journal of computational analysis and applications · "j.computational analysis and...

Volume 21, Number 2 August 2016 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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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.

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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

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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]

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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

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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]

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ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ*Correspondence author: Omar Abu arqub, e-mail: [email protected]

Existence and uniqueness of fuzzy solutions for the nonlinear

second-order fuzzy Volterra integrodifferential equations

Shaher Momani1,2

, Omar Abu Arqub3,*

, Saleh Al-Mezel2, Marwan Kutbi

2

1Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan

2Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics,

Faculty of Science, King Abdulaziz University (KAU), Jeddah 21589, Saudi Arabia 3Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt 19117, Jordan

Abstract. Formulation of uncertainty Volterra integrodifferential equations (VIDEs) is very important issue in applied

sciences and engineering; whilst the natural way to model such dynamical systems is to use the fuzzy approach. In this

work, we present and prove the existence and uniqueness of four solutions of fuzzy VIDEs based on the Hausdorff

distance under the assumption of strongly generalized differentiability for the fuzzy-valued mappings of a real variable

whose values are normal, convex, upper semicontinuous, and compactly supported fuzzy sets in . In addition to that,

we utilize and prove the characterization theorem for solutions of fuzzy VIDEs which allow us to translate a fuzzy VIDE

into a system of crisp equations. The proof methodology is based on the assumption of the generalized Lipchitz property

for each nonlinear term appears in the fuzzy equation subject to the specific metric used, while the main tools employed

in the analysis are founded on the applications of the Banach fixed point theorem and a certain integral inequality with

explicit estimate. An efficient computational algorithm is provided to guarantee the procedure and to confirm the

performance of the proposed approach.

Keywords: Fuzzy VIDE; Banach fixed point theorem; Existence and uniqueness

AMS Subject Classification: 26E50; 46S40; 34A07

1. Introduction

There is an inexhaustible supply of applications of VIDEs, especially, in characterizing many social, physical, biological,

and engineering problems. On the other aspect as well, since many real-world problems are too complex to be defined in

precise terms, uncertainty is often involved in any real-world design process. Fuzzy sets provide a widely appreciated

tool to introduce uncertain parameters into mathematical applications. In many applications, at least some of the

parameters of the model should be represented by fuzzy rather than crisp numbers. Thus, it is immensely important to

develop appropriate and applicable definitions and theorems to accomplish the mathematical construction that would

appropriately treat fuzzy VIDEs and solve them.

In this work we are interested in the following main questions; firstly, under what conditions can we be sure that

solutions of fuzzy VIDE exist; secondly, under what conditions can we be sure that there are four unique solutions; one

solution for each lateral derivative; to fuzzy VIDE, thirdly under what conditions can we be sure that fuzzy VIDE is

equivalent into system of crisp VIDEs. Anyhow, in this paper we will answered the aforementioned questions and

present an efficient computational algorithm to guarantee the procedure and to confirm the performance of the proposed

approach. More precisely, we consider the following second-order fuzzy VIDE under the assumption of strongly

generalized differentiability of the general form:

(1.1)

subject to the fuzzy initial conditions

(1.2)

where and

are continuous fuzzy-valued functions that satisfy a generalized

Lipchitz condition and .

The topics of fuzzy VIDEs which is growing interest for some time, in particular in relation to fuzzy control, fuzzy

population growth model, fuzzy oscillating magnetic fields, have been rapidly developed in recent years. Anyhow, in

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.2, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

213 Shaher Momani et al 213-227

http://science.ju.edu.jo/Home.aspx

2

this work, we are focusing our attention on second-order fuzzy VIDEs subject to given fuzzy initial conditions. At the

beginning, approaches to fuzzy IDEs and other fuzzy equations can be of three types. The first approach assumes that

even if only the initial values are fuzzy, the solution is a fuzzy function, and consequently the derivatives in the IDE

must be considered as fuzzy derivatives [1,2]. These can be done by the use of the Hukuhara derivative for fuzzy-valued

functions. Generally, this approach has a drawback; the solution becomes fuzzier as time goes, hence, the fuzzy solution

behaves quite differently from the crisp solution. In the second approach, the fuzzy IDE is transformed to a crisp one by

interpreted it as a family of differential inclusions [3,4]. The main shortcoming of using differential inclusions is that we

do not have a derivative of a fuzzy-valued function. The third approach based on the Zadeh's extension principle, where

the associated crisp problem is solved and in the solution the initial fuzzy values are substituted instead of the real

constants, and in the final solution, arithmetic operations are considered to be operations on fuzzy numbers [5,6]. The

weakness of this approach is the need to rewrite the solution in the fuzzy setting which in turn makes the methods of

solution are not user-friendly and more restricted with more computation steps. As a conclusion, to overcome the above-

mentioned shortcoming, the concept of a strongly generalized differentiability was developed and investigated in [7-14].

Anyhow, using the strongly generalized differentiability, the fuzzy IDE has locally four solutions. Indeed, with this

approach, we can find solutions for a larger class of fuzzy IDEs than using other types of differentiability.

The solvability analysis of fuzzy VIDEs has been studied by several researchers by using the strongly generalized

differentiability, the Hukuhara derivative, or the Zadeh's extension principle for the fuzzy-valued mappings of a real

variable whose values are normal, convex, upper semicontinuous, and compactly supported fuzzy sets in . The reader is

asked to refer to [15-22] in order to know more details about these analyzes, including their kinds and history, their

modifications and conditions for use, their scientific applications, their importance and characteristics, and their

relationship including the differences. But on the other aspect as well, more details about characterization theorem can

be found in [23,24].

The organization of the paper is as follows. In the next section, we present some necessary definitions and

preliminary results from the fuzzy calculus theory. The procedure of solving fuzzy VIDEs is presented in section 3. In

section 4, existence and uniqueness of four solutions are introduced. In section 5, we utilize the characterization theorem

for the solution of fuzzy VIDEs. This article ends in section 6 with some concluding remarks.

2. Excerpts of fuzzy calculus theory

Fuzzy calculus is the study of theory and applications of integrals and derivatives of uncertain functions. This branch of

mathematical analysis, extensively investigated in the recent years, has emerged as an effective and powerful tool for the

mathematical modeling of several engineering and scientific phenomena. In this section, we present some necessary

definitions from fuzzy calculus theory and preliminary results. For the concept of fuzzy derivative, we will adopt

strongly generalized differentiability, which is a modification of the Hukuhara differentiability and has the advantage of

dealing properly with fuzzy VIDEs.

Let be a nonempty set. A fuzzy set in is characterized by its membership function . Thus, is

interpreted as the degree of membership of an element in the fuzzy set for each . A fuzzy set on is called

convex if for each and , , is called upper semicontinuous if

is closed for each , and is called normal if there is such that . The support of

a fuzzy set is defined as .

Definition 2.1. [25] A fuzzy number is a fuzzy subset of the real line with a normal, convex, and upper

semicontinuous membership function of bounded support.

For each , set and , where denote the closure of .

Then, it easily to establish that is a fuzzy number if and only if is compact convex subset of for each

and [26]. Thus, if is a fuzzy number, then , where and

for each . The symbol is called the -cut representation or parametric form of a fuzzy

number . We will let denote the set of fuzzy numbers on .

The question arises here is, if we have an interval-valued function defined on , then is there a fuzzy

number such that . The next theorem characterizes fuzzy numbers through their -cut

representations.



3

Theorem 2.1. [26] Suppose that and satisfy the following conditions; first, is a bounded

increasing function and is a bounded decreasing function with ; second, for each and are

left-hand continuous functions at ; third, and are right-hand continuous functions at . Then

defined by

(2.1)

is a fuzzy number with parameterization . Furthermore, if is a fuzzy number with

parameterization , then the functions and satisfy the aforementioned conditions.

In general, we can represent an arbitrary fuzzy number by an order pair of functions which satisfy the

requirements of Theorem 2.1 Frequently, we will write simply and instead of and , respectively.

The metric structure on is given by such that for

arbitrary fuzzy numbers and , where is the Hausdorff metric between and . This metric is defined as

, where the two set

and are the -neighborhoods of and , respectively. It is shown in [27] that is a complete

metric space.

Lemma 2.1. [27] For each with the metric function satisfies the following properties:

i. ,

ii. ,

iii. ,

iv. .

For arithmetic operations on fuzzy numbers, the following results are well-known and follow from the theory of

interval analysis. If and are two fuzzy number, then for each , we have; firstly,

; secondly, ; thirdly,

; fourthly, if and only if if and only if

and . In fact, the collection of all fuzzy number with aforementioned addition and scalar multiplication

is a convex cone [28].

Let . If there exists a such that , then is called the -difference of and , denoted by

. Here, the sign " " stands always for -difference and let us remark that . Usually we

denote by , while stands for the -difference. It follows that Hukuhara differentiable function

has increasing length of support [25]. To avoid this difficulty, we consider the following definition.

Definition 2.2. [8] Let and . We say that is strongly generalized differentiable at , if there

exists an element such that either

i. for all sufficiently close to , the -differences , exist and

ii. for all sufficiently close to , the -differences , exist and

Here, the limit is taken in the metric space and at the endpoints of , we consider only one-sided

derivatives. For customizing, in Definition 2.2, the first case corresponds to the H-derivative introduced in [28], so this

differentiability concept is a generalization of the Hukuhara derivative.

Definition 2.3. [10] Let . We say that is -differentiable on if is differentiable in the sense (i)

of Definition 2.2 and its derivative is denoted . Similarly, we say that is (2)-differentiable on if is

differentiable in the sense (ii) of Definition 2.2 and its derivative is denoted .

The subsequent theorems show us a way to translate a fuzzy VIDE into a system of crisp VIDEs without the need to

consider the fuzzy setting approach. Anyhow, these theorems have many uses in the applied mathematics and the

numerical analysis fields.

Theorem 2.2. [10] Let and put for each .



4

i. if is -differentiable, then and are differentiable functions on and

,

ii. if is -differentiable, then and are differentiable functions on and

.

Next, we introduce the definitions for second fuzzy derivatives based on the selection of derivative type in each step

of differentiation. For a given fuzzy-valued function , we have two possibilities according to Definition 2.3 in order to

obtain the derivative of as follows: and

. Anyhow, for each of these two derivative, we have again two

possibilities of derivatives:

and

, respectively.

Definition 2.4. [29] Let and , we say that is -differentiable on if exist and

its -differentiable. The second derivatives of are denoted by .

Theorem 2.3. [29] Let or

, where for each :

i. if is -differentiable, then

and are differentiable functions on and

,

ii. if is -differentiable, then


,

iii. if is -differentiable, then


,

iv. if is (2)-differentiable, then


.

A fuzzy-valued function is called continuous at a point provided for arbitrary fixed ,

there exists an such that whenever for each . We say that is

continuous on if is continuous at each such that the continuity is one-sided at endpoints and .

In order to complete the expert results about the fuzzy calculus theory we finalize the present section by some

preliminary information about the fuzzy integral. Following [26], we define the integral of a fuzzy-valued function using

the Riemann integral concept.

Definition 2.5. [26] Suppose that , for each partition

of and for arbitrary points

, , let

and

. Then the definite integral of

over is defined by

provided the limit exists in the metric space .

Theorem 2.4. [26] Let be continuous fuzzy-valued function and put for each

. Then

exist, belong to , and are integrable functions on , and

.

Lemma 2.2. [30] Let be integrable fuzzy-valued functions and . Then the following are hold:

i. is integrable,

ii.

,

iii.

,

iv.

.

It should be noted that the fuzzy integral can be also defined using the Lebesgue-type approach [25] or the Henstock-

type approach [31]. However, if is continuous function, then all approaches yield the same value and results.

Moreover, the representation of the fuzzy integral using Defintion 2.5 is more convenient for numerical calculations and

computational mathematics. The reader is kindly requested to go through [25,26,30-32] in order to know more details

about the fuzzy integral, including its history and kinds, its properties and modification for use, its applications and

characteristics, its justification and conditions for use, and its mathematical and geometric properties.

3. Algorithm of solving fuzzy VIDEs

The topic of fuzzy VIDEs are one of the most important modern mathematical fields that result from modeling of

uncertain physical, engineering, and economical problems. In this section, we study fuzzy VIDEs using the concept of

strongly generalized differentiability in which fuzzy equation is converted into equivalent system of crisp equations for

each type of differentiability. Furthermore, we present an algorithm to solve the new system which consists of four crisp

VIDEs.



5

Problem formulation is normally the most important part of the process. It is the determination of -cut representation

form of nonlinear terms , the selection of the differentiability type, and the separation of fuzzy initial conditions.

Next, fuzzy VIDE (1.1) and (1.2) is first formulated as an crisp set of VIDEs subject to crisp set of initial conditions,

after that, a new discretized form of fuzzy VIDE (1.1) and (1.2) is presented. Anyhow, by considering the parametric

form for both sides of fuzzy VIDE (1.1) and (1.2), one can write

(3.1)

subject to the crisp initial conditions

(3.2)

in which the endpoints functions of

and

are given, respectively, as follows:

(3.3)

(3.4)

Definition 3.1. Let and , we say that is a -solution for fuzzy VIDE (1.1) and (1.2)

on , if and

exist on and

with

.

The object of the next algorithm is to implement a procedure to solve fuzzy VIDE in parametric form in term of its -

cut representation. To do so, let be a -solution, utilizing Theorems 2.2 and 2.3, and considering fuzzy VIDE

(1.1) and (1.2), we can thus translate it into system of crisp VIDEs, hereafter, called corresponding -system.

Anyhow, four IDEs systems are possible as given in the follow algorithm.

Algorithm 3.1: To find -solution of fuzzy VIDE (1.1) and (1.2), we discuss the following four cases:

Input: The independent interval , the unit truth interval , and the fuzzy numbers .

Output: The -differentiable solution of VIDE (1.1) and (1.2) on .

Step 1: Set

,

Set

,

Set and .

Case I. If is -differentiable, then use and

, and solving fuzzy VIDE (1.1) and (1.2)

translates into the following -system:

(3.5)


(3.6)

Case II. If is -differentiable, then use and



(3.7)



6


(3.8)

Case III. If is -differentiable, then use and



(3.9)


(3.10)

Case IV. If is -differentiable, then use and



(3.11)


(3.12)

Step 2: Solve the obtained -system of crisp VIDEs for and .

Step 3: Ensure that is -solution on the interval .

Step 4: Construct a -differentiable solution such that .

Step 5: Stop.

Sometimes, we can't decompose the membership function of the fuzzy solution as a function defined on for

each . Then, using identity (2.1) we can leave a -solution in term of its -cut representation form. To

summarize the evolution process; our strategy for solving fuzzy VIDE (1.1) and (1.2) is based on the selection of

derivatives type in the given fuzzy VIDE. The first step is to choose the type of solution and translate fuzzy VIDE into

the corresponding system of equations with coupled crisp VIDE for each type of differentiability. The second step is to

solve the obtained VIDEs system, while aim of the third step is to use the representation Theorem 2.1 in order to

construct the fuzzy solution.

Next, we construct a procedure based on Algorithm 3.1 to obtain the solutions of fuzzy VIDE (1.1) and (1.2). Here,

we discussing and considering the -differentiability in Case I of Algorithm 3.1 only; since the same procedure can

be applied directly for the remaining cases. Anyhow, without the loss of generality and for simplicity, we assume that the

function takes the form . So, based on this, fuzzy VIDE (1.1) can be

written in a new discretized form as

, in which the -cut

representation form of should be of the form

(3.13)

In order to design a scheme for solving fuzzy VIDE (1.1) and (1.2), we first replace it by the following equivalent

crisp system of VIDEs:

(3.14)


(3.15)

where the new functions are given, respectively, as



7

(3.16)

Prior to applying the analytic or the numerical methods for solving system of crisp VIDEs (3.14) and (3.15), we

suppose that the kernel function is nonnegative for and nonpositive for . Therefore, system

of crisp VIDEs (3.14) can be translated again into the following form:

(3.17)

4. Existence and uniqueness of four fuzzy solutions

It is worth stating that in many cases, since fuzzy VIDEs are often derived from problems in physical world, existence

and uniqueness are often obvious for physical reasons. Notwithstanding this, a mathematical statement about existence

and uniqueness is worthwhile. Uniqueness would be of importance if, for instance, we wished to approximate the

solutions. If two solutions passed through a point, then successive approximations could very well jump from one

solution to the other with misleading consequences.

Denote by the set of all continuous mapping from to . The supremum metric on is

defined by such that for each

, where is fixed. It is shown in [33] that is a complete metric space. On the

other aspect as well, by , we denote the set of all continuous mapping from to such that

exists as a continuous function. Anyhow, for , we define the distance function

such that . Indeed, it is shown in [33] that is

also a complete metric space.

The following lemma transforms a fuzzy VIDE into four fuzzy Volterra integral equations. Here the equivalence

between equations means that any solution of an equation is a solution too for the other one with respect to the

differentiability type used.

Lemma 4.1. The fuzzy VIDE (1.1) and (1.2), where and

are supposed to be

continuous is equivalent to one of the following fuzzy Volterra integral equations:

i.

, when is -

differentiable,

ii.

, when is

- differentiable,

iii.

, when is -

differentiable,

iv.

,

when is - differentiable.

Proof. Since and are continuous functions; so they are integrable. Now, we determine the equivalent integral forms

of fuzzy VIDE (1.1) and (1.2) under each type of strongly generalized differentiability as follows. Firstly, let us consider

is -differentiable, then the equivalent integral form of fuzzy VIDE (1.1) and (1.2) can be written by

implementation of fuzzy integration on both sides of the original equation two times as follows:

(4.1)

for and again for , one can write

(4.2)



8

Secondly, let us consider is -differentiable, then the equivalent integral form of fuzzy VIDE (1.1) and (1.2) can be

written by implementation of fuzzy integration on both sides of the original equation two times as

(4.3)

for , again for , we must have

(4.4)

Thirdly, if is -differentiable, then the equivalent form of fuzzy VIDE (1.1) and (1.2) can be written as

(4.5)

for , which is equivalent for to the integral equation of the form

(4.6)

Fourthly, since is -differentiable, then one can write

(4.7)

for and for , we can also write

(4.8)

which is equivalent to the form of part (iv).

In mathematics, the Banach fixed-point theorem; also known as the contraction mapping theorem; is an important

tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of

metric spaces, and provides a constructive method to find those fixed points. The following results (Definition 4.1 and

Theorem 4.1) were collected from [34].

Definition 4.1. Let be a metric space. A mapping is said to be a contraction mapping, if there exist a

positive real number with such that for each .

We observe that, applying to each of the two points of the space contracts the distance between them; obviously

is continuous. Anyhow, a point is called a fixed point of the mapping if . Next, we present the

Banach fixed-point theorem.

Theorem 4.1. Any contraction mapping of a nonempty complete metric space into itself has a unique fixed

point.

Lemma 4.2. The real-valued functions with represented by

(4.9)

are continuous nondecreasing functions on . Furthermore, , ,

, and .



http://en.wikipedia.org/wiki/Fixed-point_theorem

http://en.wikipedia.org/wiki/Metric_space

http://en.wikipedia.org/wiki/Fixed_point_(mathematics)

9

Proof. Clearly are continuous functions on for each . Since , ,

and

for each and ; thus, are nondecreasing functions. As a result one can

conclude that , , and . On the other aspect as well,

using the limit functions techniques it yields that

(4.10)

It should be mention here that Lemma 4.2 guarantees the existence of a unique fixed point for the next theorem. In

other word, an existence of a unique solution for fuzzy VIDE (1.1) and (1.2) for each type of differentiability.

Throughout this paper, we will try to give the results of the all theorems; however, in some cases we will switch

between the results obtained for the four type of differentiability in order not to increase the length of the paper without

the loss of generality for the remaining results. Actually, in the same manner, we can employ the same technique to

construct the proof for the omitted cases.

Theorem 4.2. Let and

are continuous fuzzy-valued functions. If there exists

such that

(4.11)

for each and . Then, the fuzzy VIDE (1.1) and (1.2) has four unique solutions

on for each type of differentiability.

Proof. Without the loss of generality, we consider the -differentiability only; actually, in the same manner, we can

employ the same technique for the remaining types. For each and define the operator and ,

respectively, as follows:

(4.12)

Thus, is continuous and . Now, we are going to show that

the operator satisfies the hypothesis of the Banach-fixed point theorem. For each , we have

(4.13)



10



11

where

. But since from Lemma

4.2, So, we can choose such that

(4.14)

Anyhow, G is a contractive mapping; whilst the unique fixed point of is in the space . Using that

is the integral of a continuous function, we conclude that it is actually in the space . Hence, by the Banach

fixed-point theorem, fuzzy VIDE (1.1) and (1.2) has a unique fixed point . That is, a contiunuous

function on satisfying . As a result, writing out, we have by Eq. (4.12)

(4.15)

On the other aspect as well, differentiate both sides Eq. (4.15) and substitute to obtain fuzzy VIDE (1.1) and

(1.2). Hence, every solution of fuzzy VIDE (1.1) and (1.2) must satisfy Eq. (4.15), and conversely. So, the proof of the

theorem is complete.

Remark 4.1: The continuous nonlinear terms and

are said to satisfy a

generalized Lipchitz condition relative to their last argument in fuzzy sense with respect to the metric space if

the conditions of Eq. (4.11) of Theorem 4.2 are hold.

5. Generalized characterization theorem

The characterization theorem shows us the following general hint on how to deal with the analytical or the numerical

solutions of fuzzy VIDEs. We can translate the original fuzzy VIDE equivalently into a system of crisp VIDEs. The

solutions techniques of the system of crisp VIDEs are extremely well studied in the literature, so any method we can

consider for the system of crisp VIDEs, since the solution will be as well solution of the fuzzy VIDE under study. As a

conclusion one does not need to rewrite the methods of solution for system of crisp VIDEs in fuzzy setting, but instead,

we can use the methods directly on the obtained crisp system.

A function is said to be equicontinuous if for any and any , we

have , whenever , and uniformly

bounded on any bounded set. Similarly, for a function defined on with the need for attention to change the

metric used on .

Theorem 5.1. Consider the fuzzy VIDE (1.1) and (1.2), where and

are such

that

i.

,

,



12

ii. and are equicontinuous functions and uniformly bounded on any bounded set,

iii. there exists real-finite constants such that

,

,

for each , , and . Then, for -differentiability, the fuzzy VIDE

(1.1) and (1.2) and the corresponding -system are equivalent.

Proof. Since the proof procedure is similar for each type of differentiability with respect to the corresponding -

system. Anyhow, we assume that is -differentiable (Case I of Algorithm 3.1) without the loss of generality. The

equicontinuity of and implies the continuity of and , respectively. Furthermore, the Lipchitz

property of condition (iii) ensures that and are satisfies a Lipchitz property in the metric space as follows:

(5.1)

Whilst on the other aspect as well, by similar fashion, it is easy to conclude that

(5.2)

By the continuity of and , from this last Lipchitz conditions of Eqs. (5.1) and (5.2), and the boundedness property of

condition (ii), it follows that fuzzy VIDE (1.1) and (1.2) has a unique solution on . Whilst, the solution of fuzzy

VIDE (1.1) and (1.2) is -differentiable and so, by Theorems 2.2 and 2.3, the functions and

are

differentiable on . As a conclusion one can obtained that is a solution of crisp VIDEs (3.5) and (3.6).

Conversely, suppose that we have a solution with is fixed, of fuzzy VIDE (1.1) and (1.2)

(note that this solution exists by property of condition (iii)). Whilst, the Lipchitz conditions of Eqs. (5.1) and (5.2)

implies the existence and uniqueness of fuzzy solution . Indeed, since is -differentiable, then and

the endpoints of are a solution of crisp VIDEs (3.5) and (3.6) (note that and are obviously valid level

sets of fuzzy-valued functions). But since the solution of crisp VIDEs (3.5) and (3.6) is unique, we have

. That is the fuzzy VIDE (1.1) and (1.2) and the system of crisp VIDEs (3.5)

and (3.6) are equivalent. This completes the proof of the theorem.

The purpose of the next corollary is not to make an essential improvement of Theorem 5.1, but rather to give alternate

conditions under which fuzzy VIDE (1.1) and (1.2) and the corresponding system of crisp VIDEs are equivalent.

Corollary 5.1. Suppose that and

are such that the condition (i) of Theorem

5.1 hold. If there exists real-finite constants such that

(5.3)

for each , , and . Then, for -differentiability, the fuzzy VIDE (1.1)

and (1.2) and the corresponding -system are equivalent.



13

Proof. Here, we consider the -differentiability only; actually, in the same manner, we can employ the same

technique for the remaining types of -differentiability. To this end, assume the hypothesis of Corollary 5.1, then

the conditions (i) and (iii) of Theorem 5.1 are clearly hold. To establish condition (ii), apply the following: fix ,

choose and , and suppose and .

Then, for each , one can write

(5.4)

Next, we want to show that are uniformly bounded on any bounded set. To do so, let be any bounded subset of

. Then there exist constants such that if , then ,

, , , and . For the conduct of proceedings in the proof, fix and

, further, let , and

, where is the support of . Suppose that and . Then one can write

(5.5)

while on the other aspect as well, the triangle inequality will gives

(5.6)

But since or , therefore is uniformly

bounded on . Similarly, ‏ is uniformly bounded on any bounded set. The same procedure can be applied directly for

. Hence, fuzzy VIDE (1.1) and (1.2) and the corresponding -system are equivalent by Theorem 5.1.

Remark 5.1. The following requirement conditions on and :

(5.7)

are fulfilled by any fuzzy-valued functions obtained from continuous real-valued functions by Zadeh’s extension

principle and Nguyen theorem [35-37]. So these conditions are not too restrictive.

6. Conclusion

Existence and uniqueness theorem is the tool which makes it possible for us to conclude that there exists only one

solution to a given problem which satisfies a constraint condition. How does it work? Why is it the case? We believe it

but it would be interesting to see the main ideas behind. To this end, in this paper we investigated and proved the

existence, uniqueness, and other properties of solutions of a certain nonlinear second-order fuzzy VIDE under strongly

generalized differentiability by considered four cases of differentiability. We make use of the standard tools of the fixed

point theorem and a certain integral inequality with explicit estimate to establish the main results. In addition to that,

some results for characterizing solution by an equivalent system of crisp VIDEs are presented and proved.

Acknowledgement

This project was funded by the deanship of scientific research of KAU under grant number (28-130-35-HiCi). The

authors, therefore, acknowledge technical and financial support of KAU.

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331.



αβ-statistical convergence and strong αβ-convergence oforder γ for a sequence of fuzzy numbers†

Zeng-Tai Gonga,∗, Xue Fenga,b

aCollege of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, ChinabSchool of Mathematics and Statistics, Qinghai University for Nationalities, Xining 810007, China

Abstract The purpose of this paper is to introduce the concepts of αβ-statistical convergence of orderγ and strong αβ-convergence of order γ for a sequence of fuzzy numbers. At the same time, someconnections between αβ-statistical convergence of order γ and strong αβ-convergence of order γ for asequence of fuzzy numbers are established. It also shows that if a sequence of fuzzy numbers is stronglyαβ-convergent of order γ then it is αβ-statistically convergent of order γ.Keywords: Fuzzy numbers; sequence of fuzzy numbers; statistical convergence.

1. Introduction

The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [1] and subsequentlyseveral authors have discussed various aspects of theory and applications of fuzzy sets. Recently Matloka[2] introduced bounded and convergent sequences of fuzzy numbers, studied some of their properties, andshowed that every convergent sequence of fuzzy numbers is bounded. In addition, sequences of fuzzynumbers have been discussed by Aytar and Pehlivan [3], Basarir and Mursaleen [4,5] and many others.The notion of statistical convergence was introduced by Fast [6] which is a very useful functional tool forstudying the convergence problems of numerical sequences. Some applications of statistical convergencein number theory and mathematical analysis can be found in [7, 8]. The idea is based on the notionof natural density of subsets of N, and the natural density of s subset A of N is denoted by δ(A) anddefined by

δ(A) = limn→∞

1n|k < n : k ∈ A|.

In 2014, Huseyin Aktuglu [9] introduced the concepts of αβ-statistically convergence and αβ-statisticallyconvergence of order γ for a sequence, which shows that αβ-statistically convergence is a non-trivialextension of ordinary and statistical convergences.

In this paper, we define the sequence spaces of αβ-statistical convergence of order γ and strong αβ-convergence of order γ, and testify some properties of these spaces. At the same time, some connectionsbetween αβ-statistical convergence of order γ and strong αβ-convergence for a sequence of order γ offuzzy numbers are established. In Section 2 we will give a brief overview about fuzzy numbers, statisticalconvergence, and present δα,β(k, γ). In Section 3 we show that αβ-statistical convergence for a sequenceof fuzzy numbers can reduce to statistical convergence, λ-statistical convergence, and lacunary statisticalconvergence. Meanwhile, strong αβ-convergence for a sequence of fuzzy numbers can reduce to strongconvergence, strong λ−convergence and strongly lacunary convergence.

2. Definitions and preliminaries

Let A ∈ F (R) be a fuzzy subset on R. If A is convex, normal, upper semi-continuous and has compactsupport, we say that A is a fuzzy number. Let Rc denote the set of all fuzzy numbers [10,11,12].

For A ∈ Rc, we write the level set of A as Aλ = x : A(x) ≥ λ and Aλ = [A−λ , A+λ ]. Let A, B ∈ Rc,

we define A+ B = C iff Aλ +Bλ = Cλ, λ ∈ [0, 1] iff A−λ +B−λ = C−λ and A+λ +B+

λ = C+λ for any λ ∈ [0, 1].

†Supported by the Natural Scientific Fund of China (11461062, 61262022).∗Corresponding Author:Zeng-Tai Gong. Tel.: +869317971430. E-mail addresses: [email protected]


228 Zeng-Tai Gong et al 228-236

Zeng-Tai Gong and Xue Feng: αβ-statistical convergence and strong αβ-convergence of...

DefineD(A, B) = sup

λ∈[0,1]d(Aλ, Bλ) = sup

λ∈[0,1]max|A−λ −B−λ |, |A

+λ −B+

λ |,

where d is the Hausdorff metric. D(A, B) is called the distance between A and B [11,13,14].Using the results of [10,11], we see that(1) (Rc, D) is a complete metric space,(2) D(u + w, v + w) = D(u, v),(3) D(ku, kv) = |k|D(u, v), k ∈ R,(4) D(u + v, w + e) ≤ D(u, w) + D(v, e),(5) D(u + v, 0) ≤ D(u, 0) + D(v, 0),(6) D(u + v, w) ≤ D(u, w) + D(v + 0),

where u, v, w, e ∈ Rc, 0(t) =

1, t = (0, 0, 0..., 0),

0, otherwise.Definitions 2.1.[15] A sequence xn of fuzzy numbers is said to be statistically convergent to a fuzzynumber x0 if for each ε > 0 the set A(ε) = n ∈ N : D(xn, x0) ≥ ε has natural density zero. The fuzzynumber x0 is called the statistical limit of the sequence xn and we write st- lim

n→∞xn = x0.

Now let α(n) and β(n) be two sequences of positive numbers satisfying the following conditions:(1) α(n) and β(n) are both non-decreasing,(2) β(n) ≥ α(n),(3) β(n)− α(n) →∞, as n →∞,

and let Λ denote the set of pairs (α, β) satisfying (1), (2) and (3).For each pair (α, β) ∈ Λ, 0 < γ ≤ 1 and K ⊂ N, we define δα,β(K, γ) in the following way:

δα,β(K, γ) = limn

|K ∩ Pα,βn |

(β(n)− α(n) + 1)γ

where Pα,βn is the closed interval [α(n), β(n)] and |S| represents the cardinality of S.

Lemma 2.1. Let K and M be two subsets of N and 0 < γ ≤ δ ≤ 1. Then for all (α, β) ∈ Λ, we have(1) δα,β(∅, γ)=0,(2) δα,β(N, 1)=1,(3) if K is a finite set, the δα,β(K, γ) = 0,(4) K ⊂ M ⇒ δα,β(K, γ) ≤ δα,β(M,γ),(5) δα,β(K, δ) ≤ δα,β(K, γ).

3. Main results

Definition 3.1. A sequence of fuzzy numbers is said to be αβ-statistically convergent of order γ to x0,if for every ε > 0,

δα,β(k : D(xk, x0) ≥ ε, γ) = limn

|k ∈ Pα,βn : D(xk, x0) ≥ ε|

(β(n)− α(n) + 1)γ= 0.

In this case, we write Sα,βγ − lim xk = x0. The set of all αβ-statistically convergent of order γ will be

denoted simply by Sα,βγ .

For γ = 1, we say that x is αβ-statistically convergent to x0 and this is denoted by Sα,β − lim xk = x0.The following example shows that Definition 3.1 is non-trivial generalization of both ordinary and

statistical convergence.Example 3.1. Taking α(n) = 1 and β(n) = n

1γ , where 0 < γ < 1 is fixed, then


|k ∈ [1, n1γ ] : D(xk, x0) ≥ ε|

n

and, in particular, for γ = 12 we have

δα,β(k : D(xk, x0) ≥ ε, 12) = lim

n

|k ∈ [1, n2] : D(xk, x0) ≥ ε|n

.




Consider the sequence of fuzzy numbers

xk(t) =

t + 1, − 1 ≤ t ≤ 0, k 6= n2,

−t + 1, 0 < t ≤ 1, k 6= n2,

t, 0 ≤ t ≤ 1, k = n2,

2− t, 1 < t ≤ 2, k = n2,

0, others;

x0(t) =

t + 1, − 1 ≤ t ≤ 0,

−t + 1, 0 < t ≤ 1,

0, others;

Obviously st− limn

xk = x0, however

δα,β(k : D(xk, x0) ≥ ε, 12) = lim

n

|k ∈ [1, n2] : D(xk, x0) ≥ ε|n

6= 0.

for all ε > 0, Sα,βγ − lim xk 6= x0.

Definition 3.2. Based on strongly αβ-convergence of order γ, for every ε > 0, we define the followingsets

Wα,βγ =

x = xk : lim

n

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(xk, x0) = 0,

Wα,βγ0 =

x = xk : lim

n

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(xk, 0) = 0,

Wα,βγ∞ =

x = xk : sup

n

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(xk, 0) < ∞,

where

0(t) =

1, t = (0, 0, 0..., 0),

0, otherwise.

If x ∈ Wα,βγ , we say that x is strongly αβ-convergent of order γ to x0 and we write Wα,β

γ − lim xk = x0.

For γ = 1, we say that x is strongly αβ-convergent to x0 and this is denoted by Wα,β − lim xk = x0.

Remark 3.1. Take α(n) = 1, β(n) = n and γ = 1, then Pα,βn = [1, n] and


|k ≤ n : D(xk, x0) ≥ ε|n

= 0.

This shows that in this case, αβ-statistical convergence of order γ reduces to statistical convergence whichwe denoted by S. Meanwhile, the sequences space Wα,β

γ reduces to W , Wα,βγ0 reduces to W0 and Wα,β

γ∞

reduces to W∞. Where W , W0 and W∞ are defined by Mursaleen and Basarir [16].

W =x = xk : lim

n

1n

n∑k=1

D(xk, x0) = 0,

W0 =x = xk : lim

n

1n

n∑k=1

D(xk, 0) = 0,

W∞ =x = xk : sup

n

1n

n∑k=1

D(xk, 0) < ∞.

Remark 3.2. Let λn be a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤λn + 1, λ1 = 1 and In = [n − λn + 1, n]. We choose α(n) = n − λn + 1, β(n) = n and γ = 1, thenPα,β

n = [n− λn + 1, n]. Moreover,


|k ∈ In : D(xk, x0) ≥ ε|λn

= 0.




This shows that in this case, αβ-statistical convergence of order γ reduces to λ-statistical convergencewhich we denoted by S(λ). Meanwhile, the sequences space Wα,β

γ reduces to W (λ), Wα,βγ0 reduces to

W0(λ) and Wα,βγ∞ reduces to W∞(λ). Where W (λ), W0(λ) and W∞(λ) are defined by Savas [17].

W (λ) =x = xk : lim

n

1λn

∑k∈In

D(xk, x0) = 0,

W0(λ) =x = xk : lim

n

1λn

∑k∈In

D(xk, 0) = 0,

W∞(λ) =x = xk : sup

n

1λn

∑k∈In

D(xk, 0) < ∞.

Remark 3.3. A lacunary sequence θ = kr is an increasing sequence such that k0 = 0, hr = kr−kr−1 →∞, r → ∞ and Ir = (kr−1, kr]. Take α(r) = kr−1 + 1, β(r) = kr and γ = 1, then Pα,β

r = [kr−1 + 1, kr].However (kr−1, kr] ∩N = [kr−1 + 1, kr] ∩N, we have

δα,β(k : D(xk, x0) ≥ ε, γ) = limr

|k ∈ Ir : D(xk, x0) ≥ ε|hr

= 0.

This shows that in this case, αβ-statistical convergence of order γ coincides with lacunary statisticalconvergence which we denoted by S(θ). Meanwhile, the sequences space Wα,β

γ reduces to W (θ), Wα,βγ0

reduces to W0(θ) and Wα,βγ∞ reduces to W∞(θ).

WhereW (θ) =

x = xk : lim

r

1hr

∑k∈Ir

D(xk, x0) = 0,

W0(θ) =x = xk : lim

r

1hr

∑k∈Ir

D(xk, 0) = 0,

W∞(θ) =x = xk : sup

r

1hr

∑k∈Ir

D(xk, 0) < ∞.

Theorem 3.1. Let x = xk, y = yk be two sequences of fuzzy numbers. We have(1) If Sα,β

γ − lim xk = x0 and c ∈ R, then Sα,βγ − lim cxk = cx0;

(2) If Sα,βγ − lim xk = x0, Sα,β

γ − lim yk = y0, then Sα,βγ − lim(xk + yk) = x0 + y0.

Proof. (1) The proof is obvious when c=0. Suppose that c 6= 0, then the proof of (1) follows from thefollowing inequality,

1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(cxk, cx0) ≥ ε| ≤ 1

(β(n)− α(n) + 1)γ|k ∈ Pα,β

n : D(xk, x0) ≥ε

|c||.

(2)Suppose that Sα,βγ − lim xk = x0, Sα,β

γ − lim yk = y0, then

limn

1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(xk, x0) ≥

ε

2| = 0,

limn

1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(yk, y0) ≥

ε

2| = 0.

Since

D(xk + yk, x0 + y0) ≤ D(xk + yk, x0 + yk) + D(x0 + yk, x0 + y0) = D(xk, x0) + D(yk, y0).

For ε > 0, we have

1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(xk + yk, x0 + y0) ≥ ε|

≤ 1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(xk, x0) ≥

ε

2|+ 1

(β(n)− α(n) + 1)γ|k ∈ Pα,β

n : D(yk, y0) ≥ε

2|




→ 0, n →∞. Hence Sα,βγ − lim(xk + yk) = x0 + y0.

Definition 3.3. The sequence of fuzzy numbers x = xk is a αβ-statistically Cauchy sequence of orderγ, if for every ε > 0 there exists a number N(= N(ε)) such that

limn

1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(xk, xN ) ≥ ε| = 0.

Theorem 3.2. Let x = xk be a sequence of fuzzy numbers. It is a αβ-statistically convergent sequenceof order γ if and only if x is a αβ-statistical Cauchy sequence of order γ.

Proof. Suppose that Sα,βγ − lim xk = x0 and let ε > 0, then

limn

1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(xk, x0) ≥ ε| = 0,

and N is choosen such that limn

1(β(n)−α(n)+1)γ |k ∈ Pα,β

n : D(xN , x0) ≥ ε| = 0,

then we have

1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(xk, xN ) ≥ ε|

≤ 1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(xk, x0) ≥ ε|+ 1

(β(n)− α(n) + 1)γ|k ∈ Pα,β

n : D(xN , x0) ≥ ε|.

Hence x = xk is a αβ-statistically Cauchy sequence of order γ.Next, assume that x = xk be αβ-statistical Cauchy sequence of order γ, then there exists a strictly

increasing sequence Np of positive integers such that limn

1(β(n)−α(n)+1)γ |k ∈ Pα,β

n : D(xk, xNp) ≥ εp| = 0,

where εp : p = 1, 2, 3, · · · is a strictly decreasing sequence of numbers converging to zero for each p and qpair (p 6= q) of positive integers, we can select Kpq such D(xKpq , xNp) < εp and D(xKpq , xNq) < εq.It follows that

D(xNp , xNq) ≤ D(xKpq , xNp) + D(xKpq , xNq) < εp + εq → 0, p, q →∞.

Hence, xNp : p = 1, 2, · · · is a Cauchy sequence and statisfies the Cauchy convergence criterion. LetxNp converge to x0. Since εp : p = 1, 2, · · · → 0, so for ε > 0, there exists p0 such that εp0 < ε

2 andD(xNp , x0) < ε

2 , p ≥ p0, then

D(xk, x0) ≤ D(xk, xNp0) + D(xNp0

, x0) ≤ D(xk, xNp0) +

ε

2,

we have

1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(xk, x0) ≥ ε| ≤ 1

(β(n)− α(n) + 1)γ|k ∈ Pα,β

n : D(xk, xNP0) ≥ ε

2|

≤ 1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(xk, xNP0

) ≥ εp0| → 0, n →∞.

This shows that x = xk is αβ-statistically convergent of order γ.Theorem 3.3. Let x = xk is a sequence of fuzzy numbers. There exsit a αβ-statistically convergentof order γ sequence y = yk such that xk = yk for almost all k according to γ, then x = xk is aαβ-statistically convergent sequence of order γ.

Proof. Let xk = yk for almost all k according to γ and Sα,βγ − lim yk = x0. Suppose ε > 0. Then for each

n,k ∈ Pα,β

n : D(xk, x0) ≥ ε ⊆ k ∈ Pα,βn : D(yk, x0) ≥ ε ∪ k ∈ Pα,β

n : xk 6= yk.Since xk = yk for almost all k according to γ, the latter set contains a fixed number of integers, sayS = S(ε). Then

limn

1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(xk, x0) ≥ ε|

≤ limn

1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(yk, x0) ≥ ε|+ lim

n

S

(β(n)− α(n) + 1)γ,




Hence Sα,βγ − lim xk = x0, i.e. x = xk is a αβ-statistically convergent sequence of order γ.

Theorem 3.4. Let 0 < γ1 ≤ γ2 ≤ 1, then Sα,βγ1 ⊆ Sα,β

γ2 .Proof. Let 0 < γ1 ≤ γ2 ≤ 1. Then we have

1(β(n)− α(n) + 1)γ2

|k ∈ Pα,βn : D(xk, x0) ≥ ε| ≤ 1

(β(n)− α(n) + 1)γ1|k ∈ Pα,β

n : D(xk, x0) ≥ ε|,

for every ε > 0 and so we get Sα,βγ1 ⊆ Sα,β

γ2 .Corollary 3.1. If a sequence x = xk of fuzzy numbers is αβ-statistically convergent of order γ, thenit is αβ-statistically convergent, for each γ ∈ (0, 1], i.e. Sα,β

γ ⊆ Sα,β.

Theorem 3.5. The sequence spaces of fuzzy numbers Wα,βγ0 , Wα,β

γ and Wα,βγ∞ satisfy the relationship:

Wα,βγ0 ⊂ Wα,β

γ ⊂ Wα,βγ∞ .

Proof. Let x = xk ∈ Wα,βγ . Note that

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(xk, 0)

≤ 1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(xk, x0) +1

(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(x0, 0)

≤ 1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(xk, x0) +1

(β(n)− α(n) + 1)γD(x0, 0),

according to the above inequality, we have supn

1(β(n)−α(n)+1)γ

∑k∈P α,β

n

D(xk, 0) < ∞, thus we get x ∈ Wα,βγ∞ .

The proof of Wα,βγ0 ⊂ Wα,β

γ is obvious.Theorem 3.6. The sequence spaces of fuzzy numbers Wα,β

γ0 , Wα,βγ and Wα,β

γ∞ are linear spaces over theset of real numbers.Proof. Let x = xk, y = yk ∈ Wα,β

γ0 , α, β ∈ R. In order to get result we need to prove the following

limn

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(αxk + βyk, 0) = 0.

Since x = xk, y = yk ∈ Wα,βγ0 , we have

limn

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(xk, 0) = 0,

limn

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(yk, 0) = 0.

And D(αxk + βyk, 0) ≤ D(αxk, 0) + D(βyk, 0) = |α|D(xk, 0) + |β|D(yk, 0),we get

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(αxk + βyk, 0) ≤ 1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

[D(αxk, 0) + D(βyk, 0)

]≤ |α|

(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(xk, x0) +|β|

(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(yk, x0) → 0, n →∞.

Thus αx + βy ∈ Wα,βγ0 . Similarly it can be shown that the other spaces are also linear spaces.

Theorem 3.7. Let 0 < γ ≤ 1. If a sequence x = xk of fuzzy number is strongly αβ-convergent oforder γ, then it is αβ-statistically convergent of order γ, i.e. Wα,β

γ ⊂ Sα,βγ .




Proof. Given ε > 0 and any sequence x = xk of fuzzy numbers, we write∑k∈P α,β

n

D(xk, x0) =∑

k∈P α,βn ,D(xk,x0)<ε

D(xk, x0) +∑

k∈P α,βn ,D(xk,x0)≥ε

D(xk, x0)

≥∑

k∈P α,βn ,D(xk,x0)≥ε

D(xk, x0) ≥ |k ∈ Pα,βn : D(xk, x0) ≥ ε| · ε

and hence

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

D(xk, x0) ≥1

(β(n)− α(n) + 1)γ|k ∈ Pα,β

n : D(xk, x0) ≥ ε| · ε.

Here, it can be easily to see that if a sequence x = xk of fuzzy number is strongly αβ-convergent oforder γ, then it is αβ-statistically convergent of order γ.Corollary 3.2. Let 0 < γ ≤ η ≤ 1. If a sequence x = xk of fuzzy number is strongly αβ-convergent oforder γ, then it is αβ-statistically convergent of order η, i.e. Wα,β

γ ⊂ Sα,βη .

Definition 3.4. Let p = pk be any sequence of strictly positive real numbers. A sequence x = xk offuzzy numbers is said to be strongly αβ(p)-convergent of order γ, if for γ ∈ (0, 1], there is a fuzzy numberx0 such that

limn

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

[D(xk, x0)

]pk = 0,

we denote the set of all strongly αβ(p)-convergent of order γ for fuzzy sequences by Wα,βγ (p). Where

Wα,βγ (p) =

x = xk : lim

n

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

[D(xk, x0)

]pk = 0,

Wα,βγ0 (p) =

x = xk : lim

n

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

[D(xk, 0)

]pk = 0,

Wα,βγ∞ (p) =

x = xk : sup

n

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

[D(xk, 0)

]pk < ∞.

It similar to the proofs of Theorem 3.5, 3.6, for strongly αβ(p)-convergent of order γ we have thefollowing results.Theorem 3.8. The sequence spaces of fuzzy numbers Wα,β

γ0 (p), Wα,βγ (p) and Wα,β

γ∞ (p) satisfy the rela-tionship: Wα,β

γ0 (p) ⊂ Wα,βγ (p) ⊂ Wα,β

γ∞ (p).Theorem 3.9. The sequence spaces of fuzzy numbers Wα,β

γ0 (p), Wα,βγ (p) and Wα,β

γ∞ (p) are linear spacesover the set of real numbers.Theorem 3.10. Let 0 < pk ≤ qk, and qk

pk be bounded. Then Wα,β

γ (q) ⊂ Wα,βγ (p).

Proof. Let x = xk ∈ Wα,βγ (q), and tk =

[D(xk, x0)

]qk , λk = pkqk

, 0 < λk ≤ 1. Let 0 < λ < λk,

and define uk =

tk, tk ≥ 1,

0, tk < 1,vk =

0, tk ≥ 1,

tk, tk < 1,then tk = uk + vk, tλk

k = uλkk + vλk

k , and

uλkk ≤ uk ≤ tk, vλk

k ≤ vλk . We have

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

[D(xk, x0)

]pk =1

(β(n)− α(n) + 1)γ

∑k∈P α,β

n

tλkk

=1

(β(n)− α(n) + 1)γ

∑k∈P α,β

n

(uλkk + vλk

k ) ≤ 1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

tk

+1

(β(n)− α(n) + 1)γ

∑k∈P α,β

n

vλk → 0, n →∞.




Since x = xk ∈ Wα,βγ (q), we have lim

n

1(β(n)−α(n)+1)γ

∑k∈P α,β

n

tk = 0. And since vk < 1, λ < 1, we

get limn

1(β(n)−α(n)+1)γ

∑k∈P α,β

n

vλk = 0. Hence, Wα,β

γ (q) ⊂ Wα,βγ (p).

In the following theorem, we shall discuss the relationship between the space Wα,βγ (p) and Sα,β

γ .Theorem 3.11. Let 0 < h = inf

kpk ≤ sup

kpk = H < ∞, l∞ be a set of all bounded sequence of fuzzy

numbers. Then(1) Wα,β

γ (p) ⊂ Sα,βγ ;

(2) If x = xk ∈ l∞ ∩ Sα,βγ , then x = xk ∈ Wα,β

γ (p);(3) l∞ ∩ Sα,β

γ = l∞ ∩ Wα,βγ (p).

Proof. (1) Let x = xk ∈ Wα,βγ (p), Note that

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

[D(xk, x0)

]pk ≥ 1(β(n)− α(n) + 1)γ

∑k∈P α,β

n ,D(xk,x0)≥ε

[D(xk, x0)

]pk

≥ 1(β(n)− α(n) + 1)γ

∑k∈P α,β

n ,D(xk,x0)≥ε

minεh, εH

=1

(β(n)− α(n) + 1)γ|k ∈ Pα,β

n : D(xk, x0) ≥ ε| ·minεh, εH,

follow from the above inequality, we have limn

1(β(n)−α(n)+1)γ |k ∈ Pα,β

n : D(xk, x0) ≥ ε| = 0. Thus we get

x = xk ∈ Sα,βγ .

(2) Let x = xk ∈ l∞ ∩ Sα,βγ , then there is a constant T > 0, such that D(xk, x0) ≤ T. Therefore

1(β(n)− α(n) + 1)γ

∑k∈P α,β

n

[D(xk, x0)

]pk

=1

(β(n)− α(n) + 1)γ

∑k∈P α,β

n ,D(xk,x0)≥ε

[D(xk, x0)

]pk +1

(β(n)− α(n) + 1)γ

∑k∈P α,β

n ,D(xk,x0)<ε

[D(xk, x0)

]pk

≤ 1(β(n)− α(n) + 1)γ

∑k∈P α,β

n ,D(xk,x0)≥ε

maxT h, TH+1

(β(n)− α(n) + 1)γ

∑k∈P α,β

n ,D(xk,x0)<ε

εpk

≤ 1(β(n)− α(n) + 1)γ

|k ∈ Pα,βn : D(xk, x0) ≥ ε| ·maxT h, TH

+maxεh, εH,

follow from the above inequality, we have limn

1(β(n)−α(n)+1)γ

∑k∈P α,β

n

[D(xk, x0)

]pk = 0. Thus we get x =

xk ∈ Wα,βγ (p).

(3) From (1) and (2), (3) is obvious.

4. Conclusion

In this article, we introduced some classes of sequences of fuzzy numbers defined by αβ-statisticallyconvergence of order γ, strong αβ-convergence of order γ, and strong αβ(p)-convergence of order γ. Wehave proved some properties and relationships of these spaces. At the same time, it also shows that if asequence of fuzzy numbers is strongly αβ-convergent of order γ then it is αβ-statistically convergent oforder γ.

References

[1] L.A. Zadeh, Fuzzy sets, Information and control 8 (1965) 338-353.




[2] M. Matloka, Sequences of fuzzy numbers, Busefal 28 (1986) 28-37.

[3] S. Aytar, S. Pehlivan, S Ttatistically monotonic and statistically bounded sequences of duzzy num-bers, Inform, Sci. 176(6) (2006) 734-774.

[4] M. Basarir, M. Mursaleen, Some sequence spaces of fuzzy numbers generated by infinite matrices,J. Fuzzy Math. 11(3) (2003) 757-764.

[5] M. Mursaleen. M. Basarir, on some new sequence spaces of fuzzy numbers, Indian J. Pure Appl.Math. 34(9) (2003) 1351-1357.

[6] H. Fast, Sur la convergence statistique, Colloquium Math. 2 (1951) 241-244.

[7] R.C. Buck, The measure theoretic approach to density. Am. J. Math. 68 (1946) 560-580.

[8] R.C. Buck, Generalized asymptotic density, Am. J. Math. 75 (1953) 335-346.

[9] Huseyin Aktuglu, Korovkin type approximation theorems proved via αβ-statistical convergence,Journal of Computational and Applied Mathematics 259 (2014) 174-181.

[10] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301-317.

[11] Wu Congxin, Gong Zengtai, On Henstock integral of fuzzy-number-valued functions (I), Fuzzy Setsand Systems 120 (2001) 523-532.

[12] Wu Congxin, Ma Ming, Embedding problem of fuzzy number space: Part II, Fuzzy Sets and Systems45 (1992) 189-202.

[13] Gong Zengtai, Wu Congxin, The Mcshane integral of fuzzy-valued functions, Southeast Asian Bull.Math. 24 (2000) 365-373.

[14] M.L. Puri, D.A. Ralescu, Differentials for fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552-558.

[15] I.J. Maddox, A new type of convergence, Math. Proc. Cambridge Philos. Soc. 83 (1978) 61-64.

[16] Mursaleen and M. Basarir, On some new sequence spaces of fuzzy numbers, Indian J. Pure Appl.Math. 34(9) (2003) 1351-1357.

[17] E. Savas, On strongly λ-summable sequence of fuzzy numbers, Information Sciences 125(2000) 181-186.



IF rough approximations based on lattices∗

Gangqiang Zhang† Yu Han‡ Zhaowen Li§

January 24, 2015

Abstract:An IF rough set, which is the result of approximation of an IF setwith respect to an IF approximation space, is an extension of fuzzy rough sets.This paper studies rough set theory within the context of lattices. First, weintroduce the concepts of IF rough sets and IF rough approximation operatorsbased on lattices. Then, we give some properties on IF rough approximationsof IF sublattices such as IF ideals and IF filters.

Keywords: Lattice; IF set; Full congruence relation; IF approximate space;IF rough set; IF sublattice; IF rough approximation.

1 Introduction

Rough set theory was originally proposed by Pawlak [11, 12] as a mathemati-cal approach to handle imprecision and uncertainty in data analysis. Usefulnessand versatility of this theory have amply been demonstrated by successful ap-plications in a variety of problems [15, 16].

The basic structure of rough set theory is an approximation space. Based onit, lower and upper approximations can be induced. Using these approximations,knowledge hidden in information systems may be revealed and expressed in theform of decision rules [11].

Intuitionistic fuzzy (IF, for short) sets were originated by Atanassov [1, 2].It is an intuitively straightforward extension of Zadeh’s fuzzy sets [19]. IF setshave played an useful role in the research of uncertainty theories. Unlike a fuzzyset, which gives a degree of which element belongs to a set, an IF set gives both

∗This work is supported by the National Natural Science Foundation of China (11461005),the Natural Science Foundation of Guangxi (2014GXNSFAA118001), Guangxi University Sci-ence and Technology Research Project, Key Laboratory of Optimization Control and En-gineering Calculation in Department of Guangxi Education and Special Funds of GuangxiDistinguished Experts Construction Engineering.

†Corresponding Author, College of Information Science and Engineering, Guangxi Univer-sity for Nationalities, Nanning, Guangxi 530006, P.R.China. [email protected]

‡College of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006,P.R.China.

§Guangxi Key Laboratory of Universities Optimization Control and Engineering Calcula-tion, and College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006,P.R.China.

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237 Gangqiang Zhang et al 237-253

a membership degree and a nonmembership degree. Thus, an IF set is moreobjective than a fuzzy set to describe the vagueness of data or information.

Recently, rough set approximation was introduced into IF sets [14, 20, 21, 22].For example, Zhou et al. [20, 21, 22] proposed a general framework for the studyof IF rough sets, Zhang et al. [24] gave a general frame for IF rough sets on twouniverses.

The purpose of this paper is to investigate IF rough approximations basedon lattices.

2 Preliminaries

Throughout this paper, “ Intuitionistic fuzzy ” is briefly written “ IF ”, Udenotes a universe, I denotes [0, 1], L denotes a lattice with the least element0L and the greatest element 1L. J = (a, b) ∈ I × I : a + b ≤ 1.

In this section, we recall some basic notions and properties.

2.1 IF sets

Definition 2.1 ([8]). Let (a, b), (c, d) ∈ I × I. Define(1) (a, b) = (c, d) ⇐⇒ a = c, b = d.(2) (a, b) t (c, d) = (a ∨ c, b ∧ d), (a, b) u (c, d) = (a ∧ c, b ∨ d).(3) (a, b)c = (b, a).Moreover, for (aα, bα) : α ∈ Γ ⊆ I × I,⊔α∈Γ

(aα, bα) = (∨

α∈Γ

aα,∧

α∈Γ

bα),d

α∈Γ

(aα, bα) = (∧

α∈Γ

aα,∨

α∈Γ

bα).

Definition 2.2 ([8]). Let (a, b), (c, d) ∈ J and let S ⊆ J × J . (a, b)S(c, d), if a≤ c and b ≥ d. We denote S by ≤.

Remark 2.3. (1) Let (J,≤) be a poset with 0J = (0, 1) and 1J = (1, 0).(2) (a, b)cc = (a, b).(3) ((a, b) t (c, d)) t (e, f) = (a, b) t ((c, d) t (e, f)),

((a, b) u (c, d)) u (e, f) = (a, b) u ((c, d) u (e, f)).(4) (a, b) t (c, d) = (c, d) t (a, b), (a, b) u (c, d) = (c, d) u (a, b).(5) ((a, b) t (c, d)) u (e, f) = ((a, b) u (e, f)) t ((c, d) u (e, f)).

((a, b) u (c, d)) t (e, f) = ((a, b) t (e, f)) u ((c, d) t (e, f)).(6) (

⊔α∈Γ

(aα, bα))c =d

α∈Γ

(aα, bα)c, (d

α∈Γ

(aα, bα))c =⊔

α∈Γ

(aα, bα)c.

Definition 2.4 ([1]). An IF set A in U is an object having the form

A = < x, µA(x), νA(x) >: x ∈ U,where µA, νA ∈ F (U) satisfying 0 ≤ µA(x) + νA(x) ≤ 1 for each x ∈ U , andµA(x), νA(x) are used to define the degree of membership and the degree of non-membership of the element x to A, respectively.

IF (U)) denotes the family of all IF sets in U .For the sake of simplicity, we give the following definition.

2



Definition 2.5. A is called an IF set in U , if A = (A∗, A∗) ∈ F (U) × F (U)and for each x ∈ U , A(x) = (A∗(x), A∗(x)) ∈ J , where A∗(x), A∗(x) are usedto define the degree of membership and the degree of non-membership of theelement x to A, respectively.

For each A ⊆ IF (U), we denote

Ac = Ac : A ∈ A,

A∗ = A∗ : A ∈ A and A∗ = A∗ : A ∈ A.For each λ ∈ J , λ represents a constant IF set which satisfies λ(x) = λ for

each x ∈ U .A ∈ IF (U) is called proper if A 6= λ for any λ ∈ J .In this paper, if we concern IF sets in U without special statements, we

always refer to the proper IF subset.Some IF relations and IF operations are defined as follows ([19]): for any

A,B ∈ IF (U) and Aα : α ∈ Γ ⊆ IF (U),(1) A = B ⇐⇒ A(x) = B(x) for each x ∈ U .(2) A ⊆ B ⇐⇒ A(x) ≤ B(x) for each x ∈ U .

(3) (⋃

α∈Γ

Aα)(x) =⊔

α∈Γ

Aα(x) for each x ∈ U .

(4) (⋂

α∈Γ

Aα)(x) =d

α∈Γ

Aα(x) for each x ∈ U .

(5) Ac(x) = A(x)c for each x ∈ U .(6) (λA)(x) = λ u (A∗(x), A∗(x)) for any x ∈ U and λ ∈ J .Obviously, A = B ⇐⇒ A∗ = B∗ and A∗ = B∗ ⇐⇒ A ⊆ B and B ⊆ A.We define a special IF sets 1y = ((1y)∗, (1y)∗) for some y ∈ U as follows:

(1y)∗(x) =

1, x = y,

0, x 6= y.(1y)∗(x) =

0, x = y,

1, x 6= y.

Remark 2.6. For each A ∈ IF (U),

A =⋃

y∈U

(A(y)1y).

Let µ ∈ IF (U) and α, β ∈ [0, 1] with α + β ≤ 1, the (α, β)-level cut set ofµ,denoted by µβ

α,is defined as follows:

µβα = x ∈ U : µ∗(x) ≥ α, µ∗(x) ≤ β.

We respectively call the sets

µα = x ∈ U : µ∗(x) ≥ α, µβ = x ∈ U : µ∗(x) ≤ βthe α-level cut set, the β-level set of membership generated by A.

3



For x ∈ U and (a, b) ∈ J − (0, 1), x(a,b) ∈ IF (U) is called an IF point if

x(a,b)(y) =

(0, 1), y 6= x,

(a, b), y = x.

It is said that the IF point x(a,b) belongs to µ ∈ IF (U), which is writtenx(a,b) ∈ µ. Obviously,

x(a,b) ∈ µ ⇐⇒ µ(x) ≥ (a, b).

IFP (U) denotes the set of all IF point of U .For µ, λ ∈ IF (U),

µ ⊆ λ ⇐⇒ ∀ x(a,b) ∈ IFP (U), x(a,b) ∈ µ implies x(a,b) ∈ λ.

2.2 Lattices

Definition 2.7. Let L be a set and let ≤ be a binary relation on L. Then ≤ iscalled a partial order on L, if

(i) a ≤ a for any a ∈ L, (ii) a ≤ b and b ≤ a imply a = b for any a, b ∈ L,(iii) a ≤ b and b ≤ c imply a ≤ c for any a, b, c ∈ L.

Moreover, the pair (L,≤) is called a partial order set (briefly, a poset).

Definition 2.8. Let (L,≤) be a poset and a, b ∈ L.(1) a is called a top (or maximal) element of L, if x ≤ a for any x ∈ L.(2) b is called a bottom (or minimal) element of L,, if b ≤ x for any x ∈ L.

If a poset L has top elements a1, a2 (resp. bottom elements b1, b2), thena1 = a2 (resp. b1 = b2). We denote this sole top element (resp. this sole bottomelement) by 1L (resp. 0L).

Definition 2.9. Let (L,≤) be a poset, S ⊆ L and a, b ∈ L.(1) a is called a above boundary in S, if x ≤ a for any x ∈ S.(2) b is called a under boundary in S, if b ≤ x for any x ∈ S.(3) a = sup S or ∨ S, if a is a minimal above boundary in S.(4) b = inf S or ∧ S, if b is a maximal under boundary in S.

Let (L,≤) be a poset and S ⊆ L. If S = a, b, then we denote ∨S = a ∨ band ∧S = a ∧ b.

Obviously, if (L,≤) is a poset and a, b ∈ L, then

a = a ∧ b ⇔ a ≤ b ⇔ b = a ∨ b.

A poset L is called a lattice, if for any a, b ∈ L, a ∨ b ∈ L and a ∧ b ∈ L.Let L be a lattice. For X ⊆ L, we denote(1) ↓ X = y ∈ L : y ≤ x for some x ∈ X,(2) ↑ X = y ∈ L : y ≥ x for some x ∈ X.Especially, ↓ x =↓ x, ↑ x =↑ x.

4



F (L)(resp. IF(L)) denotes the family of all fuzzy (resp. IF ) sets in L.µ ∈ F (L) is called a fuzzy sublattice of L, if µ(x∧y)∧µ(x∨y) ≥ µ(x)∧µ(y)

for any x, y ∈ L.Let µ be a fuzzy sublattice of L.(1) µ is a fuzzy ideal of L, if µ(x ∨ y) = µ(x) ∧ µ(y) for any x, y ∈ L.(2) µ is a fuzzy filter of L, if µ(x ∧ y) = µ(x) ∧ µ(y) for any x, y ∈ L.

2.3 Fuzzy rough approximation operators based on lat-tices

Definition 2.10 ([3]). Let θ be an equivalence relation on L. The pair (L, θ)is called Pawlak approximation space. For each µ ∈ F (L), the fuzzy lower andthe fuzzy upper approximation of µ with respect to (L, θ), denoted by θ(µ) andθ(µ), are defined as follows: for each x ∈ L,

θ(µ)(x) =∧

a∈[x]θ

A(a), θ(µ)(x) =∨

a∈[x]θ

A(a).

The pair (θ(µ), θ(µ)) is called the fuzzy rough set of µ with respect to (L, θ).θ : F (L) → F (L) and θ : F (L) → F (L) are called the fuzzy lower approx-

imation operator and the fuzzy upper approximation operator, respectively. Ingeneral, we refer to θ and θ as the fuzzy rough approximation operators.

Proposition 2.11 ([3]). Let θ be an equivalence relation on L. Then for µ, λ ∈F (L),

(1) θ(µ) ⊆ µ ⊆ θ(µ).(2) Ifµ ⊆ λ, then θ(µ) ⊆ θ(λ) and θ(µ) ⊆ θ(λ).(3) θ(µc) = (θ(µ))c and θ(µc) = (θ(µ))c.(4) θθ(µ) = θ(µ) and θθ(µ) = θ(µ).(5) θ(µ)(x) = θ(µ)(a) and θ(µ)(x) = θ(µ)(a) for any x ∈ L and a ∈ [x]θ.(6) θθ(µ) = θ(µ) and θθ(µ) = θ(µ).

Definition 2.12 ([3]). Let θ be an equivalence relation on L. Then θ is calleda full congruence relation, if (a, b) ∈ θ implies that (a ∨ x, b ∨ x) ∈ θ and(a ∧ x, b ∧ x) ∈ θ for any x ∈ L.

For a ∈ L, denote[a]θ = x ∈ L : (a, x) ∈ θ, L/θ = [a]θ : a ∈ L.

Lemma 2.13 ([3]). Let θ be a full congruence relation on L. Then for anya, b, c, d ∈ L,

(1) If (a, b), (c, d) ∈ θ, then (a ∨ c, b ∨ d), (a ∧ c, b ∧ d) ∈ θ.(2) If x ∈ [a]θ, y ∈ [b]θ, then x ∨ y ∈ [a ∨ b]θ.(3) If x ∈ [a]θ, y ∈ [b]θ, then x ∧ y ∈ [a ∧ b]θ.

Proposition 2.14 ([3]). Let θ be a full congruence relation on L.

5



(1) If µ is a fuzzy ideal, then for x, y ∈ L,

θ(µ)(x ∧ y) =∧

a∈[x]θ,b∈[y]θ

µ(a ∧ b), θ(µ)(x ∨ y) =∨

a∈[x]θ,b∈[y]θ

µ(a ∨ b).

(2) If µ is a fuzzy filter, then for x, y ∈ L,

θ(µ)(x ∨ y) =∧

a∈[x]θ,b∈[y]θ

µ(a ∨ b), θ(µ)(x ∧ y) =∨

a∈[x]θ,b∈[y]θ

µ(a ∧ b).

3 IF rough sets and IF rough approximation op-erators based on lattices

Definition 3.1. Let θ be an equivalence relation on L. The pair (L, θ) is calledPawlak approximation space. For each µ ∈ IF (L), the IF lower and the IFupper approximation of µ with respect to (L, θ), denoted by θ(µ) and θ(µ), aredefined as follows:

θ(µ) = ((θ(µ))∗, (θ(µ))∗),

θ(µ) = ((θ(µ))∗, (θ(µ))∗),

where for each x ∈ L,

(θ(µ))∗(x) =∧

a∈[x]θ

µ∗(a), (θ(µ))∗(x) =∨

a∈[x]θ

µ∗(a),

(θ(µ))∗(x) =∨

a∈[x]θ

µ∗(a), (θ(µ))∗(x) =∧

a∈[x]θ

µ∗(a).

The pair (θ(µ), θ(µ)) is called the IF rough set of µ with respect to (L, θ).θ : IF (L) → IF (L) and θ : IF (L) → IF (L) are called the IF lower ap-

proximation operator and the IF upper approximation operator, respectively. Ingeneral, we refer to θ and θ as the IF rough approximation operators.

Remark 3.2. (1) (θ(µ))∗ = θ(µ∗) (θ(µ))∗ = θ(µ∗)(2) (θ(µ))∗ = θ(µ∗) (θ(µ))∗ = θ(µ∗)

Proposition 3.3. For any x ∈ L,

θ(µ)(x) =l

a∈[x]θ

µ(a), θ(µ)(x) =⊔

a∈[x]θ

µ(a).

Proof.

θ(µ)(x) = (∧

a∈[x]θ

µ∗(a),∨

a∈[x]θ

µ∗(a))

=l

a∈[x]θ

(µ∗(a), µ∗(a))

=l

a∈[x]θ

µ(a).

6



θ(µ)(x) = (∨

a∈[x]θ

µ∗(a),∧

a∈[x]θ

µ∗(a))

=⊔

a∈[x]θ

(µ∗(a), µ∗(a))

=⊔

a∈[x]θ

µ(a).

Proposition 3.4. Let θ be an equivalence relation on L. Then for any µ, λ ∈IF (L),

(1) θ(µ) ⊆ µ ⊆ θ(µ).(2) Ifµ ⊆ λ, then θ(µ) ⊆ θ(λ) and θ(µ) ⊆ θ(λ).(3) θθ(µ) = θ(µ) and θθ(µ) = θ(µ).(4) θ(µ)(x) = θ(µ)(a) and θ(µ)(x) = θ(µ)(a) for any x ∈ L and a ∈ [x]θ(5) θθ(µ) = θ(µ) and θθ(µ) = θ(µ).

Proof. It is straightforward.

Proposition 3.5. Let θ be an equivalence relation on L. Then for any µi :i ∈ I ⊆ IF (L),

(1) θ(⊔i∈I

µi) ⊇⊔i∈I

θ(µi), θ(di∈I

µi)=di∈I

θ(µi).

(2) θ(⊔i∈I

µi) =⊔i∈I

θ(µi), θ(di∈I

µi) ⊆di∈I

θ(µi).

Proof. (1) For any x ∈ L,θ(

⊔i∈I

µi)(x)=d

a∈[x]θ

⊔i∈I

µi(a) ⊇ ⊔i∈I

da∈[x]θ

µi(a)=⊔i∈I

θ(µi)(x),

θ(di∈I

µi)(x)=d

a∈[x]θ

di∈I

µi(a)=di∈I

da∈[x]θ

µi(a)=di∈I

θ(µi)(x).

Thus, θ(⊔i∈I

µi) ⊇⊔i∈I

θ(µi), θ(di∈I

µi)=di∈I

θ(µi).

(2) The proof is similar to (1).

4 IF sublattices and IF rough approximationsbased on lattices

4.1 IF sublattices

Definition 4.1. µ ∈ IF (L) is called an IF sublattice of L, ifµ(x ∧ y) u µ(x ∨ y) ≥ µ(x) u µ(y) for any x, y ∈ L.

Definition 4.2. Let µ be an IF sublattice of L. Then(1) µ is an IF ideal of L, if µ(x ∨ y) = µ(x) u µ(y) for any x, y ∈ L.(2) µ is an IF filter of L, if µ(x ∧ y) = µ(x) u µ(y) for any x, y ∈ L.

7



Denote the set of all IF ideals of L by IFI(L).

Proposition 4.3. Let µ be an IF sublattice of L. Then(1) µ is an IF ideal of L ⇐⇒ x ≤ y implies that µ(x) ≥ µ(y) for any

x, y ∈ L.(2) µ is an IF filterof L ⇐⇒ x ≤ y implies that µ(x) ≤ µ(y) for any

x, y ∈ L.


Let µ be a proper IF ideal of L. Then(1) µ is called an IF prime ideal of L, if µ(x ∧ y) ≤ µ(x) t µ(y) for any

x, y ∈ L.(2) µ is called an IF prime filter of L, if µ(x ∨ y) ≤ µ(x) t µ(y) for any

x, y ∈ L.

4.2 IF rough approximations of some IF sublattices

Lemma 4.4. Let θ be a full congruence relation on L.(1) If µ is an IF ideal of L, then for any x, y ∈ L,

θ(µ)(x ∧ y) =l

a∈[x]θ,b∈[y]θ

µ(a ∧ b), θ(µ)(x ∨ y) =⊔

a∈[x]θ,b∈[y]θ

µ(a ∨ b).

(2) If µ is an IF filter of L, then for any x, y ∈ L,

θ(µ)(x ∨ y) =l

a∈[x]θ,b∈[y]θ

µ(a ∨ b), θ(µ)(x ∧ y) =⊔

a∈[x]θ,b∈[y]θ

µ(a ∧ b).

Proof. (1) By Lemma 2.12,∧z∈[x∧y]θ

µ∗(z) ≤ ∧a∈[x]θ,b∈[y]θ

µ∗(a ∧ b),∨

z∈[x∧y]θ

µ∗(z) ≥ ∨a∈[x]θ,b∈[y]θ

µ∗(a ∧ b).

Thenθ(µ)(x ∧ y) =

l

z∈[x∧y]θ

µ(z) ≤l

a∈[x]θ,b∈[y]θ

µ(a ∧ b).

Now assume that z ∈ [x ∧ y]θ. Then z ∨ x ∈ [x]θ, z ∨ y ∈ [y]θ.Since z ≤ (z ∨ x) ∧ (z ∨ y), by Proposition 2.14, we have

µ(z) ≥ µ(z ∨ x) ∧ (z ∨ y).

Then

µ∗(z) ≥ µ∗((z ∨ x) ∧ (z ∨ y)), µ∗(z) ≤ µ∗((z ∨ x) ∧ (z ∨ y)).

Note that∧

z∈[x∧y]θ

µ∗(z) ≥∧

a∈[x]θ,b∈[y]θ

µ∗(a ∧ b),∨

z∈[x∧y]θ

µ∗(z) ≤∨

a∈[x]θ,b∈[y]θ

µ∗(a ∧ b).

8



Thend

z∈[x∧y]θ

µ(z) ≥ da∈[x]θ,b∈[y]θ

µ(a ∧ b).

Thusθ(µ)(x ∧ y) =

l

a∈[x]θ,b∈[y]θ

µ(a ∧ b).

Note that∨

z∈[x∧y]θ

µ∗(z) ≥∨

a∈[x]θ,b∈[y]θ

µ∗(a ∨ b),∧

z∈[x∨y]θ

µ∗(z) ≥∧

a∈[x]θ,b∈[y]θ

µ∗(a ∧ b).

Thenθ(µ)(x ∨ y) =

⊔

z∈[x∨y]θ

µ(z) ≥⊔

a∈[x]θ,b∈[y]θ

µ(a ∨ b).

Now assume that z ∈ [x ∨ y]θ. Then

z ∧ x ∈ [x]θ, z ∧ y ∈ [y]θ.

Since z ≥ (z ∧ x) ∨ (z ∧ y), by Proposition 2.14, we have

µ(z) ≤ µ(z ∧ x) ∨ (z ∧ y).

Then µ∗(z) ≤ µ∗((z ∧ x) ∨ (z ∧ y)), µ∗(z) ≥ µ∗((z ∧ x) ∨ (z ∧ y)).Since∨

z∈[x∨y]θ

µ∗(z) ≤∨

a∈[x]θ,b∈[y]θ

µ∗(a ∨ b),∧

z∈[x∨y]θ

µ∗(z) ≥∧

a∈[x]θ,b∈[y]θ

µ∗(a ∨ b).

we have ⊔

z∈[x∨y]θ

µ(z) ≤⊔

a∈[x]θ,b∈[y]θ

µ(a ∨ b).

Thusθ(µ)(x ∨ y) =

⊔

a∈[x]θ,b∈[y]θ

µ(a ∨ b).


Proposition 4.5. Let θ be a full congruence relation on L. Let µ ∈ IF (L) andlet θ(µ) be an IF sublattice of L. Then

(1) If µ is an IF ideal of L, then θ(µ) is an IF ideal of L.(2) If µ is an IF filter of L, then θ(µ) is an IF filter of L.

Proof. (1) Since θ(u) is an IF sublattice of L, we conclude that for any x, y ∈ L,

θ(µ)(x ∨ y) ≥ θ(µ)(x ∧ y) u θ(µ)(x ∨ y) ≥ θ(µ)(x) u θ(µ)(y).

9



By Lemma 4.4, for any x, y ∈ L,

θ(µ)(x ∨ y) =l

z∈[x∨y]θ

µ(z)

≤l

a∈[x]θ,b∈[y]θ

µ(a ∨ b)

=l

a∈[x]θ,b∈[y]θ

(µ(a) u µ(b))

=l

a∈[x]θ,b∈[y]θ

(µ∗(a) ∧ µ∗(b), µ∗(a) ∨ µ∗(b))

= (∧

a∈[x]θ,b∈[y]θ

(µ∗(a) ∧ µ∗(b)),∨

a∈[x]θ,b∈[y]θ

(µ∗(a) ∨ µ∗(b)))

= ((∧

a∈[x]θ

µ(a)) ∧ (∧

b∈[y]θ

µ(b)), (∨

a∈[x]θ

µ(a)) ∨ (∨

b∈[y]θ

µ(b)))

= (∧

a∈[x]θ

µ(a),∨

a∈[x]θ

µ(a)) u (∧

b∈[y]θ

µ(b),∨

b∈[y]θ

µ(b))

= (l

a∈[x]θ

µ(a)) u (l

b∈[y]θ

µ(b))

= θ(µ)(x) u θ(µ)(y).


Proposition 4.6. Let θ be a full congruence relation on L. Then for µ ∈ IF (L),(1) If µ is an IF sublattice of L, then θ(µ) is an IF sublattice of L.(2) If µ is an IF ideal of L, then θ(µ) is an IF ideal of L.(3) If µ is an IF filter of L, then θ(µ) is an IF filter of L.

Proof. (1) Suppose that µ is an IF sublattice of L. Then for any x, y ∈ L,

θ(µ)(x ∧ y)d

θ(µ)(x ∨ y)= (

∨a∈[x∧y]θ

µ∗(a),∧

a∈[x∧y]θ

µ∗(a))d

(∨

b∈[x∨y]θ

µ∗(b),∧

b∈[x∨y]θ

µ∗(b))

= (∨

a∈[x∧y]θ

µ∗(a) ∧ ∨b∈[x∨y]θ

µ∗(b),∧

a∈[x∧y]θ

µ∗(a) ∨ ∧b∈[x∨y]θ

µ∗(b))

≥ (∨

a∈[x]θ,c∈[y]θ

µ∗(a ∧ c) ∧ ∨b∈[x]θ,d∈[y]θ

µ∗(b ∨ d),∧

a∈[x]θ,c∈[y]θ

µ∗(a ∧ c) ∨ ∧b∈[x]θ,d∈[y]θ

µ∗(b ∨ d))

≥ (∨

a,b∈[x]θ,c,d∈[y]θ

µ∗(a ∧ c) ∧ µ∗(b ∨ d),∧

a,b∈[x]θ,c,d∈[y]θ

µ∗(a ∧ c) ∨ µ∗(b ∨ d))

≥ (∨

a∈[x]θ,c∈[y]θ

µ∗(a ∧ c) ∧ µ∗(a ∨ c),∧

a∈[x]θ,c∈[y]θ

µ∗(a ∧ c) ∨ µ∗(a ∨ c))

≥ (∨

a∈[x]θ,c∈[y]θ

µ∗(a) ∧ µ∗(c),∧

a∈[x]θ,c∈[y]θ

µ∗(a) ∨ µ∗(c))

10



= (∨

a∈[x]θ

µ∗(a) ∧ ∨c∈[y]θ

µ∗(c),∧

a∈[x]θ,

µ∗(a) ∨ ∧c∈[y]θ

µ∗(c))

= (∨

a∈[x]θ

µ∗(a),∧

a∈[x]θ

µ∗(a))d

(∨

b∈[y]θ

µ∗(b),∧

b∈[y]θ

µ∗(b))

= θ(µ)(x)d

θ(µ)(y)Thus θ(µ) is an IF sublattice of L.(2) Suppose that µ is an IF ideal of L. Then µ is an IF sublattice of L.By (1), θ(µ) is an IF sublattice of L.For any x, y ∈ L,

θ(µ)(x ∨ y) =⊔

a∈[x]θ,b∈[y]θ

µ(a ∨ b)

=⊔

a∈[x]θ,b∈[y]θ

(µ(a) u µ(b))

=⊔

a∈[x]θ,b∈[y]θ

(µ∗(a) ∧ µ∗(b), µ∗(a) ∨ µ∗(b))

= (∨

a∈[x]θ,b∈[y]θ

(µ∗(a) ∧ µ∗(b)),∧

a∈[x]θ,b∈[y]θ

(µ∗(a) ∨ µ∗(b)))

= ((∨

a∈[x]θ

µ∗(a)) ∧ (∨

b∈[y]θ

µ∗(b)), (∧

a∈[x]θ

µ∗(a)) ∨ (∧

b∈[y]θ

µ∗(b)))

= (∨

a∈[x]θ

µ∗(a),∧

a∈[x]θ

µ∗(a)) u (∨

b∈[y]θ

µ∗(b),∧

b∈[y]θ

µ∗(b))

=⊔

a∈[x]θ

µ(a) u⊔

b∈[x]θ

µ(b)

= θ(µ)(x)l

θ(µ)(y)

Thus θ(µ) is an IF ideal of L.(3) The proof is similar to (2).

Proposition 4.7. Let θ be a full congruence relation on L and let θ(µ) is aproper IF sublattice of L.

(1) If µ ∈ IF (L) is an IF prime ideal of L, then θ(µ) is an IF prime idealof L.

(2) If µ ∈ IF (L) is an IF prime filter of L, then θ(µ) is an IF prime filterof L.

Proof. (1) Suppose that µ is an IF prime ideal of L.By Proposition 4.5, θ(µ) is an IF ideal of L.

11



By Proposition 4.3 and Lemma 4.4, for any x, y ∈ L.

θ(µ)(x ∧ y) =l

a∈[x]θ,b∈[y]θ

µ(a ∧ b)

=l

a∈[x]θ,b∈[y]θ

(µ∗(a ∧ b), µ∗(a ∧ b))

= (∧

a∈[x]θ,b∈[y]θ

µ∗(a ∧ b),∨

a∈[x]θ,b∈[y]θ

µ∗(a ∧ b))

≤ (∧

a∈[x]θ,b∈[y]θ

(µ∗(a) ∧ µ∗(b)),∨

a∈[x]θ,b∈[y]θ

(µ∗(a) ∧ µ∗(b)))

= (∧

a∈[x]θ

µ∗(a) ∨∧

b∈[y]θ

µ∗(b),∨

a∈[x]θ

µ∗(a) ∧∨

b∈[y]θ

µ∗(b))

= (∧

a∈[x]θ

µ∗(a),∨

a∈[x]θ

µ∗(a)) t (∧

b∈[y]θ

µ∗(b),∨

b∈[y]θ

µ∗(b))

=l

a∈[x]θ

µ(a) tl

b∈[y]θ

µ(b)

= θ(µ)(x) t θ(µ)(y).

Thus θ(µ) is an IF prime ideal of L.(2) The proof is similar to (1).

Definition 4.8. Let θ be a full congruence relation on L. Then(1) θ is called ∨-complete, if x ∨ y : x ∈ [a]θ, y ∈ [b]θ = [a ∨ b]θ for any

a, b ∈ L.(2) θ is called ∧-complete, if x ∧ y : x ∈ [a]θ, y ∈ [b]θ = [a ∧ b]θ for any

a, b ∈ L.(3) θ is called complete, if θ is both ∨-complete and ∧-complete.

Proposition 4.9. Let θ be a full congruence relation on L.(1) Let µ be an IF prime ideal of L and let θ be ∧-complete. If θ(µ) is proper,

then θ(µ) is an IF prime ideal of L.(2) Let µ be an IF prime filter of L and let θ be ∨-complete. If θ(µ) is proper,

then θ(µ) is an IF filter ideal.

Proof. (1) By Proposition 4.6, θ(µ) is an IF ideal of L.

12



Since θ is ∧-complete, for any x, y ∈ L, we have

θ(µ)(x ∧ y) =⊔

a∈[x]θ,b∈[y]θ

µ(a ∧ b)

≤⊔

a∈[x]θ,b∈[y]θ

µ(a) t µ(b)

=⊔

a∈[x]θ,b∈[y]θ

(µ∗(a) ∨ µ∗(b), µ∗(a) ∧ µ∗(b))

= (∨

a∈[x]θ,b∈[y]θ

(µ∗(a) ∨ µ∗(b)),∧

a∈[x]θ,b∈[y]θ

(µ∗(a) ∧ µ∗(b)))

= ((∨

a∈[x]θ

µ∗(a)) ∨ (∨

b∈[y]θ

µ∗(b)), (∧

a∈[x]θ

µ∗(a)) ∧ (∧

b∈[y]θ

µ∗(b)))

= (∨

a∈[x]θ

µ∗(a),∧

a∈[x]θ

µ∗(a)) t (∨

b∈[y]θ

µ∗(b),∧

b∈[y]θ

µ∗(b))

=⊔

a∈[x]θ

µ(a) t⊔

b∈[y]θ

µ(b)

= θ(µ)(x) t θ(µ)(y)

Thus θ(µ) is an IF prime ideal of L.(2) The proof is similar to (1)

Definition 4.10. Let µ ∈ IF (L). The least IF ideal of L containing µ is calledan IF ideal of L induced by µ. We denoted it by < µ >.

For any µ ∈ IF (L), we denote

µ¦(x) =⊔(α, β) ∈ J : x ∈ I(µβ

α) (x ∈ L).

Proposition 4.11. Let µ ∈ IF (L). Then(1) µ ⊆ µ¦.(2) µ¦ =

⋂ν ∈ IFI(L) : µ ⊆ ν , ν(0L) = 1J.Proof. (1) Consider that µβ

α = x ∈ U : µ(x) ≥ (α, β). Then

µ(x) = t(α, β) : x ∈ µβα ≤ t(α, β) : x ∈ I(µβ

α) = µ¦.

(2) Firstly, we can prove that µ¦ ∈ IFI(L).For any x, y ∈ L,

µ¦(x) =⊔(α, β) ∈ J : x ∈ I(µβ

α),µ¦(y) =

⊔(α, β) ∈ J : y ∈ I(µβα).

Put

A = (α, β) ∈ J : x ∈ I(µβα), B = (α, β) ∈ J : y ∈ I(µβ

α).

13



Suppose x ≤ y. Then A ⊆ B. So tA ≤ tB. This implies that

µ¦(y) ≤ µ¦(x).

Secondly, since 0L ∈ I(µ01), we have 1J ≤ µ¦(0L). Then µ¦(0L) = 1J .

Combined with (1), we have

µ¦ ∈ ν ∈ IFI(L) : µ ⊆ ν , ν(0L) = 1J.Then

µ¦ ⊇ ∩ν ∈ IFI(L) : µ ⊆ ν , ν(0L) = 1J.Now, we need to prove that

µ¦ ⊆ ∩ν ∈ IFI(L) : µ ⊆ ν, ν(0L) = 1J.For any ν ∈ ν ∈ IFI(L) : µ ⊆ ν , ν(0L) = 1J, we have µ ⊆ ν. This implies

µβα ⊆ νβ

α.

Then I(µβα) ⊆ I(νβ

α).Denote

C = (α, β) ∈ J : x ∈ I(νβα).

Then A ⊆ C and so µ¦(x) ≤ ν¦(x).Note that ν ∈ IFI(L). Then νβ

α ∈ IFI(L). So I(νβα) = νβ

α.This implies that

ν¦(x) = tC = t(α, β) ∈ J : x ∈ νβα = ν(x).

Thus µ¦(x) ≤ ν(x).Hence

µ¦ ⊆ ∩ν ∈ IFI(L) : µ ⊆ ν, ν(0L) = 1J.

Proposition 4.12. Let θ be a full congruence relation on L. Then for anyµ ∈ IF (L),

(1) θ(< µ >) = θ(< θ(µ) >).(2) θ(µ¦) = θ((θ(µ))

¦).

Proof. (1) Since µ ⊆< µ >, we conclude from Proposition 3.4 that

θ(µ) ⊆ θ(< µ >).

By Proposition 4.6 and Proposition 4.11,

< θ(µ) >⊆ θ(< µ >).

By Proposition 3.4,

θ(< θ(µ) >) ⊆ θ(< µ >).

14



Note that µ ⊆ θ(µ). Then < µ >⊆< θ(µ) >.By Proposition 3.4,

θ(< µ >) ⊆ θ(< θ(µ) >).

Thusθ(< µ >) = θ(< θ(µ) >).

(2) Since < µ >⊆ µ¦, by Proposition 3.4, we have θ(< µ >) ⊆ θ(µ¦).It is clear that

θ(µ¦)(0L) = 1J , < θ(µ) > ⊆ < θ(µ¦) > = θ(µ¦).

Then (θ(µ))¦ ⊆ θ(µ¦)

By Proposition 3.4,θ((θ(µ))

¦) ⊆ θ(µ¦).

Since µ¦ ⊆ (θ(µ))¦

we conclude θ(µ¦) ⊆ θ((θ(µ))¦).

Thusθ(µ¦) = θ((θ(µ))

¦).

Proposition 4.13. Let a(r,s), b(p,q) ∈ IFP (L) and µ ∈ IF (L). Then(1) θ(a(r,s)) = χ

(r,s)[a]θ

.

(2) < a(r,s) > (x) = χ(r,s)↓a and (a(r,s))¦(x) =

(1, 0) x = 0L,

(r, s) 0L 6= x,

(0, 1) otherwise.

(3) θ(< a(r,s) >)(x) =

(r, s) ↓ a ∩ [x]θ 6= ∅,(0, 1) otherwise.

θ((a(r,s))¦)(x) =

(1, 0) 0L ∈ [x]θ(r, s) ↓ a ∩ [x]θ 6= ∅,(0, 1) otherwise.

(4) < a(r,s) > ∧ < b(p,q) >=< (a ∧ b)(r,s)∧(p,q) >


Let θ be an equivalence relation on L. µ ∈ F (L) is called a fixed-point ofθ-upper (resp. θ-lower) rough approximation, if θ(µ) = µ (resp. θ(µ) = µ).

Denote

Fix(θ) = µ ∈ F (L) | θ(µ) = µ, F ix(θ) = µ ∈ F (L) | θ(µ) = µ.

15



Proposition 4.14. Let θ1 and θ2 be two equivalence relations on L. Then thefollowing are equivalent:

(1) For each µ ∈ F (L), θ1(µ) ≤ θ2(µ);(2) For each µ ∈ F (L), θ1(µ) ≥ θ2(µ);(3) Fix(θ2) ⊆ Fix(θ1);(4) Fix(θ2) ⊆ Fix(θ1).

Proof. (1) =⇒ (2). This holds by Proposition 2.10.(2) =⇒ (3) Let µ ∈ F (L) and θ1(µ

c) ≥ θ2(µc).

By Proposition 2.10, (θ1(µ))c ≥ θ2(µ))c.Thus θ1(µ) ≤ θ2(µ).Note that θ2(µ) = µ. Then µ ≤ θ1(µ) ≤ θ2(µ) = µ.It follows that θ1(µ) = µ.(3) =⇒ (1) Let µ ∈ F (L). Since θ2(µ) ∈ Fix(θ2), we have θ2(µ) ∈ Fix(θ1).Thus θ1(µ) ≤ θ1(θ2(µ)) = θ2(µ).(2) =⇒ (4) Let µ ∈ F (L) and θ2(µ) = µ. Then µ = θ2(µ) ≤ θ1(µ) = µ.It follows that θ1(µ) = µ.(4) =⇒ (2) Let µ ∈ F (L). By Proposition 3.4, θ2(µ) ∈ Fix(θ2).Then θ2(µ) ∈ Fix(θ1). Thus

θ2(µ) = θ1(θ2(µ)) ≤ θ1(µ).

References

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[12] Z.Pawlak, Rough sets: Theoretical aspects of reasoning about data, KluwerAcademic Publishers, Boston, 1991.

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17



Some results on approximating spaces ∗

Neiping Chen†

January 24, 2015

Abstract: Topology and rough set theory are widely used in research fieldof computer science. In this paper, we study properties of topologies inducedby binary relations, investigate a particular type of topological spaces whichassociate with some equivalence relation (i.e., approximating spaces) and obtainsome characteristic conditions of approximating spaces.

Keywords: Binary relation; Rough set; Topology; Approximating space

1 Introduction

Rough set theory, proposed by Pawlak [8], is a new mathematical tool fordata reasoning. It may be seen as an extension of classical set theory andhas been successfully applied to machine learning, intelligent systems, inductivereasoning, pattern recognition, mereology, image processing, signal analysis,knowledge discovery, decision analysis, expert systems and many other fields[9, 10, 11, 12].

The basic structure of rough set theory is an approximation space. Based onit, lower and upper approximations can be induced. Using these approximations,knowledge hidden in information systems may be revealed and expressed in theform of decision rules. A key notion in Pawlak rough set model is equivalencerelations. The equivalence classes are the building blocks for the constructionof these approximations. In the real world, the equivalence relation is, however,too restrictive for many practical applications. To address this issue, manyinteresting and meaningful extensions of Pawlak rough sets have been presented.Equivalence relations can be replaced by tolerance relations [15], binary relations[20] and so on.

Topological structure is an important base for knowledge extraction andprocessing. Then, an interesting research topic in rough set theory is to studyrelationships between rough sets and topologies. Many authors studied topo-logical properties of rough sets [3, 4, 7, 18, 22]. It is known that the pair oflower and upper approximation operators induced by a reflexive and transitiverelation is exactly the pair of interior and closure operators of a topology [21].

∗This work is supported by the National Natural Science Foundation of China (11461005).†Corresponding Author, S School of Mathematics and Statistics, Hunan University of

Commerce, Changsha 410205, China, [email protected]

1


254 Neiping Chen 254-263

The purpose of this paper is to investigate further approximating spaces.

2 Preliminaries

Throughout this paper, I denotes [0, 1], N is the set of natural number. Udenotes a non-empty set, 2U denotes the set of all subsets of U , |X| denotes thecardinality of X.

2.1 Binary relations

Recall that R is called a binary relation on U if R ∈ 2U×U .Let R be a binary relation on U . R is called preorder if R is reflexive and

transitive. R is called tolerance if R is both reflexive and symmetric. R is calledequivalence if R is reflexive, symmetric and transitive.

Let R be a binary relation on U . For u, v, w ∈ U , we define

Ruvw = R ∪ Suvw and Ruv =⋃

w∈U

Ruvw,

where Suvw =

(u, v), (u,w) ∈ R and (w, v) ∈ R

∅, (u,w) 6∈ R or (w, v) 6∈ R.

If Suvw 6= ∅, then

Suvw(x) =

v, x = u

∅, x 6= u.

Definition 2.1 ([4]). Let R and Rs be two binary relations on U . If for allx, y ∈ U , xRsy if and only if xRy or there exists v1, v2, . . . , vn ⊆ U such thatxθv1, v1Rv2, . . . , vnRy, then Rs is called the transmitting expression of R.

Theorem 2.2 ([4]). Let R be a binary relation on U and Rs the transmittingexpression of R. Then Rs is a transitive relation on U . Moreover,

(1) If R is reflexive, then Rs is also reflexive;(2) If R is transitive, then Rs = R;(3) If R is symmetric, then Rs is also symmetric.

2.2 Rough sets

Let R be an equivalence relation on U . Then the pair (U,R) is called aPawlak approximation space. Based on (U,R), one can define the following tworough approximations:

R∗(X) = x ∈ U : [x]R ⊆ X,R∗(X) = x ∈ U : [x]R ∩X 6= ∅.

R∗(X) and R∗(X) are called the Pawlak lower approximation and the Pawlakupper approximation of X, respectively.

2



Definition 2.3 ([19]). Let R be a binary relation on U . ∀ x ∈ U , denote

R(x) = y ∈ U : (x, y) ∈ R.

Then R(x) is called the successor neighborhood of x, the pair (U,R) is calledan approximation space. The lower and upper approximations of X ∈ 2U withregard to (U,R), denoted by R(X) and R(X) are respectively, defined as follows:

R(X) = x ∈ U : R(x) ⊆ X and R(X) = x ∈ U : R(x) ∩X 6= ∅.

Proposition 2.4. Let Rα : α ∈ Γ be a family of binary relations on U . Then∀ X ∈ 2U ,

⋂

α∈Γ

Rα(X) =⋃

α∈Γ

Rα(X).

Proof. Put R =⋃

α∈Γ

Rα. By Rβ ⊆ R for each β ∈ Γ, Rβ(X) ⊇ R(X). Then⋂

α∈Γ

Rα(X) ⊇ R(X).

Let x ∈ ⋂α∈Γ

Rα(X). Then x ∈ Rα(X) and so Rα(x) ⊆ X for each α ∈ Γ.

Thus (⋃

α∈Γ

Rα)(x) =⋃

α∈Γ

(Rα(x)) ⊆ X. So x ∈ ⋃α∈Γ

Rα(X). Hence Rβ(X) ⊆⋃

α∈Γ

Rα(X).

Therefore,⋂

α∈Γ

Rα(X) =⋃

α∈Γ

Rα(X).

Proposition 2.5. Let R be a binary relation on U . Then ∀ u, v.w ∈ U ,

Ruvw(X)− u = R(X)− u.

Proof. (1) If Ruvw = R, then Ruvw(X)− u = R(X)− u.(2) If Ruvw 6= R, then (u,w), (w, v) ∈ R and (u, v) 6∈ R.Obviously, Ruvw(X)− u ⊆ R(X)− u.For x ∈ R(X)− u, note that Suvw(x) = ∅ (x ∈ U − u), then

Ruvw(x) = (R ∪ Suvw)(x) = R(x) ∪ Suvw(x) = R(x) ⊆ X (x ∈ U − u).

So x ∈ Ruvw(X)− u. It follows Ruvw(X)− u ⊇ R(X)− u.Hence

Ruvw(X)− u = R(X)− u.

Theorem 2.6. Let R be a binary relation on U and τ a topology on U . If oneof the following conditions is satisfied, then R is preorder.

(1) R is the closure operator of τ .(2) R is the interior operator of τ .

3



Proof. (1) Let x, y, z ∈ U . Denote clτ (z1)(y) = λ.Note that R is the interior operator of τ and x ∈ clτ (x) = R(x). Then

(x, x) ∈ R. So R is reflexive.Let (x, y), (y, z) ∈ R. Then x ∈ R(y), y ∈ R(z).Note that R is the closure operator of τ . Then x ∈ cl(y), y ∈ cl(z). So

x ∈ cl(x) ⊆ cl(cl(y)) = cl(y) ⊆ cl(cl(z)) = cl(z) = R(z).This implies (x, z) ∈ R. So R is transitive.

Hence R is preorder.(2) This proof is similar to (1).

3 Topologies induced by binary relations

3.1 Topologies induced by reflexive relations

Let R be a reflexive relation on U . Denote

τR = X ∈ 2U : R(X) = X,σR = R(X) : X ∈ 2U.

Kondo [2] proved that if R is a reflexive relation on X, then τR is a topologyon X, which may be called the topology induced by R on X.

Remark 3.1. (1) If R is preorder, then τR = σR.(2) If R is equivalence, then τR = ⋃

x∈X

[x]R : X ∈ 2U.

Theorem 3.2 ([7]). Let R be a preorder relation on U . Then(1) σR is a topology on U .(2) R is an interior operator of σR.(3) R is a closure operator of σR.

Proposition 3.3. Let ρ and R be two reflexive relations on U . Then(1) ρ ⊆ R =⇒ τρ ⊇ τR.(2) If ρ and R are preorder, then τρ = τR ⇐⇒ ρ = R.

Proof. (1) ∀ X ∈ τR, R(X) = X. By ρ ⊆ R and the reflexivity of ρ,

X = R(X) ⊆ ρ(X) ⊆ X.

Then ρ(X) = X and so X ∈ τρ. Thus τρ ⊇ τR.(2) Necessity. Suppose τρ = τR. Note that ρ and R are preorder. Then

τρ = σρ = σR = τR.By Theorem 3.2(3),

(x, y) ∈ ρ ⇐⇒ x ∈ ρ(y) ⇐⇒ x ∈ clσρ(y)

⇐⇒ x ∈ clσR(y) ⇐⇒ x ∈ R(y) ⇐⇒ (x, y) ∈ R.

Then ρ = R.Sufficiency. Obviously.

4



Proposition 3.4. Let Rα : α ∈ Γ be a family of reflexive relations on U .Then

τ ⋃α∈Γ

Rα=

⋂

α∈Γ

τRα.

Proof. By Proposition 3.3(1), τ ⋃α∈Γ

Rα⊆ ⋂

α∈Γ

τRα .

Let X ∈ ⋂α∈Γ

σRα . Then ∀ α ∈ Γ, Rα(X) = X. By Proposition 2.4,

X =⋂

α∈Γ

Rα(X) =⋃

α∈Γ

Rα(X).

So X ∈ τ ⋃α∈Γ

Rα. This implies τ ⋃

α∈ΓRα⊇ ⋂

α∈Γ

τRα .

Hence τ ⋃α∈Γ

Rα=

⋂α∈Γ

τRα .

3.2 The topologies induced by some binary relations

Theorem 3.5. Let ρ, λ, R be three reflexive relations on U . If τρ = τR = τλ

and ρ ⊆ δ ⊆ λ, then τδ = τR.

Proof. By ρ ⊆ δ ⊆ λ and Proposition 3.3(1),

τR = τλ ⊆ τδ ⊆ τρ = τR.

Then τδ = τR.

Theorem 3.6. Let R be a reflexive relation on U . Then ∀ u, v.w ∈ U , τRuvw =τR = τRuv .

Proof. Obviously, Ruvw and Ruv both are reflexive.(1) 1) If u = v, then Ruvw = R and so τRuvw = τR.2) If u 6= v, Ruvw = R, we have τRuvw = τR.3) If u 6= v, Ruvw 6= R, we have (u,w) ∈ R, (w, v) ∈ R, (u, v) 6∈ R and

Suvw = (u, v).Let X ∈ σR. Then X ⊆ R(X). By Proposition 3.3(1), σR ⊇ σRuvw . By

Proposition 2.5, X − u ⊆ R(X)− u = Ruvw(X)− u.i) If u 6∈ X, then X ⊆ Ruvw(X).ii) If u ∈ X, then u ∈ R(X) and so

w ∈ R(u) ⊆ X ⊆ R(X).

We can obtain R(w) ⊆ R(X). Note that v ∈ R(w). Then v ∈ R(X). We have

Ruvw(u) = (R ∪ Suvw)(u) = R(u) ∪ Suvw(u) = R(u) ∪ v ⊆ X.

Then u ∈ Ruvw(X). Thus X ⊆ Ruvw(X). By the reflexivity of ρ, X ⊇Ruvw(X). Then Ruvw(X) = X and So X ∈ σRuvw .

By i) and ii), τR ⊆ τRuvw .

5



ThusτRuvw = τR (w ∈ U).

(2) By (1) and Proposition 3.4,

τRuv = τ ⋃w∈U

Ruvw =⋂

w∈U

τRuvw = τR.

Denote R0 = R. Rn (n ∈ ω) are defined as follows:

Rn+1 =⋃

u,v∈X

(Rn)uv.

PutR∗ = lim

n→∞Rn.

Obviously, R∗ =∞⋃

n=0Rn.

Corollary 3.7. Let R be a reflexive relation on U . Then τRn= τR = τR∗ .

Proof. This holds by Proposition 3.4 and Theorem 3.6.

Theorem 3.8. Let R be a binary relation on U . Then

R is transitive ⇐⇒ R = R1.

Proof. Necessity. Obviously.Sufficiency. Suppose that R is not translative. Then there exist x, y, z such

that (x, z), (z, y) ∈ R, (x, y) 6∈ R. So (x, y) ∈ Rxy. This implies

(x, y) ∈ R1 =⋃

u,v∈X

Ruv.

We have R1 6= R. This is a contradiction.Thus R is translative.

Corollary 3.9. If R is a preorder relation on U , then ∀ n ∈ ω, Rn = R.

Proof. This holds by Theorem 4.6.

Denote R0 = R. Rn (n ∈ ω) are defined as follows: Rn+1 =⋃

u,v∈X

(Rn)uv

DenoteR∗ = lim

n→∞Rn.

Theorem 3.10. If R is a reflexive relation on U , then R∗ is translative.

6



Proof. Let (u,w), (w, v) ∈ R∗. Then there exist n1, n2 ∈ N such that (u,w) ∈Rn1 , (w, v) ∈ Rn2 . Pick n0 = n1 + n2. Then (u,w) ∈ Rn0 , (w, v) ∈ Rn0 . So

(u, v) ∈ (Rn0)uvw ⊆ (Rn0)

uv ⊆ Rn0+1 ⊆ R∗.

So R∗ is translative.

Theorem 3.11. Let R be a reflexive relation on U . Then Rs = R∗.

Proof. Note that

(x, y) ∈ R∗. (R∗ =∞⋃

n=0

Rn)

⇐⇒ ∃n ∈ N, (x, y) ∈ Rn. (Rn =⋃

u,v∈X

(Rn−1)uv)

⇐⇒ (x, y) ∈ (Rn−1)xy. ((Rn−1)xy =⋃

w∈U

(Rn−1)xyw)

⇐⇒ ∃w2n ∈ U, (x, y) ∈ (Rn−1)xyw2n .

⇐⇒ ∃w2n ∈ U, (x,w2n), (w2n , y) ∈ Rn−1.

⇐⇒ ∃w2n−1, w2n , w2n−2 ∈ U,

(x,w2n−1), (w2n−1, w2n), (w2n , w2n−2), (w2n−2, y) ∈ Rn−2.

· · · · · · · · ·⇐⇒ ∃w2, w3, · · · , w2n ∈ U,

(x,w3), · · · , (w2n−1, w2n), (w2n , w2n−2), · · · , (w2, y) ∈ R0 = R.

⇐⇒ (x, y) ∈ Rs.

Then Rs = R∗.

Corollary 3.12. Let R be a tolerance relation on U . Then(1) Rs is equivalence.(2) τRs

= τR.(3) Rs is an interior operator of τR.(4) Rs is a closure operator of τR.

Proof. (1) This holds by Theorem 3.11.(2) This holds by Corollary 3.7 and Theorem 3.11(3) This holds by (2) and Theorem 3.2.(4) This holds by (2) and Theorem 3.2.

4 Some characteristic conditions of approximat-ing spaces

Definition 4.1 ([4]). Let (U, µ) be a topological space. If there exists an equiv-alence relation R on U such that τR = µ, then (U, τ) is called a approximatingspace.

7



Definition 4.2. Let µ be a topology on U . Define a binary relation Rµ on Uby

(x, y) ∈ Rµ ⇐⇒ x ∈ clµ(y).Then Rµ is called the binary relation induced by µ on U .

Theorem 4.3. Let (U, µ) be a topological space. Then the following are equiv-alent:

(1) (U, µ) is an approximating space;(2) There exists a tolerance relation R on U such that τR = µ;(3) There exists a tolerance relation R on U such that R is an interior

operator of µ;(4) There exists a tolerance relation R on U such that R is a closure operator

of µ;(5) There exists an equivalence relation R on U such that

µ = ⋃

x∈X

[x]R : X ∈ 2U.

Proof. (1) =⇒ (2) is obvious.(1) =⇒ (3) and (1)=⇒ (4) hold by Theorem 3.2.(1) =⇒ (5) holds by Remark 3.2.(2) =⇒ (1) Suppose that there exists a tolerance relation R on U such that

τR = µ.By Theorem 2.2 and Corollary 3.12, Rs is equivalence and τRs

= τR.Then τRs

= µ.Thus (U, µ) is an approximating space.(3) =⇒ (1) Suppose that there exists a tolerance relation R on U such that

R is an interior operator of µ. Then

X ∈ τR ⇐⇒ R(X) = X ⇐⇒ intµ(X) = X ⇐⇒ X ∈ µ.

Then τR = µ.By Theorem 2.6(2), R is preorder. So R is equivalence.Thus (U, µ) is an approximating space.(4) =⇒ (1) The proof is similar to (3) =⇒ (1).(5) =⇒ (1) holds by Remark 3.2.

Corollary 4.4. If (U, µ) is an approximating space, then Rµ is an equivalencerelation.

Proof. Obviously, Rµ is reflexive.By Theorem 4.3, there exists an equivalence relation R on U such that

µ = ⋃

x∈X

[x]R : X ∈ 2U.

By Remark 2.4, we have

(x, y) ∈ Rµ ⇒ x ∈ clµ(y) = [y]R ⇒ y ∈ [y]R = [x]R = clµ(x) ⇒ (y, x) ∈ Rµ.

8



(x, y), (y, z) ∈ Rµ =⇒ x ∈ clµ(y) = [y]R, y ∈ clµ(z) = [z]R=⇒ x ∈ [y]R = [z]R = clµ(z) =⇒ (x, z) ∈ Rµ.

Thus Rµ is equivalence.

References

[1] J.Kortelainen, On the relationship between modified sets, topologicalspaces and rough sets, Fuzzy Sets and Systems 61(1994) 91-95.

[2] M. Kondo, On the structure of generalized rough sets, Information Science,176(2006), 589-600.

[3] E.F.Lashin, A.M.Kozae, A.A.Abo Khadra, T.Medhat, Rough set theoryfor topological spaces, International Journal of Approximate Reasoning40(2005) 35-43.

[4] Z.Li, T.Xie, Q.Li, Topological structure of generalized rough sets, Comput-ers and Mathematics with Applications 63(2012) 1066-1071.

[5] G.Liu, W.Zhu, The algebraic structures of generalized rough set theory,Information Sciences 178(2008) 4105-4113.

[6] J.Mi, W.Wu, W.Zhang, Constructive and axiomatic approaches for thestudy of the theory of rough sets, Pattern Recognition Artificial Intelligence15(2002) 280-284.

[7] Z.Pei, D.Pei, L.Zheng, Topology vs generalized rough sets, InternationalJournal of Approximate Reasoning 52(2011) 231-239.

[8] Z.Pawlak, Rough sets, International Journal of Computer and InformationScience 11(1982) 341-356.

[9] Z.Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data,Kluwer Academic Publishers, Dordrecht, 1991.

[10] Z.Pawlak, A.Skowron, Rudiments of rough sets, Information Sciences177(2007) 3-27.

[11] Z.Pawlak, A.Skowron, Rough sets: some extensions, Information Sciences177(2007) 28-40.

[12] Z.Pawlak, A.Skowron, Rough sets and Boolean reasoning, Information Sci-ences 177(2007) 41-73.

[13] K.Qin, Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Setsand Systems 151(2005) 601-613.

[14] A.M. Radzikowska, E.E.Kerre, A comparative study of fuzzy rough sets,Fuzzy Sets and Systems 126(2002) 137-155.

9



[15] A.Skowron, J.Stepaniuk, Tolerance approximation spaces, Fundamenta In-formaticae 27(1996) 245-253.

[16] R.Slowinski, D.Vanderpooten, Similarity relation as a basis for rough ap-proximations, ICS Research Report 53(1995) 249-250.

[17] J.Tang, K.She, F. Min, W. Zhu, A matroidal approach to rough set theory,Theoretical Computer Science 471(2013) 1-11.

[18] Q.Wu, T.Wang, Y.Huang, J.Li, Topology theory on rough sets, IEEETransactions on Systems, Man and Cybernetics - Part B: Cybernetics38(1)(2008) 68-77.

[19] Y.Y.Yao, Relational interpretations of neighborhood operators and roughset approximation operators, Information Sciences 111(1998)239-259.

[20] Y.Y.Yao, Constructive and algebraic methods of the theory of rough sets,Information Sciences 109(1998) 21-47.

[21] Y.Y.Yao, Two views of the theory of rough sets in finite universes, Inter-national Journal of Approximate Reasoning 15(1996) 291-317.

[22] L.Yang, L.Xu, Topological properties of generalized approximation spaces,Information Sciences 181(2011) 3570-3580.

10



Divisible and strong fuzzy filters of residuated lattices

Young Bae Jun1, Xiaohong Zhang2 and Sun Shin Ahn3,∗

1 Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Korea2Department of Mathematics, College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306,

P.R. China3Department of Mathematics Education, Dongguk University, Seoul 100-715, Korea

Abstract. In a residuated lattice, divisible fuzzy filters and strong fuzzy filters are introduced, and their properties

are investigated. Characterizations of a divisible and strong fuzzy filter are discussed. Conditions for a fuzzy filter

to be divisible are established. Relations between a divisible fuzzy filter and a strong fuzzy filter are considered.

1. Introduction

In order to deal with fuzzy and uncertain informations, non-classical logic has become a formal

and useful tool. As the semantical systems of non-classical logic systems, various logical algebras

have been proposed. Residuated lattices are important algebraic structures which are basic

of MTL-algebras, BL-algebras, MV -algebras, Godel algebras, R0-algebras, lattice implication

algebras, etc. The filter theory plays an important role in studying logical systems and the

related algebraic structures, and various filters have been proposed in the literature. Zhang et

al. [8] introduced the notions of IMTL-filters (NM-filters, MV-filters) of residuated lattices, and

presented their characterizations. Ma and Hu [4] introduced divisible filters, strong filters and

n-contractive filters in residuated lattices.

In this paper, we consider the fuzzification of divisible filters and strong filters in residuated

lattices. We define divisible fuzzy filters and strong fuzzy filters, and investigate related properties.

We discussed characterizations of a divisible and strong fuzzy filter, and provided conditions for

a fuzzy filter to be divisible. We establish relations between a divisible fuzzy filter and a strong

fuzzy filter.

02010 Mathematics Subject Classification: 03G10; 06B10; 06D72.0Keywords: Residuated lattice; (Divisible, strong) filter; Fuzzy filter; Divisible fuzzy filter; Strong fuzzy filter.

∗ The corresponding author.0E-mail: [email protected] (Y. B. Jun); [email protected] (X. Zhang); [email protected]

(S. S. Ahn)


264 Young Bae Jun et al 264-276

Young Bae Jun, Xiaohong Zhang and Sun Shin Ahn

2. Preliminaries

Definition 2.1 ([1, 2, 3]). A residuated lattice is an algebra L := (L,∨,∧,⊙,→, 0, 1) of type

(2, 2, 2, 2, 0, 0) such that

(1) (L,∨,∧, 0, 1) is a bounded lattice.

(2) (L,⊙, 1) is a commutative monoid.

(3) ⊙ and → form an adjoint pair, that is,

(∀x, y, z ∈ L) (x ≤ y → z ⇔ x⊙ y ≤ z) .

In a residuated lattice L, the ordering ≤ and negation ¬ are defined as follows:

(∀x, y ∈ L) (x ≤ y ⇔ x ∧ y = x ⇔ x ∨ y = y ⇔ x→ y = 1)

and ¬x = x→ 0 for all x ∈ L.

Proposition 2.2 ([1, 2, 3, 4, 6, 7]). In a residuated lattice L, the following properties are valid.

1→ x = x, x→ 1 = 1, x→ x = 1, 0→ x = 1, x→ (y → x) = 1. (2.1)

x→ (y → z) = (x⊙ y)→ z = y → (x→ z). (2.2)

x ≤ y ⇒ z → x ≤ z → y, y → z ≤ x→ z. (2.3)

z → y ≤ (x→ z)→ (x→ y), z → y ≤ (y → x)→ (z → x). (2.4)

(x→ y)⊙ (y → z) ≤ x→ z. (2.5)

¬x = ¬¬¬x, x ≤ ¬¬x, ¬1 = 0, ¬0 = 1. (2.6)

x⊙ y ≤ x⊙ (x→ y) ≤ x ∧ y ≤ x ∧ (x→ y) ≤ x. (2.7)

x ≤ y ⇒ x⊙ z ≤ y ⊙ z. (2.8)

x→ (y ∧ z) = (x→ y) ∧ (x→ z), (x ∨ y)→ z = (x→ z) ∧ (y → z). (2.9)

x→ y ≤ (x⊙ z)→ (y ⊙ z). (2.10)

¬¬(x→ y) ≤ ¬¬x→ ¬¬y. (2.11)

x→ (x ∧ y) = x→ y. (2.12)

Definition 2.3 ([5]). A nonempty subset F of a residuated lattice L is called a filter of L if it

satisfies the conditions:

(∀x, y ∈ L) (x, y ∈ F ⇒ x⊙ y ∈ F ) . (2.13)

(∀x, y ∈ L) (x ∈ F, x ≤ y ⇒ y ∈ F ) . (2.14)




Proposition 2.4 ([5]). A nonempty subset F of a residuated lattice L is a filter of L if and only

if it satisfies:

1 ∈ F. (2.15)

(∀x ∈ F ) (∀y ∈ L) (x→ y ∈ F ⇒ y ∈ F ) . (2.16)

Definition 2.5 ([9]). A fuzzy set µ in a residuated lattice L is called a fuzzy filter of L if it

satisfies:

(∀x, y ∈ L) (µ(x⊙ y) ≥ minµ(x), µ(y)) . (2.17)

(∀x, y ∈ L) (x ≤ y ⇒ µ(x) ≤ µ(y)) . (2.18)

Theorem 2.6 ([9]). A fuzzy set µ in a residuated lattice L is a fuzzy filter of L if and only if the

following assertions are valid:

(∀x ∈ L) (µ(1) ≥ µ(x)) . (2.19)

(∀x, y ∈ L) (µ(y) ≥ minµ(x→ y), µ(x)) . (2.20)

3. Divisible and strong fuzzy filters

In what follows let L denote a residuated lattice unless otherwise specified.

Definition 3.1 ([4]). A filter F of L is said to be divisible if it satisfies:

(∀x, y ∈ L) ((x ∧ y)→ [x⊙ (x→ y)] ∈ F ) . (3.1)

Definition 3.2. A fuzzy filter µ of L is said to be divisible if it satisfies:

(∀x, y ∈ L)(µ((x ∧ y)→ [x⊙ (x→ y)]

)= µ(1)

). (3.2)

Example 3.3. Let L = 0, a, b, 1 be a chain with Cayley tables which are given in Tables 1 and

2.

Table 1. Cayley table for the “⊙”-operation

⊙ 0 a b 1

0 0 0 0 0

a 0 0 a a

b 0 a b b

1 0 a b 1

Then L := (L,∨,∧,⊙,→, 0, 1) is a residuated lattice. Define a fuzzy set µ in L by µ(1) = 0.7

and µ(x) = 0.2 for all x(= 1) ∈ L. It is routine to verify that µ is a divisible fuzzy filter of L.




Table 2. Cayley table for the “→”-operation

→ 0 a b 1

0 1 1 1 1

a a 1 1 1

b 0 a 1 1

1 0 a b 1

Example 3.4. Consider a residuated lattice L = [0, 1] in which two operations “⊙” and “→”

are defined as follows:

x⊙ y =

0 if x+ y ≤ 1

2,

x ∧ y otherwise.

x→ y =

1 if x ≤ y,(12− x)∨ y otherwise.

The fuzzy set µ of L given by µ(1) = 0.9 and µ(x) = 0.2 for all x(= 1) ∈ L is a fuzzy filter of L.But it is not divisible since

µ((0.3 ∧ 0.2)→ (0.3⊙ (0.3→ 0.2)) = µ(0.3) = µ(1).

Proposition 3.5. Every divisible fuzzy filter µ of L satisfies the following identity.

(∀x, y, z ∈ L) (µ(((x⊙ y) ∧ (x⊙ z))→ (x⊙ (y ∧ z))) = µ(1)) . (3.3)

Proof. Let x, y, z ∈ L. If we let x := x⊙ y and y := x⊙ z in (3.2), then

µ(((x⊙ y) ∧ (x⊙ z))→ ((x⊙ y)⊙ ((x⊙ y)→ (x⊙ z)))) = µ(1). (3.4)

Using (2.2) and (2.7), we have

(x⊙ y)⊙ ((x⊙ y)→ (x⊙ z)) = x⊙ y ⊙ (y → (x→ (x⊙ z)))≤ x⊙ (y ∧ (x→ (x⊙ z))),

and so

((x⊙ y) ∧ (x⊙ z))→ ((x⊙ y)⊙ ((x⊙ y)→ (x⊙ z)))≤ ((x⊙ y) ∧ (x⊙ z))→ (x⊙ (y ∧ (x→ (x⊙ z))))

by (2.3). It follows from (3.4) and (2.18) that

µ(1) = µ(((x⊙ y) ∧ (x⊙ z))→ ((x⊙ y)⊙ ((x⊙ y)→ (x⊙ z))))≤ µ(((x⊙ y) ∧ (x⊙ z))→ (x⊙ (y ∧ (x→ (x⊙ z)))))

and so that

µ(((x⊙ y) ∧ (x⊙ z))→ (x⊙ (y ∧ (x→ (x⊙ z))))) = µ(1) (3.5)




since µ(1) ≥ µ(x) for all x ∈ L. On the other hand, if we take x := x→ (x⊙ z) in (3.2) then

µ(1) = µ((y ∧ (x→ (x⊙ z)))→ ((x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y)))

≤ µ((x⊙ (y ∧ (x→ (x⊙ z))))→ (x⊙ ((x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y))))

= µ((x⊙ (y ∧ (x→ (x⊙ z))))→ (x⊙ (x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y)))

by using (2.10), (2.18) and the commutativity and associativity of ⊙. Hence

µ((x⊙ (y ∧ (x→ (x⊙ z))))→ (x⊙ (x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y))) = µ(1). (3.6)

Using (2.5), we get

(((x⊙ y) ∧ (x⊙ z))→ (x⊙ (y ∧ (x→ (x⊙ z)))))⊙((x⊙ (y ∧ (x→ (x⊙ z))))→ (x⊙ (x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y)))

≤ ((x⊙ y) ∧ (x⊙ z))→ (x⊙ (x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y)).

It follows from (2.18), (2.17), (3.5) and (3.6) that

µ(((x⊙ y) ∧ (x⊙ z))→ (x⊙ (x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y)))

≥ µ((((x⊙ y) ∧ (x⊙ z))→ (x⊙ (y ∧ (x→ (x⊙ z)))))⊙((x⊙ (y ∧ (x→ (x⊙ z))))→ (x⊙ (x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y))))

≥ minµ((((x⊙ y) ∧ (x⊙ z))→ (x⊙ (y ∧ (x→ (x⊙ z)))))),µ(((x⊙ (y ∧ (x→ (x⊙ z))))→ (x⊙ (x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y))))= µ(1)

Thus

µ(((x⊙ y) ∧ (x⊙ z))→ (x⊙ (x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y))) = µ(1). (3.7)

Since x⊙ (x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y)) ≤ x⊙ z ⊙ (z → y) ≤ x⊙ (y ∧ z), we obtain

((x⊙ y) ∧ (x⊙ z))→ (x⊙ (x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y)))

≤ ((x⊙ y) ∧ (x⊙ z))→ (x⊙ (y ∧ z)).

It follows that

µ(((x⊙ y) ∧ (x⊙ z))→ (x⊙ (y ∧ z)))≥ µ(((x⊙ y) ∧ (x⊙ z))→ (x⊙ (x→ (x⊙ z))⊙ ((x→ (x⊙ z))→ y))))

= µ(1)

and that µ(((x⊙ y) ∧ (x⊙ z))→ (x⊙ (y ∧ z))) = µ(1).

We consider characterizations of a divisible fuzzy filter.




Theorem 3.6. A fuzzy filter µ of L is divisible if and only if the following assertion is valid:

(∀x, y, z ∈ L)(µ([x→ (y ∧ z)]→ [(x→ y)⊙ ((x ∧ y)→ z)]

)= µ(1)

). (3.8)

Proof. Assume that µ is a divisible fuzzy filter of L. If we take x := x → y and y := x → z in

(3.2) and use (2.9) and (2.2), then

µ(1) = µ ([(x→ y) ∧ (x→ z)]→ [(x→ y)⊙ ((x→ y)→ (x→ z))])

= µ ([x→ (y ∧ z)]→ [(x→ y)⊙ ((x⊙ (x→ y))→ z)]) .

Using (2.4) and (2.10), we have

(x ∧ y)→ [x⊙ (x→ y)] ≤ [(x⊙ (x→ y))→ z]→ [(x ∧ y)→ z]

≤ [(x→ y)⊙ ((x⊙ (x→ y))→ z)]→ [(x→ y)⊙ ((x ∧ y)→ z)]

for all x, y, z ∈ L. Since µ is a divisible fuzzy filter of L, it follows from (3.2) and (2.18) that

µ(1) = µ((x ∧ y)→ [x⊙ (x→ y)])

≤ µ([(x→ y)⊙ ((x⊙ (x→ y))→ z)]→ [(x→ y)⊙ ((x ∧ y)→ z)])

and so from (2.19) that

µ([(x→ y)⊙ ((x⊙ (x→ y))→ z)]→ [(x→ y)⊙ ((x ∧ y)→ z)]) = µ(1)

for all x, y, z ∈ L. Using (2.5), we get([x→ (y ∧ z)]→ [(x→ y)⊙ ((x⊙ (x→ y))→ z)]

)⊙(

[(x→ y)⊙ ((x⊙ (x→ y))→ z)]→ [(x→ y)⊙ ((x ∧ y)→ z)])

≤ [x→ (y ∧ z)]→ [(x→ y)⊙ ((x ∧ y)→ z)],

and so

µ([x→ (y ∧ z)]→ [(x→ y)⊙ ((x ∧ y)→ z)]

)≥ µ

(([x→ (y ∧ z)]→ [(x→ y)⊙ ((x⊙ (x→ y))→ z)]

)⊙(

[(x→ y)⊙ ((x⊙ (x→ y))→ z)]→ [(x→ y)⊙ ((x ∧ y)→ z)]))

≥ minµ([x→ (y ∧ z)]→ [(x→ y)⊙ ((x⊙ (x→ y))→ z)]

),

µ([(x→ y)⊙ ((x⊙ (x→ y))→ z)]→ [(x→ y)⊙ ((x ∧ y)→ z)]

)= µ(1).

Therefore µ([x→ (y ∧ z)]→ [(x→ y)⊙ ((x ∧ y)→ z)]

)= µ(1) for all x, y, z ∈ L.




Conversely, let µ be a fuzzy filter that satisfies the condition (3.8). if we take x := 1 in (3.8)

and use (2.1), then we obtain (3.2).

Theorem 3.7. A fuzzy filter µ of L is divisible if and only if it satisfies:

(∀x, y ∈ L) (µ ([y ⊙ (y → x)]→ [x⊙ (x→ y)]) = µ(1)) . (3.9)

Proof. Suppose that µ is a divisible fuzzy filter of L. Note that

(x ∧ y)→ [x⊙ (x→ y)] ≤ [y ⊙ (y → x)]→ [x⊙ (x→ y)]

for all x, y ∈ L. It follows from (3.2) and (2.18) that

µ(1) = µ ((x ∧ y)→ [x⊙ (x→ y)])

≤ µ ([y ⊙ (y → x)]→ [x⊙ (x→ y)])

and that µ ([y ⊙ (y → x)]→ [x⊙ (x→ y)]) = µ(1).

Conversely, let µ be a fuzzy filter of L that satisfies the condition (3.9). Since

y → x = y → (y ∧ x) for all x, y ∈ L,

the condition (3.9) implies that

µ ([y ⊙ (y → (x ∧ y))]→ [x⊙ (x→ (x ∧ y))]) = µ(1). (3.10)

If we take y := x ∧ z in (3.10), then

µ(1) = µ ([(x ∧ z)⊙ ((x ∧ z)→ (x ∧ (x ∧ z)))]→ [x⊙ (x→ (x ∧ (x ∧ z)))])= µ ((x ∧ z)→ [x⊙ (x→ z)]) .

Therefore µ is a divisible fuzzy filter of L.

We discuss conditions for a fuzzy filter to be divisible.

Theorem 3.8. If a fuzzy filter µ of L satisfies the following assertion:

(∀x, y ∈ L) (µ((x ∧ y)→ (x⊙ y)) = µ(1)) , (3.11)

then µ is divisible.

Proof. Note that x⊙ y ≤ x⊙ (x→ y) for all x, y ∈ L. It follows from (2.3) that

(x ∧ y)→ (x⊙ y) ≤ (x ∧ y)→ (x⊙ (x→ y)).

Hence, by (3.11) and (2.18), we have

µ(1) = µ((x ∧ y)→ (x⊙ y)) ≤ µ((x ∧ y)→ (x⊙ (x→ y))),

and so µ((x ∧ y)→ (x⊙ (x→ y))) = µ(1) for all x, y ∈ L. Therefore µ is a divisible fuzzy filter

of L.





(∀x, y ∈ L) (µ((x ∧ (x→ y))→ y) = µ(1)) , (3.12)


Proof. Taking y := x⊙ y in (3.12) implies that

µ(1) = µ((x ∧ (x→ (x⊙ y)))→ (x⊙ y))≤ µ((x ∧ y)→ (x⊙ y))

and so µ((x ∧ y) → (x ⊙ y)) = µ(1) for all x, y ∈ L. It follows from Theorem 3.8 that µ is a

divisible fuzzy filter of L.


(∀x, y, z ∈ L) (µ(x→ z) ≥ minµ((x⊙ y)→ z), µ(x→ y)) , (3.13)


Proof. If we take x := x ∧ (x→ y), y := x and z := y in (3.13), then

µ((x ∧ (x→ y))→ y) ≥ minµ(((x ∧ (x→ y))⊙ x)→ y), µ((x ∧ (x→ y))→ x)= µ(1)

Thus µ((x ∧ (x → y)) → y) = µ(1) for all x, y ∈ L, and so µ is a divisible fuzzy filter of L by

Theorem 3.9.


(∀x ∈ L) (µ(x→ (x⊙ x)) = µ(1)) , (3.14)


Proof. Let µ be a fuzzy filter of L that satisfies the condition (3.14). Using (2.10) and the

commutativity of ⊙, we have x→ y ≤ (x⊙ x)→ (x⊙ y), and so

(x→ (x⊙ x))⊙ (x→ y) ≤ (x→ (x⊙ x))⊙ ((x⊙ x)→ (x⊙ y))

for all x, y ∈ L by (2.8) and the commutativity of ⊙. It follows from (2.5), (2.8) and the

commutativity of ⊙ that

((x→ (x⊙ x))⊙ (x→ y))⊙ ((x⊙ y)→ z)

≤ ((x→ (x⊙ x))⊙ ((x⊙ x)→ (x⊙ y)))⊙ ((x⊙ y)→ z)

≤ (x→ (x⊙ y))⊙ ((x⊙ y)→ z)

≤ x→ z




and so from (2.17), (2.18), (2.19) and (3.14) that

µ(x→ z) ≥ µ(((x→ (x⊙ x))⊙ (x→ y))⊙ ((x⊙ y)→ z))

≥ minµ((x→ (x⊙ x))⊙ (x→ y)), µ((x⊙ y)→ z)≥ minµ(x→ (x⊙ x)), µ(x→ y), µ((x⊙ y)→ z)= minµ(1), µ(x→ y), µ((x⊙ y)→ z)= minµ((x⊙ y)→ z), µ(x→ y)

for all x, y, z ∈ L. Therefore µ is a divisible fuzzy filter of L by Theorem 3.10.

Definition 3.12 ([4]). A filter F of L is said to be strong if it satisfies:

(∀x ∈ L) (¬¬(¬¬x→ x) ∈ F ) . (3.15)

Definition 3.13. A fuzzy filter µ of L is said to be strong if it satisfies:

(∀x ∈ L)(µ(¬¬(¬¬x→ x)

)= µ(1)

). (3.16)

Example 3.14. Consider a residuated lattice L := 0, a, b, c, d, 1 with the following Hasse

diagram (Figure 3.1) and Cayley tables (see Table 3 and Table 4).

rb r0

ra r1

r dr c

@@

@@

Figure 3.1

Table 3. Cayley table for the “⊙”-operation

⊙ 0 a b c d 1

0 0 0 0 0 0 0

a 0 a b d d a

b c b b 0 0 b

c b d 0 d d c

d b d 0 d d d

1 0 a b c d 1

Define a fuzzy set µ in L by µ(1) = 0.6 and µ(x) = 0.5 for all x(= 1) ∈ L. It is routine to check

that µ is a strong fuzzy filter of L.




Table 4. Cayley table for the “→”-operation

→ 0 a b c d 1

0 1 1 1 1 1 1

a 0 1 b c c 1

b c a 1 c c 1

c b a b 1 a 1

d b a b a 1 1

1 0 a b c d 1

We provide characterizations of a strong fuzzy filter.

Theorem 3.15. Given a fuzzy set µ of L, the following assertions are equivalent.

(1) µ is a strong fuzzy filter of L.(2) µ is a fuzzy filter of L that satisfies

(∀x, y ∈ L) (µ((y → ¬¬x)→ ¬¬(y → x) = µ(1)) . (3.17)

(3) µ is a fuzzy filter of L that satisfies

(∀x, y ∈ L) (µ((¬x→ y)→ ¬¬(¬y → x)) = µ(1)) . (3.18)

Proof. Assume that µ is a strong fuzzy filter of L. Then µ is a fuzzy filter of L. Note that

¬¬(¬¬x→ x) ≤ ¬¬((y → ¬¬x)→ (y → x))

≤ ¬¬((y → ¬¬x)→ ¬¬(y → x))

= (y → ¬¬x)→ ¬¬(y → x)

and

¬¬(¬¬x→ x) ≤ ¬¬(((¬x→ y)⊙ ¬y)→ x)

= ¬¬((¬x→ y)→ (¬y → x))

≤ ¬¬((¬x→ y)→ ¬¬(¬y → x))

= (¬x→ y)→ ¬¬(¬y → x)

for all x, y ∈ L. If follows from (3.16) and (2.18) that

µ(1) = µ(¬¬(¬¬x→ x)) ≤ µ((y → ¬¬x)→ ¬¬(y → x)) (3.19)

and

µ(1) = µ(¬¬(¬¬x→ x)) ≤ µ((¬x→ y)→ ¬¬(¬y → x)). (3.20)

Combining (2.19), (3.19) and (3.20), we have µ((y → ¬¬x)→ ¬¬(y → x)) = µ(1) and µ((¬x→y) → ¬¬(¬y → x)) = µ(1) for all x, y ∈ L. Therefore (2) and (3) are valid. Let µ be a fuzzy




filter of L that satisfies the condition (3.17). If we take y := ¬¬x in (3.17) and use (2.1), then we

can induce the condition (3.16) and so µ is a strong fuzzy filter of L. Let µ be a fuzzy filter of Lthat satisfies the condition (3.18). Taking y := ¬x in (3.18) and using (2.1) induces the condition

(3.16). Hence µ is a strong fuzzy filter of L.

We investigate relationship between a divisible fuzzy filter and a strong fuzzy filter.

Theorem 3.16. Every divisible fuzzy filter is a strong fuzzy filter.

Proof. Let µ be a divisible fuzzy filter of L. If we put x := ¬¬x and y := x in (3.2), then we have

µ((¬¬x ∧ x)→ (¬¬x⊙ (¬¬x→ x))) = µ(1). (3.21)

Using (2.4) and (2.8), we get

(¬¬x ∧ x)→ (¬¬x⊙ (¬¬x→ x)) ≤ ¬(¬¬x⊙ (¬¬x→ x))→ ¬(¬¬x ∧ x)≤ (¬¬x⊙ ¬(¬¬x⊙ (¬¬x→ x)))→ (¬¬x⊙ ¬(¬¬x ∧ x))≤ ¬(¬¬x⊙ ¬(¬¬x ∧ x))→ ¬(¬¬x⊙ ¬(¬¬x⊙ (¬¬x→ x)))

for all x ∈ L. It follows from (3.21) and (2.18) that

µ(1) = µ((¬¬x ∧ x)→ (¬¬ ⊙ (¬¬x→ x)))

≤ µ(¬(¬¬x⊙ ¬(¬¬x ∧ x))→ ¬(¬¬x⊙ ¬(¬¬x⊙ (¬¬x→ x)))).(3.22)

Combining (3.22) with (2.19), we have

µ(¬(¬¬x⊙ ¬(¬¬x ∧ x))→ ¬(¬¬x⊙ ¬(¬¬x⊙ (¬¬x→ x)))) = µ(1) (3.23)

for all x ∈ L. Using (2.2), (2.11), (2.12) and (2.6), we get

¬(¬¬x⊙ ¬(¬¬x ∧ x)) = ¬¬x→ ¬¬(¬¬x ∧ x)≥ ¬¬(x→ (¬¬x ∧ x))= ¬¬(x→ (x ∧ ¬¬x))= ¬¬(x→ ¬¬x) = ¬¬1 = 1

and so ¬(¬¬x⊙ ¬(¬¬x ∧ x)) = 1 for all x ∈ L. It follows from (3.23) and (2.20) that

µ(¬(¬¬x⊙ ¬(¬¬x⊙ (¬¬x→ x))))

≥ minµ(¬(¬¬x⊙ ¬(¬¬x ∧ x))→ ¬(¬¬x⊙ ¬(¬¬x⊙ (¬¬x→ x)))),

µ(¬(¬¬x⊙ ¬(¬¬x ∧ x)))= µ(1)




and so that

µ(1) = µ(¬(¬¬x⊙ ¬(¬¬x⊙ (¬¬x→ x))))

= µ(¬(¬¬x⊙ (¬¬x→ ¬(¬¬x→ x)))).(3.24)

Taking x := ¬¬x and y := ¬(¬¬x→ x) in (3.2) induces

µ(1) = µ((¬¬x ∧ ¬(¬¬x→ x))→ (¬¬x⊙ (¬¬x→ ¬(¬¬x→ x))))

≤ µ(¬(¬¬x⊙ (¬¬x→ ¬(¬¬x→ x)))→ ¬(¬¬x ∧ ¬(¬¬x→ x)))

by using (2.3) and (2.18). Thus

µ(¬(¬¬x⊙ (¬¬x→ ¬(¬¬x→ x)))→ ¬(¬¬x ∧ ¬(¬¬x→ x))) = µ(1). (3.25)

Since ¬(¬¬x→ x) ≤ ¬¬x for all x ∈ L, it follows from (2.19), (2.20), (3.24) and (3.25) that

µ(1) = µ(¬(¬¬x ∧ ¬(¬¬x→ x))) = µ(¬¬(¬¬x→ x))

for all x ∈ L. Therefore µ is a strong fuzzy filter of L.

Corollary 3.17. If a fuzzy filter µ of L satisfies one of conditions (3.8), (3.9), (3.11), (3.12),

(3.13) and (3.14), then µ is a strong fuzzy filter of L.

The following example shows that the converse of Theorem 3.16 may not be true in general.

Example 3.18. The strong fuzzy filter µ of L which is given in Example 3.14 is not a divisible

fuzzy filter of L since µ((a ∧ c)→ (a⊙ (a→ c))) = µ(a) = µ(1).

4. Conclusions

The filter theory plays an important role in studying logical systems and the related algebraic

structures, and various filters have been proposed in the literature. Zhang et al. [8] introduced

the notions of IMTL-filters (NM-filters, MV-filters) of residuated lattices, and presented their

characterizations. Ma and Hu [4] introduced divisible filters, strong filters and n-contractive

filters in residuated lattices.

In this paper, we have considered the fuzzification of divisible filters and strong filters in

residuated lattices. We have defined divisible fuzzy filters and strong fuzzy filters, and have

investigated related properties. We have discussed characterizations of a divisible and strong

fuzzy filter, and have provided conditions for a fuzzy filter to be divisible. We have establish

relations between a divisible fuzzy filter and a strong fuzzy filter. In a forthcoming paper, we

will study the fuzzification of n-contractive filters in residuated lattices, and apply the results to

other algebraic structures.




References

[1] R. Belohlavek, Some properties of residuated lattices, Czechoslovak Math. J. 53(123) (2003) 161–171.

[2] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets

and Systems 124 (2001) 271–288.

[3] P. Hajek, Metamathematics of Fuzzy Logic, Kluwer Academic Press, Dordrecht, 1998.

[4] Z. M. Ma and B. Q. Hu, Characterizations and new subclasses of I-filters in residuated lattices, Fuzzy Sets

and Systems 247 (2014) 92–107.

[5] J. G. Shen and X. H. Zhang, Filters of residuated lattices, Chin. Quart. J. Math. 21 (2006) 443–447.

[6] E. Turunen, BL-algebras of basic fuzzy logic, Mathware & Soft Computing 6 (1999), 49–61.

[7] E. Turunen, Boolean deductive systems of BL-algebras, Arch. Math. Logic 40 (2001) 467–473.

[8] X. H. Zhang, H. Zhou and X. Mao, IMTL(MV)-filters and fuzzy IMTL(MV)-filters of residuated lattices, J.

Intell. Fuzzy Systems 26 (2014) 589–596.

[9] Y. Q. Zhu and Y. Xu, On filter theory of residuated lattices, Inform. Sci. 180 (2010) 3614–3632.



FREQUENT HYPERCYCLICITY OF WEIGHTED COMPOSITION

OPERATORS ON CLASSICAL BANACH SPACES

SHI-AN HAN AND LIANG ZHANG∗

Abstract. In this paper we characterize the frequent hypercyclicity of weighted com-

position operators on some classical Banach spaces, such as the weighted Dirichlet space

Sv . Besides, we also discuss the frequent hypercyclicity of the weighted compositionoperators on the weighted Bergman space Apα.

1. Introduction and terminology

Let H(D) be the space of all holomorphic functions on D, where D is the open unit diskof the complex plane C. The collection of all holomorphic self-maps of D will be denoted byS(D), and let Aut(D) denote the set of all automorphisms on D. The disk algebra, denotedby A(D), consists of all functions in H(D) that are continuous up to the boundary ∂D of theunit disk D. Let dA denote the normalized Lebesegue measure on D. The space of boundedanalytic functions on D will be denoted by H∞, with the sup norm ‖ · ‖∞.

For α > −1 and 1 < p <∞, the weighted Bergman space Apα consists of analytic functionsf such that

‖f‖p =

∫D|f (z)|p dνα (z) <∞,

where dνα on D is defined by

dνα = (α+ 1)(

1− |z|2)α

dν (z)

and να (D) = 1. Under the norm ‖.‖, Apα is a separable infinite dimensional Banach space,since the set of polynomials is dense in Apα.

For each real number v, the weighted Dirichlet space Sv is the space of holomorphicfunctions f(z) =

∑∞n=0 anz

n, z ∈ D such that the following norm

‖f‖2v =∞∑n=0

|an|2(n+ 1)2v

is finite. Observe that the space Sv is Hilbert space, where the inner product is defined by

〈f, g〉 =∞∑n=0

anbn(n+ 1)2v,

where f(z) =∑∞n=0 anz

n and g(z) =∑∞n=0 bnz

n. For instance, if v = 0,−1/2, 1/2, thenSv is, respectively, the classical Hardy space H2, the Bergman space A2, and the Dirichletspace D.

By Lemma 1.2 in [5], we know the following expression

‖f‖2 =l∑i=0

|f (i)(0)|+∫D|f (l+1)(z)|2(1− |z|2)2l+1−2vdA(z)

defines an equivalent norm on Sv, where l ≥ −1 is an integer such that v < l+ 1, and whenl = −1, the first term in the right hand side above does not appear.

The work was supported in part by the National Natural Science Foundation of China (Grant Nos.11371276; 11301373; 11201331).∗Corresponding author.2010 Mathematics Subject Classification. Primary: 47A16; Secondary: 47B38, 47B33, 30H99, 46E20.Key words and phrases. Frequently hypercyclic; weighted composition operators; weighted Dirichletspace.

1


277 SHI-AN HAN et al 277-282

Han and Zhang: Frequent hypercyclicity of weighted composition operators

A bounded linear operator T acting on a separable Banach space X is said to be hyper-cyclic if there is an f ∈ X such that orbit Tnfn≥0 is dense in X. One bounded operatorT is called similar to another bounded operator S on X if there exists a bounded and in-vertible operator V on H such that TV = V S. And the similarity preserve hypercyclicity.A continuous linear operator T acting on a separable Banach space X is said to be mixing,if for any pair U, V of nonempty open subsets of X, there exists some N ≥ 0 such that

Tn (U) ∩ (V ) 6= ∅, for all n ≥ N.

The lower density of a subset A of N is defined as

dens(A) = lim infN→∞

card0 ≤ n ≤ N ;n ∈ AN + 1

.

A vector x ∈ X is called frequently hypercyclic for T , if for every non-empty open subsetU of X,

densn ∈ N, Tnx ∈ U > 0.

The operator T is called frequently hypercyclic if it possesses a frequently hypercyclic vector.It is obvious that if the operator T is frequently hypercyclic, then T is hypercyclic. Morerelated details can be founded in chapter 9 in the book [6].

Let u ∈ H(D) and ϕ ∈ S(D), the weighted composition operator uCϕ is defined as

(uCϕf)(z) = u(z)f(ϕ(z)), f ∈ H(D), z ∈ D.

And when u ≡ 1, we just have the composition operator Cϕ and when ϕ(z) = z, we get themultiplication operator Mu.

For ϕ ∈ LFT (D), we define ϕ as following:

ϕ (z) =az + b

cz + d,

where ad− bc 6= 0.Note that the linear fractional self-maps of D fall into distinct classes determined by their

fixed point properties (see [1]). There are:(a) Maps with interior fixed point. By the Schwarz Lemma the interior fixed point is

either attractive, or the map is an elliptic automorphism.(b) Parabolic maps. Its fixed point is on ∂D, and the derivative = 1 at the fixed point.(c) Hyperbolic maps with attractive fixed point on ∂D and their repulsive fixed point

outside of D. Both fixed points are on ∂D if and only if the map is the automorphism of D.In this case, the derivative < 1 at the attractive fixed point.

According to a result by P.R. Hurst [8], the composition operator Cϕ : Sv → Sv isbounded for any v ∈ R and any ϕ ∈ LFT (D). In [4], the authors partially characterizedthe frequent hypercyclicity of scalar multiples of composition operators, whose symbols arelinear fractional maps, acting on the weighted Dirichlet space Sv. E. Gallado and A. Montes[5] have furnished a complete characterization of the hypercyclicity of λCϕ on Sv in termsof λ, v, ϕ. Readers interested in related topics can refer to [3, 7, 9, 12, 13].

In this note, we will discuss the conditions of the frequent hypercyclicity of weightedcomposition operators on some classical Banach spaces, such as the weighted Dirichlet spaceSv and the weighted Bergman space Apα.

2. Frequent hypercyclicity of uCϕ on Sv

In this section, we begin to discuss the frequent hypecyclicity of the weighted compositionoperator uCϕ on Sv.

Theorem 2.1. If uCϕ is frequently hypercyclic on Sv, then ϕ is univalent and has no fixedpoint in D, and u(z) 6= 0 for every z ∈ D.

Proof. It is well known that uCϕ is hypercyclic on Sv, so by Theorem 1 in [11], we obtainit.

The following result can be found in [11, Theorem 2].




Theorem 2.2. Let v > 1/2. Then(a) No weighted composition operator on Sv is hypercyclic.(b) If ϕ has two fixed points α, β in D, and u(α) = u(β), then uCϕ is not cyclic on Sv.

Combining with the comparison principle, to discuss frequent hypecyclicity of the weight-ed composition operator uCϕ on Sv, we may assume without loss of generality that 0 ≤ v ≤12 .

2.1. The case for v = 0. In general, composition operators are bounded on H2 (see [2,Charpter 3]). Mu is also a bounded operator on Sv if u ∈ H∞. So when v = 0, ϕ ∈ S(D)and u ∈ H∞, uCϕ = MuCϕ.

According to the definition of [9], for any w ∈ ∂D and any positive number α, Lipα(w)corresponds to the class of holomorphic functions ϕ such that there is some neighborhoodG of w in ∂D and a positive constant M with

|ϕ(z)− ϕ(w)| ≤M |z − w|α , for z ∈ G.

For example, if an analytic function ϕ on D is also analytic at w ∈ ∂D, then ϕ ∈ Lipα(w)whenever 0 ≤ α ≤ 1. Moreover, if ϕ′ (ω) = 0, then ϕ ∈ Lipα(w) whenever 0 ≤ α ≤ 2.

We have the following proposition.

Proposition 2.3. Let ϕ ∈ LFT (D), w ∈ ∂D be the Denjoy-Wolff point of ϕ, u ∈ Lipα (w)∩A (D) , ‖u‖∞ = |u (w)| 6= 0. Then u(w) is an eigenvalue for uCϕ, whenever ϕ is hyperbolic

and α > 0 or ϕ is parabolic automorphism and α > 1. Moreover, if u never vanishes on D,then the eigenfunction also never vanishes.

Proof. According to the proof of Propositioin 2.4 of [9], we have that the function g(z) =∞∏n=0

u(ϕn(z))u(w) is a nonzero holomorphic function on D. Since ‖u‖∞ = |u (w)| 6= 0, then for

every j ≥ 0 and z ∈ D,∣∣∣u(ϕj(z))

u(w)

∣∣∣ ≤ 1. And note that for fixed z ∈ D,n∏j=0

∣∣∣u(ϕj(z))u(w)

∣∣∣ is

decreasing with respect to n. Therefore, ‖g‖∞ = supz∈D

∣∣∣∣ ∞∏n=0

u(ϕn(z))u(w)

∣∣∣∣ ≤ supz∈D

∣∣∣u(ϕ(z))u(w)

∣∣∣ ≤ 1.

That is, g ∈ H∞ ⊂ Sv and u (z) g (ϕ (z)) = u (w) g (z) . Thus u(w) is an eigenvalue for uCϕ.

Since u (z) 6= 0 for every z ∈ D, g (z) 6= 0 for z ∈ D.

Next, we obtain the following result.

Theorem 2.4. Let ϕ ∈ LFT (D), w ∈ ∂D be the Denjoy-Wolff point of ϕ, u ∈ Lipα (w) ∩A (D) , ‖u‖∞ = |u (w)| 6= 0 and u (z) 6= 0 for every z ∈ D, then

(a) If ϕ is hyperbolic automorphism, α > 0 and ϕ′(w)1/2 < |u(w)| < ϕ′(w)−1/2, thenuCϕ is frequently hypecyclic on H2(D).

(b) If ϕ is parabolic automorphism, α > 1 and |u(w)| = 1, then uCϕ is frequently hy-pecyclic on H2(D).

(c) If ϕ is hyperbolic non-automorphism, α > 0 and |u(w)| > ϕ′(w)1/2, then uCϕ isfrequently hypecyclic on H2(D).

Proof. By the proof of Proposition 2.4, g(z) =∞∏n=0

u(ϕn(z))u(w) 6= 0 for z ∈ D, it is easy to see

that Mg is a bounded operator on H2(D) and uCϕMg = u (w)MgCϕ. Combining with thecomparison principle, we obtain this theorem.

2.2. The case for 0 < v < 1/2. For v ∈ (0, 1/2), using the equivalent norm in Sv, we definethe Banach space Qc as follows:

Qc = f ∈ Sv : ‖f‖Qc= |f(0)|+ sup

w∈D‖f ϕw − f‖ <∞,

where c = 1− 2v and ϕw(z) = (w − z)/(1 − wz). For different p ∈ (0, 1), Qp1 ⊂ Qp2 when0 < p1 < p2 ≤ 1. In particular, Q1 = BMOA, the bounded mean oscillation space ofanalytic functions and when p > 1, Qp = B, the Bloch space on D.




Let g ∈ Q1−2v, by Corollary 2 in [10], we know that if

supζ∈∂D

∫D(ζ,r)

|g(z)|2(1− |z|)1−2vdA(z) = O(r3−2v), (2.1)

then Mg is bounded on Sv.Thus we get the following theorems.

Theorem 2.5. Let 0 < v < 1/2 and α > 0. And let ϕ ∈ LFM(D) and ϕ be a hyperbolicautomorphism of the unit disc with Denjoy-Wolff point w ∈ ∂D, u ∈ Lipα(w) and u(w) 6= 0,the function g =

∏∞i=0

1u(w)u(ϕi(w)) ∈ Q1−2v, ‖I −Mg‖Sv→Sv

< 1 and (2.1) holds, then the

following are equivalent:(a) uCϕ is frequently hypercyclic.(b) uCϕ is hypercyclic.

(c) ϕ′(w)(1−2v)/2 < |u(w)| < ϕ′(w)(2v−1)/2.

Proof. The implication (a) ⇒ (b) is trivial. If ϕ ∈ LFM(D) with Denjoy-Wolff pointw ∈ ∂D and u ∈ Lipα(w), u(w) 6= 0, as we saw in the proof of Proposition 2.4 in [9], themap g(z) =

∏∞i=0

1u(w)u(ϕi(w)) is a nonzero holomorphic function satisfying uCϕg = u(w)g.

Since g ∈ Q1−2v and (2.1) holds, we have Mg is bounded operator on Sv, so g ∈ Sv,thus the function g is an eigenfunction of uCϕ corresponding to u(w) on Sv, and uCϕMg =u(w)MgCϕ.

Note that ‖I−Mg‖Sv→Sv≤ 1+‖Mg‖Sv→Sv

. So I−Mg is also a bounded operator on Sv.Because ‖I −Mg‖Sv→Sv < 1, then Mg is a invertible operator. It is obvious that (b)⇔ (c).Besides, suppose that the condition (c) holds, by the proof of Theorem 2.6 in [4], u(w)Cϕsatisfies the Frequent Hypercyclicity Criterion. The implication (c)⇒ (a) is obvious.

2.3. The case for v = 1/2. If so, we know that Sv is the Dirichlet space D.

Theorem 2.6. Let ϕ ∈ LFT (D), α > 1, w ∈ ∂D be the Denjoy-Wolff point of ϕ, u ∈Lipα (w) ∩ A (D) , ‖u‖∞ = |u (w)| > 1 and u (z) 6= 0 for every z ∈ D. If ϕ is hyperbolicnon-automorphism, then uCϕ is frequently hypecyclic on the Dirichlet space D.

Proof. By the proof of Proposition 2.4, g(z) =∞∏n=0

u(ϕn(z))u(w) 6= 0 for z ∈ D, Since ‖u‖∞ =

|u (w)| > 1, so g ∈ H∞ ⊂ D and u (z) g (ϕ (z)) = u (w) g (z) . It is easy to see that Mg is abounded operator on the Dirichlet space D and uCϕMg = u (w)MgCϕ. By Theorem 1.8 in[5] and the comparison principle, we complete the proof.

3. Frequent hypercyclicity of uCϕ on Apα

In this section, we study in detail frequent hypercyclicity of uCϕ on the weighted Bergmanspace Apα and we suppose that the weighted composition operator uCϕ is bounded on Apα.

Proposition 3.1. Let α > −1, 1 < p < ∞ and ϕ ∈ LFT (D). If uCϕ is frequentlyhypercyclic on Apα, then

(i) ϕ has no fixed point in D and ϕ is univalent.(ii) u(z) 6= 0 for every z ∈ D.

Proof. The proof is obvious, so we omit it.

Next, we obtain the following results.

Theorem 3.2. Let α > −1, β > 0, 1 < p < ∞, ϕ ∈ LFT (D) and ϕ be a hyperbolicautomorphism and w ∈ ∂D be the Denjoy-Wolff point of ϕ, u ∈ Lipβ (w) ∩ A (D) , ‖u‖∞ =

|u (w)| 6= 0 and u (z) 6= 0 for every z ∈ D. If ϕ′ (w)(2+α)/p

< |u(w)| < ϕ′ (w)(−2−α)/p

, thenuCϕ is frequently hypecyclic on Apα.

Proof. First, since this space under consideration is unitarily invariant, we may assume that1 and −1 are fixed points of ϕ and 1 is the attractive fixed point. The change of variables

σ (z) =i(1− z)

1 + z




takes the unit disk onto the upper half plane, 1 and −1 to 0 and ∞. We obtain that ϕ isconjugate to the translation map

ψ (z) = ρz,

where 0 < ρ < 1. By using the equation σ ϕ = ψ σ, we can get

ϕ (z) =(1 + ρ) z + 1− ρ(1− ρ)z + 1 + ρ

,

where ϕ′ (1) = ρ.

Let X0 denote the subspace of polynomials vanishing m at 1, where m > 2(α+2)p . It is

obvious that X0 is dense on Apα. Fix f ∈ X0. It is similarly proved as in Theorem 3.5 in [5]that ∥∥λnCnϕf∥∥p ≤ C |λ|np ρ(α+2)n, n ∈ N,

where C is a constant independent of n. If ϕ′ (1)(2+α)/p

< |λ| < ϕ′ (1)(−2−α)/p

, we obtainthat

∞∑n=1

‖(λCϕ)nf‖ <∞, for all f ∈ X0. (3.1)

Similarly, let Y0 denote the subspace of polynomials vanishing m at −1 and Y0 is denseon Apα. We take S = (λCϕ)

−1. Observe that −1 is the attractive fixed point of ϕ−1 with(

ϕ−1)′

(−1) = 1ϕ′(−1) = ρ and ϕ′ (1)

(2+α)/p< |λ| < ϕ′ (1)

(−2−α)/p. Therefore, a similar

argument leads to∞∑n=1

‖Snf‖ <∞, for all f ∈ Y0. (3.2)

If we set X := X0 ∩ Y0, then we obtain that X is dense in Apα. Clearly (3.1) and (3.2) holdfor all f ∈ X. It is obvious that λCϕS is the identity on X. Consequently, λCϕ satisfies theFrequent Hypercyclicity Criterion. By Proposition 2.4, then uCϕ is frequently hypecyclicon Apα.

Theorem 3.3. Let α > −1, β > 0, 1 < p < ∞, ϕ ∈ LFT (D) and ϕ is a hyperbolicnon-automorphism, w ∈ ∂D be the Denjoy-Wolff point of ϕ, u ∈ Lipβ (w)∩A (D) , ‖u‖∞ =

|u (w)| 6= 0 and u (z) 6= 0 for every z ∈ D. If |u(w)| > ϕ′ (ζ)(2+α)/p

, then uCϕ is frequentlyhypecyclic on Apα.

Proof. First, we prove that if |λ| > ϕ′ (w)(2+α)/p

, then λCϕ is frequently hypecyclic on Apα.Now, we assume that w = 1 is the boundary fixed point and β is a exterior fixed point.

Upon conjugating with an appropriate map, ϕ is conjugate to

ρz + 1− ρ,where 0 < ρ < 1. Hence we may assume that ϕ (z) = ρz + 1− ρ, where ϕ′ (1) = ρ. For anyn ∈ N, we have

ϕn (z) = ρnz + 1− ρn. (3.3)

Let X0 denote the subspace of polynomials vanishing m at 1, where m is to be determinedlater on. Obviously, X0 is dense on Apα. Fix f ∈ X0. It is similarly proved as in Theorem2.11 in [5] that ∥∥λnCnϕf∥∥p ≤ C |λ|np ρmnp, n ∈ N,where C is a constant independent of n. Since 0 < ρ < 1, we can choose m large enough tohave |λρm| < 1. By the assumption, we obtain that

∞∑n=1

‖(λCϕ)nf‖ <∞, for all f ∈ X0. (3.4)

Define T = λCϕ and the inverse S = λ−1Cϕ−1 . Let Y be the set of all polynomials thatvanish m times at β where m will be suitable number. The set Y0 will be

Y0 =∞∪n=0

λ−nCnϕ−1

(Y ) =∞∪n=0

λ−nCϕ−n (Y ) .




Similarly, we obtain that for n large enough∥∥λ−nCϕ−nf∥∥p ≤ C |λ|−np ρn(α+2),

where C is a constant independent of n. By the assumption, we have∞∑n=1

‖Snf‖ <∞, for all f ∈ Y0. (3.5)

If we set X := ∪∞n=0Sn (X ∩ Y ) , then we obtain that X is dense in Apα. Clearly (3.4) and

(3.5) hold for all f ∈ X. It is obvious that λCϕS is the identity on X. Consequently, λCϕsatisfies the Frequent Hypercyclicity Criterion. By Proposition 2.4, then uCϕ is frequentlyhypecyclic on Apα.

References

[1] P.S. Bourdon, J.H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125

(1997), no. 596.

[2] C.C. Cowen, B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press,Boca Raton, FL, 1995.

[3] R.Y. Chen, Z.H. Zhou, Hypercyclicity of weighted composition operators on the unit ball of CN , J.Korean Math. Soc. 48 (5) (2011), 969-984.

[4] L.B. Gonzalez, A. Bonilla, Compositional frequent hypercyclicity on weighted Dirichlet spaces, Bull.

Belg. Math. Soc. Simon Stevin 17 (2010), 1-11.[5] E.A. Gallardo-Gutierrez, A. Montes-Rodrıguez, The role of the spectrum in the cyclic behavior of

composition operators, Mem. Amer. Math. Soc. 167 (2004), no. 791.

[6] K.G. Grosse-Erdmann, A.P. Manguillot, Linear chaos, Universitext, Springer, 2011.[7] L. Jiang, C. Ouyang, Cyclic behavior of linear fractional composition operators in the unit ball of CN ,

J. Math. Anal. Appl. 341 (2008), 601-612.

[8] P.R. Hurst, Relating composition operators on different weighted Hardy spaces, Arch. Math. 68 (1997),503-513.

[9] B. Yousefi, H. Rezaei, Hypercyclic property of weighted compocition operators, Proc. Amer. Math. Soc.

135 (10) (2007), 3263-3271.[10] C. Yuan, S. Kumar, Z.H. Zhou, Weighted composition operators on Dirichlet-type spaces and related

Qp spaces, Publ. Math. Debrecen, 80 (1-2) (2012), 79-88.[11] L. Zhang, Z.H. Zhou, Hypercyclicity of weighted composition operator on weighted Dirichlet space,

Complex Var. Elliptic Equ. 59 (7) (2014) 1043-1051.

[12] L. Zhang, Z.H. Zhou, Notes about the structure of common supercyclic vectors, J. Math. Anal. Appl.418 (2014) 336-343.

[13] L. Zhang, Z.H. Zhou, Notes about subspace-supercyclic operators, Ann. Funct. Anal. 6(2015), no. 2,

60-68.

Shi-An Han

Department of MathematicsTianjin University

Tianjin, 300072P.R. China.

E-mail address: [email protected], [email protected]

Liang ZhangSchool of Marine Science and Technology

Tianjin UniversityTianjin, 300072

P.R. China.E-mail address: [email protected]



ON THE SPECIAL TWISTED q-POLYNOMIALS

JIN-WOO PARK

Abstract. In this paper, we found some interesting identities of q-extension

of special twisted polynomials which are derive from the bosonic q-integral andfermionic q-integral on Zp.

1. Introduction

Let p be a given odd prime number. Throughout this paper, we assume that Zp,Qp and Cp will, respectively, denote the rings of p-adic integers, the fields of p-adicrational numbers and the completion of algebraic closure of Qp. The p-adic norm|·|p is normalized by |p|p = 1

p . Let UD(Zp) be the space of uniformly differentiable

functions on Zp. For f ∈ UD(Zp), the bosonic p-adic q-integral on Zp is defined byT. Kim to be

Iq(f) =

∫Zp

f(x)dµq(x) = limN→∞

1

[pN ]q

pN−1∑x=0

f(x)qx, (see [9, 10]), (1.1)

and the fermionic p-adic q-integral on Zp is also defined by Kim to be

I−q(f) =

∫Zp

f(x)dµ−q(x) = limN→∞

1

[pN ]−q

pN−1∑x=0

f(x)(−q)x, (see [9, 11]). (1.2)

Let f1(x) = f(x+ 1). Then, by (1.1) and (1.2), we get

qIq(f1)− Iq(f) = (q − 1)f(0) +q − 1

log qf′(0), (1.3)

and

qI−q(f1) + I−q(f) = [2]qf(0), (1.4)

where f′(0) = d

dxf(x)∣∣x=0

(see [9, 10, 11]).It is well known that the q-Bernoulli polynomials are defined by the generating

function to beq − 1 + (q−1)t

log q

qet − 1ext =

∞∑n=0

Bn,q(x)tn

n!, (1.5)

and the q-Euler polynomials are given by

[2]qqet + 1

ext =∞∑n=0

En,q(x)tn

n!. (1.6)

1991 Mathematics Subject Classification. 11S80, 11B68, 05A30.Key words and phrases. twisted q-Daehee polynomials of order r, twisted q-Changhee polyno-

mials of order r, q-Cauchy polynomials of order r.

1


283 JIN-WOO PARK 283-292

2 JIN-WOO PARK

When x = 0, Bn,q = Bn,q(0)(En,q = En,q(0)) are called the nth q-Bernoullinumbers(nth q-Euler numbers, respectively)(see [7, 8, 14, 16]).

The Stirling numbers of the first kind are defined by

(x)n = x(x− 1) · · · (x− n+ 1) =n∑l=0

S1(n, l)xl, (n ≥ 0),

and the Stirling numbers of the second kind are defined by

(et − 1)n = n!∞∑l=n

S2(l, n)tl

l!, (see [1, 12]).

The Daehee polynomials of the first kind are defined by the generating functionto be (

log(1 + t)

t

)(1 + t)x =

∞∑n=0

Dn(x)tn

n!(see [4, 5]).

Recently, the q-Daehee polynomials are defined by the generating function to be(1− q + 1−q

log q

1− q − qt

)(1 + t)x =

∞∑n=0

Dn,q(x)tn

n!, (see [2, 13]), (1.7)

and the q-Changhee polynomials are defined by the generating function to be

[2]q[2]q + qt

(1 + t)x =∞∑n=0

Chn,q(x)tn

n!, (see [3]) (1.8)

where t ∈ Cp with |t|p < p−1

p−1 . When x = 0, Dn,q = Dn,q(0)(Chn,q = Chn,q(0))are called the nth q-Daehee numbers(nth q-Changhee numbers, respectively).

The Daehee polynomials and Changhee polynomials are introduced by T. Kimet. al. in [4, 6], and found interesting identities in [2, 4, 5, 6, 13, 15, 16]. Inthis paper, we found some interesting identities of q-extension of special twistedpolynomials which are derive from the bosonic q-integral and fermionic q-integralon Zp.

2. Twisted q-Daehee numbers and polynomials of higher-order

For n ∈ N, let Tp be the p-adic locally constant space defined by

Tp = ∪∪∪n≥1

Cpn = limn→∞

Cpn ,

where Cpn =ω|ωpn = 1

is the cyclic group of order pn.

In this section, we assume that t ∈ Cp with |t|p < p−1

p−1 . We define the higherorder q-Beroulli polynomials as follows:(

q − 1 + q−1log q t

qet − 1

)rext =

∞∑n=0

B(r)n,q(x)

tn

n!. (2.1)

When x = 0, B(r)n,q(0) = B

(r)n,q are called the higher order q-Bernoulli numbers.

For ε ∈ Tp, we consider the twisted q-Daehee polynomials of order r as follows:(q − 1 + q−1

log q log(1 + εt)

qεt+ q − 1

)r(1 + εt)x =

∞∑n=0

D(r)n,ε,q(x)

tn

n!. (2.2)

When x = 0, D(r)n,ε,q(0) = D

(r)n,ε,q are called twisted q-Daehee numbers of order r.



ON THE SPECIAL TWISTED q-POLYNOMIALS 3

From (1.1), we can obtain the equation:∫Zp

· · ·∫Zp

(1 + εt)x1+···+xr+xdµq(x1) · · · dµq(xr)

=

(q − 1 + q−1

log q log(1 + εt)

qεt+ q − 1

)r(1 + εt)x

=∞∑n=0

D(r)n,ε,q(x)

tn

n!.

(2.3)

By (2.3), we get

∫Zp

· · ·∫Zp

εn(x1 + · · ·+ xr + x

n

)dµq(x1) · · · dµq(xr) =

D(r)n,ε,q(x)

n!(n ≥ 0). (2.4)

By replacing t by 1ε (et − 1) in (2.3), we have

∞∑n=0

D(r)n,ε,q(x)

(1ε (et − 1)

)nn!

=

(q − 1 + q−1

log q t

qet − 1

)rext =

∞∑n=0

B(r)n,q(x)

tn

n!(2.5)

and

∞∑n=0

D(r)n,ε,q(x)

1

εnn!

(et − 1

)n=∞∑n=0

D(r)n,ε,q(x)

1

εnn!n!∞∑m=n

S2(m,n)tm

m!

=∞∑n=0

(n∑

m=0

D(r)m,ε,q(x)S2(n,m)

εn

)tn

n!.

(2.6)

Thus, by (2.5) and (2.6), we have

B(r)n,q(x) =

n∑m=0


εn. (2.7)

Therefore, by (2.4) and (2.7), we obtain the following theorem.

Theorem 2.1. For n ≥ 0, we have

B(r)n,q(x) =

n∑m=0


εn

and

D(r)n,ε,q(x)

n!=

∫Zp

· · ·∫Zp

εn(x1 + · · ·+ xr + x

n

)dµq(x1) · · · dµq(xr)

where S2(m,n) is the Stirling number of the second kind.



4 JIN-WOO PARK

From (2.1), by replacing t by log(1 + εt), we have(q − 1 + q−1

log q log(1 + εt)

qεt+ q − 1

)r(1 + εt)x

=∞∑n=0

B(r)n,q(x)

1

n!(log(1 + εt))

=∞∑n=0

B(r)n,q(x)

1

n!n!∞∑m=n

S1(m,n)(εt)m

m!

=∞∑m=0

(m∑n=0

εmS1(m,n)B(r)n,q(x)

)tm

m!,

(2.8)

where S1(m,n) is the Stirling number of the first kind. Thus, by (2.2) and (2.8),we obtain the following theorem.


D(r)n,ε,q(x) =

m∑n=0

εmS1(m,n)B(r)n,q(x).

Now, we consider the q-Changhee polynomials of order r which are defined bythe generating function as follows:

[2]qqεt+ [2]q

(1 + εt)x =∞∑n=0

Ch(r)n,ε,q(x)tn

n!. (2.9)

In the special case x = 0, Ch(r)n,ε,q(0) = Ch

(r)n,ε,q are called the q-Changhee numbers

of order r.From (1.2), we note that∫

Zp

· · ·∫Zp

(1 + εt)x1+···+xr+xdµ−q(x1) · · · dµ−q(xr)

=

([2]q

qεt+ [2]q

)r(1 + εt)x.

(2.10)

By (2.10), we have

∫Zp

· · ·∫Zp

εn(x1 + · · ·+ xr + x

n

)dµ−q(x1) · · · dµ−q(xr) =

Ch(r)n,ε,q(x)

n!. (2.11)

In view of (1.6), we define the higher order q-Euler polynomials by generatingfunction to be (

[2]qqet + 1

)rext =

∞∑n=0

E(r)n,q(x)

tn

n!. (2.12)




From (2.10), we note that∫Zp

(1 + εt)x1+···+xr+xdµ−q(x1) · · · dµ−q(xr)

=

([2]q

qelog(1+εt) + 1

)rex log(1+εt)

=∞∑n=0

E(r)n,q

1

n!(log(1 + εt))

n

=∞∑n=0

E(r)n,q(x)

1

n!n!∞∑m=n

S1(m,n)(εt)m

m!

=∞∑m=0

(m∑n=0

εmE(r)n,q(x)S1(m,n)

)tm

m!.

(2.13)

Hence, by (2.11) and (2.13), we obtain the following theorem.

Theorem 2.3. For n ≥ 0, we have∫Zp

· · ·∫Zp

εn(x1 + · · ·+ xr + x

n

)dµ−q(x1) · · · dµ−q(xr)

=Ch

(r)n,ε,q(x)

n!=

1

n!

n∑m=0

εnE(r)m,q(x)S1(n,m).

By replacing t by 1ε (et − 1) in (2.9), we have

∞∑n=0

Ch(r)n,ε,q(x)(et − 1)n

εnn!=

([2]q

qet + 1

)rext (2.14)

and

∞∑n=0

ε−nCh(r)n,ε,q(x)1

n!

(et − 1

)n=

∞∑n=0

ε−nCh(r)n,ε,q(x)

∞∑m=n

S2(m,n)tm

m!

=

∞∑m=0

( ∞∑n=0

ε−nCh(r)n,ε,q(x)S2(m,n)

)tm

m!

(2.15)

By (2.12), (2.14) and (2.15), we obtain the following theorem.


E(r)n,q(x) =

n∑m=0

ε−mCh(r)m,ε,q(x)S2(n,m).

From now on, we consider the q-analogue of the twisted Cauchy polynomials oforder r, which are defined by the generating function to be(

q(1 + εt)− 1

(q − 1) + q−1log q log(1 + εt)

)r(1 + εt)x =

∞∑n=0

C(r)n,ε,q(x)

tn

n!. (2.16)



6 JIN-WOO PARK

In the special case x = 0, C(r)n,ε,q(0) = C

(r)n,ε,q are called the twisted Cauchy numbers

of order r. Note that

limq→1

(q(1 + εt)− 1

(q − 1) + q−1log q log(1 + εt)

)r(1 + r)x

=

(εt

log(1 + εt)

)r(1 + t)x =

∞∑n=0

C(r)n,ε(x)

tn

n!,

(2.17)

where C(r)n,ε are called the Cauchy polynomials of order r.

By (2.2), we can derive the followings:

(1 + εt)x =

(q(1 + εt)− 1

(q − 1) + q−1log q log(1 + εt)

)r(1 + εt)x

((q − 1) + q−1

log q log(1 + εt)

q(1 + εt)− 1

)r

=

( ∞∑k=0

C(r)n,ε,q

)( ∞∑m=0

D(r)m,ε,q

tm

m!

)

=

∞∑n=0

(n∑l=0

(n

l

)C

(r)l,ε,q(x)D

(r)n−l,ε,q

)tn

n!

(2.18)

and

(1 + εt)x =∞∑n=0

εn(x)ntn

n!. (2.19)

By (2.18) and (2.19), we obtain the following theorem.

Theorem 2.5. For n ≥ 0, we have(x

n

)=

1

εnn!

n∑l=0

(n

l

)C

(r)l,ε,q(x)D

(r)n−l,ε,q.

Let n be a given nonnegative integer. In [2], authors defined q-analogue of the

Bernoulli-Euler mixed-type polynomials of order (r, s) BE(r,s)n,q (x), and derived the

following equation.

∞∑n=0

BE(r,s)n,q (x)

tn

n!=

([2]q

qet + 1

)s(q − 1 + q−1log q t

qet − 1

)rext. (2.20)

By replacing t by log(1 + εt), we get∞∑n=0

BE(r,s)n,q (x)

(log(1 + εt))n

n!

=

([2]q

q(1 + εt) + 1

)s(q − 1 + q−1log q log(1 + εt)

q(1 + εt)− 1

)r(1 + εt)x

=

( ∞∑m=0

Ch(s)m,ε,q(x)tm

m!

)( ∞∑l=0

D(r)l,ε,q

tl

l!

)

=∞∑n=0

(n∑l=0

(n

l

)Ch

(s)l,ε,q(x)D

(r)n−l,ε,q

)tn

n!,

(2.21)




and∞∑n=0

BE(r,s)n,q (x)

(log(1 + εt))n

n!=∞∑n=0

(εn

n∑m=0

BE(r,s)m,q (x)S1(n,m)

)tm

m!. (2.22)

Thus, by (2.21) and (2.22), we obtain the following theorem.

Theorem 2.6. For n ≥ 0, we haven∑l=0

(n

l

)Ch

(s)l,ε,q(x)D

(r)n−l,ε,q = εn

n∑m=0

BE(r,s)m,q (x)S1(n,m).

From now on, we consider the q-analogue of the twisted Daehee-Changhee mixed-type polynomials of order (r, s) as follows:

DC(r,s)n,ε,q(x) =

∫Zp

· · ·∫Zp

D(r)n,ε,q(x+ y1 + · · ·+ ys)dµ−q(y1) · · · dµ−q(ys) (2.23)

where n is a given nonnegative integer.By (2.23), we get∞∑n=0

DC(r,s)n,ε,q(x)

tn

n!

=

∫Zp

· · ·∫Zp

D(r)n,ε,q(x+ y1 + · · ·+ ys)

tn

n!dµ−q(y1) · · · dµ−q(ys)

=

(q − 1 + q−1

log q log(1 + εt)

qεt+ q − 1

)r ∫Zp

· · ·∫Zp

(1 + εt)x+y1+···+ysdµ−q(y1) · · · dµ−q(ys)

=

(q − 1 + q−1

log q log(1 + εt)

qεt+ q − 1

)r ([2]q

qεt+ [2]q

)s(1 + εt)x

=

( ∞∑n=0

D(r)n,ε,q(x)

tn

n!

)( ∞∑m=0

Ch(s)m,ε,qtm

m!

)

=∞∑n=0

( ∞∑m=0

(n

m

)D(r)m,ε,q(x)Ch

(s)m−n,ε,q

)tn

n!

(2.24)

and∞∑n=0

DC(r,s)n,ε,q(x)

(1ε (et − 1)

)nn!

=

(q − 1 + q−1

log q t

qet − 1

)r ([2]q

qet + 1

)sext

=∞∑n=0

(n∑

m=0

(n

m

)B(r)m,q(x)E

(s)n−m,q

)tn

n!.

(2.25)

Note that∞∑n=0

DC(r,s)m,ε,q(x)

(1ε (et − 1)

)nn!

=∞∑n=0

DC(r,s)n,ε,q(x)

1

εnn!n!∞∑m=n

S2(m,n)tm

m!

=∞∑n=0

(n∑

m=0

ε−mDC(r,s)m,ε,q(x)S2(n,m)

)tn

n!.

(2.26)

Hence, by (2.24), (2.25) and (2.26), we obtain the following theorem.



8 JIN-WOO PARK

Theorem 2.7. For n ≥ 0, we get

DC(r,s)n,ε,q(x) =

∞∑m=0

(n

m

)D(r)m,ε,q(x)Ch

(s)m−n,ε,q

andn∑

m=0

(n

m

)B(r)m,q(x)E

(s)n−m,q =

n∑m=0

ε−mDC(r,s)m,ε,q(x)S2(n,m).

Now, we consider the q-analogue of the twisted Cauchy-Changhee mixed-typepolynomials of order (r, s) as follows:

CC(r,s)n,ε,q(x) =

∫Zp

· · ·∫Zp

C(r)n,ε,q(x+ y1 + · · ·+ ys)dµ−q(y1) · · · dµ−q(ys) (2.27)

where n is a given nonnegative integer.By (2.27), we have

∞∑n=0

CC(r,s)n,ε,q(x)

tn

n!

=

∫Zp

· · ·∫Zp

∞∑n=0

C(r)n,ε,q(x+ y1 + · · ·+ ys)

tn


=

(q(1 + εt)− 1

(q − 1) + q−1log q log(1 + εt)

)r ([2]q

qεt+ [2]q

)s(1 + εt)x

=

∞∑n=0

( ∞∑m=0

(n

m

)C(r)m,ε,q(x)Ch

(s)m−n,ε,q

)tn

n!

(2.28)

and∞∑n=0

CC(r,s)n,ε,q(x)

(1ε (et − 1)

)nn!

=

(qet − 1

q − 1 + q−1log q t

)r ([2]q

qet + 1

)sext

=∞∑n=0

(n∑

m=0

(n

m

)B(−r)m,q (x)E

(s)n−m,q

)tn

n!.

(2.29)

Now, we observe that∞∑n=0

CC(r,s)m,ε,q(x)

(1ε (et − 1)

)nn!

=∞∑n=0

CC(r,s)n,ε,q(x)

1

εnn!n!∞∑m=n

S2(m,n)tm

m!

=∞∑n=0

(n∑

m=0

ε−mCC(r,s)m,ε,q(x)S2(n,m)

)tn

n!.

(2.30)

Therefore, by (2.28), (2.29) and (2.30), we obtain the following theorem.


CC(r,s)n,ε,q(x) =

∞∑m=0

(n

m

)C(r)m,ε,q(x)Ch

(s)m−n,ε,q

andn∑

m=0

(n

m

)B(−r)m,q (x)E

(s)n−m,q =

n∑m=0

ε−mCC(r,s)m,ε,q(x)S2(n,m).




From now on, we consider the q-analogue of twisted Cauchy-Daehee mixed-typepolynomials of order (r, s) as follows:

CD(r,s)n,ε,q(x) =

∫Zp

· · ·∫Zp

C(r)n,ε,q(x+ y1 + · · ·+ ys)dµq(y1) · · · dµq(ys) (2.31)

where n is a given nonnegative integer.By (2.31), we have

∞∑n=0

CD(r,s)n,ε,q(x)

tn

n!

=

∫Zp

· · ·∫Zp

∞∑n=0

C(r)n,ε,q(x+ y1 + · · ·+ ys)

tn


=

(q(1 + εt)− 1

(q − 1) + q−1log q log(1 + εt)

)r (q − 1 + q−1

log q log(1 + εt)

qεt+ q − 1

)s(1 + εt)x

=

∑∞n=0 C

(r−s)n,ε,q (x) t

n

n! if r > s,∑∞n=0D

(s−r)n,ε,q (x) t

n

n! if r < s,∑∞n=0(x)n

tn

n! if r = s.

(2.32)

Thus, by (2.32), we obtain the following theorem.


CD(r,s)n,ε,q =

C

(r−s)n,ε,q (x) if r > s,

D(s−r)n,ε,q (x) if r < s,

(x)n if r = s.

References

[1] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974.

[2] D. V. Dolgy, D. S. Kim, T. Kim and S. H. Kim, Some identities of special q-polynomials, J.Inequal. Appl. 2014, 2014:438, 17 pp.

[3] D. Kim, T. Mansour, S. H. Rim and J. J. Seo, A Note on q-Changhee Polynomials and

Numbers, Adv.Studies Theor. Phys., Vol. 8, 2014, no. 1, 35-41.[4] D. S. Kim and T. Kim, Daehee numbers and polynomials, Appl. Math. Sci., 7 (2013), no.

120, 5969-5976.

[5] D. S. Kim, T. Kim, S. H. Lee and J. J. Seo, Higher-order Daehee numbers and polynomials,Int. J. Math. Anal., 8 (2014), no. 6, 273-283.

[6] D. S. Kim, T. Kim, and J. J. Seo, Higher-order Daehee numbers and polynomials, Adv.

Studies Theor. Phys., 7 (2013), no. 20, 993-1003.[7] J. H. Jin, T. Mansour, E. J. Moon and J. W. Park, On the (r, q)-Bernoulli and (r, q)-Euler

numbers and polynomials, J. Comput. Anal. Appl., 19 (2015), no. 2, 250-259.

[8] H. M. Kim, D. S. Kim, T. Kim, S. H. Lee, D. V. Dolgy and B. Lee, Identities for the Bernoulliand Euler numbers arising from the p-adic integral on Zp, Proc. Jangjeon Math. Soc., 15

(2012), no. 2, 155-161.[9] T. Kim, q-Volkenvorn integration, Russ. J. Math. Phys., 19 (2002), 288-299.

[10] T. Kim, An invariant p-adic integral associated with Daehee numbers, Integral Transforms

Spec. Funct., 13 (2002), no. 1, 65-69.[11] T. Kim, On q-analogye of the p-adic log gamma functions and related integral, J. Number

Theory, 76 (1999), no. 2, 320-329.

[12] T. Kim, D. S. Kim, T. Mansour, S,-H. Rim and M. Schork Umbral calculus and Sheffersequences of polynomials, J. Math. Phys. 54, 083504 (2013); doi:10.1063/1.4817853.



10 JIN-WOO PARK

[13] T. Kim, and T. Mansour, A note on q-Daehee polynomials and numbers, Adv. Stud. Con-temp. Math., 24 (2014), no. 2, 131-139.

[14] H. Ozden, I. N. Cangul and Y. Simsek, Remarks on q-Bernoulli numbers associated with

Daehee numbers, Adv. Stud. Contemp. Math., 18 (2009), no. 1, 41-48.[15] J. W. Park, On the q-analogue of λ-Daehee polynomials, J. Comput. Anal. Appl., 19 (2015),

no. 6, 966-974.

[16] J. W. Park, S. H. Rim and J. Kwom, The twisted Daehee numbers and polynomials, Adv.Difference Equ., 2014, 2014:1.

[17] Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials associatedwith their interpolation functions, Adv. Stud. Contemp. Math., 16 (2008), no. 2, 271-278.

Department of Mathematics Education, Daegu University, Gyeongsan-si, Gyeongsangbuk-

do, 712-714, Republic of Korea.

E-mail address: [email protected]



Equicontinuity of Maps on [0, 1)

Kesong Yan, Fanping Zeng and Bin Qin*

School of Information and StatisticsGuangxi University of Finance and Economics

Nanning, Guangxi, 530003, P.R China

Abstract: We mainly study the equicontinuity of maps on [0,1). Let f : X → X bea continuous map on X = [0, 1). We show that if f is an equicontinuous map withF (f) nonempty, then one of the following two conditions holds: (1) F (f) consists ofa single point and F (f2) =

⋂∞n=1 fn(X);(2) F (f) =

⋂∞n=1 fn(X). Last we construct

two examples to show that the converse result doesn’t hold.

Keywords: Interval , Equicontinous, Periodic point.

Mathematics Subject Classification (2000): Primary: 54B20, 54E40

1 Introduction

Let (X, d) be a metric space with the metric d (not necessary compact) and f : X → X bea continuous map. For every nonnegative integer n define fn inductively by fn = ffn−1,where f0 is the identity map on X. A point x of X is said to be a periodic point of f ifthere is a positive integer n such that fn(x) = x. The least such n is called the period ofx. A point of period one is called a fixed point. Let F (f) denote the fixed point set of fand P (f) the set of periodic points of f .

If x ∈ X then the trajectory (or orbit) of x is the sequence orb(x, f) = fn(x) : n > 0and the ω−limit set of x is

ω(x, f) =⋂

m>0

⋃n>m

fn(x).

Equivalently, y ∈ ω(x, f) if and only if y ∈ X is a limit point of the trajectory orb(x, f),i.e., fnk(x) → y for some sequence of integers nk →∞.

The map f is said to be equicontinuous (in some terminology also Lyapunov stable)if given ε > 0 there exists a δ > 0 such that d(f i(x), f i(y)) < ε whenever d(x, y) < δ forall x, y ∈ X and all i > 1.

In 1982, J. Cano [4] proved the following theorem on equicontinuous map for theclosed interval I.

The authors are supported by NNSF of China (11161029, 11401288) and NSF for Young Scholarof Guangxi Province (2013GXNSFBA019020). The first author is partially supported by Project ofOutstanding Young Teachers in Higher Education Institutions of Guangxi Province (GXQG022014080)and Project of Key Laboratory of Quantity Economics of Guangxi University of Finance and Economics(2014SYS03).

∗ Corresponding author.Email address: [email protected] (K. Yan), [email protected] (F. Zeng), bin-

[email protected](B. Qin)


293 Kesong Yan et al 293-298

2 K. Yan, F. Zeng AND B. Qin

Theorem 1.1 Let f : I → I be an equicontinuous map. Then F (f) is connected andif it is non-degenerate then F (f) = P (f).

The next theorem was due to Bruckner and Hu [3]. This result was also proved byBlokh in [2].

Theorem 1.2 Let f : I → I be a continuous map. Then f is equicontinuous if andonly if

⋂∞i=1 fn(I) = F (f2).

In [9], Valaristos described the characters of equicontinuous circle maps: A continuousmap f of the unit circle S1 to itself is equicontinuous if and only if one of the followingfour statements holds: (1) f is topologically conjugate to a rotation; (2) F (f) containsexactly two points and F (f2) = S1; (3) F (f) contains exactly one point and F (f2) =⋂∞

n=1 fn(S1); (4) F (f) =⋂∞

n=1 fn(S1). In 2000, Sun [8] obtained some necessary andsufficient conditions of equicontinuous σ-maps. Later, Mai [6] studied the structure ofequicontinuous maps of general metric spaces, and given some still simpler necessary andsufficient conditions of equicontinuous graph maps.

In [5], Gu showed that a map on Warsaw circle W is equicontinuous if and only ifF (f) consists of a single point and F (f2) =

⋂∞n=1 fn(X) or F (f) =

⋂∞n=1 fn(X).

Warsaw circle W is simple connected but not locally connected, and it often appearsas an example of circle-like and non arc-like in the theory of continuum (see [7]). Inaddition, Warsaw circle is not a continuous image of the closed interval. So it is not aPeano continuum. However, it is easily to see that there is a continuous bijective mapφ : [0, 1) → X. Moreover, if f is a continuous self-map of Warsaw circle W , then there isunique continuous map f : [0, 1) → [0, 1) such that φ f = f φ (see [10]). Note that φis not a homeomorphism since [0, 1) is not compact but Warsaw circle W is compact. Itfollows that f and f are not topologically conjugate. So, it may be that there are somedifferent dynamical properties between maps on [0, 1) and on Warsaw circle.

In this paper we shall deal with the problem of equicontinuity of maps on [0, 1). Ourmain results are the following theorems.

Theorem 1.3 Let X = [0, 1) and f : X → X be an equicontinuous map. If F (f) 6= ∅,then every periodic point of f has periodic 1 or 2, both F (f2) and F (f) are connected.Furthermore, if F (f) is non-degenerate then F (f) = P (f).

Theorem 1.4 Let X = [0, 1) and f : X → X be a continuous map with F (f) 6= ∅.If f is equicontinuous, then one of the following two conditions holds:

(1) F (f) consists of a single point and F (f2) =⋂∞

n=1 fn(X);

(2) F (f) =⋂∞

n=1 fn(X).

Moreover, it is equivalent whenever f is uniformly continuous.

In Section 3, we will construct two examples to show that the converse result ofTheorem 1.4 doesn’t hold.

2 Proof of Theorem 1.3 and 1.4

In this section, we mainly prove Theorem 1.3 and 1.4.



Equicontinuity of Maps on [0, 1) 3

2.1 Some lemmas

In this section, we give some lemmas which are needed in proof of Theorem 1.3 and 1.4.

Lemma 2.1 Let f : X → X be an continuous map of X = [0, 1). If F (f2) = X,then f is the identity map on X.

Proof It is not hard to see that f(X) = X. Assume there exist x ∈ X such thatf(x) 6= x. Then we can choose 0 6 p1 < p2 < 1 such that f(p1) = p2 and f(p2) = p1.Let m = maxx∈[0,p2] f(x). It is clear that p2 6 m < 1. For each x ∈ (p2, 1), we havef(x) < p2 or there exists q ∈ (p2, 1) such that f(q) = p2 by the continuity of f , whichcontradicts to q ∈ F (f2). Hence f(X) = [0,m]. This also contradicts to f(X) = X.Therefore, f is the identity map on X.

Lemma 2.2 Let X = [0, 1) and f : X → X be an equicontinuous map with a fixedpoint p. Suppose J is a component of F (f4) containing p. Then ω(x, f) ⊂ J for everyx ∈ X.

Proof Without loss of generality, we may assume that J is a proper subset of X(note that it is clearly hold whenever J = X). Firstly, we prove that there is a connectedopen subset K ⊃ J such that ω(x, f) ⊂ J for every x ∈ K.

Case 1 J = p. Let ε = (1/2)min(p, 1 − p). By the equicontinuity of f , there is anopen interval K of p such that |fn(x)− p| < ε for every x in K and every positiveinteger n. Let L =

⋃j>0 f j(K). Then L is a closed proper invariant interval

of X. It follows from Theorem 1.1 and 1.2 that the fixed point set of f |L andf2|L is connected, and therefore it is p. Moreover, all periodic points of f |Lhave period 1 or 2. But the fixed point p is the only periodic point of f in L.Therefore P (f |L) = F (f |L) and by Proposition 15 in [1, p. 78] the ω-limit pointscoincide with the fixed points. Hence p is the only ω-limit point of f in L. Thusω(x, f) = p = J for every x ∈ L. Since K ⊂ L, we have ω(x, f) = J for everyx ∈ K.

Case 2 J = [q1, q2] is a closed interval of X. For every i = 1, 2 we consider the or-bit qi, f(qi), f2(qi), f3(qi) of qi. Let ε = (1/2)min(f j(qi), 1 − f j(qi)). By theequicontinuity of f , there is an open interval Kij containing f j(qi) such that|f4n(x) − f j(qi)| < ε for every x ∈ Kij and every positive integer n. LetKi =

⋂3j=0 f−j(Kij), define L =

⋃∞j=0 f j(K1 ∪ J ∪K2). Then L is a closed proper

invariant interval of X. We know from Theorem 1.1 and 1.2 that fixed point set off |L and f2|L is connected and therefore, it is contained in J . Moreover, all periodicpoints of f |L have period 1 and 2. Since P (f |L) is closed, by Proposition 15 in [1,p. 78], it coincides with the set of ω-limits points. Therefore, ω(x, f) ⊂ J for eachx ∈ L. Let K = K1 ∪ J ∪K2. Then K ⊃ J and ω(x, f) ⊂ J for each x ∈ K.

Case 3 J = [q, 1), where 0 < q < 1. Obviously, f(J) ⊂ J . By Lemma 2.1, we haveJ ⊂ F (f). Then limx→1 f(x) = 1. Let g : [0, 1] → [0, 1] such that g|X = fand g(1) = 1. So g is a equicontinuous map on [0, 1]. It follows from Theorem1.1 and 1.2 that the fixed point set of g2 is connected and P (g) = F (g2). HenceP (g) = [q, 1]. By Proposition 15 in [1, p. 78], ω(x, g) ⊂ [q, 1] for each x ∈ [0, 1].Let K = X, then ω(x, f) ⊂ J for each x ∈ K.




Secondly, we show that ω(x, f) ⊂ J for each x ∈ X. Let

S = x ∈ X : ω(x, f) ⊂ J.

Note that S is a nonempty set since K ⊂ S. Let y ∈ S. Then there is a positive integer msuch that fm(y) ∈ K. By the continuity of fm, there exists an open subset U containingy such that fm(U) ⊂ K. Hence U ⊂ S and S is an open set. Let T be the component ofS containing J and therefore K as well. Then T is open and connected. It is sufficient toshow that T = X. Suppose that T 6= X. Let ε = (1/2) min|x− y| : x ∈ J, y ∈ X − T.Then ε > 0. Assume that z is an endpoint of X − T . Then we have fn(z) /∈ T for eachpositive integer n. On the other hand, for any δ > 0 we can choose x ∈ T such that|x− z| < δ. Since ω(x, f) ⊂ J, there is a positive integer m such that fm(x) ∈ B(J, ε/2).Hence |fm(x)− fm(z)| > ε/2. This is a contradiction. Therefore, T = X and the proofis completed.

The following two lemmas are obviously facts on any compact metric space.

Lemma 2.3 Let f : X → X be a continuous map, where X is a compact metricspace. Let k be a positive integer and g = fk. Then f is equicontinuous if and only if gis equicontinuous.

Lemma 2.4 Let f : X → X be a continuous map, where X is a compact metricspace. If f |f(X) is equicontinuous then f is equicontinuous.

2.2 Proof of Theorem 1.3

Let X = [0, 1) and f : X → X be an equicontinuous map. If p is a fixed point of f andJ is a component of F (f4) containing p, then we consider the following three case.

Case 1 J = p. By Lemma 2.2, ω(x, f) ⊂ p for each x ∈ X. This shows that p is aunique periodic point of f . Hence F (f) = F (f2) = p is connected.

Case 2 J = [q1, q2]. By Lemma 2.2, ω(x, f) ⊂ J for each x ∈ X. This shows thatP (f) ⊂ J . Hence P (f) = F (f4) = J and F (f4) is connected. Applying Theorem1.1 to f |J , we know that all periodic points of f have period 1 or 2, both F (f) andF (f2) are connected. Furthermore, if F (f) is non-degenerate then F (f) = P (f).

Case 3 J = [q, 1). By Lemma 2.2, ω(x, f) ⊂ J for each x ∈ X. This shows that P (f) ⊂ J.Hence P (f) = F (f4) = J and F (f4) is connected. Applying Lemma 2.1 to f |J , wehave P (f) = F (f) = J is connected.

This complete the proof of Theorem 1.3.

2.3 Proof of Theorem 1.4

Let X = [0, 1) and f : X → X be a continuous map. We suppose that f is equicontinuous.By Theorem 1.3, both F (f2) and F (f) are connected.

(1) If F (f) consists a single point p then F (f2) = P (f). Moreover by Lemma 2.2, wehave ω(x, f) ⊂ F (f2) for every x ∈ X.



Equicontinuity of Maps on [0, 1) 5

Case 1 If F (f2) = [q1, q2], where 0 6 q1 6 q2 < 1. Similar the proof of Lemma 2.2, thereexists an open, connected subset K containing F (f2) such that L =

⋃∞j=0 f j(K) ⊂

X is a closed and invariant interval. Fixed ε > 0, there is δ > 0 such that |x−y| < δimplies |fn(x)− fn(y)| < ε for all n > 0. For x ∈ X, since ω(x, f) ⊂ F (f2) ⊂ K ⊂L, there exists a positive integer Nx such that fNx(x) ∈ K. Then fm(x) ∈ L foreach m > Nx. By the continuity of fNx , there is an open neighborhood Vx of xsuch that fNx(Vx) ⊂ K and hence fm(Vx) ⊂ L for every m > Nx. Note that thecollection Vxx∈Iδ

forms an open cover of Iδ = [0, 1 − δ]. By the compactness ofIδ, there is a finite subcover Vx1 , . . . , Vxs

. Set N = maxNx1 , . . . , Nxs. Then

fm(Vxi) ⊂ L for every m > N and any 1 6 i 6 s. Thus, fm(Iδ) ⊂ L for everym > N , and hence fm(X) ⊂ B(L, ε) for all m > N , where B(L, ε) = y ∈ X :d(x, y) < ε for some x ∈ L. By the arbitrary of ε, we can get

⋂∞n=1 fn(X) ⊂

L. Using Theorem 1.2, we have⋂∞

n=1 fn(L) = F (f2). It follows that F (f2) =⋂∞n=1 fn(L) =

⋂∞n=1 fn(X), i.e., (1) holds.

Case 2 If F (f2) = [q, 1). Applying Lemma 2.1 to f |[q,1), we have f(x) = x for all x ∈ [q, 1),i.e., [q, 1) ⊂ F (f). This contradicts to F (f) consists a single point.

(2) If F (f) is non-degenerate, then F (f) = P (f) by Theorem 1.3. Similar to theabove argument we can get F (f) =

⋂∞n=1 fn(X) whenever F (f) = [q1, q2]. Now we

assume F (f) = [q, 1) for some 0 6 q < 1. Define g : [0, 1] → [0, 1] as g|X = f andg(1) = 1. So g is a equicontinuous map on [0, 1]. It follows from Theorem 1.1 and 1.2that

⋂∞n=1 gn([0, 1]) = F (g2) = F (g). Thus,

⋂∞n=1 fn(X) = F (f).

3 Examples

In this section, we will construct two examples to show that the converse result of The-orem 1.4 doesn’t hold.

Example 3.1 Let I = [0, 1) and let an = 1 − 1/2n for every n = 1, 2, · · · . Now wedefine a piecewise linear continuous map f : I → I as follows (See Figure 1):

(1) f(x) = 1− x for each x ∈ [0, 1/2];

(2) f(a2n) = 1/2 and f(a2n−1) = 0 for all n = 1, 2, · · · .

It is easily to see that F (f) consists of a single point and F (f2) =⋂∞

n=1 fn(I) =[0, 1/2]. However, f is not equicontinuous since |an+1 − an| = 1

2n+1 → 0 but |f(an+1)−f(an)| = 1/2 for all n > 1.

Figure 1




Example 3.2 Let I = [0, 1) and let an = 1 − 1/2n for every n = 1, 2, · · · . Now wedefine a piecewise linear continuous map f : I → I as follows (See Figure 2):

(1) f(x) = x for each x ∈ [0, 1/2];

(2) f(a2n) = 0 and f(a2n−1) = 1/2 for all n = 1, 2, · · · .

It is easily to see that F (f) is non-degenerate and F (f) =⋂∞

n=1 fn(I) = [0, 1/2].However, f is not equicontinuous since |an+1 − an| = 1

2n+1 → 0 but |f(an+1)− f(an)| =1/2 for all n > 1.

Figure 2

Acknowledgments. Part of this work was carried out when Yan visited NCTS, thefirst author sincerely appreciates the warm hospitality of the host.

References

[1] L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics,1513, Springer-Verlag, Berlin, 1992.

[2] A. M. Blokh, The set of all iterates is nonwhere dense in C([0, 1], [0, 1]),Trans. Amer. Math.Soc., 333 (1992), 787–798.

[3] A. M. Bruckner and T. Hu, Equiconitinuity of iterates of an interval map, Tamkang, J.Math., 21 (1990), 287–294.

[4] J. Cano, Common fixed points for a class of commuting mappings on an interval, Trans.Amer. Math. Soc., 86 (1982), 336–338.

[5] R.B. Gu, Equicontinuity of maps on Warsaw circle, Journal of Mathematical Study, 35(2002), 249–256.

[6] J. Mai, The sturcture of equicontinuous maps, Trans. Amer. Math. Soc., 355 (2003), 4127–4136.

[7] Sam B. Nadler, Jr., Continuum Theory: An Introduction, Pure and Applied Mathematics,158, Marcel Dekker Inc., New York, 1992.

[8] T. Sun, Equicontinuity of σ-maps, Pure and Applied Math., 16 (2000), 9–14. (in Chinese)

[9] A. Valaristos, Equiconitinuity of iterates of circle maps, Internat. J. Math. and Math. Sci.,3(1998), 453–458.

[10] J. Xiong, X. Ye, Z. Zhang and J. Huang, Some dynamical properties for continuous mapson Warsaw circle, Acta Math. Sinica, 3 (1996), 294–299. (in Chinese)



On mixed type Riemann-Liouville and Hadamard fractional

integral inequalities

Weerawat Sudsutad, 1 S.K. Ntouyas 2,3 and Jessada Tariboon 1

1Nonlinear 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] (W. Sudsutad), [email protected] (J. Tariboon)2Department 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 Arabia

Abstract

In this paper, some new mixed type Riemann-Liouville and Hadamard fractional integral in-equalities are established, in the case where the functions are bounded by integrable functions.Moreover, mixed type Riemann-Liouville and Hadamard fractional integral inequalities of Cheby-shev type are presented.

Key words and phrases: Fractional integral; fractional integral inequalities; Riemann-Liouville frac-tional integral; Hadamard fractional integral; Chebyshev inequalities.AMS (MOS) Subject Classifications: 26D10; 26A33.

1 Introduction

The study of mathematical inequalities play very important role in classical differential and integralequations which has applications in many fields. Fractional inequalities are important in studying theexistence, uniqueness and other properties of fractional differential equations. Recently many authorshave studied integral inequalities on fractional calculus using Riemann-Liouville and Caputo derivative,see [1], [2], [3], [4], [5], [6] and the references therein.

Another kind of fractional derivative that appears in the literature is the fractional derivative dueto Hadamard introduced in 1892 [7], which differs from the Riemann-Liouville and Caputo derivativesin the sense that the kernel of the integral contains logarithmic function of arbitrary exponent. Detailsand properties of Hadamard fractional derivative and integral can be found in [8, 9, 10, 11, 12, 13].Recently in the literature, were appeared some results on fractional integral inequalities using Hadamardfractional integral; see [14, 15, 16].

Recently, we have been established some new Riemann-Liouville fractional integral inequalities in[17], and some fractional integral inequalities via Hadamard’s fractional integral in [18]. In the presentpaper we combine the results of [17] and [18] and obtain some new mixed type Riemann-Liouville andHadamard fractional integral inequalities. In Section 3, we consider the case where the functions arebounded by integrable functions and are not necessary increasing or decreasing as are the synchronousfunctions. In Section 4, we establish mixed type Riemann-Liouville and Hadamard fractional integralinequalities of Chebyshev type, concerning the integral of the product of two functions and the productof two integrals. As applications, in Section 5, we present a way for constructing the four boundingfunctions, and use them to give some estimates of Chebyshev type inequalities of Riemann-Liouvilleand Hadamard fractional integrals for two unknown functions.


299 Weerawat Sudsutad et al 299-314

W. SUDSUTAD, S. K. NTOUYAS AND J. TARIBOON

2 Preliminaries

In this section, we give some preliminaries and basic properties used in our subsequent discussion. Thenecessary background details are given in the book by Kilbas et al. [8].

Definition 2.1 The Riemann-Liouville fractional integral of order α > 0 of a continuous functionf : (a,∞) → R is defined by

Iαa f(t) =

1Γ(α)

∫ t

a

(t − s)α−1f(s)ds,

provided the right-hand side is point-wise defined on (a,∞), where Γ is the gamma function.

Definition 2.2 The Hadamard fractional integral of order α ∈ R+ of a function f(t), for all 0 < a <t < ∞, is defined as

Jαa f(t) =

1Γ(α)

∫ t

a

(log

t

s

)α−1

f(s)ds

s,

provided the integral exists.

From Definitions (2.1) and (2.2), we derive the following properties:

Iαa Iβ

a f(t) = Iα+βa f(t) = Iβ

a Iαa f(t),

Jαa Jβ

a f(t) = Jα+βa f(t) = Jβ

a Jαa f(t),

for α, β > 0 and

Iαa (tγ) =

Γ(γ + 1)Γ(γ + α + 1)

(t − a)γ+α,

Jαa (log t)γ =

Γ(γ + 1)Γ(γ + α + 1)

(log

t

a

)γ+α

,

for α > 0, γ > −1, t > a > 0.

3 Inequalities Involving Mixed Type of Riemann-Liouville andHadamard Fractional Integral for Bounded Functions

In this section we obtain some new inequalities of mixed type for Riemann-Liouville and Hadamardfractional integral in the case where the functions are bounded by integrable functions and are notnecessary increasing or decreasing as are the synchronous functions.

Theorem 3.1 Let f be an integrable function on [a,∞), a > 0. Assume that:

(H1) There exist two integrable functions ϕ1, ϕ2 on [a,∞) such that

ϕ1(t) ≤ f(t) ≤ ϕ2(t), for all t ∈ [a,∞), a > 0. (1)

Then for 0 < a < t < ∞ and α, β > 0, the following two inequalities hold:

(A1) Jαa ϕ2(t)Iβ

a f(t) + Jαa f(t)Iβ

a ϕ1(t) ≥ Jαa ϕ2(t)Iβ

a ϕ1(t) + Jαa f(t)Iβ

a f(t),

(B1) Iαa ϕ2(t)Jβ

a f(t) + Iαa f(t)Jβ

a ϕ1(t) ≥ Iαa ϕ2(t)Jβ

a ϕ1(t) + Iαa f(t)Jβ

a f(t).



RIEMANN-LIOUVILLE & HADAMARD FRACTIONAL INTEGRAL INEQUALITIES

Proof. From condition (H1), for all τ, ρ > a, we have

(ϕ2(τ) − f(τ))(f(ρ) − ϕ1(ρ)) ≥ 0,

which impliesϕ2(τ)f(ρ) + ϕ1(ρ)f(τ) ≥ ϕ1(ρ)ϕ2(τ) + f(τ)f(ρ). (2)

Multiplying both sides of (2) by (log(t/τ))α−1/τΓ(α), τ ∈ (a, t), we get

f(ρ)(log(t/τ))α−1

τΓ(α)ϕ2(τ) + ϕ1(ρ)

(log(t/τ))α−1

τΓ(α)f(τ)

≥ ϕ1(ρ)(log(t/τ))α−1

τΓ(α)ϕ2(τ) + f(ρ)

(log(t/τ))α−1

τΓ(α)f(τ). (3)

Integrating both sides of (3) with respect to τ on (a, t), we obtain

f(ρ)1

Γ(α)

∫ t

a

(log

t

τ

)α−1

ϕ2(τ)dτ

τ+ ϕ1(ρ)

1Γ(α)

∫ t

a

(log

t

τ

)α−1

f(τ)dτ

τ

≥ ϕ1(ρ)1

Γ(α)

∫ t

a

(log

t

τ

)α−1

ϕ2(τ)dτ

τ+ f(ρ)

1Γ(α)

∫ t

a

(log

t

τ

)α−1

f(τ)dτ

τ,

which yieldsf(ρ) Jα

a ϕ2(t) + ϕ1(ρ) Jαa f(t) ≥ ϕ1(ρ) Jα

a ϕ2(t) + f(ρ) Jαa f(t). (4)

Multiplying both sides of (4) by (t − ρ)β−1/Γ(β), ρ ∈ (a, t), we have

Jαa ϕ2(t)

(t − ρ)β−1

Γ(β)f(ρ) + Jα

a f(t)(t − ρ)β−1

Γ(β)ϕ1(ρ)

≥ Jαa ϕ2(t)

(t − ρ)β−1

Γ(β)ϕ1(ρ) + Jα

a f(t)(t − ρ)β−1

Γ(β)f(ρ). (5)

Integrating both sides of (5) with respect to ρ on (a, t), we get

Jαa ϕ2(t)

1Γ(β)

∫ t

a

(t − ρ)β−1f(ρ)dρ + Jαa f(t)

1Γ(β)

∫ t

a

(t − ρ)β−1ϕ1(ρ)dρ

≥ Jαa ϕ2(t)

1Γ(β)

∫ t

a

(t − ρ)β−1 ϕ1(ρ)dρ + Jαa f(t)

1Γ(β)

∫ t

a

(t − ρ)β−1 f(ρ)dρ. (6)

Hence, we get the desired inequality in (A1). The inequality (B1), is proved by similar arguments. ¤

Corollary 3.2 Let f be an integrable function on [a,∞), a > 0 satisfying m ≤ f(t) ≤ M, for allt ∈ [a,∞) and m,M ∈ R. Then for 0 < a < t < ∞ and α, β > 0, the following two inequalities hold:

(A2) M(log t

a )α

Γ(α + 1)Iβa f(t) + m

(t − a)β

Γ(β + 1)Jα

a f(t) ≥ mM(log t

a )α(t − a)β

Γ(α + 1)Γ(β + 1)+ Jα

a f(t)Iβa f(t),

(B2) M(t − a)α

Γ(α + 1)Jβ

a f(t) + m(log t

a )β

Γ(β + 1)Iαa f(t) ≥ mM

(t − a)α(log ta )β

Γ(α + 1)Γ(β + 1)+ Iα

a f(t)Jβa f(t).

Theorem 3.3 Let f be an integrable function on [a,∞), a > 0 and θ1, θ2 > 0 satisfying 1/θ1+1/θ2 = 1.In addition, suppose that the condition (H1) holds. Then for 0 < a < t < ∞ and α, β > 0, the followingtwo inequalities hold:




(A3) Jαa ϕ2(t)Iβ

a ϕ1(t) + Jαa f(t)Iβ

a f(t) +1θ1

(t − a)β

Γ(β + 1)Jα

a (ϕ2 − f)θ1(t) +1θ2

(log ta )α

Γ(α + 1)Iβa (f − ϕ1)θ2(t)

≥ Jαa ϕ2(t)Iβ

a f(t) + Jαa f(t)Iβ

a ϕ1(t),

(B3) Iαa ϕ2(t)Jβ

a ϕ1(t), +Iαa f(t)Jβ

a f(t) +1θ1

(log ta )β

Γ(β + 1)Iαa (ϕ2 − f)θ1(t) +

1θ2

(t − a)α

Γ(α + 1)Jβ

a (f − ϕ1)θ2(t)

≥ Iαa ϕ2(t)Jβ

a f(t) + Iαa f(t)Jβ

a ϕ1(t).

Proof. Firstly, we recall the well-known Young’s inequality as

1θ1

xθ1 +1θ2

yθ2 ≥ xy, ∀x, y ≥ 0, θ1, θ2 > 0,

where 1/θ1 + 1/θ2 = 1. By setting x = ϕ2(τ) − f(τ) and y = f(ρ) − ϕ1(ρ), τ, ρ > a, we have

1θ1

(ϕ2(τ) − f(τ))θ1 +1θ2

(f(ρ) − ϕ1(ρ))θ2 ≥ (ϕ2(τ) − f(τ))(f(ρ) − ϕ1(ρ)). (7)

Multiplying both sides of (7) by (log(t/τ))α−1(t − ρ)β−1/τΓ(α)Γ(β), τ, ρ ∈ (a, t), we get

1θ1

(log t/τ)α−1(t − ρ)β−1

τΓ(α)Γ(β)(ϕ2(τ) − f(τ))θ1 +

1θ2

(log t/τ)α−1(t − ρ)β−1

τΓ(α)Γ(β)(f(ρ) − ϕ1(ρ))θ2

≥ (log t/τ)α−1

τΓ(α)(ϕ2(τ) − f(τ))

(t − ρ)β−1

Γ(β)(f(ρ) − ϕ1(ρ)).

Double integrating the above inequality with respect to τ and ρ from a to t, we have

1θ1

Jαa (ϕ2 − f)θ1(t)Iβ

a (1)(t) +1θ2

Jαa (1)(t)Iβ

a (f − ϕ1)θ2(t) ≥ Jαa (ϕ2 − f)(t)Iβ

a (f − ϕ1)(t),

which implies the result in (A3). By using the similar method, we obtain the inequality in (B3). ¤

Corollary 3.4 Let f be an integrable function on [a,∞), a > 0 satisfying m ≤ f(t) ≤ M, θ1 = θ2 = 2for all t ∈ [a,∞) and m,M ∈ R. Then for 0 < a < t < ∞ and α, β > 0, the following two inequalitieshold:

(A4) (m + M)2(log t

a )α(t − a)β

Γ(α + 1)Γ(β + 1)+

(t − a)β

Γ(β + 1)Jα

a f2(t) +(log t

a )α

Γ(α + 1)Iβa f2(t) + 2Jα

a f(t)Iβa f(t)

≥ 2(m + M)(

(log ta )α

Γ(α + 1)Iβa f(t) +

(t − a)β

Γ(β + 1)Jα

a f(t))

,

(B4) (m + M)2(t − a)α(log t

a )β

Γ(α + 1)Γ(β + 1)+

(t − a)α

Γ(α + 1)Jβ

a f2(t) +(log t

a )β

Γ(β + 1)Iαa f2(t) + 2Jβ

a f(t)Iαa f(t)

≥ 2(m + M)

((log t

a )β

Γ(β + 1)Iαa f(t) +

(t − a)α

Γ(α + 1)Jβ

a f(t)

).

Theorem 3.5 Let f be an integrable function on [a,∞), a > 0 and θ1, θ2 > 0 satisfying θ1 + θ2 = 1. Inaddition, suppose that the condition (H1) holds. Then for 0 < a < t < ∞, and α, β > 0, the followingtwo inequalities hold:

(A5) θ1(t − a)β

Γ(β + 1)Jα

a ϕ2(t) + θ2

(log ta )α

Γ(α + 1)Iβa f(t)

≥ Jαa (ϕ2 − f)θ1(t)Iβ

a (f − ϕ1)θ2(t) + θ1(t − a)β

Γ(β + 1)Jα

a f(t) + θ2

(log ta )α

Γ(α + 1)Iβa ϕ1(t),




(B5) θ1

(log ta )β

Γ(β + 1)Iαa ϕ2(t) + θ2

(t − a)α

Γ(α + 1)Jβ

a f(t)

≥ Iαa (ϕ2 − f)θ1(t)Jβ

a (f − ϕ1)θ2(t) + θ1

(log ta )β

Γ(β + 1)Iαa f(t) + θ2

(t − a)α

Γ(α + 1)Jβ

a ϕ1(t).

Proof. From the well-known weighted AM-GM inequality

θ1x + θ2y ≥ xθ1yθ2 , ∀x, y ≥ 0, θ1, θ2 > 0,

where θ1 + θ2 = 1, and setting x = ϕ2(τ) − f(τ) and y = f(ρ) − ϕ1(ρ), τ, ρ > a, we have

θ1(ϕ2(τ) − f(τ)) + θ2(f(ρ) − ϕ1(ρ)) ≥ (ϕ2(τ) − f(τ))θ1(f(ρ) − ϕ1(ρ))θ2 . (8)

Multiplying both sides of (8) by (log(t/τ))α−1(t − ρ)β−1/τΓ(α)Γ(β), τ, ρ ∈ (a, t), we obtain

θ1(log(t/τ))α−1(t − ρ)β−1

τΓ(α)Γ(β)(ϕ2(τ) − f(τ)) + θ2

(log(t/τ))α−1(t − ρ)β−1

τΓ(α)Γ(β)(f(ρ) − ϕ1(ρ))

≥ (log t/τ)α−1

τΓ(α)(ϕ2(τ) − f(τ))θ1

(t − ρ)β−1

Γ(β)(f(ρ) − ϕ1(ρ))θ2 .

Double integration the above inequality with respect to τ and ρ from a to t, we have

θ1Jαa (ϕ2 − f)(t)Iβ

a (1)(t) + θ2Jαa (1)(t)Iβ

a (f − ϕ1)(t) ≥ Jαa (ϕ2 − f)θ1(t)Iβ

a (f − ϕ1)θ2(t).

Therefore, we deduce the inequality in (A5). By using the similar method, we obtain the desired boundin (B5). ¤

Corollary 3.6 Let f be an integrable function on [a,∞), a > 0 satisfying m ≤ f(t) ≤ M , θ1 = θ2 = 1/2for all 0 < a < t < ∞ and m,M ∈ R. Then for 0 < a < t < ∞ and α, β > 0, the following twoinequalities hold:

(A6) M(log t

a )α(t − a)β

Γ(α + 1)Γ(β + 1)+

(log ta )α

Γ(α + 1)Iβa f(t)

≥ m(log t

a )α(t − a)β

Γ(α + 1)Γ(β + 1)+

(t − a)β

Γ(β + 1)Jα

a f(t) + 2Jαa (M − f)1/2(t)Iβ

a (f − m)1/2(t),

(B6) M(t − a)α(log t

a )β

Γ(α + 1)Γ(β + 1)+

(t − a)α

Γ(α + 1)Jβ

a f(t)

≥ m(t − a)α(log t

a )β

Γ(α + 1)Γ(β + 1)+

(log ta )β

Γ(β + 1)Iαa f(t) + 2Iα

a (M − f)1/2(t)Jβa (f − m)1/2(t).

Lemma 3.7 [19] Assume that a ≥ 0, p ≥ q ≥ 0, and p 6= 0. Then, we have

aq/p ≤(

q

pk(q−p)/pa +

p − q

pkq/p

).

Theorem 3.8 Let f be an integrable function on [a,∞), a > 0 and constants p ≥ q ≥ 0, p 6= 0. Inaddition, assume that the condition (H1) holds. Then for any k > 0, 0 < a < t < ∞, α > 0, thefollowing two inequalities hold:

(A7) Jαa (ϕ2 − f)q/p(t)Iα

a (f − ϕ1)q/p(t) +q

pk(q−p)/p (Jα

a ϕ2(t)Iαa ϕ1(t) + Jα

a f(t)Iαa f(t))

≤ q

pk(q−p)/p (Jα

a ϕ2(t)Iαa f(t) + Jα

a f(t)Iαa ϕ1(t)) +

p − q

pkq/p (t − a)α(log t

a )α

Γ2(α + 1),




(B7) Iαa (ϕ2 − f)q/p(t)Jα

a (f − ϕ1)q/p(t) +q

pk(q−p)/p (Iα

a ϕ2(t)Jαa ϕ1(t) + Iα

a f(t)Jαa f(t))

≤ q

pk(q−p)/p (Iα

a ϕ2(t)Jαa f(t) + Iα

a f(t)Jαa ϕ1(t)) +

p − q


a )α

Γ2(α + 1).

Proof. From condition (H1) and Lemma 3.7, for p ≥ q ≥ 0, p 6= 0, it follows that

((ϕ2(τ) − f(τ))(f(ρ) − ϕ1(ρ)))q/p ≤ q

pk(q−p)/p(ϕ2(τ) − f(τ))(f(ρ) − ϕ1(ρ)) +

p − q

pkq/p, (9)

for any k > 0. Multiplying both sides of (9) by (log(t/τ))α−1/τΓ(α), τ ∈ (a, t), and integrating theresulting identity with respect to τ from a to t, one has

(f(ρ) − ϕ1(ρ))q/pJαa (ϕ2 − f)q/p(t)

≤ q

pk(q−p)/p(f(ρ) − ϕ1(ρ))Jα

a (ϕ2 − f)(t) +p − q

pkq/p (log t

a )α

Γ(α + 1). (10)

Multiplying both sides of (10) by (t−ρ)α−1/Γ(α), ρ ∈ (a, t), and integrating the resulting identity withrespect to ρ from a to t, we obtain

Jαa (ϕ2 − f)q/p(t)Iα

a (f − ϕ1)(t)q/p

≤ q

pk(q−p)/pJα

a (ϕ2 − f)(t)Iαa (f − ϕ1)(t) +

p − q


a )α

Γ2(α + 1),

which leads to inequality in (A6). Using the similar arguments, we get the required inequality in (B6).¤

Corollary 3.9 Let f be an integrable function on [a,∞), a > 0 satisfying m ≤ f(t) ≤ M for allt ∈ [a,∞), constants q = 1, p = 2, k = 1 and m, M ∈ R. Then for 0 < a < t < ∞ and α > 0, thefollowing two inequalities hold:

(A8) 2Jαa (M − f)1/2(t)Iα

a (f − m)1/2(t) + Jαa f(t)Iα

a f(t)

≤M(log t

a )α

Γ(α + 1)Iαa f(t) +

m(t − a)α

Γ(α + 1)Jα

a f(t) + (1 − mM)(t − a)α(log t

a )α

Γ2(α + 1),

(B8) 2Iαa (M − f)1/2(t)Jα

a (f − m)1/2(t) + Iαa f(t)Jα

a f(t)

≤ M(t − a)α

Γ(α + 1)Jα

a f(t) +m(log t

a )α

Γ(α + 1)Iαa f(t) + (1 − mM)

(t − a)α(log ta )α

Γ2(α + 1).

4 Chebyshev Type Inequalities for Riemann-Liouville and HadamardFractional Integrals

In this section, we establish our main fractional integral inequalities of Chebyshev type, concerning theintegral of the product of two functions and the product of two integrals, with the help of the followinglemma.

Lemma 4.1 Let f be an integrable function on [a,∞), a > 0 and ϕ1, ϕ2 are two integrable functionson [a,∞). Assume that the condition (H1) holds. Then for 0 < a < t < ∞, and α, β > 0, the followingtwo equalities hold:

(A9)(t − a)α

Γ(α + 1)Jα

a f2(t) +(log t

a )α

Γ(α + 1)Iαa f2(t) − 2Jα

a f(t)Iαa f(t)

= Jαa (f − ϕ1)(t)Iα

a (ϕ2 − f)(t) + Jαa (ϕ2 − f)(t)Iα

a (f − ϕ1)(t)




+(t − a)α

Γ(α + 1)(Jα

a (ϕ1f + ϕ2f − ϕ1ϕ2)(t) − Jαa ((ϕ2 − f)(f − ϕ1))(t))

+(log t

a )α

Γ(α + 1)(Iα

a (ϕ1f + ϕ2f − ϕ1ϕ2)(t) − Iαa ((ϕ2 − f)(f − ϕ1))(t))

+ Jαa ϕ1(t)Iα

a (ϕ2 − f)(t) + Jαa ϕ2(t)Iα

a (ϕ1 − f)(t) − Jαa f(t)Iα

a (ϕ1 + ϕ2)(t),

(B9)(t − a)β

Γ(β + 1)Jα

a f2(t) +(log t

a )α

Γ(α + 1)Iβa f2(t) − 2Jα

a f(t)Iβa f(t)

= Jαa (f − ϕ1)(t)Iβ

a (ϕ2 − f)(t) + Jαa (ϕ2 − f)(t)Iβ

a (f − ϕ1)(t)

+(t − a)β

Γ(β + 1)(Jα

a (ϕ1f + ϕ2f − ϕ1ϕ2)(t) − Jαa ((ϕ2 − f)(f − ϕ1))(t))

+(log t

a )α

Γ(α + 1)(Iβa (ϕ1f + ϕ2f − ϕ1ϕ2)(t) − Iβ

a ((ϕ2 − f)(f − ϕ1))(t))

+ Jαa ϕ1(t)Iβ

a (ϕ2 − f)(t) + Jαa ϕ2(t)Iβ

a (ϕ1 − f)(t) − Jαa f(t)Iβ

a (ϕ1 + ϕ2)(t).

Proof. For any 0 < a < τ, ρ < t < ∞, we have

(ϕ2(ρ) − f(ρ))(f(τ) − ϕ1(τ)) + (ϕ2(τ) − f(τ))(f(ρ) − ϕ1(ρ))− (ϕ2(τ) − f(τ))(f(τ) − ϕ1(τ)) − (ϕ2(ρ) − f(ρ))(f(ρ) − ϕ1(ρ))

= f2(τ) + f2(ρ) − 2f(τ)f(ρ) + ϕ2(ρ)f(τ) + ϕ1(τ)f(ρ) − ϕ1(τ)ϕ2(ρ) (11)+ ϕ2(τ)f(ρ) + ϕ1(ρ)f(τ) − ϕ1(ρ)ϕ2(τ) − ϕ2(τ)f(τ) + ϕ1(τ)ϕ2(τ)− ϕ1(τ)f(τ) − ϕ2(ρ)f(ρ) + ϕ1(ρ)ϕ2(ρ) − ϕ1(ρ)f(ρ).

Multiplying (11) by (log(t/τ))α−1/τΓ(α), τ ∈ (a, t), 0 < a < t < ∞, and integrating the resultingidentity with respect to τ from a to t, we get

(ϕ2(ρ) − f(ρ))(Jαa f(t) − Jα

a ϕ1(t)) + (Jαa ϕ2(t) − Jα

a f(t))(f(ρ) − ϕ1(ρ))

− Jαa ((ϕ2 − f)(f − ϕ1))(t) − (ϕ2(ρ) − f(ρ))(f(ρ) − ϕ1(ρ))

(log ta )α

Γ(α + 1)

= Jαa f2(t) + f2(ρ)

(log ta )α

Γ(α + 1)− 2f(ρ)HIα

a f(t) + ϕ2(ρ)HIαa f(t) + f(ρ)Jα

a ϕ1(t) (12)

− ϕ2(ρ)Jαa ϕ1(t) + f(ρ)Jα

a ϕ2(t) + ϕ1(ρ)Jαa f(t) − ϕ1(ρ)Jα

a ϕ2(t)

− Jαa ϕ2f(t) + Jα

a ϕ1ϕ2(t) − Jαa ϕ1f(t) − ϕ2(ρ)f(ρ)

(log ta )α

Γ(α + 1)

+ ϕ1(ρ)ϕ2(ρ)(log t

a )α

Γ(α + 1)− ϕ1(ρ)f(ρ)

(log ta )α

Γ(α + 1).

Multiplying (12) by (t − ρ)α−1/Γ(α), ρ ∈ (a, t), 0 < a < t < ∞, and integrating the resulting identitywith respect to ρ from a to t, we have

(Jαa f(t) − Jα

a ϕ1(t))(Iαa ϕ2(t) − Iα

a f(t)) + (Jαa ϕ2(t) − Jα

a f(t))(Iαa f(t) − Iα

a ϕ1(t))

− Jαa ((ϕ2 − f)(f − ϕ1))(t)

(t − a)α

Γ(α + 1)− Iα

a ((ϕ2 − f)(f − ϕ1))(t)(log t

a )α

Γ(α + 1)

= Jαa f2(t)

(t − a)α

Γ(α + 1)+ Iα

a f2(t)(log t

a )α

Γ(α + 1)− 2Jα

a f(t)Iαa f(t)

+ Jαa f(t)Iα

a ϕ2(t) + Jαa ϕ1(t)Iα

a f(t) − Jαa ϕ1(t)Iα

a ϕ2(t)+ Jα

a ϕ2(t)Iαa f(t) + Jα

a f(t)Iαa ϕ1(t) − Jα

a ϕ2(t)Iαa ϕ1(t)

− Jαa ϕ2f(t)

(t − a)α

Γ(α + 1)+ Jα

a ϕ1ϕ2(t)(t − a)α

Γ(α + 1)− Jα

a ϕ1f(t)(t − a)α

Γ(α + 1)




− Iαa ϕ2f(t)

(log ta )α

Γ(α + 1)+ Iα

a ϕ1ϕ2(t)(log t

a )α

Γ(α + 1)− Iα

a ϕ1f(t)(log t

a )α

Γ(α + 1).

Therefore, the desired equality (A9) is proved. The equality (B9) is derived by using the similararguments. ¤

Let now g be an integrable function on [a,∞), a > 0 satisfying the assumption:

(H2) There exist ψ1 and ψ2 integrable functions on [a,∞) such that

ψ1(t) ≤ g(t) ≤ ψ2(t) for 0 < a < t < ∞.

Theorem 4.2 Let f and g be two integrable functions on [a,∞), a > 0 and ϕ1, ϕ2, ψ1 and ψ2 arefour integrable functions on [a,∞) satisfying the conditions (H1) and (H2) on [a,∞). Then for all0 < a < t < ∞ and α > 0, the following inequality holds:∣∣∣ (t − a)α

Γ(α + 1)Jα

a fg(t) +(log t

a )α

Γ(α + 1)Iαa fg(t) − Jα

a f(t)Iαa g(t) − Iα

a f(t)Jαa g(t)

∣∣∣≤ |K(f, ϕ1, ϕ2)|1/2 |K(g, ψ1, ψ2)|1/2

. (13)

where K(u, v, w) is defined by

K(u, v, w) =(t − a)α

Γ(α + 1)Jα

a (uw + uv − vw)(t) +(log t

a )α

Γ(α + 1)Iαa (uw + uv − vw)(t) − 2Jα

a u(t)Iαa u(t).

Proof. Let f and g be two integrable functions defined on [a,∞) satisfying (H1) and (H2), respectively.We define a function H for 0 < a < t < ∞ as follows

H(τ, ρ) := (f(τ) − f(ρ))(g(τ) − g(ρ)), τ, ρ ∈ (a, t). (14)

Multiplying both sides of (14) by (log(t/τ))α−1(t − ρ)α−1/τΓ2(α), τ, ρ ∈ (a, t), and double integratingthe resulting identity with respect to τ and ρ from a to t, we have

1Γ2(α)

∫ t

a

∫ t

a

(log

t

τ

)α−1

(t − ρ)α−1H(τ, ρ)dρ

dτ

τ

=(t − a)α

Γ(α + 1)Jα

a fg(t) +(log t

a )α



a f(t)Jαa g(t). (15)

Applying the Cauchy-Schwarz inequality to (15), we have((t − a)α

Γ(α + 1)Jα

a fg(t) +(log t

a )α



a f(t)Jαa g(t)

)2

≤

(1

Γ2(α)

∫ t

a

∫ t

a

(log

t

τ

)α−1

(t − ρ)α−1 (f(τ) − f(ρ))2dρdτ

τ

)

×

(1

Γ2(α)

∫ t

a

∫ t

a

(log

t

τ

)α−1

(t − ρ)α−1 (g(τ) − g(ρ))2dρdτ

τ

)

=

((t − a)α

Γ(α + 1)Jα

a f2(t) +(log t

a )α

Γ(α + 1)Iαa f2(t) − 2Jα

a f(t)Iαa f(t)

)

×

((t − a)α

Γ(α + 1)Jα

a g2(t) +(log t

a )α

Γ(α + 1)Iαa g2(t) − 2Jα

a g(t)Iαa g(t)

). (16)




Since (ϕ2(t) − f(t))(f(t) − ϕ1(t)) ≥ 0 and (ψ2(t) − f(t))(f(t) − ψ1(t)) ≥ 0 for t ∈ [a,∞), we get

(t − a)α

Γ(α + 1)Jα

a ((ϕ2 − f)(f − ϕ1))(t) ≥ 0,

(log ta )α

Γ(α + 1)Iαa ((ϕ2 − f)(f − ϕ1))(t) ≥ 0,

(t − a)α

Γ(α + 1)Jα

a ((ψ2 − g)(g − ψ1))(t) ≥ 0,

(log ta )α

Γ(α + 1)Iαa ((ψ2 − g)(g − ψ1))(t) ≥ 0.

Thus, from Lemma 4.1, we obtain

(t − a)α

Γ(α + 1)Jα

a f2(t) +(log t

a )α

Γ(α + 1)Iαa f2(t) − 2Jα

a f(t)Iαa f(t)

≤Jαa (f − ϕ1)(t)Iα

a (ϕ2 − f)(t) + Jαa (ϕ2 − f)(t)Iα

a (f − ϕ1)(t)

+(t − a)α

Γ(α + 1)Jα

a (ϕ2f + ϕ1f − ϕ1ϕ2)(t) +(log t

a )α

Γ(α + 1)Iαa (ϕ2f + ϕ1f − ϕ1ϕ2)(t) (17)

+ Jαa ϕ1(t)Iα

a (ϕ2 − f)(t) + Jαa ϕ2(t)Iα

a (ϕ1 − f)(t) − Jαa f(t)Iα

a (ϕ1 + ϕ2)(t)

=(t − a)α

Γ(α + 1)Jα

a (ϕ2f + ϕ1f − ϕ1ϕ2)(t) +(log t

a )α

Γ(α + 1)Iαa (ϕ2f + ϕ1f − ϕ1ϕ2)(t) − 2Jα

a f(t)Iαa f(t)

= K(f, ϕ1, ϕ2),

and

(t − a)α

Γ(α + 1)Jα

a g2(t) +(log t

a )α

Γ(α + 1)Iαa g2(t) − 2Jα

a g(t)Iαa g(t)

≤Jαa (g − ψ1)(t)Iα

a (ψ2 − g)(t) + Jαa (ψ2 − g)(t)Iα

a (g − ψ1)(t)

+(t − a)α

Γ(α + 1)Jα

a (ψ2g + ψ1g − ψ1ψ2)(t) +(log t

a )α

Γ(α + 1)Iαa (ψ2g + ψ1g − ψ1ψ2)(t) (18)

+ Jαa ψ1(t)Iα

a (ψ2 − g)(t) + Jαa ψ2(t)Iα

a (ψ1 − g)(t) − Jαa g(t)Iα

a (ψ1 + ψ2)(t),

=(t − a)α

Γ(α + 1)Jα

a (ϕ2g + ϕ1g − ϕ1ϕ2)(t) +(log t

a )α

Γ(α + 1)Iαa (ϕ2g + ϕ1g − ϕ1ϕ2)(t) − 2Jα

a g(t)Iαa g(t)

= K(g, ψ1, ψ2).

From (16), (17) and (18), the required inequality in (13) is proved. ¤

Corollary 4.3 If K(f, ϕ1, ϕ2) = K(f,m,M) and K(g, ψ1, ψ2) = K(g, p, P ), m,M, p, P ∈ R, theninequality (13) reduces to the following fractional integral inequality:∣∣∣∣ (t − a)α

Γ(α + 1)Jα

a fg(t) +(log t

a )α



a f(t)Jαa g(t)

∣∣∣∣≤1

4

[ (Jα

a f(t) − Iαa f(t) + M

(t − a)α

Γ(α + 1)− m

(log ta )α

Γ(α + 1)

)2

+(

Iαa f(t) − Jα

a f(t) + M(log t

a )α

Γ(α + 1)− m

(t − a)α

Γ(α + 1)

)2]1/2

×

[ (Jα

a g(t) − Iαa g(t) + P

(t − a)α

Γ(α + 1)− p

(log ta )α

Γ(α + 1)

)2




+(

Jαa g(t) − Iα

a g(t) + p(t − a)α

Γ(α + 1)− P

(log ta )α

Γ(α + 1)

)2]1/2

.

Theorem 4.4 Let f and g be two integrable function on [a,∞), a > 0. Assume that there exist fourintegrable functions ϕ1, ϕ2, ψ1 and ψ2 satisfying the conditions (H1) and (H2) on [a,∞). Then for all0 < a < t < ∞ and α, β > 0, the following inequality holds:∣∣∣ (t − a)β

Γ(β + 1)Jα

a fg(t) +(log t

a )α

Γ(α + 1)Iβa fg(t) − Jα

a f(t)Iβa g(t) − Iβ

a f(t)Jαa g(t)

∣∣∣≤ |K1(f, ϕ1, ϕ2)|1/2 |K1(g, ψ1, ψ2)|1/2

, (19)

where K1(u, v, w) is defined by

K1(u, v, w) =(t − a)β

Γ(β + 1)Jα

a (uw + uv − vw)(t) +(log t

a )α

Γ(α + 1)Iβa (uw + uv − vw)(t) − 2Jα

a u(t)Iβa u(t).

Proof. Multiplying both sides of (14) by (log(t/τ))α−1(t − ρ)β−1/τΓ(α)Γ(β), τ, ρ ∈ (a, t), and doubleintegrating with respect to τ and ρ from a to t we get

1Γ(α)Γ(β)

∫ t

a

∫ t

a

(log

t

τ

)α−1

(t − ρ)β−1H(τ, ρ)dρ

dτ

τ

=(t − a)β

Γ(α + 1)Jα

a fg(t) +(log t

a )α



a f(t)Jαa g(t). (20)

By using the Cauchy-Schwarz inequality for double integrals, we have∣∣∣∣ (t − a)α

Γ(α + 1)Jα

a fg(t) +(log t

a )α



a f(t)Jαa g(t)

∣∣∣∣≤

[1

Γ(α)Γ(β)

∫ t

a

∫ t

a

(log

t

τ

)α−1

(t − ρ)β−1f2(τ)dρ

dτ

τ

+1

Γ(α)Γ(β)

∫ t

a

∫ t

a

(log

t

τ

)α−1

(t − ρ)β−1f2(ρ)dρ

dτ

τ

− 2Γ(α)Γ(β)

∫ t

a

∫ t

a

(log

t

τ

)α−1

(t − ρ)β−1f(τ)f(ρ)dρ

dτ

τ

]1/2

×

[1

Γ(α)Γ(β)

∫ t

a

∫ t

a

(log

t

τ

)α−1

(t − ρ)β−1g2(τ)dρ

dτ

τ

+1

Γ(α)Γ(β)

∫ t

a

∫ t

a

(log

t

τ

)α−1

(t − ρ)β−1g2(ρ)dρ

dτ

τ

− 2Γ(α)Γ(β)

∫ t

a

∫ t

a

(log

t

τ

)α−1

(t − ρ)β−1g(τ)g(ρ)dρ

dτ

τ

]1/2

.

Therefore, we get∣∣∣∣ (t − a)α

Γ(α + 1)Jα

a fg(t) +(log t

a )α



a f(t)Jαa g(t)

∣∣∣∣≤

[(t − a)β

Γ(β + 1)Jα

a f2(t) +

(log t

τ

)α

Γ(α + 1)Iβa f2(t) − 2Jα

a f(t)Iβa f(t)

]1/2

×

[(t − a)β

Γ(β + 1)Jα

a g2(t) +

(log t

τ

)α

Γ(α + 1)Iβa g2(t) − 2Jα

a g(t)Iβa g(t)

]1/2

.




Thus, from Lemma 4.1, we get

(t − a)β

Γ(β + 1)Jα

a f2(t) +(log t

a )α

Γ(α + 1)Iβa f2(t) − 2Jα

a f(t)Iβa f(t)

≤Jαa (f − ϕ1)(t)Iβ

a (ϕ2 − f)(t) + Jαa (ϕ2 − f)(t)Iβ

a (f − ϕ1)(t)

+(t − a)β

Γ(β + 1)Jα

a (ϕ2f + ϕ1f − ϕ1ϕ2)(t) +(log t

a )α

Γ(α + 1)Iβa (ϕ2f + ϕ1f − ϕ1ϕ2)(t) (21)

+ Jαa ϕ1(t)Iβ

a (ϕ2 − f)(t) + Jαa ϕ2(t)Iβ

a (ϕ1 − f)(t) − Jαa f(t)Iβ

a (ϕ1 + ϕ2)(t)

=(t − a)β

Γ(β + 1)Jα

a (ϕ2f + ϕ1f − ϕ1ϕ2)(t) +(log t

a )α

Γ(α + 1)Iβa (ϕ2f + ϕ1f − ϕ1ϕ2)(t) − 2Jα

a f(t)Iβa f(t)

= K1(f, ϕ1, ϕ2),

and

(t − a)β

Γ(β + 1)Jα

a g2(t) +(log t

a )α

Γ(α + 1)Iβa g2(t) − 2Jα

a g(t)Iβa g(t)

≤Jαa (g − ψ1)(t)Iβ

a (ψ2 − g)(t) + Jαa (ψ2 − g)(t)Iβ

a (g − ψ1)(t)

+(t − a)β

Γ(β + 1)Jα

a (ψ2g + ψ1g − ψ1ψ2)(t) +(log t

a )α

Γ(α + 1)Iβa (ψ2g + ψ1g − ψ1ψ2)(t) (22)

+ Jαa ψ1(t)Iβ

a (ψ2 − g)(t) + Jαa ψ2(t)Iβ

a (ψ1 − g)(t) − Jαa g(t)Iβ

a (ψ1 + ψ2)(t),

=(t − a)β

Γ(β + 1)Jα

a (ϕ2g + ϕ1g − ϕ1ϕ2)(t) +(log t

a )α

Γ(α + 1)Iβa (ϕ2g + ϕ1g − ϕ1ϕ2)(t) − 2Jα

a g(t)Iβa g(t)

= K1(g, ψ1, ψ2).

From (15), (21) and (22), we obtain the desired bound in (19). ¤

Corollary 4.5 If K(f, ϕ1, ϕ2) = K(f,m,M) and K(g, ψ1, ψ2) = K(g, p, P ), m,M, p, P ∈ R, theninequality (13) reduces to the following fractional integral inequality:∣∣∣∣ (t − a)β

Γ(β + 1)Jα

a fg(t) +(log t

a )α



a f(t)Jαa g(t)

∣∣∣∣≤1

4

[ (Jα

a f(t) − Iβa f(t) + M

(t − a)β

Γ(β + 1)− m

(log ta )α

Γ(α + 1)

)2

+(

Iβa f(t) − Jα

a f(t) + M(log t

a )α

Γ(α + 1)− m

(t − a)β

Γ(β + 1)

)2]1/2

×

[ (Jα

a g(t) − Iαa g(t) + P

(t − a)β

Γ(β + 1)− p

(log ta )α

Γ(α + 1)

)2

+(

Jαa g(t) − Iβ

a g(t) + p(t − a)β

Γ(β + 1)− P

(log ta )α

Γ(α + 1)

)2]1/2

.

5 Applications

In this section we present a way for constructing the four bounding functions, and use them to givesome estimates of Chebyshev type inequalities of Riemann-Liouville and Hadamard fractional integralsfor two unknown functions.

From the Definitions 2.1 and 2.2, for 0 < a = t0 < t1 < t2 < · · · < tp < tp+1 = T , we define two




notations of sub-integrals for Riemann-Liouville and Hadamard fractional integrals as

Iαtj ,tj+1

f(T ) =1

Γ(α)

∫ tj+1

tj

(T − τ)α−1f(τ)dτ, j = 0, 1, . . . , p. (23)

and

Jαtj ,tj+1

f(T ) =1

Γ(α)

∫ tj+1

tj

(log

T

τ

)α−1

f(τ)dτ

τ, j = 0, 1, . . . , p. (24)

Note that

Iαa f(T ) =

p∑j=0

Iαtj ,tj+1

f(T )

=1

Γ(α)

∫ t1

a

(T − τ)α−1f(τ)dτ +1

Γ(α)

∫ t2

t1

(T − τ)α−1f(τ)dτ

+ · · · + 1Γ(α)

∫ T

tp

(T − τ)α−1f(τ)dτ,

and

Jαa f(T ) =

p∑j=0

Jαtj ,tj+1

f(T )

=1

Γ(α)

∫ t1

a

(log

T

τ

)α−1

f(τ)dτ +1

Γ(α)

∫ t2

t1

(log

T

τ

)α−1

f(τ)dτ

+ · · · + 1Γ(α)

∫ T

tp

(log

T

τ

)α−1

f(τ)dτ.

Let u be a unit step function defined by

u(t) =

1, t > 0,0, t ≤ 0,

(25)

and let ua(t) be the Heaviside unit step function defined by

ua(t) = u(t − a) =

1, t > a,0, t ≤ a.

(26)

Let ϕ1 be a piecewise continuous functions on [0, T ] defined by

ϕ1(t) =m1(u0(t) − ut1(t)) + m2(ut1(t) − ut2(t)) + m3(ut2(t) − ut3(t)) + . . . + mp+1utp(t)= m1u0(t) + (m2 − m1)ut1(t) + (m3 − m2)ut2(t) + . . . + (mp+1 − mp)utp(t)

=p∑

j=0

(mj+1 − mj)utj (t), (27)

where m0 = 0 and 0 < a = t0 < t1 < t2 < · · · < tp < tp+1 = T.Analogously, we define the functions ϕ2, ψ1 and ψ2 as

ϕ2(t) =p∑

j=0

(Mj+1 − Mj)utj (t), (28)

ψ1(t) =p∑

j=0

(nj+1 − nj)utj (t), (29)




ψ2(t) =p∑

j=0

(Nj+1 − Nj)utj(t), (30)

where the constants n0 = N0 = M0 = 0. If there is an integrable function f on [a, T ] satisfying condition(H1) then we get mj+1 ≤ f(t) ≤ Mj+1 for each t ∈ (tj , tj+1], j = 0, 1, 2, . . . , p. In particular, p = 4, thetime history of f can be shown as in figure 1.

Figure 1: Functions f , ϕ1 and ϕ2.

Proposition 5.1 Let f and g be two integrable functions on [a, T ], a > 0. Assume that the functionsϕ1, ϕ2, ψ1 and ψ2 are defined by (27), (28), (29) and (30), respectively, satisfying (H1)-(H2). Then forα > 0, the following inequality holds:∣∣∣∣ (t − a)α

Γ(α + 1)Jα

a fg(t) +(log t

a )α



a f(t)Jαa g(t)

∣∣∣∣ (31)

≤ |K∗(f, ϕ1, ϕ2)|1/2 |K∗(g, ψ1, ψ2)|1/2,

where

K∗(u, v, w)(T )

≤ (T − a)α

Γ(α + 1)

p∑j=0

wJα

ti,tj+1u(T ) + vJα

ti,tj+1u(T ) − vw

[(log

T

tj

)α

−(

logT

tj+1

)α]

+(log T

a )α

Γ(α + 1)

p∑j=0

wIα

ti,tj+1u(T ) + vIα

ti,tj+1u(T ) − vw [(T − tj)α − (T − tj+1)α]

− 2

p∑j=0

Jαti,tj+1

u(T )

p∑j=0

Iαti,tj+1

u(T )

.

Proof. Since

Iαtj ,tj+1

(1)(T ) =1

Γ(α)

∫ tj+1

tj

(T − τ)α−1dτ

=1

Γ(α + 1)[(T − tj)

α − (T − tj+1)α] ,

Jαtj ,tj+1

(1)(T ) =1

Γ(α)

∫ tj+1

tj

(log

T

τ

)α−1dτ

τ

=1

Γ(α + 1)

[(log

T

tj

)α

−(

logT

tj+1

)α],




we have

Iαa (ϕ1ϕ2)(T ) =

p∑j=0

mj+1Mj+1

Γ(α + 1)[(T − tj)

α − (T − tj+1)α] ,

Jαtj ,tj+1

(ψ1ψ2)(T ) =p∑

j=0

nj+1Nj+1

Γ(α + 1)

[(log

T

tj

)α

−(

logT

tj+1

)α].

Therefore, two functional K∗(f, ϕ1, ϕ2)(T ) and K∗(g, ψ1, ψ2)(T ) can be expressed by

K∗(f, ϕ1, ϕ2)(T ) ≤ (t − a)α

Γ(α + 1)

p∑j=0

Mj+1J

αti,tj+1

f(T ) + mj+1Jαti,tj+1

f(T )

− mj+1Mj+1

[(log

T

tj

)α

−(

logT

tj+1

)α]

+(log t

a )α

Γ(α + 1)

p∑j=0

Mj+1I

αti,tj+1

f(T ) + mj+1Iαti,tj+1

f(T )

− mj+1Mj+1 [(T − tj)α − (T − tj+1)α]

− 2

p∑j=0

Jαti,tj+1

f(T )

p∑j=0

Iαti,tj+1

f(T )

,

and

K∗(g, ψ1, ψ2)(T ) ≤ (t − a)α

Γ(α + 1)

p∑j=0

Nj+1J

αti,tj+1

g(T ) + nj+1Jαti,tj+1

g(T )

− nj+1Mj+1

[(log

T

tj

)α

−(

logT

tj+1

)α]

+(log t

a )α

Γ(α + 1)

p∑j=0

Nj+1I

αti,tj+1

g(T ) + nj+1Iαti,tj+1

g(T )

− nj+1Nj+1 [(T − tj)α − (T − tj+1)α]

− 2

p∑j=0

Jαti,tj+1

g(T )

p∑j=0

Iαti,tj+1

g(T )

.

By applying Theorem (4.2), the required inequality (31) is established. ¤

Proposition 5.2 Let f and g be two integrable functions on [a, T ], a > 0. Assume that the functionsϕ1, ϕ2, ψ1 and ψ2 are defined by (27), (28), (29) and (30), respectively, satisfying (H1)-(H2). Then forα, β > 0, the following inequality holds:∣∣∣∣ (t − a)β

Γ(β + 1)Jα

a fg(t) +(log t

a )α



a f(t)Jαa g(t)

∣∣∣∣ (32)

≤ |K∗1 (f, ϕ1, ϕ2)|1/2 |K∗

1 (g, ψ1, ψ2)|1/2,

where

K∗1 (u, v, w)(T ) ≤ (T − a)β

Γ(β + 1)

p∑j=0

wJα

ti,tj+1u(T ) + vJα

ti,tj+1u(T ) − vw

[(log

T

tj

)α

−(

logT

tj+1

)α]




+(log T

a )α

Γ(α + 1)

p∑j=0

wIβ

ti,tj+1u(T ) + vIβ

ti,tj+1u(T ) − vw

[(T − tj)β − (T − tj+1)β

]

− 2

p∑j=0

Jαti,tj+1

u(T )

p∑j=0

Iβti,tj+1

u(T )

.

Proof. By direct computations, we have

K∗1 (f, ϕ1, ϕ2)(T ) ≤ (t − a)β

Γ(β + 1)

p∑j=0

Mj+1J

αti,tj+1

f(T ) + mj+1Jαti,tj+1

f(T )

− mj+1Mj+1

[(log

T

tj

)α

−(

logT

tj+1

)α]

+(log t

a )α

Γ(α + 1)

p∑j=0

Mj+1I

αti,tj+1

f(T ) + mj+1Iβti,tj+1

f(T )

− mj+1Mj+1

[(T − tj)β − (T − tj+1)β

]

− 2

p∑j=0

Jαti,tj+1

f(T )

p∑j=0

Iβti,tj+1

f(T )

,

and

K∗1 (g, ψ1, ψ2)(T ) ≤ (t − a)β

Γ(β + 1)

p∑j=0

Nj+1J

αti,tj+1

g(T ) + nj+1Jαti,tj+1

g(T )

− nj+1Mj+1

[(log

T

tj

)α

−(

logT

tj+1

)α]

+(log t

a )α

Γ(α + 1)

p∑j=0

Nj+1I

βti,tj+1

g(T ) + nj+1Iβti,tj+1

g(T )

− nj+1Nj+1

[(T − tj)β − (T − tj+1)β

]

− 2

p∑j=0

Jαti,tj+1

g(T )

p∑j=0

Iβti,tj+1

g(T )

,

By applying Theorem (4.4), the required inequality (32) is established. ¤

References

[1] G. Anastassiou, Opial type inequalities involving fractional derivatives of two functions and appli-cations, Comput. Math. Appl. 48 (2004), 1701-1731.

[2] Z.Denton, A, Vatsala, Fractional integral inequalities and applications, Comput. Math. Appl. 59(2010), 1087-1094.

[3] S. Belarbi, Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math.10 (2009), Article ID 86.

[4] Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci. 9 (2010), 493-497.




[5] W.T. Sulaiman, Some new fractional integral inequalities, J. Math. Anal. 2 (2011), 23-28.

[6] M.Z. Sarikaya, H. Ogunmez, On new inequalities via Riemann-Liouville fractional integration,Abstract Appl. Anal. 2012 (2012), Article ID 428983.

[7] J. Hadamard, Essai sur l’etude des fonctions donnees par leur developpment de Taylor, J. Mat.Pure Appl. Ser. 8 (1892) 101-186.

[8] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Dif-ferential Equations, vol. 204 of North-Hooland Mathematics Studies, Elsevier, Amsterdam, TheNetherlands, 2006.

[9] 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.

[10] 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.

[11] 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.

[12] A.A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001), 1191- 1204.

[13] A.A. Kilbas, J.J. Trujillo, Hadamard-type integrals as G-transforms, Integral Transform. Spec.Funct. 14 (2003), 413-427.

[14] V. L. Chinchane, D. B. Pachpatte, A note on some integral inequalities via Hadamard integral, J.Fractional Calculus Appl. 4 (2013), 1-5.

[15] V. L. Chinchane, D. B. Pachpatte, On some integral inequalities using Hadamard fractional inte-gral, Malaya J. Math. 1 (2012), 62-66.

[16] B. Sroysang, A study on Hadamard fractional integral, Int. Journal of Math. Analysis, 7 (2013),1903-1906

[17] J. Tariboon, S.K. Ntouyas, W. Sudsutad, Some new Riemann-Liouville fractional integral inequal-ities Inter. J. Math. & Math. Sci., Volume 2014, Article ID 869434, 6 pages.

[18] W. Sudsutad, S.K. Ntouyas, J. Tariboon, Fractional integral inequalities via Hadamard’s fractionalintegral, Abstr. Appl. Anal. Volume 2014, Article ID 563096, 11 pages.

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Weighted composition operatorsfrom F (p, q, s) spaces to nth weighted-Orlicz spaces

Haiying Li and Zhitao GuoHenan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control,

School of Mathematics and Information Science,Henan Normal University, Xinxiang 453007.

Email address: [email protected](Haiying Li)

AbstractThe boundedness and compactness of the weighted composition operator from F (p, q, s) spaces to nth weighted-

Orlicz spaces are characterized in this paper.

Keywords: weighted composition operator, F (p, q, s) spaces, nth weighted-Orlicz spaces.

1 IntroductionLet H(D) be the space of all holomorphic functions on the open unit disk D in the complex plane C, N0 the set of allnonnegative integers, N the set of all positive integers, and dA the Lebesgue measure on D normalized so that A(D) =1. Let u ∈ H(D), the weighted composition operator uCφ is defined by (uCφf)(z) = u(z)f(φ(z)), f ∈ H(D), formore details, see, [1, 3, 16, 18].

For 0 < p, s <∞, −2 < q <∞, a function f ∈ H(D) is said to belong to the general function space F (p, q, s) if

‖f‖F (p,q,s) = |f(0)|p + supz∈D

∫D

|f ′(z)|p(1− |z|2)q(1− |ψa(z)|2)sdA(z) <∞,

where ψa(z) = (a− z)/(1− az), a ∈ D. The space F (p, q, s) was introduced by Zhao in [14]. Since for q+ s ≤ −1,F (p, q, s) is the space of constant functions, we assume that q + s > −1. For some results on F (p, q, s) space see, forexample, [4, 5, 7, 8, 10, 11, 15, 16, 17, 18].

Let µ be a positive continuous function on [0,1). We say that µ is normal if there exist two positive numbers a andb with 0 < a < b, and δ ∈ [0, 1) such that (see [6])

µ(r)(1− r)a

is decreasing on [δ, 1), limr→1

µ(r)(1− r)a

= 0;µ(r)

(1− r)bis increasing on [δ, 1), lim

r→1

µ(r)(1− r)b

= ∞.

Let µ(z) = µ(|z|) be a normal function on D. The nth weighted-type space on D, denoted by W(n)µ = W(n)

µ (D)which was introduced by Stevic in [9], consists of all f ∈ H(D) such that

bW(n)µ

(f) = supz∈D

µ(z)|f (n)(z)| <∞.

For n = 0 the space becomes the weighted-type space H∞µ (D), for n = 1 the Bloch-type space Bµ(D) and for n = 2

the Zygmund-type space Zµ(D). From now on, we will assume that n ∈ N. Set

‖f‖W(n)µ

=n−1∑j=0

|f (j)(0)|+ bW(n)µ

(f).

With this norm the nth weighted-type space becomes a Banach space.Recently, Fernandz in [2] uses Young’s functions to define the Bloch-Orlicz space. More precisely, let ϕ : [0,∞) →

[0,∞) be a strictly increasing convex function such that ϕ(0) = 0 and limt→∞ ϕ(t) = ∞.The Bloch-Orlicz spaceassociated 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 the Minkowski’s functional

‖f‖bϕ = infk > 0 : Sϕ

(f ′

k

)≤ 1


315 Haiying Li et al 315-323

2 Haiying Li and Zhitao Guo

defines a seminorm for Bϕ, which, in this case, is known as Luxemburgs seminorm, where

Sϕ(f) = supz∈D

(1− |z|2)ϕ(|f(z)|).

In fact, it can be shown that Bϕ is a Banach space with the norm ‖f‖Bϕ = |f(0)|+‖f‖bϕ . For more details, see [2]. Wealso have that the Bloch-Orlicz space is isometrically equal to the µ-Bloch space, where µ(z) = 1

ϕ−1( 11−|z|2

), z ∈ D.

Inspired by this, now we define the nth weighted-Orlicz space, which is denoted byW(n)ϕ , as the class of all analytic

function f in D such thatsupz∈D

(1− |z|2)ϕ(λ|f (n)(z)|) <∞

for some λ > 0 depending on f . Same as the Bloch-Orlicz space, it is not difficult to see that the Minkowski’sfunctional

‖f‖wϕ = infk > 0 : Sϕ

(f (n)

k

)≤ 1

defines a seminorm for W(n)

ϕ . Furthermore, it can be shown that W(n)ϕ is a Banach space with the norm

‖f‖W(n)ϕ

=n−1∑j=0

|f (j)(0)|+ ‖f‖wϕ .

In the same way as in the case Bϕ, for any f ∈ W(n)ϕ \0, the relation

Sϕ

(f (n)

‖f‖W(n)ϕ

)≤ 1

holds. Also, as a direct consequence of this, we have that the nth weighted-Orlicz space is isometrically equal to thenth weighted-type space, where µ(z) = 1

ϕ−1( 11−|z|2

), z ∈ D. Thus, for any f ∈ W(n)

ϕ , we have

‖f‖W(n)ϕ

=n−1∑j=0

|f (j)(0)|+ supz∈D

|f (n)(z)|ϕ−1( 1

1−|z|2 ).

Clearly, for n = 1, the nth weighted-Orlicz space W(n)ϕ becomes the Bloch-Orlicz space, and for n = 2 the Zygmund-

Orlicz space. In this paper, we are devoted to investigating the boundedness and compactness of the weighted com-position operator uCφ from F (p, q, s) spaces to nth weighted-Orlicz spaces. In what follows, we use the letter C todenote a positive constant whose value may change its value at each occurrence.

2 Auxiliary ResultsIn this section we formulate some auxiliary results which will be used in the proof of the main results. Lemma 1 andLemma 2 can be found in [5].

Lemma 1. Assume that f ∈ F (p, q, s), 0 < p, s < ∞, −2 < q < ∞, q + s > −1. Then, for each n ∈ N, there is apositive constant C, independent of f such that ‖f‖

B2+q

p≤ C‖f‖F (p,q,s) and

|f (n)(z)| ≤C‖f‖F (p,q,s)

(1− |z|2)2+q−p

p +n, z ∈ D.

Lemma 2. Let α > 0 and f ∈ Bα. Then,

|f(z)| ≤

C‖f‖Bα , 0 < α < 1,C log 2

1−|z|2 ‖f‖Bα , α = 1,C

(1−|z|2)α−1 ‖f‖Bα , α > 1.



Weighted composition operators from F (p, q, s) toW(n)ϕ 3

Lemma 3 and Lemma 4 can be found in [12].

Lemma 3. Assume a > 0 and

Dn+1 =

∣∣∣∣∣∣∣∣∣1 1 · · · 1a a+ 1 · · · a+ n...

......∏n−1

j=0 (a+ j)∏n−1

j=0 (a+ j + 1) · · ·∏n−1

j=0 (a+ j + n)

∣∣∣∣∣∣∣∣∣ .Then, Dn+1 =

∏nj=1 j!.

Lemma 4. Assume n ∈ N, u, f ∈ H(D) and φ is an analytic self-map of D. Then,(u(z)f(φ(z))

)(n) =n∑

k=0

f (k)(φ(z))n∑

l=k

Clnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

),

where

Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)=

∑k1,··· ,kl

l!k1!, · · · , kl!

l∏j=1

(φ(j)(z)j!

)kj

, (1)

and the sum in (1) is overall non-negative integer k1, · · · , kl satisfying k1+k2+· · ·+kl = k and k1+2k2+· · ·+lkl = l.

The next characterization of compactness is proved in a standard way (see, e.g., the proofs of [1], Prop 3.11]).Hence we omit it. The following Lemma 6 can be found in [13].

Lemma 5. Suppose that u ∈ H(D), n ∈ N, φ is an analytic self-map of D. Then, uCφ : F (p, q, s) → W(n)ϕ is

compact if and only if uCφ : F (p, q, s) → W(n)ϕ is bounded and for any bounded sequence fkk∈N in F (p, q, s)

which converges to zero uniformly on compact subsets of D as k →∞, we have ‖uCφf‖W(n)ϕ

→ 0 as k →∞.

Lemma 6. Fix 0 < α < 1 and let fkk∈N be a bounded sequence in Bα which converges to zero uniformly oncompact subsets of D as k →∞. Then we have

limk→∞

supz∈D

|fk(z)| = 0. (2)

3 The Boundedness of uCφ : F (p, q, s) →W (n)ϕ

Theorem 7. Let u ∈ H(D), 0 < p, s <∞, −2 < q <∞, q + s > −1, n ∈ N and φ be an analytic self-map of D.(a) If 2 + q < p, then uCφ : F (p, q, s) →W(n)

ϕ is bounded if and only if

M0 = supz∈D

|u(n)(z)|ϕ−1( 1

1−|z|2 )<∞, (3)

and

Mk = supz∈D

∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣ϕ−1( 1

1−|z|2 )(1− |φ(z)|2)2+q−p

p +k<∞, (4)

where k = 1, 2, · · · , n.(b) If 2 + q = p, then uCφ : F (p, q, s) →W(n)

ϕ is bounded if and only if (4) holds and

M ′0 = sup

z∈D

|u(n)(z)| log 21−|φ(z)|2

ϕ−1( 11−|z|2 )

<∞. (5)

(c) If 2 + q > p, then uCφ : F (p, q, s) →W(n)ϕ is bounded if and only if (4) holds and

M ′′0 = sup

z∈D

|u(n)(z)|ϕ−1( 1

1−|z|2 )(1− |φ(z)|2)2+q−p

p

<∞. (6)




Proof. If 2 + q < p. Assume that (3) and (4) hold, then for each f ∈ W(n)ϕ \0, by Lemma 1, Lemma 2 and Lemma

4, we have

Sϕ

((uCφf)(n)(z)C‖f‖F (p,q,s)

)≤ sup

z∈D(1− |z|2) ·

ϕ

( |u(n)(z)||f(φ(z))|+∣∣ ∑n

k=1 f(k)(φ(z))

∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣C‖f‖F (p,q,s)

)≤ sup

z∈D(1− |z|2) ·

ϕ

(ϕ−1( 1

1−|z|2 )M0|f(φ(z))|+ ϕ−1( 11−|z|2 )

∑nk=1Mk(1− |φ(z)|2)

2+q−pp +k

∣∣f (k)(φ(z))∣∣

C‖f‖F (p,q,s)

)

≤ supz∈D

(1− |z|2)ϕ(CM0‖f‖

B2+q

p+

∑nk=1 CkMk‖f‖F (p,q,s)

C‖f‖F (p,q,s)ϕ−1

(1

1− |z|2

))≤ sup

z∈D(1− |z|2)ϕ

(∑nj=0 CjMj

Cϕ−1

(1

1− |z|2

))≤ 1.

HereCj(j = 0, 1, · · · , n) are all constants, andC ≥∑n

j=0 CjMj . Now, we can conclude that there exists a constantC

such that ‖uCφf‖W(n)ϕ

≤ C‖f‖F (p,q,s) and the weighted composition operator uCφ : F (p, q, s) →W(n)ϕ is bounded.

If 2 + q = p, or 2 + q > p, from (4) (5), or (4) (6), we can get uCφ : F (p, q, s) →W(n)ϕ is bounded similarly.

Conversely, suppose that uCφ : F (p, q, s) →W(n)ϕ is bounded, that is, for all f ∈ F (p, q, s), there exists a constant

C such that ‖uCφf‖W(n)ϕ

≤ C. For ω ∈ D, and constants c0, c1, · · · , cn, set

fω(z) =n∑

j=0

cj(1− |ω|2)j+1

(1− ωz)α+j, (7)

where α = 2+qp . It is well known that fω ∈ F (p, q, s), and

fω(ω) =1

(1− |ω|2)α−1

n∑j=0

cj , (8)

f (l)ω (ω) =

ωl

(1− |ω|2)α−1+l

n∑j=0

cj

l−1∏r=0

(α+ j + r), l = 1, 2, · · · , n. (9)

We claim that for each k ∈ 1, 2, · · · , n, there are constants c0, c1, · · · , cn such that∑n

j=0 cj 6= 0 and

f (k)ω (ω) =

ωk

(1− |ω|2)α−1+k, f (t)

ω (ω) = 0, t ∈ 0, 1, 2, · · · , n\k. (10)

In fact, by (8) and (9), (10) is equivalent to the following system of liner equations

c0 + c1 + · · ·+ cn = 0,c0α+ c1(α+ 1) + · · ·+ cn(α+ n) = 0,c0α(α+ 1) + c1(α+ 1)(α+ 2) + · · ·+ cn(α+ n)(α+ n+ 1) = 0,· · · · · ·c0

∏k−1r=0(α+ r) + c1

∏k−1r=0(α+ 1 + r) + · · ·+ cn

∏k−1r=0(α+ n+ r) = 1,

· · · · · ·c0

∏n−1r=0 (α+ r) + c1

∏n−1r=0 (α+ 1 + r) + · · ·+ cn

∏n−1r=0 (α+ n+ r) = 0.

(11)




By using Lemma 3, we obtain that the determinant of system of linear Eq.(11) is different from zero, from whichthe claim follows. For each k ∈ 1, 2, · · · , n, we choose the corresponding family of functions that satisfy (10) anddenote it by fω,k. Then, from Lemma 4 and the boundedness of uCφ : F (p, q, s) → W(n)

ϕ , for ω ∈ D such that|φ(ω)| > 1

2 ,

1 ≥ Sϕ

((uCφfφ(ω),k)(n)(z)

C

)≥ sup|φ(ω)|> 1

2

(1− |ω|2)ϕ( |(uCφfφ(ω),k)(n)(ω)|

C

)

= sup|φ(ω)|> 1

2

(1− |ω|2)ϕ( |φ(ω)|k

∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣C(1− |φ(ω)|2)

2+q−pp +k

)It follows that

sup|φ(ω)|> 1

2

∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣ϕ−1( 1

1−|z|2 )(1− |φ(z)|2)2+q−p

p +k<∞. (12)

By the test functions fk(z) = zk(k = 1, 2, · · · , n), use the mathematical induction as in [12], we can get that

supz∈D

∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣ϕ−1( 1

1−|z|2 )<∞.

Then, for each k ∈ 1, 2, · · · , n,

sup|φ(ω)|≤ 1

2

∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣ϕ−1( 1

1−|z|2 )(1− |φ(z)|2)2+q−p

p +k<∞. (13)

Combining (12) with (13), we obtain that (4) is necessary for all cases.If 2 + q < p, taking f(z) = 1, then (uCφf)(z) = u(z), by the boundedness of uCφ : F (p, q, s) →W(n)

ϕ , we have

Sϕ

((uCφf)(n)(z)

C

)= sup

z∈D(1− |z|2)ϕ

(|u(n)(z)|

C

)≤ 1.

It follows that (3) holds.If 2 + q = p, for a fixed ω ∈ D, set

gω(z) = log2

1− ωz.

Then it is easy to see that gω ∈ F (p, q, s) and we have

gω(ω) = log2

1− |ω|2, g(k)

ω (ω) = (k − 1)!ωk

(1− |ω|2)k, k = 1, 2, · · · , n.

From Lemma 4 and the boundedness of uCφ : F (p, q, s) →W(n)ϕ , we have

1 ≥ Sϕ

((uCφgφ(ω))(n)(z)

C

)≥ sup

ω∈D(1− |ω|2)ϕ

( |(uCφfφ(ω))(n)(ω)|C

)≥ sup

ω∈D(1− |ω|2) ·

ϕ

(∣∣u(n)(ω) log 21−|φ(ω)|2

∣∣C

−n∑

k=1

|φ(ω)|k∣∣ ∑n

l=k Clnu

(n−l)(z)Bl,k

(φ′(z), · · · , φ(l−k+1)(z)

)∣∣C(1− |φ(ω)|2)k

).

By Mk <∞ and the boundedness of φ(ω), it follows that

supω∈D

|u(n)(ω)| log 21−|φ(ω)|2

ϕ−1( 11−|ω|2 )

≤ C +n∑

k=1

∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣ϕ−1( 1

1−|z|2 )(1− |φ(z)|2)k<∞.




If 2 + q > p, using the function in (7), and in the system of linear Eq.(11), we can also find c0, c1, · · · , cn anddenote the corresponding function hω(z) such that

hω(ω) =1

(1− |ω|2)2+q−p

p

, h(k)ω (ω) = 0, k = 1, 2, · · · , n.

Then from Lemma 4 and the boundedness of uCφ : F (p, q, s) →W(n)ϕ , we have

1 ≥ Sϕ

((uCφhφ(ω))(n)(z)

C

)≥ sup

ω∈D(1− |ω|2)ϕ

( |(uCφhφ(ω))(n)(ω)|C

)= sup

ω∈D(1− |ω|2)ϕ

(|u(n)(ω)|

C(1− |φ(ω)|2)2+q−p

p

),

from which we can see that (6) holds.

4 The Compactness of uCφ : F (p, q, s) →W (n)ϕ

Theorem 8. Let u ∈ H(D), 0 < p, s <∞, −2 < q <∞, q + s > −1, n ∈ N and φ be an analytic self-map of D.(a) If 2 + q < p, then uCφ : F (p, q, s) →W(n)

ϕ is compact if and only if uCφ : F (p, q, s) →W(n)ϕ is bounded and

lim|φ(z)|→1

∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣ϕ−1( 1

1−|z|2 )(1− |φ(z)|2)2+q−p

p +k= 0, (14)

where k = 1, 2, · · · , n.(b) If 2 + q = p, then uCφ : F (p, q, s) → W(n)

ϕ is compact if and only if uCφ : F (p, q, s) → W(n)ϕ is bounded,

(14) holds and

lim|φ(z)|→1

|u(n)(z)| log 21−|φ(z)|2

ϕ−1( 11−|z|2 )

= 0. (15)

(c) If 2 + q > p, then uCφ : F (p, q, s) → W(n)ϕ is compact if and only if uCφ : F (p, q, s) → W(n)

ϕ is bounded,(14) holds and

lim|φ(z)|→1

|u(n)(z)|ϕ−1( 1

1−|z|2 )(1− |φ(z)|2)2+q−p

p

= 0. (16)

Proof. Let fii∈N be a sequence in F (p, q, s) with ‖fi‖F (p,q,s) ≤ L, and fi converges to zero uniformly on compactsubsets of D as i → ∞. To prove that uCφ : F (p, q, s) → W(n)

ϕ is compact, by Lemma 5, we only need to showlimi→∞ ‖uCφfi‖W(n)

ϕ= 0.

If 2 + q < p, suppose that uCφ : F (p, q, s) →W(n)ϕ is bounded and (14) holds, then for given ε > 0, there exists a

δ ∈ (0, 1), when δ < |φ(z)| < 1, we have∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣ϕ−1( 1

1−|z|2 )(1− |φ(z)|2)2+q−p

p +k< ε, k = 1, 2, · · · , n. (17)

By the proof of the boundedness, we know that M0 <∞ and

supz∈D

∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣ϕ−1( 1

1−|z|2 )≤ C, k = 1, 2, · · · , n.




Let K = z ∈ D, |φ(z)| ≤ δ, then by Lemma 1 and (17), we have

supz∈D

∣∣(uCφfi)(n)(z)∣∣

ϕ−1( 11−|z|2 )

≤ supz∈D

|u(n)(z)|ϕ−1( 1

1−|z|2 )|fi(φ(z))|

+n∑

k=1

supz∈K

∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣ϕ−1( 1

1−|z|2 )|f (k)

i (φ(z))|

+n∑

k=1

supz∈D\K

∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣ϕ−1( 1

1−|z|2 )|f (k)

i (φ(z))|

≤ M0|fi(φ(z))|+ Cn∑

k=1

supz∈K

|f (k)i (φ(z))|

+n∑

k=1

supz∈D\K

∣∣ ∑nl=k C

lnu

(n−l)(z)Bl,k

(φ′(z), φ′′(z), · · · , φ(l−k+1)(z)

)∣∣‖fk‖F (p,q,s)

ϕ−1( 11−|z|2 )(1− |φ(z)|2)

2+q−pp +k

≤ M0|fi(φ(z))|+ nC sup|ω|≤δ

|f (k)i (ω)|+ nLε.

Since fk ∈ F (p, q, s) ⊂ B2+q

p , by Lemma 6, we have limi→∞ supz∈D |fi(φ(z))| = 0. By Cauchy’s estimate, weknow sup|ω|≤δ |f

(k)i (ω)| → 0, as i → ∞. On the other hand, since φ(0) is also compact subset of D, we have∑n−1

j=0 |f(j)i (0)| → 0, as i→∞. So ‖uCφfi‖W(n)

ϕ→ 0, as i→∞. Hence uCφ : F (p, q, s) →W(n)

ϕ is compact.

If 2 + q = p or 2 + q > p, assume that uCφ : F (p, q, s) → W(n)ϕ is bounded, (14), (15) or (14), (16) hold

respectively. Then given ε > 0, there exists a δ ∈ (0, 1), when δ < |φ(z)| < 1, we have

|u(n)(z)| log 21−|φ(z)|2

ϕ−1( 11−|z|2 )

< ε or|u(n)(z)|

ϕ−1( 11−|z|2 )(1− |φ(z)|2)

2+q−pp

< ε.

Then by Lemma 1, Lemma 2 and Lemma 5 and similar to the above, we can easily get that uCφ : F (p, q, s) →W(n)ϕ

is compact.Conversely, assume that uCφ : F (p, q, s) → W(n)

ϕ is compact, then it is clear that uCφ : F (p, q, s) → W(n)ϕ is

bounded. Let zii∈N be a sequence in D such that |φ(zi)| → 1, as i→∞. (If such a sequence does not exist, then thecondition in (14), (15), (16) automatically hold.) Let fω,k(z)(k = 1, 2, · · · , n) be as defined in the proof of Theorem7. Then the sequence fφ(zi),k are bounded in F (p, q, s) and converge to zero uniformly on compact subsets of D asi→∞. By Lemma 5 and the compactness of uCφ : F (p, q, s) →W(n)

ϕ , we have

limi→∞

‖uCφfφ(zi),k‖W(n)ϕ

= 0. (18)

Then

1 ≥ Sϕ

((uCφfφ(zi),k)(n)(zi)‖uCφfφ(zi),k‖W(n)

ϕ

)

≥ (1− |zi|2)ϕ( |φ(zi)|k

∣∣ ∑nl=k C

lnu

(n−l)(zi)Bl,k

(φ′(zi), φ′′(zi), · · · , φ(l−k+1)(zi)

)∣∣‖uCφfφ(zi),k‖W(n)

ϕ(1− |φ(zi)|2)

2+q−pp +k

)

It follows that

|φ(zi)|k∣∣ ∑n

l=k Clnu

(n−l)(zi)Bl,k

(φ′(zi), φ′′(zi), · · · , φ(l−k+1)(zi)

)∣∣ϕ−1( 1

1−|zi|2 )(1− |φ(zi)|2)2+q−p

p +k≤ ‖uCφfφ(zi),k‖W(n)

ϕ.




lim|φ(zi)|→1

∣∣ ∑nl=k C

lnu

(n−l)(zi)Bl,k

(φ′(zi), φ′′(zi), · · · , φ(l−k+1)(zi)

)∣∣ϕ−1( 1

1−|zi|2 )(1− |φ(zi)|2)2+q−p

p +k

= limi→∞

|φ(zi)|k∣∣ ∑n

l=k Clnu

(n−l)(zi)Bl,k

(φ′(zi), φ′′(zi), · · · , φ(l−k+1)(zi)

)∣∣ϕ−1( 1

1−|zi|2 )(1− |φ(zi)|2)2+q−p

p +k= 0,

which implies that (14) is necessary for all cases. If 2 + q = p, set

gi(z) =(

log2

1− φ(zi)z

)2(log

21− |φ(zi)|2

)−1

.

Then gi(z) is a bounded sequence in F (p, q, s) and converges to zero uniformly on compact subsets of D, and wehave

gi(φ(zi)) = log2

1− |φ(zi)|2, g

(k)i (φ(zi)) =

2(k − 1)!φ(zi)k

(1− |φ(zi)|2)k+ Ck

φ(zi)k

(1− |φ(zi)|2)k

(log

21− |φ(zi)|2

)−1

,

where Ck(k = 1, 2, · · · , n) is constants about k. By Lemma 5 and the compactness of uCφ : F (p, q, s) → W(n)ϕ , we

have

limi→∞

‖uCφgi‖W(n)ϕ

= 0. (19)

Then

1 ≥ Sϕ

((uCφgi)(n)(zi)‖uCφgi‖W(n)

ϕ

)≥ (1− |zi|2) ·

ϕ

( |u(n)(zi)| log 21−|φ(zi)|2

‖uCφgi‖W(n)ϕ

−n∑

k=1

C|φ(zi)|k∣∣ ∑n

l=k Clnu

(n−l)(zi)Bl,k

(φ′(zi), φ′′(zi), · · · , φ(l−k+1)(zi)

)∣∣‖uCφgi‖W(n)

ϕ(1− |φ(zi)|2)

2+q−pp +k

).

It follows that

|u(n)(zi)| log 21−|φ(zi)|2

ϕ−1( 11−|zi|2 )

≤ ‖uCφgi‖W(n)ϕ

+n∑

k=1

C∣∣ ∑n

l=k Clnu

(n−l)(zi)Bl,k

(φ′(zi), φ′′(zi), · · · , φ(l−k+1)(zi)

)∣∣ϕ−1( 1

1−|zi|2 )(1− |φ(zi)|2)2+q−p

p +k.

Then by (14) and (19), we can get (15) holds.If 2 + q > p, let hω(z) be as defined in the proof of Theorem 7. Then the sequence hφ(zi) is bounded in F (p, q, s)and converges to zero uniformly on compact subsets of D as i → ∞. By Lemma 5 and the compactness of uCφ :F (p, q, s) →W(n)

ϕ , we have

limi→∞

‖uCφhφ(zi)‖W(n)ϕ

= 0. (20)

Then

1 ≥ Sϕ

((uCφhφ(zi))

(n)(zi)‖uCφhφ(zi)‖W(n)

ϕ

)≥ (1− |zi|2)ϕ

(|u(n)(zi)|

‖uCφhφ(zi)‖W(n)ϕ

(1− |φ(zi)|2)2+q−p

p

).

It follows that

|u(n)(zi)|ϕ−1( 1

1−|zi|2 )(1− |φ(zi)|2)2+q−p

p

≤ ‖uCφhφ(zi)‖W(n)ϕ

from which we can get (16) holds by (20).

Acknowledgement. This work is supported by the NNSF of China (Nos.11201127;11271112) and IRTSTHN(14IRTSTHN023).




References[1] C. C. Cowen and B. D. MacCluer. Composition operators on spaces of analytic functions, CRC Press, Boca Roton, 1995.

[2] J. C. Ramos Fernandez. Composition operators on Bloch-Orlicz type spaces. Applied Mathematics and Computation, 217(7)(2010) 3392-3402.

[3] S. Li and S. Stevic. Weighted composition operators from Zygmund spaces into Bloch spaces. Applied Mathematics andComputation, 206(2) (2008) 825-831.

[4] S. Li and S. Stevic. Compactness of Riemann-Stieltjes operators between F (p, q, s) and α-Bloch spaces, Publ. Math. Debre-cen 72(1-2) (2008) 111-128 .

[5] Y. Liu and Y. Yu. The multiplication operator from F (p, q, s) spaces to nth weighted-type spaces on the unit disk. Journal ofFunction Spaces and Applications, Volume 2012, Article ID 343194, 21 pages.

[6] A. Shields and D. Williams. Bounded projections, duality, and multipliers in spaces of analytic functions. Trans. Amer. Math.Soc., 162 (1971) 287-302.

[7] S. Stevic. On some integral-type operators between a general space and Bloch-type spaces. Applied Mathematics and Com-putation, 218(6) (2011) 2600-2618.

[8] S. Stevic. Boundedness and compactness of an integral-type operator from Bloch-type spaces with normal weights toF (p, q, s) space. Applied Mathematics and Computation, 218(9) (2012) 5414-5421.

[9] S. Stevic. Composition operators from the weighted Bergman space to the nth weighted spaces on the unit disc. DiscreteDynamics in Nature and Society, vol. 2009, Article ID 742019, 11 pages, 2009.

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[12] L. Zhang and H. Zeng. Weighted differentiation composition operators from weighted Bergman space to nth weighted spaceon the unit disk. Journal of Inequalities and Applications, 2011, 2011:65.

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[14] R. Zhao. On a general family of function spaces. Ann. Acad. Sci. Fenn. Math. Diss. No. 105 (1996), 56 pp.

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[17] X. Zhu. Composition operators from Bloch type spaces to F (p, q, s) spaces. Filomat, 21(2) (2007) 11-20.

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MODIFIED q-DAEHEE NUMBERS AND POLYNOMIALS

DONGKYU LIM

Abstract. The p-adic q-integral was defined by T. Kim to be

Iq(f) =

∫Zp


pN−1∑x=0

qx

[pN ]qf(x) (see [9, 10]).

From p-adic q-integrals’ equations, we can derive various q-extension of Bernoullipolynomials and numbers (see [1-20]). In [4], T. Kim have studied Daehee polyno-mials and numbers and their applications. Recently, many properties and valuableidentities related to Daehee polynomials and numbers are introduced by severalauthors (see [1-20]). In [11], T. Kim et al. introduced the q-analogue of Daeheenumbers and polynomials which are called q-Daehee numbers and polynomials.In this paper, we consider the modified q-Daehee numbers and polynomials whichare different the q-Daehee numbers and polynomials of T. Kim et al. and givesome useful properties and identities of those polynomials which are derived thenew p-adic q-integral equations.

MSC: 11B68, 11S40, 11S80

Keywords and phrases. Modified q-Daehee number; Modified q-Daehee poly-nomial; Modified q-Bernoulli number; p-adic q-integral

1. Introduction

The q-Daehee polynomials Dn,q(x) are defined and studied by T. Kim et al., thegenerating function to be

(1)1− q + 1−q

log q log(1 + t)

1− q − qt(1 + t)x =

∞∑n=0

Dn,q(x)tn

n!(see [11]).

This generating function for Dn,q(x) is related with p-adic q-integral on Zp definedby T. Kim(see [9, 10]).

In this paper, we consider modified p-adic q-integration on Zp which are used bymany authors(see [1-20]). We define modified q-Daehee polynomials Dn(x|q) frommodified p-adic q−integrals, and relate Dn(x|q) with modified q-Bernoulli polyno-mials Bn(x|q).

Throughout this paper, we denote the ring of p-adic integers, the field of p-adicnumbers and the completion of algebraic closure of Qp by Zp, Qp and Cp, respec-tively. The p-adic norm | · |p is normalized by |p|p = 1

p . We denote the space of

uniformly differentiable function on Zp by UD[Zp]. The q-Haar measure is defined

as (see [9, 10]) µq(a+pmZp) = qa

[Pm]q, where [x]q = 1−qx

1−q . For a function f in UD[Zp],

the modified p-adic q-integral on Zp is given by

(2) Iq(f) =

∫Zp


pN−1∑x=0

qx

[pN ]qf(x) (see [9-20]).

1


324 DONGKYU LIM 324-330

2 DONGKYU LIM

The bosonic integral on Zp is given by I1(f) = limq→1 Iq(f).From (2), we have the following integral identity.

(3) qIq(f1)− Iq(f) =q − 1

log qf ′(0) + (q − 1)f(0),

where f1(x) = f(x+ 1) and f ′(x) = ddxf(x).

In special case, we apply f(x) = q−xetx on (3), we have

(et − 1)

∫Zp

q−xextdµq(x) =q − 1

log qt.

Thus

(4)

∫Zp

q−xextdµq(x) =q − 1

log q

t

et − 1.

The q-analogue Bernoulli numbers Bn(q) are known as follows:

(5)

∞∑n=0

Bn(q)tn

n!=q − 1

log q

t

et − 1(see [3, 5, 9]).

Indeed if q → 1, we have limq→1Bn(q) = Bn. So we call Bn(x|q) as the nthmodified q-Bernoulli polynomials and the generating function to be

(6)q − 1

log q

t

et − 1ext =

∞∑n=0

Bn(x|q) tn

n!.

When x = 0, Bn(0|q) = Bn(q) are the nth modified q-Bernoulli numbers.From (3) and (6), we have

Bn(x|q) =

∫Zp

q−y(x+ y)ndµq(y).

From (6), we note that

(7) Bn(x|q) =

n∑l=0

(n

l

)Bl(q)x

n−l.

For the case |t|p ≤ p−1

p−1 , the Daehee polynomials are defined as follows:

(8)∞∑n=0

Dn(x)tn

n!=

log(1 + t)

t(1 + t)x (see [11]).

From p-adic q-integrals’ equations, we can derive various q-extension of Bernoullipolynomials and numbers(see [1-20]). In [4], T. Kim have studied Daehee polyno-mials and numbers and their applications. Recently, many properties and valuableidentities related to Daehee polynomials and numbers are introduced by severalauthors(see [1-20]). In [11], T. Kim et al. introduced the q-analogue of Daeheenumbers and polynomials which are called q-Daehee numbers and polynomials. Inthis paper, we consider the modified q-Daehee numbers and polynomials which aredifferent the q-Daehee numbers and polynomials of T. Kim et al. and give some use-ful properties and identities of those polynomials which are derived the new p-adicq-integral equations.



MODIFIED q-DAEHEE NUMBERS AND POLYNOMIALS 3

2. Modified q-Daehee numbers and polynomials

Let us now consider the p-adic q-integral representation as follows:

(9)

∫Zp

q−y(x+ y)ndµq(y) (n ∈ Z+ = N ∪ 0),

where (x)n is known as the Pochhammer symbol(or decreasing fractorial) defined by

(10) (x)n = x(x− 1) · · · (x− n+ 1) =n∑

k=0

S1(n, k)xk

and here S1(n, k) is the Stirling number of the first kind (see [4, 11]).From (9), we have

(11)

∞∑n=0

(∫Zp

q−y(x+ y)ndµq(y)

)tn

n!=

∫Zp

q−y( ∞∑

n=0

(x+ y

n

)tn)dµq(y)

=

∫Zp

q−y(1 + t)x+ydµq(y),

where t ∈ Cp with |t|p < p− 1

p−1 .

For t ∈ Cp with |t|p < p− 1

p−1 , from (3), we have

(12)

∫Zp

q−y(1 + t)x+ydµq(y) =q − 1

log q

log(1 + t)

t(1 + t)x.

Let

(13) Fq(x, t) =q − 1

log q

log(1 + t)

t(1 + t)x =

∞∑n=0

Dn(x|q) tn

n!.

In here the polynomial Dn(x|q) is called modified nth q-Daehee polynomials ofthe first kind. Moreover, we have

(14) Dn(x|q) =

∫Zp

q−y(x+ y)ndµq(y).

When x = 0, Dn(0|q) = Dn(q) is called modified the n-th q-Daehee numbers.Notice that Fq(x, t) seems to be a new q-extension of the generating function for

Daehee polynomials of the first kind. Therefore, from (8) and the following fact,

limq→1

Fq(x, t) =log(1 + t)

t(1 + t)x.

On the other hand, we can derive

(15)∞∑n=0

(∫Zp

q−x(x)ndµq(x)

)tn

n!=q − 1

log q

log(1 + t)

t=∞∑n=0

Dn(q)tn

n!.

From (10) and (15), we have

(16)q − 1

log qDn(x) = Dn(x|q).



4 DONGKYU LIM


(17)

Dn(x|q) =

∫Zp

q−y(x+ y)ndµq(y)

=

n∑k=0

S1(n, k)Bk(x|q).

Bk(x|q) are the modified q-Bernoulli polynomials introduced in (6).Thus we have the following theorem, which relates modified q-Bernoulli polyno-

mials and modified q-Daehee polynomials.

Theorem 1. For n,m ∈ Z+, we have the following equalities.

Dn(x|q) =n∑

k=0

S1(n, k)Bk(x|q)

and

Dn(q) =

n∑k=0

S1(n, k)Bk(q).

From the generating function of modified q-Daehee polynomials in Dn(x|q) in(13), by replacing t to et − 1, we have

(18)

∞∑n=0

Dn(x|q)(et − 1)n

n!=∞∑n=0

Bn(x|q) tn

n!

=∞∑

m=0

Dm(x|q)∞∑n=0

S2(n,m)tn

n!.

Thus by comparing the coefficients of tn, we have

Bn(x|q) =n∑

m=0

Dm(x|q)S2(n,m).

In here, S2(n,m) is the Stirling number of the second kind defined by the followinggenerating series:

(19)∞∑

n=m

S2(n,m)tn

n!=

(et − 1)m

m!cf.[4, 11].

Therefore, we obtan the following theorem.

Theorem 2. For n,m ∈ Z+, we have the following identity.

Bn(x|q) =

n∑m=0

Dm(x|q)S2(n,m).

The increasing factorial sequence is known as

x(n) = x(x+ 1)(x+ 2) · · · (x+ n− 1) (n ∈ Z+).

Let us define the modified q-Daehee numbers of the second kind as follows:

(20) Dn(q) =

∫Zp

q−y(−y)ndµq(y) (n ∈ Z+).




It is easy to observe that

(21) x(n) = (−1)n(−x)n =n∑

k=0

S1(n, k)(−1)n−kxk.


(22)

Dn(q) =

∫Zp

q−y(−y)ndµq(y)

=

∫Zp

q−yy(n)(−1)ndµq(y)

=n∑

k=0

S1(n, k)(−1)kBk(q).

Thus, we state the following theorem.

Theorem 3. The following holds true:

Dn(q) =

n∑k=0

S1(n, k)(−1)kBk(q).

Let us now consider the generating function of the modified q-Daehee numbers ofthe second kind as follows:

(23)

∞∑n=0

Dn(q)tn

n!=∞∑n=0

(∫Zp

q−y(−y)ndµq(y)

)tn

n!

=

∫Zp

q−y( ∞∑

n=0

(−yn

)tn)dµq(y)

=

∫Zp

q−y(1 + t)−ydµq(y).

From (23), we denote the generating function for the modified q-Daehee numbersof the second as follows:

(24) Fq(t) =q − 1

log q

log(1 + t)

t(1 + t).

Let us consider the modified q-Daehee polynomials of the second kind as follows:

(25)q − 1

log q

log(1 + t)

t(1 + t)x+1 =

∞∑n=0

Dn(x|q) tn

n!.

It follows from (25) that

(26)

∫Zp

q−y(1 + t)x−ydµq(y) =

∞∑n=0

Dn(x|q) tn

n!.

From (26) gives

(27)

Dn(x|q) =

∫Zp

q−y(x− y)ndµq(y)

= q−1n∑

k=0

|S1(n, k)|Bk(x+ 1|q−1),



6 DONGKYU LIM

where n ≥ 0 and |S1(n, k)| is the unsigned stirling numbers of the first kind.Then, by (27), we have the following theorem.

Theorem 4. For n ≥ 0, the following are true.

Dn(x|q) = q−1n∑

k=0

|S1(n, k)|Bk(x+ 1|q−1).

From the modified q-Bernoulli polynomials in (6),

(28)

q∞∑n=0

Bn(x|q−1) tn

n!=q − 1

log q

t

et − 1e(1−x)t

=

∞∑n=0

Bn(1− x|q) tn

n!.

Thus, we have

(29) q(−1)nBn(x|q−1) = Bn(1− x|q).From (29), the value at x = 1, we have

q(−1)nBn(1|q−1) = Bn(q).

On the other hand, we can check easily the following

(30) (x+ y)n = (−1)n(−x− y + n− 1)n

and

(31)(x+ y)n

n!= (−1)n

(−x+ y + n− 1

n

).

From (13), (27), (30) and (31), we have

(32)

(−1)nDn(x|q)n!

=

∫Zp

q−y(−x− y + n− 1

n

)dµq(y)

=n∑

m=0

(n− 1

n−m

)∫Zp

q−y(−x− ym

)dµq(y)

=n∑

m=1

(n− 1

m− 1

)Dm(−x|q)

m!

and

(33)

(−1)nDn(x|q)n!

= (−1)n∫Zp

q−y(−x+ y

n

)dµq(y)

=

∫Zp

q−y(−x+ y + n− 1

n

)dµq(y)

=

n∑m=0

(n− 1

n−m

)∫Zp

q−y(−x+ y

m

)dµq(y)

=n∑

m=1

(n− 1

m− 1

)Dm(−x|q)

m!.

Therefore, we get the following theorem, which relates modified q-Daehee poly-nomials of the first and the second kind.




Theorem 5. For n ∈ N, the following equlity hold true.

(−1)nDn(x|q)n!

=n∑

m=1

(n− 1

m− 1

)Dm(−x|q)

m!

and

(−1)nDn(x|q)n!

=

n∑m=1

(n− 1

m− 1

)Dm(−x|q)

m!.

References

[1] S. Araci, M. Acikgoz, A. Esi, A note on the q-Dedekind-type Daehee-Changhee sums with weightα arising from modified q-Genocchi polynomials with weight α, J. Assam Acad. Math. 5 (2012),47-54.

[2] A. Bayad, Modular properties of elliptic Bernoulli and Euler functions, Adv. Stud. Contemp.Math. (Kyungshang) 20 (2010), no. 3, 389-401.

[3] D. V. Dolgy, T. Kim, S.-H. Rim, S. H. Lee, Symmetry identities for the generalized higher-order q-Bernoulli polynomials under S3 arising from p-adic Volkenborn ingegral on Zp, Proc.Jangjeon Math. Soc. 17 (2014), no. 4, 645-650.

[4] D. S. Kim, T. Kim, Daehee numbers and polynomials, Appl. Math. Sci. (Ruse) 7 (2013), no.120, 5969-5976.

[5] D. S. Kim, T. Kim, q-Bernoulli polynomials and q-umbral calculus, Sci. China Math. 57 (2014),no. 9, 1867-1874.

[6] D. S. Kim, T. Kim, T. Komatsu, S.-H. Lee, Barnes-type Daehee of the first kind and poly-Cauchy of the first kind mixed-type polynomials, Adv. Difference Equ. 2014:140 (2014).

[7] D. S. Kim, T. Kim, S.-H. Lee, J.-J. Seo, Higher-order Daehee numbers and polynomials, Int.J. Math. Anal. (Ruse) 8 (2014), no. 6, 273-283.

[8] D. S. Kim, T. Kim, J. J. Seo, Higher-order Daehee polynomials of the first kind with umbralcalculus, Adv. Stud. Contemp. Math. (Kyungshang) 24 (2014), no. 1, 5-18.

[9] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), no. 3, 288-299.[10] T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,

Russ. J. Math. Phys. 15 (2008), no. 1, 51-57.[11] T. Kim, S.-H. Lee, T. Mansour, J.-J. Seo, A Note on q-Daehee polynomials and numbers, Adv.

Stud. Contemp.Math. 24 (2014), no. 2, 155-160.[12] J. Kwon, J.-W. Park, S.-S. Pyo, S.-H. Rim, A note on the modified q-Euler polynomials, JP J.

Algebra Number Theory Appl. 31 (2013), no. 2, 107-117.[13] E.-J. Moon, J.-W. Park, S.-H. Rim, A note on the generalized q-Daehee numbers of higher

order, Proc. Jangjeon Math. Soc. 17 (2014), no. 4, 557-565.[14] H. Ozden, I. N. Cangul, Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee

numbers, Adv. Stud. Contemp. Math. (Kyungshang) 18 (2009), no. 1, 41-48.[15] J.-W. Park, On the twisted Daehee polynomials with q-parameter, Adv. Difference Equ.

2014:304 (2014).[16] J.-W. Park, S.-H.Rim, J. Kwon, The twisted Daehee numbers and polynomials, Adv. Difference

Equ. 2014:1 (2014).[17] C. S. Ryoo, T. Kim, A new identities on the q-Bernoulli numbers and polynomials, Adv. Stud.

Contemp. Math. (Kyungshang) 21 (2011), no. 2, 161-169.[18] J. J. Seo, S. H. Rim, T. Kim, S. H. Lee, Sums products of generalized Daehee numbers, Proc.

Jangjeon Math. Soc. 17 (2014), no. 1, 1-9.[19] Y. Simsek, S.-H. Rim, L.-C. Jang, D.-J. Kang, J.-J. Seo, A note on q-Daehee sums, J. Anal.

Comput. 1 (2005), no. 2, 151-160.[20] H.M. Srivastava, T. Kim, Y.Simsek, q-Bernoulli numbers and polynomials associated with mul-

tiple q-zeta functions and basic L-series, Russ. J. Math. Phys. 12 (2005), no. 2, 241-268.

School of Mathematical Sciences, Nankai University, Tianjin Ciy, 300071, ChinaE-mail address: [email protected]



A class of BVPS for second-order impulsive

integro-differential equations of mixed type in

Banach space∗

Jitai Liang1 Liping Wang2 Xuhuan Wang2,3

1Yunnan Normal University Business School,Kunming, Yunnan, 650106, P.R. China2Department of Mathematics, Pingxiang University, Pingxiang, Jiangxi, 337000, P.R. China

3Department of Mathematics, Baoshan University, Baoshan, Yunnan, 678000, P.R. China

September 15, 2015

This paper is concerned with a class of boundary value problems for the non-linear mixed impulsive integro-differential equations with the derivative u

′and

deviating arguments in Banach space by using the cone theory and upper andlower solutions method together with monotone iterative technique. Sufficientconditions are established for the existence of extremal solutions of the givenproblem.

Keywords Integro-differential equations; cone; upper and lower solutions;monotone iterative technique; Impulsive

Mathematics Subject Classifications (2000) 34B15, 34B37.

1 Introduction

Impulsive differential equations have become more important in recent years insome mathematical models of real processes and phenomena studied in physics,chemical technology, population dynamics, biotechnology and economics etc.and there have appeared many papers (see [1-28]) and the references therein).There has been a significant development in impulse theory. Especially, thereis an increasing interest in the study of nonlinear mixed integro-differential

∗Project supported by the scientific research of yunnan province department of educa-tion(Grant No. 2013Y493), the NSF of Yunnan Province (Grant No.2013FD054) and theNSF of Jiangxi Province (Grant No.20144BAB2020010). Email: [email protected];[email protected].

1


331 Jitai Liang et al 331-344

equations with deviating arguments and multipiont BVPS[7-14] for impulsivedifferential equations.

In this article, we are concerned with the following BVPS for the nonlin-ear mixed impulsive integro-differential equations with the derivative u

′and

deviating arguments in Banach space E:u′′(t) = f(t, u(t), u(α(t)), u′(t), Tu, Su) t 6= tk, t ∈ J = [0, 1]∆u(tk) = Qku′(tk) k = 1, 2, · · · ,m∆u′(tk) = Ik(u′(tk), u(tk)) k = 1, 2, · · · ,mu(0) = λ1u(1) + k1 u′(0) = λ2u

′(1) + λ3

∫ 1

0w(s, u(s))ds + µu′(η) + k2

(1.1)where 0 = t0 < t1 < t2 < · · · < tk < · · · < tm < tm+1 = 1, f ∈ C(J ×

E5, E), Ik ∈ C(E × E,E), Qk ≥ 0, (Tu)(t) =∫ β(t)

0

k(t, s)u(γ(s))ds, (Su)(t) =∫ 1

0

h(t, s)u(δ(s))ds, D = (t, s) ∈ J2 | 0 ≤ s ≤ β(t), k ∈ C(D,R+), and h ∈

C(J2, R+), w ∈ (J×E,E), α, β, γ, δ ∈ C(J, J), ∆u(tk) = u(t+k )−u(t−k ),∆u′(tk) =u′(t+k )− u′(t−k ), 0 ≤ η ≤ 1, 0 ≤ µ, 0 < λ1, λ2 < 1, 0 ≤ λ3, k1, k2 ∈ E.

The article is organized as follow. In section 2, we establish comparison prin-ciples and lemmas. In Section 3, we prove the existence of the result of minimaland maximal solutions for the first order impulsive differential equations, whichnonlinearly involve the operator A by using upper and lower solutions, i.e. The-orem 3.1. In Section 4, we obtain the main results (Theorem4.1) by applyingTheorem 3.1, that is the existence of the theorem of minimal and maximalsolutions of (1.1).

2 Preliminaries and lemmas

Let PC(J,E) = x : J → E;x(t) is continuous everywhere expect for some tk atwhich x(t+k ) and x(t−k ) exist and x(tk) = x(t−k ), k = 1, 2, · · · ,m ;PC1(J,E) =x ∈ PC(J,E) : x′(t) is continuous everywhere expect for some tk at whichx′(t+k ) and x′(t−k ) exist and x′(tk) = x′(t−k ), k = 1, 2, · · · ,m. Evidently,PC(J,E)and PC1(J,E) are Banach spaces with the norms ‖ x ‖PC= sup|x(t)| : t ∈ Jand ‖ x ‖PC1= max‖ x ‖PC , ‖ x′ ‖PC. Let J− = J\tk, k = 1, 2, · · · ,m,Ω = PC1(J,E) ∩ C2(J−, E).

If P is a normal cone in E, then Pc = x ∈ PC(J,E) | x(t) ≥ θ,∀t ∈ J is anormal cone in PC(J,E), P ∗ = f ∈ E∗ | f(x) ≥ 0,∀x ∈ P denotes the dualcone of P.

A function x ∈ Ω is called a solution of BVPS (1.1) if it satisfies Eq.(1.1).In this paper, we always assume that E is a real Banach space and P is a regularcone in E,and denote K0 = maxk(t, s), (t, s) ∈ D and H0 = maxh(t, s), (t, s) ∈J2.

We consider the following first order impulsive differential equation in Ba-

2



nach space E:x′(t) = f(t, Ax(t), Ax(α(t)), x(t), TAx, SAx) t 6= tk, t ∈ J = [0, 1]∆x(tk) = Ik(Ax(tk), x(tk)) k = 1, 2, · · · ,mx(0) = λ2x(1) + λ3

∫ 1

0w(s,Ax(s))ds + µx(η) + k2

(2.1)where f, Ik, T, S, w,Qk, tk, λ2, λ3, µ, k2 are difined as (1.1) and

Ax(t) =k1

1− λ1+

∫ 1

0

G(t, s)x(s)ds +m∑

k=1

G(t, tk)Qkx(tk)

with

G(t, s) =

1

1− λ1, 0 ≤ s ≤ t ≤ 1

λ1

1− λ1, 0 ≤ t ≤ s ≤ 1

Lemma 2.1 Suppose x ∈ PC(J,E) ∩ C1(J−, E) satisfies x′(t) + Mx(t) + M1Bx + M2Bx(α(t)) + M3TBx + M4SBx ≤ 0 t 6= tk, t ∈ J = [0, 1]∆x(tk) ≤ −LkBx(tk) k = 1, 2, · · · ,mx(0) ≤ λ2x(1)

(2.2)where

Bx(t) =∫ 1

0

G(t, s)x(s)ds +m∑

k=1

G(t, tk)Qkx(tk)

0 < λ1, λ2 < 1, Lk ≥ 0 and constants M,Mi(i = 1, 2, 3, 4) satisfy

M > 0, Mi ≥ 0, M+(M1+M2+M3K0+M4H0+m∑

k=1

Lk)(1

1− λ1+

m∑k=1

Qk

1− λ1) ≤ λ2.

(2.3)then x(t) ≤ θ for t ∈ J .(θ denotes the zero elment of E )Proof. For any given g ∈ P ∗, let y(t) = g(x(t)), then y ∈ PC(J,R)∩C1(J−, R)and y′(t) = g(x′(t)).In view of (2.2), we get y′(t) + My(t) + M1By + M2By(α(t)) + M3TBy + M4SBy ≤ 0 t 6= tk, t ∈ J = [0, 1]

∆y(tk) ≤ −LkBy(tk) k = 1, 2, · · · ,my(0) ≤ λ2y(1)

(2.4)We will show that y(t) ≤ 0, t ∈ J.

(i)Suppose to contrary that y(t) ≥ 0, y(t) 6≡ 0 for t ∈ J,In view of the first inequality of (2.4),we get y′(t) ≤ 0. And by the second one in(2.4),we obtain that y(t) is decreasing in J. Then 0 ≤ y(1) ≤ y(t) ≤ y(0). By thethird inequality of(2.4), we have y(1) > 0 and λ2 ≥ 1, which is contradiction.

3



(ii) Suppose there are t, t ∈ J such that y(t) > 0 and y(t) < 0.Let y(t∗) = min

t∈Jy(t) = −λ, then λ > 0.By (2.4),we get

y′(t) ≤ M + (M1 + M2)(∫ 1

0G(t, s)ds +

∑mk=1 G(t, tk)Qk)

+M3(∫ β(t)

0K(t, s)[

∫ 1

0G(s, r)dr +

∑mk=1 G(s, tk)Qk]ds)

+M4(∫ 1

0H(t, s)[

∫ 1

0G(s, r)dr +

∑mk=1 G(s, tk)Qk]ds)λ

≤ M + (M1 + M2)(1

1− λ1+

m∑k=1

Qk

1− λ1)

+M3K0(1

1− λ1+

m∑k=1

Qk

1− λ1) + M4H0(

11− λ1

+m∑

k=1

Qk

1− λ1)λ

≤ [M + (M1 + M2 + M3K0 + M4H0)(1

1− λ1+

m∑k=1

Qk

1− λ1)]λ t 6= tk

∆y(tk) ≤ −LkBy(tk)) ≤ λLk(1

1− λ1+

m∑k=1

Qk

1− λ1) k = 1, 2, · · · ,m

Case 1 If t∗ ∈ [0, t), integrating from t∗ to t,we get

0 < y(t) = y(t∗) +∫ t

t∗

y′(s)ds +∑

t∗≤tk<t

∆y(tk)

≤ −λ + [M + (M1 + M2 + M3K0 + M4H0)

(1

1− λ1+

m∑k=1

Qk

1− λ1)]λ−

∑t∗≤tk<t

LkBy(tk)

≤ −λ + [M + (M1 + M2 + M3K0 + M4H0)

(1

1− λ1+

m∑k=1

Qk

1− λ1)]λ + λ

m∑k=1

Lk(1

1− λ1+

m∑k=1

Qk

1− λ1)

Hence

1 < [M+(M1+M2+M3K0+M4H0)(1

1− λ1+

m∑k=1

Qk

1− λ1)]+

m∑k=1

Lk(1

1− λ1+

m∑k=1

Qk

1− λ1)

It is contradiction to (2.3).Case 2 If t∗ ∈ [t, 1], we have

0 < y(t) = y(0) +∫ t

0

y′(s)ds +∑

0<tk<t

∆y(tk)

≤ y(0) +∫ t

0

[M + (M1 + M2 + M3K0 + M4H0)

(1

1− λ1+

m∑k=1

Qk

1− λ1)]λds + λ

∑0<tk<t

Lk(1

1− λ1+

m∑k=1

Qk

1− λ1)

4



y(1) = u(t∗) +∫ 1

t∗

y′(s)ds +∑

t∗≤tk<1

∆y(tk)

≤ −λ +∫ 1

t∗

[M + (M1 + M2 + M3K0 + M4H0)

(1

1− λ1+

m∑k=1

Qk

1− λ1)]λds + λ

∑t∗≤tk<1

Lk(1

1− λ1+

m∑k=1

Qk

1− λ1)

Hence

−λ +1λ2

∫ 1

t∗

[M + (M1 + M2 + M3K0 + M4H0)

(1

1− λ1+

m∑k=1

Qk

1− λ1)]λds +

1λ2

λ∑

t∗≤tk<1

Lk(1

1− λ1+

m∑k=1

Qk

1− λ1)

> −λ +∫ 1

t∗

[M + (M1 + M2 + M3K0 + M4H0)

(1

1− λ1+

m∑k=1

Qk

1− λ1)]λds + λ

∑t∗≤tk<1

Lk(1

1− λ1+

m∑k=1

Qk

1− λ1)

≥ y(1) ≥ 1λ2

y(0)

> − 1λ2

∫ t

0

y′(s)ds− 1λ2

∑0<tk<t

∆y(tk)

≥ − 1λ2

∫ t

0

[M + (M1 + M2 + M3K0 + M4H0)

(1

1− λ1+

m∑k=1

Qk

1− λ1)]λds− 1

λ2λ

∑0<tk<t

Lk(1

1− λ1+

m∑k=1

Qk

1− λ1)

≥ − 1λ2

λ

∫ t∗

0

([M + (M1 + M2 + M3K0 + M4H0)

(1

1− λ1+

m∑k=1

Qk

1− λ1)]ds− 1

λ2λ

∑0<tk<t∗

Lk(1

1− λ1+

m∑k=1

Qk

1− λ1).

We obtain that M +(M1+M2+M3K0+M4H0+m∑

k=1

Lk)(1

1− λ1+

m∑k=1

Qk

1− λ1) >

λ2 which is contradiction.Since g ∈ P ∗ is arbitrary, we have x(t) ≤ θ, ∀t ∈ J.We complete the proof.

Lemma 2.2 Assume that (2.3) is satisfied.Let ek, a ∈ E, σ ∈ PC(J,E).Then the linear problem x′(t) = −Mx(t)−M1Ax−M2Ax(α(t))−M3TAx−M4SAx + σ(t) t 6= tk, t ∈ J = [0, 1]

∆x(tk) = −LkAx(tk) + ek k = 1, 2, · · · ,mx(0) = λ2x(1) + a

(2.5)

5



has a unique solution x ∈ PC1(J,E) if and only if x ∈ PC(J,E) is a solutionof the integral equation:

x(t) = aDe−Mt +∫ 1

0H(t, s)(σ(s)−M1Ax(s)−M2Ax(α(s))

−M3TAx(s)−M4SAx(s))ds +m∑

k=1

H(t, tk)(−LkAx(tk) + ek),

(2.6)where D = (1− λ2e

−M )−1,

H(t, s) =

De−M(t−s), 0 ≤ s ≤ t ≤ 1,

Dλ2e−M(1+t−s), 0 ≤ t ≤ s ≤ 1.

(2.7)

Proof. First, differentiating (2.6), we have

x′(t) = (aDe−Mt +∫ 1

0H(t, s)(σ(s)−M1Ax(s)−M2Ax(α(s))

−M3TAx(s)−M4SAx(s))ds +m∑

k=1

H(t, tk)(−LkAx(tk) + ek))′

= −M(t)[aDe−Mt +∫ 1

0H(t, s)(σ(s)−M1Ax(s)

−M2Ax(α(s))−M3TAx(s)−M4SAx(s))ds

+m∑

k=1

H(t, tk)(−LkAx(tk) + ek)]−M1Ax

−M2Ax(α(t))−M3TAx−M4SAx + σ(t)= −M(t)x(t)−M1Ax(t)−M2Ax(α(t))−M3TAx(t)−M4SAx(t) + σ(t)

∆x(tk) = x(t+k )− x(t−k )=

∑0<tj<tk

∆x(tj)−∑

0<tj<t−k

∆x(tj)

=k∑

j=1

(−LjAx(tj) + ej)−k−1∑j=1

(−LjAx(tj) + ej)

= −LkAx(tk) + ek.

Also

x(0) = λ2D∫ 1

0e−M(1−s)(σ(s)−M1Ax(s)−M2Ax(α(s))

−M3TAx(s)−M4SAx(s))ds + λ2D∑m

k=1 e−M(1−tk)∆x(tk) + aD

x(1) = D∫ 1

0e−M(1−s)(σ(s)−M1Ax(s)−M2Ax(α(s))

−M3TAx(s)−M4SAx(s))ds + e−MaD∑m

k=1 e−M(1−tk)∆x(tk) + aD.

It is easy to check that x(0) = λ2x(1) + a.Hence, we know that (2.6) is a solution of (2.5).Next we show that the solution of (2.5) is unique. Let x1, x2 are the solutionsof (2.5) and set p = x1 − x2, we get

p′ = x′1 − x′2= −Mx1(t)−M1Ax1 −M2Ax1(α(t))−M3TAx1 −M4SAx1 + σ(t)

−(−Mx2(t)−M1Ax2 −M2Ax2(α(t))−M3TAx2 −M4SAx2 + σ(t))= −Mp−M1Ap−M2Ap(α(t))−M3TAp−M4SAp,

6



∆p(tk) = ∆x1 −∆x2

= −LkAx1(tk) + ek − (−LkAx2(tk) + ek)= −LkAp(tk),

p(0) = x1(0)− x2(0)= λ2x1(T ) + a− (λ2x2(1) + a)= λ2p(1).

In view of Lemma 2.1, we get p ≤ θ which implies x1 ≤ x2. Similarly, we havex1 ≥ x2. Hence x1 = x2. The proof is complete.

3 Results for first order impulsive differentialequation

For convenience, let us list the following conditions:(H1) There exit x0, y0 ∈ PC1(J,E) satisfying

x′0(t) ≤ f(t, Ax0(t), Ax0(α(t)), x0(t), TAx0, SAx0) t 6= tk, t ∈ J = [0, 1]∆x0(tk) ≤ Ik(Ax0(tk), x0(tk)) k = 1, 2, · · · ,mx0(0) ≤ λ2x0(1) + λ3

∫ 1

0w(s,Ax0(s))ds + µx0(η) + k2

y′0(t) ≥ f(t, Ay0(t), Ay0(α(t)), y0(t), TAy0, SAy0) t 6= tk, t ∈ J = [0, 1]∆y0(tk) ≥ Ik(Ay0(tk), y0(tk)) k = 1, 2, · · · ,my0(0) ≥ λ2y0(T ) + λ3

∫ 1

0w(s,Ay0(s))ds + µy0(η) + k2

(3.1)(H2)

f(t, x, y, z, u, v)− f(t, x, y, z, u, v)≥ −M1(x− x)−M2(y − y)−M(z − z)−M3(u− u)−M4(v − v) (3.2)

Ik(x, z)− Ik(x, z) ≥ −Lk(x− x) (3.3)

Where Ax0 ≤ x ≤ x ≤ Ay0, Ax0(α(t)) ≤ y ≤ y ≤ Ay0(α(t)), x0 ≤ z ≤ z ≤ y0,TAx0 ≤ u ≤ u ≤ TAy0, SAx0 ≤ v ≤ v ≤ SAy0, ∀t ∈ J.(H3) Constants Lk,M,Mi, i = 1, 2, 3, 4 satisfy (2.3).(H4) Assume that a(t) is non-negative integral function, such that

w(t, Au)− w(t, Au) ≥ a(t)(Au−Au) (3.4)

Where x0 ≤ u ≤ u ≤ y0.If x0, y0 ∈ PC1(J,E) and x0 ≤ y0, t ∈ J ,then the interval [x0, y0] denotes theset

x ∈ PC1(J,E) : x0(t) ≤ x(t) ≤ y0(t), t ∈ JTheorem 3.1 Assume the hypotheses (H1) − (H4) hold. Then Eq.(2.1) hasthe extremal solutions x∗(t), y∗(t) ∈ [x0, y0]. Moreover there exist two iterativesequences xn and yn satisfying

x0 ≤ x1 ≤ · · · ≤ xn ≤ · · · ≤ yn ≤ · · · ≤ y1 ≤ y0 (3.5)

7



such that xn,yn uniformly converge in PC(J,E)⋂

C1(J−, E) to x∗, y∗, re-spectively.Proof. For z ∈ [x0, y0], considering (2.5) withσ(t) = f(t, Az(t), Az(α(t)), z, TAz, SAz)+M(t)z(t)+M1Az(t)+M2Az(α(t))+M3TAz + M4SAz,ek = Ik(Az(tk), z(tk)) + LkAz(tk),a = λ3

∫ T

0w(s,Az(s)) + µz(η)ds + k2.

By Lemma 2.2, the BVPS has a unique solution z ∈ [x0, y0] .We define an operator ϕ by x = ϕz, then ϕ is an operator from [x0, y0] toPC(J,E).We claim that(a) x0 ≤ ϕx0 , ϕy0 ≤ y0,(b) ϕ is nondecreasing on [x0, y0].We prove (a), let x1 = ϕx0, p(t) = x0(t)− x1(t)

p′ = x′0 − x′1≤ f(t, Ax0(t), Ax0(α(t)), x0(t), TAx0, SAx0)− [f(t, Ax0(t), Ax0(α(t)), x0(t), TAx0, SAx0)

+M(x0(t)− x1(t)) + M1(Ax0(t)−Ax1(t)) + M2(Ax0(α(t))−Ax1(α(t)))+M3(TAx0 − TAx1) + M4(SAx0 − SAx1)]

= −Mp(t)−M1Ap−M2Ap(α(t))−M3TAp−M4SAp,

∆p(tk) = ∆x0(tk)−∆x1(tk)≤ Ik(Ax0(tk), x0(tk))− [Ik(Ax0(tk), x0(tk))− Lk(Ax1 −Ax0)]= −LkAp(tk),

p(0) = x0(0)− x1(0)≤ λ2x0(1) + µu0(η) + λ3

∫ 1

0w(s,Ax0(s))ds + k2

−(λ2u1(1) + µu0(η) + λ3

∫ 1

0w(s,Ax0(s))ds + k2)

= λ1p(1).

By Lemma 2.1, we have p ≤ θ. That is x0 ≤ ϕx0. Similarly, we can proveϕy0 ≤ y0.To prove (b), let x1 = ϕx0, y1 = ϕy0, p = x1 − y1, then

p′(t) = x′1 − y′1= f(t, Ax0(t), Ax0(α(t)), x0(t), TAx0, SAx0) + M(x0(t)− x1(t))

+M1(Ax0(t)−Ax1(t)) + M2(Ax0(α(t))−Ax1(α(t)))+M3(TAx0 − TAx1) + M4(SAx0 − SAx1)−[f(t, Ay0(t), Ay0(α(t)), y0(t), TAy0, SAy0)+M(y0(t)− y1(t)) + M1(Ay0(t)−Ay1(t))+M2(Ay0(α(t))−Ay1(α(t))) + M3(TAy0 − TAy1) + M4(SAy0 − SAy1)]

≤ −Mp(t)−M1Ap−M2Ap(α(t))−M3TAp−M4SAp,

∆p(tk) = ∆x1(tk)−∆y1(tk)= −LkAx1 + Ik(Ax0(tk), x0(tk)) + LkAx0

−(−LkAy1 + Ik(Ay0(tk), y0(tk)) + LkAy0)≤ −Lk(Ax0 −Ay0) + LkAx0 − LkAy0 − LkAp≤ −LkAp(tk),

8



p(0) = x1(0)− y1(0)≤ λ2x1(1) + µx0(η) + λ3

∫ 1

0w(s,Ax0(s))ds + k2

−(λ2y1(1) + µy0(η) + λ3

∫ 1

0w(s,Ay0(s))ds + k2)

= λ2p(1) + µ(x0(η)− y0(η)) + λ3

∫ 1

0a(s)(Ax0(s)−Ay0(s))ds

≤ λ1p(1).

In view of Lemma 2.1 , we know ϕx0 ≤ ϕy0. Hence (b) holds.We define two sequences xn, yn

xn+1 = ϕxn, yn+1 = ϕyn, (n = 0, 1, 2, · · ·)

By (a) and (b), we know that (3.5) holds.And each xn, yn satisfies

x′n(t) = f(t, Axn−1(t), Axn−1(α(t)), xn−1(t), TAxn−1, SAxn−1)+M(xn−1(t)− x1(t)) + M1(Axn−1(t)−Axn(t)) + M2(Axn−1(α(t))−Axn(α(t)))+M3(TAxn−1 − TAxn) + M4(SAxn−1 − SAxn) t 6= tk, t ∈ J = [0, 1]∆xn(tk) = −LkAxn(tk) + Ik(Axn−1(tk), xn−1(tk)) + LkAxn−1(tk) k = 1, 2, · · · ,mxn(0) = λ2xn(1) + λ3

∫ 1

0w(s,Axn−1(s))ds + µxn−1(η) + k2

(3.6)y′n(t) = f(t, Ayn−1(t), Ayn−1(α(t)), yn−1(t), TAyn−1, SAyn−1)+M(yn−1(t)− y1(t)) + M1(Ayn−1(t)−Ayn(t)) + M2(Ayn−1(α(t))−Ayn(α(t)))+M3(TAyn−1 − TAyn) + M4(SAyn−1 − SAyn) t 6= tk, t ∈ J = [0, 1]∆yn(tk) = −LkAyn(tk) + Ik(Ayn−1(tk), yn−1(tk)) + LkAyn−1(tk) k = 1, 2, · · · ,myn(0) = λ2yn(1) + λ3

∫ 1

0w(s,Ayn−1(s))ds + µyn−1(η) + k2.

(3.7)By virtue of the regularity of the cone P, we obtain that there exist x∗, y∗ ∈[x0, y0] such that

limn−→∞

xn(t) = x∗(t) limn−→∞

yn(t) = y∗(t) (3.8)

and xn|n = 1, 2, · · · is a bounded subset in PC(J,E).Let X = xn|n = 1, 2, · · ·, X(t) = xn(t)|n = 1, 2, · · · t ∈ J, in view of (3.8)we get

α(X(t)) = 0 t ∈ J

which implies that X(t) is relatively compact for t ∈ J.For any z ∈ [x0, y0], by (H1) (H2) we have

x′0(t) + Mx0(t) + M1Ax0(t) + M2Ax0(α(t)) + M3TAx0 + M4SAx0

≤ f(t, Ax0(t), Ax0(α(t)), x0(t), TAx0, SAx0) + Mx0(t)+M1Ax0(t) + M2Ax0(α(t)) + M3TAx0 + M4SAx0

≤ f(t, Az0(t), Az0(α(t)), z0(t), TAz0, SAz0) + Mz0(t)+M1Az0(t) + M2Az0(α(t)) + M3TAz0 + M4SAz0

≤ f(t, Ay0(t), Ay0(α(t)), y0(t), TAy0, SAy0) + My0(t)+M1Ay0(t) + M2Ay0(α(t)) + M3TAy0 + M4SAy0

≤ y′0(t) + My0(t) + M1Ay0(t) + M2Ay0(α(t)) + M3TAy0 + M4SAy0.

9



In view of the normality of the cone Pc, we get that there exists a constantC > 0, such that

‖ f(t, Az0(t), Az0(α(t)), z0(t), TAz0, SAz0) + Mz0(t)+M1Az0(t) + M2Az0(α(t)) + M3TAz0 + M4SAz0 ‖≤ C,

∀z ∈ [x0, y0], t ∈ J. From (3.5) (3.6), it is obviously to show that x′n|n =1, 2, · · · is a bounded subset in PC(J,E). It follows in view of the mean valuetheorem that X is equicontinuous on Jk, k = 0, 1, 2, · · · ,m. So we obtain byvirtue of Ascoli-Arzela’s theorem and α(X(t)) = 0 that α(X) = supt∈J α(X(t)) =0 which implies X is relatively compact in PC(J,E) and so there exists a se-quence of xn(t) which converges uniformly on J to x∗(t). Since xn|n =1, 2, · · · is nondecreasing and the cone Pc is normal, we get that xn|n =1, 2, · · · itself converges uniformly on J to x∗(t), which implies x∗ ∈ PC(J,E).By the lemma 2.2 and (3.6), we see that x∗ satisfies (2.1).

Similarly, we also can prove that yn converges uniformly on J to y∗(t), andy∗ satisfies (2.1).

Finally, we assert that if z ∈ [x0, y0] is any solution of Eq.(2.1),then x∗(t) ≤z(t) ≤ y∗(t) on J. We will prove that if xn ≤ z ≤ yn, for n = 0, 1, 2, · · · , thenxn+1(t) ≤ z(t) ≤ yn+1(t).

Letting p(t) = xn+1(t)− z(t), then p′(t) ≤ −Mp(t)−M1Ap−M2Ap(α(t))−M3TAp−M4SAp ≤ 0 t 6= tk, t ∈ J = [0, 1]∆p(tk) ≤ −LkAp(tk) k = 1, 2, · · · ,mp(0) ≤ λ2p(1)

By Lemma 2.1 , we have p(t) ≤ θ for all t ∈ J, that is xn+1(t) ≤ z(t). Simi-larly , we can prove z(t) ≤ yn+1(t). for all t ∈ J. Thus xn+1(t) ≤ z(t) ≤ yn+1(t)for all t ∈ J, which implies x∗(t) ≤ z(t) ≤ y∗(t).The proof is complete.Remark In (2.1), if w(s,Ax(s)) = a(s)Ax(s), where a(t) is non-negative in-tegral function ,then (H4) is not required in Theorem 3.1, and we have thefollowing theorem.Theorem 3.2 Suppose that conditions (H1)− (H3)are satisfied. In additionalthat x0, y0 ∈ PC1(J,E) be such that x0 ≤ y0. Then the conclusion of Theorem3.1 holds.The proof is almost similar to theorem 3.1, so we omit it.

4 Results for second order impulsive differentialequation

In this section, we prove the existence theorem of maximal and minimal solutionsof (1.1) by applying Theorem 3.1 in Section 3.

Let us list other conditions for convenience.

10



(G1) There exists u0, v0 ∈ Ω, satisfying u0(t) ≤ v0(t), u′0(t) ≤ v′0(t),u′′0(t) ≤ f(t, u0(t), u0(α(t)), u0(t), Tu0, Su0) t 6= tk, t ∈ J = [0, 1]∆u0(tk) = Qku′0(tk)∆u′0(tk) ≤ Ik(u0(tk), u′0(tk)) k = 1, 2, · · · ,mu0(0) = λ1u0(1) + k1

u′0(0) ≤ λ2u′0(1) + λ3

∫ 1

0w(s, u0(s))ds + µu′0(η) + k2

(4.1)

and v0 satisfies inverse inequalities of (4.1)(G2)

f(t, x, y, z, u, SAv)− f(t, x, y, z, u, v)≥ −M1(x− x)−M2(y − y)−M(z − z)−M3(u− u)−M4(v − v) (4.2)

Ik(x, z)− Ik(x, z) ≥ −Lk(x− x) (4.3)

Where u0 ≤ x ≤ x ≤ v0, u0(α(t)) ≤ y ≤ y ≤ v0(α(t)), u′0 ≤ z ≤ z ≤ v′0,Tu0 ≤ u ≤ u ≤ Tv0, Su0 ≤ v ≤ v ≤ Sv0, ∀t ∈ J.(G3) Constants Lk,M,Mi, i = 1, 2, 3, 4 satisfy (2.3).(G4) Assume that a(t) is non-negative integral function, such that

w(t, u)− w(t, u) ≥ a(t)(u− u) (4.4)

Where u0 ≤ u ≤ u ≤ v0.Let Λ = z ∈ [x0, y0]

⋂PC1(J,E) | u′0(t) ≤ z′(t) ≤ v′0(t).

Theorem 4.1 Assume the conditions (G1)− (G4) hold.Then Eq.(1.1) has min-imal and maximal solutions u∗, v∗ ∈ Ω in Λ.Proof. In Eq.(1.1),let u′(t) = x(t). Then (1.1) is equivalent to the followingsystem:

u′(t) = x(t)x′(t) = f(t, u, u(α), x, Tu(t), Su(t))∆u(tk) = Qkx(tk)∆x(tk) = Ik(x(tk), u(tk))u(0) = λ1u(1) + k1

x(0) = λ2x(1) + λ3

∫ 1

0w(s, u(s))ds + µx(η) + k2

(4.5)

For x ∈ PC(J,E), the system u′(t) = x(t)∆u(tk) = Qkx(tk)u(0) = λ1u(1) + k1

(4.6)

has a unique solution x ∈ PC(J,E)⋂

C1(J−, E), which satisfies

u(t) =k1

1− λ1+

∫ 1

0

G(t, s)x(s)ds +m∑

k=1

G(t, tk)Qkx(tk) (4.7)

11



It is easy to prove, so we omit it .Define an operator A by u = Ax(t), t ∈ J. It is easy to show thatA : PC(J,E)

⋂C1(J−, E) −→ Ω is continuous and nondecreasing .

Hence, from (4.5)-(4.7), Eq.(1.1) is transformed into first order boundary valueproblem (2.1).

Let x0 = u′0, y0 = v′0, by (G1) we have x0 ≤ y0 and

u0(t) =k1

1− λ1+

∫ 1

0

G(t, s)x0(s)ds +m∑

k=1

G(t, tk)Qkx0(tk) (4.8)

v0(t) =k1

1− λ1+

∫ 1

0

G(t, s)y0(s)ds +m∑

k=1

G(t, tk)Qky0(tk) (4.9)

which imply that u0 = Ax0, v0 = Ay0, and x0, y0 satisfies (H1).By the condition (G2) (G4) it is easy to see that (H2) (H4) hold .Therefore, it follows from Theorem 3.1 that (2.1) has minimal and maximalsolutions x∗, y∗ ∈ PC(J,E)

⋂C1(J−, E) in [x0, y0].

Let u∗ = Ax∗, v∗ = Ay∗, then u∗, v∗ ∈ Ω and

u∗(t) =k1

1− λ1+

∫ 1

0

G(t, s)x∗(s)ds +m∑

k=1

G(t, tk)Qkx∗(tk) (4.10)

In view of (4.10), we have u∗′(t) = x∗(t)

∆u∗(tk) = Qkx∗(tk)u∗(0) = λ1u

∗(1) + k1

(4.11)

The fact that x∗ satisfies (2.1) and u∗ satisfies (4.11) implies u∗ is a solution of(1.1). Similarly, we can prove v∗ is a solution of (1.1).It is easy to show that u∗, v∗ ∈ Ω are minimal and maximal solutions for (1.1)in Λ. We complete the proof.Remark In (1.1), if w(s, x(s)) = a(s)x(s), where a(t) is non-negative integralfunction, then (H4) is not required in Theorem 4.1, and we have the followingtheorem.Theorem 4.2 Suppose that conditions (G1) − (G3)are satisfied. Then theconclusion of Theorem 4.1 holds.The proof is almost similar to theorem 4.1, so we omit it.

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14



DISTRIBUTION AND SURVIVAL FUNCTIONS WITHAPPLICATIONS IN INTUITIONISTIC RANDOM LIE C∗-ALGEBRAS

AFRAH A. N. ABDOU, YEOL JE CHO*, AND REZA SAADATI

Abstract. In this paper, first, we consider the distribution and survival functions andwe define intuitionistic random Lie C∗-algebras. As an application, using the fixed pointmethod, we approximate the derivations on intuitionistic random Lie C∗-algebras for thethe following additive functional equation

m∑i=1

f(mxi +

m∑j=1, j =i

xj

)+ f

( m∑i=1

xi

)= 2f

( m∑i=1

mxi

)for all m ∈ N with m ≥ 2.

1. Introduction

Distribution and survival functions are important in probability theory. In this pa-per, we use these functions to define intuitionistic random Lie C∗-algebras and find anapproximation of an m-variable functional equation.

2. Preliminaries

Now, we give some definitions and lemmas for our main results in this paper.

Definition 2.1. A function µ : R → [0, 1] is called a distribution function if it is leftcontinuous on R, non-decreasing and

inft∈R

µ(t) = 0, supt∈R

µ(t) = 1.

We denote by D the family of all measure distribution functions and by H a specialelement of D defined by

H(t) =

0, if t ≤ 0,

1, if t > 0.

Forward, µ(x) is denoted by µx.

Definition 2.2. A function ν : R → [0, 1] is called a survival function if it is rightcontinuous on R, non-increasing and

inft∈R

ν(t) = 0, supt∈R

ν(t) = 1.

2010 Mathematics Subject Classification. Primary 39A10, 39B52, 39B72, 46L05, 47H10, 46B03.Key words and phrases. intuitionistic random normed spaces, additive functional equation, fixed point,

derivations, random C∗-algebras, random Lie C∗-algebras, approximation.*The corresponding author.

1


345 ABDOU et al 345-354

2 AFRAH A. N. ABDOU, YEOL JE CHO, AND REZA SAADATI

We denote by B the family of all survival functions and by G a special element of Bdefined by

G(t) =

1, if t ≤ 0,

0, if t > 0.

Forward, ν(x) is denoted by νx.

Lemma 2.3. ([1]) Consider the set L∗ and the operation ≤L∗ defined by:

L∗ = (x1, x2) : (x1, x2) ∈ [0, 1]2 and x1 + x2 ≤ 1,

(x1, x2) ≤L∗ (y1, y2)⇐⇒ x1 ≤ y1, x2 ≥ y2

for all (x1, x2), (y1, y2) ∈ L∗. Then (L∗,≤L∗) is a complete lattice.

We denote the bottom and the top elements of lattices by 0L∗ = (0, 1) and 1L∗ = (1, 0).Classically, the triangular norm ∗ = T on [0, 1] is defined as an increasing, commutativeand associative mapping T : [0, 1]2 −→ [0, 1] satisfying

T (1, x) = 1 ∗ x = x

for all x ∈ [0, 1]. The triangular conorm S = ⋄ is defined as an increasing, commutative,associative mapping S : [0, 1]2 −→ [0, 1] satisfying S(0, x) = 0 ⋄ x = x for all x ∈ [0, 1].

Using the lattice (L∗,≤L∗), these definitions can be straightforwardly extended.

Definition 2.4. ([1]) A triangular norm (t–norm) on L∗ is a mapping T : (L∗)2 −→ L∗

satisfying the following conditions:(1) for all x ∈ L∗, T (x, 1L∗) = x (: boundary condition);(2) for all (x, y) ∈ (L∗)2, T (x, y) = T (y, x) (: commutativity);(3) for all (x, y, z) ∈ (L∗)3, T (x, T (y, z)) = T (T (x, y), z) (: associativity);(4) for all (x, x′, y, y′) ∈ (L∗)4, x ≤L∗ x′ and y ≤L∗ y′ =⇒ T (x, y) ≤L∗ T (x′, y′) (:

monotonicity).

In this paper, (L∗,≤L∗ , T ) has an Abelian topological monoid with the top element 1L∗

and so T is a continuous t–norm.

Definition 2.5. A continuous t–norm T on L∗ is said to be continuous representablet-norm if there exist a continuous t–norm ∗ and a continuous t–conorm ⋄ on [0, 1] suchthat, for all x = (x1, x2), y = (y1, y2) ∈ L∗,

T (x, y) = (x1 ∗ y1, x2 ⋄ y2).

For example,

T (a, b) = (a1b1,mina2 + b2, 1)and

M(a, b) = (mina1, b1,maxa2, b2)for all a = (a1, a2), b = (b1, b2) ∈ L∗ are the continuous representable t-norm.

Definition 2.6. (1) A negator on L∗ is any decreasing mapping N : L∗ −→ L∗ satisfyingN (0L∗) = 1L∗ and N (1L∗) = 0L∗ .

(2) If N (N (x)) = x for all x ∈ L∗, then N is called an involutive negator.


346 ABDOU et al 345-354

DISTRIBUTION AND SURVIVAL FUNCTIONS WITH APPLICATIONS 3

(3) A negator on [0, 1] is a decreasing mapping N : [0, 1] −→ [0, 1] satisfying N(0) = 1and N(1) = 0, where Ns denotes the standard negator on [0, 1] defined by

Ns(x) = 1− x

for all x ∈ [0, 1].

Definition 2.7. Let µ and ν be a distribution function and a survival function fromX × (0,+∞) to [0, 1] such that µx(t) + νx(t) ≤ 1 for all x ∈ X and t > 0. The 3-tuple (X,Pµ,ν , T ) is said to be an intuitionistic random normed space (briefly, IRN-space)if X is a vector space, T is a continuous representable t-norm and Pµ,ν is a mappingX × (0,+∞)→ L∗ satisfying the following conditions: for all x, y ∈ X and t, s > 0,

(1) Pµ,ν(x, 0) = 0L∗ ;(2) Pµ,ν(x, t) = 1L∗ if and only if x = 0;(3) Pµ,ν(αx, t) = Pµ,ν(x,

tα) for all α = 0;

(4) Pµ,ν(x+ y, t+ s) ≥L∗ T (Pµ,ν(x, t),Pµ,ν(y, s)).

In this case, Pµ,ν is called an intuitionistic random norm, where

Pµ,ν(x, t) = (µx(t), νx(t)).

Note that, if (X,Pµ,ν , T ) is an IRN-space and define Pµ,ν(x−y, t) =Mµ,ν(x, y, t), then

(X,Mµ,ν , T )

is an intuitionistic Menger spaces.

Example 2.8. Let (X, ∥ · ∥) be a normed space. Let T (a, b) = (a1b1,mina2 + b2, 1)for all a = (a1, a2), b = (b1, b2) ∈ L∗ and µ, ν be a distribution function and a survivalfunction defined by

Pµ,ν(x, t) = (µx(t), νx(t)) =( t

t+ ∥x∥,∥x∥

t+ ∥x∥

)for all t ∈ R+. Then (X,Pµ,ν , T ) is an IRN-space.

Definition 2.9. (1) A sequence xn in an IRN-space (X,Pµ,ν , T ) is called a Cauchysequence if, for any ε > 0 and t > 0, there exists n0 ∈ N such that

Pµ,ν(xn − xm, t) >L∗ (Ns(ε), ε)

for all n,m ≥ n0, where Ns is the standard negator.(2) A sequence xn in an IRN-space (X,Pµ,ν , T ) is said to be convergent to a point

x ∈ X (denoted by xnPµ,ν−→ x) if Pµ,ν(xn − x, t) −→ 1L∗ as n −→∞ for all t > 0.

(3) An IRN-space (X,Pµ,ν , T ) is said to be complete if every Cauchy sequence in X isconvergent to a point x ∈ X.

Definition 2.10. A intuitionistic random normed algebra (X,Pµ,ν , T , T ′) is a IRN-space(X,Pµ,ν , T ) with algebraic structure such that

(4) Pµ,ν(xy, ts) ≥ T ′(Pµ,ν(x, t),Pµ,ν(y, s)) for all x, y ∈ X and t, s > 0, in which T ′ isa continuous representable t-norm.


347 ABDOU et al 345-354


Every normed algebra (X, ∥ ·∥) defines a random normed algebra (X,µ, TM , TP ), where

Pµ,ν(x, t) =( t

t+ ∥x∥,∥x∥

t+ ∥x∥

)for all t > 0 if and only if

∥xy∥ ≤ ∥x∥∥y∥+ s∥y∥+ t∥x∥

for all x, y ∈ X and t, s > 0. This space is called the induced random normed algebra (see[6]). For more properties and example of theory of random normed spaces, we refer to[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].

Definition 2.11. Let (U ,Pµ,ν , T , T ′) be an intuitionistic random Banach algebra. Aninvolution on U is a mapping u→ u∗ from U into U satisfying the following conditions:

(1) u∗∗ = u for all u ∈ U ;(2) (αu+ βv)∗ = αu∗ + βv∗ for all u, v ∈ U and α, β ∈ C;(3) (uv)∗ = v∗u∗ for all u, v ∈ U .If, in addition, νu∗u(ts) = T ′(νu(t), νu(s)) for all u ∈ U and t, s > 0, then U is an

intuitionistic random C∗–algebra.

Now, we recall a fundamental result in fixed point theory.

Let Ω be a set. A function d : Ω× Ω→ [0,∞] is called a generalized metric on Ω if dsatisfies the following conditions:

(1) d(x, y) = 0 if and only if x = y for all x, y ∈ Ω;(2) d(x, y) = d(y, x) for all x, y ∈ Ω;(3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ Ω.

Theorem 2.12. ([2]) Let (Ω, d) be a complete generalized metric space and let J : Ω→ Ωbe a contractive mapping with Lipschitz constant L < 1. Then, for each given elementx ∈ Ω, either d(Jnx, Jn+1x) =∞ for all nonnegative integers n or there exists a positiveinteger n0 such that

(1) d(Jnx, Jn+1x) <∞ for all n ≥ n0;(2) the sequence Jnx converges to a fixed point y∗ of J ;(3) y∗ is the unique fixed point of J in the set Γ = y ∈ Ω | d(Jn0x, y) <∞;(4) d(y, y∗) ≤ 1

1−Ld(y, Jy) for all y ∈ Γ.

In this paper, using the fixed point method, we approximate the derivations on intu-itionistic random Lie C∗-algebras for the the following additive functional equation

m∑i=1

f(mxi +

m∑j=1, j =i

xj

)+ f( m∑

i=1

xi

)= 2f

( m∑i=1

mxi

)(2.1)

for all m ∈ N with m ≥ 2.

3. Approximation of derivations in intuitionistic random Lie C∗-algebras

In this section, we approximate the derivations on intuitionistic random Lie C∗-algebras(see also [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44]).


348 ABDOU et al 345-354


For any mapping f : A→ A, we define

Dωf(x1, · · · , xm) :=m∑i=1

µf(mxi +

m∑j=1, j =i

xj

)+ f(µ

m∑i=1

xi

)− 2f

(µ

m∑i=1

mxi

)for all ω ∈ T1 := ξ ∈ C : |ξ| = 1 and x1, · · · , xm ∈ A.

Note that a C-linear mapping δ : A→ A is called a derivation on intuitionistic randomC∗-algebras if δ satisfies δ(xy) = yδ(x) + xδ(y) and δ(x∗) = δ(x)∗ for all x, y ∈ A.

Now, we approximate the derivations on intuitionistic random Lie C∗-algebras for thefunctional equation Dωf(x1, · · · , xm) = 0.

Theorem 3.1. Let f : A→ A be a mapping for which there are functions φ : Am → L∗,ψ : A2 → L∗ and η : A→ L∗ such that

Pµ,ν(Dωf(x1, · · · , xm), t) ≥L φ(x1, · · · , xm, t), (3.1)

limj→∞

φ(mjx1, · · · ,mjxm,mjt) = 1L, (3.2)

Pµ,ν(f(xy)− xf(y)− xf(y), t) ≥L ψ(x, y, t), (3.3)

limj→∞

ψ(mjx,mjy,m2jt) = 1L, (3.4)

Pµ,ν(f(x∗)− f(x)∗, t) ≥L η(x, t), (3.5)

limj→∞

η(mjx,mjt) = 1L (3.6)

for all ω ∈ T1, x1, · · · , xm, x, y ∈ A and t > 0. If there exists R < 1 such that

φ(mx, 0, · · · , 0,mRt) ≥L φ(x, 0, · · · , 0, t) (3.7)

for all x ∈ A and t > 0, then there exists a unique derivation δ : A→ A such that

Pµ,ν(f(x)− δ(x), t) ≥L φ(x, 0, · · · , 0, (m−mR)t) (3.8)

for all x ∈ A and t > 0.

Proof. Consider the set X := g : A → A and introduce the generalized metric on Xdefined by

d(g, h) = infC ∈ R+ : Pµ,ν(g(x)− h(x), Ct) ≥L φ(x, 0, · · · , 0, t), ∀x ∈ A, t > 0.It is easy to show that (X, d) is complete.

Now, we consider the linear mapping J : X → X such that

Jg(x) :=1

mg(mx)

for all x ∈ A. By Theorem 3.1 of [46],

d(Jg, Jh) ≤ Rd(g, h)

for all g, h ∈ X. Letting ω = 1, x = x1 and x2 = · · · = xm = 0 in (3.1), we have

Pµ,ν(f(mx)−mf(x), t) ≥L φ(x, 0, · · · , 0, t) (3.9)

for all x ∈ A and t > 0 and so

Pµ,ν(f(x)−1

mf(mx), t) ≥L φ(x, 0, · · · , 0,mt)


349 ABDOU et al 345-354


for all x ∈ A and t > 0. Hence d(f, Jf) ≤ 1m. By Theorem 2.12, there exists a mapping

δ : A→ A such that(1) δ is a fixed point of J , i.e.,

δ(mx) = mδ(x) (3.10)

for all x ∈ A. The mapping δ is a unique fixed point of J in the set

Y = g ∈ X : d(f, g) <∞.This implies that δ is a unique mapping satisfying (3.10) such that there exists C ∈ (0,∞)satisfying

Pµ,ν(δ(x)− f(x), Ct) ≥L φ(x, 0, · · · , 0, t)for all x ∈ A and t > 0.

(2) d(Jnf, δ)→ 0 as n→∞. This implies the equality

limn→∞

f(mnx)

mn= δ(x) (3.11)

for all x ∈ A.(3) d(f, δ) ≤ 1

1−Rd(f, Jf), which implies the inequality d(f, δ) ≤ 1

m−mR. This implies

that the inequality (3.8) holds.

Thus it follows from (3.1), (3.2) and (3.11) that

Pµ,ν

( m∑i=1

δ(mxi +

m∑j=1, j =i

xj

)+ δ( m∑

i=1

xi

)− 2δ

( m∑i=1

mxi

), t)

= limn→∞

Pµ,ν

( m∑i=1

f(mn+1xi +

m∑j=1, j =i

mnxj

)+ f( m∑

i=1

mnxi

)− 2f

( m∑i=1

mn+1xi

),mnt

)≤L lim

n→∞φ(mnx1, · · · ,mnxm,m

nt)

= 1L

for all x1, · · · , xm ∈ A and t > 0 and som∑i=1

δ(mxi +

m∑j=1, j =i

xj

)+ δ( m∑

i=1

xi

)= 2δ

( m∑i=1

mxi

)for all x1, · · · , xm ∈ A.

By a similar method to above, we get

ωδ(mx) = δ(mωx)

for all ω ∈ T1 and x ∈ A. Thus one can show that the mapping H : A→ A is C-linear.Also, it follows from (3.3), (3.4) and (3.11) that

Pµ,ν(δ(xy)− yδ(x)− xδ(y), t)= lim

n→∞Pµ,ν(f(m

nxy)−mnyf(mnx)−mnxf(mny),mnt)

≤ limn→∞

ψ(mnx,mny,m2nt)

= 1L

for all x, y ∈ A and soδ(xy) = yδ(x) + xδ(y)

for all x, y ∈ A. Thus δ : A→ A is a derivation satisfying (3.7), as desired.


350 ABDOU et al 345-354


Also, Similarly, by (3.5), (3.6) and (3.11), we have δ(x∗) = δ(x)∗. This completes theproof.

4. Approximation of derivations on intuitionistic random Lie C∗-algebras

An intuitionistic random C∗-algebra C, endowed with the Lie product

[x, y] :=xy − yx

2

in C, is called a intuitionistic random Lie C∗-algebra.

Definition 4.1. Let A and B be intuitionistic random Lie C∗-algebras. A C-linearmapping δ : A→ A is called an intuitionistic random Lie C∗-algebra derivation if

δ([x, y]) = [δ(x), y] + [x, δ(y)]

for all x, y ∈ A.

Throughout this Section, assume that A is an intuitionistic random Lie C∗-algebra withnorm Pµ,ν .

Now, we approximate the derivations on intuitionistic random Lie C∗-algebras for thefunctional equation

Dωf(x1, · · · , xm) = 0.

Theorem 4.2. Let f : A → A be a mapping for which there are functions φ : Am → L∗

and ψ : A2 → L∗ such that

limj→∞

φ(mjx1, · · · ,mjxm,mjt) = 1L, (4.1)

Pµ,ν(Dωf(x1, · · · , xm), t) ≥L φ(x1, · · · , xm, t), (4.2)

Pµ,ν(f([x, y])− [f(x), y]− [x, f(y)], t) ≥L ψ(x, y, t), (4.3)

limj→∞

ψ(mjx,mjy,m2jt) = 1L (4.4)

for all ω ∈ T1, x1, · · · , xm, x, y ∈ A and t > 0. If there exists R < 1 such that

φ(mx, 0, · · · , 0,mx) ≥L φ(x, 0, · · · , 0, t)

for all x ∈ A and t > 0, then there exists a unique homomorphism δ : A→ A such that

Pµ,νf(x)− δ(x), t) ≥L φ(x, 0, · · · , 0, (m−mR)t) (4.5)

for all x ∈ A and t > 0.

Proof. By the same reasoning as the proof of Theorem 3.1, we can find the mappingδ : A→ A given by

δ(x) = limn→∞

f(mnx)

mn


351 ABDOU et al 345-354


for all x ∈ A. It follows from (4.3) that

Pµ,ν(δ([x, y])− [δ(x), y]− [x, δ(y)], t)

= limn→∞

Pµ,ν(f(m2n[x, y])− [f(mnx), ·mny]− [mnx, f(mny)],m2nt)

≥L limn→∞

ψ(mnx,mny,m2nt) = 1L

for all x, y ∈ A and t > 0 and so

δ([x, y]) = [δ(x), y] + [x, δ(y)]

for all x, y ∈ A. Thus δ : A → B is an intuitionistic random Lie C∗-algebra derivationsatisfying (4.5). This completes the proof.

Corollary 4.3. Let 0 < r < 1 and θ be nonnegative real numbers and f : A → A be amapping such that

Pµ,ν(Dωf(x1, · · · , xm), t)

≥L

( t

t+ θ(∥x1∥rA + ∥x2∥rA + · · ·+ ∥xm∥rA),

θ(∥x1∥rA + ∥x2∥rA + · · ·+ ∥xm∥rA)t+ θ(∥x1∥rA + ∥x2∥rA + · · ·+ ∥xm∥rA)

),

Pµ,ν(f([x, y])− [f(x), y]− [x, f(y)], t)

≥L

( t

t+ θ · ∥x∥rA · ∥y∥rA,

θ · ∥x∥rA · ∥y∥rAt+ θ · ∥x∥rA · ∥y∥rA

)for all ω ∈ T1, x1, · · · , xm, x, y ∈ A and t > 0. Then there exists a unique derivationδ : A→ A such that

Pµ,ν(f(x)− δ(x), t) ≤L

( t

t+ θm−mr ∥x∥rA

,θ

m−mr ∥x∥rAt+ θ

m−mr ∥x∥rA

)for all x ∈ A and t > 0.

Proof. The proof follows from Theorem 4.2 by taking

φ(x1, · · · , xm, t)

=( t

t+ θ(∥x1∥rA + ∥x2∥rA + · · ·+ ∥xm∥rA),

θ(∥x1∥rA + ∥x2∥rA + · · ·+ ∥xm∥rA)t+ θ(∥x1∥rA + ∥x2∥rA + · · ·+ ∥xm∥rA)

),

ψ(x, y, t) :=( t

t+ θ · ∥x∥rA · ∥y∥rA,

θ · ∥x∥rA · ∥y∥rAt+ θ · ∥x∥rA · ∥y∥rA

)and

R = mr−1

for all x1, · · · , xm, x, y ∈ A and t > 0. This completes the proof.


352 ABDOU et al 345-354


Acknowledgement

This project was funded by the Deanship of Scientific Research (DSR), King AbdulazizUniversity, under grant no. (18-130-36-HiCi). The authors, therefore, acknowledge withthanks DSR technical and financial support. Also, Yeol Je Cho was supported by BasicScience Research Program through the National Research Foundation of Korea (NRF)funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100)

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mappings, Proc. Amer. Math. Soc. 126 (1998), 425–430.[44] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Applications to Nonlinear Analysis,

Internat. J. Math. Math. Sci. 19(1996), 219–228.[45] C. Park, Homomorphisms between Poisson JC∗-algebras, Bull. Braz. Math. Soc. 36(2005), 79–97.[46] L. Cadariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal.

Pure Appl. Math. 4 (2003), no. 1, Art. ID 4.

(Afrah A. N. Abdou) Department of Mathematics, King Abdulaziz University, Jeddah21589, Saudi Arabia


(Yeol Je Cho) Department of Mathematics Education and the RINS, Gyeongsang Na-tional University, Chinju 660-701, Korea, and Department of Mathematics, King Abdu-laziz University, Jeddah 21589, Saudi Arabia


(Reza Saadati) Department of Mathematics, Iran University of Science and Technology,Tehran, Iran



354 ABDOU et al 345-354

CUBIC ρ-FUNCTIONAL INEQUALITY AND QUARTIC ρ-FUNCTIONAL

INEQUALITY

CHOONKIL PARK, JUNG RYE LEE∗, AND DONG YUN SHIN

Abstract. In this paper, we solve the following cubic ρ-functional inequality

‖f(2x+ y) + f(2x− y)− 2f(x+ y)− 2f(x− y)− 12f(x)‖

≤∥∥∥ρ(4f

(x+

y

2

)+ 4f

(x− y

2

)− f(x+ y)− f(x− y)− 6f(x)

)∥∥∥ , (0.1)

where ρ is a fixed complex number with |ρ| < 2, and the quartic ρ-functional inequality

‖f(2x+ y) + f(2x− y)− 4f(x+ y)− 4f(x− y)− 24f(x) + 6f(y)‖ (0.2)

≤∥∥∥ρ(8f

(x+

y

2

)+ 8f

(x− y

2

)− 2f(x+ y)− 2f(x− y)− 12f(x) + 3f(y)

)∥∥∥ ,where ρ is a fixed complex number with |ρ| < 2.

Using the direct method, we prove the Hyers-Ulam stability of the cubic ρ-functional in-equality (0.1) and the quartic ρ-functional inequality (0.2) in complex Banach spaces.

1. Introduction and preliminaries

The stability problem of functional equations originated from a question of Ulam [17] con-cerning the stability of group homomorphisms. Hyers [8] gave a first affirmative partial answerto the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for ad-ditive mappings and by Rassias [12] for linear mappings by considering an unbounded Cauchydifference. A generalization of the Rassias theorem was obtained by Gavruta [5] by replac-ing the unbounded Cauchy difference by a general control function in the spirit of Rassias’approach.

In [9], Jun and Kim considered the following cubic functional equation

f(2x+ y) + f(2x− y) = 2f(x+ y) + 2f(x− y) + 12f(x). (1.1)

It is easy to show that the function f(x) = x3 satisfies the functional equation (1.1), which iscalled a cubic functional equation and every solution of the cubic functional equation is said tobe a cubic mapping. We can define the following Jensen type cubic functional equation

4f

(x+

y

2

)+ 4f

(x− y

2

)= f(x+ y) + f(x− y) + 6f(x).

Note that if f(2x) = 8f(x) then the Jensen type cubic functional equation is equivalent to thecubic functional equation (1.1).

In [10], Lee et al. considered the following quartic functional equation

f(2x+ y) + f(2x− y) = 4f(x+ y) + 4f(x− y) + 24f(x)− 6f(y). (1.2)

It is easy to show that the function f(x) = x4 satisfies the functional equation (1.2), which iscalled a quartic functional equation and every solution of the quartic functional equation is said

2010 Mathematics Subject Classification. Primary 39B62, 39B52.Key words and phrases. Hyers-Ulam stability; cubic ρ-functional inequality; quartic ρ-functional inequality;

complex Banach space.∗Corresponding author (AMAT 2015).


355 CHOONKIL PARK et al 355-362

C. PARK, J. R. LEE, AND D. Y. SHIN

to be a quartic mapping. We can define the following Jensen type quartic functional equation

8f

(x+

y

2

)+ 8f

(x− y

2

)= 2f(x+ y) + 2f(x− y) + 12f(x)− 3f(y).

Note that if f(2x) = 16f(x) then the Jensen type quartic functional equation is equivalent tothe quartic functional equation (1.2).

Recently, considerable attention has been increasing to the problem of the Hyers-Ulam sta-bility of functional equations. Several Hyers-Ulam stability results concerning Cauchy, Jensen,quadratic, cubic and quartic functional equations have been investigated in [1, 3, 13, 14, 15,16, 18].

In [6], Gilanyi showed that if f satisfies the functional inequality

‖2f(x) + 2f(y)− f(xy−1)‖ ≤ ‖f(xy)‖ (1.3)

then f satisfies the Jordan-von Neumann functional equation

2f(x) + 2f(y) = f(xy) + f(xy−1).

Gilanyi [7] and Fechner [4] proved the Hyers-Ulam stability of the functional inequality (1.3).Park, Cho and Han [11] proved the Hyers-Ulam stability of additive functional inequalities.

In Section 3, we solve the cubic ρ-functional inequality (0.1) and prove the Hyers-Ulamstability of the cubic ρ-functional inequality (0.1) in complex Banach spaces.

In Section 4, we solve the quartic ρ-functional inequality (0.2) and prove the Hyers-Ulamstability of the quartic ρ-functional inequality (0.2) in complex Banach spaces.

Throughout this paper, assume that X is a complex normed space and that Y is a complexBanach space.

2. Cubic ρ-functional inequality (0.1)

Throughout this section, assume that ρ is a fixed complex number with |ρ| < 2.In this section, we solve and investigate the cubic ρ-functional inequality (0.1) in complex

normed spaces.

Lemma 2.1. Let X and Y be vector spaces. A mapping f : X → Y satisfies f(2x) = 8f(x)and

4f

(x+

y

2

)+ 4f

(x− y

2

)= f(x+ y) + f(x− y) + 6f(x)

if and only if the mapping f : X → Y is a cubic mapping.

Proof. One can easily prove it. We omit the proof.

Lemma 2.2. If a mapping f : X → Y satisfies

‖f(2x+ y) + f(2x− y)− 2f(x+ y)− 2f(x− y)− 12f(x)‖

≤∥∥∥∥ρ(4f

(x+

y

2

)+ 4f

(x− y

2

)− f(x+ y)− f(x− y)− 6f(x)

)∥∥∥∥ (2.1)

for all x, y ∈ X, then f : X → Y is cubic.

Proof. Assume that f : X → Y satisfies (2.1).Letting x = y = 0 in (2.1), we get ‖ − 14f(0)‖ ≤ |ρ|‖0‖ = 0. So f(0) = 0.Letting y = 0 in (2.1), we get ‖2f (2x) − 16f(x)‖ ≤ 0 and so f(2x) = 8f(x) for all x ∈ X.

Thus

f

(x

2

)=

1

8f(x) (2.2)

for all x ∈ X.



CUBIC AND QURATIC ρ-FUNCTIONAL INEQUALITIES

It follows from (2.1) and (2.2) that

‖f(2x+ y) + f(2x− y)− 2f(x+ y)− 2f(x− y)− 12f(x)‖

≤∥∥∥∥ρ(4f

(x+

y

2

)+ 4f

(x− y

2

)− f(x+ y)− f(x− y)− 6f(x)

)∥∥∥∥=|ρ|2‖f(2x+ y) + f(2x− y)− 2f(x+ y)− 2f(x− y)− 12f(x)‖

and so

f(2x+ y) + f(2x− y) = 2f(x+ y) + 2f(x− y) + 12f(x)

for all x, y ∈ X, since |ρ| < 2. So f : X → Y is cubic.

We prove the Hyers-Ulam stability of the cubic ρ-functional inequality (2.1) in complexBanach spaces.

Theorem 2.3. Let ϕ : X2 → [0,∞) be a function and let f : X → Y be a mapping such that

Ψ(x, y) :=∞∑j=1

8jϕ(x

2j,y

2j) <∞, (2.3)

‖f(2x+ y) + f(2x− y)− 2f(x+ y)− 2f(x− y)− 12f(x)‖ (2.4)

≤∥∥∥∥ρ(4f

(x+

y

2

)+ 4f

(x− y

2

)− f(x+ y)− f(x− y)− 6f(x)

)∥∥∥∥+ ϕ(x, y)

for all x, y ∈ X. Then there exists a unique cubic mapping C : X → Y such that

‖f(x)− C(x)‖ ≤ 1

16Ψ(x, 0) (2.5)

for all x ∈ X.

Proof. Letting y = 0 in (2.4), we get

‖2f(2x)− 16f(x)‖ ≤ ϕ(x, 0) (2.6)

and so∥∥f(x)− 8f

(x2

)∥∥ ≤ 12ϕ(x2 , 0

)for all x ∈ X. So∥∥∥∥8lf ( x2l

)− 8mf

(x

2m

)∥∥∥∥ ≤ m∑j=l+1

∥∥∥∥8jf ( x2j)− 8j+1f

(x

2j+1

)∥∥∥∥≤ 1

16

m∑j=l+1

8jϕ

(x

2j, 0

)(2.7)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.7) that thesequence 8nf( x

2n ) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence8nf( x

2n ) converges. So one can define the mapping C : X → Y by

C(x) := limn→∞

8nf(x

2n)

for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (2.7), we get (2.5).




It follows from (2.3) and (2.4) that

‖C(2x+ y) + C(2x− y)− 2C(x+ y)− 2C(x− y)− 12C(x)‖

= limn→∞

8n∥∥∥∥f (2x+ y

2n

)+ f

(2x− y

2n

)− 2f

(x+ y

2n

)− 2f

(x− y

2n

)− 12f

(x

2n

)∥∥∥∥≤ lim

n→∞8n|ρ|

∥∥∥∥4f (2x+ y

2n+1

)+ 4f

(2x− y2n+1

)− f

(x+ y

2n

)− f

(x− y

2n

)− 6f

(x

2n

)∥∥∥∥+ limn→∞

8nϕ

(x

2n,y

2n

)=

∥∥∥∥ρ(4C

(x+

y

2

)+ 4C

(x− y

2

)− C(x+ y)− C(x− y)− 6C(x)

)∥∥∥∥for all x, y ∈ X. So

‖C(2x+ y) + C(2x− y)− 2C(x+ y)− 2C(x− y)− 12C(x)‖

≤∥∥∥∥ρ(4C

(x+

y

2

)+ 4C

(x− y

2

)− C(x+ y)− C(x− y)− 6C(x)

)∥∥∥∥for all x, y, z ∈ X. By Lemma 2.2, the mapping C : X → Y is cubic.

Now, let T : X → Y be another cubic mapping satisfying (2.5). Then we have

‖C(x)− T (x)‖ =

∥∥∥∥8qC ( x2q)− 8qT

(x

2q

)∥∥∥∥≤

∥∥∥∥8qC ( x2q)− 8qf

(x

2q

)∥∥∥∥+

∥∥∥∥8qT ( x2q)− 8qf

(x

2q

)∥∥∥∥≤ 2

16· 8qΨ

(x

2q, 0

),

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that C(x) = T (x) for allx ∈ X. This proves the uniqueness of C. Thus the mapping C : X → Y is a unique cubicmapping satisfying (2.5).

Corollary 2.4. Let r > 3 and θ be nonnegative real numbers, and let f : X → Y be a mappingsuch that

‖f(2x+ y) + f(2x− y)− 2f(x+ y)− 2f(x− y)− 12f(x)‖ (2.8)

≤∥∥∥∥ρ(4f

(x+

y

2

)+ 4f

(x− y

2

)− f(x+ y)− f(x− y)− 6f(x)

)∥∥∥∥+ θ(‖x‖r + ‖y‖r)


‖f(x)− C(x)‖ ≤ θ

2r+1 − 16‖x‖r

for all x ∈ X.

Theorem 2.5. Let ϕ : X2 → [0,∞) be a function and let f : X → Y be a mapping satisfying(2.4) and

Ψ(x, y) :=∞∑j=0

1

8jϕ(2jx, 2jy) <∞


‖f(x)− C(x)‖ ≤ 1

16Ψ(x, 0) (2.9)

for all x ∈ X.




Proof. It follows from (2.6) that∥∥∥∥f(x)− 1

8f(2x)

∥∥∥∥ ≤ 1

16ϕ(x, 0)

for all x ∈ X. Hence∥∥∥∥ 1

8lf(2lx)− 1

8mf(2mx)

∥∥∥∥ ≤ m−1∑j=l

∥∥∥∥ 1

8jf(2jx

)− 1

8j+1f(2j+1x

)∥∥∥∥≤ 1

16

m−1∑j=l

1

8jϕ(2jx, 0) (2.10)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.10) that thesequence 1

8n f(2nx) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence

18n f(2nx) converges. So one can define the mapping C : X → Y by

C(x) := limn→∞

1

8nf(2nx)

for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (2.10), we get (2.9).The rest of the proof is similar to the proof of Theorem 2.3.

Corollary 2.6. Let r < 3 and θ be positive real numbers, and let f : X → Y be a mappingsatisfying (2.8). Then there exists a unique cubic mapping C : X → Y such that

‖f(x)− C(x)‖ ≤ θ

16− 2r+1‖x‖r (2.11)

for all x ∈ X.

Remark 2.7. If ρ is a real number such that −2 < ρ < 2 and Y is a real Banach space, thenall the assertions in this section remain valid.

3. Quartic ρ-functional inequality (0.2)

Throughout this section, assume that ρ is a fixed complex number with |ρ| < 2.In this section, we solve and investigate the quartic ρ-functional inequality (0.2) in complex

normed spaces.

Lemma 3.1. Let X and Y be vector spaces. An even mapping f : X → Y satisfies

8f

(x+

y

2

)+ 8f

(x− y

2

)= 2f(x+ y) + 2f(x− y) + 12f(x)− 3f(y) (3.1)

if and only if the mapping f : X → Y is a quartic mapping.

Proof. Sufficiency. Assume that f : X → Y satisfies (3.1)Letting x = y = 0 in (3.1), we have 16f(0) = 13f(0). So f(0) = 0.Letting x = 0 in (3.1), we get 16f

(y2

)= f(y) for all y ∈ X. So f : X → Y satisfies the

quartic functional equation.Necessity. Assume that f : X → Y is a quartic mapping. Then f(2x) = 16f(x) for all

x ∈ X. So f : X → Y satisfies (3.1).

Lemma 3.2. If a mapping f : X → Y satisfies


≤∥∥∥∥ρ(8f

(x+

y

2

)+ 8f

(x− y

2

)− 2f(x+ y)− 2f(x− y)− 12f(x) + 3f(y)

)∥∥∥∥for all x, y ∈ X, then the mapping f : X → Y is quartic.




Proof. Assume that f : X → Y satisfies (3.2).Letting x = y = 0 in (3.2), we get ‖24f(0)‖ ≤ |ρ|‖3f(0)‖. So f(0) = 0.Letting y = 0 in (3.2), we get

‖2f (2x)− 32f(x)‖ ≤ 0 (3.3)

and so

f

(x

2

)=

1

16f(x) (3.4)

for all x ∈ X.It follows from (3.2) and (3.4) that

‖f(2x+ y) + f(2x− y)− 4f(x+ y)− 4f(x− y)− 24f(x) + 6f(y)‖

≤∥∥∥∥ρ(8f

(x+

y

2

)+ 8f

(x− y

2

)− 2f(x+ y)− 2f(x− y)− 12f(x) + 3f(y)

)∥∥∥∥=|ρ|2‖f(2x+ y) + f(2x− y)− 4f(x+ y)− 4f(x− y)− 24f(x) + 6f(y)‖

and so

f(2x+ y) + f(2x− y) = 4f(x+ y) + 4f(x− y) + 24f(x)− 6f(y)

for all x, y ∈ X, since |ρ| < 2. So f : X → Y is quartic.

We prove the Hyers-Ulam stability of the quartic ρ-functional inequality (3.2) in complexBanach spaces.

Theorem 3.3. Let ϕ : X2 → [0,∞) be a function and let f : X → Y be a mapping satisfyingf(0) = 0,

Ψ(x, y) :=∞∑j=1

16jϕ(x

2j,y

2j) <∞,


≤∥∥∥∥ρ(8f

(x+

y

2

)+ 8f

(x− y

2

)− 2f(x+ y)− 2f(x− y)− 12f(x) + 3f(y)

)∥∥∥∥+ ϕ(x, y)

for all x, y ∈ X. Then there exists a unique quartic mapping Q : X → Y such that

‖f(x)−Q(x)‖ ≤ 1

32Ψ(x, 0) (3.6)

for all x ∈ X.

Proof. Letting y = 0 in (3.5), we get

‖2f(2x)− 32f(x)‖ ≤ ϕ(x, 0) (3.7)

and so∥∥f(x)− 16f

(x2

)∥∥ ≤ 12ϕ(x2 , 0

)for all x ∈ X. So∥∥∥∥16lf

(x

2l

)− 16mf

(x

2m

)∥∥∥∥ ≤ m∑j=l+1

∥∥∥∥16jf

(x

2j

)− 16j+1f

(x

2j+1

)∥∥∥∥≤ 1

32

m∑j=l+1

16jϕ

(x

2j, 0

)(3.8)




for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.8) that thesequence 16nf( x

2n ) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence16nf( x

2n ) converges. So one can define the mapping Q : X → Y by

Q(x) := limn→∞

16nf(x

2n)

for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (3.8), we get (3.6).The rest of the proof is similar to the proof of Theorem 2.3.

Corollary 3.4. Let r > 4 and θ be nonnegative real numbers, and let f : X → Y be a mappingsuch that


≤∥∥∥∥ρ(8f

(x+

y

2

)+ 8f

(x− y

2

)− 2f(x+ y)− 2f(x− y)− 12f(x) + 3f(y)

)∥∥∥∥+θ(‖x‖r + ‖y‖r)


‖f(x)−Q(x)‖ ≤ θ

2r+1 − 32‖x‖r

for all x ∈ X.

Proof. Letting x = y = 0 in (3.9), we get ‖24f(0)‖ ≤ |ρ|‖3f(0)‖, So f(0) = 0. Lettingϕ(x, y) := θ(‖x‖r + ‖y‖r) in Theorem 3.3, we obtain the desired result.

Theorem 3.5. Let ϕ : X2 → [0,∞) be a function and let f : X → Y be a mapping satisfyingf(0) = 0, (3.5) and

Ψ(x, y) :=∞∑j=0

1

16jϕ(2jx, 2jy) <∞


‖f(x)−Q(x)‖ ≤ 1

32Ψ(x, 0) (3.10)

for all x ∈ X.

Proof. It follows from (3.7) that∥∥∥∥f(x)− 1

16f(2x)

∥∥∥∥ ≤ 1

32ϕ(x, 0)

for all x ∈ X. Hence∥∥∥∥ 1

16lf(2lx)− 1

16mf(2mx)

∥∥∥∥ ≤ m−1∑j=l

∥∥∥∥ 1

16jf(2jx

)− 1

16j+1f(2j+1x

)∥∥∥∥≤ 1

32

m−1∑j=l

1

16jϕ(2jx, 0) (3.11)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.11) that thesequence 1

16n f(2nx) is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence

116n f(2nx) converges. So one can define the mapping Q : X → Y by

Q(x) := limn→∞

1

16nf(2nx)

for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (3.11), we get (3.10).




The rest of the proof is similar to the proof of Theorem 2.3.

Corollary 3.6. Let r < 4 and θ be positive real numbers, and let f : X → Y be a mappingsatisfying (3.9). Then there exists a unique quartic mapping Q : X → Y such that

‖f(x)−Q(x)‖ ≤ θ

32− 2r+1‖x‖r

for all x ∈ X.

Remark 3.7. If ρ is a real number such that −2 < ρ < 2 and Y is a real Banach space, thenall the assertions in this section remain valid.

References

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222–224.[9] K. Jun, H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal.

Appl. 274 (2002), 867–878.[10] S. Lee, S. Im and I. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005), 387–394.[11] C. Park, Y. Cho, M. Han, Functional inequalities associated with Jordan-von Neumann-type additive func-

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Choonkil ParkResearch Institute for Natural Sciences, Hanyang University, Seoul 133-791, KoreaE-mail address: [email protected]

Jung Rye LeeDepartment of Mathematics, Daejin University, Kyeonggi 487-711, KoreaE-mail address: [email protected]

Dong Yun ShinDepartment of Mathematics, University of Seoul, Seoul 130-743, KoreaE-mail address: [email protected]



Complex Valued Gb-Metric Spaces

Ozgur EGE

Celal Bayar University

Department of Mathematics

45140 Manisa, Turkey

E-mail: [email protected]

February 3, 2015

Abstract

In this paper, we introduce the concept of complex valued Gb-metric spaces. We also prove Banach contrac-tion principle and Kannan's xed point theorem in this space. Our result generalizes some well-known resultsin the xed point theory.

Keywords: Complex valued Gb-metric space, xed point, Banach contraction principle, Kannan's xed point

theorem.2010 Mathematics Subject Classication. 47H10,54H25.

1 Introduction

The concept of a metric space was introduced by Frechet [11]. Then many mathematicians study of xed points of contractivemappings. After the introduction of Banach contraction principle, the study of existence and uniqueness of xed pointsand common xed points have been a major area of interest. In a number of generalized metric spaces, many researchersproved the Banach xed point theorem.

Bakhtin [6] presented b-metric spaces as a generalization of metric spaces. He also proved generalized Banach contractionprinciple in b-metric spaces. After that, many papers related to variational principle for single-valued and multi-valuedoperators have studied in b-metric spaces (see [7, 8, 9, 10, 18]). Azam et al. [4] dened the notion of complex valued metricspaces and gave common xed point result for mappings. Rao et al. [21] introduced the complex valued b-metric spaces.Mustafa and Sims [13] presented the notion of G-metric spaces. Many researchers [1, 2, 3, 12, 14, 15, 19, 20, 22, 23, 25]obtained common xed point results for G-metric spaces. The concept of Gb-metric space was given in [5]. Mustafa etal. [16] prove some coupled coincidence xed point theorems for nonlinear (ψ,ϕ)-weakly contractive mappings in partiallyordered Gb-metric spaces. Other important studies on Gb-metric spaces, see [17, 24].

In this work, our aim is to prove Banach contraction principle and Kannan's xed point theorem in complex valuedGb-metric spaces. For this purpose, we give new denitions and additional theorems with proofs.

2 Preliminaries

In this section, we recall some properties of Gb-metric spaces.

Denition 2.1. [5]. Let X be a nonempty set and s ≥ 1 be a given real number. Suppose that a mapping G : X×X×X →R+ satises:

(Gb1) G(x, y, z) = 0 if x = y = z;

(Gb2) 0 < G(x, x, y) for all x, y ∈ X with x 6= y;

(Gb3) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with y 6= z;

(Gb4) G(x, y, z) = G(px, y, z), where p is a permutation of x, y, z;

(Gb5) G(x, y, z) ≤ s(G(x, a, a) +G(a, y, z)) for all x, y, z, a ∈ X (rectangle inequality).

Then, G is called a generalized b-metric and (X,G) is called a generalized b-metric or a Gb-metric space.

Note that each G-metric space is a Gb-metric space with s = 1.

Proposition 2.2. [5]. Let X be a Gb-metric space. Then for each x, y, z, a ∈ X it follows that:

1


363 EGE 363-368

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(i) if G(x, y, z) = 0 then x = y = z,

(ii) G(x, y, z) ≤ s(G(x, x, y) +G(x, x, z)),

(iii) G(x, y, y) ≤ 2sG(y, x, x),

(iv) G(x, y, z) ≤ s(G(x, a, z) +G(a, y, z)).

Denition 2.3. [5]. Let X be a Gb-metric space. A sequence xn in X is said to be:• Gb-Cauchy if for each ε > 0, there exists a positive integer n0 such that for all m,n, l ≥ n0, G(xn, xm, xl) < ε,• Gb-convergent to a point x ∈ X if for each ε > 0, there exists a positive integer n0 such that for all m,n ≥ n0,G(xn, xm, x) < ε.

Proposition 2.4. [5]. Let X be a Gb-metric space.

(1) The sequence xn is Gb-Cauchy.(2) For any ε > 0, there exists n0 ∈ N such that G(xn, xm, xm) < ε, for all m,n ≥ n0.

Proposition 2.5. [5]. Let X be a Gb-metric space. The following are equivalent:

(1) xn is Gb-convergent to x.(2) G(xn, xn, x)→ 0 as n→∞.

(3) G(xn, x, x)→ 0 as n→∞.

Denition 2.6. [5]. A Gb-metric space X is called complete if every Gb-Cauchy sequence is Gb-convergent in X.

The complex metric space was initiated by Azam et al. [4]. Let C be the set of complex numbers and z1, z2 ∈ C. Denea partial order - on C as follows:

z1 - z2 if and only if Re(z1) ≤ Re(z2) and Im(z1) ≤ Im(z2).

It follows that z1 - z2 if one of the following conditions is satised:

(C1) Re(z1) = Re(z2) and Im(z1) = Im(z2),

(C2) Re(z1) < Re(z2) and Im(z1) = Im(z2),

(C3) Re(z1) = Re(z2) and Im(z1) < Im(z2),

(C4) Re(z1) < Re(z2) and Im(z1) < Im(z2).

Particularly, we write z1 z2 if z1 6= z2 and one of (C2), (C3) and (C4) is satised and we write z1 ≺ z2 if only (C4) issatised. The following statements hold:

(1) If a, b ∈ R with a ≤ b, then az ≺ bz for all z ∈ C.(2) If 0 - z1 z2, then |z1| < |z2|.(3) If z1 - z2 and z2 ≺ z3, then z1 ≺ z3.

3 Complex Valued Gb-Metric Spaces

In this section, we dene the complex valued Gb-metric space.

Denition 3.1. Let X be a nonempty set and s ≥ 1 be a given real number. Suppose that a mapping G : X×X×X → Csatises:

(CGb1) G(x, y, z) = 0 if x = y = z;

(CGb2) 0 ≺ G(x, x, y) for all x, y ∈ X with x 6= y;

(CGb3) G(x, x, y) - G(x, y, z) for all x, y, z ∈ X with y 6= z;

(CGb4) G(x, y, z) = G(px, y, z), where p is a permutation of x, y, z;

(CGb5) G(x, y, z) - s(G(x, a, a) +G(a, y, z)) for all x, y, z, a ∈ X (rectangle inequality).

Then, G is called a complex valued Gb-metric and (X,G) is called a complex valued Gb-metric space.

From (CGb5), we have the following proposition.

Proposition 3.2. Let (X,G) be a complex valued Gb-metric space. Then for any x, y, z ∈ X,• G(x, y, z) - s(G(x, x, y) +G(x, x, z)),• G(x, y, y) - 2sG(y, x, y).


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Denition 3.3. Let (X,G) be a complex valued Gb-metric space, let xn be a sequence in X.(i) xn is complex valued Gb-convergent to x if for every a ∈ C with 0 ≺ a, there exists k ∈ N such that G(x, xn, xm) ≺ afor all n,m ≥ k.(ii) A sequence xn is called complex valued Gb-Cauchy if for every a ∈ C with 0 ≺ a, there exists k ∈ N such thatG(xn, xm, xl) ≺ a for all n,m, l ≥ k.(iii) If every complex valued Gb-Cauchy sequence is complex valued Gb-convergent in (X,G), then (X,G) is said to becomplex valued Gb-complete.

Proposition 3.4. Let (X,G) be a complex valued Gb-metric space and xn be a sequence in X. Then xn is complexvalued Gb-convergent to x if and only if |G(x, xn, xm)| → 0 as n,m→∞.

Proof. (⇒) Assume that xn is complex valued Gb-convergent to x and let

a =ε√2

+ iε√2

for a real number ε > 0. Then we have 0 ≺ a ∈ C and there is a natural number k such that G(x, xn, xm) ≺ a for alln,m ≥ k. Thus, |G(x, xn, xm)| < |a| = ε for all n,m ≥ k and so |G(x, xn, xm)| → 0 as n,m→∞.(⇐) Suppose that |G(x, xn, xm)| → 0 as n,m → ∞. For a given a ∈ C with 0 ≺ a, there exists a real number δ > 0 suchthat for z ∈ C

|z| < δ ⇒ z ≺ a.Considering δ, we have a natural number k such that |G(x, xn, xm)| < δ for all n,m ≥ k. This means that G(x, xn, xm) ≺ afor all n,m ≥ k, i.e., xn is complex valued Gb-convergent to x.

From Propositions 3.2 and 3.4, we can prove the following theorem.

Theorem 3.5. Let (X,G) be a complex valued Gb-metric space, then for a sequence xn in X and point x ∈ X, thefollowing are equivalent:

(1) xn is complex valued Gb-convergent to x.

(2) |G(xn, xn, x)| → 0 as n→∞.

(3) |G(xn, x, x)| → 0 as n→∞.

(4) |G(xm, xn, x)| → 0 as m,n→∞.

Proof. (1)⇒ (2) It is clear from Proposition 3.4.(2)⇒ (3) By Proposition 3.2, we have

G(xn, x, x) - s(G(xn, xn, x) +G(xn, xn, x))

and using (2), we get|G(xn, x, x)| → 0

as n→∞.(3)⇒ (4) If we use (CGb4) and Proposition 3.2, then

G(xm, xn, x) = G(x, xm, xn) - s(G(x, x, xm) +G(x, x, xn))

= s(G(xm, x, x) +G(xn, x, x))

and |G(xm, xn, x)| → 0 as m,n→∞.(4)⇒ (1) We will use the equivalence in Proposition 3.4, (CGb3) and (CGb4). Since

G(x, xn, xm) = G(xm, x, xn) - s(G(xm, xm, x) +G(xm, xm, xn))

- s(G(xm, xn, x))

and |G(xm, xn, x)| → 0 as m,n→∞, we obtain |G(x, xn, xm)| → 0 and this completes the proof.

Theorem 3.6. Let (X,G) be a complex valued Gb-metric space and xn be a sequence in X. Then xn is complex valuedGb-Cauchy sequence if and only if |G(xn, xm, xl)| → 0 as n,m, l→∞.

Proof. (⇒) Let xn be complex valued Gb-Cauchy sequence and

b =ε√2

+ iε√2

where ε > 0 is a real number. Then 0 ≺ b ∈ C and there is a natural number k such that G(xn, xm, xl) ≺ b for all n,m, l ≥ k.Therefore, we get |G(xn, xm, xl)| < |b| = ε for all n,m, l ≥ k and the required result.(⇐) Assume that |G(xn, xm, xl)| → 0 as n,m, l→∞. Then there exists a real number γ > 0 such that for z ∈ C

|z| < γ implies z ≺ b

where b ∈ C with 0 ≺ b. For this γ, there is a natural number k such that |G(xn, xm, xl)| < γ for all n,m, l ≥ k. Thismeans that G(xn, xm, xl) ≺ b for all n,m, l ≥ k. Hence xn is complex valued Gb-Cauchy sequence.


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We prove the contraction principle in complex valued Gb-metric spaces as follows:

Theorem 3.7. Let (X,G) be a complete complex valued Gb-metric space with coecient s > 1 and T : X → X be amapping satisfying:

G(Tx, Ty, Tz) - kG(x, y, z) (3.1)

for all x, y, z ∈ X, where k ∈ [0, 1s). Then T has a unique xed point.

Proof. Let T satisfy (3.1), x0 ∈ X be an arbitrary point and dene the sequence xn by xn = Tnx0. From (3.1), we obtain

G(xn, xn+1, xn+1) - kG(xn−1, xn, xn). (3.2)

Using again (3.1), we haveG(xn−1, xn, xn) - kG(xn−2, xn−1, xn−1)

and by (3.2), we getG(xn, xn+1, xn+1) - k2G(xn−2, xn−1, xn−1).

If we continue in this way, we ndG(xn, xn+1, xn+1) - knG(x0, x1, x1). (3.3)

Using (CGb5) and (3.3) for all n,m ∈ N with n < m,

G(xn, xm, xm) - s[G(xn, xn+1, xn+1) +G(xn+1, xm, xm)]

- s[G(xn, xn+1, xn+1)] + s2[G(xn+1, xn+2, xn+2)

+G(xn+2, xm, xm)]

- s[G(xn, xn+1, xn+1)] + s2[G(xn+1, xn+2, xn+2)]+

s3[G(xn+2, xn+3, xn+3)] + . . .+ sm−nG(xm−1, xm, xm)]

- (skn + s2kn+1 + s3kn+2 + . . .+ sm−nkm−1)G(x0, x1, x1)

- skn[1 + sk + (sk)2 + (sk)3 + . . .+ (sk)m−n−1]G(x0, x1, x1)

-skn

1− skG(x0, x1, x1).

Thus, we obtain

|G(xn, xm, xm)| ≤ skn

1− sk |G(x0, x1, x1)|.

Since k ∈ [0, 1s) where s > 1, taking limits as n→∞, then

skn

1− sk |G(x0, x1, x1)| → 0.

This means that|G(xn, xm, xm)| → 0.

By Proposition 3.2, we getG(xn, xm, xl) - G(xn, xm, xm) +G(xl, xm, xm)

for n,m, l ∈ N. Thus,|G(xn, xm, xl)| ≤ |G(xn, xm, xm)|+ |G(xl, xm, xm)|.

If we take limit as n,m, l→∞, we obtain |G(xn, xm, xl)| → 0. So xn is complex valued Gb-Cauchy sequence by Theorem3.6. Completeness of (X,G) gives us that there is an element u ∈ X such that xn is complex valued Gb-convergent to u.

To prove Tu = u, we will assume the contrary. From (3.1), we obtain

G(xn+1, Tu, Tu) - kG(xn, u, u)

and|G(xn+1, Tu, Tu)| ≤ k|G(xn, u, u)|.

If we take the limit as n ≥ ∞, we get|G(u, Tu, Tu)| ≤ k|G(u, u, u)|,

which is a contradiction since k ∈ [0, 1s). As a result, Tu = u.

Lastly, we prove the uniqueness. Let w 6= u be another xed point of T . Using (3.1),

G(z, w,w) = G(Tz, Tw, Tw) - kG(z, w,w).

and|G(z, w,w)| ≤ k|G(z, w,w)|.

Since k ∈ [0, 1s), we have |G(z, w,w)| ≤ 0. Thus, u = w and so u is a unique xed point of T .


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Example 3.8. Let X = [−1, 1] and G : X ×X ×X → C be dened as follows:

G(x, y, z) = |x− y|+ |y − z|+ |z − x|

for all x, y, z ∈ X. (X,G) is complex valued G-metric space [12]. Dene

G∗(x, y, z) = G(x, y, z)2.

G∗ is a complex valued Gb-metric with s = 2 (see [5]). If we dene T : X → X as Tx = x3, then T satises the following

condition for all x, y, z ∈ X:

G(Tx, Ty, Tz) = G(x

3,y

3,z

3) =

1

3G(x, y, z) - kG(x, y, z)

where k ∈ [ 13, 1s), s > 1. Thus x = 0 is the unique xed point of T in X.

We will prove Kannan's xed point theorem for complex valued Gb-metric spaces.

Theorem 3.9. Let (X,G) be a complete complex valued Gb-metric space and the mapping T : X → X satises for everyx, y ∈ X

G(Tx, Ty, Ty) - α[G(x, Tx, Tx) +G(y, Ty, Ty)] (3.4)

where α ∈ [0, 12). Then T has a unique xed point.

Proof. Let x0 ∈ X be arbitrary. We dene a sequence xn by xn+1 = Txn for all n ≥ 0. We shall show that xn isGb-Cauchy sequence. If xn = xn+1, then xn is the xed point of T . Thus, suppose that xn 6= xn+1 for all n ≥ 0. SettingG(xn, xn+1, xn+1) = Gn, it follows from (3.4) that

G(xn, xn+1, xn+1) = G(Txn−1, Txn, Txn)

- α[G(xn−1, Txn−1, Txn−1) +G(xn, Txn, Txn)]

- α[G(xn−1, xn, xn) +G(xn, xn+1, xn+1)]

- α[Gn−1 +Gn]

Gn -α

1− αGn−1 = βGn−1,

where β = α1−α < 1 as α ∈ [0, 1

2). If we repeat this process, then we get

Gn - βnG0. (3.5)

We can also suppose that x0 is not a periodic point. If xn = x0, then we have

G0 - βnG0.

Since β < 1, then 1− βn < 1 and(1− βn)|G0| ≤ 0 ⇒ |G0| = 0.

It follows that x0 is a xed point of T . Therefore in the sequel of proof we can assume Tnx0 6= x0 for n = 1, 2, 3, . . . Frominequality (3.4), we obtain

G(Tnx0, Tn+mx0, T

n+mx0) - α[G(Tn−1x0, Tn+mx0, T

n+mx0)

+G(Tn+m−1x0, Tn+mx0, T

n+mx0)]

- α[βn−1G(x0, Tx0, Tx0) + βn+m−1G(x0, Tx0, Tx0)].

So, |G(xn, xn+m, xn+m)| → 0 as n→∞. It implies that xn is a Gb-Cauchy in X. By the completeness of X, there existsu ∈ X such that xn → u. From (CGb5), we get

G(Tu, u, u) - s[G(Tu, Tn+1x0, Tn+1x0) +G(Tn+1x0, u, u)]

- s(α[G(u, Tu, Tu) +G(Tnx0, Tn+1x0, T

n+1x0)]) + sG(Tn+1x0, u, u)

- sα[G(u, Tu, Tu) + sαG(Tnx0, Tn+1x0, T

n+1x0)]) + sG(Tn+1x0, u, u).

Letting n→∞, since sα < 1 and xn → u, we have |G(Tu, u, u)| → 0, i.e., u = Tu.Now we show that T has a unique xed point. For this, assume that there exists another point v in X such that v = Tv.

Now,G(v, u, u) - G(Tv, Tu, Tu)

- α[G(v, Tv, Tv) +G(u, Tu, Tu)]

- α[G(v, v, v) +G(u, u, u)]

- 0.

Hence, we conclude that u = v.


367 EGE 363-368

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References

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[17] Z. Mustafa, J.R. Roshan and V. Parvaneh, Existence of tripled coincidence point in ordered Gb-metric spaces andapplications to a system of integral equations, J. Inequalities Appl, 2013:453, (2013).

[18] V. Parvaneh, J.R. Roshan and S. Radenovic, Existence of tripled coincidence points in ordered b-metric spaces and anapplication to a system of integral equations, Fixed Point Theory and Applications, 2013:130, (2013).

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[20] K.P.R. Rao, K. BhanuLakshmi, Z. Mustafa and V.C.C. Raju, Fixed and Related Fixed Point Theorems for ThreeMaps in G-Metric Spaces, J Adv Studies Topology, 3(4), 1219 (2012).

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368 EGE 363-368

Finite Difference approximations for the Two-side

Space-time Fractional Advection-diffusion

Equations∗

Yabin Shao1,2 †, Weiyuan Ma 2

1. Department of Applied Mathematics

Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

2. College of Mathematics and Computer Science

Northwest University for Nationalities, Lanzhou, 730030, China

February 4, 2015

Abstract

Fractional order advection-diffusion equation is viewed as generaliza-tions of classical diffusion equations, treating super-diffusive flow pro-cesses. In this paper, we present a new weighted finite difference ap-proximation for the equation with initial and boundary conditions in afinite domain. Using mathematical induction, we prove that the weightedfinite difference approximation is conditionally stable and convergent. Nu-merical computations are presented which demonstrate the effectivenessof the method and confirm the theoretical claims.Keywords: Fractional order advection-diffusion equation; Weighted fi-nite difference approximation; Stability; Convergence.

1 INTRODUCTION

In recent years, fractional differential equations have attracted much attention.Many important phenomena in physics [1, 2, 3], finance [4, 5], hydrology [6],engineering [7], mathematics [8] and material science are well described by dif-ferential equations of fractional order. These fractional order models tend to bemore appropriate than the traditional integer-order models. So, the fractionalderivatives are considered to be a very powerful and useful tool.

The fractional advection-diffusion equation provides a useful description oftransport dynamics in complex systems which are governed by anomalous diffu-sion and non-exponential relaxation [9]. In this paper, we consider a special case

∗The authors would like to thank National Natural Science Foundation of China(No.11161041).†Corresponding author. E-mail: [email protected](Y. Shao)

1


369 Yabin Shao et al 369-379

of anomalous diffusion, the two-sided space-time fractional advection-diffusionequation can be written in the following way

∂βu(x,t)∂tβ

= −v(x)∂u(x,t)∂x + d+(x)∂

αu(x,t)∂+xα

+ d−(x)∂αu(x,t)∂−xα

+ f(x, t), x ∈ [L,R], t ∈ (0, T ], (1)

u(L, t) = 0, u(R, t) = ϕ(t), t ∈ [0, T ], (2)

u(x, 0) = u0(x), x ∈ (L,R], (3)

where α and β are parameters describing the order of the space- and time-fractional derivatives, respectively, physical considerations restrict 0 < β <1, 1 < α < 2. The functions v(x, t), d+(x, t) and d−(x, t) are all non-negative,bounded and d+(x, t), d−(x, t) ≥ v(x, t).

The left-sided (+) and the right-sided (−) Riemann-Liouville fractional deriva-tives of order α of a function u(x, t) are defined as follows

∂αu(x, t)

∂+xα=

1

Γ(n− α)

∂n

∂xn

∫ x

L

u(ξ, t)

(x− ξ)α+1−n dξ (4)

and

∂αu(x, t)

∂−xα=

(−1)n

Γ(n− α)

∂n

∂xn

∫ R

x

u(ξ, t)

(x− ξ)α+1−n dξ, (5)

where n is an integer such that n − 1 < α ≤ n. The time derivative ∂βu(x,t)∂tβ

isgiven by a Caputo fractional derivative

∂βu(x, t)

∂tβ=

1

Γ(1− β)

∫ t

0

(t− η)−β∂u(x, η)

∂ηdη, (6)

where Γ(·) is the gamma function.As is well known, the fractional order differential operator is a nonlocal op-

erator, which requires more involved computational schemes for its handling.Finite difference schemes for fractional partial differential equations are morecomplex than partial differential equations [1, 2, 4, 10, 11, 12, 13, 14]. It shouldnote the following work for fractional advection-diffusion equation. Su et al. [13]presented a Crank-Nicolson type finite difference scheme for two-sided spacefractional advection-diffusion equation. Liu et al. [14] considered a space-time fractional advection-diffusion with Caputo time fractional derivative andRiemann-Liouville space fractional derivatives. In this paper, we present a newweighted finite difference approximation for the equation.

The rest of the paper is as follows. In Section 2, we derive the new weightedfinite difference approximation (NWFDM) for the fractional advection-diffusionequation. The convergence and stability of the finite difference scheme is givenin Section 3, where we apply discrete energy method. In Section 4, numericalresults are shown which confirm that the numerical method is effective.

2



2 NEW WEIGHTED FINITE DIFFERENCESCHEME

To present the numerical approximation scheme, we give some notations: τ isthe time step, unj be the numerical solution at (xi, tn) for xj = L + ih, tn =nτ, j = 0, 1, · · · , J, n = 0, 1, · · · , N .

The shifted Grunwald formula is applied to discretize the left-handed frac-tional derivative and right-handed fractional derivative [15],

∂αu(xi, tn)

∂+xα=

1

hα

i+1∑j=0

gju(xi − (j − 1)h, tn) + o(h), (7)

∂αu(xi, tn)

∂−xα=

1

hα

N−i+1∑j=0

gju(xi + (j − 1)h, tn) + o(h), (8)

where the Grunwald coefficients are defined by

g0 = 1, gj = (1− α+ 1

j)gj−1, j = 1, 2, 3, · · · .

Adopting the discrete scheme in [15], we discretize the Caputo time fractionalderivative as,

∂βu(xi, tn)

∂tβ=

τ1−β

Γ(2− β)

n∑j=0

u(xi, tn+1−j)− u(xi, tn−j)

τσj + o(τ),

where σj = (j + 1)1−β − j1−β .Now we replace (1) with the following weighted finite difference approxima-

tion:

τ1−β

Γ(2− β)

n∑j=0

un+1−ji − un−ji

τσj = −vi[θ

uni+1 − uni−1

2h

+(1− θ)un+1i+1 − u

n+1i−1

2h] +

d+i

hα[θi+1∑k=0

gkuni−k+1

+(1− θ)i+1∑k=0

gkun+1i−k+1] +

d−ihα

[θN−i+1∑k=0

gkuni+k−1

+(1− θ)N−i+1∑k=0

gkun+1i+k−1] + θfni + (1− θ)fn+1

i , (9)

for i = 1, 2, · · · , J − 1, n = 0, 1, · · · , N − 1, where θ is the weighting parametersubjected to 0 ≤ θ ≤ 1. When θ = 0, 1, 1

2 , we get the space-time fractionalimplicit, explicit, Crank-Nicolson type difference scheme, respectively.

3



The above equation (9) can be simplified, for n = 0,

−(1− θ)(ξi + ηig2 + ζi)u1i−1 − (1− θ)ηi

i+1∑k=3

gku1i−k+1

−(1− θ)ζiJ−i+1∑k=3

gku1i+k−1 + (1− θ)(ξi − ηi − ζig2)u1

i+1

+[1− (1− θ)(ηig1 + ζig1)]u1i = θ(ξi + ηig2 + ζi)u

0i−1

+[1 + θ(ηig1 + ζig1)]u0i + θ(−ξi + ηi + ζig2)u0

i+1

+θηi

i+1∑k=3

gku0i−k+1 + θζi

J−i+1∑k=3

gku0i+k−1

+Γ(1− β)τβ(θf0i + (1− θ)f1

i ), (10)

and for n > 0,

−(1− θ)(ξi + ηig2 + ζi)un+1i−1 − (1− θ)ηi

i+1∑k=3

gkun+1i−k+1

−(1− θ)ζiJ−i+1∑k=3

gkun+1i+k−1 + (1− θ)(ξi − ηi − ζig2)un+1

i+1

+[1− (1− θ)(ηig1 + ζig1)]un+1i = θ(ξi + ηig2 + ζi)u

ni−1

+[2− 21−β + θ(ηig1 + ζig1)]uni + θ(−ξi + ηi + ζig2)uni+1

+θηi

i+1∑k=3

gkuni−k+1 + θζi

J−i+1∑k=3

gkuni+k−1 +

n−1∑j=1

djun−ji

+u0iσn + Γ(1− β)τβ(θfni + (1− θ)fn+1

i ), (11)

and Dirichlet boundary conditions

un0 = 0, unJ = ϕ(tn), n = 1, 2, · · · , N − 1,

and initial conditions

u0i = u0(xi), i = 0, 1, · · · , J,

where ξi = viτβΓ(2−β)

2h , ηi = d+iτβΓ(2−β)hα , ζi = d−iτ

βΓ(2−β)hα and dj = σj+1 −

σj , j = 1, 2, · · · , n− 1.The numerical method (10) and (11) can be written in the matrix form:

AU1 = B0U0 +Q0,

AUn+1 = BUn + d1Un−1 + · · ·+ dn−1U

1 + σnU0 +Qn,

where

Un = (un1 , un2 , · · · , unJ−1)T ,

4



U0 = [u0(x1), u0(x2), · · · , u0(xJ−1)]T ,

b = (ηJ−1 + ζJ−1g2)[(1− θ)un+1J + θunJ ],

Fn = (fn1 , fn2 , · · · , fnJ−1 + b)T ,

E = (ζ1gJ , ζ2gJ−1, · · · , ζJ−1g2)T ,

Qn = Γ(2− β)τβ(θFn + (1− θ)Fn+1)

+(1− θ)Un+1J E + θUnJE,

and matrix A = (Aij)(J−1)×(J−1) is defined as follows:

Aij =

−(1− θ)(ξi + ηig2 + ζi), j = i− 1,

1− (1− θ)(ηig1 + ζig1), j = i,(1− θ)(ξ − ηi − ζig2), j = i+ 1,−(1− θ)ηigi+1−j , j = 1, 2, · · · , i− 2,

−(1− θ)ζigj+1−i, j = i+ 2, i+ 3, · · · , J − 1.

It is obvious that matrix A is strictly dominant, the system defined by (10) and(11) has unique solution.

3 STABILITY AND CONVERGENCE

In this section, we investigate the stability and convergence of the numericalscheme (9).

Theorem 1 For

θαΓ(2− β)τβ

hαmaxx∈[L,R]

(d+(x) + d−(x)) ≤ 2− 21−β , (12)

the weighted finite difference scheme (9) for solving equation (1)-(3) is stable.

Proof. Let uni , uni (i = 1, 2, · · · , J, n = 0, 1, 2, · · · , N − 1) be the numerical so-

lutions of (9) corresponding to the initial data u0i and u0

i , respectively. Letεni = uni − uni , the stability condition is equivalent to

‖En‖∞ ≤ ‖E0‖∞, n = 0, 1, · · · , N − 1, (13)

where En = (εk1 , εk2 , · · · , εkJ−1). We will use mathematical induction to get the

above result.For n = 0, we have

−(1− θ)[(ξi + ηig2 + ζi)ε1i−1 + ηi

i+1∑k=3

gkε1i−k+1

+ζi

J−i+1∑k=3

gkε1i+k−1 − (ξi − ηi − ζig2)ε1

i+1]

5



+[1− (1− θ)(ηig1 + ζig1)]ε1i = θ[(ξi + ηig2 + ζi)ε

0i−1

+ζi

J−i+1∑k=3

gkε0i+k−1 + (−ξi + ηi + ζig2)ε0

i+1

+ηi

i+1∑k=3

gkε0i−k+1] + [1 + θ(ηig1 + ζig1)]ε0

i , (14)

for n > 0,

−(1− θ)[(ξi + ηig2 + ζi)εn+1i−1 + ηi

i+1∑k=3

gkεn+1i−k+1

+ζi

J−i+1∑k=3

gkεn+1i+k−1 − (ξi − ηi − ζig2)εn+1

i+1 ]

+[1− (1− θ)(ηig1 + ζig1)]εn+1i =

n−1∑j=1

djεn−ji

+σnε0i + θ[(−ξi + ηi + ζig2)εni+1 + ηi

i+1∑k=3

gkεni−k+1

+ζi

J−i+1∑k=3

gkεni+k−1 + (ξi + ηig2 + ζi)ε

ni−1]

+[2− 21−β + θ(ηig1 + ζig1)]εni . (15)

Note that d+(x, t), d−(x, t) ≥ v(x, t), we have

ξi − ηi − ζig2 ≤ 0. (16)

In fact, if n = 0, suppose |ε1l | = max1≤i≤J−1 |ε1

i |, note that ξi, ηi, ζi > 0 andfor any integer number m,

∑mj=0 gj < 0, from (12), (16), we derive

‖E1‖∞ = |ε1l | ≤ −(1− θ)ηl

l+1∑k=0

gk|ε1l |+ |ε1

l | − (1− θ)ζlJ−l+1∑k=0

|ε1l |

≤ | − (1− θ)[(ξl + ηlg2 + ζl)ε1l−1 + ζl

J−l+1∑k=3

glε1l+k−1

+(ηl + ζlg2 − ξl)ε1l+1 + ηl

l+1∑k=3

gkε1l−k+1]

+[1− (1− θ)(ηlg1 + ζlg1)]ε1l |

≤ θ[(ξl + ηlg2 + ζl)|ε0l−1|+ ζl

J−l+1∑k=3

gk|ε0l+k−1|

6



+(ηl + ζlg2)|ε0l+1|+ ηl

l+1∑k=3

gk|ε0l−k+1|]

+[1− θ(ξl − ηlg1 − ζlg1)]|ε0l | ≤ ‖E0‖∞,

Suppose that ‖En‖∞ ≤ ‖E0‖∞, n = 1, 2, · · · , s, then when n = s + 1, let|εs+1l | = max1≤i≤J−1 |εs+1

i |. Similar to former estimate, we obtain

‖Es+1‖∞ ≤ | − (1− θ)[(ξl + ηlg2 + ζl)εn+1l−1 + ηl

l+1∑k=3

gkεn+1l−k+1

+ζl

J−l+1∑k=3

gkεn+1l+k−1 − (ξl − ηl − ζlg2)εn+1

l+1 ]

+[1− (1− θ)(ηlg1 + ζlg1)]εn+1l |

≤ θ(ξl + ηlg2 + ζl)|εsl−1|+ θ(−ξl + ηl + ζlg2)|εsl+1|

+[2− 21−β + θ(ηlg1 + ζlg1)]|εsl |+ θηl

l+1∑k=3

gk|εsl−k+1|

+θζl

J−l+1∑k=3

gk|εsl+k−1|+s−1∑j=1

dj |εs−jl |+ σs|ε0l |

≤ ‖E0‖∞.

Hence, ‖Es+1‖∞ ≤ ‖E0‖∞. The proof is completed.

Theorem 2 Suppose that u(x, t) is the sufficiently smooth solution of (1)-(3)and uki is the difference solution of difference scheme (9). If the condition (12)is satisfied, then

‖u(xi, tn)− uni ‖∞ ≤Mσ−1n−1(τ1+β + τβh),

where M is a positive constant.

Proof. Define eni = u(xi, tn) − uni and en = (en1 , en2 , · · · , enJ−1). Notice that

e0j = 0, we have: when n = 0,

−(1− θ)[(ξi + ηig2 + ζi)e1i−1 + ηi

i+1∑k=3

gke1i−k+1

+ζi

J−i+1∑k=3

gke1i+k−1 − (ξi − ηi − ζig2)e1

i+1]

+[1− (1− θ)(ηig1 + ζig1)]e1i = R1

i , (17)

when n > 0,

−(1− θ)[(ξi + ηig2 + ζi)en+1i−1 + ηi

i+1∑k=3

gken+1i−k+1

7



+ζi

J−i+1∑k=3

gken+1i+k−1 − (ξi − ηi − ζig2)en+1

i+1 ]

+[1− (1− θ)(ηig1 + ζig1)]en+1i − θηi

i+1∑k=3

gkeni−k+1

−[2− 21−β + θ(ηig1 + ζig1)]eni − θζiJ−i+1∑k=3

gkeni+k−1

−θ(ξi + ηig2 + ζi)eni−1 −

n−1∑j=1

djen−ji

−θ(−ξi + ηi + ζig2)eni+1 = Rn+1i , (18)

where Rn+1i is the truncation error of difference scheme (9). Furthermore, there

exists a positive constant M independent of step sizes such that |Rn+1i | ≤

M(τ1+β + τβh).We will prove by inductive method. Let |e1

l | = max1≤i≤J−1 |e1i |. If k = 1,

subject to the condition (12), based on (17), we have

‖e1‖∞ ≤ | − (1− θ)[(ξi + ηig2 + ζi)e1i−1 + ηi

i+1∑k=3

gke1i−k+1

+ζi

J−i+1∑k=3

gke1i+k−1 − (ξi − ηi − ζig2)e1

i+1]

+[1− (1− θ)(ηig1 + ζig1)]e1i |

≤M(τ1+β + τβh) = σ−10 M(τ1+β + τβh).

Assume that ‖en‖∞ ≤ Mσ−1n−1(τ1+β + τβh), n = 1, 2, · · · , s, then when n =

s+ 1, let |es+1l | = max1≤i≤J−1|es+1

i |, notice that σ−1j < σ−1

k , j = 0, 1, · · · , k− 1.Similarly, we obtain

‖es+1‖∞ ≤ d1‖es‖∞ +n−1∑j=1

dj‖es−j‖∞ +M(τ1+β + τβh)

≤ (d1σ−1s−1 + d2σ

−1s−1 + · · ·+ dsσ

−10 + 1)M(τ1+β + τβh)

≤ σ−1s M(τ1+β + τβh).

Thus, the proof is completed.In additional, since

limn→∞

σ−1n

nβ= limn→∞

n−1

(1− β)n−1=

1

1− β.

there is a constant C1 for which

‖en‖∞ ≤ C1nβ(τ1+β + τβh).

and nτ ≤ T is finite, we obtain the following result.

8



Theorem 3 Under the conditions of Theorem 2, then numerical solution con-verges to exact solution as h and τ tend to zero. Furthermore there exists positiveconstant C > 0, such that

‖u(xi, tn)− uni ‖ ≤ C(τ + h),

where i = 1, 2, · · · , J − 1;n = 1, 2, · · · , N.

4 NUMERICAL RESULTS

In this section, the following two-sided space-time space-time fractional advection-diffusion equation in a bounded domain is considered in [15]:

∂0.6u(x, t)

∂t0.6= −∂u(x, t)

∂x+ d+(x, t)

∂1.6u(x, t)

∂+x1.6

+d−(x, t)∂1.6u(x, t)

∂−x1.6+ f(x, t), (x, t) ∈ [0, 1]× [0, 1]

u(0, t) = 0, u(1, t) = 1 + 4t2, t ∈ [0, 1],

u(x, 0) = x2, x ∈ [0, 1],

where d+(x, t) = 25Γ(0.4)x0.6, d−(x, t) = 5Γ(0.4)(1−x)1.6, and f(x, t) = 100

7Γ(0.4)x2t1.4+

(1 + 4t2)(−25x2 + 40x− 12). The exact solution is u(x, t) = (1 + 4t2)x2.

Table 1: The error max |uki − u(xi, tk)| for the IWFDMs with θ = 1

N J State The error

10 10 Divergence 1.1305e+019

100 10 Divergence 2.3237e+163

10000 10 Divergence Infinity

30000 10 Convergence 1.3230

Table 1 shows the maximum absolute numerical error between the exactsolution and the numerical solution obtained by NWFDM with θ = 1. FromTable 1, it can see that our scheme is conditionally stable.

Table 2 and Table 3 show the maximum absolute error, at time t = 1.0,between the exact analytical solution and the numerical solution obtained byNWFDM with θ = 1/2 and θ = 0, respectively.

Table 4 and Table 5 show the comparison of maximum absolute numericalerror of the weighted finite difference scheme in [12] (WFDM) and new weightedfinite difference (NWFDM). We can see that the NNWDM is more accuratethan WFDM at θ = 0, but at θ = 0.4 is opposite. From the above five tables,it can seen that the numerical tests are in excellent agreement with theoreticalanalysis.

9



Table 2: The error and convergence rate for the scheme with θ = 1/2

N J Maximum error Convergence rate

200 200 0.0809 -

400 400 0.0486 1.6646

800 800 0.0298 1.6309

1600 1600 0.0055 1.6022

Table 3: The error and convergence rate for the scheme with θ = 0

N J Maximum error Convergence rate

200 200 0.0415 -

400 400 0.0209 1.9378

800 800 0.0107 1.9533

1600 1600 0.0054 1.9815

References

[1] E. Sousa, Finite difference approximations for a fractional advection diffu-sion problem, Journal of Computational Physics, 228, 4038-4054 (2009).

[2] F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergenceof the difference methods for the space-time fractional advection-diffusionequation, Applied Mathematics and Computation, 191, 2-20 (2007).

[3] P. Zhuang, F. Liu, V. Anh, I. Turner, New solution and analytical tech-niques of the implicit numerical method for the anomalous subdiffusionequation, SIAM Journal on Numerical Analysis, 46, 1079-1095 (2008).

[4] M. Raberto, E. Scalas, F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study, Physica A: Statistical Me-chanics and its Applications, 314, 749-755 (2002).

[5] L. Sabatelli, S. Keating, J. Dudley, P. Richmond, Waiting time distributionsin financial markets, The European Physical Journal B, 27, 273-275 (2002).

[6] L. Galue , S. L. Kalla, B. N. Al-Saqabi, Fractional extensions of the temper-ature field problems in oil strata, Applied Mathematics and Computation,186, 35-44 (2007).

[7] X. Li, M. Xu, J. Xiang, Homotopy perturbation method to time-fractionaldiffusion equation with a moving boundary, Applied Mathematics and Com-putation, 208, 434-439 (2009).

10



Table 4: The comparison of two schemes with θ = 0

N J NWFDM WFDM

50 50 0.1514 0.1522

100 100 0.0783 0.0797

150 150 0.0533 0.0545

200 200 0.0405 0.0415

Table 5: The comparison of two schemes with θ = 0.4

N J NWFDM WFDM

50 50 0.2198 0.1498

100 100 0.1242 0.0785

150 150 0.0899 0.0537

200 200 0.0717 0.0409

[8] Z. Odibat, S. Momani, V. S. Erturk, Generalized differential transformmethod: application to differential equations of fractional order, AppliedMathematics and Computation, 197, 467-477 (2008).

[9] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: afractional dynamics approach, Physics Reports, 339, 1-77 (2000).

[10] Y. Zhang, A finite difference method for fractional partial differential equa-tion, Applied Mathematics and Computation, 215, 524-529 (2009).

[11] Z. Q. Ding, A. G. Xiao, M. Li, Weighted finite difference methods for a classof space fractional partial differential equations with variable coefficients,Journal of Computational and Applied Mathematics, 233, 1905-1914 (2010).

[12] Y. M. Lin, C. J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics, 225, 1533-1552 (2007).

[13] L. J. Su, W. Q. Wang, Z. X. Yang, Finite difference approximations for thefractional advection-diffusion equation, Physics Letters: A, 373, 4405-4408(2009).

[14] F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and conver-gence of difference methods for the space-time fractional advection-diffusionequation, Applied Mathematics and Computation, 191, 12-20 (2007).

[15] I. Podlubny, Fractional differential equations, Academic Press, New York,1999.

11



A modified Newton-Shamanskii method for anonsymmetric algebraic Riccati equation∗

Jian-Lei Lia†, Li-Tao Zhangb, Xu-Dong Lic, Qing-Bin LibaCollege of Mathematics and Information Science, North China University of,

Water Resources and Electric Power, Zhengzhou, Henan, 450011, PR China.bDepartment of Mathematics and Physics, Zhengzhou Institute of,

Aeronautical Industry Management, Zhengzhou, Henan, 450015, PR China.cInvestment Projects Construction Agent Center of Luohe City People’s,

Government, Luohe, Henan, 462001, PR China.

AbstractThe non-symmetric algebraic Riccati equation arising in transport theory

can be rewritten as a vector equation and the minimal positive solution of thenon-symmetric algebraic Riccati equation can be obtained by solving the vectorequation. In this paper, based on the Newton-Shamanskii method, we propose anew iterative method called modified Newton-Shamanskii method for solving thevector equation. Some convergence results are presented. The convergence analy-sis shows that sequence of vectors generated by the modified Newton-Shamanskiimethod is monotonically increasing and converges to the minimal positive solu-tion of the vector equation. Finally, numerical experiments are presented toillustrate the performance of the modified Newton-Shamanskii method.

Key words: non-symmetric algebraic Riccati equation; M -matrix; transporttheory; minimal positive solution; modified Newton-Shamanskii method.

AMSC(2000): 49M15, 65H10, 15A24

1 Introduction

For convenience, firstly, we give some definitions and notations. For any matricesA = [ai,j] and B = [bi,j] ∈ Rm×n, we write A ≥ B(A > B) if ai,j ≥ bi,j(ai,j > bi,j)

∗This research was supported by the natural science foundation of Henan Province(14B110023),Growth Funds for Scientific Research team of NCWU(320009-00200), Youth Science and TechnologyInnovation talents of NCWU(70491), Doctoral Research Project of NCWU(201119), National NaturalScience Foundation of China(11501525,11501200).

†Corresponding author. E-mail:[email protected].

1


380 Jian-Lei Li et al 380-393

Jian-Lei Li etal: A modified Newton-Shamanskii method for Riccati equation

holds for all i, j. The Hadamard product of A and B is defined by A B = [ai,j · bi,j].I denotes the identity matrix with appropriate dimension. The superscript T denotesthe transpose of a vector or a matrix. We denote the norm by ‖ · ‖ for a vector or amatrix.

In this paper we are interested in iteratively solving the following nonsymmetricalgebraic Riccati equation (NARE) arising in transport theory (see [3–5, 21] and thereferences cited therein):

XCX −XE − AX + B = 0, (1.1)

where A,B,C,E ∈ Rn×n have the following special form:

A = ∆− eqT , B = eeT , C = qqT , E = D − qeT . (1.2)

Here and in the following, e = (1, 1, ..., 1)T , q = (q1, q2, ..., qn)T with qi = ci/2ωi,

∆ = diag(δ1, δ2, ..., δn) with δi =1

cωi(1 + α),

D = diag(d1, d2, ..., dn) with di =1

cωi(1− α),

(1.3)

and0 < c ≤ 1, 0 ≤ α < 1, 0 < ωn < ... < ω2 < ω1 < 1 (1.4)

∑ni=1 ci = 1, ci > 0, i = 1, 2, ..., n.The form of the Riccati equation (1.1) arises in Markov models [22] and in nuclear

physics [3, 24], and it has many positive solutions in the componentwise sense. Therehave been a lot of studies about algebraic properties [11, 21] and iterative methodsfor the nonnegative solution of the nonsymmetric algebraic Riccati equations (1.1),including the basic fixed-point iterations [5–8,19], the doubling algorithm [9], the Schurmethod [23,28], the Matrix Sign Function method [13,25] and the alternately linearizedimplicit iteration method [15], and so on; see related references therein. The existenceof positive solutions of (1.1) has been shown in [3] and [4], but only the minimalpositive solution is physically meaningful. So it is important to develop some effectiveand efficient procedures to compute the minimal positive solution of Equation (1.1).

Recently, Lu [10] has shown that the matrix equation (1.1) is equivalent to a vectorequation and has developed a simple and efficient iterative procedure to compute theminimal positive solution of (1.1). The fixed-point iteration methods were furtherstudied in [14, 16] for solving the vector equation. In [14] Bai, Gao and Lu proposedtwo nonlinear splitting iteration methods: the nonlinear block Jacobi and the nonlinearblock Gauss-Seidel iteration methods. In [16] Bao, Lin and Wei proposed a modifiedsimple iteration method for solving the vector equation. Furthermore, the convergencerates of various fixed-point iterations [10,14,16] were determined and compared in [20].




The Newton method has been presented and analyzed by Lu for solving the vectorequation in [12]. It has been shown that the Newton method for the vector equation ismore simple and efficient than using the corresponding Newton method directly for theoriginal Riccati equation (1.1). Li, Huang and Zhang present a relaxed Newton-likemethod [17] for solving the vector equation. Especially, in [18] Lin and Bao appliedthe Newton-Shamanskii method [2,26] to solve the vector equation.

Based on the Newton-Shamanskii method [18], in this paper, we propose a modi-fied Newton-Shamanskii method to solve the vector equation. The convergence analysisshows that the sequence of vectors generated by the new iterative method is monoton-ically increasing and converges to the minimal positive solution of the vector equation,which can be used to obtain the minimal positive solution of the original Riccati equa-tion. Our method extends the recent work done by Lu [12] and Lin and Bao [18].

Now, we give the definition of Z-matrix and M -matrix, and also give the followingtwo Lemmas which will be used later.

Definition 1 [1] A real square matrix A is called a Z-matrix if all its off-diagonalelements are non-positive. Any Z-matrix A can be written as A = sI −B with B ≥ 0,s > 0.

Definition 2 [1] Any matrix A of the form A = sI − B for which s > ρ(B), thespectral radius of B, is called an M-matrix.

Lemma 1.1 [1] For a Z-matrix A, the following statements are equivalent:(1) A is a nonsingular M-matrix;(2) A is nonsingular and A−1 ≥ 0;(3) Av > 0 for some vector v ≥ 0.

Lemma 1.2 [1] Let A ∈ Rn×n be a nonsingular M-matrix. If B ∈ Rn×n is a Z-matrixand satisfies the relation B ≥ A, then B ∈ Rn×n is also a nonsingular M-matrix.

The rest of the paper is organized as follows. In Section 2, we review the Newton-Shamanskii method and some useful results, and present the modified Newton-Shamanskiimethod. Some convergence results are given in Section 3. Section 4 and 5 give numer-ical experiments and conclusions, respectively.

2 The modified Newton-Shamanskii method

It has been shown in [10,12] that the solution of (1.1) must have the following form:

X = T (uvT ) = (uvT ) T,




where T = [ti,j] = [1/(δi + dj)] and u, v are two vectors, which satisfy the vectorequations:

u = u (Pv) + e,

v = v (P u) + e,(2.1)

where P = [pi,j] = [qj/(δi + dj)], P = [pi,j] = [qj/(δj + di)]. Define w = [uT , vT ]T . Theequation (2.1) can be rewritten equivalently as

f(w) = w − w Pw − e = 0, (2.2)

where

P =

[0 P

P 0

].

The minimal positive solution of (1.1) can be obtained via computing the minimalpositive solution of the vector equation (2.2).

The Newton method presented by Lu in [12] for the vector equation (2.2) is thefollowing:

wk+1 = wk − f ′(wk)−1f(wk), k = 0, 1, 2...

where for any w ∈ R2n, the Jacobian matrix f ′(w) of f(w) is given by

f ′(w) = I2n −G(w), with G(w) =

[G1(v) H1(u)H2(v) G2(u)

](2.3)

where G1(v) = diag(Pv), G2(u) = diag(P u), H1(u) = [u p1, u p2, ..., u pn] andH2(v) = [v p1, v p2, ..., v pn]. For i = 1, 2, ..., n, pi and pi are the ith column of Pand P , respectively. Obviously, when w > 0, G(w) ≥ 0 and f ′(w) is a Z-matrix.

The Newton-Shamanskii method for solving the vector equation (2.2) is given in [18]as follows:

Algorithm 2.1 (Newton-Shamanskii method) For a given m ≥ 1 and k = 0, 1, 2, ...,

wk,1 = wk − f ′(wk)−1f(wk),

wk,p+1 = wk,p − f ′(wk)−1f(wk,p), 1 ≤ p ≤ m− 1,

wk+1 = wk,m.

(2.4)

It has been shown in [18] that the Newton-Shamanskii method has a better conver-gence than the Newton method [12]. However, if the inversion of the Jacobian matrixf ′(w) is difficult to compute, the Newton-Shamanskii method may converge slowly.Hence, based on the Newton-Shamanskii method, we propose the following modifiedNewton-Shamanskii method:




Algorithm 2.2 (Modified Newton-Shamanskii method) For a given m ≥ 1 and k =0, 1, 2, ..., the Modified Newton-Shamanskii method is defined as follows:

wk,1 = wk − T−1k f(wk),

wk,p+1 = wk,p − T−1k f(wk,p), 1 ≤ p ≤ m− 1,

wk+1 = wk,m.

(2.5)

where Tk is a Z-matrix and Tk ≥ f ′(wk).

Remark 2.1 When Tk = f ′(wk), the modified Newton-Shamanskii method becomesthe Newton-Shamanskii method [18]. When m = 1 and Tk = f ′(wk), the modifiedNewton-Shamanskii method becomes the Newton method [12].

Before we give the convergence analysis of the Modified Newton-Shamanskii method,let us now state some results which are indispensable for our subsequent discussions.

Lemma 2.1 [18] For any vectors w1, w2 ∈ R2n, f ′(w1) − f ′(w2) = G(w2 − w1). Fur-thermore, if w2 > w1, we have f ′(w1)− f ′(w2) = G(w2 − w1) ≥ 0.

Here and in the subsequent section, for convenience, [f ′′(w)y]y is define as f ′′(w)y2.Let

f ′′(w)y = [L1y, L2y, ..., L2ny]T ∈ R2n×2n,

where Li ∈ R2n×2n, y ∈ R2n and for k = 1, 2, ..., n,

Lk =

[0 (−ekP

Tk )

(−ekPTk )T 0

], Ln+k =

[0 (−Pke

Tk )

(−PkeTk )T 0

]

with eTk = (0, ..., 0, 1, 0, ...), P T

k and PkT

are the kth rows of the matrices P and P ,respectively.

Lemma 2.2 [12] For any vectors w+, w ∈ R2n, we have

f(w+) = f(w) + f ′(w)(w+ − w) +1

2f ′′(w)(w+ − w, w+ − w). (2.6)

In particular, if w+ = w∗, the minimal positive solution of (2.2), then

0 = f(w) + f ′(w)(w∗ − w) +1

2f ′′(w)(w∗ − w, w∗ − w). (2.7)

Furthermore, for any y > 0 or y < 0,

f ′′(w)y2 < 0 (2.8)




and f ′′(w)y2 is independent of w.Because of the independence, in the following, we denote the operator f ′′(w) by L ,

i.e., L (y, y) = f ′′(w)(y, y) for any y ∈ R2n. By (2.7), we have

f(w) = f ′(w)(w − w∗)− 1

2L (w − w∗, w − w∗), (2.9)

f ′(w)(w − w∗) = f(w) +1

2L (w − w∗, w − w∗). (2.10)

Lemma 2.3 [12] If 0 ≤ w < w∗ and f(w) < 0, then f ′(w) is a nonsingular M-matrix.

3 Convergence analysis of the Modified Newton-

Shamanskii method

Now, we analyse convergence of the modified Newton-Shamanskii method (2.5).

Theorem 3.1 Given a vector wk ∈ R2n. wk,1, wk,2, ..., wk,m, wk+1 are obtained by themodified Newton-Shamanskii method (2.5). If wk < w∗ and f(wk) < 0, then, f ′(wk) isa nonsingular M-matrix, moreover,

(1) wk < wk,1 < wk,2 < ... < wk,m = wk+1 < w∗;(2) f(wk,p) < 0 for p = 1, 2, ..., m;(3) f ′(wk,p) is a nonsingular M-matrix for p = 1, 2, ..., m.

Therefore, wk+1 < w∗, f(wk+1) < 0 and f ′(wk+1) is a nonsingular M-matrix.

Proof. Since wk < w∗ and f(wk) < 0, by Lemma 2.3, we can easily obtain thatf ′(wk) is a nonsingular M -matrix. By Lemma 1.2, we can conclude that Tk is also anonsingular M -matrix. Now, we prove the theorem by mathematical induction. Definethe error vectors ek,i = wk,i − w∗ and ek = wk − w∗, then ek < 0. For p = 1, we havewk,1 = wk − T−1

k f(wk). Since f(wk) < 0 and Tk is also a nonsingular M -matrix, thenwk,1 > wk by Lemma 1.1.

By Eqs. (2.5) and (2.9), we obtain

ek,1 = ek − T−1k f(wk)

= ek − T−1k [f ′(wk)ek − 1

2L (ek, ek)]

= T−1k [Tk − f ′(wk)]ek +

1

2T−1

k L (ek, ek) < 0. (3.1)

Thus, wk,1 < w∗.




By Eq. (2.6) and Lemma 1.1, we have

f(wk,1) = f(wk − T−1k f(wk))

= f(wk)− f ′(wk)T−1k f(wk) +

1

2L (T−1

k f(wk), T−1k f(wk))

= [Tk − f ′(wk)]T−1k f(wk) +

1

2L (T−1

k f(wk), T−1k f(wk)) < 0. (3.2)

By Lemma 2.3, it can be concluded that f ′(wk,1) is a nonsingular M -matrix. Therefore,the results hold for p = 1.

Assume the results are true for 1 ≤ p ≤ t. Then, for p = t + 1, we have wk,t+1 =wk,t − T−1

k f(wk,t). Since f(wk,t) < 0 and Tk is a nonsingular M -matrix, then wk,t+1 >wk,t.

Since wk < wk,1 < wk,2 < ... < wk,t, by Lemma 2.1, we have f ′(wk) > f ′(wk,1) >f ′(wk,2) > ... > f ′(wk,t). Therefore,

Tk − f ′(wk,t) > ... > Tk − f ′(wk,1) > Tk − f ′(wk) ≥ 0.

By Eqs. (2.5) and (2.9), we have the following error vectors equation

ek,t+1 = ek,t − T−1k f(wk,t)

= ek,t − T−1k [f ′(wk,t)ek,t − 1

2L (ek,t, ek,t)]

= T−1k [Tk − f ′(wk,t)]ek,t +

1

2T−1

k L (ek,t, ek,t) < 0. (3.3)

Therefore, wk,t+1 < w∗.Similarly, by Eq. (2.6) and Lemma 1.1, we have

f(wk,t+1) = f(wk,t − T−1k f(wk,t))

= f(wk,t)− f ′(wk,t)T−1k f(wk,t) +

1

2L (T−1

k f(wk,t), T−1k f(wk,t))

= [Tk − f ′(wk,t)]T−1k f(wk,t) +

1

2L (T−1

k f(wk,t), T−1k f(wk,t)) < 0. (3.4)

By Lemma 2.3, we have that f ′(wk,t+1) is a nonsingular M -matrix. Therefore, theresults hold for p = t+1. Hence, by the principle of mathematical induction, the proofof the theorem is completed. ¤

In practical computation, we should choose Tk such that the iteration step (2.5)is less expensive to implement. For any wk ∈ R2n, according to the structure of theJacobian f ′(wk), Tk may be chosen as

Tk = I2n −[

G1(vk) 00 G2(uk)

](3.5)




or

Tk = I2n −[

G1(vk) H1(uk)0 G2(uk)

]. (3.6)

Another choice for Tk is

Tk = I2n −[

G1(vk) 0H2(vk) G2(uk)

].

Numerical experiments show that the performance for this choice is almost the sameas that for Tk given by (3.6).

The following theorem provides some results concerning the convergence of themodified Newton-Shamanskii method for the vector equation (2.2).

Theorem 3.2 Let w∗ be the minimal positive solution of the vector equation (2.2). Thesequence of the vector sets wk, wk,1, wk,2, ..., wk,m obtained by the modified Newton-Shamanskii method (2.5) with the initial vector w0 = 0 is well defined. For all k ≥ 0and 1 ≤ p ≤ m, we have

(1) f(wk) < 0 and f(wk,p) < 0;(2) f ′(wk) and f ′(wk,p) are nonsingular M-matrices;(3) w0 < w0,1 < w0,2 < ... < w0,m = w1 < w1,1 < w1,2 < ... < w1,m = w2 < ... <

wk−1,m = wk < wk,1 < ... < wk,m = wk+1 < ... < w∗.Furthermore, we have

limk→∞

wk = w∗.

Proof. This theorem can also be proved by mathematical induction. The proof issimilar to that of the Theorem 1 in [18]. Therefore, it is omitted. ¤

4 Numerical experiments

In this section, we give numerical experiments to illustrate the performance of themodified Newton-Shamanskii method presented in Section 3 with two different choicesof the matrix Tk. Let NS denote the Newton-Shamanskii iterative method [18], MNS1and MNS2 denote the modified Newton-Shamanskii iterative method (2.5) with Tk

given by(3.5) and (3.6), respectively. In order to show numerically the performanceof the modified Newton-Shamanskii iterative method, we list the number of iterationsteps (denoted as IT), the CPU time in seconds (denoted as CPU), and relative residualerror (denoted as ERR). The residual error is defined by

ERR = max

‖uk+1 − uk‖2

‖uk+1‖2

,‖vk+1 − vk‖2

‖vk+1‖2

,




0 5 10 15 201

2

3

4

5

6

7CPU time with (c, α)=(0.999,0.001)

m

CP

U ti

me

(sec

onds

)

NSMNS1MNS2

0 5 10 15 200

50

100

150

200

250

300

350

400IT numbers with (c, α)=(0.999,0.001)

m

IT n

umbe

rs

NSMNS1MNS2

Figure 1: CPU time and IT numbers for (c, α) = (0.999, 0.001) andn = 512 with different m. Left: CPU time; right: IT numbers

1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2CPU time with (c, α)=(0.5,0.5)

m

CP

U ti

me

(sec

onds

)

NSMNS1MNS2

1 2 3 4 5 6 7 8 92

4

6

8

10

12

14

16IT numbers with (c, α)=(0.5,0.5)

m

IT n

umbe

rs

NSMNS1MNS2

Figure 2: CPU time and IT numbers for (c, α) = (0.5, 0.5) and n = 512with different m. Left: CPU time; right: IT numbers

where ‖ · ‖2 is the 2-norm for a vector. For comparison, every experiment is repeated5 times, and the average of the 5 CPU times is shown here. All the experiments arerun in MATLAB 7.0 on a personal computer with Intel(R) Pentium(R) D 3.00GHzCPU and 0.99 GB memory, and all iterations are terminated once the current iteratesatisfies ERR ≤ n · eps, where eps = 1× 10−16.

In the test example, the constants ci and wi, i = 1, 2, ...n, are given by the numericalquadrature formula on the interval [0, 1], which are obtained by dividing [0, 1] into n

4

subintervals of equal length and applying a Gauss-Legendre quadrature [27] with 4nodes to each subinterval; see the Example 5.2 in [6]

We test several different values (c, α). In Table 1, for n = 512 with different m andpairs of (c, α), and in Table 2, for the fixed (c, α) = (0.99, 0.01) with different n, we listITs, CPUs and ERRs for the NS method and MNS methods, respectively. Figure 1 and




Table 1: Numerical results for n = 512 and different pairs of (c, α)

m method(c, α)

(0.999, 0.001) (0.99, 0.01) (0.9, 0.1) (0.5, 0.5)

1

NSIT 10 9 7 5

CPU 2.9380 2.6100 2.2810 1.6090ERR 2.1776e-15 1.5433e-15 1.4280e-15 1.5773e-14

MNS1IT 376 130 43 16

CPU 5.9370 2.0630 0.7820 0.2810ERR 4.7938e-14 4.3618e-14 2.7158e-14 7.0829e-15

MNS2IT 195 69 24 10

CPU 3.7500 1.3430 0.5150 0.2190ERR 4.7717e-014 3.3654e-14 1.6087e-14 1.7311e-15

3

NSIT 6 5 5 4

CPU 2.5310 2.0630 2.0780 1.7190ERR 5.3953e-15 4.6570e-14 1.2318e-15 1.0553e-15

MNS1IT 132 46 16 6

CPU 2.5780 0.9370 0.3130 0.1410ERR 4.3397e-14 3.7357e-14 1.0270e-14 2.6302e-14

MNS2IT 69 25 10 5

CPU 1.8440 0.6720 0.2810 0.1410ERR 4.4170e-14 2.9843e-14 8.7831e-16 1.5640e-16

6

NSIT 5 4 4 3

CPU 2.9220 2.2340 2.2810 1.7350ERR 1.9497e-15 1.7274e-15 1.3919e-15 1.0832e-15

MNS1IT 68 24 9 4

CPU 1.9060 0.6100 0.2340 0.1100ERR 4.7883e-14 4.0900e-14 3.3139e-15 2.9604e-16

MNS2IT 36 14 6 3

CPU 1.3280 0.5160 0.2340 0.1250ERR 4.7025e-14 8.5873e-15 5.5104e-16 2.1332e-15

12

NSIT 4 4 3 3

CPU 3.4530 3.5160 2.5630 2.6410ERR 1.9512e-15 1.6885e-15 1.3243e-15 1.1225e-15

MNS1IT 36 13 5 3

CPU 1.2660 0.4680 0.1880 0.1250ERR 2.9584e-14 2.6812e-14 1.9980e-14 1.64101e-16

MNS2IT 20 8 4 3

CPU 1.2190 0.4530 0.2180 0.2030ERR 1.1402e-14 4.8204e-15 5.5981e-16 1.6410e-16




Table 2: Numerical results for (c, α) = (0.99, 0.01) and different n, m

m methodn

64 128 256 512 1024

1

NSIT 9 9 9 9 9

CPU 0.0310 0.0620 0.4220 2.6100 16.9060ERR 6.8597e-16 9.5495e-16 1.1845e-15 1.5433e-15 2.7143e-15

MNS1IT 140 136 133 130 126

CPU 0.0630 0.0940 0.4850 2.0630 7.4530ERR 5.3918e-15 1.2247e-14 2.3157e-14 4.3618e-14 1.0212e-14

MNS2IT 73 72 70 69 67

CPU 0.0160 0.0630 0.2970 1.3430 4.7180ERR 6.1723e-15 9.44380e-15 2.1990e-14 3.3654e-14 7.8688e-14

5

NSIT 5 5 5 5 5

CPU 0.0160 0.0630 0.4220 2.3750 14.6560ERR 8.1022e-16 8.6594e-16 1.2226e-15 1.6581e-15 2.1565e-15

MNS1IT 31 30 29 29 28

CPU 0.0150 0.0310 0.1410 0.6410 2.3590ERR 2.7024e-15 7.6059e-15 2.2028e-14 2.1877e-14 6.3491e-14

MNS2IT 17 17 16 16 16

CPU 0.0160 0.0310 0.0930 0.5160 1.9530ERR 2.4804e-15 2.4814e-15 2.0690e-14 2.0656e-14 2.0698e-14

10

NSIT 4 4 4 4 4

CPU 0.0160 0.0780 0.5000 2.7500 16.6090ERR 7.2604e-16 8.1207e-16 1.1571e-15 1.5460e-15 2.2290e-15

MNS1IT 16 16 16 15 15

CPU 0.0150 0.0150 0.0780 0.4680 1.7350ERR 6.0508e-15 5.8374e-15 6.0658e-15 4.9045e-14 4.8935e-14

MNS2IT 10 10 9 9 9

CPU 0.0160 0.0320 0.0630 0.4380 1.6400ERR 5.6442e-16 4.2340e-16 1.4346e-14 1.3941e-14 1.3698e-14




Figure 2 describe the CPU time and IT numbers of those methods when n = 512 for(c, α) = (0.999, 0.001) and (c, α) = (0.5, 0.5). From these Tables and Figures, we cansee that the optimal choice of m for the modified Newton-Shamanskii method is largerwhen (c, α) = (0.999, 0.001), compared with (c, α) = (0.5, 0.5). Obviously, comparedwith the Newton-Shamanskii iterative method, though the iterations number of themodified Newton-Shamanskii iterative method is more, according to the CPU time,we can find that the modified Newton-Shamanskii iterative method outperforms theNewton-Shamanskii iterative method. Among these methods, the MNS2 method is thebest one.

5 Conclusion

In this paper, based on the Newton-Shamanskii method, we have proposed a modi-fied Newton-Shamanskii method for solving the minimal positive solution of the non-symmetric algebraic Riccati equation arising in transport theory and have given theconvergence analysis. The convergence analysis shows that the iteration sequence gen-erated by the modified Newton-Shamanskii method is monotonically increasing andconverges to the minimal positive solution of the vector equation. Numerical experi-ments show that the modified Newton-Shamanskii method has a better performancethan the Newton-Shamanskii method for the nonsymmetric algebraic Riccati equa-tion. We find that when Tk is chosen as the block triangular of the Jacobian matrix,the modified Newton-Shamanskii method has a better convergence rate. The choiceof the matrix Tk impacts the convergence rate of the modified Newton-Shamanskiimethod, hence, the determination of the optimum matrix Tk such that the modifiedNewton-Shamanskii method has a better convergence rate needs further to be studied.

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Hesitant fuzzy filters and hesitant fuzzy G-filters in

residuated lattices

G. Muhiuddin

Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Abstract.

Characterizations of a hesitant fuzzy filter in a residuated lattice are considered. Given a

hesitant fuzzy set, a new hesitant fuzzy filter of a residuated lattice is constructed. The notion

of a hesitant fuzzy G-filter of a residuated lattice is introduced, and its characterizations are

discussed. Conditions for a hesitant fuzzy filter to be a hesitant fuzzy G-filter are provided.

Finally, the extension property of a hesitant fuzzy G-filter is established.

1. Introduction

The notions of Atanassov’s intuitionistic fuzzy sets, type 2 fuzzy sets and fuzzy multisets etc.

are a generalization of fuzzy sets. As another generalization of fuzzy sets, Torra and Narukawa

[5] and Torra [6] introduced the notion of hesitant fuzzy sets and discussed the relationship

between hesitant fuzzy sets and intuitionistic fuzzy sets. Xia and Xu [11] studied hesitant fuzzy

information aggregation techniques and their application in decision making. They developed

some hesitant fuzzy operational rules based on the interconnection between the hesitant fuzzy

set and the intuitionsitic fuzzy set. Xu and Xia [12] proposed a variety of distance measures for

hesitant fuzzy sets, and investigated the connections of the aforementioned distance measures

and further developed a number of hesitant ordered weighted distance measures and hesitant

ordered weighted similarity measures. Xu and Xia [13] defined the distance and correlation

measures for hesitant fuzzy information and then considered their properties in detail. Wei

[9] investigated the hesitant fuzzy multiple attribute decision making problems in which the

attributes are in different priority level.

Residuated lattices are a non-classical logic system which is a formal and useful tool for

computer science to deal with uncertain and fuzzy information. Filter theory, which is an

important notion, in residuated lattices is studied by Shen and Zhang [4] and Zhu and Xu [15].

Wei [10] introduced the notion of hesitant fuzzy (implicative, regular and Boolean) filters in

residuated lattice, and discussed its properties.

2010 Mathematics Subject Classification: 06F35, 03G25, 06D72.

Keywords: Hesitant fuzzy filter, Hesitant fuzzy G-filter.

E-mail: [email protected]


394 Muhiuddin 394-404

2 G. Muhiuddin

In this paper, we deal with further properties of a hesitant fuzzy filter in a residuated lattice.

We consider characterizations of a hesitant fuzzy filter in a residuated lattice. Given a hesitant

fuzzy set, we construct a new hesitant fuzzy filter of a residuated lattice. We introduce the

notion of a hesitant fuzzy G-filter of a residuated lattice, and discuss its characterizations. We

provide conditions for a hesitant fuzzy filter to be a hesitant fuzzy G-filter. Finally, we establish

the extension property of a hesitant fuzzy G-filter.

2. Preliminaries

Definition 2.1 ([1, 2, 3]). A residuated lattice is an algebra (L,∨,∧,,→, 0, 1) of type (2, 2, 2, 2, 0, 0)

such that

(1) (L,∨,∧, 0, 1) is a bounded lattice.

(2) (L,, 1) is a commutative monoid.

(3) and → form an adjoint pair, that is,

(∀x, y, z ∈ L) (x ≤ y → z ⇔ x y ≤ z) .

In a residuated lattice L, the ordering ≤ and negation ¬ are defined as follows:

(∀x, y ∈ L) (x ≤ y ⇔ x ∧ y = x ⇔ x ∨ y = y ⇔ x→ y = 1)

and ¬x = x→ 0 for all x ∈ L.

Proposition 2.2 ([1, 2, 3, 7, 8]). In a residuated lattice L, the following properties are valid.

1→ x = x, x→ 1 = 1, x→ x = 1, 0→ x = 1, x→ (y → x) = 1.(2.1)

y ≤ (y → x)→ x.(2.2)

x ≤ y → z ⇔ y ≤ x→ z.(2.3)

x→ (y → z) = (x y)→ z = y → (x→ z).(2.4)

x ≤ y ⇒ z → x ≤ z → y, y → z ≤ x→ z.(2.5)

z → y ≤ (x→ z)→ (x→ y), z → y ≤ (y → x)→ (z → x).(2.6)

(x→ y) (y → z) ≤ x→ z.(2.7)

x y ≤ x ∧ y.(2.8)

x ≤ y ⇒ x z ≤ y z.(2.9)

y → z ≤ x ∨ y → x ∨ z.(2.10)

(x ∨ y)→ z = (x→ z) ∧ (y → z).(2.11)



Hesitant fuzzy filters and hesitant fuzzy G-filters in residuated lattices 3

Definition 2.3 ([4]). A nonempty subset F of a residuated lattice L is called a filter of L if it

satisfies the conditions:

(∀x, y ∈ L) (x, y ∈ F ⇒ x y ∈ F ) .(2.12)

(∀x, y ∈ L) (x ∈ F, x ≤ y ⇒ y ∈ F ) .(2.13)

Proposition 2.4 ([4]). A nonempty subset F of a residuated lattice L is a filter of L if and

only if it satisfies:

1 ∈ F.(2.14)

(∀x ∈ F ) (∀y ∈ L) (x→ y ∈ F ⇒ y ∈ F ) .(2.15)

3. Hesitant fuzzy filters

Let E be a reference set. A hesitant fuzzy set on E (see [6]) is defined in terms of a function

h that when applied to E returns a subset of [0, 1], that is, h : E →P([0, 1]).

In what follows, we take a residuated lattice L as a reference set.

Definition 3.1 ([10]). A hesitant fuzzy set h on L is called a hesitant fuzzy filter of L if it

satisfies:

(∀x, y ∈ L) (x ≤ y ⇒ h(x) ⊆ h(y)) ,(3.1)

(∀x, y ∈ L) (h(x) ∩ h(y) ⊆ h(x y)) .(3.2)

Example 3.2. Let L = [0, 1] be a subset of R. For any a, b ∈ L, define

a ∨ b = maxa, b, a ∧ b = mina, b,

a→ b =

1 if a ≤ b,(1− a) ∨ b otherwise,

and

a b =

0 if a+ b ≤ 1,

a ∧ b otherwise.

Then (L,∨,∧,,→, 0, 1) is a residuated lattice (see [15]). We define a hesitant fuzzy set

h : L→P([0, 1]), x 7→

(0.2, 0.7) if x ∈ (c, 1] where 0.5 ≤ c ≤ 1,

(0.3, 0.6] otherwise.

It is routine to verify that h is a hesitant fuzzy filter of L.

Example 3.3. Let L = 0, a, b, c, d, 1 be a set with the lattice diagram appears in Figure 1.



4 G. Muhiuddin

r

rrrr ra

1

0

d

c

b

HHH

H

HHHH

Figure 1

Consider two operation ‘’ and ’→’ shown in Table 1 and Table 2, respectively.

Table 1. Cayley table for the binary operation ‘’

0 a b c d 1

0 0 0 0 0 0 0

a 0 a c c 0 a

b 0 c b c d b

c 0 c c c 0 c

d 0 0 d 0 0 d

1 0 a b c d 1

Table 2. Cayley table for the binary operation ‘→’

→ 0 a b c d 1

0 1 1 1 1 1 1

a d 1 b b d 1

b 0 a 1 a d 1

c d 1 1 1 d 1

d a 1 1 1 1 1

1 0 a b c d 1

Then (L,∨,∧,,→, 0, 1) is a residuated lattice. We define a hesitant fuzzy set

h : L→P([0, 1]), x 7→

[0.2, 0.9) if x ∈ 1, a,(0.3, 0.8] otherwise.

It is routine to verify that h is a hesitant fuzzy filter of L.

Wei [10] provided a characterization of a hesitant fuzzy filter as follows.




Lemma 3.4 ([10]). A hesitant fuzzy set h on L is a hesitant fuzzy filter of L if and only if it

satisfies

(∀x ∈ L) (h(x) ⊆ h(1)) .(3.3)

(∀x, y ∈ L) (h(x) ∩ h(x→ y) ⊆ h(y)) .(3.4)

We provide other characterizations of a hesitant fuzzy filter.

Theorem 3.5. A hesitant fuzzy set h on L is a hesitant fuzzy filter of L if and only if it

satisfies:

(∀x, y, z ∈ L) (x ≤ y → z ⇒ h(x) ∩ h(y) ⊆ h(z)) .(3.5)

Proof. Assume that h is a hesitant fuzzy filter of L. Let x, y, z ∈ L be such that x ≤ y → z.

Then h(x) ⊆ h(y → z) by (3.1), and so

h(z) ⊇ h(y) ∩ h(y → z) ⊇ h(x) ∩ h(y)

by (3.4).

Conversely let h be a hesitant fuzzy set on L satisfying (3.5). Since x ≤ x→ 1 for all x ∈ L,it follows from (3.5) that

h(1) ⊇ h(x) ∩ h(x) = h(x)

for all x ∈ L. Since x→ y ≤ x→ y for all x, y ∈ L, we have

h(y) ⊇ h(x) ∩ h(x→ y)

for all x, y ∈ L. Hence h is a hesitant fuzzy filter of L.

Theorem 3.6. A hesitant fuzzy set h on L is a hesitant fuzzy filter of L if and only if h satisfies

the condition (3.3) and

(∀x, y, z ∈ L) (h(x→ (y → z)) ∩ h(y) ⊆ h(x→ z)) .(3.6)

Proof. Assume that h is a hesitant fuzzy filter of L. Then the condition (3.3) is valid. Using

(2.4) and (3.4), we have

h(x→ z) ⊇ h(y) ∩ h(y → (x→ z))

= h(y) ∩ h(x→ (y → z))

for all x, y, z ∈ L.Conversely, let h be a hesitant fuzzy set on L satisfying (3.3) and (3.6). Taking x := 1 in

(3.6) and using (2.1), we get

h(z) = h(1→ z) ⊇ h(1→ (y → z)) ∩ h(y)

= h(y → z) ∩ h(y)

for all y, z ∈ L. Thus h is a hesitant fuzzy filter of L by Lemma 3.4.



6 G. Muhiuddin

Lemma 3.7. Every hesitant fuzzy filter h on L satisfies the following condition:

(∀a, x ∈ L) (h(a) ⊆ h((a→ x)→ x)) .(3.7)

Proof. If we take y = (a→ x)→ x and x = a in (3.4), then

h((a→ x)→ x) ⊇ h(a) ∩ h(a→ ((a→ x)→ x))

= h(a) ∩ h((a→ x)→ (a→ x))

= h(a) ∩ h(1) = h(a).

This completes the proof.

Theorem 3.8. A hesitant fuzzy set h on L is a hesitant fuzzy filter of L if and only if it satisfies

the following conditions:

(∀x, y ∈ L) (h(x) ⊆ h(y → x)) ,(3.8)

(∀x, a, b ∈ L) (h(a) ∩ h(b) ⊆ h((a→ (b→ x))→ x)) .(3.9)

Proof. Assume that h is a hesitant fuzzy filter of L. Using (2.1), (3.3) and (3.4), we have

h(y → x) ⊇ h(x) ∩ h(x→ (y → x)) = h(x) ∩ h(1) = h(x)

for all x, y ∈ L. Using (3.6) and (3.7), we get

h((a→ (b→ x))→ x) ⊇ h((a→ (b→ x))→ (b→ x)) ∩ h(b) ⊇ h(a) ∩ h(b)

for all a, b, x ∈ L.

Conversely, let h be a hesitant fuzzy set on L satisfying two conditions (3.8) and (3.9). If we

take y := x in (3.8), then h(x) ⊆ h(x→ x) = h(1) for all x ∈ L. Using (3.9) induces

h(y) = h(1→ y) = h((x→ y)→ (x→ y))→ y) ⊇ h(x→ y) ∩ h(x)

for all x, y ∈ L. Therefore h is a hesitant fuzzy filter of L by Lemma 3.4.

Theorem 3.9. A hesitant fuzzy set h on L is a hesitant fuzzy filter of L if and only if the set

hτ := x ∈ L | τ ⊆ h(x)

is a filter of L for all τ ∈P([0, 1]) with hτ 6= ∅.

Proof. Assume that h is a hesitant fuzzy filter of L. Let x, y ∈ L and τ ∈ P([0, 1]) be such

that x ∈ hτ and x→ y ∈ hτ . Then τ ⊆ h(x) and τ ⊆ h(x→ y). It follows from (3.3) and (3.4)

that h(1) ⊇ h(x) ⊇ τ and h(y) ⊇ h(x) ∩ h(x → y) ⊇ τ and so that 1 ∈ hτ and y ∈ hτ . Hence

hτ is a filter of L by Proposition 2.4.

Conversely, suppose that hτ is a filter of L for all τ ∈P([0, 1]) with hτ 6= ∅. For any x ∈ L,

let h(x) = δ. Then x ∈ hδ and hδ is a filter of L. Hence 1 ∈ hδ and so h(x) = δ ⊆ h(1). For




any x, y ∈ L, let h(x) = δx and h(x → y) = δx→y. If we take δ = δx ∩ δx→y, then x ∈ hδ and

x→ y ∈ hδ which imply that y ∈ hδ. Thus

h(x) ∩ h(x→ y) = δx ∩ δx→y = δ ⊆ h(y).

Therefore h is a hesitant fuzzy filter of L by Lemma 3.4.

Theorem 3.10. For a hesitant fuzzy set h on L, let h be a hesitant fuzzy set on L defined by

h : L→P([0, 1]), x 7→

h(x) if x ∈ hτ ,∅ otherwise,

where τ ∈P([0, 1]) \ ∅. If h is a hesitant fuzzy filter of L, then so is h.

Proof. Suppose that h is a hesitant fuzzy filter of L. Then hτ is a filter of L for all τ ∈P([0, 1])

with hτ 6= ∅ by Theorem 3.9. Thus 1 ∈ hτ , and so h(1) = h(1) ⊇ h(x) ⊇ h(x) for all x ∈ L.

Let x, y ∈ L. If x ∈ hτ and x→ y ∈ hτ , then y ∈ hτ . Hence

h(x) ∩ h(x→ y) = h(x) ∩ h(x→ y) ⊆ h(y) = h(y).

If x /∈ hτ or x→ y /∈ hτ , then h(x) = ∅ or h(x→ y) = ∅. Thus

h(x) ∩ h(x→ y) = ∅ ⊆ h(y).

Therefore h is a hesitant fuzzy filter of L.

Theorem 3.11. If h is a hesitant fuzzy filter of L, then the set

Γa := x ∈ L | h(a) ⊆ h(x)

is a filter of L for every a ∈ L.

Proof. Since h(1) ⊇ h(a) for all a ∈ L, we have 1 ∈ Γa. Let x, y ∈ L be such that x ∈ Γa and

x → y ∈ Γa. Then h(x) ⊇ h(a) and h(x → y) ⊇ h(a). Since h is a hesitant fuzzy filter of L, it

follows from (3.4) that

h(y) ⊇ h(x) ∩ h(x→ y) ⊇ h(a)

so that y ∈ Γa. Hence Γa is a filter of L by Proposition 2.4.

Theorem 3.12. Let a ∈ L and let h be a hesitant fuzzy set on L. Then

(1) If Γa is a filter of L, then h satisfies the following condition:

(∀x, y ∈ L) (h(a) ⊆ h(x) ∩ h(x→ y) ⇒ h(a) ⊆ h(y)).(3.10)

(2) If h satisfies (3.3) and (3.10), then Γa is a filter of L.



8 G. Muhiuddin

Proof. (1) Assume that Γa is a filter of L. Let x, y ∈ L be such that

h(a) ⊆ h(x) ∩ h(x→ y).

Then x→ y ∈ Γa and x ∈ Γa. Using (2.15), we have y ∈ Γa and so h(y) ⊇ h(a).

(2) Suppose that h satisfies (3.3) and (3.10). From (3.3) it follows that 1 ∈ Γa. Let x, y ∈ Lbe such that x ∈ Γa and x → y ∈ Γa. Then h(a) ⊆ h(x) and h(a) ⊆ h(x → y), which imply

that h(a) ⊆ h(x) ∩ h(x → y). Thus h(a) ⊆ h(y) by (3.10), and so y ∈ Γa. Therefore Γa is a

filter of L by Proposition 2.4.

Definition 3.13 ([14]). A nonempty subset F of L is called a G-filter of L if it is a filter of L

that satisfies the following condition:

(∀x, y ∈ L) ((x x)→ y ∈ F ⇒ x→ y ∈ F ) .(3.11)

We consider the hesitant fuzzification of G-filters.

Definition 3.14. A hesitant fuzzy set h on L is called a hesitant fuzzy G-filter of L if it is a

hesitant fuzzy filter of L that satisfies:

(∀x, y ∈ L) (h((x x)→ y) ⊆ h(x→ y)) .(3.12)

Note that the condition (3.12) is equivalent to the following condition:

(∀x, y ∈ L) (h(x→ (x→ y)) ⊆ h(x→ y)) .(3.13)

Example 3.15. The hesitant fuzzy filter h in Example 3.3 is a hesitant fuzzy G-filter of L.

Lemma 3.16. Every hesitant fuzzy filter h of L satisfies the following condition:

(∀x, y, z ∈ L) (h(x→ (y → z)) ∩ h(x→ y) ⊆ h(x→ (x→ z))) .(3.14)

Proof. Let x, y, z ∈ L. Using (2.4) and (2.6), we have

x→ (y → z) = y → (x→ z) ≤ (x→ y)→ (x→ (x→ z)).

It follows from Theorem 3.5 that

h(x→ (y → z)) ∩ h(x→ y) ⊆ h(x→ (x→ z)).

This completes the proof.

Theorem 3.17. Let h be a hesitant fuzzy set on L. Then h is a hesitant fuzzy G-filter of L if

and only if it is a hesitant fuzzy filter of L that satisfies the following condition:

(∀x, y, z ∈ L) (h(x→ (y → z)) ∩ h(x→ y) ⊆ h(x→ z)) .(3.15)




Proof. Assume that h is a hesitant fuzzy G-filter of L. Then h is a hesitant fuzzy filter of L.

Note that x ≤ 1 = (x → y) → (x → y), and thus x → y ≤ x → (x → y) for all x, y ∈ L. It

follows from (3.1) that h(x→ y) ⊆ h(x→ (x→ y)). Combining this and (3.13), we have

h(x→ y) = h(x→ (x→ y))(3.16)

for all x, y ∈ L. Using (3.14) and (3.16), we have

h(x→ (y → z)) ∩ h(x→ y) ⊆ h(x→ z)

for all x, y, z ∈ L.

Conversely, let h be a hesitant fuzzy filter of L that satisfies the condition (3.15). If we put

y = x and z = y in (3.15) and use (2.1) and (3.3), then

h(x→ y) ⊇ h(x→ (x→ y)) ∩ h(x→ x)

= h(x→ (x→ y)) ∩ h(1)

= h(x→ (x→ y))

for all x, y ∈ L. Therefore h is a hesitant fuzzy G-filter of L.

Theorem 3.18. Let h be a hesitant fuzzy filter of L. Then h is a hesitant fuzzy G-filter of L

if and only if the following condition holds:

(∀x ∈ L) (h(x→ (x x)) = h(1)) .(3.17)

Proof. Assume that h satisfies the condition (3.17) and let x, y ∈ L. Since

x→ (x→ y) = (x x)→ y ≤ (x→ (x x))→ (x→ y)

by (2.4) and (2.6), it follows from (3.1) that

h(x→ (x→ y)) ⊆ h((x→ (x x))→ (x→ y)).

Hence, we have

h(x→ y) ⊇ h((x→ (x x))→ (x→ y)) ∩ h(x→ (x x))

⊇ h(x→ (x→ y)) ∩ h(x→ (x x))

= h(x→ (x→ y)) ∩ h(1)

= h(x→ (x→ y))

by using (3.4), (3.17) and (3.3). Hence h is a hesitant fuzzy G-filter of L.

Theorem 3.19. (Extension property) Let h and g be hesitant fuzzy filters of L such that h ⊆ g,

i.e., h(x) ⊆ g(x) for all x ∈ L and h(1) = g(1). If h is a hesitant fuzzy G-filter of L, then so is

g.



10 G. Muhiuddin

Proof. Assume that h is a hesitant fuzzy G-filter of L. Using (2.4) and (2.1), we have

x→ (x→ ((x→ (x→ y))→ y)) = (x→ (x→ y))→ (x→ (x→ y)) = 1

for all x, y ∈ L. Thus

g(x→ ((x→ (x→ y))→ y)) ⊇ h(x→ ((x→ (x→ y))→ y))

= h(x→ (x→ ((x→ (x→ y))→ y)))

= h(1) = g(1)

by hypotheses and (3.16), and so

g(x→ ((x→ (x→ y))→ y)) = g(1)

for all x, y ∈ L by (3.3). Since g is a hesitant fuzzy filter of L, it follows from (3.4), (2.4) and

(3.3) that

g(x→ y) ⊇ g(x→ (x→ y)) ∩ g((x→ (x→ y))→ (x→ y))

= g(x→ (x→ y)) ∩ g(x→ ((x→ (x→ y))→ y))

= g(x→ (x→ y)) ∩ g(1)

= g(x→ (x→ y))

for all x, y ∈ L. Therefore g is a hesitant fuzzy G-filter of L.

Acknowledgements

The author wishes to thank the anonymous reviewers for their valuable suggestions. Also, the

author would like to acknowledge financial support for this work, from the Deanship of Scientific

Research (DRS), University of Tabuk, Tabuk, Saudi Arabia, under grant no. S/0123/1436.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO. 2, 2016

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αβ-Statistical Convergence and Strong αβ-Convergence of Order γ for a Sequence of Fuzzy Numbers, Zeng-Tai Gong, and Xue Feng,…………………………………………………228

IF Rough Approximations Based On Lattices, Gangqiang Zhang, Yu Han, and Zhaowen Li,237

Some Results on Approximating Spaces, Neiping Chen,……………………………………254

Divisible and Strong Fuzzy Filters of Residuated Lattices, Young Bae Jun, Xiaohong Zhang, and Sun Shin Ahn,……………………………………………………………………………….264

Frequent Hypercyclicity of Weighted Composition Operators on Classical Banach Spaces, Shi-An Han, and Liang Zhang,………………………………………………………………277

On The Special Twisted q-Polynomials, Jin-Woo Park,…………………………………….283

Equicontinuity of Maps on [0, 1), Kesong Yan, Fanping Zeng, and Bin Qin,………………293

On Mixed Type Riemann-Liouville and Hadamard Fractional Integral Inequalities, Weerawat Sudsutad, S.K. Ntouyas, and Jessada Tariboon,…………………………………………….299

Weighted Composition Operators From F(p, q, s) Spaces To nth Weighted-Orlicz Spaces, Haiying Li, and Zhitao Guo,…………………………………………………………………315

Modified q-Daehee Numbers and Polynomials, Dongkyu Lim,…………………………….324

A Class of BVPS for Second-Order Impulsive Integro-Differential Equations of Mixed Type In Banach Space, Jitai Liang, Liping Wang, and Xuhuan Wang,………………………………331

Distribution and Survival Functions With Applications in Intuitionistic Random Lie C*-Algebras, Afrah A. N. Abdou, Yeol Je Cho, and Reza Saadati,…………………………345

Cubic ρ-Functional Inequality and Quartic ρ-Functional Inequality, Choonkil Park, Jung Rye Lee, and Dong Yun Shin,…………………………………………………………………….355

Complex Valued Gb-Metric Spaces, Ozgur Ege,…………………………………………….363

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO. 2, 2016

(continued)

Finite Difference approximations for the Two-side Space-time Fractional Advection-diffusion Equations, Yabin Shao, and Weiyuan Ma,…………………………………………………369

A Modified Newton-Shamanskii Method for a Nonsymmetric Algebraic Riccati Equation, Jian-Lei Li, Li-Tao Zhang, Xu-Dong Li, Qing-Bin Li,………………………………..……380

Hesitant Fuzzy Filters and Hesitant Fuzzy G-Filters in Residuated Lattices, G. Muhiuddin,394

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