). Then there exists a unique additive mapping A : X → Y such that
‖f(x)−A(x)‖ ≤ 2θ
2β1r − 2β2‖x‖r (2.12)
for all x ∈ X.
Proof. Letting x = y = 0 in (2.4), we get ‖2f(0)‖ ≤ |ρ|β2‖2f(0)‖. So f(0) = 0.Letting y = x in (2.4), we get
‖f(2x)− 2f(x)‖ ≤ 2θ‖x‖r (2.13)
for all x ∈ X. So∥∥f(x)− 2f
(x2
)∥∥ ≤ 22β1r
θ‖x‖r for all x ∈ X. Hence∥∥∥∥2lf ( x2l)− 2mf
(x
2m
)∥∥∥∥ ≤ m−1∑j=l
∥∥∥∥2jf ( x2j)− 2j+1f
(x
2j+1
)∥∥∥∥ ≤ 2
2β1r
m−1∑j=l
2β2j
2β1rjθ‖x‖r (2.14)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.14) thatthe sequence 2nf( x
2n ) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence2nf( x
2n ) converges. So one can define the mapping A : X → Y by
A(x) := limn→∞
2nf(x
2n)
for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (2.14), we get (2.12).Since f : X → Y is odd, the mapping A : X → Y is odd.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
901 SUNGSIK YUN et al 897-909
S. YUN, G. A. ANASTASSIOU, C. PARK
It follows from (2.4) that
‖A(x+ y) +A(x− y)− 2A(x)−A(y)−A(−y)‖
= limn→∞
2β2n∥∥∥∥f (x+ y
2n
)+ f
(x− y
2n
)− 2f
(x
2n
)− f
(y
2n
)− f
(−y2n
)∥∥∥∥≤ lim
n→∞2β2n|ρ|β2
(∥∥∥∥2f (x+ y
2n+1
)+ 2f
(x− y2n+1
)− 3
2f
(x
2n
)+
1
2f
(−x2n
)−1
2f
(y
2n
)− 1
2f
(−y2n
)∥∥∥∥)+ limn→∞
2β2nθ
2β1nr(‖x‖r + ‖y‖r)
= |ρ|β2∥∥∥∥2A(x+ y
2
)+ 2A
(x− y
2
)− 3
2A(x) +
1
2A(−x)− 1
2A(y)− 1
2A(−y)
∥∥∥∥for all x, y ∈ X. So
‖A(x+ y) +A(x− y)− 2A(x)−A(y)−A(−y)‖
≤∥∥∥∥ρ(2A
(x+ y
2
)+ 2A
(x− y
2
)− 3
2A(x) +
1
2A(−x)− 1
2A(y)− 1
2A(−y)
)∥∥∥∥for all x, y ∈ X. By Lemma 2.5, the mapping A : X → Y is additive.
Now, let T : X → Y be another additive mapping satisfying (2.12). Then we have
‖A(x)− T (x)‖ = 2β2n∥∥∥∥A( x
2n
)− T
(x
2n
)∥∥∥∥≤ 2β2n
(∥∥∥∥A( x
2n
)− f
(x
2n
)∥∥∥∥+
∥∥∥∥T ( x
2n
)− f
(x
2n
)∥∥∥∥)≤ 4 · 2β2n
(2β1r − 2β2)2β1nrθ‖x‖r,
which tends to zero as n→∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X.This proves the uniqueness of A. Thus the mapping A : X → Y is a unique additive mappingsatisfying (2.12).
Theorem 2.8. Let r < β2β1
and θ be nonnegative real numbers, and let f : X → Y be an odd
mapping satisfying (2.4). Then there exists a unique additive mapping A : X → Y such that
‖f(x)−A(x)‖ ≤ 2θ
2β2 − 2β1r‖x‖r (2.15)
for all x ∈ X.
Proof. It follows from (2.13) that∥∥∥f(x)− 1
2f(2x)∥∥∥ ≤ 2θ
2β2‖x‖r for all x ∈ X. Hence∥∥∥∥ 1
2lf(2lx)− 1
2mf(2mx)
∥∥∥∥ ≤ m−1∑j=l
∥∥∥∥ 1
2jf(2jx)− 1
2j+1f(2j+1x)
∥∥∥∥ ≤ m−1∑j=l
2β1rj
2β2j2θ
2β2‖x‖r (2.16)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.16) that thesequence 1
2n f(2nx) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence
12n f(2nx) converges. So one can define the mapping A : X → Y by
A(x) := limn→∞
1
2nf(2nx)
for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (2.16), we get (2.15).The rest of the proof is similar to the proof of Theorem 2.7.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
902 SUNGSIK YUN et al 897-909
ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES
By the triangle inequality, we have
‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖
−∥∥∥∥ρ(2f
(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
)∥∥∥∥≤ ‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)
−ρ(
2f
(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
)∥∥∥∥ .As corollaries of Theorems 2.3, 2.4, 2.7 and 2.8, we obtain the Hyers-Ulam stability results for theadditive-quadratic ρ-functional equation (2.3) in β-homogeneous complex Banach spaces.
Corollary 2.9. Let r > 2β2β1
and θ be nonnegative real numbers, and let f : X → Y be an even
mapping such that
‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y) (2.17)
−ρ(
2f
(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
)∥∥∥∥ ≤ θ(‖x‖r + ‖y‖r)
for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y satisfying (2.5).
Corollary 2.10. Let r < 2β2β1
and θ be nonnegative real numbers, and let f : X → Y be an even
mapping satisfying (2.17). Then there exists a unique quadratic mapping Q : X → Y satisfying(2.8).
Corollary 2.11. Let r > β2β1
and θ be nonnegative real numbers, and let f : X → Y be an odd
mapping satisfying (2.17). Then there exists a unique additive mapping A : X → Y satisfying(2.12).
Corollary 2.12. Let r < β2β1
and θ be nonnegative real numbers, and let f : X → Y be an odd
mapping satisfying (2.17). Then there exists a unique additive mapping A : X → Y satisfying(2.15).
Remark 2.13. If ρ is a real number such that −1 < ρ < 1 and Y is a β2-homogeneous real Banachspace, then all the assertions in this section remain valid.
3. Additive-quadratic ρ-functional inequality (0.2)
Throughout this section, assume that ρ is a fixed complex number with |ρ| < 12 .
In this section, we investigate the additive-quadratic ρ-functional inequality (0.2) in β-homogeneouscomplex Banach spaces.
Lemma 3.1. An even mapping f : X → Y satisfies∥∥∥∥2f (x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
∥∥∥∥≤ ‖ρ(f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y))‖ (3.1)
for all x, y ∈ X if and onlt if f : X → Y is quadratic.
Proof. Assume that f : X → Y satisfies (3.1).Letting x = y = 0 in (3.1), we get ‖2f(0)‖ ≤ |ρ|β2‖2f(0)‖. So f(0) = 0.Letting y = 0 in (3.1), we get ∥∥∥∥4f (x2
)− f(x)
∥∥∥∥ ≤ 0 (3.2)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
903 SUNGSIK YUN et al 897-909
S. YUN, G. A. ANASTASSIOU, C. PARK
for all x ∈ X. So f(x2
)= 1
4f(x) for all x ∈ X.It follows from (3.1) and (3.2) that
1
2β2‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖
=
∥∥∥∥2f (x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
∥∥∥∥≤ |ρ|β2‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖
and sof(x+ y) + f(x− y) = 2f(x) + 2f(y)
for all x, y ∈ X.The converse is obviously true.
Corollary 3.2. An even mapping f : X → Y satisfies
2f
(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
= ρ (f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)) (3.3)
for all x, y ∈ X if and only if f : X → Y is quadratic.
The functional equation (3.3) is called an additive-quadratic ρ-functional equation.We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (3.1) in
β-homogeneous complex Banach spaces for an even mapping case.
Theorem 3.3. Let r > 2β2β1
and θ be nonnegative real numbers, and let f : X → Y be an even
mapping such that
‖2f(x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)‖ (3.4)
≤ ‖ρ(f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y))‖+ θ(‖x‖r + ‖y‖r)for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that
‖f(x)−Q(x)‖ ≤ 2β1rθ
2β1r − 4β2‖x‖r (3.5)
for all x ∈ X.
Proof. Letting x = y = 0 in (3.4), we get ‖2f(0)‖ ≤ |ρ|β2‖2f(0)‖. So f(0) = 0.Letting y = 0 in (3.4), we get ∥∥∥∥4f (x2
)− f(x)
∥∥∥∥ ≤ θ‖x‖r (3.6)
for all x ∈ X. So∥∥∥∥4lf ( x2l)− 4mf
(x
2m
)∥∥∥∥ ≤ m−1∑j=l
∥∥∥∥4jf ( x2j)− 4j+1f
(x
2j+1
)∥∥∥∥ ≤ m−1∑j=l
4β2j
2β1rjθ‖x‖r (3.7)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.7) that thesequence 4nf( x
2n ) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence4nf( x
2n ) converges. So one can define the mapping Q : X → Y by
Q(x) := limn→∞
4nf(x
2n)
for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (3.7), we get (3.5).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
904 SUNGSIK YUN et al 897-909
ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES
Since f : X → Y is even, the mapping Q : X → Y is even.It follows from (3.4) that∥∥∥∥2Q(x+ y
2
)+ 2Q
(x− y
2
)− 3
2Q(x) +
1
2Q(−x)− 1
2Q(y)− 1
2Q(−y)
∥∥∥∥= lim
n→∞4β2n
(∥∥∥∥2f (x+ y
2n+1
)+ 2f
(x− y2n+1
)− 3
2f
(x
2n
)+
1
2f
(−x2n
)− 1
2f
(y
2n
)− 1
2f
(−y2n
)∥∥∥∥)≤ lim
n→∞4β2n
∥∥∥∥ρ(f (x+ y
2n
)+ f
(x− y
2n
)− 2f
(x
2n
)− f
(y
2n
)− f
(−y2n
))∥∥∥∥+ limn→∞
4β2nθ
2β1nr(‖x‖r + ‖y‖r)
= ‖ρ(Q(x+ y) +Q(x− y)− 2Q(x)−Q(y)−Q(−y))‖for all x, y ∈ X. So∥∥∥∥2Q(x+ y
2
)+ 2Q
(x− y
2
)− 3
2Q(x) +
1
2Q(−x)− 1
2Q(y)− 1
2Q(−y)
∥∥∥∥≤ ‖ρ(Q(x+ y) +Q(x− y)− 2Q(x)−Q(y)−Q(−y))‖
for all x, y ∈ X. By Lemma 3.1, the mapping Q : X → Y is quadratic.Now, let T : X → Y be another quadratic mapping satisfying (3.5). Then we have
‖Q(x)− T (x)‖ = 4β2n∥∥∥∥Q( x
2n
)− T
(x
2n
)∥∥∥∥≤ 4β2n
(∥∥∥∥Q( x
2n
)− f
(x
2n
)∥∥∥∥+
∥∥∥∥T ( x
2n
)− f
(x
2n
)∥∥∥∥)≤ 2 · 4β2n · 2β1r
(2β1r − 4β2)2β1nrθ‖x‖r,
which tends to zero as n→∞ for all x ∈ X. So we can conclude that Q(x) = T (x) for all x ∈ X.This proves the uniqueness of Q. Thus the mapping Q : X → Y is a unique quadratic mappingsatisfying (3.5).
Theorem 3.4. Let r < 2β2β1
and θ be nonnegative real numbers, and let f : X → Y be an even
mapping satisfying (3.4). Then there exists a unique quadratic mapping Q : X → Y such that
‖f(x)−Q(x)‖ ≤ 2β1rθ
4β2 − 2β1r‖x‖r (3.8)
for all x ∈ X.
Proof. It follows from (3.6) that∥∥∥f(x)− 1
4f(2x)∥∥∥ ≤ 2β1rθ
4β2‖x‖r for all x ∈ X. Hence∥∥∥∥ 1
4lf(2lx)− 1
4mf(2mx)
∥∥∥∥ ≤ m−1∑j=l
∥∥∥∥ 1
4jf(2jx)− 1
4j+1f(2j+1x)
∥∥∥∥ ≤ 2β1rθ
4β2
m−1∑j=l
2β1rj
4β2j‖x‖r (3.9)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.9) that thesequence 1
4n f(2nx) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence
14n f(2nx) converges. So one can define the mapping Q : X → Y by
Q(x) := limn→∞
1
4nf(2nx)
for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (3.9), we get (3.8).The rest of the proof is similar to the proof of Theorem 3.3.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
905 SUNGSIK YUN et al 897-909
S. YUN, G. A. ANASTASSIOU, C. PARK
Lemma 3.5. An odd mapping f : X → Y satisfies (3.1) if and only if f : X → Y is additive.
Proof. Assume that f : X → Y satisfies (3.1).Letting x = y = 0 in (3.1), we get ‖2f(0)‖ ≤ |ρ|β2‖2f(0)‖. So f(0) = 0.Letting y = 0 in (3.1), we get ∥∥∥∥4f (x2
)− 2f(x)
∥∥∥∥ ≤ 0 (3.10)
for all x ∈ X. So f(x2
)= 1
2f(x) for all x ∈ X.It follows from (3.1) and (3.10) that
1
2β2‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖
=
∥∥∥∥2f (x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
∥∥∥∥≤ |ρ|β2‖f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)‖
and so
f(x+ y) + f(x− y) = 2f(x)
for all x, y ∈ X.The converse is obviously true.
Corollary 3.6. An odd mapping f : X → Y satisfies (3.3) if and only if f : X → Y is additive.
We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (3.1) inβ-homogeneous complex Banach spaces for an odd mapping case.
Theorem 3.7. Let r > β2β1
and θ be nonnegative real numbers, and let f : X → Y be an odd
mapping satisfying (3.4). Then there exists a unique additive mapping A : X → Y such that
‖f(x)−A(x)‖ ≤ 2β1rθ
(2β1r − 2β2)2β2‖x‖r (3.11)
for all x ∈ X.
Proof. Letting x = y = 0 in (3.4), we get ‖2f(0)‖ ≤ |ρ|β2‖2f(0)‖. So f(0) = 0.Letting y = 0 in (3.4), we get ∥∥∥∥4f (x2
)− 2f(x)
∥∥∥∥ ≤ θ‖x‖r (3.12)
for all x ∈ X. So∥∥∥∥2lf ( x2l)− 2mf
(x
2m
)∥∥∥∥ ≤ m−1∑j=l
∥∥∥∥2jf ( x2j)− 2j+1f
(x
2j+1
)∥∥∥∥ ≤ m−1∑j=l
2β2j
2β1rjθ
2β2‖x‖r (3.13)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.13) thatthe sequence 2nf( x
2n ) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence2nf( x
2n ) converges. So one can define the mapping A : X → Y by
A(x) := limn→∞
2nf(x
2n)
for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (3.13), we get (3.11).Since f : X → Y is odd, the mapping A : X → Y is odd.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
906 SUNGSIK YUN et al 897-909
ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES
It follows from (3.4) that∥∥∥∥2A(x+ y
2
)+ 2A
(x− y
2
)− 3
2A(x) +
1
2A(−x)− 1
2A(y)− 1
2A(−y)
∥∥∥∥= lim
n→∞2β2n
(∥∥∥∥2f (x+ y
2n+1
)+ 2f
(x− y2n+1
)− 3
2f
(x
2n
)+
1
2f
(−x2n
)− 1
2f
(y
2n
)− 1
2f
(−y2n
)∥∥∥∥)≤ lim
n→∞2β2n
∥∥∥∥ρ(f (x+ y
2n
)+ f
(x− y
2n
)− 2f
(x
2n
)− f
(y
2n
)− f
(−y2n
))∥∥∥∥+ limn→∞
2β2nθ
2β1nr(‖x‖r + ‖y‖r)
= ‖ρ(A(x+ y) +A(x− y)− 2A(x)−A(y)−A(−y))‖for all x, y ∈ X. So∥∥∥∥2A(x+ y
2
)+ 2A
(x− y
2
)− 3
2A(x) +
1
2A(−x)− 1
2A(y)− 1
2A(−y)
∥∥∥∥≤ ‖ρ(A(x+ y) +A(x− y)− 2A(x)−A(y)−A(−y))‖
for all x, y ∈ X. By Lemma 3.5, the mapping A : X → Y is additive.Now, let T : X → Y be another additive mapping satisfying (3.11). Then we have
‖A(x)− T (x)‖ = 2β2n∥∥∥∥A( x
2n
)− T
(x
2n
)∥∥∥∥≤ 2β2n
(∥∥∥∥A( x
2n
)− f
(x
2n
)∥∥∥∥+
∥∥∥∥T ( x
2n
)− f
(x
2n
)∥∥∥∥)≤ 2 · 2β2n · 2β1r
(2β1r − 2β2)2β1nrθ
2β2‖x‖r,
which tends to zero as n→∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X.This proves the uniqueness of A. Thus the mapping A : X → Y is a unique additive mappingsatisfying (3.11).
Theorem 3.8. Let r < β2β1
and θ be nonnegative real numbers, and let f : X → Y be an odd
mapping satisfying (3.4). Then there exists a unique additive mapping A : X → Y such that
‖f(x)−A(x)‖ ≤ 2β1rθ
(2β2 − 2β1r)2β2‖x‖r (3.14)
for all x ∈ X.
Proof. It follows from (3.12) that∥∥∥f(x)− 1
2f(2x)∥∥∥ ≤ 2β1rθ
4β2‖x‖r for all x ∈ X. Hence∥∥∥∥ 1
2lf(2lx)− 1
2mf(2mx)
∥∥∥∥ ≤ m−1∑j=l
∥∥∥∥ 1
2jf(2jx)− 1
2j+1f(2j+1x)
∥∥∥∥ ≤ 2β1rθ
4β2
m−1∑j=l
2β1rj
2β2j‖x‖r (3.15)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.15) that thesequence 1
2n f(2nx) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence
12n f(2nx) converges. So one can define the mapping A : X → Y by
A(x) := limn→∞
1
2nf(2nx)
for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (3.15), we get (3.14).The rest of the proof is similar to the proof of Theorem 3.7.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
907 SUNGSIK YUN et al 897-909
S. YUN, G. A. ANASTASSIOU, C. PARK
By the triangle inequality, we have∥∥∥∥2f (x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
∥∥∥∥−‖ρ (f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y))‖
≤∥∥∥∥2f (x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y)
−ρ (f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y))‖ .As corollaries of Theorems 3.3, 3.4, 3.7 and 3.8, we obtain the Hyers-Ulam stability results for theadditive-quadratic ρ-functional equation (3.3) in β-homogeneous complex Banach spaces.
Corollary 3.9. Let r > 2β2β1
and θ be nonnegative real numbers, and let f : X → Y be an even
mapping such that∥∥∥∥2f (x+ y
2
)+ 2f
(x− y
2
)− 3
2f(x) +
1
2f(−x)− 1
2f(y)− 1
2f(−y) (3.16)
−ρ (f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y))‖ ≤ θ(‖x‖r + ‖y‖r)for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y satisfying (3.5).
Corollary 3.10. Let r < 2β2β1
and θ be nonnegative real numbers, and let f : X → Y be an even
mapping satisfying (3.16). Then there exists a unique quadratic mapping Q : X → Y satisfying(3.8).
Corollary 3.11. Let r > β2β1
and θ be nonnegative real numbers, and let f : X → Y be an odd
mapping satisfying (3.16). Then there exists a unique additive mapping A : X → Y satisfying(3.11).
Corollary 3.12. Let r < β2β1
and θ be nonnegative real numbers, and let f : X → Y be an odd
mapping satisfying (3.16). Then there exists a unique additive mapping A : X → Y satisfying(3.14).
Remark 3.13. If ρ is a real number such that −12 < ρ < 1
2 and Y is a β2-homogeneous realBanach space, then all the assertions in this section remain valid.
References
[1] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50–59.[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.[3] J. Bae and W. Park, Approximate bi-homomorphisms and bi-derivations in C∗-ternary algebras, Bull. Korean
Math. Soc. 47 (2010), 195–209.[4] L. Cadariu, L. Gavruta and P. Gavruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl.
6 (2013), 60–67.[5] A. Chahbi and N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci.
Appl. 6 (2013), 198–204.[6] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86.[7] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation,
Aequationes Math. 71 (2006), 149–161.[8] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math.
Anal. Appl. 184 (1994), 431–436.[9] A. Gilanyi, Eine zur Parallelogrammgleichung aquivalente Ungleichung, Aequationes Math. 62 (2001), 303–309.
[10] A. Gilanyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710.[11] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224.[12] C. Park, Y. Cho and M. Han, Functional inequalities associated with Jordan-von Neumann-type additive func-
tional equations, J. Inequal. Appl. 2007 (2007), Article ID 41820, 13 pages.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
908 SUNGSIK YUN et al 897-909
ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES
[13] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C∗-algebras, J.Comput. Anal. Appl. 15 (2013), 365-368.
[14] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J.Comput. Anal. Appl. 15 (2013), 452–462.
[15] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978),297–300.
[16] J. Ratz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66(2003), 191–200.
[17] S. Rolewicz, Metric Linear Spaces, PWN-Polish Scientific Publishers, Warsaw, 1972.[18] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach,
J. Nonlinear Sci. Appl. 4 (2011), 37–49.[19] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary
Frechet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114.[20] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary
Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105.[21] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C∗-homomorphisms, J. Comput.
Anal. Appl. 16 (2014), 964–973.[22] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J∗-homomorphisms and J∗-derivations
for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134.[23] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129.[24] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.[25] L. G. Wang and B. Liu, The Hyers-Ulam stability of a functional equation deriving from quadratic and cubic
functions in quasi-β-normed spaces, Acta Math. Sin., Engl. Ser. 26 (2010), 2335–2348.[26] Z. H. Wang and W. X. Zhang, Fuzzy stability of quadratic-cubic functional equations, Acta Math. Sin., Engl.
Ser. 27 (2011), 2191–2204.[27] T. Z. Xu, J. M. Rassias and W. X. Xu, Generalized Hyers-Ulam stability of a general mixed additive-cubic
functional equation in quasi-Banach spaces, Acta Math. Sin., Engl. Ser. 28 (2012), 529–560.
Sungsik YunDepartment of Financial Mathematics, Hanshin University, Gyeonggi-do 447-791, KoreaE-mail address: [email protected]
George A. AnastassiouDepartment of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USAE-mail address: [email protected]
Choonkil ParkResearch Institute for Natural Sciences, Hanyang University, Seoul 133-791, KoreaE-mail address: [email protected]
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
909 SUNGSIK YUN et al 897-909
A note on stochastic functional differential equations
driven by G-Brownian motion with discontinuous drift
coefficients
Faiz Faizullah∗, Aamir Mukhtar1, M. A. Rana1
∗Department of BS and H, College of E and ME, National University
of Sciences and Technology (NUST) Pakistan.1Department of Basic Sciences, Riphah International University, Islamabad, Pakistan.
March 27, 2015
Abstract
In the fields of sciences and engineering, the role of discontinuous functions is of immense
importance. Heaviside function, for instance, describes the switching process of voltage in
an electrical circuit through mathematical process. The current paper aims at exploring the
existence theory for stochastic functional differential equations driven by G-Brownian motion
(G-SFDEs) whose drift coefficients may not be continuous. It is ascertain that G-SFDEs with
discontinuous drift coefficients have more than one bounded and continuous solutions.
Key words: Stochastic functional differential equations, discontinuous drift coefficints,
G-Brownian motion, existence.
1 Introduction
For the purpose of analysis and formulation of systems pertaining to engineering, economics and
social sciences, stochastic dynamical systems play an important role. Through these equations,
while considering the present status, one reconstructs the history and predicts the future of the
dynamical systems. On the other hand, in several applications, analysis of the modeling system
predicts that the change rate of the system’s existing status depends not only on the state that
is prevalent but also on the precedent record of the system. This leads to stochastic functional
differential equations. The stochastic functional differential equations driven by G-Brownian motion
(G-SFDEs) with Lipschitz continuous coefficients was initiated by Ren et.al. [12]. Afterwards,
Faizullah used the Caratheodory approximation scheme for developing the existence and uniqueness
of solution for G-SFDEs with continuous coefficients [3]. On the other hand, in this case, we study
∗Corresponding author, E-mail: faiz¯
1
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
910 Faiz Faizullah et al 910-919
the existence theory for G-SFDEs with discontinuous drift coefficients, such as in the following
G-SFDE
dX(t) = H(Xt)dt+ d〈B〉(t) + dB(t),
where H : R→ R is the Heaviside function defined by
H(x) =
0, if x < 0 ;
1, if x ≥ 0.
The above mentioned equations arise, when we take into account the effects of background noise
switching systems with delays [5]. For more details on SDEs with discontinuous drift coefficients
see [4, 7]. The following stochastic functional differential equation driven by G-Brownian motion
(G-SFDE) with finite delay is considered
dX(t) = α(t,Xt)dt+ β(t,Xt)d〈B,B〉(t) + σ(t,Xt)dB(t), 0 ≤ t ≤ T, (1.1)
where X(t) is the value of stochastic process at time t and Xt = X(t + θ) : −τ ≤ θ ≤ 0is a BC([−τ, 0];R)-valued stochastic process, which represents the family of bounded continuous
R-valued functions ϕ defined on [−τ, 0] having norm ‖ϕ‖ = sup−τ≤θ≤0
| ϕ(θ) | . Let α : [0, T ] ×
BC([−τ, 0];R) → R, β : [0, T ] × BC([−τ, 0];R) → R and σ : [0, T ] × BC([−τ, 0];R) → R are
Borel measurable. The condition ξ(0) ∈ R is given , 〈B,B〉(t), t ≥ 0 is the quadratic variation
process of G-Brownian motion B(t), t ≥ 0 and α, β, σ ∈M2G([−τ, T ];R). Let L2 denote the space
of all Ft-adapted process X(t), 0 ≤ t ≤ T , such that ‖ X ‖L2= sup−τ≤t≤T
|X(t)| < ∞. We define the
initial condition of equation (1.1) as follows;
Xt0 =ξ = ξ(θ) : −τ < θ ≤ 0 is F0 −measurable, BC([−τ, 0];R)− valued
random variable such that ξ ∈M2G ([−τ, 0];R) .
(1.2)
G-SFDEs (1.1) with initial condition (1.2) can be written in the following integral form;
X(t) = ξ(0) +
∫ t
0α(s,Xs)ds+
∫ t
0β(s,Xs)d〈B,B〉(s) +
∫ t
0σ(s,Xs)dB(s).
Consider the following linear growth and Lipschitz conditions respectively.
(i) For any t ∈ [0, T ], |α(t, x)|2 + |β(t, x)|2 + |σ(t, x)|2 ≤ K(1 + |x|2),K > 0.
(ii) For all x, y ∈ (BC[−τ, 0];R) and t ∈ [0, T ], |α(t, x)−α(t, y)|2 + |β(t, x)−|β(t, y)|2 + |σ(t, x)−σ(t, y)|2 ≤ K(x− y)2,K > 0.
The above G-SFDE has a unique solution X(t) ∈ M2G ([−τ, T ];R) if all the coefficients α, β and
σ satisfy the Linear growth and Lipschitz conditions [3, 12]. However, we suppose that the drift
coefficient α does not need to be continuous. The solution of equation 1.1 with initial condition 1.2
is an R valued stochastic processes X(t), t ∈ [−τ, T ] if
2
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
911 Faiz Faizullah et al 910-919
(i) X(t) is path-wise continuous and Ft-adapted for all t ∈ [0, T ];
(ii) α(t,Xt) ∈ L1([o, T ];R) and β(t,Xt), σ(t,Xt) ∈ L2([o, T ];R);
(iii) X0 = ξ and for each t ∈ [0, T ], dX(t) = α(t,Xt)dt+ β(t,Xt)d〈B,B〉(t) + σ(t,Xt)dB(t) q.s.
In the subsequent section, some preliminaries are given whereas in section 3, the comparison
theorem is developed. The last section, shows that under some suitable conditions, the G-SFDE
(1.1), provides more than one solutions.
2 Basic concepts and notions
In this section, we give some notions and basic definitions of the sublinear expectation [1, 2, 10, 11,
13]. Let Ω be a (non-empty) basic space and H be a linear space of real valued functions defined
on Ω such that any arbitrary constant c ∈ H and if X ∈ H then |X| ∈ H. We consider that H is
the space of random variables.
Definition 2.1. A functional E : H → R is called sub-linear expectation, if ∀ X,Y ∈ H, c ∈ R and
λ ≥ 0 it satisfies the following properties
(1) (Monotonicity): If X ≥ Y then E[X] ≥ E[Y ].
(2) (Constant preserving): E[c] = c.
(3) (Sub-additivity): E[X + Y ] ≤ E[X] + E[Y ].
(4) (Positive homogeneity): E[λX] = λE[X].
The triple (Ω,H,E) is called a sublinear expectation space. Consider the space of random
variables H such that if X1, X2, ..., Xn ∈ H then ϕ(X1, X2, ..., Xn) ∈ H for each ϕ ∈ Cl.Lip(Rn),
where Cl.Lip(Rn) is the space of linear functions ϕ defined as the following
Cl.Lip(Rn) =ϕ : Rn → R | ∃C ∈ [0,∞) : ∀ x, y ∈ Rn,|ϕ(x)− ϕ(y)| ≤ C(1 + |x|C + |y|C)|x− y|.
G-expectation and G-Brownian Motion. Let Ω = C0([0,∞)), that is, the space of all
R-valued continuous paths (wt)t∈[0,∞) with w0 = 0 equipped with the distance
ρ(w1, w2) =∞∑k=1
1
2k( maxt∈[0,k]
|w1t − w2
t | ∧ 1),
and consider the canonical process Bt(w) = wt for t ∈ [0,∞), w ∈ Ω then for each fixed T ∈ [0,∞)
we have
Lip(ΩT ) = ϕ(Bt1 , Bt2 , ..., Btn) : t1, ..., tn ∈ [0, T ], ϕ ∈ Cl.Lip(Rn), n ∈ N,
3
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
912 Faiz Faizullah et al 910-919
where Lip(Ωt) ⊆ Lip(ΩT ) for t ≤ T and Lip(Ω) = ∪∞m=1Lip(Ωm).
Consider a sequence ξi∞i=1 of random variables on a sublinear expectation space (Ω, Hp, E)
such that ξi+1 is independent of (ξ1, ξ2, ..., ξi) for each i = 1, 2, ... and ξi is G-normally distributed
for each i ∈ 1, 2, .... Then a sublinear expectation E[.] defined on Lip(Ω) is introduced as follows.
For 0 = t0 < t1 < ... < tn <∞ (t0, t1, ..., tn ∈ [t,∞)) [13], ϕ ∈ Cl.Lip(Rn) and each
X = ϕ(Bt1 −Bt0 , Bt2 −Bt1 , ..., Btn −Btn−1) ∈ Lip(Ω),
E[ϕ(Bt1 −Bt0 , Bt2 −Bt1 , ..., Btn −Btn−1)]
= E[ϕ(√t1 − t0ξ1, ...,
√tn − tn−1ξn)].
The conditional sublinear expectation of X ∈ Lip(Ωt) is defined by
E[X|Ωt] = E[ϕ(Bt1 , Bt2 −Bt1 , ..., Btm −Btm−1)|Ωt]
= ψ(Bt1 , Bt2 −Bt1 , ..., Btj −Btj−1),
where
ψ(x1, ..., xj) = E[ϕ(x1, ..., xj ,√tj+1 − tjξj+1, ...,
√tn − tn−1ξn)].
Definition 2.2. The sublinear expectation E : Lip(Ω)→ R defined above is called a G-expectation
and the corresponding canonical process Bt, t ≥ 0 is called a G-Brownian motion.
The completion of Lip(Ω) under the norm ‖X‖p = (E[|X|p])1/p [11, 13] for p ≥ 1 is denoted
by LpG(Ω) and LpG(Ωt) ⊆ LpG(ΩT ) ⊆ LpG(Ω) for 0 ≤ t ≤ T < ∞. The filtration generated by the
canonical process (Bt)t≥0 is denoted by Ft = σBs, 0 ≤ s ≤ t, F = Ftt≥0.
Ito’s Integral of G-Brownian motion. For any T ∈ [0,∞), a finite ordered subset πT =
t0, t1, ..., tN such that 0 = t0 < t1 < ... < tN = T is a partition of [0, T ] and
µ(πT ) = max|ti+1 − ti| : i = 0, 1, ..., N − 1.
A sequence of partitions of [0, T ] is denoted by πNT = tN0 , tN1 , ..., tNN such that limN→∞
µ(πNT ) = 0.
Consider the following simple process:
Let p ≥ 1 be fixed. For a given partition πT = t0, t1, ..., tN of [0, T ],
ηt(w) =
N−1∑i=0
ξi(w)I[ti,ti+1](t), (2.1)
where ξi ∈ LpG(Ωti), i = 0, 1, ..., N − 1. The collection containing the above type of processes,
that is, containing ηt(w) is denoted by Mp,0G (0, T ). The completion of Mp,0
G (0, T ) under the norm
‖η‖ = ∫ T0 E[|ηu|p]du1/p is denoted by Mp
G(0, T ) and for 1 ≤ p ≤ q, MpG(0, T ) ⊃M q
G(0, T ).
4
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
913 Faiz Faizullah et al 910-919
Definition 2.3. For each ηt ∈M2,0G (0, T ), the Ito’s integral of G-Brownian motion is defined as
I(η) =
∫ T
0ηudBu =
N−1∑i=0
ξi(Bti+1 −Bti).
Definition 2.4. An increasing continuous process 〈B〉t : t ≥ 0 with 〈B〉0 = 0, defined by
〈B〉t = B2t − 2
∫ t
0BudBu,
is called the quadratic variation process of G-Brownian motion.
3 Comparison theorem for G-SFDEs
The purpose of this section is to establish comparison result for problem (1.1) with initial data
(1.2). Consider the following two stochastic functional differential equations
X (t) = ξ1(0) +
∫ t
t0
α1(s,Xs)ds+
∫ t
t0
β(s,Xs)d〈B,B〉(s) +
∫ t
t0
σ(s,Xs)dB(s), t ∈ [0, T ], (3.1)
X (t) = ξ2(0) +
∫ t
t0
α2(s,Xs)ds+
∫ t
t0
β(s,Xs)d〈B,B〉(s) +
∫ t
t0
σ(s,Xs)dB(s), t ∈ [0, T ]. (3.2)
Theorem 3.1. Assume that:
(i) X1 and X2 are unique strong solutions of problems (3.1) and (3.2) respectively.
(ii) α1(s,Xs) ≤ α2(s,Xs) componentwise for all t ∈ [t0, T ], x ∈ BC([−τ, 0];Rd) and ξ1 ≤ ξ2.
(iii) α1 or α2 is increasing such that f(t, x) ≤ f(t, y) when x ≤ y for all x, y ∈ C([−τ, 0];R).
Then for all t > 0 we have X1 ≤ X2 q.s.
Proof. First, we define an operator q(., .) : C([−τ, 0];R)× C([−τ, 0];R)→ C([−τ, 0];R) such that
q(x, y) = max[x, y].
Obviously, y → q(x, y) satisfies the linear growth and Lipschitz conditions. Now we suppose that
α2 is increasing and consider the following equation
Y (t) = ξ2(0) +
∫ t
t0
α2(s, q(Xs1, Ys))ds+
∫ t
t0
β(s, q(Xs1, Ys)d〈B,B〉(s)
+
∫ t
t0
σ(s, q(Xs1, Ys)dB(s), t0 ≤ t ≤ T.
(3.3)
5
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
914 Faiz Faizullah et al 910-919
Thus it is easy to see that the coefficients satisfy the linear growth and Lipschitz conditions, so
3.3 has a unique solution Y (t). We shall prove that Y (t) ≥ X1s q.s. We define the following two
stopping times. For more details on stopping times we refere the reader to [7, 8].
τ1 = inft ∈ [t0, T ] : X1s − Y (t) > 0 where τ1 < T,
τ2 = inft ∈ [τ1, T ] : X1s − Y (t) < 0.
Contrary suppose that there exist an interval (τ1, τ2) ⊂ [t0, T ] such that Y (τ1) = X1(τ1) = ξ∗(0)
and Y (t) ≤ X1(t) for all t ∈ (τ1, τ2). Then,
Y (t)−X1(t) = ξ∗(0) +
∫ t
τ1
α2(s, q(Xs1, Ys))ds+
∫ t
τ1
β(s, q(Xs1, Ys))d〈B,B〉(s)
+
∫ t
τ1
σ(s, q(Xs1, Ys))dB(s)− ξ∗(0)−
∫ t
τ1
α1(s,Xs1)ds
−∫ t
τ1
β(s,Xs1)d〈B,B〉(s)−
∫ t
τ1
σ(s,Xs1)dB(s), t ∈ (τ1, τ2).
Y (t)−X1(t) =
∫ t
τ1
[α2(s, q(Xs1, Ys))− α1(s,Xs
1)]ds
+
∫ t
τ1
[β(s, q(Xs1, Ys))− β(s,Xs
1)]d〈B,B〉(s)
+
∫ t
τ1
[σ(s, q(Xs1, Ys))− σ(s,Xs
1)]dB(s), t ∈ (τ1, τ2).
But our supposition Y (t) ≤ X1(t) yields q(X1, Y ) = max[X1, Y ] = X1. So, we have
Y (t)−X1(t) =
∫ t
τ1
[α2(s,Xs1)− α1(s,Xs
1)]ds
+
∫ t
τ1
[β(s,Xs1)− β(s,Xs
1)]d〈B,B〉(s)
+
∫ t
τ1
[σ(s,Xs1)− σ(s,Xs
1)]dB(s)
Y (t)−X1(t) =
∫ t
τ1
[α2(s,Xs1)− α1(s,Xs
1)]ds ≥ 0,
because α2(t, x) ≥ α1(t, x). Which gives contradiction. So, our supposition Y (t) ≤ X1(t) for all
t ∈ (τ1, τ2) is wrong. Thus Y (t) ≥ X1(t) q.s. and so p(X1, Y ) = Y . It means that Y = X2 ≥ X1
because G-SFDE (3.3) has a unique solution X2.
4 G-SFDEs with discontinuous drift coefficients
We now suppose that α is left continuous, increasing and α(t, x) ≥ 0 for all (t, x) ∈ [0, T ] ×BC([−τ, 0];R) but not continuous. Consider the following sequence of problems.
Xn(t) = ξ(0) +
∫ t
0α(s,Xn−1
s )ds+
∫ t
0β(s,Xn
s )d〈B,B〉(s) +
∫ t
0σ(s,Xn
s )dB(s), t ∈ [0, T ], (4.1)
6
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
915 Faiz Faizullah et al 910-919
where X0 = Lt, Lt is the unique solution of the following problem
Lt = ξ +
∫ t
0β(s, Ls)d〈B,B〉(s) +
∫ t
0σ(s, Ls)dB(s), t ∈ [0, T ]. (4.2)
Thus using the comparison theorem and the fact that α(t, x) ≥ 0, we have X1 ≥ Lt. So, we can
see that Xn is an increasing sequence. Now we shall prove that Xn is bounded in L2 norm.
Lemma 4.1. Suppose Xn(t) be a solution of problem (4.1) then there exists a positive constant C
independent of n such that,
E
(sup
−τ≤s≤T|Xn(s)|2
)≤ C.
Proof. For any n ≥ 1 we define the following stopping time in a similar way as given in [9]
τm = T ∧ inft ∈ [t0, T ] : ‖Xnt ‖ ≥ m.
We have τm ↑ T and define Xn,m(t) = Xn(t ∧ τm) for t ∈ (−τ, T ). Then for t ∈ [0, T ],
Xn,m(t) = ξ(0) +
∫ t
0α(t,Xn−1,m
t )I[o,τm]dt+
∫ t
0β(t,Xn,m
t )I[o,τm]d〈B,B〉t +
∫ t
0σ(t,Xn,m
t )I[o,τm]dBt.
|Xn,m(t)|2 = |ξ(0) +
∫ t
0α(t,Xn−1,m
t )I[0,τm]dt+
∫ t
0β(t,Xn,m
t )I[0,τm]d〈B,B〉t
+
∫ t
0σ(t,Xn,m
t )I[0,τm]dBt|2
≤ 4|ξ(0)|2 + 4|∫ t
0α(t,Xn−1,m
t )I[0,τm]dt|2 + 4|∫ t
0β(t,Xn,m
t )I[0,τm]d〈B,B〉t|2
+ 4|∫ t
0σ(t,Xn,m
t )I[0,τm]dBt|2
Taking G-expectation, using properties of G-integral, G-quadratic variation process [10, 11] and
linear growth condition we get
E[|Xn,m(t)|2] ≤ 4E|ξ(0)|2 + 4C1
∫ t
0[1 + E|Xn−1,m
t |2]dt+ 4C2
∫ t
0[1 + E|Xn,m
t |2]dt|
+ 4C3
∫ t
0[1 + E|Xn,m
t |2]dt
≤ 4E|ξ(0)|2 + 4C1
∫ t
0dt+ 4C1
∫ t
0E|Xn−1,m
t |2dt+ 4C2
∫ t
0dt+ 4C2
∫ t
0E|Xn,m
t |2dt
+ 4C3
∫ t
0dt+ 4C3
∫ t
0E|Xn,m
t |2dt
= 4E|ξ(0)|2 + 4C1T + 4C1
∫ t
0E|Xn−1,m
t |2dt+ 4C2T + 4C2
∫ t
0E|Xn,m
t |2dt
+ 4C3T + 4C3
∫ t
0E|Xn,m
t |2dt.
7
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
916 Faiz Faizullah et al 910-919
Then for any k ∈ N we have,
max1≤n≤k
E[|Xn,m(t)|2] ≤ C4+4C1
∫ t
0max1≤n≤k
E|Xn−1,mt |2dt+4C2
∫ t
0max1≤n≤k
E|Xn,mt |2dt+4C3
∫ t
0max1≤n≤k
E|Xn,mt |2dt,
where C4 = 4[E|ξ|2 + C1T + C2T + C3T ] and thus using Doob’s martingale inequality for any
n,m ∈ N we have,
E[ sup0≤s≤t
|Xn,m(s)|2] ≤ C4 + C5
∫ t
0E|Xn,m
s |2dt, (4.3)
where C5 = 4(C1 + C2 + C3). One can observe the fact [9],
sup−τ≤s≤t
|Xn,m(s)|2 ≤ ‖ξ‖+ sup0≤s≤t
|Xn,m(s)|2,
and thus 4.3 yields
E[ sup−τ≤s≤t
|Xn,m(s)|2] ≤ E[‖ξ‖] + C4 + C5
∫ t
0E|Xn,m
s |2dt
≤ C6 + C5
∫ t
0E[ sup−τ≤r≤s
|Xn,m(r)|2]dt,
where C6 = E[‖ξ‖] + C4. So, using the Gronwall inequality and taking m→∞ we have,
E[ sup−τ≤s≤t
|Xn(s)|2] ≤ C6eC4t.
Letting t = T we get the desired result,
E[ sup−τ≤s≤T
|Xn(s)|2] ≤ C∗, C∗ = C4eCT .
Theorem 4.2. Suppose that:
(i) The coefficient α be left continuous and increasing in the second variable x.
(ii) For all (t, x) ∈ [0, T ]×BC([−τ, 0];R), α(t, x) ≥ 0.
Then the G-SFDE (1.1) has more than one solution X(t) ∈M2G ([−τ, T ];R).
Proof. By theorem 3.1 we know that Xn is increasing and by Lemma 4.1 it is bounded in L2.
Then by Dominated Convergence theorem we can deduce that Xn converges in L2. Denoting the
limit of Xn by X and thus for almost all w, we get
α(t,Xn(t))→ α(t,X(t)) as n→∞,
and
|α(t,Xn(t))| ≤ K(1 + supn|Xn(t)|) ∈ L1([t0, T ]).
8
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
917 Faiz Faizullah et al 910-919
Thus, for almost all w and uniformly in t∫ t
0α(s,Xn(s))ds→
∫ t
0α(s,X(s))ds, n→∞.
By the properties of β, σ and by the continuity properties of G-integral and its quadratic variation
process we have,
sup0≤t≤T
∣∣∣∣∫ t
0β(s,Xn(s))d〈B,B〉s −
∫ t
0β(s,X(s))d〈B,B〉s
∣∣∣∣→ 0 (q.s), n→∞.
sup0≤t≤T
∣∣∣∣∫ t
0σ(s,Xn(s))dB(s)−
∫ t
0σ(s,X(s))dB(s)
∣∣∣∣→ 0 (q.s), n→∞.
It is easy to conclude that Xn converges uniformly to X in t, hence X is continuous. Taking limit in
equation (4.1), we get that X is the desired solution for stochastic functional differential equation
(1.1) with initial condition (1.2).
5 Acknowledgments
We are very grateful to Dr. Ali Anwar for his careful reading and some useful suggestions. First
author acknowledge the financial support of National University of Sciences and Technology (NUST)
Pakistan for this research.
References
[1] Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation:
Application to G-Brownian motion paths. Potential Anal., 2010; 34: 139-161.
[2] Faizullah F. A note on the Caratheodory approximation scheme for stochastic differential
equations under G-Brownian motion. Zeitschrift fr Naturforschung A. 2012; 67a: 699-704.
[3] Faizullah F. Existence of solutions for G-SFDEs with Caratheodory Approximation Scheme.
Abstract and Applied Analysis. http://dx.doi.org/10.1155/2014/809431, 2014; volume 2014:
pages 8.
[4] Faizullah F, Piao D. Existence of solutions for G-SDEs with upper and lower solu- tions in the
reverse order. International Journal of the Physical Sciences. 2012; 16(12): 1820-1829.
[5] Halidias N, Ren Y. An existence theorem for stochastic functional differential equations with
delays under weak conditions. Statistics and Probability Letters. 2008; 78: 2864-2867.
[6] Halidias N, Kloeden P. A note on strong solutions for stochastic differential equations with
discontinuous drift coefficient. J. Appl. Math. Stoch. Anal. doi:10.1155/JAMSA/2006/73257.
Article ID 73257., 2006; 78: 1-6.
9
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
918 Faiz Faizullah et al 910-919
[7] Hu M, Peng S. Extended conditional G-expectations and related stopping times.
arXiv:1309.3829v1[math.PR] 16 Sep 2013.
[8] Li X, Peng S. Stopping times and related Ito’s calculus with G-Brownian motion. Stochastic
Processes and thier Applications. 2011; 121: 1492-1508.
[9] Mao X. Stochastic differential equations and their applications. Horwood Publishing Chichester
1997.
[10] Peng S. G-expectation, G-Brownian motion and related stochastic calculus of Ito’s type. The
abel symposium 2005, Abel symposia 2, edit. benth et. al., Springer-vertag. 2006; 541-567.
[11] Peng S. Multi-dimentional G-Brownian motion and related stochastic calculus under G-
expectation. Stochastic Processes and thier Applications. 2008; 12: 2223-2253.
[12] Ren Y, Bi Q, Sakthivel R. Stochastic functional differential equations with infinite delay driven
by G-Brownian motion. Mathematical Methods in the Applied Sciences, 2013; 36(13): 1746-
1759.
[13] Song Y. Properties of hitting times for G-martingale and their applications. Stochastic Pro-
cesses and thier Applications. 2011; 8(121): 1770-1784.
10
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
919 Faiz Faizullah et al 910-919
SUBCLASSES OF JANOWSKI-TYPE FUNCTIONSDEFINED BY CHO-KWON-SRIVASTAVA OPERATOR
SAIMA MUSTAFA, TEODOR BULBOACA, AND BADR S. ALKAHTANI
Abstract. We introduce a new subclass of analytic functions inthe unit disk U defined by using Cho-Kwon Srivastava integraloperator. Inclusion results radius problem and integral preservingproperties are investigated.
1. Introduction
Let Ap be the class of analytic functions in U of the form
(1.1) f(z) = zp +∞∑n=1
ap+nzp+n, z ∈ U, (p ∈ N) ,
where N := 1, 2, . . . . For p = 1 we denotes A := A1. Note that theclass Ap is closed under the convolution (or Hadamard) product, thatis
f(z) ∗ g(z) := zp +∞∑n=1
ap+nbp+nzp+n, z ∈ U, (p ∈ N) ,
where f is given by (1.1) and g(z) = zp +∑∞
n=1 bp+nzp+n, z ∈ U.
The operator Lp(d, e) : Ap → Ap is defined by using the Hadamard(convolution) product, that is
(1.2) Lp(d, e)f(z) := f(z) ∗ ϕp(d, e; z),
where
ϕp(d, e; z) := zp +∞∑n=1
(d)n(e)n
zp+n,(d ∈ C, e ∈ C \ Z−0
),
and (d)n = d(d + 1) . . . (d + n − 1), with (d)0 = 1, represents thewell-known Pochhammer symbol.
From (1.2) it follows immediately that
z (Lp(d, e)f(z))′ = dLp(d+ 1, e)f(z)− (d− p)Lp(d, e)f(z).
The operator Lp(d, e) was introduced by Saitoh [16] and this is anextension of the operator L(d, e) which was defined by Carlson andShaffer [2].
2010 Mathematics Subject Classification. 30C45, 30C50.Key words and phrases. Analytic functions, univalent functions, Cho-Kwon-
Srivastava operator.1
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
920 SAIMA MUSTAFA et al 920-933
2 S. MUSTAFA, T. BULBOACA, AND B. S. ALKAHTANI
Analogous to the Lp(d, e) operator, Cho et. al. [4] introduced theoperator Ipµ(d, e) : Ap → Ap defined by
(1.3) Ipµ(d, e)f(z) := f(z) ∗ ϕ†p(d, e; z),
where
ϕ†p(d, e; z) := zp +∞∑n=1
(µ+ p)n(e)nn!(d)n
zp+n,(d, e ∈ C \ Z−0 , µ > −p
).
We notice that
ϕ†p(d, e; z) ∗ ϕp(d, e; z) =zp
(1− z)µ+p, z ∈ U.
From (1.3), the following identities can be easily obtained [4]:
z(Ipµ(d+ 1, e)f(z)
)′= dIpµ(d, e)f(z)− (d− p) Ipµ(d+ 1, e)f(z),(1.4)
z(Ipµ(d, e)f(z)
)′= (µ+ p)Ipµ+1(d, e)f(z)− µIpµ(d, e)f(z).(1.5)
We may easily remark the following relations
Ip1 (p+ 1, 1)f(z) = f(z), Ip1 (p, 1)f(z) =zf ′(z)
p,
and remark that the operator I1µ(a + 2, 1), with µ > −1 and a > −2,was studied in [5].
If f and g are two analytic functions in U, we say that f is subordinateto g, written symbolically as f(z) ≺ g(z), if there exists a Schwarzfunction w, which (by definition) is analytic in U, with w(0) = 0,and |w(z)| < 1, z ∈ U, such that f(z) = g(w(z)), for all z ∈ U.Furthermore, if the function g is univalent in U, then we have thefollowing equivalence, (cf., e.g., [10], see also [11, p. 4]):
f(z) ≺ g(z)⇔ f(0) = g(0) and f(U) ⊂ g(U).
Definition 1.1. 1. Like in [3], for arbitrary fixed numbers A, B andβ, with −1 ≤ B < A ≤ 1 and 0 ≤ β < 1, let P [A,B, β] denote thefamily of functions p that are analytic in U, with p(0) = 1, and suchthat
p(z) ≺ 1 + [(1− β)A+ βB] z
1 +Bz.
We will use the notations P [A,B] := P [A,B, 0] and P (0) := P [1,−1, 0].2. Let Pl[A,B, β] denote the class of functions p that are analytic in
U, with p(0) = 1, that are represented by
(1.6) p(z) =
(l
4+
1
2
)K1(z)−
(l
4− 1
2
)K2(z),
where K1, K2 ∈ P [A,B, β], −1 ≤ B < A ≤ 1, 0 ≤ β < 1, and l ≥ 2.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
921 SAIMA MUSTAFA et al 920-933
SUBCLASSES OF JANOWSKI-TYPE FUNCTIONS 3
Remarks 1.1. (i) Remark that the class Pl(β) := Pl[1,−1, β] was de-fined and studied in [12], while for l = 2 and β = 0 the above classwas introduced by Janowski [8]. Moreover, the class Pl := Pl[1,−1, 0]is the well-known class of Pinchuk [15].
Also, we see that Pl[A,B, β] ⊂ Pl(β), where β =1− A1
1−Band A1 =
(1− β)A+ βB.(ii) Notice that, if g is analytic in U with g(0) = 1, then there exit
functions g1 and g2 analytic in U with g1(z) = g2(z) = 1, such that thefunction g could be written in the form (1.6). For example, taking
g1(z) =g(z)− 1
k+g(z) + 1
2and g1(z) =
g(z) + 1
2− g(z)− 1
k,
then g1 and g2 are analytic in U, and g1(z) = g2(z) = 1.
We will assume throughout our discussion, unless otherwise stated,that λ > 0, d, e ∈ R\Z−0 , µ > −p, −1 ≤ B < A ≤ 1, ϑ ≥ 0, and p ∈ N.Moreover, all the powers are the principal ones.
Using the Cho-Kwon-Srivastava integral operator Ipµ(d, e) defined by(1.4), we will define the following subclasses of Ap.Definition 1.2. Let d, e ∈ R \ Z−0 , λ > 0, µ > −p, 0 ≤ β < 1,and ϑ ≥ 0. For the function f ∈ Ap, p ∈ N, we say that f ∈N λ,ϑl,p (d, e;µ; β,A,B), with l ≥ 2, if and only if
(1 + ϑ)
(zp
Ipµ(d, e)f(z)
)λ−ϑ
Ipµ+1(d, e)f(z)
Ipµ(d, e)f(z)
(zp
Ipµ(d, e)f(z)
)λ∈ Pl[A,B, β].
We need to notice that, since the left-hand side function from theabove definition need to be analytic in U, we implicitly assumed thatIpµ(d, e)f(z) 6= 0 for all z ∈ U.
Remarks 1.2. We remark the following special cases of the above classes:(i) for β = 0 and l = 2 we obtain the subclass of non-Bazilevic
functions defined by [18];(ii) for µ = 0, l = 2, ϑ = B = −1, A = 1 and λ > 0, the above class
reduces to the class Q(λ) of p–valent non-Bazilevic functions (see [14]).
2. Preliminaries
The following definitions and lemmas will be required in our presentinvestigation.
Lemma 2.1. [7] Let h be a convex function in U with h(0) = 1. Sup-pose also that the function p given by
p(z) = 1 + cnzn + cn+1z
n+1 + . . . , z ∈ U,
is analytic in U. Then
p(z) +zp′(z)
γ≺ h(z) (Re γ ≥ 0, γ 6= 0) ,
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
922 SAIMA MUSTAFA et al 920-933
4 S. MUSTAFA, T. BULBOACA, AND B. S. ALKAHTANI
implies
(2.1) p(z) ≺ q(z) =γ
nz−
γn
∫ z
0
tγn−1h(t) d t ≺ h(z),
and q is the best dominant of (2.1).
For real or complex numbers a, b and c, the Gauss hypergeometricfunction is defined by
2F1(a, b, c; z) = 1 +a · bc
z
1!+a(a+ 1) · b(b+ 1)
c(c+ 1)
z2
2!+ . . .
=∞∑k=0
(a)k(b)k(c)k
zk
k!, a, b ∈ C, c ∈ C \ 0,−1,−2, . . . ,(2.2)
where (d)k is the previously recalled Pochhammer symbol. The series(2.2) converges absolutely for z ∈ U, hence it represents an analyticfunction in U (see [19, Chapter 14]).
Each of the following identities are fairly well-known:
Lemma 2.2. [19, Chapter 14] For all real or complex numbers a, band c, with c 6= 0,−1,−2, . . . , , the next equalities hold:∫ 1
0
tb−1(1− t)c−b−1(1− tz)−a d t =Γ(b)Γ(c− b)
Γ(c)2F1(a, b, c; z)(2.3)
where Re c > Re b > 0,
(2.4) 2F1(a, b, c; z) = (1− z)−a 2F1
(a, c− b, c; z
z − 1
),
and
(2.5) 2F1(a, b, c; z) =2 F1(b, a, c; z).
Lemma 2.3. [17] Let f(z) =∞∑k=0
akzk be analytic in U and g(z) =
∞∑k=0
bkzk be analytic and convex in U. If f(z) ≺ g(z), then
|ak| ≤ |b1| , k ∈ N.
3. Main results for the class N λ,ϑl,p (d, e;µ; β,A,B)
In this section, some properties of the class N λ,ϑl,p (d, e;µ; β,A,B)
such as inclusion results, integral preserving property, radius problem,coefficient bound will be discussed.
Theorem 3.1. 1. If f ∈ N λ,ϑl,p (d, e;µ; β,A,B), then(zp
Ipµ(d, e)f(z)
)λ∈ Pl[A,B, β].
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
923 SAIMA MUSTAFA et al 920-933
SUBCLASSES OF JANOWSKI-TYPE FUNCTIONS 5
2. Moreover, if f ∈ N λ,ϑl,p (d, e;µ; β, γ) with ϑ 6= 0, then(
zp
Ipµ(d, e)f(z)
)λ∈ Pl(β1),
where
β1 := β + (1− β)ϑ1
and
ϑ1 := ϑ1(p, λ, ϑ, µ;A,B) =AB
+(1− A
B
)(1−B)−1 2F1
(1, 1, λ(µ+p)
ϑ+ 1, B
B−1
), B 6= 0,
1− λ(µ+p)λ(µ+p)+ϑ
A, B = 0.
(All the powers are the principal ones).
Proof. Since the implication is obvious for ϑ = 0, suppose that ϑ > 0.Letting
(3.1) K(z) =
(zp
Ipµ(d, e)f(z)
)λ.
It follows that K is analytic in U, with K(0) = 1, and according to thepart (ii) of Remarks 1.1 the function K could be written in the form
(3.2) K(z) =
(k
4+
1
2
)K1(z)−
(k
4− 1
2
)K2(z),
where K1 and K2 are analytic in U, with K1(z) = K2(z) = 1.From the part 2. of Definition 1.1 we have that K ∈ Pl[A,B, β], if
and only if the function K has the representation given by the aboverelation, where K1, K2 ∈ P [A,B, β]. Consequently, supposing that Kis of the form (3.2), we will prove that K1, K2 ∈ P [A,B, β].
Differentiating the relation (3.1) and using the identity (1.5), we have
zK ′(z)
λ(µ+ p)= K(z)−
Ipµ+1(d, e)f(z)
Ipµ(d, e)f(z)
(zp
Ipµ(d, e)f(z)
)λ,
and from this relation we deduce that
(1 + ϑ)
(zp
Ipµ(d, e)f(z)
)λ− ϑ
Ipµ+1(d, e)f(z)
Ipµ(d, e)f(z)
(zp
Ipµ(d, e)f(z)
)λ=
K(z) +ϑ
λ(µ+ p)zK ′(z).
Since f ∈ N λ,ϑl,p (d, e;µ; β,A,B), from the above relation it follows that
K(z) +ϑ
λ(µ+ p)zK ′(z) ∈ Pl[A,B, β],
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
924 SAIMA MUSTAFA et al 920-933
6 S. MUSTAFA, T. BULBOACA, AND B. S. ALKAHTANI
and according to the second part of the Definition 1.1, this is equivalentto
Ki(z) +ϑ
λ(µ+ p)zK ′i(z) ∈ P [A,B, β], (i = 1, 2),
that is
1
1− β
[Ki(z) +
ϑ
λ(µ+ p)zK ′i(z)− β
]∈ P [A,B], (i = 1, 2).
Writing
(3.3) Ki(z) = (1− β)pi(z) + β, (i = 1, 2),
from the previous relation we have
pi(z) +ϑ
λ(µ+ p)zp′i(z) ∈ P [A,B], (i = 1, 2).
By using Lemma 2.1 for γ =λ(µ+ p)
ϑand n = 1, from the above
relation we deduce that
pi(z) ≺ q(z) ≺ 1 + Az
1 +Bz, (i = 1, 2),
where
q(z) =λ(µ+ p)
ϑz−
λ(µ+p)ϑ
∫ z
0
tλ(µ+p)ϑ−1 1 + At
1 +Btd t
is the best dominant for pi, i = 1, 2.
Since pi(z) ≺ 1 + Az
1 +Bz, i = 1, 2, from (3.3) it follows that Ki ∈
P [A,B, β], i = 1, 2, and according to (3.1) we conclude that K ∈Pl[A,B, β], which proves the first part of the theorem.
For the second part of our result, we distinguish the following twocases:
(i) For B = 0, a simple computation shows that
pi(z) ≺ q(z) = 1 +λ(µ+ p)
λ(µ+ p) + ϑAz, (i = 1, 2).
(ii) For B 6= 0, making the change of variables s = zt, followed bythe use of the identities (2.3), (2.4) and (2.5) of Lemma 2.2, we obtain
pi(z) ≺ q(z) =λ(µ+ p)
ϑ
∫ 1
0
sλ(µ+p)ϑ−1 1 + Asz
1 +Bszd s =
A
B+
(1− A
B
)(1 +Bz)−1 2F1
(1, 1,
λ(µ+ p)
ϑ+ 1,
Bz
Bz + 1
), (i = 1, 2).
Now, it is sufficient to show that
(3.4) inf Re q(z) : z ∈ U = q(−1).
We may easily show that
Re1 + Az
1 +Bz≥ 1− Ar
1−Br, for |z| ≤ r < 1.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
925 SAIMA MUSTAFA et al 920-933
SUBCLASSES OF JANOWSKI-TYPE FUNCTIONS 7
Denoting G(t, z) =1 + Atz
1 +Btzand dµ(t) =
λ(µ+ p)
ϑtλ(µ+p)ϑ−1 d t, which
is a positive measure on [0, 1], we have
q(z) =
∫ 1
0
G(t, z) dµ(t),
hence it follows
Re q(z) ≥∫ 1
0
1− Atr1−Btr
dµ(t) = q(−r), for |z| ≤ r < 1.
By letting r → 1− we obtain (3.4), and from (3.3) and (3.1) we concludethat K ∈ Pl(β1), which completes our proof.
Theorem 3.2. If 0 ≤ ϑ1 < ϑ2, then
N λ,ϑ2l,p (d, e;µ; β,A,B) ⊂ N λ,ϑ1
l,p (d, e;µ; β,A,B)
Proof. The first part of Theorem 3.1 shows that the above inclusionholds whenever ϑ1 = 0.
If 0 < ϑ1 < ϑ2, for an arbitrary f ∈ N λ,ϑ2l,p (d, e;µ; β,A,B) let denote
U1(z) = (1 + ϑ1)
(zp
Ipµ(d, e)f(z)
)λ− ϑ1
Ipµ+1(d, e)f(z)
Ipµ(d, e)f(z)
(zp
Ipµ(d, e)f(z)
)λand
U0(z) =
(zp
Ipµ(d, e)f(z)
)λ.
A simple computation shows that
(1 + ϑ1)
(zp
Ipµ(d, e)f(z)
)λ− ϑ1
Ipµ+1(d, e)f(z)
Ipµ(d, e)f(z)
(zp
Ipµ(d, e)f(z)
)λ=(
1− ϑ1
ϑ2
)U0(z) +
ϑ1
ϑ2
U2(z).
Since Pl[A,B, β] is a convex set, from the first part of Theorem 3.1,according to the above notations it follows that(
1− ϑ1
ϑ2
)U0(z) +
ϑ1
ϑ2
U2(z) ∈ Pl[A,B, β],
that is f ∈ N λ,ϑ1l,p (d, e;µ; β,A,B).
Theorem 3.3. If f ∈ N λ,0l,p (d, e;µ; β, 1,−1), then f(ρz) ∈ N λ,ϑ
l,p (d, e;µ; β, 1,−1),where ρ is given by
(3.5) ρ =−(β + ϑ
λ(µ+p)
)+
√(β + ϑ
λ(µ+p)
)2+ 1− β2
1 + β.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
926 SAIMA MUSTAFA et al 920-933
8 S. MUSTAFA, T. BULBOACA, AND B. S. ALKAHTANI
Proof. For an arbitrary f ∈ N λ,0l,p (d, e;µ; β, 1,−1), let denote(
zp
Ipµ(d, e)f(z)
)λ= K(z) =
(l
4+
1
2
)K1(z)−
(l
4− 1
2
)K2(z),
where K1, K2 ∈ P [1,−1, β], which in equivalent to K1(0) = K2(0) = 1and ReK1(z) > β, ReK2(z) > β, z ∈ U .
With this notation, like in the proof of Theorem 3.1 we obtain
(1 + ϑ)
(zp
Ipµ(d, e)f(z)
)λ− ϑ
Ipµ+1(d, e)f(z)
Ipµ(d, e)f(z)
(zp
Ipµ(d, e)f(z)
)λ=
K(z) +ϑ
λ(µ+ p)zK ′(z) =(
l
4+
1
2
)[K1(z) +
ϑ
λ(µ+ p)zK ′1(z)
]−(l
4− 1
2
)[K2(z) +
ϑ
λ(µ+ p)zK ′2(z)
].
In order to have f(ρz) ∈ N λ,ϑl,p (d, e;µ; β, 1,−1), according to the above
formula, we need to find the (bigger) value of ρ, such that
Re
[Ki(z) +
ϑ
λ(µ+ p)zK ′i(z)
]> β, |z| < ρ, (i = 1, 2).
From the well-known estimates for the class P (0) (see, eq., [6]) wehave
|K ′i(z)| ≤ 2 ReKi(z)
1− r2, |z| ≤ r < 1, (i = 1, 2),
ReKi(z) ≥ 1− r1 + r
, |z| ≤ r < 1, (i = 1, 2),
thus, we deduce that
Re
[Ki(z) +
ϑ
λ(µ+ p)zK ′i(z)
]≥ ReKi(z)− ϑ
λ(µ+ p)|zK ′i(z)| ≥
ReKi(z)
[1− ϑ
λ(µ+ p)
2r
1− r2
], |z| ≤ r < 1, (i = 1, 2).(3.6)
A simple computation shows that
(3.7) 1− ϑ
λ(µ+ p)
2r
1− r2≥ 0,
for 0 ≤ r ≤ r0, where
r0 :=−ϑ+
√ϑ2 + λ2(µ+ p)2
λ(µ+ p)∈ (0, 1).
Now, from the inequality (3.6) we have
Re
[Ki(z) +
ϑ
λ(µ+ p)zK ′i(z)
]≥ 1− r
1 + r
[1− ϑ
λ(µ+ p)
2r
1− r2
], |z| ≤ r0 < 1,
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
927 SAIMA MUSTAFA et al 920-933
SUBCLASSES OF JANOWSKI-TYPE FUNCTIONS 9
for i = 1, 2. It is easy to check that
1− r1 + r
[1− ϑ
λ(µ+ p)
2r
1− r2
]> β
for 0 ≤ r < ρ, where ρ is given by (3.5). Moreover, since the aboveinequality is equivalent to
1− ϑ
λ(µ+ p)
2r
1− r2>
1 + r
1− rβ
for r ∈ [0, 1), if follows that (3.7) holds for r ∈ [0, ρ), and our theoremis completely proved.
Next we will consider some properties of generalized p–valent Ber-nardi integral operator. Thus, for f ∈ Ap, let Fη,p : Ap → Ap bedefined by
(3.8) Fη,pf(z) =η + p
zη
∫ z
0
f(t)tη−1 d t, (η > −p).
We will give a short proof that this operator is well-defined, as follows.If the function f ∈ Ap is of the form (1.1), then the definition relation(3.8) could be written as
Fη,pf(z) =η + p
zη
∫ z
0
f(t)tη−1 d t = (η + p)Iη,pf(z),
where
Iη,pf(z) =1
zη
∫ z
0
f(t)tη−1 d t.
We see that integral operator Iη,p defined above is similar to that ofLemma 1.2c. of [11]. According to this lemma, it follows that Iη,p is ananalytic integral operator for any function f of the form (1.1) wheneverRe η > −p, and Fη,pf ∈ Ap has the form
Fη,pf(z) = zp + (η + p)∞∑n=1
ap+np+ n+ η
zp+n, z ∈ U.
The operator defined in (3.8) is called the generalized p–valent Ber-nardi integral operator, and for special case p = 1 we get the generalizedBernardi integral operator. Thus, for p = 1 and η ∈ N, the operatorFη := Fη,1 was introduced by Bernardi [1], and in particular, if η = 1it reduces to the operator F1 that was earlier introduced by Livingston[9].
Theorem 3.4. Let f ∈ Ap and F = Fη,pf , where Fη,p is given by(3.8). If(3.9)
(1 + ϑ)
(zp
Ipµ(d, e)F (z)
)λ−ϑ
Ipµ(d, e)f(z)
Ipµ(d, e)F (z)
(zp
Ipµ(d, e)F (z)
)λ∈ Pl[A,B, β],
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
928 SAIMA MUSTAFA et al 920-933
10 S. MUSTAFA, T. BULBOACA, AND B. S. ALKAHTANI
where d, e ∈ R \ Z−0 , λ > 0, µ > −p, 0 ≤ β < 1, ϑ ≥ 0 and l ≥ 2, then(zp
Ipµ(d, e)F (z)
)λ∈ Pl[A,B, β].
(All the powers are the principal ones).
Proof. Like we mentioned after the Definition 1.2, since the left-handside function from the relation (3.9) need to be analytic in U, we im-plicitly assumed that Ipµ(d, e)F (z) 6= 0 for all z ∈ U.
The implication is obvious for ϑ = 0, hence suppose that ϑ > 0.Letting
(3.10)
(zp
Ipµ(d, e)F (z)
)λ= K(z),
from the assumption (3.9) it follows that K is analytic in U, withK(0) = 1.
It is easy to check that, if f, g ∈ Ap, then
(3.11)z
p(f(z) ∗ g(z))′ =
(z
pf ′(z)
)′∗ g(z).
Moreover, since F = Fη,pf , where Fη,p is given by (3.8), a simple dif-ferentiation shows that
(3.12) z(Ipµ(d, e)F (z)
)′= (η + p)Ipµ(d, e)f(z)− ηIpµ(d, e)F (z).
Taking the logarithmical differentiation of (3.10), we have
λ
[p−
z(Ipµ(d, e)F (z)
)′Ipµ(d, e)F (z)
]=zK ′(z)
K(z),
and using the relations (3.11) and (3.12), it follows that
Ipµ(d, e)f(z)
Ipµ(d, e)F (z)= 1− 1
λ(η + p)
zK ′(z)
K(z),
and thus
(1 + ϑ)
(zp
Ipµ(d, e)F (z)
)λ− ϑ
Ipµ(d, e)f(z)
Ipµ(d, e)F (z)
(zp
Ipµ(d, e)F (z)
)λ=
K(z) +ϑ
λ(p+ η)zK ′(z).
From the assumption (3.9), the above relation gives that
K(z) +ϑ
λ(p+ η)zK ′(z) ∈ Pl[A,B, β],
and using a similar proof with those of the first part of Theorem 3.1we obtain that K ∈ Pl[A,B, β], which proves our result.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
929 SAIMA MUSTAFA et al 920-933
SUBCLASSES OF JANOWSKI-TYPE FUNCTIONS 11
Theorem 3.5. If
(3.13) f(z) = zp +∞∑n=1
ap+nzp+n ∈ N λ,ϑ
l,p (d, e;µ; β,A,B) ,
where d, e ∈ R \ Z−0 , λ > 0, µ > −p, 0 ≤ β < 1, ϑ ≥ 0 and l ≥ 2, then
(3.14) |ap+1| ≤∣∣∣∣de∣∣∣∣ (1− β)(A−B)
|ϑ+ λ(µ+ p)|.
The inequality (3.14) is sharp.
Proof. If we let(3.15)
(1 + ϑ)
(zp
Ipµ(d, e)f(z)
)λ− ϑ
Ipµ+1(d, e)f(z)
Ipµ(d, e)f(z)
(zp
Ipµ (d, e) f(z)
)λ= p(z),
using the fact that
Ipµ(d, e)f(z) = zp +∞∑n=1
(µ+ p)n(e)nn!(d)n
ap+nzp+n,
we have
(1 + ϑ)
(zp
Ipµ(d, e)f(z)
)λ− ϑ
Ipµ+1(d, e)f(z)
Ipµ(d, e)f(z)
(zp
Ipµ(d, e)f(z)
)λ=
1−(
1 +ϑ
λ(µ+ p)
)(µ+ p)1(e)1
1!(d)1λ ap+1z + · · · =
1− e
d[ϑ+ λ(µ+ p)] ap+1z + . . . , z ∈ U.(3.16)
Since f ∈ N λ,ϑl,p (d, e;µ; β,A,B), it follows that the function p defined
by (3.15) is of the form
p(z) =
(l
4+
1
2
)p1(z)−
(l
4− 1
2
)p2(z),
where p1, p2 ∈ P [A,B, β]. It follows that
pi(z) ≺ 1 + [(1− β)A+ βB] z
1 +Bz(i = 1, 2),
and from the above relation we deduce that
(3.17) p(z) ≺ 1 + [(1− β)A+ βB] z
1 +Bz.
According to (3.16), from the subordination (3.17) we obtain
1− e
d
ϑ+ λ(µ+ p)
1− βap+1z + · · · = p(z)− β
1− β≺ 1 + Az
1 +Bz,
and from Lemma 2.3 we conclude that∣∣∣∣−ed ϑ+ λ(µ+ p)
1− β
∣∣∣∣ |ap+1| ≤ |A−B|,
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
930 SAIMA MUSTAFA et al 920-933
12 S. MUSTAFA, T. BULBOACA, AND B. S. ALKAHTANI
which proves our result.To prove that the inequality (3.14) is sharp we need to show that
there exists a function f ∈ N λ,ϑl,p (d, e;µ; β,A,B) of the form (3.13),
such that for this function we have equality in (3.14).
Thus, we will prove that there exists f ∈ N λ,ϑl,p (d, e;µ; β,A,B), such
that the identity (3.15) holds for the special case
p(z) =1 + [(1− β)A+ βB] z
1 +Bz.
Setting
(3.18) K(z) =
(zp
Ipµ(d, e)f(z)
)λ,
like in the proof of Theorem 3.1 we deduce that the relation (3.15) isequivalent to
(3.19) K(z) +1
γzK ′(z) = p(z), where γ :=
λ(µ+ p)
ϑ.
(i) If ϑ = 0, the above differential equation has the solution K = p.(ii) If ϑ > 0, then γ > 0 whenever λ > 0 and µ > −p. Since the
function p is convex in the unit disk U, according to Lemma 2.1 itfollows that this differential equation has the solution
K(z) =γ
zγ
∫ z
0
tγ−1p(t) d t ≺ p(z).
It is easy to check that p(z) 6= 0 for all z ∈ U, and from the above
subordination we get that K(z) 6= 0, z ∈ U.Now, if we define the function K0 by
K0(z) =
p(z), if ϑ = 0,
K(z), if ϑ > 0,
then K0 is the analytic solution of the differential equation (3.19), andmoreover K0(z) 6= 0, z ∈ U.
Thus, for K = K0 the relation (3.18) is equivalent to
Ipµ(d, e)f(z) = zpK−1/λ0 (z),
and this equation has the solution
(3.20) f0(z) := ψp(d, e; z) ∗(zpK
−1/λ0 (z)
),
where
ψp(d, e; z) := zp +∞∑n=1
n!(d)n(µ+ p)n(e)n
zp+n.(d, e ∈ C \ Z−0 , µ > −p
),
Consequently, for the function f0 defined by (3.20) we get equality in(3.14), hence the sharpness of our result is proved.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
931 SAIMA MUSTAFA et al 920-933
SUBCLASSES OF JANOWSKI-TYPE FUNCTIONS 13
As a special case, for l = 2 and β = 0 we obtain the correspondingresult for the class N λ,ϑ
2,p (d, e;µ; 0, A,B) (see [18] for n = 1).
Acknowledgement. The work of the second author (T. Bulboaca)was entirely supported by the grant given by Babes-Bolyai University,dedicated for Supporting the Excellence Research 2015.
The third author B. S. Alkahtani is grateful to King Saud University,Deanship of Scientific Research, College of Science Research Center, forsupporting this project.
References
[1] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math.Soc., 135(1969), 429–446.
[2] B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric func-tions, SIAM J. Math. Anal., 159(1984), 737–745.
[3] M. Caglar, Y. Polatoglu, E. Yavuz, λ–Fractional properties of generalizedJanowski functions in the unit disc, Mat. Vesnik, 50(2008), 165–171.
[4] N. E. Cho, O. S. Kwon and H. M. Srivastava, Inclusion relationships andargument properties for certain subclasses of multivalent functions associatedwith a family of linear operators, J. Math. Anal. Appl., 292(2004), 470–483.
[5] J. H. Choi, M. Saigo and H. M. Srivastava, Some inclusion properties of certainfamily of integral operators, J. Math. Anal. Appl., 276(2002) 432–444.
[6] A. W. Goodman, Univalent Functions, Vol. I, II, Polygonal Publishing House,Washington, New Jersey, 1983.
[7] D. J. Hallenbeck and St. Ruscheweyh, Subordination by convex functions, Proc.Amer. Math. Soc., 52(1975), 191–195.
[8] W. Janowski, Some extremal problems for certain families of analytic func-tions, Ann. Polon. Math., 28(1973), 297–327.
[9] A. E. Livingston, On the radius of univalence of certain analytic functions,Proc. Amer. Math. Soc., 17(1966), 352–357.
[10] S. S. Miller and P. T. Mocanu, Second order differential inequalities in thecomplex plane, J. Math. Anal. Appl., 65(1978), 289–305.
[11] S. S. Miller and P. T. Mocanu, Differential Subordinations. Theory and Ap-plications, Marcel Dekker Inc., New York Basel, 2000.
[12] K. I. Noor and M. A. Noor, On certain classes of analytic functions defined byNoor integral operator, J. Math. Anal. Appl., 281(2003), 244-252.
[13] K. I. Noor, Applications of certain operators to the classes related with gener-alized Janowski functions, Integral Transforms Spec. Funct., 2(2010), 557–567.
[14] K. I. Noor, M. Ali, M. Arif, On a class of p–valent non-Bazilevic functions,Gen. Math., 18(2)(2010), 31–46
[15] B. Pinchuk, Functions with bounded boundary rotation, Israel J. Math.,10(1971), 7–16.
[16] H. Saitoh, A linear operator and its applications of first order differential sub-ordinations, Math. Japon., 44(1996), 31–38.
[17] W. Rogosinski, On the coefficients of subordinate functions, Proc. LondonMath. Soc. (Ser. 2), 48(1943), 48–82.
[18] Z.-G. Wang, H.-T. Wang and Y. Sun, A class of multivalent non-Bazilevicfunctions involving the Cho-Kwon-Srivastava operator, Tamsui Oxf. J. Inf.Math. Sci., 26(1)(2010), 1–19.
[19] E. T. Whittaker and G. N. Watson, A Course on Modern Analysis: An Intro-duction to the General Theory of Infinite Processes and of Analytic Functions;
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
932 SAIMA MUSTAFA et al 920-933
14 S. MUSTAFA, T. BULBOACA, AND B. S. ALKAHTANI
With an Account of the Principal Transcendental Functions, Fourth Edition,Cambridige Univ. Press, Cambridge, 1927.
Department of Mathematics, Pir Mehr Ali Shah Arid AgricultureUniversity, Rawalpindi, Pakistan
E-mail address: [email protected]
Faculty of Mathematics and Computer Science, Babes-Bolyai Uni-versity, 400084 Cluj-Napoca, Romania
E-mail address: [email protected]
Mathematics Department, College of Science, King Saud Univer-sity, P.O.Box 1142, Riyadh 11989, Saudi Arabia
E-mail address: [email protected]
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
933 SAIMA MUSTAFA et al 920-933
ON A PRODUCT-TYPE OPERATOR FROM MIXED-NORM SPACES TOBLOCH-ORLICZ SPACES
HAIYING LI AND ZHITAO GUO
ABSTRACT. The boundedness and compactness of a product-type operator DMuCψfrom mixed-norm spaces to Bloch-Orlicz spaces are characterized in this paper.
1. INTRODUCTION
Let D denote the unit disk in the complex plane C, H(D) the class of all analytic func-tions on D and N the set of nonnegative integers.
A positive continuous function φ on [0,1) is called normal if there exist two positivenumbers s and t with 0 < s < t, and δ ∈ [0, 1) such that (see [19])
φ(r)(1− r)s
is decreasing on [δ, 1), limr→1
φ(r)(1− r)s
= 0;
φ(r)(1− r)t
is increasing on [δ, 1), limr→1
φ(r)(1− r)t
= ∞.
For p, q ∈ (0,∞) and φ normal, the mixed-norm space H(p, q, φ)(D) = H(p, q, φ) isthe space of all functions f ∈ H(D) such that
‖f‖H(p,q,φ) =( ∫ 1
0
Mpq (f, r)
φp(r)1− r
dr
) 1p
<∞,
where
Mq(f, r) =(
12π
∫ 2π
0
|f(reiθ)|qdθ) 1
q
.
For 1 ≤ p, q < ∞, H(p, q, φ), equipped with the norm ‖f‖H(p,q,φ), is a Banach space,while for the other vales of p and q, ‖ · ‖H(p,q,φ) is a quasinorm on H(p, q, φ), H(p, q, φ)
is a Frechet space but not a Banach space. Note that if φ(r) = (1− r)α+1
p , then H(p, q, φ)is equivalent to the weighted Bergman space Apα(D) = Apα defined for 0 < p < ∞ andα > −1, as the spaces of all f ∈ H(D) such that
‖f‖pAp
α= (α+ 1)
∫D|f(z)|p(1− |z|2)αdm(z) <∞,
where dm(z) = 1π rdrdθ is the normalized Lebesgue area measure on D ([8, 12, 18, 25,
27, 33, 35, 48, 51]). For more details on the mixed-norm space on various domains andoperators on them, see, e.g., [1, 7, 10, 20, 22, 23, 24, 28, 29, 34, 36, 37, 38, 41, 42, 43, 44,46, 47, 54].
2000 Mathematics Subject Classification. Primary 47B33.Key words and phrases. A product-type operator, mixed-norm spaces, Bloch-Orlicz spaces, boundedness,
compactness.1
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
934 HAIYING LI et al 934-945
2 HAIYING LI AND ZHITAO GUO
For every 0 < α < ∞, the α-Bloch space, denoted by Bα, consists of all functionsf ∈ H(D) such that
supz∈D
(1− |z|2)α|f ′(z)| <∞.
Bα is a Banach space under the norm
‖f‖Bα = |f(0)|+ supz∈D
(1− |z|2)α|f ′(z)|.
For α = 1 is obtained the Bloch space. α-Bloch space is introduced and studied bynumerous authors. Recently, many authors studied different classes of Bloch-type spaces,where the typical weight function, ω(z) = 1 − |z|2(z ∈ D) is replaced by a boundedcontinuous positive function µ defined on D. More precisely, a function f ∈ H(D) iscalled a µ-Bloch function, denoted by f ∈ Bµ, if
‖f‖µ = supz∈D
µ(z)|f ′(z)| <∞.
Clearly, if µ(z) = ω(z)α with α > 0, Bµ is just the α-Bloch space Bα. It is readily seenthat Bµ is a Banach space with the norm
‖f‖Bµ = |f(0)|+ ‖f‖µ.For some information on the Bloch, α-Bloch and Bloch-type spaces, as well as some op-erators on them see, e.g., [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 23, 25,26, 27, 29, 30, 31, 32, 34, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55].
Recently, Fernandez in [17] used Young’s functions to define the Bloch-Orlicz space.More precisely, let ϕ : [0,∞) → [0,∞) be a strictly increasing convex function such thatϕ(0) = 0 and limt→∞ ϕ(t) = ∞. The Bloch-Orlicz space associated with the function ϕ,denoted by Bϕ, is the class of all analytic functions f in D such that
supz∈D
(1− |z|2)ϕ(λ|f ′(z)|) <∞
for some λ > 0 depending on f . Also, since ϕ is convex, it is not hard to see that theMinkowski’s functional
‖f‖ϕ = infk > 0 : Sϕ
(f ′
k
)≤ 1
define a seminorm for Bϕ, which, in this case, is known as Luxemburg’s seminorm, where
Sϕ(f) = supz∈D
(1− |z|2)ϕ(|f(z)|)
We know that Bϕ is a Banach space with the norm ‖f‖Bϕ = |f(0)|+ ‖f‖ϕ. We also havethat the Bloch- Orlicz space is isometrically equal to µ-Bloch space, where
µ(z) =1
ϕ−1( 11−|z|2 )
, z ∈ D.
Thus for any f ∈ Bϕ, we have
‖f‖Bϕ = |f(0)|+ supz∈D
µ(z)|f ′(z)|.
It is well known that the differentiation operator D is defined by
(Df)(z) = f ′(z), f ∈ H(D).
Let u ∈ H(D), then the multiplication operator Mu is defined by
(Muf)(z) = u(z)f(z), f ∈ H(D).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
935 HAIYING LI et al 934-945
ON A PRODUCT-TYPE OPERATOR FROM MIXED-NORM SPACES TO BLOCH-ORLICZ SPACES 3
Let ψ be an analytic self-map of D. The composition operator Cψ is defined by
(Cψf)(z) = f(ψ(z)), f ∈ H(D).
Investigation of products of these and integral-type operators attracted a lot of atten-tion recently (see, e.g., [2]-[49], [51]-[55]). For example, in [3] and [17], the authorsinvestigated bounded superposition operators between Bloch-Orlicz and α-Bloch spacesand composition operators on Bloch-Orlicz type spaces. In [37] and [38], S. Stevic in-vestigated extended Cesaro operators between mixed-norm spaces and Bloch-type spacesand an integral-type operator from logarithmic Bloch-type spaces to mixed-norm spaceson the unit ball. In [36] and [41], S. Stevic investigated an integral-type operator fromlogarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces and weighted dif-ferentiation composition operators from mixed-norm spaces to weighted-type spaces. In[42] and [46], S. Stevic investigated an integral-type operator from Zygmund-type spacesto mixed-norm spaces on the unit ball and weighted differentiation composition operatorsfrom the mixed-norm space to the nth weighted-type space on the unit disk. S. Stevic in[34] gave the properties of products of integral-type operators and composition operatorsfrom the mixed norm space to Bloch-type spaces. In [47], S. Stevic investigated weightedradial operator from the mixed-norm space to the nth weighted-type space on the unit ball.In [54], X. Zhu studied extended Cesaro operators from mixed-norm spaces to Zygmundtype spaces.
Motivated, among others, by these papers, we will study here the boundedness andcompactness of the following operator, which is also a product-type one,
(DMuCψf)(z) = u′(z)f(ψ(z)) + u(z)ψ′(z)f ′(ψ(z)), f ∈ H(D),
from H(p, q, φ) to Bϕ.In what follows,
µ(z) =1
ϕ−1( 11−|z|2 )
,
and we use the letter C to denote a positive constant whose value may change at eachoccurrence.
2. THE BOUNDEDNESS AND COMPACTNESS OF DMuCφ : H(p, q, φ) → Bϕ
In this section, we will give our main results and proofs. In order to prove our mainresults, we need some auxiliary results. Our first lemma characterizes compactness in termsof sequential convergence. Since the proof is standard, it is omitted here (see, Proposition3.11 in [4]).
Lemma 1. Suppose u ∈ H(D), ψ is an analytic self-map of D, 0 < p, q < ∞ and φis normal. Then the operator DMuCψ : H(p, q, φ) → Bϕ is compact if and only if it isbounded and for each sequence fnn∈N which is bounded in H(p, q, φ) and convergesto zero uniformly on compact subsets of D as n →∞, we have ‖DMuCψfn‖Bϕ → 0 asn→∞.
The following lemma can be found in [36].
Lemma 2. Assume 0 < p, q <∞, ψ is normal and f ∈ H(p, q, φ). Then for every n ∈ N,there is a positive constant C independent of f such that
|f (n)(z)| ≤C‖f‖H(p,q,φ)
φ(|z|)(1− |z|2)1q +n
.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
936 HAIYING LI et al 934-945
4 HAIYING LI AND ZHITAO GUO
Theorem 3. Let u ∈ H(D), ψ be an analytic self-map of D, 0 < p, q < ∞ and φ benormal. Then DMuCψ : H(p, q, φ) → Bϕ is bounded if and only if
k1 = supz∈D
µ(z)|u′′(z)|φ(|ψ(z)|)(1− |ψ(z)|2)
1q
<∞, (1)
k2 = supz∈D
µ(z)|2u′(z)ψ′(z) + u(z)ψ′′(z)|φ(|ψ(z)|)(1− |ψ(z)|2)
1q +1
<∞, (2)
k3 = supz∈D
µ(z)|u(z)||ψ′(z)|2
φ(|ψ(z)|)(1− |ψ(z)|2)1q +2
<∞. (3)
Proof. Assume that (1), (2) and (3) hold. By Lemma 2, then we get
|f(ψ(z))| ≤C1‖f‖H(p,q,φ)
φ(|ψ(z)|)(1− |ψ(z)|2)1q
,
|f ′(ψ(z))| ≤C2‖f‖H(p,q,φ)
φ(|ψ(z)|)(1− |ψ(z)|2)1q +1
,
|f ′′(ψ(z))| ≤C3‖f‖H(p,q,φ)
φ(|ψ(z)|)(1− |ψ(z)|2)1q +2
.
Then for each f ∈ H(p, q, φ) \ 0, we have:
Sϕ
((DMuCψf)′(z)C‖f‖H(p,q,φ)
)
≤ supz∈D
(1− |z|2)ϕ[(
k1φ(|ψ(z)|)(1− |ψ(z)|2)1q |f(ψ(z))|
Cµ(z)‖f‖H(p,q,φ)
)+
(k2φ(|ψ(z)|)(1− |ψ(z)|2)
1q +1|f ′(ψ(z))|
Cµ(z)‖f‖H(p,q,φ)
)+
(k3φ(|ψ(z)|)(1− |ψ(z)|2)
1q +2|f ′′(ψ(z))|
Cµ(z)‖f‖H(p,q,φ)
)]≤ sup
z∈D(1− |z|2)ϕ
[k1C1 + k2C2 + k3C3
Cµ(z)
]≤ sup
z∈D(1− |z|2)ϕ
(ϕ−1
(1
1− |z|2
))= 1
where C is a constant such that C ≥ k1C1 + k2C2 + k3C3. Now, we can conclude thatthere exists a constant C such that ‖DMuCψf‖Bϕ ≤ C‖f‖H(p,q,φ) for all f ∈ H(p, q, φ),so the product-type operator DMuCψ : H(p, q, φ) → Bϕ is bounded.
Conversely, suppose that DMuCψ : H(p, q, φ) → Bϕ is bounded, i.e., there existsC > 0 such that ‖DMuCψf‖Bϕ ≤ C‖f‖H(p,q,φ) for all f ∈ H(p, q, φ). Taking thefunction f(z) = 1 ∈ H(p, q, φ), and ‖f‖H(p,q,φ) ≤ C, then
Sϕ
((DMuCψf)′(z)
C
)= Sϕ
(u′′(z)C
)= sup
z∈D(1− |z|2)ϕ
(|u′′(z)|C
)≤ 1.
It follows thatsupz∈D
µ(z)|u′′(z)| <∞. (4)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
937 HAIYING LI et al 934-945
ON A PRODUCT-TYPE OPERATOR FROM MIXED-NORM SPACES TO BLOCH-ORLICZ SPACES 5
Taking the function f(z) = z ∈ H(p, q, φ), and ‖f‖H(p,q,φ) ≤ C, then
Sϕ
((DMuCψf)′(z)
C
)= sup
z∈D(1− |z|2)ϕ
(|u′′(z)ψ(z) + (2u′(z)ψ′(z) + u(z)ψ′′(z))|
C
)≤ 1.
Hencesupz∈D
µ(z)|u′′(z)ψ(z) + 2u′(z)ψ′(z) + u(z)ψ′′(z)| <∞.
By (4) and the boundedness of ψ(z), we can see that
supz∈D
µ(z)|2u′(z)ψ′(z) + u(z)ψ′′(z)| <∞. (5)
Taking the function f(z) = z2
2 ∈ H(p, q, φ), similarly, we can get
supz∈D
µ(z)|u(z)||ψ′(z)|2 <∞. (6)
For a fixed ω ∈ D, set
fψ(ω)(z) =A(1− |ψ(ω)|2)t+1
φ(|ψ(ω)|)(1− ψ(ω)z)1q +t+1
+B(1− |ψ(ω)|2)t+2
φ(|ψ(ω)|)(1− ψ(ω)z)1q +t+2
+(1− |ψ(ω)|2)t+3
φ(|ψ(ω)|)(1− ψ(ω)z)1q +t+3
, (7)
where the constant t is from the definition of the normality of the function φ.Then supω∈D ‖fψ(ω)‖H(p,q,φ) <∞, and we have
fψ(ω)(ψ(ω)) =A+B + 1
φ(|ψ(ω)|)(1− |ψ(ω)|2)1q
,
f ′ψ(ω)(ψ(ω)) =(AM1 +BM2 +M3)ψ(ω)
φ(|ψ(ω)|)(1− |ψ(ω)|2)1q +1
,
f ′′ψ(ω)(ψ(ω)) =(AM1M2 +BM2M3 +M3M4)ψ(ω)
2
φ(|ψ(ω)|)(1− |ψ(ω)|2)1q +2
.
where Mi = 1q + t+ i, i = 1, 2, 3, 4.
To prove (1), we choose the corresponding function in (7) with
A =M3
M1, B = −2M3
M2,
and denote it by fψ(ω), then we have
fψ(ω)(ψ(ω)) =P
φ(|ψ(ω)|)(1− |ψ(ω)|2)1q
, f ′ψ(ω)(ψ(ω)) = f ′′ψ(ω)(ψ(ω)) = 0, (8)
where P = M3M1
− 2M3M2
+ 1.By the boundedness of DMuCψ : H(p, q, φ) → Bϕ, we have ‖DMuCψfψ(ω)‖Bϕ ≤
C, then
1 ≥ Sϕ
((DMuCψfψ(ω))′(z)
C
)≥ sup
w∈D(1− |ω|2)ϕ
(P |u′′(ω)|
Cφ(|ψ(ω)|)(1− |ψ(ω)|2)1q
),
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
938 HAIYING LI et al 934-945
6 HAIYING LI AND ZHITAO GUO
from which we can get (1). To prove (2), we choose the corresponding function in (7) with
A =−2M2 −M1M2 +M3M4
2M2, B =
M1M2 −M3M4
2M2,
and denote it by gψ(ω), then we have
g′ψ(ω)(ψ(ω)) =Eψ(ω)
φ(|ψ(ω)|)(1− |ψ(ω)|2)1q +1
, gψ(ω)(ψ(ω)) = g′′ψ(ω)(ψ(ω)) = 0, (9)
where
E =−2M1M2 −M2
1M2 +M1M3M4
2M2+M1M2 −M3M4
2+M3.
By the boundedness of DMuCψ : H(p, q, φ) → Bϕ, we have ‖DMuCψgψ(ω)‖Bϕ ≤C, then
1 ≥ Sϕ
((DMuCψgψ(ω))′(z)
C
)≥ sup
12<|ψ(ω)|<1
(1− |ω|2)ϕ( |(DMuCψgψ(ω))′(ω)|
C
)= sup
12<|ψ(ω)|<1
(1− |ω|2)ϕ(E|ψ(ω)||2u′(ω)ψ′(ω) + u(ω)ψ′′(ω)|
Cφ(|ψ(ω)|)(1− |ψ(ω)|2)1q +1
).
It follows that
sup12<|ψ(ω)|<1
µ(ω)|2u′(ω)ψ′(ω) + u(ω)ψ′′(ω)|φ(|ψ(ω)|)(1− |ψ(ω)|2)
1q +1
≤ 2 sup12<|ψ(ω)|<1
µ(ω)|ψ(ω)||2u′(ω)ψ′(ω) + u(ω)ψ′′(ω)|φ(|ψ(ω)|)(1− |ψ(ω)|2)
1q +1
<∞. (10)
Since φ is normal, and using (5), we have
sup|ψ(ω)|≤ 1
2
µ(ω)|2u′(ω)ψ′(ω) + u(ω)ψ′′(ω)|φ(|ψ(ω)|)(1− |ψ(ω)|2)
1q +1
≤ C sup|ψ(ω)|≤ 1
2
µ(ω)|2u′(ω)ψ′(ω) + u(ω)ψ′′(ω)| <∞. (11)
From (10) and (11), we can get (2). To prove (3), we choose the corresponding functionin (7) with
A = 1, B = −2,
and denote it by hψ(ω), then we have
hψ(ω)(ψ(ω)) = h′ψ(ω)(ψ(ω)) = 0, h′′ψ(ω)(ψ(ω)) =Fψ(ω)
2
φ(|ψ(ω)|)(1− |ψ(ω)|2)1q +2
, (12)
where F = M1M2 − 2M2M3 +M3M4.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
939 HAIYING LI et al 934-945
ON A PRODUCT-TYPE OPERATOR FROM MIXED-NORM SPACES TO BLOCH-ORLICZ SPACES 7
By the boundedness of DMuCψ : H(p, q, φ) → Bϕ, we have ‖DMuCψhψ(ω)‖Bϕ ≤C, then
1 ≥ Sϕ
((DMuCψhψ(ω))′(z)
C
)≥ sup
12<|ψ(ω)|<1
(1− |ω|2)ϕ( |(DMuCψhψ(ω))′(ω)|
C
)= sup
12<|ψ(ω)|<1
(1− |ω|2)ϕ(
F |ψ(ω)|2|u(ω)||ψ′(ω)|2
Cφ(|ψ(ω)|)(1− |ψ(ω)|2)1q +2
).
It follows that
sup12<|ψ(ω)|<1
µ(ω)|u(ω)||ψ′(ω)|2
φ(|ψ(ω)|)(1− |ψ(ω)|2)1q +2
≤ 4 sup12<|ψ(ω)|<1
µ(ω)|ψ(ω)|2|u(ω)||ψ′(ω)|2
φ(|ψ(ω)|)(1− |ψ(ω)|2)1q +2
<∞. (13)
Since φ is normal, and using (6), we have
sup|ψ(ω)|≤ 1
2
µ(ω)|u(ω)||ψ′(ω)|2
φ(|ψ(ω)|)(1− |ψ(ω)|2)1q +2
≤ C sup|ψ(ω)|≤ 1
2
µ(ω)|u(ω)||ψ′(ω)|2 <∞. (14)
From (13) and (14), we can get (3), finishing the proof of the theorem.
Theorem 4. Let u ∈ H(D), ψ be an analytic self-map of D, 0 < p, q <∞ and φ be nor-mal. Then DMuCψ : H(p, q, φ) → Bϕ is compact if and only if DMuCψ : H(p, q, φ) →Bϕ is bounded and
lim|ψ(z)|→1
µ(z)|u′′(z)|φ(|ψ(z)|)(1− |ψ(z)|2)
1q
= 0, (15)
lim|ψ(z)|→1
µ(z)|2u′(z)ψ′(z) + u(z)ψ′′(z)|φ(|ψ(z)|)(1− |ψ(z)|2)
1q +1
= 0, (16)
lim|ψ(z)|→1
µ(z)|u(z)||ψ′(z)|2
φ(|ψ(z)|)(1− |ψ(z)|2)1q +2
= 0. (17)
Proof. Suppose that DMuCψ : H(p, q, φ) → Bϕ is compact. It is clear that DMuCψ :H(p, q, φ) → Bϕ is bounded. Let znn∈N be a sequence in D such that |ψ(zn)| → 1 asn→∞. Set
fn(z) = fψ(zn)(z), gn(z) = gψ(zn)(z), hn(z) = hψ(zn)(z).
Then by the proof of Theorem 3,
supn∈N
‖fn‖H(p,q,φ) <∞, supn∈N
‖gn‖H(p,q,φ) <∞, supn∈N
‖hn‖H(p,q,φ) <∞.
Moreover, we can see that fn, gn, hn converges to 0 uniformly on compact subsets of D.Since DMuCψ : H(p, q, φ) → Bϕ is compact, by Lemma 1, we get
limn→∞
‖DMuCψfn‖Bϕ = limn→∞
‖DMuCψgn‖Bϕ = limn→∞
‖DMuCψhn‖Bϕ = 0.
By (8) we have
fn(ψ(zn)) =P
φ(|ψ(zn)|)(1− |ψ(zn)|2)1q
, f ′n(ψ(zn)) = f ′′n (ψ(zn)) = 0,
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
940 HAIYING LI et al 934-945
8 HAIYING LI AND ZHITAO GUO
Then
1 ≥ Sϕ
((DMuCψfn)′(zn)‖DMuCψfn‖Bϕ
)≥ (1− |zn|2)ϕ
(P |u′′(zn)|
φ(|ψ(zn)|)(1− |ψ(zn)|2)1q ‖DMuCψfn‖Bϕ
).
It follows that
µ(zn)∣∣u′′(zn)∣∣
φ(|ψ(zn)|)(1− |ψ(zn)|2)1q
≤ C‖DMuCψfn‖Bϕ .
Therefore
lim|ψ(zn)|→1
µ(zn)|u′′(zn)|φ(|ψ(zn)|)(1− |ψ(zn)|2)
1q
= limn→∞
µ(zn)|u′′(zn)|φ(|ψ(zn)|)(1− |ψ(zn)|2)
1q
= 0. (18)
So (15) follows. By (9) we have
g′n(ψ(zn)) =E · ψ(zn)
φ(|ψ(zn)|)(1− |ψ(zn)|2)1q +1
, gn(ψ(zn)) = g′′n(ψ(zn)) = 0,
Then
1 ≥ Sϕ
((DMuCψgn)′(zn)‖DMuCψgn‖Bϕ
)≥ (1− |zn|2)ϕ
(E|ψ(zn)||2u′(zn)ψ′(zn) + u(zn)ψ′′(zn)|
φ(|ψ(zn)|)(1− |ψ(zn)|2)1q +1‖DMuCψgn‖Bϕ
).
It follows that
µ(zn)|ψ(zn)||2u′(zn)ψ′(zn) + u(zn)ψ′′(zn)|φ(|ψ(zn)|)(1− |ψ(zn)|2)
1q +1
≤ C‖DMuCψgn‖Bϕ .
Therefore
lim|ψ(zn)|→1
µ(zn)|2u′(zn)ψ′(zn) + u(zn)ψ′′(zn)|φ(|ψ(zn)|)(1− |ψ(zn)|2)
1q +1
= limn→∞
µ(zn)|ψ(zn)||2u′(zn)ψ′(zn) + u(zn)ψ′′(zn)|φ(|ψ(zn)|)(1− |ψ(zn)|2)
1q +1
= 0. (19)
So (16) follows. By (12), we have
hn(ψ(zn)) = h′n(ψ(zn)) = 0, h′′n(ψ(zn)) =F · ψ(zn)
2
φ(|ψ(zn)|)(1− |ψ(zn)|2)1q +2
.
Then
1 ≥ Sϕ
((DMuCψhn)′(zn)‖DMuCψhn‖Bϕ
)≥ (1− |zn|2)ϕ
(F |ψ(zn)|2|u(zn)||ψ′(zn)|2
φ(|ψ(zn)|)(1− |ψ(zn)|2)1q +2‖DMuCψhn‖Bϕ
).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
941 HAIYING LI et al 934-945
ON A PRODUCT-TYPE OPERATOR FROM MIXED-NORM SPACES TO BLOCH-ORLICZ SPACES 9
It follows that
µ(zn)|ψ(zn)|2|u(zn)||ψ′(zn)|2
φ(|ψ(zn)|)(1− |ψ(zn)|2)1q +2
≤ C‖DMuCψhn‖Bϕ .
Therefore
lim|ψ(zn)|→1
µ(zn)|u(zn)||ψ′(zn)|2
φ(|ψ(zn)|)(1− |ψ(zn)|2)1q +2
= limn→∞
µ(zn)|ψ(zn)|2|u(zn)||ψ′(zn)|2
φ(|ψ(zn)|)(1− |ψ(zn)|2)1q +2
= 0.
So (17) follows.Conversely, suppose DMuCψ : H(p, q, φ) → Bϕ is bounded and (15), (16), (17) hold.
Then (4), (5), (6) hold by Theorem 3 and for every ε > 0, there is a δ ∈ (0, 1) such that
µ(z)|u′′(z)|φ(|ψ(z)|)(1− |ψ(z)|2)
1q
< ε, (20)
µ(z)|2u′(z)ψ′(z) + u(z)ψ′′(z)|φ(|ψ(z)|)(1− |ψ(z)|2)
1q +1
< ε, (21)
µ(z)|u(z)||ψ′(z)|2
φ(|ψ(z)|)(1− |ψ(z)|2)1q +2
< ε. (22)
whenever δ < |ψ(z)| < 1.Assume that tnn∈N is a sequence in H(p, q, φ) such that supn∈N ‖tn‖H(p,q,φ) ≤ L,
and tn converges to 0 uniformly on compact subsets of D as n → ∞. Let K = z ∈D : |ψ(z)| ≤ δ. Then by Lemma 2, (4), (5), (6), (21), (22) and (23), we have
supz∈D
µ(z)|(DMuCψtn)′(z)|
≤ supz∈D
µ(z)|u′′(z)||tn(ψ(z))|+ supz∈D
µ(z)|2u′(z)ψ′(z) + u(z)ψ′′(z)||t′n(ψ(z))|
+supz∈D
µ(z)|u(z)||ψ′(z)|2|t′′n(ψ(z))|
≤ supz∈K
µ(z)|u′′(z)||tn(ψ(z))|+ C1 supz∈D\K
µ(z)|u′′(z)|‖tn‖H(p,q,φ)
φ(|ψ(z)|)(1− |ψ(z)|2)1q
+ supz∈K
µ(z)|2u′(z)ψ′(z) + u(z)ψ′′(z)||t′n(ψ(z))|
+C2 supz∈D\K
µ(z)|2u′(z)ψ′(z) + u(z)ψ′′(z)|‖tn‖H(p,q,φ)
φ(|ψ(z)|)(1− |ψ(z)|2)1q +1
+ supz∈K
µ(z)|u(z)||ψ′(z)|2|t′′n(ψ(z))|+ C3 supz∈D\K
µ(z)|u(z)||ψ′(z)|2‖tn‖H(p,q,φ)
φ(|ψ(z)|)(1− |ψ(z)|2)1q +2
≤ C
(sup|ω|≤δ
|tn(ω)|+ sup|ω|≤δ
|t′n(ω)|+ sup|ω|≤δ
|t′′n(ω)|)
+ 3Lε.
So we obtain
‖DMuCψtn‖Bϕ = |u′(0)tn(ψ(0)) + u(0)ψ′(0)t′n(ψ(0))|+ supz∈D
µ(z)|(DMuCψtn)′(z)|
≤ |u′(0)||tn(ψ(0))|+ |u(0)||ψ′(0)||t′n(ψ(0))|
+C(
sup|ω|≤δ
|tn(ω)|+ sup|ω|≤δ
|t′n(ω)|+ sup|ω|≤δ
|t′′n(ω)|)
+ 3Lε. (23)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
942 HAIYING LI et al 934-945
10 HAIYING LI AND ZHITAO GUO
Since tn converges to 0 uniformly on compact subsets of D as n → ∞, Cauchy’s estima-tion gives that t′n, t′′n also do as n→∞. In particular, since ω : |ω| ≤ δ and ψ(0) arecompact it follows that
limn→∞
|u′(0)||tn(ψ(0))|+ |u(0)||ψ′(0)||t′n(ψ(0))| = 0,
limn→∞
sup|ω|≤δ
|tn(ω)| = limn→∞
sup|ω|≤δ
|t′n(ω)| = limn→∞
sup|ω|≤δ
|t′′n(ω)| = 0.
Hence, letting n→∞ in (24), we get
limn→∞
‖DMuCψtn‖Bϕ = 0.
Employing Lemma 1 the implication follows.
Acknowledgement. This work is supported in part by the National Natural ScienceFoundation of China (Nos.11201127;11271112) and IRTSTHN (14IRTSTHN023).
REFERENCES
[1] K. Avetisyan and S. Stevic. Holomorphic functions on the mixed norm spaces on the polydiscII, J. Comput. Anal. Appl. 11 (2) (2009), 239-251.
[2] K. Avetisyan and S. Stevic. Extended Cesaro operators between different Hardy spaces, Appl.Math. Comput. 207 (2009), 346-350.
[3] R. E. Castillo, J. C. Ramos Fernandez and M.Salazar. Bounded superposition operators betweenBloch-Orlicz and α-Bloch spaces. Appl. Math. Comput. 218 (7) (2011), 3441-3450.
[4] C. C. Cowen and B. D. MacCluer. Composition operators on spaces of analytic functions, Stud-ies in Advanced Mathematics, CRC, Boca Raton, 1995.
[5] S. G. Krantz and S. Stevic. On the iterated logarithmic Bloch space on the unit ball. NonlinearAnal. 71 (2009), 1772-1795.
[6] S. Li. On an integral-type operator from the Bloch space into the QK(p, q) space. Filomat, 26(2) (2012) 331-339.
[7] S. Li and S. Stevic. Integral type operators from mixed-norm spaces to α-Bloch spaces, IntegralTransforms Spec. Funct. 18 (7) (2007), 485-493.
[8] S. Li and S. Stevic. Weighted composition operators from Bergman-type spaces into Blochspaces, Proc. Indian Acad. Sci. Math. Sci. 117 (3) (2007), 371-385.
[9] S. Li and S. Stevic. Weighted composition operators from H∞ to the Bloch space on the poly-disc, Abstr. Appl. Anal. Vol. 2007, Article ID 48478, (2007), 12 pp.
[10] S. Li and S. Stevic. Composition followed by differentiation from mixed-norm spaces to α-Bloch spaces, Sb. Math. 199 (12) (2008), 1847-1857.
[11] S. Li and S. Stevic. Generalized composition operators on Zygmund spaces and Bloch typespaces, J. Math. Anal. Appl. 338 (2) (2008), 1282-1295.
[12] S. Li and S. Stevic. Riemann-Stieltjes operators between different weighted Bergman spaces,Bull. Belg. Math. Soc. Simon Stevin 15 (4) (2008), 677-686.
[13] S. Li and S. Stevic. Weighted composition operators from Zygmund spaces into Bloch spaces.Appl. Math. Comput. 206 (2) (2008), 825-831.
[14] S. Li and S. Stevic. Integral-type operators from Bloch-type spaces to Zygmund-type spaces.Appl. Math. Comput. 215 (2) (2009), 464-473.
[15] S. Li and S. Stevic. Composition followed by differentiation between H∞ and α-Bloch spaces,Houston J. Math. 35 (1) (2009), 327-340.
[16] K. Madigan and A. Matheson. Compact composition operators on the Bloch space, Trans.Amer. Math. Soc. 347(7) (1995), 2679-2687.
[17] J. C. Ramos Fernandez. Composition operators on Bloch-Orlicz type spaces, Applied Mathe-matics and Computation, 217 (7) (2010), 3392-3402.
[18] A. K. Sharma. Products of composition multiplication and differentiation between Bergmanand Bloch type spaces. Turk. J. Math. 35(2011), 275-291.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
943 HAIYING LI et al 934-945
ON A PRODUCT-TYPE OPERATOR FROM MIXED-NORM SPACES TO BLOCH-ORLICZ SPACES 11
[19] A. Shields and D. Williams. Bounded projections, duality, and multipliers in spaces of analyticfunctions, Trans. Amer. Math. Soc., 162 (1971), 287-302.
[20] S. Stevic. Boundedness and compactness of an integral operator on mixed norm spaces on thepolydisc, Siberian Math. J. 48 (3) (2007), 559-569.
[21] S. Stevic. Essential norms of weighted composition operators from the α-Bloch space to aweighted-type space on the unit ball, Abstr. Appl. Anal. Vol. 2008, Article ID 279691, (2008),11 pp.
[22] S. Stevic. Generalized composition operators between mixed norm space and some weightedspaces, Numer. Funct. Anal. Optimization 29 (7-8) (2008), 959-978.
[23] S. Stevic. Generalized composition operators from logarithmic Bloch spaces to mixed-normspaces, Util. Math. 77 (2008), 167-172.
[24] S. Stevic. Holomorphic functions on the mixed norm spaces on the polydisc, J. Korean Math.Soc. 45 (1) (2008), 63-78.
[25] S. Stevic. Norms of some operators from Bergman spaces to weighted and Bloch-type space,Util. Math. 76 (2008), 59-64.
[26] S. Stevic, On a new operator from H∞ to the Bloch-type space on the unit ball, Util. Math. 77(2008), 257-263.
[27] S. Stevic. On a new integral-type operator from the weighted Bergman space to the Bloch-typespace on the unit ball, Discrete Dyn. Nat. Soc. Vol. 2008, Art. ID 154263, (2008), 14 pp.
[28] S. Stevic. Boundedness and compactness of an integral operator between H∞ and a mixednorm space on the polydisk, Siberian Math. J. 50 (3) (2009), 495-497.
[29] S. Stevic. Integral-type operators from a mixed norm space to a Bloch-type space on the unitball, Siberian Math. J. 50 (6) (2009), 1098-1105.
[30] S. Stevic. Norm and essential norm of composition followed by differentiation from α-Blochspaces to H∞µ , Appl. Math. Comput. 207 (2009), 225-229.
[31] S. Stevic. On an integral operator from the Zygmund space to the Bloch-type space on the unitball, Glasg. J. Math. 51 (2009), 275-287.
[32] S. Stevic. On a new integral-type operator from the Bloch space to Bloch-type spaces on theunit ball, J. Math. Anal. Appl. 354 (2009), 426-434.
[33] S. Stevic. Products of composition and differentiation operators on the weighted Bergmanspace, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), 623-635.
[34] S. Stevic. Products of integral-type operators and composition operators from the mixed normspace to Bloch-type spaces, Siberian Math. J. 50 (4) (2009), 726-736.
[35] S. Stevic. Weighted composition operators from weighted Bergman spaces to weighted-typespaces on the unit ball, Appl. Math. Comput. 212 (2009), 499-504.
[36] S. Stevic. Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. Appl. Math. Comput. 211 (1) (2009), 222-233.
[37] S. Stevic. Extended Cesaro operators between mixed-norm spaces and Bloch-type spaces in theunit ball. Houston J. Math. 36 (3) (2010), 843-858.
[38] S. Stevic. On an integral-type operator from logarithmic Bloch-type spaces to mixed-normspaces on the unit ball. Appl. Math. Comput. 215 (11) (2010), 3817-3823.
[39] S. Stevic. Characterizations of composition followed by differentiation between Bloch-typespaces. Appl. Math. Comput. 218 (8) (2011), 4312-4316.
[40] S. Stevic. Boundedness and compactness of an integral-type operator from Bloch-type spaceswith normal weights to F (p, q, s) space. Appl. Math. Comput. 218 (9) (2012), 5414-5421.
[41] S. Stevic. On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces toBloch-type spaces. Nonlinear Anal. 71 (12) (2009), 6323-6342.
[42] S. Stevic. On an integral-type operator from Zygmund-type spaces to mixed-norm spaces onthe unit ball. Abstr. Appl. Anal. 2010, Art. ID 198608, 7 pp.
[43] S. Stevic. On operator P gϕ from the logarithmic Bloch-type space to the mixed-norm space on
unit ball, Appl. Math. Comput. 215 (2010), 4248-4255.[44] S. Stevic. Norm estimates of weighted composition operators between Bloch-type spaces. Ars
Combin. 93 (2009), 161-164.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
944 HAIYING LI et al 934-945
12 HAIYING LI AND ZHITAO GUO
[45] S. Stevic. Weighted differentiation composition operators from H∞ and Bloch spaces to nthweigthed-type spaces on the unit disk, Appl. Math. Comput. 216 (2010), 3634-3641.
[46] S. Stevic. Weighted differentiation composition operators from the mixed-norm space to thenth weighted-type space on the unit disk. Abstr. Appl. Anal. 2010, Art. ID 246287, 15 pp.
[47] S. Stevic. Weighted radial operator from the mixed-norm space to the nth weighted-type spaceon the unit ball. Appl. Math. Comput. 218 (18) (2012), 9241-9247.
[48] S. Stevic and A. K. Sharma. Integral-type operators between weighted Bergman spaces on theunit disk. J. Comput. Anal. Appl. 14 (7) (2012), 1339-1344.
[49] S. Stevic and S. Ueki. Integral-type operators acting between weighted-type spaces on the unitball, Appl. Math. Comput. 215 (2009), 2464-2471.
[50] K. Zhu. Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23 (3) (1993), 1143-1177.
[51] X. Zhu. Generalized weighted composition operators from Bloch-type spaces to weightedBergman spaces, Indian J. Math. 49 (2) (2007), 139-149.
[52] X. Zhu. Multiplication followed by differentiation on Bloch-type spaces, Bull. Allahbad. Math.Soc. 23 (1) (2008), 25-39.
[53] X. Zhu. Volterra type operators from logarithmic Bloch spaces to Zygmund type space, Inter. J.Modern Math. 3 (3) (2008), 327-336.
[54] X. Zhu. Extended Cesaro operators from mixed norm spaces to Zygmund type spaces. TamsuiOxf. J. Math. Sci. 26 (4) (2010), 411-422.
[55] X. Zhu, Generalized weighted composition operators from Bers-type spaces into Bloch-typespaces Math. Inequal. Appl. 17 (1) (2014), 187-195.
HAIYING LI, HENAN ENGINEERING LABORATORY FOR BIG DATA STATISTICAL ANALYSIS AND OPTI-MAL CONTROL, SCHOOL OF MATHEMATICS AND INFORMATION SCIENCE, HENAN NORMAL UNIVERSITY,453007 XINXIANG, P.R.CHINA
E-mail address: [email protected]
ZHITAO GUO, HENAN ENGINEERING LABORATORY FOR BIG DATA STATISTICAL ANALYSIS AND OPTI-MAL CONTROL, SCHOOL OF MATHEMATICS AND INFORMATION SCIENCE, HENAN NORMAL UNIVERSITY,453007 XINXIANG, P.R.CHINA
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
945 HAIYING LI et al 934-945
A SHORT NOTE ON INTEGRAL INEQUALITY OF TYPE
HERMITE-HADAMARD THROUGH CONVEXITY
MUHAMMAD IQBAL, SHAHID QAISAR, AND MUHAMMAD MUDDASSAR*
Abstract. In this short note, a Riemann-Liouville fractional integral identity
including first order derivative of a given function is established. With the help
of this fractional-type integral identity, some new Hermite-Hadamard-type in-equality involving Riemann-Liouville fractional integrals for(m,h1, h2)−convex
function are considered. Our method considered here may be a stimulant forfurther investigations concerning Hermite-Hadamard-type inequalities involv-
ing fractional integrals.
1. Introduction and Defintions
Many inequalities have been established for convex functions but the most famousis the Hermite-Hadamarad inequality, due to its rich geometrical significance andapplications, which is stated as in [12]
Let f : I ⊂ R→ R be a convex function defined on the closed interval I of realnumbers and a, b ∈ I with a < b
f
(a+ b
2
)≤ 1
b− a
∫ b
a
f (t)dt ≤ f (a) + f (b)
2
Both the inequalities hold in reversed direction if f is concave. We recall somepreliminary concepts about convex functions:
Definition 1. [7]. A function f : [0,∞) → R is said to be s-convex function orf belongs to the class Ki
s if for all x, y ∈ [0,∞) and µ, ν ∈ [0, 1], the followinginequality holds
f (µx+ νy) ≤ µsf (x) + νsf (y)
for some fixed α ∈ (0, 1].
Note that, if µs + νs = 1, the above class of convex functions is called s-convexfunctions in first sense and represented by K1
s and if µ + ν = 1 the above class iscalled s-convex in second sense and represented by K2
s .
Definition 2. [11]. A function f : [0, b] → R is said to be (α,m)-convex, where(α,m) ∈ [0, 1]2 , if for every x, y ∈ [0, b] and for λ ∈ [0, 1], the following inequalityholds
f (λy +m (1− λ)x) ≤ λαf (y) +m (1− λα) f (x)
where (α,m) ∈ [0, 1]2 and for some fixed m ∈ (0, 1].
Date: October 20, 2015.
2010 Mathematics Subject Classification. 26D15; 26A33; 26A51.Key words and phrases. Hermite-Hadamard-type inequality; (m,h1, h2)−convex function;
Holder’s integral inequality; Riemann-Liouville fractional integrals.
corresponding Author *.
1
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
946 IQBAL et al 946-953
2 M. IQBAL, S. QAISAR, AND M. MUDDASSAR
Theorem 1. [3]. Let f : I ⊂ R→ R be a differentiable function on I (interior ofI) where a, b ∈ I with a < b. If |f ′| is convex on [a, b], then we have∣∣∣∣∣f(a) + f(b)
2− 1
b− a
∫ b
a
f(x)dx
∣∣∣∣∣ ≤ b− a8
[|f ′(a)|+ |f ′(b)|] .
Theorem 2. [8]. Let f : I ⊂ R → R be a differentiable function on Io (interiorof I) where a, b ∈ I with a < b. If the mapping |f ′|q is convex on [a, b], for someq ≥ 1, then the following inequality holds:∣∣∣∣∣f(a) + f(b)
2− 1
b− a
∫ b
a
f(x)dx
∣∣∣∣∣ ≤ b− a4
[|f ′(a)|q + |f ′(b)|q
2
] 1q
Theorem 3. [13].Let f : I ⊂ [0,∞)→ R be differentiable mapping on Io a, b ∈ Iowith a < b, and If |f ′|q is quasi-convex on [a, b], p > 1. Then the following inequalityholds: ∣∣∣∣∣f(a) + f(b)
2− 1
b− a
∫ b
a
f(x)dx
∣∣∣∣∣ ≤ (b− a)
16
(4
1 + p
) 1p
[|f ′(a)|
pp−1 + 3 |f ′(b)|
pp−1
]1− 1p
+[3 |f ′(a)|
pp−1 + |f ′(b)|
pp−1
]1− pp−1
Theorem 4. [9]. Let f : I ⊂ [0,∞) → R be a differentiable function on I0 wherea, b ∈ Io with a < b such that f ′ ∈ L [a, b] , where a, b ∈ I with a < b. If the mapping|f ′′|q is s-convex on [a, b] for some fixed s ∈ (0, 1] and q ≥ 1, then the followinginequality holds:∣∣∣∣∣f(a) + f(b)
2− 1
b− a
∫ b
a
f(x)dx
∣∣∣∣∣ ≤ b− a2
1q
[s+
(12
)s(s+ 1) (s+ 2)
] 1q [|f ′(a)|q + |f ′(b)|q
] 1q
Theorem 5. [4]. Suppose that f : [0,∞) → [0,∞) is a convex function in thesecond sensewhere α ∈ (0, 1) and let a, b ∈ [0,∞) , a < b. if f ∈ L1 ([a, b]) , thenthe following inequality holds
2α−1f
(a+ b
2
)≤ 1
b− a
b∫a
f (x)dx ≤ f (a) + f (b)
α+ 1.
Fraction calculus [2, 6, 1, 5] was introduced at the end of the nineteenth centuryby Riemann and Liouville the subject of which has become a rapidly growing areaand has found applications in diverse fields ranging from physical sciences andengineering to biological sciences and economics. We recall some definitions andpreliminary facts of fractional calculus theory which will be used in this paper.
Definition 3. [6].Let f ∈ L1 [a, b] . The Riemann-Liouville fractional integralsJαα+f and Jαb−f of order α > 0 with α ≥ 0 are defined by
Jαα+f (x) =1
Γ (α)
∫ x
a
(x− t)α−1f (t) dt, (a < x) ,
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
947 IQBAL et al 946-953
HERMITE-HADAMARD’S TYPE FRACTIONAL INTEGRAL INEQUALITIES 3
and
Jαb−f (x) =1
Γ (α)
∫ b
x
(t− x)α−1
f (t) dt, (b > x) ,
respectively.
Here Γ (α) =∞∫0
e−uuα−1du. Here is J0a+f (x) = J0
b−f (x) = f (x) .
In case of α = 1, the fractional integral reduces to the classical integral. Theaim of this paper is to establish Hermite-Hadamard type inequalities based on(m,h1, h2)− convexity. Using these results we obtained new inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.
2. Main Results
Before proceeding to our main results, we present some necessary definition andlemma which are used further in this paper.
Definition 4. Let f : I ⊆ R0 → R, h1, h2 : [0, 1] → R0, and m ∈ (0, 1] , then fis said to be (m,h1, h2)− convex. if f is non–negative and the following inequality
f (λx+m (1− λ) y) ≤ h1 (λ) f (x) +mh2 (1− λ) f (y) ,
holds for all x, y ∈ I and λ ∈ [0, 1].
If the above inequality is reversed then f is said to be (m,h1, h2)− concave.
Mα (a, b) = 14
[f (a) + 1
2
(f(
3a+b4
)+ f
(a+3b
4
))+ f (b)
]−Γ(α+1)4α−1
(b−a)α
[Jαα+f
(3a+b
4
)+ Jα3a+b
4
+f(a+3b
4
)+ Jαa+3b
4
+f (b)
].
Specially, when α = 1, we have
M1 (a, b) =1
4
[f (a) +
1
2
(f
(3a+ b
4
)+ f
(a+ 3b
4
))+ f (b)
]− 1
b− a
∫ b
a
f (t) dt.
Lemma 1. Suppose f : [a, b] → R is a differentiable mapping on (a, b) . If f ′ ∈L1 ([a, b]), then we have the following identity
Mα (a, b) =b− a
16×
∫ 1
0(0− λα)f ′
(λa+ (1− λ) 3a+b
4
)dλ
+∫ 1
0
(12 − λ
α)f ′(λ 3a+b
4 + (1− λ) a+3b4
)dλ
+∫ 1
0(1− λα)f ′
(λa+3b
4 + (1− λ) b)dλ
Proof. By integrating, and by making use of the substitution u = λa+(1− λ) 3a+b
4we have
b−a16
∫ 1
0(−λα)f ′
(λa+ (1− λ) 3a+b
4
)dλ
= 14
[f (a)− α
∫ 1
0(−λα)f
(λa+ (1− λ) 3a+b
4
)λα−1dλ
]= 1
4f (a)− α4α−1
(b−a)α∫ 3a+b
4
a
(3a+b
4 − u)α−1
f (u) du
= 14f (a)− Γ(α+1)4α−1
(b−a)α Jαα+f(
3a+b4
)b−a16
∫ 1
0
(12 − λ
α)f ′(λ 3a+b
4 + (1− λ) a+3b4
)dλ
= 18
[(f(
3a+b4
)+ f
(a+3b
4
))]= b−a
16
∫ 1
0
(12 − λ
α)f ′(λ 3a+b
4 + (1− λ) a+3b4
)dλ
= 18
[(f(
3a+b4
)+ f
(a+3b
4
))]− Γ(α+1)4α−1
(b−a)α Jαα+f(a+3b
4
)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
948 IQBAL et al 946-953
4 M. IQBAL, S. QAISAR, AND M. MUDDASSAR∫ 1
0(1− λα)f ′
(λa+3b
4 + (1− λ) b)dλ
= 14f (b)− Γ(α+1)4α−1
(b−a)α Jαa+3b4
+f (b)
This proves as required.
Theorem 6. Suppose f : [a, b]→ R is a differentiable mapping on (a, b) with a < b.such thatf ′ ∈ L1 ([a, b])for 0 < a < b. If |f ′|q is (m,h1, h2)−convex on [a, b] forsome fixed q > 1 and h1, h2 ∈ L1 ([a, b]), then we have the following inequalityMα (a, b) ≤ b−a
16 ×(
q−1αq+q−1
)1−1/q
×(|f ′ (a)|q ‖h1‖1 +m
∣∣f ′ ( 3a+b4m
)∣∣q ‖h2‖1)1/q
+ 12
(α2(α−1)/α−α+1
α+1
)1−1/q
×(∣∣f ′ ( 3a+b
4
)∣∣q ‖h1‖1 +m∣∣f ′ (a+3b
4m
)∣∣q ‖h2‖1)1/q
+(
1αB
(2q−1q−1 ,
1α
))1−1/q
×(∣∣f ′ (a+3b
4
)∣∣q ‖h1‖1 +m∣∣f ′ ( bm)∣∣q ‖h2‖1
)1/q
Suppose ‖h1‖p =
(∫ 1
0hp1 (λ) dλ
)1/p
for p ≥ 1 with B (x, y) is the classical Beta
function which may be defined B (x, y) =∫ 1
0λx−1 (1− λ)
y−1, x > 0, y > 0. for
0 < a < b.
Proof. H older integral inequality and Lemma 1 together implies with (m,h1, h2)-convexity of |f ′|q
Mα (a, b) ≤ b−a16 ×
(∫ 1
0λαq/q−1dλ
)1−1/q
×(|f ′ (a)|q ‖h1‖1 +m
∣∣f ′ ( 3a+b4m
)∣∣q ‖h2‖1)1/q
+(12
)1/q (∫ 1
0
∣∣ 12−λ
α∣∣ dλ)1−1/q
×(∣∣f ′ (3a+b4
)∣∣q ‖h1‖1+m∣∣f ′ (a+3b
4m
)∣∣q ‖h2‖1)1/q
+(∫ 1
0(1−λα)q/q−1
dλ)1−1/q
×(∣∣f ′ (a+3b
4
)∣∣q ‖h1‖1+m∣∣f ′ ( bm)∣∣q ‖h2‖1
)1/q
Therefore,∫ 1
0λαq/q−1dλ = q−1
αq+q−1 ,∫ 1
0(1− λα)
q/q−1dλ = 1
αB(
2q−1q−1 ,
1α
),∫ 1
0
∣∣ 12 − λ
α∣∣ dλ = α2(α−1)/α−α+1
α+1
This completes the proof .
Corollary 1. In Theorem 6, if we choose h1 (λ) = h (λ) , h2 (λ) = h (1− λ) , thenwe have,
Mα (a, b) ≤ (b−a)‖h1‖1/q1
16 ×(
q−1αq+q−1
)1−1/q
×(|f ′ (a)|q +m
∣∣f ′ ( 3a+b4m
)∣∣q)1/q
+ 12
(α2(α−1)/α−α+1
α+1
)1−1/q
×(∣∣f ′ ( 3a+b
4
)∣∣q +m∣∣f ′ (a+3b
4m
)∣∣q)1/q
+(
1αB
(2q−1q−1 ,
1α
))1−1/q
×(∣∣f ′ (a+3b
4
)∣∣q +m∣∣f ′ ( bm)∣∣q)1/q
Furthermore if we choose m = 1, we have
Mα (a, b) ≤ (b−a)‖h1‖1/q1
16 ×(
q−1αq+q−1
)1−1/q
×(|f ′ (a)|q +
∣∣f ′ ( 3a+b4
)∣∣q)1/q
+ 12
(α2(α−1)/α−α+1
α+1
)1−1/q
×(∣∣f ′ ( 3a+b
4
)∣∣q +∣∣f ′ (a+3b
4
)∣∣q)1/q
+(
1αB
(2q−1q−1 ,
1α
))1−1/q
×(∣∣f ′ (a+3b
4
)∣∣q + |f ′ (b)|q)1/q
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
949 IQBAL et al 946-953
HERMITE-HADAMARD’S TYPE FRACTIONAL INTEGRAL INEQUALITIES 5
Corollary 2. Under the conditions of Corollary 1, if we chooseh1 (λ) = h (λ) =λs,m = 1, we have the
Mα (a, b) ≤ (b−a)16
(1s+1
)1/q
×(
q−1αq+q−1
)1−1/q
×(|f ′ (a)|q +
∣∣f ′ ( 3a+b4
)∣∣q)1/q
+ 12
(α2(α−1)/α−α+1
α+1
)1−1/q
×(∣∣f ′ ( 3a+b
4
)∣∣q +∣∣f ′ (a+3b
4
)∣∣q)1/q
+(
1αB
(2q−1q−1 ,
1α
))1−1/q
×(∣∣f ′ (a+3b
4
)∣∣q + |f ′ (b)|q)1/q
Specially, if we choose α = s = m = 1, we have the
Mα (a, b) ≤ (b− a)
16
(1
2
)1/q
×
(q−12q−1
)1−1/q
×(|f ′ (a)|q +
∣∣f ′ ( 3a+b4
)∣∣q)1/q
+(
12
)2−1/q ×(∣∣f ′ ( 3a+b
4
)∣∣q +∣∣f ′ (a+3b
4
)∣∣q)1/q
+(q−12q−1
)1−1/q
×(∣∣f ′ (a+3b
4
)∣∣q + |f ′ (b)|q)1/q
Corollary 3. Under the conditions of Theorem 6, if we choose h1 (λ) = λα1 , h2 (λ) =1− λα1 , we have the
Mα (a, b) ≤ (b−a)16
(1
α1+1
)1/q
×(
q−1αq+q−1
)1−1/q
×(|f ′ (a)|q +mα1
∣∣f ′ ( 3a+b4m
)∣∣q)1/q
+ 12
(α2(α−1)/α−α+1
α+1
)1−1/q
×(∣∣f ′ ( 3a+b
4
)∣∣q +m∣∣f ′ (a+3b
4m
)∣∣q)1/q
+(
1αB
(2q−1q−1 ,
1α
))1−1/q
×(∣∣f ′ (a+3b
4
)∣∣q +mα1
∣∣f ′ ( bm)∣∣q)1/q
Specially, if we choose m = 1, we have the,
Mα (a, b) ≤ (b−a)16
(1
α1+1
)1/q
×(
q−1αq+q−1
)1−1/q
×(|f ′ (a)|q + α1
∣∣f ′ ( 3a+b4
)∣∣q)1/q
+ 12
(α2(α−1)/α−α+1
α+1
)1−1/q
×(∣∣f ′ ( 3a+b
4
)∣∣q +∣∣f ′ (a+3b
4
)∣∣q)1/q
+(
1αB
(2q−1q−1 ,
1α
))1−1/q
×(∣∣f ′ (a+3b
4
)∣∣q + α1 |f ′ (b)|q)1/q
Theorem 7. Suppose f : [a, b]→ R is a differentiable mapping on (a, b) with a < b.such thatf ′ ∈ L1 ([a, b])for 0 < a < b. If |f ′|q is and (m,h1, h2)− convex on [a, b]for q ≥ 1, and h1, h2 ∈ L1 ([a, b]), then we have the following inequality
Mα (a, b) ≤ (b−a)16
(1
α+1
)1−1/q
×
[12 |f′ (a)|q
(1
2α+1 + ‖h1‖22)
+ m2
∣∣f ′ ( 3a+b4m
)∣∣q ( 12α+1 + ‖h2‖22
)]1/q+ 1
2
(α2(α−1)/α−α+1
α+1
)1−1/q
×(∣∣f ′ ( 3a+b
4
)∣∣q ‖h1‖1 +m∣∣f ′ (a+3b
4m
)∣∣q ‖h2‖1)1/q
+α1−1/q
12
∣∣f ′ (a+3b4
)∣∣q ( 2α2
(2α+1)(α+1) + ‖h1‖22)
+m2
∣∣f ′ ( bm)∣∣q ( 2α2
(2α+1)(α+1) + ‖h1‖22) 1/q
Where 1
p + 1q = 1.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
950 IQBAL et al 946-953
6 M. IQBAL, S. QAISAR, AND M. MUDDASSAR
Proof. Using Holder’s inequality and by Lemma 1, and (m,h1, h2)− convexity of|f ′|q , we get
Mα (a, b) ≤ (b−a)16 ×
(∫ 1
0λαdλ
)1−1/q[|f ′ (a)|q
∫ 1
0λαh1(λ) dλ+m
2
∣∣f ′ (3a+b4m
)∣∣q∫ 1
0λαh2(λ) dλ
]1/q+(12
)1/q (∫ 1
0
∣∣ 12−λ
α∣∣ dλ)1−1/q
×(∣∣f ′ (3a+b4
)∣∣q ∫ 1
0h1 (λ) dλ+m
∣∣f ′ (a+3b4m
)∣∣q∫ 1
0h2 (λ) dλ
)1/q
+(∫ 1
0(1− λα) dλ
)1−1/q[ ∣∣f ′ (a+3b
4
)∣∣q ∫ 1
0(1− λα)h1 (λ) dλ
+m∣∣f ′( bm)∣∣q ∫ 1
0(1−λα)h2 (λ) dλ
]1/q
Mα (a, b) ≤ (b−a)
16 ×
(∫ 1
0λαdλ
)1−1/q [|f ′(a)|q
∫ 1
0λ2α+h2
1(λ)2 dλ+m
∣∣f ′ (3a+b4m
)∣∣q∫ 1
0λ2α+h2
2(λ)2 dλ
]1/q+(12
)1/q(∫ 1
0
∣∣ 12−λ
α∣∣ dλ)1−1/q
×(∣∣f ′ (3a+b4
)∣∣q∫ 1
0h1 (λ) dλ+m
∣∣f ′ (a+3b4m
)∣∣q∫ 1
0h2 (λ) dλ
)1/q
+(∫ 1
0(1−λα) dλ
)1−1/q[ ∣∣f ′ (a+3b
4
)∣∣q∫ 1
0(1−λα)2+h2
1(λ)2 dλ
+m∣∣f ′ ( bm)∣∣q∫ 1
0(1−λα)2+h2
2(λ)2 dλ
]1/q
= (b−a)
16
(1
α+1
)1−1/q
×
[12 |f′ (a)|q
(1
2α+1 + ‖h1‖22)
+ m2
∣∣f ′ ( 3a+b4m
)∣∣q ( 12α+1 + ‖h2‖22
)]1/q+ 1
2
(α2(α−1)/α−α+1
α+1
)1−1/q
×(∣∣f ′ ( 3a+b
4
)∣∣q ‖h1‖1 +m∣∣f ′ (a+3b
4m
)∣∣q ‖h2‖1)1/q
+α1−1/q
12
∣∣f ′ (a+3b4
)∣∣q ( 2α2
(2α+1)(α+1) + ‖h1‖22)
+m2
∣∣f ′ ( bm)∣∣q ( 2α2
(2α+1)(α+1) + ‖h1‖22) 1/q
This completes the proof .
Corollary 4. In Theorem 7, if we choose h1 (λ) = λα1 , h2 (λ) = 1− λα1 , we have
Mα (a, b) ≤ (b−a)16
(1
α+1
)1−1/q
×
[12 |f′ (a)|q
(1
2α+1 + 12α1+1
)+ m
2
∣∣f ′ ( 3a+b4m
)∣∣q ( 12α+1 +
2α21
(2α1+1)(α1+1)
)]1/q+ 1
2
(α2(α−1)/α − α+ 1
)1−1/q ×(∣∣f ′ ( 3a+b
4
)∣∣q 1α1+1 +m
∣∣f ′ (a+3b4m
)∣∣q α1
α1+1
)1/q
+α1−1/q
12
∣∣f ′ (a+3b4
)∣∣q ( 2α2
(2α+1)(α+1) + 12α1+1
)+m
2
∣∣f ′ ( bm)∣∣q ( 2α2
(2α+1)(α+1) +2α2
1
(2α1+1)(α1+1)
) 1/q
If we choose h1 (λ) = h (λ) , h2 (λ) = h (1− λ) , m = 1 we have
Mα (a, b) ≤ (b−a)16 ×
(1
α+1
)1−1/q [12
(1
2α+1 + ‖h1‖22)
+(|f ′ (a)|q +
∣∣f ′ ( 3a+b4
)∣∣q)]1/q+ 1
2
(α2(α−1)/α − α+ 1
)1−1/q ×(‖h1‖1
(∣∣f ′ ( 3a+b4
)∣∣q +∣∣f ′ (a+3b
4
)∣∣q))1/q
+α1−1/q
12
(2α2
(2α+1)(α+1) + ‖h1‖22)
×(∣∣f ′ (a+3b
4
)∣∣q + |f ′ (b)|q) 1/q
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
951 IQBAL et al 946-953
HERMITE-HADAMARD’S TYPE FRACTIONAL INTEGRAL INEQUALITIES 7
If we choose h1 (λ) = h (λ) = λs, h2 (λ) = h (1− λ) , m = 1 we have the,
Mα (a, b) ≤ (b−a)16 ×
(1
α+1
)1−1/q [12
(1
2α+1 + 12s+1
)+(|f ′ (a)|q +
∣∣f ′ ( 3a+b4
)∣∣q)]1/q+ 1
2
(α2(α−1)/α − α+ 1
)1−1/q ×(
1s+1
(∣∣f ′ ( 3a+b4
)∣∣q +∣∣f ′ (a+3b
4
)∣∣q))1/q
+α1−1/q
12
(2α2
(2α+1)(α+1) + 12s+1
)×(∣∣f ′ (a+3b
4
)∣∣q + |f ′ (b)|q) 1/q
In Theorem 7, if we choose h1 (λ) = h (λ) = λs, h2 (λ) = h (1− λ) , α = s =
m = 1, we have
Mα (a, b) ≤ (b− a)
16×
(
12
)1−1/q[
13
(|f ′ (a)|q +
∣∣f ′ ( 3a+b4
)∣∣q)]1/q+ 1
2
(12
(∣∣f ′ ( 3a+b4
)∣∣q +∣∣f ′ (a+3b
4
)∣∣q))1/q
+[
13
(∣∣f ′ (a+3b4
)∣∣q + |f ′ (b)|q)]1/q
.
3. Acknowledgments
The authors are grateful to Dr S. M. Junaid Zaidi, Rector, COMSATS Instituteof Information Technology, Pakistan for providing excellent research facilities. Theauthor S. Qaisar was partially supported by the Higher Education Commission ofPakistan [grant number No. 21-52/SRGP/R&D/HEC /2014.
References
[1] Z. Dahmani, New inequalities in fractional integrals, International Journal of Nonlinear Sci-ence, 9(4) (2010), 493-497..
[2] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathemat-
ics, Springer, Berlin, 2010.[3] S. S. Dragomir, R. P. Agarwal. Two inequalities for differentiable mappings and applications
to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(5) (1998)
91-95.[4] S. S. Dragomir, S. Fitzpatrick. The Hadamard’s inequality for s-convex functions in the second
sense, Demonstratio Math. 32 (4) (1999), 687-696.
[5] E.K. Godunova and V. I. Levin, Neravenstva dlja funkeii sirokogo klassa, zascego vypuk-lye, monotonnye i nekotorye drugie vidy funkii, Vyscislitel. Mat. i. Fiz. Mezvuzov. Sb.
Nauc.Trudov, MGPI, Moskva, 1985, 138-142.
[6] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractionalorder, Springer Verlag, Wien (1997), 223-276.
[7] H. Hudzik, L. Maligrada. Some remarks on s-convex functions, Aequationes Math. 48(1994)100-111.
[8] U.S. Kirmaci. Inequalities for differentiable mappings and applications to special means of
real numbers and to midpoint formula, Appl. Math. Comp. 147(1)(2004), 137-146.
[9] U. S. Kirmaci, & Ozdemir, M. E. 2004a. On some inequalities for differentiable mappings and
applications to special means of real numbers and to midpoint formula, Applied Mathematics
and Computation 153: 361-368.[10] G. Maksa, Z. S. Pales, The equality case in some recent convexity inequalities, Opuscula
Math., 31(2011), no.2, 269-277.[11] V. G. Mihesan, A generalization of the convexity, Seminar on Functional Equations, Ap-
prox.and Convex,Cluj-Napoca (Romania) (1993).
[12] C. Niculescu, L. E. Persson, Convex functions and their application, Springer, Berlin Heidel-berg New York, (2004).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
952 IQBAL et al 946-953
8 M. IQBAL, S. QAISAR, AND M. MUDDASSAR
[13] C. E. M. Pearce, & J. Pecric 2000a. Inequalities for differentiable mappings with applicationto special means and quadrature formula, Applied Mathematics Letters 13: 51-55.
[14] S. Qaisar, C. He. S. Hussain. On New Inequalities of Hermite-Hadamard type for generalized
Convex Functions. Italian journal of pure and applied mathematics. In Press.[15] S. Qaisar, S. Hussain, Some results on Hermite-Hadamard type inequality through convex-
ity.Turkish J.Anal.Num.Theo Vol. 2(2), 53-59 (2014).
[16] S. Qaisar, C. He. S. Hussain, New integral inequalities through invexity with applications.International Journal of Analysis and Applications.(In press).
[17] S. Qaisar, He. C, S. Hussain, On New Inequalities Of Hermite-Hadamard Type ForFunctions Whose Third Derivative Absolute Values Are Quasi-Convex With Applica-
tions,JournalofEgyptianMathematicalSociety.doi.org/10.1016/j.joems.2013.05.01
[18] M.Z. Sarikaya, E. Set, H. Yaldiz, and N. Basak, Hermite–Hadamard’s inequalities for frac-tional integrals and related fractional inequalities, Math. Comput. Model, 57(2013), 2403-
2407.
[19] S. Varosanec, On h-convexity, J. Math. Anal. Appl., 326 (2007), 303-311.
E-mail address: [email protected]
Government College of Technology, Sahiwal, Pakistan
E-mail address: [email protected]
Comsats Institute of Information Technology, Sahiwal, Pakistan
E-mail address: [email protected]
University of Engineering and Technology, Taxila, Pakistan
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
953 IQBAL et al 946-953
DEGENERATE POLY-BERNOULLI POLYNOMIALS OF THE
SECOND KIND
DMITRY V. DOLGY, DAE SAN KIM, TAEKYUN KIM, AND TOUFIK MANSOUR
Abstract. In this paper, we introduce the degenerate poly-Bernoulli polynomialsof the second kind, which reduce in the limit to the poly-Bernoulli polynomials of thesecond kind. We present several explicit formulas and recurrence relations for thesepolynomials. Also, we establish a connection between our polynomials and severalknown families of polynomials.
1. Introduction
The Korobov polynomials of the first kind Kn(λ, x) with λ 6= 0 introduced by Korobov(actually he defined the polynomials 1
n!Kn(λ, x)) (see [13, 14, 18]) are given by
λt
(1 + λt)λ − 1(1 + t)x =
∑n≥0
Kn(λ, x)tn
n!.(1.1)
When x = 0, we define Kn(λ) = Kn(λ, 0). These are what would have been called thedegenerate Bernoulli polynomials of the second kind, since limλ→0Kn(λ, x) = bn(x),where bn(x) is the nth Bernoulli polynomial of the second kind (see [15]) given by
t
log(1 + t)(1 + t)x =
∑n≥0
bn(x)tn
n!.
On the other hand, the poly-Bernoulli polynomials of the second kind Pb(k)n (x) (of index
k) are introduced in [12] (see also [5, 7, 10]) and given by
Lik(1− e−t)
log(1 + t)(1 + t)x =
∑n≥0
Pb(k)n (x)tn
n!,(1.2)
where Lik(x) (k ∈ Z) is the classical polylogarithm function given by Lik(x) =∑
n≥1xn
nk.
In this paper, we introduce the degenerate poly-Bernoulli polynomials of the second
kind Pb(k)n (λ, x) with λ 6= 0 (of index k) (see [3, 6, 8]) which are given by
λLik(1− e−t)
(1 + t)λ − 1(1 + t)x =
∑n≥0
Pb(k)n (λ, x)tn
n!.(1.3)
When x = 0, Pb(k)n (λ, 0) are called the degenerate poly-Bernoulli numbers of the second
kind. Clearly, Pb(1)n (λ, x) = Kn(λ, x) and limλ→0 Pb
(k)n (λ, x) = Pb
(k)n (x).
2010 Mathematics Subject Classification. 05A19, 05A40, 11B83.Key words and phrases. Degenerate Poly-Bernoulli polynomials, Umbral calculus.
1
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
954 DOLGY et al 954-966
2 DMITRY V. DOLGY, DAE SAN KIM, TAEKYUN KIM, AND TOUFIK MANSOUR
Recall here that the λ-Daehee polynomials of the first kind Dn,λ(x) (see [9]) are givenby
λ log(1 + t)
(1 + t)λ − 1(1 + t)x =
∑n≥0
Dn,λ(x)tn
n!.(1.4)
When x = 0, Dn,λ = Dn,λ(0) are called the λ-Daehee numbers of the first kind. Note
that, as λ log(1+t)(1+t)λ−1
Lik(1−e−t)
log(1+t)(1 + t)x =
∑n≥0 Pb
(k)n (λ, x) t
n
n!, the degenerate poly-Bernoulli
polynomials of the second kind are mixed-type of the λ-Daehee polynomials of the firstkind and the poly-Bernoulli polynomials of the second kind.
The goal of this paper is to use umbral calculus to obtain several new and interestingidentities of degenerate poly-Bernoulli polynomials of the second kind. To do thatwe refer the reader to umbral algebra and umbral calculus as given in [16, 17]. Moreprecisely, we give some properties, explicit formulas, recurrence relations and identitiesabout the degenerate poly-Bernoulli polynomials of the second kind. Also, we establisha connection between our polynomials and several known families of polynomials.
2. Explicit formulas
In this section we present several explicit formulas for the degenerate poly-Bernoulli
polynomials of the second kind, namely Pb(k)n (λ, x). It is immediate from (1.3) that
the degenerate poly-Bernoulli polynomials of the second kind are given by the Sheffersequence for the pair
Pb(k)n (λ, x) ∼ (gk(t), f(t)) ≡
(eλt − 1
λLik(1− e1−et), et − 1
).(2.1)
To do so, we recall that Stirling numbers S1(n, k) of the first kind can be defined bymeans of exponential generating functions as
∑ℓ≥j
S1(ℓ, j)tℓ
ℓ=
1
j!logj(1 + t)(2.2)
and can be defined by means of ordinary generating functions as
(x)n =
n∑m=0
S1(n,m)xm ∼ (1, et − 1),(2.3)
where (x)n = x(x− 1)(x− 2) · · · (x− n + 1) with (x)0 = 1.
Theorem 2.1. For all n ≥ 0,
Pb(k)n (λ, x) =
n∑j=0
(n∑
ℓ=j
(n
ℓ
)S1(ℓ, j)Pb
(k)n−ℓ(λ, 0)
)xj
=
n∑j=0
(n∑
ℓ=j
n−ℓ∑m=0
(n
ℓ
)(n− ℓ
m
)S1(ℓ, j)Pb(k)m Dn−ℓ−m,λ
)xj .
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
955 DOLGY et al 954-966
DEGENERATE POLY-BERNOULLI POLYNOMIALS OF THE SECOND KIND 3
Proof. By applying the fact that
sn(x) =n∑
j=0
1
j!〈g(f(t))−1f(t)j | xn〉xj ,(2.4)
for any sn(x) ∼ (g(t), f(t) (see [16, 17]) in the case of degenerate poly-Bernoulli poly-nomials of the second kind (see (2.1)), we have
1
j!〈gk(f(t))
−1f(t)j | xn〉
=1
j!
⟨λLik(1− e−t)
(1 + t)λ − 1(log(1 + t))j | xn
⟩=
⟨λLik(1− e−t)
(1 + t)λ − 1|logj(1 + t)
j!xn
⟩,(2.5)
which, by (2.3), we have
1
j!〈gk(f(t))
−1f(t)j | xn〉
=
⟨λLik(1− e−t)
(1 + t)λ − 1|∑ℓ≥j
S1(ℓ, j)tℓ
ℓ!xn
⟩=
n∑ℓ=j
(n
ℓ
)S1(ℓ, j)
⟨λLik(1− e−t)
(1 + t)λ − 1| xn−ℓ
⟩
=n∑
ℓ=j
(n
ℓ
)S1(ℓ, j)Pb
(k)n−ℓ(λ, 0),
which completes the proof of the first equality.
Now let us calculate aj =1j!〈gk(f(t))
−1f(t)j | xn〉 in another way. By (2.5), we have
aj =n∑
ℓ=j
(n
ℓ
)S1(ℓ, j)
⟨λ log(1 + t)
(1 + t)λ − 1|Lik(1− e−t)
log(1 + t)xn−ℓ
⟩,
which, by (1.3) and (1.4), implies
aj =
n∑ℓ=j
n−ℓ∑m=0
(n
ℓ
)(n− ℓ
m
)S1(ℓ, j)Pb(k)m
⟨λ log(1 + t)
(1 + t)λ − 1| xn−ℓ−m
⟩
=
n∑ℓ=j
n−ℓ∑m=0
(n
ℓ
)(n− ℓ
m
)S1(ℓ, j)Pb(k)m Dn−ℓ−m,λ.
Thus,
Pb(k)n (λ, x) =
n∑j=0
(n∑
ℓ=j
n−ℓ∑m=0
(n
ℓ
)(n− ℓ
m
)S1(ℓ, j)Pb(k)m Dn−ℓ−m,λ
)xj ,
as required.
Theorem 2.2. For all n ≥ 0,
Pb(k)n (λ, x) =
n∑m=0
(n
m
)Dn−m,λPb(k)m (x) =
n∑m=0
(n
m
)Pb
(k)n−mDm,λ(x).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
956 DOLGY et al 954-966
4 DMITRY V. DOLGY, DAE SAN KIM, TAEKYUN KIM, AND TOUFIK MANSOUR
Proof. By (1.3), we have
Pb(k)n (λ, y) =
⟨λ log(1 + t)
(1 + t)λ − 1|Lik(1− e−t)
log(1 + t)(1 + t)yxn
⟩,
which, by (1.2), we obtain
Pb(k)n (λ, y) =n∑
m=0
(n
m
)Pb(k)m (y)
⟨λ log(1 + t)
(1 + t)λ − 1| xn−m
⟩.
Therefore, by (1.4), we obtain the first equality. To obtain the second equality, wereverse the order, namely we use at first (1.4) and then (1.2), to obtain
Pb(k)n (λ, y) =n∑
m=0
(n
m
)Dm,λ(y)
⟨Lik(1− e−t)
log(1 + t)| xn−m
⟩=
n∑m=0
(n
m
)Dm,λ(y)Pb
(k)n−m,
which completes the proof.
Note that it was shown in [9] that Dn,λ(x) is given by∑n
j=0 S1(n, j)λjBj(x/λ), where
Bm(x) is the mth Bernoulli polynomial. Thus, for x = 0, we have
Dn,λ =n∑
j=0
S1(n, j)λjBj,
where Bm is the mth Bernoulli number. Hence, we obtain
Pb(k)n (λ, x) =n∑
m=0
(n−m∑ℓ=0
(n
m
)S1(n−m, ℓ)λℓBℓ
)Pb(k)m (x)
=
n∑m=0
(n∑
ℓ=m
(n
ℓ
)S1(ℓ,m)λmPb
(k)n−ℓ
)Bm(x/λ).
Note that Stirling number S2(n, k) of the second kind can be defined by the exponentialgenerating functions as
∑n≥k
S2(n, k)xn
n!=
(et − 1)k
k!.(2.6)
Theorem 2.3. For all n ≥ 1,
Pb(k)n (λ, x) =
n∑r=0
(n−r∑ℓ=0
n−ℓ−r∑m=0
(n− 1
ℓ
)(n− ℓ
r
)B
(n)ℓ Pb(k)m S2(n− ℓ− r,m)λr
)Br(x/λ).
Proof. By 2.1, xn ∼ (1, t), and the transfer formula (see [16,17]), we obtain, for n ≥ 1,
eλt − 1
λLik(1− e1−et)Pb(k)n (λ, x)
= xtn
(et − 1)nx−1xn = x
∑ℓ≥0
B(n)ℓ
tℓ
ℓ!xn−1 =
n−1∑ℓ=0
(n− 1
ℓ
)B
(n)ℓ xn−ℓ.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
957 DOLGY et al 954-966
DEGENERATE POLY-BERNOULLI POLYNOMIALS OF THE SECOND KIND 5
Thus,
Pb(k)n (λ, x) =
n−1∑ℓ=0
(n− 1
ℓ
)B
(n)ℓ
λt
eλt − 1
λLik(1− e−s)
log(1 + s)|s=et−1 x
n−ℓ,
which, by (1.2) and (2.6), implies
Pb(k)n (λ, x) =n−1∑ℓ=0
((n− 1
ℓ
)B
(n)ℓ
λt
eλt − 1
∑m≥0
Pb(k)m
(et − 1)m
m!xn−ℓ
)
=n−1∑ℓ=0
n−ℓ∑m=0
n−ℓ∑r=m
(n− 1
ℓ
)B
(n)ℓ Pb(k)m S2(r,m)
(λt
eλt − 1
tr
r!xn−ℓ
)
=
n−1∑ℓ=0
n−ℓ∑m=0
n−ℓ∑r=m
(n− 1
ℓ
)(n− ℓ
r
)B
(n)ℓ Pb(k)m S2(r,m)
(λt
eλt − 1xn−ℓ−r
)
=n−1∑ℓ=0
n−ℓ∑m=0
n−ℓ∑r=m
(n− 1
ℓ
)(n− ℓ
r
)B
(n)ℓ Pb(k)m S2(r,m)λn−ℓ−rBn−ℓ−r(x/λ).
Here we used the following fact: 1g(λt)
xn = λnsn(x/λ) for any sn(x) ∼ (g(t), t) and
λ 6= 0. Indeed, 〈tk | 1/g(λt)xn〉 = λ−k〈(λt)k/g(λt)|xn〉 = λ−k〈tk/g(t)|λnxn〉 = λn−k〈tk |1/g(t)xn〉 = λn〈(t/λ)k | sn(x)〉 = 〈tk | λnsn(x/λ)〉.
By exchanging the indices of the summations, we obtain that
Pb(k)n (λ, x) =
n−1∑ℓ=0
n−ℓ∑m=0
n−ℓ−m∑r=0
(n− 1
ℓ
)(n− ℓ
r
)B
(n)ℓ Pb(k)m S2(n− ℓ− r,m)λrBr(x/λ)
=n∑
r=0
(n−r∑ℓ=0
n−ℓ−r∑m=0
(n− 1
ℓ
)(n− ℓ
r
)B
(n)ℓ Pb(k)m S2(n− ℓ− r,m)λr
)Br(x/λ),
as claimed.
Theorem 2.4. For all n ≥ 0,
Pb(k)n (λ, x) =n∑
r=0
(n∑
ℓ=r
ℓ−r∑m=0
(ℓ
r
)S1(n, ℓ)S2(ℓ− r,m)λrPb(k)m
)Br(x/λ).
Proof. By (2.1) we have that eλt−1
λLik(1−e1−e
t )Pb
(k)n (λ, x) ∼ (1, et − 1). Thus, by (2.3), we
obtain
Pb(k)n (λ, x) =λLik(1− e1−et)
eλt − 1(x)n =
n∑ℓ=0
S1(n, ℓ)λLik(1− e1−et)
eλt − 1xℓ.(2.7)
By replacing the function λLik(1−e1−e
t
)eλt−1
by
λt
eλt − 1
λLik(1− e−s)
log(1 + s)|s=et−1,
and by using very similar arguments as in the proof of Theorem 2.3, one can completethe proof.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
958 DOLGY et al 954-966
6 DMITRY V. DOLGY, DAE SAN KIM, TAEKYUN KIM, AND TOUFIK MANSOUR
Note that Li2(1 − e−t) =∫ t
0y
ey−1dy =
∑j≥0Bj
1j!
∫ t
0yjdy =
∑j≥0
Bjtj+1
j!(j+1). For general
k ≥ 2, the function Lik(1− e−t) has integral representation as
Lik(1− e−t) =
∫ t
0
1
ey − 1
∫ y
0
1
ey − 1
∫ y
0
· · ·1
ey − 1
∫ y
0︸ ︷︷ ︸(k−2) times
y
ey − 1dy · · · dydydy,
which, by induction on k, implies
Lik(1− e−t) =∑j1≥0
· · ·∑
jk−1≥0
tj1+···+jk−1+1
k−1∏i=1
Bji
ji!(j1 + · · ·+ ji + 1).(2.8)
Theorem 2.5. For all n ≥ 0 and k ≥ 2,
Pb(k)n (λ, x) =
n∑ℓ=0
(n)ℓKn−ℓ(λ, x)
∑
j1+...+jk−1=ℓ
k−1∏i=1
Bji
ji!(j1 + · · ·+ ji + 1)
.
Proof. By (2.1), we have
Pb(k)n (λ, y) =
⟨λLik(1− e−t)
(1 + t)λ − 1(1 + t)y | xn
⟩
=
⟨Lik(1− e−t)
t|
λt
(1 + t)λ − 1(1 + t)yxn
⟩,
which, by (1.1), implies
Pb(k)n (λ, y) =
⟨Lik(1− e−t)
t|∑ℓ≥0
Kℓ(λ, y)tℓ
ℓ!xn
⟩
=
n∑ℓ=0
(n
ℓ
)Kℓ(λ, y)
⟨Lik(1− e−t)
t| xn−ℓ
⟩.
Thus, by (2.8), we complete the proof.
3. Recurrences
Note that, by (1.3) and the fact that (x)n ∼ (1, et−1), we obtain the following identity.
Pb(k)n (λ, x+ y) =
∑nj=0
(nj
)Pb
(k)j (λ, x)(y)n−j. Moreover, in the next results, we present
several recurrences for the degenerate poly-Bernoulli polynomials, namely Pb(k)n (x).
Theorem 3.1. For all n ≥ 1, Pb(k)n (λ, x+ 1) = Pb
(k)n (λ, x) + nPb
(k)n−1(λ, x),
Proof. It is well-known that f(t)sn(x) = nsn−1(x) for all sn(x) ∼ (g(t), f(t)) (see
[16, 17]). Thus, by (2.1), we have (et − 1)Pb(k)n (λ, x) = nPb
(k)n−1(λ, x), which gives
Pb(k)n (λ, x+ 1)− Pb
(k)n (λ, x) = nPb
(k)n−1(λ, x), as required.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
959 DOLGY et al 954-966
DEGENERATE POLY-BERNOULLI POLYNOMIALS OF THE SECOND KIND 7
Theorem 3.2. For all n ≥ 1,
Pb(k)n+1(λ, x) = xPb(k)n (λ, x− 1)
−n∑
m=0
m+1∑ℓ=0
m+1∑j=ℓ
S1(n,m)S2(j, ℓ)
m+ 1
(m+ 1
j
)Pb
(k)ℓ (λ, 0)λm+1−jBm+1−j(
x+ λ− 1
λ)
+n∑
m=0
m+1∑ℓ=0
m+1∑j=ℓ
S1(n,m)S2(j, ℓ)
m+ 1
(m+ 1
j
)PB
(k−1)ℓ λm+1−jBm+1−j(
x
λ).
Proof. It is well-known that that sn+1(x) = (x − g′(t)/g(t)) 1f ′(t)
sn(x) for all sn(x) ∼
(g(t), f(t)) (see [16, 17]). Thus, by 2.1, we have
(x− g′k(t)/gk(t))1
f ′(t)= xe−t − e−tg′k(t)/gk(t),
which gives
Pb(k)n+1(λ, x) = xPb(k)n (λ, x− 1)− e−tg′k(t)/gk(t)Pb(k)n (λ, x),(3.1)
where
e−t g′k(t)
gk(t)= e−t
(λeλt
eλt − 1−
1
Lik(1− e1−et)
Lik−1(1− e1−et)
1− e1−etete1−et
)
=1
t
(λte(λ−1)t
eλt − 1
λLik(1− e1−et)
eλt − 1−
λt
eλt − 1
Lik−1(1− e1−et)
eet−1 − 1
)eλt − 1
λLik(1− e1−et).
Note that the order te−t g′
k(t)
gk(t)
is at least one, and by (2.7) we have eλt−1
λLik(1−e1−e
t )Pb
(k)n (λ, x) =∑n
m=0 S1(n,m)xm. Thus, by (1.3), we have
e−t g′k(t)
gk(t)Pb(k)n (λ, x)
=n∑
m=0
S1(n,m)
m+ 1
(λte(λ−1)t
eλt − 1
λLik(1− e1−et)
eλt − 1−
λt
eλt − 1
Lik−1(1− e1−et)
eet−1 − 1
)xm+1.
Therefore, by (1.2) and (1.3), we have
e−t g′k(t)
gk(t)Pb(k)n (λ, x)
=
n∑m=0
S1(n,m)
m+ 1
(λte(λ−1)t
eλt − 1
λLik(1− e−s)
(1 + s)λ − 1−
λt
eλt − 1
Lik−1(1− e−s)
es − 1
∣∣∣∣s=et−1
)xm+1
=
n∑m=0
S1(n,m)
m+ 1
λt
eλt − 1
(e(λ−1)t
∑ℓ≥0
Pb(k)ℓ (λ, 0)
(et − 1)ℓ
ℓ!−∑ℓ≥0
PB(k−1)ℓ
(et − 1)ℓ
ℓ!
)xm+1,
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
960 DOLGY et al 954-966
8 DMITRY V. DOLGY, DAE SAN KIM, TAEKYUN KIM, AND TOUFIK MANSOUR
where PB(k)ℓ are the poly-Bernoulli numbers(of index k). So with help of (2.6), we
obtain
e−t g′(t)
g(t)Pb(k)n (λ, x)
=
n∑m=0
m+1∑ℓ=0
m+1∑j=ℓ
S1(n,m)S2(j, ℓ)
m+ 1
(m+ 1
j
)(Pb
(k)ℓ (λ, 0)
λte(λ−1)t
eλt − 1− PB
(k−1)ℓ
λt
eλt − 1
)xm+1−j
=
n∑m=0
m+1∑ℓ=0
m+1∑j=ℓ
S1(n,m)S2(j, ℓ)
m+ 1
(m+ 1
j
)Pb
(k)ℓ (λ, 0)λm+1−jBm+1−j(
x+ λ− 1
λ)
−n∑
m=0
m+1∑ℓ=0
m+1∑j=ℓ
S1(n,m)S2(j, ℓ)
m+ 1
(m+ 1
j
)PB
(k−1)ℓ λm+1−jBm+1−j(
x
λ).
Therefore, by changing the summation on j, then substituting into (3.1), we completethe proof.
In next theorem, we find expression for ddxPb
(k)n (λ, x).
Theorem 3.3. For all n ≥ 0,
d
dxPb(k)n (λ, x) = n!
n−1∑ℓ=0
(−1)n−ℓ−1
ℓ!(n− ℓ)Pb(ℓ)n (λ, x).
Proof. It is well-known that ddxsn(x) =
∑n−1ℓ=0
(nℓ
)〈f(t) | xn−ℓ〉sℓ(x), for all sn(x) ∼
(g(t), f(t)). Thus, in the case of degenerate poly-Bernoulli polynomials of the secondkind (see (2.1)), we have
〈f(t) | xn−ℓ〉 = 〈log(1 + t) | xn−ℓ〉 = (−1)n−ℓ−1(n− ℓ− 1)!.
Thusd
dxPb(k)n (λ, x) =
n−1∑ℓ=0
(n
ℓ
)(−1)n−ℓ−1(n− ℓ− 1)!Pb(ℓ)n (λ, x),
which completes the proof.
Theorem 3.4. For all n ≥ 1,
Pb(k)n (λ, x)− xPb(k)n−1(λ, x− 1)
=1
n
n∑m=0
(n
m
)(Pb(k−1)
m (λ, x)Bn−m − Pb(k)m (λ, x+ λ− 1)Kn−m(λ)).
Proof. By (1.3), we have, for n ≥ 1,
Pb(k)n (λ, y) =
⟨λLik(1− e−t)
(1 + t)λ − 1(1 + t)y | xn
⟩
=
⟨λLik(1− e−t)
(1 + t)λ − 1
d
dt(1 + t)y | xn−1
⟩(3.2)
+
⟨d
dt
λLik(1− e−t)
(1 + t)λ − 1(1 + t)y | xn−1
⟩.(3.3)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
961 DOLGY et al 954-966
DEGENERATE POLY-BERNOULLI POLYNOMIALS OF THE SECOND KIND 9
The term in (3.2) is given by
y
⟨λLik(1− e−t)
(1 + t)λ − 1(1 + t)y−1) | xn−1
⟩= yPb
(k)n−1(λ, y − 1).(3.4)
For the term in (3.3), we observe that ddt
λLik(1−e−t)
(1+t)λ−1= 1
t(A−B), where
A =t
et − 1
λLik−1(1− e−t)
(1 + t)λ − 1, B =
λt
(1 + t)λ − 1
λLik(1− e−t)
(1 + t)λ − 1(1 + t)λ−1.
Note that the expression A − B has order at least 1. Now, we ready to compute theterm in (3.3). By (1.3), we have⟨
d
dt
λLik(1− e−t)
(1 + t)λ − 1(1 + t)y | xn−1
⟩=
⟨1
t(A− B)(1 + t)y | xn−1
⟩
=1
n〈A(1 + t)y | xn〉 −
1
n〈B(1 + t)y | xn〉
=1
n
⟨t
et − 1|∑m≥0
Pb(k−1)m (λ, y)
tm
m!xn
⟩
−1
n
⟨λt
(1 + t)λ − 1|∑m≥0
Pb(k)m (λ, y + λ− 1)tm
m!xn
⟩
=1
n
n∑m=0
(n
m
)Pb(k−1)
m (λ, y)
⟨t
et − 1| xn−m
⟩
−1
n
n∑m=0
(n
m
)Pb(k)m (λ, y + λ− 1)
⟨λt
(1 + t)λ − 1| xn−m
⟩
=1
n
n∑m=0
(n
m
)(Pb(k−1)
m (λ, y)Bn−m − Pb(k)m (λ, y + λ− 1)Kn−m(λ)).(3.5)
Thus, if we replace (3.2) by (3.4) and (3.3) by (3.5), we obtain
Pb(k)n (λ, x)− xPb(k)n−1(λ, x− 1)
=1
n
n∑m=0
(n
m
)(Pb(k−1)
m (λ, x)Bn−m − Pb(k)m (λ, x+ λ− 1)Kn−m(λ)),
as claimed.
4. Connections with families of polynomials
In this section, we present a few examples on the connections with families of polyno-mials. To do that we use the following fact from [16, 17]: For sn(x) ∼ (g(t), f(t)) andrn(x) ∼ (h(t), ℓ(t)), let sn(x) =
∑nk=0 cn,krk(x). Then we have
cn,k =1
k!
⟨h(f(t))
g(f(t))(ℓ(f(t)))k|xn
⟩.(4.1)
We start with the connection to Korobov polynomials Kn(λ, x) of the first kind.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
962 DOLGY et al 954-966
10 DMITRY V. DOLGY, DAE SAN KIM, TAEKYUN KIM, AND TOUFIK MANSOUR
Theorem 4.1. For all n ≥ 0,
Pb(k)n (λ, x) =
n∑m=0
((n
m
) n−m∑ℓ=0
1
n−m− ℓ+ 1
(n−m
ℓ
)PB
(k)ℓ
)Km(λ, x)
and
Pb(k)n (λ, x) =1
n + 1
n∑m=0
(n−m+1∑ℓ=0
(−1)n−m+1−ℓ
(n+ 1
m
)ℓ!
ℓkS2(n−m+ 1, ℓ)
)Km(λ, x).
Proof. By (1.1), we have that Kn(λ, x) ∼(
eλt−1λ(et−1)
, et − 1). Let
Pb(k)n (λ, x) =n∑
m=0
cn,mKm(λ, x).
Thus, by (2.1) and (4.1), we obtain
cn,m =1
m!
⟨(1 + t)λ − 1
λt
λLik(1− e−t)
(1 + t)λ − 1tm|xn
⟩=
1
m!
⟨Lik(1− e−t)
t|tmxn
⟩
=
(n
m
)⟨et − 1
t|Lik(1− e−t)
et − 1xn−m
⟩=
(n
m
)⟨et − 1
t|∑ℓ≥0
PB(k)ℓ
tℓ
ℓ!xn−m
⟩
=
(n
m
) n−m∑ℓ=0
(n−m
ℓ
)PB
(k)ℓ
⟨et − 1
t|xn−m−ℓ
⟩
=
(n
m
) n−m∑ℓ=0
(n−m
ℓ
)PB
(k)ℓ
∫ 1
0
un−m−ℓdu
=
(n
m
) n−m∑ℓ=0
1
n−m− ℓ+ 1
(n−m
ℓ
)PB
(k)ℓ ,
which completes the proof of the first identity. Note that we can compute cn,m inanother way, as follows. By using
cn,m =
(n
m
)⟨Lik(1− e−t)
t|xn−m
⟩,
and Lik(1− e−t) =∑
ℓ≥1(1−e−t)ℓ
ℓktogether with (2.6), we obtain that
cn,m =1
n+ 1
n−m+1∑ℓ=0
(−1)n−m+1−ℓ
(n + 1
m
)ℓ!
ℓkS2(n−m+ 1, ℓ),
which leads to the second identity.
Theorem 4.2. For all n ≥ 0,
Pb(k)n (λ, x) =n∑
m=0
((n
m
) n−m∑ℓ=0
(n−m
ℓ
)Kℓ(λ)Dn−m−ℓ
)Pb(k)m (x),
where Dn is the nth Daehee number defined bylog(1+t)
t=∑
n≥0Dntn
n!.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
963 DOLGY et al 954-966
DEGENERATE POLY-BERNOULLI POLYNOMIALS OF THE SECOND KIND 11
Proof. By (1.2), we have that Pb(k)n (x) ∼
(t
Lik(1−e1−e
t ), et − 1
). Let
Pb(k)n (λ, x) =
n∑m=0
cn,mPb(k)m (x).
Thus, by (2.1) and (4.1), we obtain
cn,m =1
m!
⟨log(1 + t)
Lik(1− e−t)
λLik(1− e−t)
(1 + t)λ − 1tm|xn
⟩=
(n
m
)⟨log(1 + t)
t|
λt
(1 + t)λ − 1xn−m
⟩
=
(n
m
)⟨log(1 + t)
t|∑ℓ≥0
Kℓ(λ)tℓ
ℓ!xn−m
⟩
=
(n
m
) n−m∑ℓ=0
(n−m
ℓ
)Kℓ(λ)
⟨log(1 + t)
t|xn−m−ℓ
⟩
=
(n
m
) n−m∑ℓ=0
(n−m
ℓ
)Kℓ(λ)Dn−m−ℓ
which completes the proof.
We start with the connection to Bernoulli polynomials B(s)n (x) of order s. Recall that
the Bernoulli polynomials B(s)n (x) of order s are defined by the generating function(
t
et − 1
)s
ext =∑n≥0
B(s)n (x)
tn
n!,
or equivalently,
B(s)n (x) ∼
((et − 1
t
)s
, t
)(4.2)
(see [2,4]). In the next result, we express our polynomials Pb(k)n (x) in terms of Bernoulli
polynomials of order s. To do that, we recall that Bernoulli numbers of the second kind
b(s)n of order s are defined as
ts
logs(1 + t)=∑n≥0
b(s)n
tn
n!.(4.3)
Theorem 4.3. For all n ≥ 0,
Pb(k)n (λ, x) =
n∑m=0
(n∑
ℓ=m
n−ℓ∑j=0
(n
ℓ
)(n− ℓ
j
)S1(ℓ,m)Pb
(k)j (λ, 0)b
(s)n−ℓ−j
)B(s)
m (x).
Proof. Let Pb(k)n (λ, x) =
∑nm=0 cn,mB
(s)m (x). By (2.1), (4.1) and (4.2), we have
cn,m =1
m!
⟨(t
log(1 + t)
)sλLik(1− e−t)
(1 + t)λ − 1(log(1 + t))m | xn
⟩,
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
964 DOLGY et al 954-966
12 DMITRY V. DOLGY, DAE SAN KIM, TAEKYUN KIM, AND TOUFIK MANSOUR
which, by (2.3) and (1.3), implies
cn,m =
⟨(t
log(1 + t)
)sλLik(1− e−t)
(1 + t)λ − 1|∑ℓ≥m
S1(ℓ,m)tℓ
ℓ!xn
⟩
=
n∑ℓ=m
(n
ℓ
)S1(ℓ,m)
⟨(t
log(1 + t)
)s
|λLik(1− e−t)
(1 + t)λ − 1xn−ℓ
⟩
=
n∑ℓ=m
(n
ℓ
)S1(ℓ,m)
⟨(t
log(1 + t)
)s
|∑j≥0
Pb(k)j (λ, 0)
tj
j!xn−ℓ
⟩
=
n∑ℓ=m
n−ℓ∑j=0
(n
ℓ
)(n− ℓ
j
)S1(ℓ,m)Pb
(k)j (λ, 0)
⟨(t
log(1 + t)
)s
| xn−ℓ−j
⟩.
Thus, by (4.3), we obtain
cn,m =
n∑ℓ=m
n−ℓ∑j=0
(n
ℓ
)(n− ℓ
j
)S1(ℓ,m)Pb
(k)j (λ, 0)b
(s)n−ℓ−j,
which completes the proof.
Similar techniques as in the proof of the previous theorem, we can express our poly-
nomials Pb(k)n (λ, x) in terms of other families. For instance, we can express our poly-
nomials Pb(k)n (λ, x) in terms of Frobenius-Euler polynomials (we leave the proof to
the interested reader). Note that the Frobenius-Euler polynomials H(s)n (x|µ) of order
s are defined by the generating function(
1−µet−µ
)sext =
∑n≥0H
(s)n (x|µ) t
n
n!, (µ 6= 1), or
equivalently, H(s)n (x|µ) ∼
((et−µ1−µ
)s, t)(see [1, 2, 4, 11]).
Theorem 4.4. For all n ≥ 0,
Pb(k)n (λ, x) =n∑
m=0
(n∑
ℓ=m
n−ℓ∑r=0
(n
ℓ
)(n− ℓ
r
)s!S1(ℓ,m)Pb
(k)r (λ, 0)
(1− µ)n−ℓ−r(s+ ℓ+ r − n)!
)H(s)
m (x|µ).
References
[1] Araci, S. and Acikgoz, M., A note on the Frobenius-Euler numbers and polynomials associatedwith Bernstein polynomials, Adv. Stud. Contemp. Math. 22(3) (2012), 399–406.
[2] Bayad A. and Kim, T., Identities involving values of Bernstein, q-Bernoulli, and q-Euler polyno-mials, Russ. J. Math. Phys. 18(2) (2011), 133–143.
[3] Carlitz, L., Degenerate stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979) 51–88.[4] Ding, D. and Yang, J., Some identities related to the Apostol-Euler and Apostol-Bernoulli poly-
nomials, Adv. Stud. Contemp. Math. 20(1) (2010), 7–21.[5] Jolany, H. and Faramarzi, H., Generalization on poly-Eulerian numbers and polynomials, Scientia
Magna 6 (2010), 9–18.[6] Kim, D. S. and Kim, T., A note on degenerate poly-Bernoulli numbers and polynomials,
arXiv:1503.08418 (2015).[7] Kim, D. S. and Kim, T., A note on poly-Bernoulli and higher-order poly-Bernoulli polynomilas,
Russ. J. Math. Phys. 22 (2015), 26–33.[8] Kim, D. S., Kim, T. and Dolgy, D. V., A note on degenerate Bernoulli numbers and polynomials
associated with p-adic invariant integral on Zp, Appl. Math. Comput. 295 (2015), 198–204.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
965 DOLGY et al 954-966
DEGENERATE POLY-BERNOULLI POLYNOMIALS OF THE SECOND KIND 13
[9] Kim, D. S., Kim, T., Lee, S. H. and Seo, J. J., A Note on the λ-Daehee Polynomials, Int. Journalof Math. Analysis 7 (2013), 3069–3080.
[10] Kim, D. S., Kim, T., Mansour, T. and Dolgy, D. V., On poly-Bernoulli polynomials of the secondkind with umbral calculus viewpoint, Adv. Difference Equ. 2015 (2015), 2015:27.
[11] Kim, T., Indentities involving Frobenius-Euler polynomials arising from non-linear differentialequations, J. Number Theory 132(12) (2012), 2854–2865.
[12] Kim, T., Kwon, H. I., Lee, S. H. and Seo, J. J., A note on poly-Bernoulli numbers and polynomialsof the second kind, Adv. Difference Equ. 2014 (2014), 2014:219.
[13] Korobov, N. M., On some properties of special polynomials, Proceedings of the IV InternationalConference “Modern Problems of Number Theory and its Applications” (Russian) (Tula, 2001),vol. 1, 2001, pp. 40–49.
[14] Korobov, N. M., Special polynomials and their applications, in: Diophantine Approximations.Mathematical Notes (Russian), vol. 2, Izd. Moskov. Univ., Moscow, 1996, pp. 77–89.
[15] Prabhakar, T. R. and Gupta, S., Bernoulli polynomials of the second kind and general order,Indian J. Pure Appl. Math. 11 (1980), 1361–1368.
[16] Roman, S., More on the umbral calculus, with emphasis on the q-umbral calculus, J. Math. Anal.Appl. 107 (1985), 222–254.
[17] Roman, S., The umbral calculus, Dover Publ. Inc. New York, 2005.[18] Ustinov, A. V., Korobov polynomials and umbral analysis, (Russian) Chebyshevskii Sb. 4 (2003),
137–152.
Institute of Natural Sciences, Far Eastern Federal University, Vladivostok, 690950,
Russia
E-mail address : d−[email protected]
Department of Mathematics, Sogang University, Seoul 121-742, S. Korea
E-mail address : [email protected]
Department of Mathematics, Kwangwoon University, Seoul 139-701, S. Korea
E-mail address : [email protected]
University of Haifa, Department of Mathematics, 3498838 Haifa, Israel
E-mail address : [email protected]
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
966 DOLGY et al 954-966
Some results for meromorphic functions of several
variables
Yue Wang*
School of Information, Renmin University of China, Beijing, 100872, China
Abstract: Using the Nevanlinna theory of the value distribution of meromorphic functions, we
investigate the value distribution of complex partial q-difference polynomials of meromorphic
functions of zero order, and also investigate the existence of meromorphic solutions of some
types of systems of complex partial q-difference equations in Cn. Some existing results are
improved and generalized, and some new results are obtained. Examples show that our results
are precise.
Keywords: value distribution; meromorphic solution; complex partial q-difference polynomi-
als; complex partial q-difference equations
§1 Introduction
In this paper, we assume that the reader is familiar with the standard notation and basic
results of the Nevanlinna theory of meromorphic functions, see, for example [1].
The reference related to notations of this section are referred to Tu[2].
Let M be a connected complex manifold of dimension n and let
A(M) =2m∑n=0
An(M)
be the graded ring of complex valued differential forms on M . Each set An(M) can be split
into a direct sum
An(M) =∑
p+q=n
Ap,q(M),
where Ap,q(M) is the forms of type p, q. The differential operators d and dc on A(M) are
defined as
d := ∂ + ∂ and dc :=1
4πi(∂ − ∂).
where
∂ : Ap,q(M) −→ Ap+1,q(M),
*Corresponding authorE-mail addresses: [email protected](Y. Wang)2010 MR Subject Classification: 30D35.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
967 Yue Wang 967-979
2
∂ : Ap,q(M) −→ Ap,q+1(M).
Let z = (z1, ..., zn) ∈ Cn, and let r ∈ R+. We define
ωn(z) := ddc log |z|2 and σn(z) := dc log |z|2 ∧ ωn−1n (z),
where z ∈ Cn\0 and |z|2 := |z1|2 + · · ·+ |zn|2.Let Cn < r >= z ∈ Cn : |z| = r, Cn(r) = z ∈ Cn : |z| < r, Cn[r] = z ∈ Cn : |z| ≤ r.
Then σn(z) defines a positive measure on Cn < r > with total measure one. In addition, by
defining
νn(z) := ddc|z|2 and ρn(z) := νnn(z),
for all z ∈ Cn, it follows that ρn(z) is the Lebesgue measure on Cn normalized such that Cn(r)
has measure r2n.
Let w be a meromorphic function on Cn in the sense that w can be written as a quotient of
two relatively prime holomorphic functions. We will write w = (w0, w1) where w0 6≡ 0, thus w
can be regarded as a meromorphic map w : Cn −→ P1 such that w−1(∞) 6= Cn.Let P1 be the Riemann sphere. For a, b ∈ P1, the chordal distance from a to b is denoted
by ‖ a, b ‖, ‖ a,∞ ‖= 1√1+|a|2
, ‖ a, b ‖= |a−b|√1+|a|2·
√1+|b|2
, a, b ∈ C, where ‖ a, a ‖= 0 and
0 ≤‖ a, b ‖=‖ b, a ‖≤ 1. If a ∈ P1 and w−1(a) 6= Cn, then we define the proximity function as
m(r, w, a) =
∫|z|=r
log1
‖ a,w(z) ‖σn ≥ 0, r > 0.
Let ν be a divisor on Cn. We identify ν with its multiplicity function, define
ν(r) = z ∈ Cn : |z| < r⋂suppν, r > 0.
The pre-counting function of ν is defined by
n(r, ν) =∑z∈ν(r)
ν(z), if n = 1, n(r, ν) = r2−2n∫ν(r)
ννn−1n , if n > 1.
The counting function of ν is defined by
N(r, ν) =
∫ r
s
n(t, ν)dt
t, r > s.
Let w be a meromorphic function on Cn. If a ∈ P1 and w−1(a) 6= Cn, the a-divisor
ν(w, a) ≥ 0 is defined, and its pre-counting function and counting function will be denoted by
n(r, w, a) and N(r, w, a), respectively.
For a divisor ν on Cn, let
n(r, ν) =∑z∈ν(r)
1, if n = 1, n(r, ν) = r2−2n∫ν(r)
νn−1n , if n > 1.
N(r, ν) =
∫ r
s
n(t, ν)dt
t, r > s. N(r, w, a) = N(r, ν(w, a)).
For 0 < s < r, the characteristic of w is defined by
T (r, w) =
∫ r
s
1
t2n−1
∫Cn[t]
w∗(ω) ∧ νn−1n dt =
∫ r
s
1
t
∫Cn[t]
w∗(ω) ∧ ωn−1n dt.
where the pullback w∗(ω) satisfies w∗(ω) = ddc log(|w0|2 + |w1|2).
The First Main Theorem states
T (r, w) = N(r, w, a) +m(r, w, a)−m(s, w, a).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
968 Yue Wang 967-979
Y.W Some results for meromorphic functions of several variables 3
In 2012, Korhonen R has investigated the difference analogues of the lemma on the Log-
arithmic Derivate and of the Second Main Theorem of Nevanlinna theory for meromorphic
functions of several variables, see [3]. Particularly, in 2013, Cao T B, see [4], using different
method obtains difference analogues of the second main theorem for meromorphic functions
in several complex variables from which difference analogues of Picard-type theorems are also
obtained. His results are improvements or extensions of some results of Korhonen R.
Similarly, in 2014, Wen Z T has investigated the q-difference theory for meromorphic func-
tions of several variables, see [5]. Some results that we will use in this paper are as follows.
Theorem A [5] Let w be a meromorphic function in Cn of zero order such that w(0) 6= 0,∞,
and let q ∈ Cn\0. Then,
m(r,w(qz)
w(z)) = o(T (r, w)),
on a set of logarithmic density 1.
Theorem B [5] Let w be a meromorphic function in Cn of zero order such that w(0) 6= 0,∞,
and let q ∈ Cn\0. Then,
T (r, w(qz)) = T (r, w(z)) + o(T (r, w)),
on a set of logarithmic density 1.
Remark: From the proof of Theorem B in [5], we have
N(r, w(qz)) = N(r, w(z)) + o(N(r, w)).
The remainder of the paper is organized as follows. In §2, we discuss Theorem A’s appli-
cations to complex partial q-difference equations. We present q-shift analogues of the Clunie
lemmas which can be used to study value distribution of zero-order meromorphic solutions of
large classes of complex partial q-difference equations. In §3, we study the existence of mero-
morphic solutions of complex partial q-difference equation of several variables, and obtain four
theorems, and then we give some examples, which show that the results obtained in §3 are, in
a sense, the best possible. And finally, we prove these four theorems by a series of lemmas.
§2 Value distribution of complex partial q-difference polynomials
Recently, Laine I, Halburd R G, Korhonen R J, Barnett D, Morgan W, investigate complex
q-difference theory, and have obtained some results,see [6,7,8,9]. Especially, in 2007, Barnett D
C, Halburd R G have obtained a theorem which is analogous to the Clunie Lemma as follows
Theorem C [7] Let w(z) be a non-constant zero-order meromorphic solution of
wn1(z)P1(z, w) = Q1(z, w),
where P1(z, w) and Q1(z, w) are complex q-difference polynomials in w(z) of the form
P1(z, w) =∑λ1∈I′1
aλ1(z)w(z)l
10(w(q1z))
l11 · · · (w(qνz))l1ν ,
Q1(z, w) =∑γ1∈J′
1
bγ1(z)w(z)l20(w(q1z))
l21 · · · (w(qµz))l2µ .
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
969 Yue Wang 967-979
4
If the degree of Q1(z, w) as a polynomial in w(z) and its q-shifts is at most n1, then
m(r, P1(z, w)) = S(r, w) = oT (r, w),for all r on a set of logarithmic density 1.
We will investigate the problem of value distribution of complex partial q-difference poly-
nomials (2.1), (2.2) and (2.3), where z = (z1, ..., zn) ∈ Cn.
P (z, w) =∑λ∈I1
aλ(z)w(z)lλ0 (w(qλ1z))lλ1 · · · (w(qλσλ z))
lλσλ . (2.1)
Q(z, w) =∑µ∈J1
bµ(z)w(z)mµ0 (w(qµ1z))mµ1 · · · (w(qµτµ z))
mµτµ . (2.2)
U(z, w) =∑ν∈K1
cν(z)w(z)nν0 (w(qν1z))nν1 · · · (w(qνυν z))
nνυν . (2.3)
where coefficients aλ(z), bµ(z), cν(z) are small functions of w(z). I1, J1,K1 are three
finite sets of multi-indices, qj ∈ Cn\0, (j ∈ λ1, . . . , λσλ , µ1, . . . , µτµ , ν1, . . . , νυν .).We will prove
Theorem 2.1. Let w be a meromorphic function in Cn, and be a non-constant meromorphic
solution of zero order of a complex partial q-difference equation of the form
U(z, w)P (z, w) = Q(z, w),
where complex partial q-difference polynomials P (z, w), Q(z, w) and U(z, w) are respectively
as the form of (2.1), (2.2), (2.3), the total degree degU(z, f) = n1 in w(z) and its shifts, and
degQ(z, f) ≤ n1. Moreover, we assume that U(z, w) contains just one term of maximal total
degree in w(z) and its shifts. Then, we have
m(r, P (z, w)) = S(r, w) = oT (r, w),for all r on a set of logarithmic density 1.
Corollary 2.1. Let w be a meromorphic function in Cn, and be a non-constant transcendental
meromorphic solution of zero order of a complex partial q-difference equation of the form
H(z, w)P (z, w) = Q(z, w),
where H(z, w) is a complex partial q-difference product of total degree n1 in w(z) and its shifts,
and where P (z, w), Q(z, w) are complex partial q-difference polynomials such that the total degree
of Q(z, w) is at most n1. Then, we obtain
m(r, P (z, w)) = S(r, w) = oT (r, w),for all r on a set of logarithmic density 1.
Proof of Theorem 2.1 As the proof of Theorem 1 in [10], we rearrange the expression
for the complex partial q-difference polynomial U(z, w) by collecting together all terms having
the same total degree and then writing U(z, w) as follows
U(z, w) =
n1∑j=0
dj(z)wj(z),
where dj(z) =∑ν=j
cν(z)(w(qν1z)
w(z))nν1 · · · (
w(qνυν z)
w(z))nνυν , j = 0, 1, · · · , n1. Since degU(z, w) =
n1 in w(z) and its shifts, and U(z, w) contains just one term of maximal total degree n1 in w(z)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
970 Yue Wang 967-979
Y.W Some results for meromorphic functions of several variables 5
and its shifts, therefore, dn1(z) contains just one product of the described form.
By Theorem A, for all r on a set of logarithmic density 1, we have
m(r, dj(z)) = S(r, w) = oT (r, w), j = 0, 1, · · · , n1.
It follows from the assumption that dn1(z) has just one term of maximal total degree in
U(z, w), thus, for all r on a set of logarithmic density 1, we get
m(r,1
dn1(z)
) = S(r, w) = oT (r, w).
Let
A(z) = max1≤j≤n1
1, 2 | dn1−j
dn1
|1j .
Then
m(r,A(z)) ≤n1∑j=0
m(r, dn1−j) +m(r,1
dn1
) +O(1) = S(r, w) = oT (r, w).
Let
E1 = z ∈ Cn < r >: | w(z) |≤ A(z), E2 = Cn < r > \E1.
Thus
m(r, P (z, w)) =
∫E1
log+ |P (z, w)|σn(z) +
∫E2
log+ |P (z, w)|σn(z). (2.4)
Next we estimate respectively
∫E1
log+ |P (z, w)|σn(z) and
∫E2
log+ |P (z, w)|σn(z) in (2.4).
When z ∈ E1, we have
| P (z, w) | = |∑λ∈I1
aλ(z)w(z)lλ0 (w(qλ1z))lλ1 · · · (w(qλσλ z))
lλσλ |
≤∑λ∈I1
| aλ(z) || w(z) |lλ0 | w(qλ1z) |lλ1 · · · | w(qλσλ z) |
lλσλ
=∑λ∈I1
| aλ(z) || w(z) |lλ | w(qλ1z)
w(z)|lλ1 · · · |
w(qλσλ z)
w(z)|lλσλ
≤∑λ∈I1
| aλ(z) || A(z) |lλ | w(qλ1z)
w(z)|lλ1 · · · |
w(qλσλ z)
w(z)|lλσλ ,
where lλ = lλ0+ lλ1
+ · · ·+ lλσλ . By Theorem A, for all r on a set of logarithmic density 1, we
have ∫E1
log+ |P (z, w)|σn(z) = S(r, w) = oT (r, w). (2.5)
When z ∈ E2, we obtain
| w(z) |> A(z) ≥ 2 | dn1−j
dn1
|1j , (j = 1, 2, . . . , n1),
i.e.| w(z) |j
2j≥| dn1−j
dn1
|, (j = 1, 2, . . . , n1).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
971 Yue Wang 967-979
6
It follows from U(z, w) =
n1∑j=0
dj(z)wj that
| U(z, w) | ≥ | dn1|| w |n1 −(| dn1−1 || w |n1−1 + | dn1−2 || w |n1−2 + . . .
+ | d1 || w | + | d0 |)
= | dn1|| w |n1 − | dn1
|| w |n1 (| dn1−1 || dn1 || w |
+| dn1−2 || dn1 || w |2
+ · · ·+ | d1 || dn1
|| w |n1−1+
| d0 || dn1
|| w |n1)
= | dn1 || w |n1 − | dn1 || w |n1 (
n1∑j=1
| dn1−j || dn1 || w |j
)
≥ | dn1 || w |n1 (1−n1∑j=1
1
2j)
=| dn1
|| w |n1
2n1.
Since z ∈ E2 , then
| w(z) |> A(z) ≥ 1,
that is1
| w(z) |< 1.
Using U(z, w)P (z, w) = Q(z, w) and the total degree of Q(z, w) is at most n1, we obtain
| P (z, w) | = | Q(z, w)
U(z, w)|
≤ 2n1
| dn1|| w |n1
∑µ∈J1
| bµ(z) || w(z) |mµ0 | w(qµ1z) |mµ1
· · · | w(qµτµ z) |mµτµ
≤ 2n1
| dn1 |∑µ∈J1
| bµ(z) || w(qµ1z)
w(z)|mµ1 · · · |
w(qµτµ z)
w(z)|mµτµ .
From Theorem A, for all r on a set of logarithmic density 1, we have∫E2
log+ |P (z, w)|σn(z) = S(r, w) = oT (r, w). (2.6)
Combining (2.4),(2.5),(2.6), yields
m(r, P (z, w)) =
∫E1
log+ |P (z, w)|σn(z) +
∫E2
log+ |P (z, w)|σn(z)
= S(r, w) = oT (r, w).This completes the proof of Theorem 2.1.
§3 Applications to complex partial q-difference equations
Recently, many authors, such as Chiang Y M, Halburd R G, Korhonen R J, Chen Zongxuan,
Gao Lingyun have studied solutions of some types of complex difference equation, and systems
of complex difference equations, and also obtained many important results, see[11, 12, 13, 14,
15, 16, 17].
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
972 Yue Wang 967-979
Y.W Some results for meromorphic functions of several variables 7
Let w be a non-constant meromorphic function of zero order, if meromorphic function g
satisfies T (r, g) = o T (r, w), for all r outside of a set of upper logarithmic density 0, i.e.
outside of a set E such that lim supr→∞
∫E∩[1,r]
dtt
log r= 0. The complement of E has lower logarithmic
density 1, then g is called small function of w.
Let qj ∈ Cn\0, j = 1, . . . , n2, wi : Cn → P1, i = 1, 2. z = (z1, ..., zn) ∈ Cn, I, J, I, Jare four finite sets of multi-indices, complex partial q-difference polynomials Ω1(z, w1, w2),
Ω2(z, w1, w2),Ω3(z, w1, w2),Ω4(z, w1, w2) can be expressed as
Ω1(z, w1, w2) =∑(i)∈I
a(i)(z)2∏k=1
wik0k (wk(q1z))ik1 · · · (wk(qn2z))
ikn2 ,
Ω2(z, w1, w2) =∑(j)∈J
b(j)(z)2∏k=1
wjk0k (wk(q1z))jk1 · · · (wk(qn2
z))jkn2 ,
Ω3(z, w1, w2) =∑(i)∈I
c(i)
(z)2∏k=1
wik0k (wk(q1z))ik1 · · · (wk(qn2
z))ikn2 ,
Ω4(z, w1, w2) =∑(j)∈J
d(j)(z)
2∏k=1
wjk0k (wk(q1z))
jk1 · · · (wk(qn2z))jkn2 ,
where coefficients a(i)(z), b(j)(z), c(i)(z), d(j)(z) are small functions of w1, w2.
Let Φ1 =Ω1(z, w1, w2)
Ω2(z, w1, w2), Φ2 =
Ω3(z, w1, w2)
Ω4(z, w1, w2), for Φ1, we denote λ11 = max
(i)n2∑l=0
i1l,
λ12 = max(i)n2∑l=0
i2l, λ21 = max(j)n2∑l=0
j1l, λ22 = max(j)n2∑l=0
j2l, λ1 = maxλ11, λ21, λ2 =
maxλ12, λ22. For Φ2, we denote similarly λ1, λ2.
We will investigate the existence of meromorphic solutions of complex partial q-difference
equation of several variables (3.1) and systems of complex partial q-difference equations of
several variables (3.2) and (3.3), where z = (z1, ..., zn) ∈ Cn.n2∑j=1
w(qjz) = R (z, w(z)) =a0(z) + a1(z)w(z) + · · ·+ ap(z)w
p(z)
b0(z) + b1(z)w(z) + · · ·+ bq(z)wq(z), (3.1)
where q1, . . . , qn2∈ Cn\0, R(z, w(z)) is irreducible rational function in w(z), a0(z), . . . , ap(z),
b0(z), . . . , bq(z) are rational functions.Ω1(z, w1, w2) = R1(z, w1) =
a0(z) + a1(z)w1(z) + · · ·+ ap1(z)wp11 (z)
b0(z) + b1(z)w1(z) + · · ·+ bq1(z)wq11 (z),
Ω2(z, w1, w2) = R2(z, w2) =c0(z) + c1(z)w2(z) + · · ·+ cp2(z)wp22 (z)
d0(z) + d1(z)w2(z) + · · ·+ dq2(z)wq22 (z),
(3.2)
where coefficients ai(z), bj(z) are small functions of w1, cl(z), dm(z) are small functions
of w2. ap1bq1 6= 0, cp2dq2 6= 0. The definition of Ω1(z, w1, w2) and Ω2(z, w1, w2) is as before.Φ1 = R1(z, w1, w2),
Φ2 = R2(z, w1, w2).(3.3)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
973 Yue Wang 967-979
8
where Rj(j = 1, 2) are irreducible rational functions with the meromorphic coefficients.
Definition 3.1. Let w1 and w2 be meromorphic functions in Cn.
S(r) =∑
T (r, a(i)) +∑
T (r, b(j)) +∑
T (r, c′
(i)) +
∑T (r, d(j)) +
∑T (r, d
′
(j)),
where∑T (r, d
′
(j)) means the sum of characteristic functions of all coefficients in Rj(j = 1, 2).
(w1(z), w2(z)) be a set of meromorphic solutions of (3.2) or (3.3). If one (Let be w1(z)) of
meromorphic solutions (w1(z), w2(z)) of (3.2) or (3.3) satisfies S(r) = o T (r, w1), outside a
possible exceptional set with finite logarithmic measure, then we say w1(z) is admissible.
We will prove
Theorem 3.1. Let w be a meromorphic function in Cn. If the q-difference equation (3.1)
admits a transcendental meromorphic solution of zero order, then
maxp, q ≤ n2.
Remark 3.1. If we replace the left side of (3.1) by
n2∏j=1
w(qjz), then the same assertion that
maxp, q ≤ n2 holds.
Theorem 3.2. Let w1 and w2 be meromorphic functions in Cn, and (w1(z), w2(z)) be a set of
zero order meromorphic solution of (3.2). If
maxp1, q1 > λ11,maxp2, q2 > λ22,
and both w1 and w2 are admissible, then
[maxp1, q1 − λ11][maxp2, q2 − λ22] ≤ λ12λ21.
Example 3.1. (w1, w2) = (z1z2,1
z1z2) is a set of zero order admissible meromorphic solution
of the following system of complex partial q-difference equationsw2
2(−2z1,−2z2) =1
16w21
,
w21(
1
3z1,
1
3z2) =
1
81w22
.
Easily, we obtain
λ11 = 0, λ22 = 0, λ12 = 2, λ21 = 2,maxp1, q1 = 2,maxp2, q2 = 2.
Thus
[maxp1, q1 − λ11] [maxp2, q2 − λ22] = 4 = λ12λ21.
This example shows the upper bound in Theorem 3.2 can be reached.
Example 3.2. For a system of complex partial q-difference equationsw2
1(−2z1,−2z2)w2(−1
2z1,−
1
2z2) =
w41 − ( 7
2z1z2 −4916 )w2
1 − 458 z1z2 −
6516
w21 − 2w1 − z21z22 + 2
,
w1(−2z1,−2z2)w22(−1
2z1,−
1
2z2) =
1256w
22 + 1
64z1z2w3
2
3z1z2
w2 − 3z1z2 + 1.
(w1, w2) = (z1z2 + 1, z21z22) is a set of non-admissible solutions.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
974 Yue Wang 967-979
Y.W Some results for meromorphic functions of several variables 9
Clearly, we know
λ11 = 2, λ22 = 2, λ12 = 1, λ21 = 1,maxp1, q1 = 4,maxp2, q2 = 3.
Thus
[maxp1, q1 − λ11] [maxp2, q2 − λ22] = 2 > 1 = λ12λ21.
This example shows that we can not omit ’admissible’ in Theorem 3.2.
Theorem 3.3. Let w1 and w2 be meromorphic functions in Cn. Let (w1, w2) be a set of zero
order meromorphic solution of (3.2). If one of the following conditions is satisfied
(i) maxp1, q1 > λ11, (ii) maxp2, q2 > λ22,
then both w1 and w2 are admissible or none of w1 and w2 is admissible.
Theorem 3.4. Let w1 and w2 be meromorphic functions in Cn. Let (w1, w2) be a set of zero
order meromorphic solution of (3.3). If one of the following conditions is satisfied
(i) p1 > λ1, q2 > λ2, (ii) p2 > λ2, q1 > λ1,
then both w1 and w2 are admissible or none of w1 and w2 is admissible, where p1 and p2 are
the highest degree of w1 and w2 in R1(z, w1, w2), we denote similarly q1, q2 in R2(z, w1, w2).
Example 3.3. For a system of complex partial q-difference equations
w21(− 1
2z1,−12z2)w2( 1
2z1,12z2)
w1(3z1, 3z2) + w2(−√
3z1,−√
3z2)=
5w31w
22 + 3w2
1w22 − w2
1w2 + 2z1z2
w1w22 + 4w1w2 − w2 + 1
5− 21z1z2
w31w
32 + 37w1w2
2 − 23z1z2w21w
22 − 4w2
1w2
,
(1− z1)(1− z2)w31( 1√
2z1,
1√2z2)
w2(√
3z1,√
3z2)=
( 8z2
+ 8z1
)w41w2 + 8w3
1 − 8w21
3w41w
32 − 7w4
1w22 + 2w2
1w22 + 5w2
1w2 + 4w2 + 2,
admits a non-admissible meromorphic solution (w1, w2) = (− 1
z1z2, z21z
22). Clearly, we obtain
λ1 = 2, λ2 = 1, p1 = 3, p2 = 3. λ1 = 3, λ2 = 1, q1 = 4, q2 = 3.
In this case
p1 > λ1, q2 > λ2, p2 > λ2, q1 > λ1.
This example shows that Theorem 3.4 holds.
To prove theorems, we need some lemmas as follows.
Lemma 3.1. [2] Let R (z, w(z)) =a0(z) + a1(z)w(z) + · · ·+ ap′(z)w
p′(z)
b0(z) + b1(z)w(z) + · · ·+ bq′(z)wq′(z)
be an irreducible
rational function in w(z) with the meromorphic coefficients ai(z) and bj(z). If w(z) is a
meromorphic function in Cn, then
T (r,R(z, w)) = maxp′, q′T (r, w) +O∑
T (r, ai) +∑
T (r, bj).
Lemma 3.2. Let w1 and w2 be non-constant meromorphic functions of zero order in Cn,
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
975 Yue Wang 967-979
10
qi ∈ Cn\0, i = 1, ..., n2. If
Ω1(z, w1, w2) =∑(i)∈I
a(i)(z)2∏k=1
wik0k (wk(q1z))ik1 · · · (wk(qn2
z))ikn2 ,
a(i)(z) is a small function of w1 and w2. λ1k = maxn2∑l=0
ikl(k = 1, 2), then
T (r,Ω1(z, w1, w2)) ≤ λ11T (r, w1) + λ12T (r, w2) + S(r, w1) + S(r, w2) + S(r).
Proof It is easy to prove by Theorem B.
As the proof of Theorem 2.1 in [18], we have
Lemma 3.3. Let w1 and w2 be nonconstant meromorphic functions in Cn. If
limr→∞
supr 6∈I1
S(r)
T (r, w1)= 0, T (r, w2) = O S(r) (r 6∈ I2),
then
limr→∞
supr 6∈I1∪I2
T (r, w2)
T (r, w1)= 0,
where I1, I2 are both exceptional sets with upper logarithmic density 0.
Lemma 3.4. Let w1 and w2 be non-constant meromorphic functions of zero order in Cn,
qi ∈ Cn\0, i = 1, ..., n2. Let Φ1 =Ω1(z, w1, w2)
Ω2(z, w1, w2), a(i)(z), b(j)(z) are both small functions
of w1 and w2. If
λ1 = maxλ11, λ21, λ2 = maxλ12, λ22,then
T (r,Φ1) ≤ λ1T (r, w1) + λ2T (r, w2) + S(r, w1) + S(r, w2).
Proof Let Cn < r >= z = (z1, ..., zn) ∈ Cn : |z1|2 + · · ·+ |zn|2 = r2.
Firstly, we estimate m (r,Φ1). Set
u(z) = max |Ω1(z, w1, w2)|, |Ω2(z, w1, w2)| ,we have
log+ |Φ1| = log u(z)− log |Ω2(z, w1, w2)|,thus ∫
Cn<r>log+ |Φ1|σn =
∫Cn<r>
log u(z)σn −∫Cn<r>
log |Ω2(z, w1, w2)|σn.
As the proof of Lemma 3.3 in [18], and using Theorem A and Theorem B, we have
T (r,Φ1) ≤ λ1T (r, w1) + λ2T (r, w2) + S(r, w1) + S(r, w2).
This completes the proof of Lemma 3.4.
Proof of Theorem 3.1 Let w be a meromorphic function in Cn, and w(z) be a transcen-
dental meromorphic solution of zero order of (3.1). It follows from Lemma 3.1 and Theorem B
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
976 Yue Wang 967-979
Y.W Some results for meromorphic functions of several variables 11
thatmaxp, qT (r, w(z)) = T (r,R(z, w)) + S(r, w)
= T (r,
n2∑j=1
w(qjz)) + S(r, w)
≤ n2T (r, w) + S(r, w).
Thus, we have
maxp, q ≤ n2.
This completes the proof of Theorem 3.1.
Proof of Theorem 3.2 Let w1 and w2 be meromorphic functions in Cn. Let (w1, w2) be
a set of admissible meromorphic function of (3.2). From the first and the second equation of
(3.2), and also using Lemma 3.1 and Lemma 3.2, we obtain
maxp1, q1T (r, w1) ≤ λ11T (r, w1) + λ12T (r, w2) + S(r, w1) + S(r, w2) + S(r). (3.4)
maxp2, q2T (r, w2) ≤ λ21T (r, w1) + λ22T (r, w2) + S(r, w1) + S(r, w2) + S(r). (3.5)
By (3.4) and (3.5), we have
[maxp1, q1 − λ11 + o(1)]T (r, w1) ≤ (λ12 + o(1))T (r, w2). (3.6)
[maxp2, q2 − λ22 + o(1)]T (r, w2) ≤ (λ21 + o(1))T (r, w1). (3.7)
Combining (3.6) and (3.7), we obtain
[maxp1, q1 − λ11] [maxp2, q2 − λ22] ≤ λ12λ21.
This completes the proof of Theorem 3.2.
Proof of Theorem 3.3 Let w1 and w2 be nonconstant meromorphic functions of zero
order in Cn. It follows from Lemma 3.1 and Lemma 3.2 that
maxp1, q1T (r, w1) ≤ λ11T (r, w1) + λ12T (r, w2) + S(r, w1) + S(r, w2) + S(r). (3.8)
maxp2, q2T (r, w2) ≤ λ21T (r, w1) + λ22T (r, w2) + S(r, w1) + S(r, w2) + S(r). (3.9)
If w1 is admissible and w2 is non-admissible, then the inequality (3.8) becomes
maxp1, q1 ≤ λ11 + (λ12 + o(1))T (r, w2)
T (r, w1)+
S(r)
T (r, w1),
using Lemma 3.3, we get
maxp1, q1 ≤ λ11,outside of a set with upper logarithmic density 0. It is in contradiction with the condition (i).
If w2 is admissible and w1 is non-admissible, then the inequality (3.9) becomes
maxp2, q2 ≤ λ22 + (λ21 + o(1))T (r, w1)
T (r, w2)+
S(r)
T (r, w2),
using Lemma 3.3, we get
maxp2, q2 ≤ λ22,outside of a set with upper logarithmic density 0. It is in contradiction with the condition (ii).
Thus both w1 and w2 are admissible or none of w1 and w2 is admissible.
This proves Theorem 3.3.
Proof of Theorem 3.4 Let w1 and w2 be meromorphic functions in Cn. Let (w1, w2) be
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
977 Yue Wang 967-979
12
a zero order meromorphic solution of (3.3). Using Lemma 3.1 and Lemma 3.4, we obtain
p1T (r, w1) +O T (r, w2) ≤ λ1T (r, w1) + λ2T (r, w2) + S(r, w1) + S(r, w2) + S(r). (3.10)
p2T (r, w2) +O T (r, w1) ≤ λ1T (r, w1) + λ2T (r, w2) + S(r, w1) + S(r, w2) + S(r). (3.11)
q1T (r, w1) +O T (r, w2) ≤ λ1T (r, w1) + λ2T (r, w2) + S(r, w1) + S(r, w2) + S(r). (3.12)
q2T (r, w2) +O T (r, w1) ≤ λ1T (r, w1) + λ2T (r, w2) + S(r, w1) + S(r, w2) + S(r). (3.13)
If w1 is admissible and w2 is non-admissible, then the inequality (3.10) becomes
p1 +O T (r, w2)T (r, w1)
≤ λ1 + (λ2 + o(1))T (r, w2)
T (r, w1)+
S(r)
T (r, w1).
Using Lemma 3.3, we get
p1 ≤ λ1,outside of a set with upper logarithmic density 0. It is in contradiction with the first inequality
of (i).
If w2 is admissible and w1 is non-admissible, then the inequality (3.13) becomes
q2 +O T (r, w1)T (r, w2)
≤ λ2 +(λ1 + o(1)
) T (r, w1)
T (r, w2)+
S(r)
T (r, w2).
Using Lemma 3.3, we get
q2 ≤ λ2,outside of a set with upper logarithmic density 0. It is in contradiction with the second inequality
of (i).
Similarly, we can prove for conditions (ii).
Thus both w1 and w2 are admissible or none of w1 and w2 is admissible.
This completes the proof of Theorem 3.4.
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
All authors drafted the manuscript, read and approved the final manuscript.
Acknowledgements
The work is supported by the Outstanding Innovative Talents Cultivation Funded Programs
2014 of Renmin Univertity of China.
References
[1] Yi, HX, Yang, CC: Theory of the Uniqueness of Meromorphic Function, Science Press, Bei-
jing(1995)(in Chinese)
[2] Tu, ZH: Some Malmquist type theorems of partial differential equations on Cn. J. Math. Anal.
Appl. 179, 41-60(1993)
[3] Korhonen, R: A difference Picard theorem for meromorphic functions of several variables. Com-
putational Methods and Function Theory, 12, 343-361(2012)
[4] Cao, TB: Difference analogues of the second main theorem for meromorphic functions in several
complex variables. Math. Nachr. 287, 530-545(2014)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
978 Yue Wang 967-979
Y.W Some results for meromorphic functions of several variables 13
[5] Wen, ZT: The q-difference theorems for meromorphic functions of several variables. Abstr. Appl.
Anal. Article ID: 736021(2014)
[6] Zhang, JL, Korhonen, R: On the Nevanlinna characteristic of f(qz) and its applications. J.
Math. Anal. Appl. 369, 537-544(2010)
[7] Barnett, DC, Halburd, RG, Korhone, RJ, Morgan, W: Nevanlinna theory for the q-difference
operator and meromorphic solutions of q-difference equations. Royal Society of Edinburgh, 137A,
457-474(2007)
[8] Laine, I, Yang, CC: Clunie theorems for difference and q-difference polynomials. J.London Math.
Soc. 76, 556-566(2007)
[9] Morgan, W: Nevanlinna theory for the q-difference equations. Proc. Roy. SocEdinburgh. 137,
457-474(2007)
[10] Laine, I, Yang, CC: Value distribution of difference polynomials. Proc. Japan Acad. Ser. A
Math. Sci. 83, 148-151(2007)
[11] Chiang, YM, Feng, SJ: On the Nevanlinna characteristic of f(z + η) and difference equations in
the complex plane. The Ramanujan J. 16, 105-129(2008)
[12] Halburd, RG, Korhonen, RJ: Difference analogue of the Lemma on the logarithmic derivative
with applications to difference equations. J. Math. Anal. Appl. 314, 477-487(2006)
[13] Chen, ZX: On difference equations relating to Gamma function. Acta Mathematica Scientia,B,
31, 1281-1294(2011)
[14] Gao, LY: Solutions of complex higher-order difference equations. Acta Mathematica Sinica, 56,
451-458(2013)
[15] Gao, LY: On meromorphic solutions of a type of difference equations. Chinese Contemporary
Math, 35, 163-170(2014)
[16] Gao, LY: The growth order of solutions of systems of complex difference equations. Acta Math-
ematica Scientia, 33B, 814-820(2013)
[17] Gao, LY: Systems of complex difference equations of Malmquist type. Acta Mathematica Sinica,
55, 293-300(2012)
[18] Gao, LY: On admissible solution of two types of systems of complex differential equations. Acta
Math.Sinica, 43, 149-156(2000)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
979 Yue Wang 967-979
Nonstationary refinable functions based on
generalized Bernstern polynomials
Ting Cheng and Xiaoyuan Yang
Department of Mathematics, Beihang University,LMIB of the Ministry of Education, Beijing 100191, China
E-mail:[email protected]
Corresponding author:[email protected]
Abstract In this paper, we introduce a new family of nonstationary refinable functions from
Generalized Bernstein Polynomials, which include a class of nonstationary refinable functions
generated from the family of masks for the pseudo splines of type II (see [17]). Furthermore, a
proof of the convergence of nonstationary cascade algorithms associated with the new masks is
completed. We then construct symmetric compacted supported nonstationary C∞ tight wavelet
frames in L2(R) with the spectral frame approximation order.
Keywords Nonstationary tight wavelet frames; Nonstationary refinable functions; Nonsta-
tionary cascade algorithms; Generalized Bernstein Polynomials; Spectral frame approximation
order.
Mathematics Subject Classification (2000) 42C40 · 41A15 · 46B15.
1 Introduction
In frame systems, because tight wavelet frames (in the stationary case) can not satisfy
compactly supported C∞ properties, the nonstationary case was considered to obtain C∞ tight
wavelet frames with compacted support. Recently, the development of nonstationary tight
wavelet frames has attracted a considerable amount of attention.
In 2008, Han and Shen [17] obtained symmetric compactly supported C∞ tight wavelet
frames in L2(R) with the spectral frame approximation order based on pseudo-splines of type
II. In 2009, compactly supported nonstationary C∞ tight wavelet frames in L2(Rs) with the
spectral frame approximation order from pseudo box splines were constructed in [22]. Li and
1
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
980 Ting Cheng et al 980-993
NONSTATIONARY REFINABLE FUNCTIONS BASED ON GENERALIZED
Shen [22] generalized univariate pseudo-splines to the multivariate setting and got a new class
of refinable functions named pseudo box splines. Next, in [18] and [23], the analysis of char-
acterization of nonstationary tight wavelet frames in Sobolev spaces was given. Han and Shen
[18] characterized Sobolev spaces of an arbitrary order of smoothness using nonstationary tight
wavelet frames for L2(Rs). Also, approximation order of nonstationary tight wavelet frames
in Sobolev spaces was obtained in [23]. Recently, the nonstationary subdivision scheme, which
nonstationary cascade algorithms is closely related to, was studied in [2–10, 12, 15, 21, 24, 26].
In particular, in [14] and [20], the properties of nonstationary subdivision scheme were per-
formed. Daniel et al.[14] and Jeonga et al.[20] showed C2 approximating and Holder regularities
of nonstationary subdivision scheme, respectively.
This paper is concerned with the study of symmetric C∞ nonstationary tight wavelet frames
in L2(R) with compacted support and the spectral frame approximation order, which gener-
alize nonstationary tight wavelet frames from pseudo-splines of type II in [17]. We discover
a new extensive function based on Generalized Bernstein polynomials [1]. Furthermore, exis-
tence of L2-solutions of nonstationary refinable functions from the new extensive function is
implemented. At last, we prove the convergence of nonstationary cascade algorithms of the new
family of nonstationary refinable functions.
The remainder of this paper is organized as follows: Section 2 collects some notations.
Section 3 elaborates on existence of L2-solutions of nonstationary refinable functions. Section 4
implements convergence of nonstationary cascade algorithms. Section 5 constructs symmetric
C∞ nonstationary tight wavelet frames in L2(R) with compacted support and the spectral
frame approximation order. Section 6 gives the conclusion.
2 Preliminaries
For the convenience of the readers, we review some definitions about nonstationary refinable
functions in this section.
Generalized Bernstein polynomials [1] are defined as
S(n)k (t) =
(nk
) t(t + α) · · · (t + [k − 1]α)(1− t)(1− t + α) · · · (1− t + [n− k − 1]α)(1 + α)(1 + 2α) · · · (1 + [n− 1]α)
, (2.1)
where α ≥ 0. We apply (2.1) to marks of new refinable functions by substituting t = sin2(ω2 ),
n = m + l in (2.1) and the summation of l + 1 terms of them as follows:
τm,l,α0 (ω) :=
l∑
k=0
(m+l
k
)(
k−1∏
i=0
(sin2(ω
2) + iα)
m+l−k−1∏
i=0
(cos2(ω
2) + iα)
)/
m+l−1∏
i=1
(1 + iα). (2.2)
Let τm,l,α0,j (ω) = τ
mj ,lj ,αj
0 (ω) (j ∈ N) be defined in (2.2), we obtain
τm,l,α0,j (ω) :=
lj∑
k=0
(mj+lj
k
)
k−1∏
i=0
(sin2(ω
2) + iαj)
mj+lj−k−1∏
i=0
(cos2(ω
2) + iαj)
/
mj+lj−1∏
i=1
(1 + iαj),(2.3)
2
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
981 Ting Cheng et al 980-993
TING CHENG AND XIAOYUAN YANG
where two positive integers lj , mj and αj (j ∈ N) satisfy lj < mj − 5,∞∑
j=1
2−jmj < ∞,
limj→∞
mj = ∞ and 0 ≤ αj < 13(mj + lj)− 7 .
A class of 2π-periodic trigonometric polynomials aj , j ∈ N, and their associated nonstation-
ary refinable functions φj−1, j ∈ N, defined by
φj−1(ω) := aj(ω/2)φj(ω/2) =∞∏
n=1
an+j−1(2−nω) ω ∈ R, j ∈ N, (2.4)
where the 2π-periodic trigonometric polynomials aj , j ∈ N, are called refinement masks. Here
the Fourier transform f of a function f ∈ L1(R) is defined to be f(ω) :=∫R f(t)e−itωdt and
can be naturally extended to square integrable functions.
The notation T was introduced in [25], which is defined by
T := R/[2πZ].
Denote deg(a) the smallest nonnegative integer such that its Fourier coefficients of a vanish
outside [-deg(a),deg(a)]. deg(a) here is the minimal integer k such that [−k, k] contains the
support of the Fourier coefficients of both a and a(−·), which was introduced in [17].
In the following, we will adopt some of the notations from [19]. The transition operator Ta
for 2π-periodic functions a and f can be defined as
[Taf ](w) := |a(ω/2)|2f(ω/2) + |a(ω/2 + π)|2f(ω/2 + π), ω ∈ R.
For τ ∈ R, a quantity is defined by
ρτ (a,∞) := lim supn→∞
∥∥∥Tna
(∣∣∣sin(ω
2
)∣∣∣τ)∥∥∥
1/n
L∞(T).
The notation ρ(a) is defined by
ρ(a) := infρτ (a,∞) : |a(ω + π)|2| sin(ω/2)|τ ∈ L∞(T) and τ ≥ 0.
A function f ∈ W ν2 (R) if it satisfys
‖f‖2W ν2 (R) :=
∫
R(1 + |ω|2ν)|f(ω)|2dω < ∞.
As [17], let aj∞j=1 be a sequence of 2π-periodic measurable functions. Define fn∞j=1 by
fn(ω) := χ[−π,π](2−nω)n∏
j=1
aj(2−nω), ω ∈ R, n ∈ N, (2.5)
where χ[−π,π] denotes the characteristic function of the interval [−π, π]. This can be understood
as a representation of the nonstationary cascade algorithm associated with the masks aj∞j=1
in the frequency domain. For a sequence of masks aj∞j=1 and an initial function f ∈ W ν2 (R),
we say that the (nonstationary) cascade algorithm associated with masks aj∞j=1 and an initial
3
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
982 Ting Cheng et al 980-993
NONSTATIONARY REFINABLE FUNCTIONS BASED ON GENERALIZED
function f converges in the Sobolev space W ν2 (R), if fn ∈ W ν
2 (R) for all n ∈ N and the sequence
fn∞j=1 is convergent in W ν2 (R).
Denote ρ(a) the spectral radius of the square matrix (c2j−k)−K≤j,k≤K and define ν2(a) :=
−1/2−log2
√ρ(a). It is known ([[16], Theorem 4.3 and Proposition 7.2] and [[19], Theorem 2.1])
that the stationary cascade algorithm associated with a 2π-periodic trigonometric polynomial
mask a converges in f ∈ W ν2 (R) if and only if ν2(a) > ν.
For a sequence φn∞n=0 of functions in L2(R), we define the linear operators Pn(f), n ∈ N0,
by
Pn(f) :=∑
k∈Z〈f, φn;n,k〉φn;n,k, f ∈ L2(R) with φn;n,k := 2n/2φn(2n · −k). (2.6)
Wavelet functions ψ`j−1, j ∈ N and ` ∈ 1, · · · ,Jj, are obtained from φj by
ψ`j−1(ω) := b`
j(ω/2)φj(ω/2), ` ∈ 1, · · · ,Jj, (2.7)
where Jj are positive integers and each b`j , ` = 1, · · · ,Jj , is called a (high-pass) wavelet mask.
Denote N0 := N⋃0. We say that φ0
⋃ψ`j : j ∈ N0, ` = 1, · · · ,Jj+1 generates a nonsta-
tionary tight wavelet frame in L2(R) if
φ0(· − k) : k ∈ Z⋃ψ`
j;j,k := 2j/2ψ`j(2
j · −k) : j ∈ N0, k ∈ Z, ` = 1, · · · ,Jj+1 (2.8)
is a tight frame of L2(R).
Finally, we note that the 2π-periodic trigonometric polynomial wavelet masks b`j , j ∈ N
and ` ∈ 1, · · · , γ, can be constructed from the masks aj by many ways provided that the
refinement masks aj , j ∈ N, satisfy |aj(ω)|2 + |aj(ω + π)|2 ≤ 1, a.e.ω ∈ R. Define
b1j (ω) := e−iωaj(ω + π),
b2j (ω) := 2−1[Aj(ω) + e−iωAj(ω)],
b3j (ω) := 2−1[Aj(ω) + e−iωAj(ω)],
(2.9)
where Aj is a π-periodic trigonometric polynomial with real coefficients such that
|Aj(ω)|2 = 1− |aj(ω)|2 − |aj(ω + π)|2.
Then, aj , b1j , b2
j and b3j , j ∈ N, satisfy
|aj(ω)|2 +Jj∑
`=1
|b`j(ω)|2 = 1 and aj(ω)aj(ω + π) +
Jj∑
`=1
b`j(ω)b`
j(ω + π) = 0, (2.10)
with J = 3. Thus, the wavelet system in (2.8) is a compactly supported tight wavelet frame in
L2(R) (see, [17], Theorem1.1). Furthermore, the corresponding wavelets defined by (2.7) using
masks in (2.9) are symmetric or antisymmetric whenever φj is symmetric.
4
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
983 Ting Cheng et al 980-993
TING CHENG AND XIAOYUAN YANG
3 Existence of L2-solutions of nonstationary refinable func-tions
In this section, demonstration of the existence of L2-solutions of nonstationary refinable
functions is given. For notational simplicity, we will introduce the following two definitions:
Bk,j(ω) :=
k−1∏
i=0
(sin2(ω
2) + iαj)
mj+lj−k−1∏
i=1
(cos2(ω
2) + iαj)
/
mj+lj−1∏
i=1
(1 + iαj), j ∈ N.
T0,j(ω) :=l∑
k=0
(mj+lj
k
)
k−1∏
i=0
(sin2(ω
2) + iαj)
mj+lj−k−1∏
i=1
(cos2(ω
2) + iαj)
/
mj+lj−1∏
i=1
(1+iαj), j ∈ N.
Two lemmas about the relations of the quantities ρτ (τm,l,α0,j (ω),∞), j ∈ N associated with
masks (2.3) will be provided in the following.
Lemma 3.1 ([19], Theorem 4.1) Let a be a 2π-periodic measurable function such that |a|2 ∈Cβ(T) with |a|2(0) 6= 0 and β > 0. If |a(ω)|2 = |1 + e−iω|2τ |A(ω)|2 a.e. ω ∈ R for some τ ≥ 0
such that A(ω) ∈ L∞(T), then
ρ2τ (a,∞) = infn∈N
‖Tna 1‖
1n
L∞(T) = limn→∞
‖Tna 1‖
1n
L∞(T) = ρ0(A,∞).
Lemma 3.2 ([19], Theorem 4.3) Let a and c be 2π-periodic measurable functions such that
|a(ω)| ≤ |c(ω)|
for almost every ω ∈ R. Then
ρτ (a,∞) ≤ ρτ (c,∞), τ ∈ R.
The following two lemmas are necessary for proving existence of L2-solutions of nonstation-
ary refinable functions.
Lemma 3.3 ([17], Lemma 2.1) Let aj , j ∈ N be a 2π-periodic trigonometric polynomials such
that supj∈N ‖aj‖L∞(R) < ∞. If∑∞
j=1 2−jdeg(aj) < ∞ holds and∑∞
j=1 |aj(0) − 1| < ∞, then
the infinite product (2.4) converges uniformly on every compact set of R and all φj , j ∈ N0, in
(2.4) are well-defined compactly supported tempered distributions.
Lemma 3.4 ([17], Lemma 2.2) Let aj , j ∈ N, be 2π-periodic measurable functions satisfying
|aj(ω)|2 + |aj(ω + π)|2 ≤ 1, a.e.ω ∈ R for each j ∈ N. Assume that, for every j ∈ N0,
φj(ω) := limN→∞∏N
n=1an+j(2−nω) is well defined for almost every ω ∈ R; that is, the infinite
product in (2.4) exists for almost every point in R. Then [φj , φj ](ω) :=∑
k∈Z |φj(ω + 2πk)|2 ≤1, a.e.ω ∈ R∀j ∈ N0 holds and consequently, φj ∈ L2(R) with ‖φj‖L2(R) ≤ 1 for every j ∈ N0.
A useful condition of establishing existence of L2-solutions of nonstationary refinable func-
tions is described in the following lemma.
5
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
984 Ting Cheng et al 980-993
NONSTATIONARY REFINABLE FUNCTIONS BASED ON GENERALIZED
Lemma 3.5 For two positive integers lj ,mj, lj < mj − 5, j ∈ N, if
0 ≤ αj <1
3(mj + lj)− 7(j ∈ N), (3.1)
then
maxω∈T
Bk,j(ω) ≤ (12)mj+lj−1, k = 1, 2, . . . , lj , (3.2)
Proof. For j ∈ N, k = 1, 2, . . . , lj , it is obvious that
Bk,j(ω) =cos2(ω
2 ) + (mj + lj − 1− j)αj
sin2(ω2 ) + jαj
Bk+1,j(ω).
We claim thatBk,j(ω)
Bk+1,j(ω)=
cos2(ω2 ) + (mj + lj − 1− j)αj
sin2(ω2 ) + jαj
> 1. (3.3)
Since lj < mj − 5, for k = 1, 2, . . . , lj , it holds that
k < mj + lj − 1− k. (3.4)
There are two cases to consider:
Case I: Suppose that cos(ω) ≥ 0. By (3.1) and (3.4) , it is easy to see that
αj > 0 >− cos(ω)
mj + lj − 1− 2k.
Then
cos2(ω
2) + (mj + lj − 1− k)αj > sin2(
ω
2) + kαj . (3.5)
This implies Condition (3.3).
Case II: Suppose that cos(ω) < 0. Note that since
(22lj − 2−1)(mj − 1− lj)− ljlj(mj − 1− lj)
>0.5(mj − 1− lj)− lj
lj(mj − 1− lj)> 0,
we can obtain that
2mj+lj−1 − 2−1
lj− 1
mj − lj − 1=
(2mj+lj−1 − 2−1)(mj − 1− lj)− ljlj(mj − 1− lj)
>(22lj − 2−1)(mj − 1− lj)− lj
lj(mj − 1− lj)> 0.
By (3.1), then
αj ≥ 2mj+lj−1 − 2−1
lj>
1mj − lj − 1
=1
mj + lj − 1− 2lj≥ | cos(ω)|
mj + lj − 1− 2k=
− cos(ω)mj + lj − 1− 2k
,
for k = 1, 2, . . . , lj . Then (3.5) holds. This concludes the claim (3.3).
By using (3.1), one gets
(4(mj + lj − 2)− (mj + lj − 1))αj <4(mj + lj − 2)− (mj + lj − 1)
3(mj + lj)− 7= 1.
Then(mj + lj − 2)αj
(1 + αj)(1 + (mj + lj − 1)αj)<
(mj + lj − 2)αj
1 + (mj + lj − 1)αj<
14. (3.6)
6
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
985 Ting Cheng et al 980-993
TING CHENG AND XIAOYUAN YANG
Since lj < mj − 5, we have
(3(mj + lj)− 7)− (mj + lj − 4) = 2mj + 2lj − 3 > 0.
Then
13(mj + lj)− 7)
<1
mj + lj − 4.
Thus
(2(mj + lj − 3)− (mj + lj − 2))αj <2(mj + lj − 3)− (mj + lj − 2)
mj + lj − 4= 1.
Similarly, one has
(mj + lj − 3)αj
1 + (mj + lj − 2))αj<
12. (3.7)
For any x, Notice that
(x
1 + (1 + x))′ > 0 (3.8)
and B1,j(ω) which is a continuous function on [−π, π] and is differentiable on (−π, π), has the
maximum value at ω = π. The reason as follow:
The equation [B1,j(ω)]′ = 0 has three zeros, at ω = 0,±π. Since [B1,j(ω)]′′ > 0, B1,j(0)
is the minimum of B1,j(ω) on [−π, π]. Thus B1,j(±π) is the maximum of B1,j(ω) on [−π, π].
Therefore, applying (3.3), (3.6), (3.7), (3.8) and
B1,j(ω) =
sin2(
ω
2)
mj+lj−2∏
i=1
(cos2(ω
2) + iαj)
/
mj+lj−1∏
i=1
(1 + iαj)
≤mj+lj−2∏
i=1
iαj/
mj+lj−1∏
i=1
(1 + iαj)
≤mj+lj−3∏
i=1
iαj
1 + (i + 1)αj· (mj + lj − 2)αj
(1 + αj)(1 + (mj + lj − 1)αj)
≤ ((mj + lj − 3)αj
1 + (mj + lj − 2)αj)mj+lj−3 · 1
4
= (12)mj+lj−1,
we get the inequality (3.2). ¶
Theorem 3.1 Let τm,l,α0,j (ω) be the mark (2.3), which are defined in (2.4), then the infinite
product in (2.4) converges uniformly on every compact set of R.
Proof. Since τm,l,α0,j (ω) = τm,l,α
0,j (ω + 2π), j ∈ N, we obtain τm,l,α0,j (ω) are 2π-periodic trigono-
7
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
986 Ting Cheng et al 980-993
NONSTATIONARY REFINABLE FUNCTIONS BASED ON GENERALIZED
metric polynomials. Applying Lemma 3.5, one has
|τm,l,α0,j (ω)| = |
lj∑
k=0
(mj+lj
k
)
k−1∏
i=0
(sin2(ω
2) + iαj)
mj+lj−k−1∏
i=0
(cos2(ω
2) + iαj)
/
mj+lj−1∏
i=1
(1 + iαj)|
≤∣∣∣∣∣∣1 +
(maxω∈T
Bk,j(ω)) lj∑
k=1
(mj+lj
k
)∣∣∣∣∣∣
= 1 + (12)mj+lj−1|
lj∑
k=1
(mj+lj
k
)|
≤ 1 + (12)mj+lj−1 · 2mj+lj = 3.
Thus, there exists M = 3, for any j ∈ N, we derive that τm,l,α0,j (ω)| ≤ 3 holds.
For τm,l,α0,j (0) = 1, we obtain
∞∑
j=1
|τm,l,α0,j (0)− 1| = 0 < ∞.
By using lj < mj − 5,∑∞
j=1 2−jmj < ∞, ones get
∞∑
j=1
2−jdeg(τm,l,α0,j (ω)) =
∞∑
j=1
2−j (2(mj + lj) + 1)
<∞∑
j=1
2−j(4mj − 9) < ∞.
Therefore, by Lemma 3.3, the infinite product in (2.4) converges uniformly on every compact
set of R. ¶
Theorem 3.2 Let τm,l,α0,j (ω) be the mark (2.3), which are defined in (2.4), then the correspond-
ing nonstationary refinable functions φj ∈ L2(R), j ∈ N0.
Proof. By Theorem 3.1, we obtain that
φj(ω) = limN→∞
N∏n=1
τm,l,α0,n+j−1(ω)(2−nω)
is well defined for almost every ω ∈ R. In the following, we claim that
|τm,l,α0,j (ω)|2 + |τm,l,α
0,j (ω + π)|2 ≤ 1, a.e.ω ∈ R. (3.9)
There are two cases to consider:
Case I: Suppose that ω = 0. One has
|τm,l,α0,j (0)|2 + |τm,l,α
0,j (π)|2 = 0 + 1 = 1, a.e.ω ∈ R.
Case II: Suppose that ω 6= 0. Set E0 = 0, for ω ∈ R \ E0, let t = 2ω, ones get
|τm,l,α0,j (ω)|2 = 2−4(1 + e−it)4|T0,j(t)|2
= (1 + e−it)4|2−2T0,j(t)|2.
8
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
987 Ting Cheng et al 980-993
TING CHENG AND XIAOYUAN YANG
Applying
B0,j(t) =m+l−1∏
i=0
(cos2(t
2) + iαj)/
m+l−1∏
i=1
(1 + iαj)
≤m+l−1∏
i=1
(1 + iαj)/m+l−1∏
i=1
(1 + iαj) = 1
and Lemma 3.5, we obtain
maxt∈T0,ג
2|2−2Tm,l,α0 (t)|2 = max
t∈T2−3
∣∣∣∣∣∣B0,j(t) +
l∑
j=0
(m+l
j
)Bk,j(t)
∣∣∣∣∣∣
2
< maxt∈T
2−3
∣∣∣∣∣1 +(
maxt∈[−π,π]
Bk,j(t)) l∑
k=1
(m+l
k
)∣∣∣∣∣
2
< maxt∈T
2−3
∣∣∣∣(1 + (12)m+l−1(2)m+l−1)
∣∣∣∣2
=12. (3.10)
Bringing Lemma 3.1 and Lemma 3.2 together yields
ρ(τm,l,α0,j (t)) 6 ρ4(τ
m,l,α0,j (t),∞) = ρ0(2−2T(t),∞) < 1.
For
ρ0(2−2T0,j(t),∞) = lim supn→∞
‖Tnτm,l,α0,j (t)
‖1/nL∞(T)
> Tτm,l,α0,j (t)
= |τm,l,α0,j (ω)|2 + |τm,l,α
0,j (ω + π)|2.
Thus,
|τm,l,α0,j (ω)|2 + |τm,l,α
0,j (ω + π)|2 < 1.
This concludes the claim (3.9).
By Lemma 3.4, the corresponding nonstationary refinable functions φj ∈ L2(R), j ∈ N0. ¶
4 Convergence of nonstationary cascade algorithms
In this section, demonstration of the convergence of nonstationary cascade algorithms in
the Sobolev space W ν2 (R) is given. We will show a lemma about a sufficient condition on the
convergence of a nonstationary cascade algorithm which is necessary for the following theorem.
Lemma 4.1 ([17], Proposition 2.6) Let aj and bj(j ∈ N) be 2π-periodic measurable functions
such that for all j ∈ N,
|aj(ω)| ≤ |bj(ω)|, a.e. ω ∈ R. (4.1)
Let η ∈ W ν2 (R) such that limj→∞ η(2−jω) = 1 for almost every ω ∈ R. Define
fn(ω) := η(2−nω)n∏
j=1
aj(2−nω) and gn(ω) := η(2−nω)n∏
j=1
bj(2−nω), ω ∈ R.
9
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
988 Ting Cheng et al 980-993
NONSTATIONARY REFINABLE FUNCTIONS BASED ON GENERALIZED
Assume that f∞(ω) := limn→∞∏n
j=1 aj(2−nω) and g∞(ω) := limn→∞∏n
j=1 bj(2−nω) are well
defined for almost every ω ∈ R. Then, limn→∞ ‖gn − g∞‖W ν2 (R) = 0 implies limn→∞ ‖fn −
f∞‖W ν2 (R) = 0. In particular, suppose that there are a positive integer J and a 2π-periodic
measurable function b such that
|aj(ω)| ≤ |b(ω)|, a.e.ω ∈ R, ∀j > J and aj ∈ L∞(R), 1 ≤ j ≤ J. (4.2)
For n ∈ N, define hn(ω) := η(2−nω)∏n
j=1 bj(2−nω). If hn∞n=1 converges in W ν2 (R), then fn
converges to fn in W ν2 (R), i.e., limn→∞ ‖fn − f∞‖W ν
2 (R) = 0.
Theorem 4.1 Let τm,l,α0,j (ω) be the mark (2.3), which are defined in (2.4), then, for every
n ∈ N0, the nonstationary cascade algorithm (2.5) associated with τmj ,lj ,α0,j+n (ω)∞j=1 converges
in W ν2 (R), for any ν ≥ 0. Consequently, the nonstationary refinable functions φ
mj ,lj ,αj , j ∈ N0,
in (2.4) must be well-defined compactly supported C∞(R) functions.
Proof. Since deg(τm,l,α0,j (ω)(ω)) ≤ 2(mj + lj) + 1 < 4mj − 9, and τm,l,α
0,j (ω)) = 1, applying∑∞
j=1 2−jmj < ∞, we get
∞∑
j=1
2−jdeg(τm,l,α0,j (ω)) ≤
∞∑
j=1
2−j(4mj − 9) ≤ 4∞∑
j=1
2−jmj < ∞.
Moreover, by (3.9), ones obtain |τm,l,α0,j (ω)| ≤ 1. By using Lemma (3.3), we can derive that
φmj ,lj ,α0,j , j ∈ N0 are well defined compactly supported functions.
Because τm,l,α0,j (ω) in the case α = 0 have ν2(τ
m,l,α0,j (ω)) → ∞ ([11, 13]). The same proof is
carried out for any α. So, there exists a positive integer J such that ν2(τm,l,α0,j (ω)) ≥ ν + 2. By
limj→∞mj = ∞, there exists a positive integer N such that mj > J and it is obtained that
ν2(τm,l,α0,j (ω)) ≥ ν + 2. (4.3)
Let b be the unique 2π-periodic trigonometric polynomial such that τm,l,α0,j (ω) = 2−1(1 +
e−iω)b(ω). Bringing the definition of ν2(b) and (4.3) together yields ν2(b) = ν2(τm,l,α0,j (ω))−1 ≥
ν + 1 > ν, thus, the stationary cascade algorithm associated with the mask b converges in
W ν2 (R) (see [([16]), Theorem 4.3]). Since |τm,l,α
0,j (ω)| ≤ |b(ω)|, applying Lemma (4.1), we derive
that the nonstationary cascade algorithm associated with masks τm,l,α0,j (ω) converges in W ν
2 (R).
The same proof is carried out for every φmj ,lj ,αn and for the nonstationary cascade algorithm
associated with masks τm,l,α0,j (ω)∞j=1. ¶
Remark 4.1 Notice that when αj = 0 (j ∈ N), the Theorem 4.1 in this paper is the same as
corresponding Theorem 2.8 given in [17].
5 Construction of nonstationary tight wavelet frames
In this section, we shall construct the symmetric C∞ tight wavelet frames in L2(R) with
compact support and the spectral frame approximation order based on the masks (2.3). The
10
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
989 Ting Cheng et al 980-993
TING CHENG AND XIAOYUAN YANG
following two lemmas analyze the approximation properties of the operators Pn and relations
of τm,l,α0,j (ω), respectively, which are useful for construction of C∞ tight wavelet frames.
Lemma 5.1 ([17], Theorem 3.2) Let aj , j ∈ N be 2π-periodic measurable functions such that
|aj(ω)|2 + |aj(ω + π)|2 ≤ 1, a.e.ω ∈ R holds for all j ∈ N and for every n ∈ N0, the function
φn(ω) := limJ→∞∏J
j=1 aj+n(2−jω) is well defined for almost every ω ∈ R. Let ν ≥ 0. If, for
n ∈ N, ∣∣∣1− |φn(ω)|2∣∣∣2
≤ Cφn|ω|2ν , a.e. ω ∈ [−π, π],∑
k∈Z\0|φn(ω)|2|φn(ω + 2πk)|2 ≤ C
φn|ω|2ν , a.e. ω ∈ [−π, π], (5.1)
where Cφn is a constant depending only on φn, then for the linear operators Pn in (2.6),
‖f − Pn(f)‖L2(R) ≤ max(2,√
Cφn)2−νn|f |W ν
2 (R) ∀f ∈ W ν2 (R) and n ∈ N. (5.2)
In particular, (5.1) is satisfied if
1− |φn(ω)|2 ≤ Cφn|ω|2ν . (5.3)
Lemma 5.2 Let τm,l,α0,j (ω) be the mark (2.3), which are defined in (2.4). For any 0 < ρ ≤ 1
and ν ≥ 0, there exist a positive integer N and a positive constant C, (both of them depend
only on ρ and ν), such that for all N ≤ ρm < l ≤ m,
0 ≤ 1− |τm,l,α0,j (ω)|2 ≤ C|ω|2ν ∀ω ∈ [−π, π]. (5.4)
Proof. Suppose that α = 0, the case holds (see [17], Lemma 3.3). Assume that (5.4) holds
for α = k − 1. Then suppose that α = k, since τm,l,α0,j (ω) increases as α increases, we have
0 ≤ 1− |τm,l,α0,j (ω)|2 ≤ 1− |τm,l,α
0,j (ω)|2 ≤ C|ω|2ν .
This completes the claim (5.4). ¶
Theorem 5.1 Let τm,l,α0,j (ω) be the mark (2.3). For j ∈ N, define φj−1 as in (2.4) and ψ1
j−1,
ψ2j−1, ψ3
j−1 as in (2.7) with the wavelet masks b1j , b2
j and b3j being derived from aj in (2.8).
Then the following hold:
(1) Each nonstationary refinable function φj , j ∈ N0, is a compactly supported C∞ real-valued
function that is symmetric about the origin: φj(−·) = φj.
(2) Each wavelet function ψ`j , ` = 1, 2, 3 and j ∈ N0, is a compactly supported C∞ real-valued
function with lj+1 vanishing moments and satisfies ψ`j(1 − ·) = ψ`
j for ` = 1, 2 and
ψ3j (1− ·) = −ψ3
j .
(3) The system φ0(· − k) : k ∈ Z⋃ψ`j;j,k := 2j/2ψ`
j(2j · −k) : j ∈ N0, k ∈ Z, ` = 1, 2, 3 is a
compactly supported symmetric C∞tight wavelet frame in L2(R).
11
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
990 Ting Cheng et al 980-993
NONSTATIONARY REFINABLE FUNCTIONS BASED ON GENERALIZED
(4) If in addition lim infj→∞ lj/mj > 0, then the tight wavelet frame in item (3) has the spectral
frame approximation order.
Proof. For item (1), by using Theorem 4.1, it is derived that all φj , j ∈ N0 are compactly
supported functions in C∞(R). Combining all the masks aj , which are 2π-periodic trigonometric
polynomials with real coefficients and are symmetric about the origin: aj = aj , and the definition
of φj in (2.4) yields,
φj(ω) = φj(ω),
which φj are real-valued. By the definition of τm,l,α0,j (ω) (j ∈ N) in (2.3), we get
1− |τm,l,α0,j (ω)|2 = O(|ω|2l), ω → 0. (5.5)
b`j = O(|ω|lj ), ω → 0 follows from the fact that (2.10) and (5.5). Thus, ψ`
j has lj+1 vanishing
moments.
For item (3), notice that the the definition of b`j in (2.9), we can straightforward obtain
that (2.10). Therefore, the wavelet system in (2.8) is a compactly supported tight wavelet
frame in L2(R) (see [17], Theorem1.1). For item (4), let ν be an arbitrary positive integer
and denote dj := | τm,l,α0,j (ω)|2. For lim infj→∞ lj/mj > 0, there exist a positive integer N and
0 < ρ < lim infj→∞ lj/mj such that 2ν < N < ρmj < lj ≤ mj for all j ≥ N . By using Lemma
5.2, it is to see that (5.4) holds. That is, there exists a positive constant C, independent of j,
such that
0 ≤ 1− dj(ω) ≤ C|ω|2ν , ω ∈ [−π, π] and j ≥ N.
For n ≥ N and ` ∈ N, since d`+n(0) = 1, ones get
|d`+n(0)− d`+n(2−`ω)| = |1− d`+n(2−`ω)| ≤ C2−2ν`|ω|2ν , ∀ω ∈ [−π, π].
Since d`+n(0) = 1, 0 ≤ d`+n(ω) ≤ 1 and (3.9), we derive that
0 ≤ 1− |φn(ω)|2 ≤∞∑
`=1
|d`+n(0)− d`+n(2−`ω)| ω ∈ R. (5.6)
Applying (5.6), it is obtained that
1− |φn(ω)|2 ≤ C|ω|2ν∞∑
`=1
2−2ν`, ω ∈ [−π, π].
Thus, (5.3) holds with
Cφn:= C
∞∑
`=1
2−2ν` =C
1− 2−2ν< ∞.
Combining Qn = Pn and Theorem 5.1, one has
‖f −Qn(f)‖L2(R) ≤ C12−νn|f |W ν2 (R), ∀f ∈ W ν
2 (R) and n ∈ N,
where C1 := max(2,√
C1−2−2ν ) is independent of f and n. Since ν is arbitrary, the tight wavelet
frame has the desired spectral frame approximation order. ¶
12
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
991 Ting Cheng et al 980-993
TING CHENG AND XIAOYUAN YANG
Remark 5.1 Under the condition αj = 0 (j ∈ N) of Theorem 5.1, one can derive that it is
consistent with the claim of Theorem 1.2 given in [17].
Acknowledgments
The authors would like to thank the referees for their valuable comments and suggestions
that have improved the paper immeasurably. The authors are grateful to the support of the Na-
tional Natural Science Foundation of China (Grant NO. 61271010)and Beijing Natural Science
Foundation4152029.
References
[1] R.P. Boyer and L.C. Thiel, Generalized Bernstein Polynomials and Symmetric Functions,Adv. Appl. Math., 28,17-39(2001).
[2] C. Costanza, Stationary and nonstationary affine combination of subdivision masks, Math.Comput. Simulation, 81,623-635(2010) .
[3] A. Cohen and N. Dyn, Nonstationary Subdivision Schemes and Multiresolution Analysis,SIAM J. Math. Anal., 27,1745-1769(1996).
[4] C. Conti, L. Gori and F. Pitolli, Totally positive functions through nonstationary subdivi-sion schemes, J. Comput. Appl. Math., 200,255-265 (2007) .
[5] W. S. Chen, On the convergence of multidimensional nonstationary subdivision schemes,Appl. Anal., 72,179-189(1999).
[6] W. S. Chen, C. Xu and W. Lin, Spectral approximation orders of multidimensional non-tationary biorthogonal semi-multiresolution analysis in sobolev space, J. Comput. Math.,24,81-90(2006).
[7] M. Cecchi and M. Fornasier, Fast homogenization algorithm based on asymptotic theoryand multiscale schemes, Numerical algorithms, 40,171-186(2005).
[8] N. Dyn, D. Levin and J. Yoon, Analysis of Univariate Nonstationary Subdivision Schemeswith Application to Gaussian-Based Interpolatory Schemes, SIAM J. Math. Anal., 39,470-488(2007).
[9] N. Dyn, Exponentials Reproducing Subdivision Schemes, Found Comput. Math., 3,187(2004).
[10] N. Dyn and D. Levin, Subdivision schemes in geometric modelling, Acta Numerica, 11,73-144(2002).
[11] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. Appl. Math. 61,SIAM, Philadelphia,1992.
[12] G. Derfel, N. Dynnd and D. Levin, Generalized Refinement Equations and SubdivisionProcesses, J. Approx. Theory, 80,272-297(1995).
[13] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl.Math., 41,909-996 (1988).
[14] S. Daniel and P. Shunmugaraj, An approximating C2 non-stationary subdivision scheme,Computer Aided Geometric Design, 26,810-821(2009).
[15] K. Gitta and S. Tomas, Adaptive directional subdivision schemes and shearlet multireso-lution analysis, SIAM J. Math. Anal., 41,1436-1471(2009).
[16] B. Han, Vector cascade algorithms and refinable function vectors in Sobolev spaces, J.Approx. Theory, 124,44-88(2003).
13
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
992 Ting Cheng et al 980-993
NONSTATIONARY REFINABLE FUNCTIONS BASED ON GENERALIZED
[17] B. Han and Z. Shen, Compactly supported symmetric C∞ wavelets with spectral approx-imation order, SIAM J. Math. Anal., 40,905-938(2008).
[18] B. Han and Z.W. Shen, Characterization of Sobolev spaces of arbitrary smoothness usingnonstationary tight wavelet frames, Israe J. Math., 172,371-398(2009).
[19] B. Han, Refinable functions and cascade algorithms in weighted spaces with Holder con-tinuous masks, SIAM J. Math. Anal., 40,70-102(2008).
[20] B. Jeonga, Y. J. Leea and J. Yoon, A family of non-stationary subdivision schemes repro-ducing exponential polynomials, J. Math. Anal. Appl., 402,207-219(2013).
[21] L. Kobbelt, Using the Discrete Fourier Transform to Analyze the Convergence of Subdivi-sion Schemes, Appl. Comput. Harmon. A., 1,68-91(1998).
[22] S. Li and Y. Shen, Pseudo box splines, J. Appl. Comput. Harmon. Anal., 26,344-356(2009).
[23] L.R. Liang, Approximation order of nonstationary tight wavelet frames in Sobolev spaces,Int. J. Pure Appl. Math., 72,481-489(2011).
[24] G. Laura and P. Francesca, Multiresolution analyses originated from nonstationary subdi-vision schemes, J.Comput. Appl. MAth., 221,406-415(2008).
[25] Y. Shen and S. Li, Cascade algorithm associated with Hlolder continuous masks. Appl.Math. Lett., 22,1213-1216(2009).
[26] L. Yue, S. Yang and Z. Xia, Multiscale decomposition of divided difference operators andnonstationary subdivision schemes of B-spline series, Numer. Math. J. Chinese Univ., 26,1-11(2004).
14
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
993 Ting Cheng et al 980-993
The Order and Type of Meromorphic Functions and EntireFunctions of Finite Iterated Order ∗†
Jin Tu1,∗, Yun Zeng1, Hong-Yan Xu2
1.College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China2. Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China
Abstract
In this paper, the authors investigate the p-iterated order and p-iterated type of f1 +f2, f1f2, where f1(z), f2(z) are meromorphic functions or entire functions with the samep-iterated order and different p-iterated type, and we obtain some results which improve andextend some previous results.Key words: meromorphic function; entire function; iterated order; iterated typeAMS Subject Classification(2010): 30D35, 30D15
1. Introduction and Notations
In this paper, we assume that readers are familiar with the fundamental results and thestandard notations of the Nevanlinna value distribution theory of meromorphic functions in thecomplex plane (e.g. [4, 6-8, 10, 14, 15]). Throughout this paper, by a meromorphic function f(z),we mean a meromorphic function in the complex plane. We use T (r, f) and M(r, f) to denotethe characteristic function of a meromorphic function and the maximum modulus of an entirefunction. In the following, we will recall some notations about meromorphic functions and entirefunctions.Definition 1.1. (see [4, 8, 10]) The order of a meromorphic function f(z) is defined by
σ(f) = limr→∞
log T (r, f)
log r. (1.1)
Especially, if 0 < σ(f) <∞, then the type of f(z) is defined by
ψ(f) = limr→∞
T (r, f)
rσ(f). (1.2)
Definition 1.2. (see [4, 6− 8, 10]) The order of an entire function f(z) is defined by
∗Corresponding author:[email protected]†This project is supported by the National Natural Science Foundation of China (11561031, 11561033) the
Natural Science Foundation of Jiangxi Province in China (Grant No.20132BAB211002, 20122BAB211005) and theFoundation of Education Bureau of Jiangxi Province in China (GJJ14271, GJJ14272).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
994 Jin Tu et al 994-1003
2 J. Tu, Y. Zeng, H. Y. Xu
σ(f) = limr→∞
log T (r, f)
log r= lim
r→∞
log logM(r, f)
log r. (1.3)
Especially, if 0 < σ(f) <∞, then the type of f(z) is defined by
τ(f) = limr→∞
logM(r, f)
rσ(f). (1.4)
The order and type are two important indicators in revealing the growth of meromorphic functionsin the complex plane or analytic functions in the unit disc. Many authors have investigated thegrowth ofmeromorphic functions or analytic functions in the unit disc(e.g. [3,4,7-10,12-15]) andobtain a lot of classical results in the following.Theorem A. (see [4, 10, 14, 15]) Let f1(z) and f2(z) be meromorphic functions of finite order
satisfying σ(f1) = σ1 and σ(f2) = σ2. Then
σ(f1 + f2) ≤ maxσ1, σ2, σ(f1f2) ≤ maxσ1, σ2.
Furthermore, if σ1 = σ2, then σ(f1 + f2) = σ(f1f2) = maxσ1, σ2.
Theorem B. (see [15]) Let f1(z) and f2(z) be meromorphic functions of finite order. Thenµ(f1 + f2) ≤ maxσ(f1), µ(f2), µ(f1f2) ≤ maxσ(f1), µ(f2).
Theorem C. (see [11]) Let f1(z) and f2(z) be meromorphic functions of finite order satisfyingσ(f1) < µ(f2), then µ(f1 + f2) = µ(f1f2) = µ(f2).
Theorem D. (see [7]) Let f1(z) and f2(z) be entire functions of finite order satisfying σ(f1) =σ(f2) = σ. Then the following two statements hold:(i) If τ(f1) = 0 and 0 < τ(f2) <∞, then σ(f1f2) = σ, τ(f1f2) = τ(f2).(ii) If τ(f1) <∞ and τ(f2) =∞, then σ(f1f2) = σ, τ(f1f2) =∞.
Theorem E. (see [4, 14, 15]) Let f(z) be a meromorphic function of finite order, then σ(f ′) =σ(f).
From Theorems A−E, a natural question is : can we get the similar results for entire functionsand meromorphic functions of infinite order (i.e. finite iterated order)? In the following, we recallsome notations and definitions of finite iterated order. For r ∈ (0,+∞), we define exp1 r = er
and expi+1 = exp(expi r), i ∈ N; for sufficiently large r ∈ (0,+∞), we define log1 r = log rand logi+1 r = log(logi r), r ∈ N; we also denote exp0 r = r = log0 r and exp−1 r = log1 r.Throughout this paper, we use p to denote a positive integer. We denote the linear measure of aset E ⊂ (0,+∞) by mE =
∫E dt and the logarithmic measure of E ⊂ (0,+∞) by mlE =
∫E
dtt .
Definition 1.3. (see [1, 5, 11]) The p-iterated order and p-iterated lower-order of a meromorphic
function f(z) are respectively defined by
σp(f) = limr→∞
logp T (r, f)
log r; µp(f) = lim
r→∞
logp T (r, f)
log r. (1.5)
Definition 1.4. (see [1, 5, 11]) The p-iterated order and p-iterated lower-order of an entire
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
995 Jin Tu et al 994-1003
The Order and Type of Meromorphic Functions and Entire Functions of Finite Iterated Order 3
function f(z) are respectively defined by
σp(f) = limr→∞
logp+1M(r, f)
log r= lim
r→∞
logp T (r, f)
log r; (1.6)
µp(f) = limr→∞
logp+1M(r, f)
log r= lim
r→∞
logp T (r, f)
log r. (1.7)
Definition 1.5. Let f(z) be a meromorphic function satisfying 0 < σp(f) = σ < ∞ or 0 <
µp(f) = µ < ∞. Then the p-iterated type of order and the p-iterated lower-type of lower-orderof f(z) are respectively defined by
ψp(f) = limr→∞
logp−1 T (r, f)
rσ; ψ
p(f) = lim
r→∞
logp−1 T (r, f)
rµ. (1.8)
Definition 1.6. Let f(z) be an entire function satisfying 0 < σp(f) = σ < ∞ or 0 < µp(f) =
µ < ∞. Then the p-iterated type of order and the p-iterated lower-type of lower-order of f(z)are respectively defined by
τp(f) = limr→∞
logpM(r, f)
rσ; τp(f) = lim
r→∞
logpM(r, f)
rµ. (1.9)
From the above definitions, we can easily obtain the following propositions:
(i) If f1(z) and f2(z) are meromorphic functions with σp(f1) = σ1 < ∞ and σp(f2) = σ2 <∞, then σp(f1 + f2) ≤ maxσ1, σ2, σp(f1f2) ≤ maxσ1, σ2. Furthermore, if σ1 = σ2, thenσp(f1 + f2) = σp(f1f2) = maxσ1, σ2.
(ii) If f1(z) and f2(z) are meromorphic functions with σp(f1) < ∞ and µp(f2) < ∞, thenmaxµp(f1 + f2), µp(f1f2) ≤ maxσp(f1), µp(f2) or maxµp(f1 + f2), µp(f1f2) ≤ maxµp(f1),σp(f2).
(iii) If f1(z) and f2(z) are meromorphic functions satisfying σp(f1) < µp(f2) ≤ ∞, thenµp(f1 + f2) = µp(f1f2) = µp(f2).
(iv) If f(z) is an entire function with 0 < σp(f) <∞, then ψp(f) = τp(f), ψp(f) = τp(f) for
p ≥ 2 and ψ(f) ≤ τ(f), ψ(f) ≤ τ(f) for p = 1.
2. Main Results
Here our first question is : can we get the similar results as Theorem D for meromorphicfunction or entire function of finite iterated order? Since it is easy to see σp(f
′) = σp(f)(p ≥ 1)for meromorphic function f(z) of finite iterated order; our second question is : can we getψp(f
′) = ψp(f) or τp(f′) = τp(f) for meromorphic function or entire function of finite iterated
order? In fact, we obtain the following results.
Theorem 2.1. Let f1(z) and f2(z) be meromorphic functions satisfying 0 < σp(f1) = σp(f2) =σ <∞, 0 ≤ ψp(f1) < ψp(f2) ≤ ∞. Set α = ψ(f2)− ψ(f1), β = ψ(f1) + ψ(f2), then(i) σp(f1 + f2) = σp(f1f2) = σ(p ≥ 1); (ii) If p > 1, we have ψp(f1 + f2) = ψp(f1f2) = ψp(f2);(iii) If p = 1, we have α ≤ ψ(f1 + f2) ≤ β, α ≤ ψ(f1f2) ≤ β.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
996 Jin Tu et al 994-1003
4 J. Tu, Y. Zeng, H. Y. Xu
Theorem 2.2. Let f1(z) and f2(z) be entire functions satisfying 0 < σp(f1) = σp(f2) = σ <∞, 0 ≤ τp(f1) < τp(f2) ≤ ∞. Then the following statements hold:(i) If p ≥ 1, then σp(f1 + f2) = σ, τp(f1 + f2) = τp(f2);(ii) If p > 1, then σp(f1f2) = σ, τp(f1f2) = τ(f2).
Remark 2.1. When p = 1, (ii) of Theorem 2.2 does not hold. For example, set f1 = e−z, f2 = e2z
satisfy τ(f1) = 1 < τ(f2) = 2, but τ(f1f2) = 1 < τ(f2) = 2.
Theorem 2.3. Let f1(z) and f2(z) be meromorphic functions satisfying σp(f1) = µp(f2) =µ (0 < µ <∞), 0 ≤ ψp(f1) < ψ
p(f2) ≤ ∞. Then the following statements hold:
(i) µp(f1 + f2) = µp(f1f2) = µ(p ≥ 1); (ii) If p > 1, then ψp(f1 + f2) = ψ
p(f1f2) = ψ
p(f2);
(iii) If p = 1, then ψ(f2)− ψ(f1) ≤ maxψ(f1 + f2), ψ(f1f2) ≤ ψ(f1) + ψ(f2).
Theorem 2.4. Let f1(z) and f2(z) be entire functions satisfying σp(f1) = µp(f2) = µ (0 < µ <∞), 0 ≤ τp(f1) < τp(f2) ≤ ∞. Then the following statements hold:(i) µp(f1 + f2) = µp(f1f2) = µ(p ≥ 1); (ii) If p ≥ 1, then τp(f1 + f2) = τp(f2);(iii) If p > 1, then τp(f1f2) = τp(f2); if p = 1, then τ(f2)− τ(f1) ≤ τ(f1f2) ≤ τ(f1) + τ(f2).
Theorem 2.5. Let p > 1, f(z) be a meromorphic function satisfying 0 < σp(f) < ∞. Thenψp(f
′) = ψp(f).
Theorem 2.6. Let p ≥ 1, f(z) be an entire function satisfying 0 < σp(f) < ∞. Then τp(f ′) =τp(f).
3. Preliminary Lemmas
Lemma 3.1. (see [11]) Let f(z) be an entire function of p-iterated order satisfying 0 < σp(f) =σ < ∞, 0 < τp(f) = τ ≤ ∞. Then for any given β < τ, there exists a set E ⊂ [1, +∞) havinginfinite logarithmic measure such that for all r ∈ E, we have
logpM(r, f) > βrσ.
Lemma 3.2. (see [7]) Let f(z) be an analytic function in the circle |z| ≤ R and has no zeros inthis circle, and if f(0) = 1, then its modulus in the circle |z| ≤ r < R satisfies the inequality
ln |f(z)| ≥ − 2r
R− rlnM(R, f).
Lemma 3.3. (see [2]) Given any number H > 0 and complex numbers a1, a2, · · · , an, there is a
system of circles in the complex plane, with the sum of the radii equal to 2H, such that for eachpoint z lying outside these circles one has the inequality
n∏k=1
|z − ak| ≥(H
e
)n
.
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
997 Jin Tu et al 994-1003
The Order and Type of Meromorphic Functions and Entire Functions of Finite Iterated Order 5
Lemma 3.4. Let f(z) be an analytic function in the circle |z| ≤ βR (β > 1) with f(0) = 1, andlet ε be an arbitrary positive number not exceeding 2. Then inside the circle |z| ≤ R, but outsideof a family of excluded circles the sum of whose radii is not greater than 2εeβR, we have
ln |f(z)| > −(
2
β − 1+
ln 2− ln ε
lnβ
)lnM(β2R, f).
Proof. By the similar proof in [7, p.21], constructing the function
h(z) =(−βR)n
a1a2 · · · an
n∏k=1
βR(z − ak)(βR)2 − akz
,
where a1, a2, · · · , an are the zeros of f(z) in the circle |z| < βR, then we have h(0) = 1 and
|h(βReiθ)| = (βR)n
|a1a2···an| > 1, then function g(z) = f(z)h(z) has no zeros in the circle |z| < βR.
Therefore, by Lemma 3.2, for any z satisfying |z| ≤ R < βR, we have
ln |g(z)| ≥ − 2R
βR−RlnM(βR, g)
=− 2
β − 1(lnM(βR, f)− ln |h(βReiθ)|)
≥− 2
β − 1lnM(βR, f). (3.1)
For |z| ≤ βR, we get∏n
k=1 |(βR)2−akz| ≤ [2(βR)2]n = 2n(βR)2n. By Lemma 3.3, we get outsideof a family of excluded circles the sum of whose radii are not greater than 2εeβR, we have
n∏k=1
|βR(z − ak)| > (βR)n(βεR)n = εn(βR)2n,
where n = n(βR) denotes the number of zeros of f(z) in |z| < βR. So we have
|h(z)| ≥ (βR)n
|a1a2 · · · an|εn(βR)2n
2n(βR)2n≥(ε2
)n. (3.2)
Since 0 < ε < 2, by (3.2), we have
ln |ψ(z)| ≥ −n ln 2
ε. (3.3)
On the other hand, by Jensen’s formula, we have
M(β2R, f) ≥ exp
1
2π
∫ 2π
0log |f(β2Reiθ)|dθ
=
n(β2R)∏k=1
β2R
|ak|≥
n(βR)∏k=1
β2R
|ak|≥ βn.
Therefore,
n ≤ lnM(β2R, f)
lnβ. (3.4)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
998 Jin Tu et al 994-1003
6 J. Tu, Y. Zeng, H. Y. Xu
By (3.1), (3.3) and (3.4), we have
ln |f(z)| ≥ − 2
β − 1lnM(βR, f)− ln 2− ln ε
lnβlnM(β2R, f)
≥ −(
2
β − 1+
ln 2− ln ε
lnβ
)lnM(β2R, f). (3.5)
Lemma 3.5. Let p > 1, f(z) be an entire function satisfying 0 < σp(f) = σ <∞ and τp(f) <∞.Then for any given ε > 0, there exists a set E ⊂ (0,+∞) having finite logarithmic measure, suchthat for all |z| = r ∈ E, we have
exp− expp−1(τp(f) + ε)rσ < |f(z)| < expp(τp(f) + ε)rσ.
Proof. By (1.9), for any given ε > 0 and for sufficiently large r, we have
|f(z)| < expp(τp(f) + ε)rσ, M(β2r, f) < expp(τp(f) +ε
2)β2σrσ (β > 1). (3.6)
For any given ε(0 < ε < 2) and any β > 1, we choose ε, β satisfying (τp(f) +ε2)β
2σ < τp(f) + ε,by Lemma 3.4 and (3.5), there exists a set E ⊂ (0, +∞) having finite logarithmic measure, suchthat for all |z| = r ∈ E, we have
|f(z)| > exp− expp−1(τp(f) + ε)rσ (p > 1). (3.7)
By (3.6), (3.7), we obtain that Lemma 3.5 holds.
Lemma 3.6. (see [4, 14]) Let f(z) be a meromorphic function satisfying f(0) = ∞. Then forany τ > 1 and r > 0, we have
T (r, f) < CτT (τr, f′) + log+(τr) + 4 + log+ |f(0)|,
where Cτ > 0 is a constant related to τ .
Lemma 3.7. (see [6]) Let g : (0,∞) → R and h : (0,∞) → R be monotone non-decreasingfunctions such that g(r) ≤ h(r) outside of an exceptional set E of finite linear measure. Then forany α > 1, there exists r0 > 0 such that g(r) ≤ h(αr) for all r > r0.
4. Proofs of Theorems 2.1 - 2.6
Proof of Theorem 2.1. (i) Without loss of generality, set 0 ≤ ψp(f1) < ψp(f2) < ∞, by(1.8), for any ε > 0 and for sufficiently large r, we have
T (r, f1 + f2) ≤ T (r, f1) + T (r, f2) + ln 2
≤ expp−1(ψp(f1) + ε)rσ+ expp−1(ψp(f2) + ε)rσ≤ 2 expp−1(ψp(f2) + ε)rσ (4.1)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
999 Jin Tu et al 994-1003
The Order and Type of Meromorphic Functions and Entire Functions of Finite Iterated Order 7
By (4.1), we get σp(f1+f2) ≤ σ. On the other hand, by (1.8), for any ε(0 < 2ε < ψp(f2)−ψp(f1)),there exists a sequence rn∞n=1 tending to ∞ such that
T (rn, f1 + f2) ≥ T (rn, f2)− T (rn, f1)− ln 2
≥ expp−1(ψp(f2)− ε)rσn − expp−1(ψp(f1) + ε)rσn (4.2)
holds for sufficiently large rn. By (4.2), we get σp(f1+ f2) ≥ σ; therefore σp(f1+ f2) = σ (p ≥ 1).(ii)-(iii) By (4.1) and (4.2), we have ψp(f1 + f2) = ψp(f2) for p > 1 and α ≤ ψ(f1 + f2) ≤ β
for p = 1. Since
T (r, f1f2) ≤ T (r, f1) + T (r, f2), T (r, f1f2) ≥ T (r, f2)− T (r, f1)− o(1)
and by the similar proof in (4.1) and (4.2), we can easily obtain that the conclusions in cases(ii)-(iii) holds.
By the above proof, we can easily obtain that Theorem 2.1 also holds for 0 ≤ ψp(f1) <ψp(f2) =∞.
Proof of Theorem 2.2. (i) Set 0 ≤ τp(f1) < τp(f2) < ∞. By (1.9), for any ε > 0 and forsufficiently large r, we have
M(r, f1 + f2) ≤M(r, f1) +M(r, f2)
≤ expp(τp(f1) + ε)rσ+ expp(τp(f2) + ε)rσ≤ 2 expp(τp(f2) + ε)rσ, (4.3)
by (4.3), we get σp(f1 + f2) ≤ σ, τp(f1 + f2) ≤ τp(f2). On the other hand, by (1.9), for anyε (0 < 2ε < τp(f2)− τp(f1) there exists a sequence rn∞n=1 tending to ∞ such that
M(rn, f1) < expp(τp(f1) + ε)rσn, M(rn, f2) > expp(τp(f2)− ε)rσn (4.4)
holds for sufficiently large rn. In each circle |z| = rn(n = 1, 2, · · · ), we choose a sequence zn∞n=1satisfying |f2(zn)| =M(rn, f2) such that
M(rn, f1 + f2) ≥ |f1(zn) + f2(zn)| ≥ |f2(zn)| − |f1(zn)|≥M(rn, f2)−M(rn, f1)
≥ expp(τp(f2)− ε)rσn − expp(τp(f1) + ε)rσn
≥ 1
2expp(τp(f2)− ε)rσn (rn →∞), (4.5)
by (4.5), we get σp(f1+f2) ≥ σ, τp(f1+f2) ≥ τp(f2); therefore σp(f1+f2) = σ, τp(f1+f2) = τp(f2).(ii) By (1.9), for any ε > 0 and for sufficiently large r, we have
M(r, f1f2) ≤M(r, f1)M(r, f2)
≤ expp(τp(f1) + ε)rσ expp(τp(f2) + ε)rσ≤ expp(τp(f2) + 2ε)rσ (p > 1), (4.6)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
1000 Jin Tu et al 994-1003
8 J. Tu, Y. Zeng, H. Y. Xu
by (4.6), we get σp(f1f2) ≤ σ, τp(f1f2) ≤ τp(f2)(p > 1). On the other hand, by Lemma 3.1, forany ε > 0 there exists a set E1 having infinite logarithmic measure such that for all r ∈ E1, wehave
M(r, f2) > expp(τp(f2)− ε)rσ. (4.7)
By Lemma 3.5, for any ε > 0, there exists a set E2 having finite logarithmic measure such that|z| = r ∈ E2, we have
|f1(z)| > exp− expp−1(τp(f1) + ε)rσ (p > 1). (4.8)
Therefore, by (4.7) and (4.8), for any ε > 0 and for all |z| = r ∈ E1\E2, we have
M(r, f1f2) ≥M(r, f2) exp− expp−1(τp(f1) + ε)rσ≥ expp(τp(f2)− ε)rσ exp− expp−1(τp(f1) + ε)rσ. (4.9)
Since τp(f1) < τp(f2), we choose ε satisfying τp(f2)− ε > τp(f1)+ ε. By (4.9), for any ε(0 < 2ε <τp(f2)− τp(f1)) and for all r ∈ E1\E2, we have
M(r, f1f2) > expp(τp(f2)− 2ε)rσ (p > 1). (4.10)
By (4.10), we have σp(f1f2) ≥ σ, τp(f1f2) ≥ τp(f2) for p > 1; Therefore σp(f1f2) = σ, τp(f1f2) =τp(f2) for p > 1.
By the similar proof in cases (i) and (ii), we can easily obtain that Theorem 2.2 holds for0 ≤ τp(f1) < τp(f2) =∞.
Proof of Theorem 2.3. Set 0 ≤ ψp(f1) < ψp(f2) < ∞. By (1.8), for any ε > 0 and for
sufficiently large r, we have
T (r, f1) < expp−1(ψp(f1) + ε)rµ, T (r, f2) > expp−1(ψp(f2)− ε)rµ. (4.11)
By (4.11), for any ε (0 < 2ε < ψp(f2)− ψp(f1)) and for sufficiently large r, we have
T (r, f1 + f2) ≥ T (r, f2)− T (r, f1)− ln 2
≥ expp−1(ψp(f2)− ε)rµ − expp−1(ψp(f1) + ε)rµ. (4.12)
By (4.12), we have
µp(f1 + f2) ≥ µ(p ≥ 1), ψp(f1 + f2) ≥ ψp
(f2)(p > 1), ψ(f1 + f2) ≥ ψ(f2)− ψ(f1). (4.13)
On the other hand, by (1.8), for any ε > 0 there exists a sequence rn∞n=1 tending to ∞ suchthat
T (rn, f2) < expp−1(ψp(f2) + ε)rµn (4.14)
for sufficiently large rn. By (4.11) and (4.14), for any ε > 0 and for sufficiently large rn, we have
T (r, f1 + f2) ≤ T (rn, f1) + T (rn, f2) + ln 2
≤ expp−1(ψp(f1) + ε)rµn+ expp−1(ψp(f2) + ε)rµn. (4.15)
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
1001 Jin Tu et al 994-1003
The Order and Type of Meromorphic Functions and Entire Functions of Finite Iterated Order 9
By (4.15), we have
µp(f1 + f2) ≤ µ(p ≥ 1), ψp(f1 + f2) ≤ ψp
(f2)(p > 1), ψ(f1 + f2) ≤ ψ(f2) + ψ(f1). (4.16)
Therefore, by (4.13) and (4.16), the conclusions of Theorem 2.3 hold for f1+f2. Since T (r, f1f2) ≤T (r, f1)+T (r, f2), T (r, f1f2) ≥ T (r, f2)−T (r, f1)−o(1), we can easily obtain that the conclusionsof Theorem 2.3 hold for f1f2.
Theorem 2.3 also holds for 0 ≤ ψp(f1) < ψp(f2) =∞ by the above proof.
Proof of Theorem 2.4. We can obtain the conclusions of Theorem 2.4 by the similar proofin Theorem 2.2 and Theorem 2.3.
Proof of Theorem 2.5. By the Lemma of logarithmic derivative, we have that
T (r, f ′) ≤ 3T (r, f) +Olog rholds outside of an exceptional set E of finite linear measure. By Lemma 3.7, there existsα > 1 such that T (r, f ′) ≤ 3T (αr, f) + Olog(αr) holds for sufficiently large r, so we haveτp(f
′) ≤ τp(f) (p > 1). On the other hand, by Lemma 3.6, for any τ > 1, there exists a constantCτ such that
T (r, f) < CτT (τr, f′) + log+(τr) + 4 + log+ |f(0)|.
Set τ → 1+, by the above inequality, we have τp(f′) ≥ τp(f)(p > 1). Therefore τp(f
′) = τp(f) forp > 1.
Proof of Theorem 2.6. For an entire function f(z), we have
f(z) = f(0) +
∫ z
0f ′(ζ)dζ, (4.17)
where the integral route is a line from 0 to z. By (4.17), we have
M(r, f) ≤ |f(0)|+ |∫ z
0f ′(ζ)dζ| ≤ |f(0)|+ rM(r, f ′), (4.18)
By (4.18), we have
M(r, f ′) ≥ M(r, f)− |f(0)|r
. (4.19)
By (1.9) and (4.19), we have τp(f′) ≥ τp(f). On the other hand, by Cauchy’s integral formula,
we have
f ′(z) =1
2πi
∫C
f(ζ)
(ζ − z)2dζ, (4.20)
where integral curve is the circle |ζ| = R. By (4.20), for any |z| = r < R,
M(r, f ′) ≤ 1
2π
∫C
max
∣∣∣∣ f(ζ)
(ζ − z)2
∣∣∣∣ |dζ| ≤ 1
2π
M(R, f)
(R− r)2· 2πR. (4.21)
Set R = βr (β > 1), then by (4.21), we have
M(r, f ′) ≤ β
(β − 1)2rM(βr, f) ≤ β
(β − 1)2rexpp(τp(f) + ε)(βr)σp(f), r →∞. (4.22)
Since σp(f′) = σp(f), set β → 1, we have τp(f
′) ≤ τp(f); therefore τp(f ′) = τp(f).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
1002 Jin Tu et al 994-1003
10 J. Tu, Y. Zeng, H. Y. Xu
References
[1] L.G. Bernal, On Growth k-order of Solutions of a Complex Homongeneous Linear Differential Equan-tials, Proc. Amer. Math. Soc. 101 (1987) 317-322.
[2] H. Cartan, Sur les systemes de fonctions holomorphes a varietes lineaires lacunaires et leurs applica-tions, Ann. Sci.Ecole Norm. Sup. (3) 45 (1928), 255-346.
[3] A. Edrei and W. H. J. Fuchs, On the growth of meromorphic functions with several deficient values,Trans. Amer. Math. Soc. 93 (2) (1959), 292-328.
[4] W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
[5] L. Kinnunen, Linear differential equations with solutions of finite iterated order, Southeast Asian Bull.Math. (4) 22 (1988), 385-405.
[6] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
[7] B. Ja. Levin, Distribution of Zeros of Entire Functions, revised edition, Transl. Math. Monographs Vol.5, Amer. Math. Soc. Providence, 1980.
[8] R. Nevanlinna, Analytic Function, Springer-Verlag, New-York, 1970.
[9] W. J. Pugh and S. M. Shah, On the growth of entire functions of bounded index, Pacific J. Math. 33(1) (1970), 191-201.
[10] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, New York, 1975, reprint of the 1959edition.
[11] J. Tu, H. Y. Xu, C. Y. Zhang, On the zeros of solutions of any order of derivative of second orderlinear differential equations taking small functions, E. J. Qualitative Theory of Diff. Equ., No. 23(2011), 1-17.
[12] J. Tu, H. X. Huang, H. Y. Xu, C. F. Chen, The order and type of of meromorphic functions andanalytic functions in the unit disc. J. Jiangxi Norm. Univ., Nat. Sci. 37 (5), (2013), 52-55.
[13] J. Tu, J. S. Wei, H. Y. Xu, The order and type of meromorphic functions and analytic functions of[p,q]-φ(r) order in the unit disc, J. Jiangxi Norm. Univ., Nat. Sci. 39 (2), (2015), 207-211.
[14] L. Yang, Vaule Distribution Theory and Its New Research, Science Press, Beijing, 1982(in Chinese).
[15] H. X. Yi, C. C.Yang, Uniqueness Theory of Meromorphic Functions, Science Press/Kluwer Academic,Beijing/New York (1995/2003).
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
1003 Jin Tu et al 994-1003
1004
TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO. 5, 2016
Mixed Problems of Fractional Coupled Systems of Riemann-Liouville Differential Equations and Hadamard Integral Conditions, S. K. Ntouyas, Jessada Tariboon, and Phollakrit Thiramanus,…………………………………………………………………………………813
Ternary Jordan Ring Derivations on Banach Ternary Algebras: A Fixed Point Approach, Madjid Eshaghi Gordji, Shayan Bazeghi, Choonkil Park, and Sun Young Jang,…………………829
Initial Value Problems for a Nonlinear Integro-Differential Equation of Mixed Type in Banach Spaces, Xiong-Jun Zheng, and Jin-Ming Wang,……………………………………………835
Solving the Multicriteria Transportation Equilibrium System Problem with Nonlinear Path Costs, Chaofeng Shi, and Yingrui Wang,………………………………………………………….848
Barnes' Multiple Frobenius-Euler and Hermite Mixed-Type Polynomials, Dae San Kim, Dmitry V. Dolgy, and Taekyun Kim,………………………………………………………………856
Robust Stability and Stabilization of Linear Uncertain Stochastic Systems with Markovian Switching, Yifan Wu,………………………………………………………………………871
On Interval Valued Functions and Mangasarian Type Duality Involving Hukuhara Derivative, Izhar Ahmad, Deepak Singh, Bilal Ahmad Dar, and S. Al-Homidan,……………………..881
Additive-Quadratic ρ-Functional Inequalities in β-Homogeneous Normed Spaces, Sungsik Yun, George A. Anastassiou, and Choonkil Park,……………………………………………….897
A Note on Stochastic Functional Differential Equations Driven By G-Brownian Motion with Discontinuous Drift Coefficients, Faiz Faizullah, Aamir Mukhtar, and M. A. Rana,……..910
Subclasses of Janowski-Type Functions Defined By Cho-Kwon-Srivastava Operator, Saima Mustafa, Teodor Bulboaca, and Badr S. Alkahtani,………………………………………920
On A Product-Type Operator from Mixed-Norm Spaces to Bloch-Orlicz Spaces, Haiying Li, and Zhitao Guo,………………………………………………………………………………..934
A Short Note on Integral Inequality of Type Hermite-Hadamard through Convexity, Muhammad Iqbal, Shahid Qaisar, and Muhammad Muddassar,…………………………………..……946
Degenerate Poly-Bernoulli Polynomials of The Second Kind, Dmitry V. Dolgy, Dae San Kim, Taekyun Kim, And Toufik Mansour,………………………………………………………954
TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO. 5, 2016
(continued)
Some Results for Meromorphic Functions of Several Variables, Yue Wang,………………967
Nonstationary Refinable Functions Based On Generalized Bernstern Polynomials, Ting Cheng and Xiaoyuan Yang,…………………………………………………………………………980
The Order and Type of Meromorphic Functions and Entire Functions of Finite Iterated Order, Jin Tu, Yun Zeng, and Hong-Yan Xu,………………………………………………………994