democratic people's republic of algeria - univ-tebessa · democratic people's republic of...

Democratic People's Republic of Algeria Ministry of Higher Education and Scientific Research

University BADJI Mokhtar Annaba Laboratory of Advanced Materials

The Algerian-Turkish International days on Mathematics 2012

TOPICS

Nonlinear Analysis Functional Analysis Differential Equations Applied Mathematics Sequence Spaces Summability Theory Algebraic Coding Theory and Cryptography Mathematical Chemistry and Environment

9 ‐ 11 October 2012, Annaba, Algeria

i

Welcome

The “Algerian-Turkish International days on Mathematics 2012 ATIM’2012”

jointly organized by Laboratory of Advanced Materials, Badji Mokhtar Annaba

University and Fatih University, Istanbul, Turkey, will be held on 9-11 October

2012 in Annaba, Algeria. The aim of this conference is to provide a platform for

scientific expertise in mathematics to present their recent works, exchange

ideas and new methods in this important area and to bring together

mathematicians to improve collaboration between local and international

participants. We are looking forward to meeting you in Annaba at ATIM’2012.

Topics

Nonlinear Analysis

Functional Analysis

Differential Equations

Applied Mathematics

Sequence Spaces

Summability Theory

Algebraic Coding Theory and Cryptography

Mathematical Chemistry and Environment

ii

Honorary Committee

Prof. Abdelkrim KADI, Rector of the University Badji Mokhtar Annaba

Prof. Djelloul MESSADI, Director of Laboratory LASEA, Annaba

iii

Scientific Committee

Chairship: Prof. Assia Guezane-Lakoud, Badji Mokhtar University, Annaba, Algeria

Co-Chairship: Prof. Feyzi Basar, Fatih University, Turkey

Prof. Ravi P Agarwal, Texas A & M University Kingsville, USA

Prof. Allaberen Ashyralyev, Fatih University, Turkey

Prof. Eberhard Malkowsky, Fatih University, Turkey & Universitat Giessen, Germany

Prof. Mahmoud Abdel-Aty, Bahrain University, Bahrain

Prof. Mustafa Bayram, Yildiz Technical University, Turkey

Prof. Mouffak Benchohra, Université de Sidi Bel-Abbès, Algeria

Prof Ali Bentrad, University Reims, France

Prof. Rifat Colak, Firat University, Turkey

Prof. Smail Djebali, ENS, Algiers, Algeria

Prof. Rabah Khaldi, Badji Mokhtar university,Annaba,Algeria

Prof. Ismail kelaiaia, Badji Mokhtar university,Annaba,Algeria

Prof. M. Mursaleen, Aligarh Muslim University, India

Prof. Ekrem Savas, Istanbul Commerce University, Turkey

Prof. Reza Saadati, IslamicAzad University, Iran

Prof. Irfan Siap,Yildiz Technical University, Turkey

Prof. Cemil Tunc, Yuzuncu yil University, Turkey

Prof. Ahcen Djoudi, Badji Mokhtar university, Annaba, Algeria

Prof. Kamel Haouam, University Tebessa, Algeria

iv

Organizing committee

Chair: Prof. Abdelhamid Souahi, Director of Laboratory of Advanced Materials,

Annaba University

Prof. Assia Guezane-Lakoud

Prof. Seddik Bouras

Prof. Rabah Khaldi

Prof. Smail Kelaiaia

Prof. Abdelrani Messalhi

Dr. Fateh Ellagoune

Dr. Mohamed Faouzi Harket

Dr. Mohamed Tahar Bouaza

Dr. Farid Gheldane

Dr. Abderrezak Chaoui

Dr. Lamine Sahari

Mrs. Salima Bensebaa

Mr. Mohamed Boulakraa

Mrs. Nacira Hamidane

Mrs. Assia Frioui

Mrs. Fatima Aissaoui

v

Invited speakers

Feyzi Basar (Fatih University, Büyükçekmece, Istanbul, Turkey) Survey on the domain of triangles in the sequence spaces

Eberhard Malkowsky (Büyükçekmece, Istanbul, Turkey; Arndtstr. 2, D-35392 Giessen, Germany) Characterisation of Compact Linear Operators between Certain BK Spaces

Ali Bentrad (Laboratoire de Maths, Université de Reims) On the solutions of Cauchy problem for a class pf PDE with double characteristic at point

Irfan Siap (Yildiz Technical University Turkey) Recent Studies on DNA Error Correcting Codes

Cemil Tunç (Yüzüncü Yıl University, Van –Turkey) Stability to Vector Lienard Equation with Constant Deviating Argument

Ekrem Savas (Istanbul Commerce University, Turkey) On Generalized Statistical Convergence in 2-normed space Via Ideals

Rifat Colak (Firat University, Turkey) Statistical convergence and its some generalizations

Mahmoud Abdel-Aty (Bahrain University, Bahrain) Aspects of Mathematical Model of Nanomechanical Resonators interacting with Superconducting Materials

Smail Djebali (ENS, Algiers, Algeria) ELEMENTS OF FIXED POINT THEORY: Old and New

M. Mursaleen (Aligarh Muslim University, India)

Mustafa Bayram (Yildiz Technical University, Turkey)

M Benchohra (University of Sidi Bel-Abbès, Algeria)

Ravi P Agarwal (Texas A & M University Kingsville, USA)

vi

Contents

Feyzi Basar Survey on the domain of triangles in the sequence spaces

1

Cemil Tunç Stability to Vector Lienard Equation with Constant Deviating Argument

2

Rifat Colak Statistical convergence and its some generalizations

6

Eberhard Malkowsky Characterisation of Compact Linear Operators between Certain BK Spaces

9

Ekrem Savas On Generalized Statistical Convergence in 2-normed space Via Ideals

10

Smail Djebali ELEMENTS OF FIXED POINT THEORY: Old and New

12

Najet Abada, R.P.AGARWAL, M. BENCHOHRA & H.HAMMOUCHE Extrapolation method and some Nondensely defined impulsive semilinear neutral partial functional differential inclusions

15

Adimi Hadjer, Abdenacer Makhlouf Computing Index of Graded Filiform and Quasi Filiform Lie Algebras

16

Aissaoui Fatima, Assia Guezane Lakoud New Čebyšev type inequalities for fractional integrals

18

Abdelouaheb Ardjouni, A. Djoudi Stability in nonlinear neutral integro-differential equations with variable delay

19

Arrache Saïda, Ouafi Rachid Solving the traffic assignment problem using a new version of the Frank-Wolfe algorithm

20

RAHIMA ATMANIA Local and extremal solutions of some fractional integrodifferential equation with impulses

21

Salih AYTAR A Neighbourhood System of Fuzzy Numbers and its Topology

22

vii

Aicha Batoul, Kenza Guenda, and T. Aaron Gulliver Construction of Self-dual and Isodual Cyclic Codes over Finite chain Rings

23

A. Bensayah, S. Nicaise, A. Ghezal and D. A. Chacha Asymptotic analysis of linearly elastostatic Signorini problem with Coulomb friction of shallow shell

24

Bensebaa Salima, Guezane-Lakoud Assia Positive solutions for a fractional boundary value problem with fractional derivative condition

25

Berrebah Fatima, A.Lakmeche Impulsive Differential Equations with Variable Times

26

Cennet Bolat, Ahmet İPEK The solutions of the octonionic equations in the form α(xβ)=ρ, (αx)β=ρ ve αxα=ρ

27

Bouakkaz Ahlème, A. DJOUDI Stability of nonlinear differential equation with delay via Schauder's and Krasnoselskii's theorems

28

Boualem Alleche Multivalued mixed variational inequalities with locally cocoercive multivalued mappings

29

Hafsia Deham, Djoudi Ahcene Periodic solutions for neutral nonlinear system of di¤erential equations with two functional delays

30

Asma Djerrai, Djellit Ilhem Some pecular dynamical properties of three dimensional maps

31

Erdinç Dündar, Feyzi Başar On the Fine Spectrum of the Upper Triangle Double Band Matrix Δ⁺ on the Sequence Space c₀

33

Amir Elhaffaf, Mostepha Naceri triple positive solutions for system of nonlinear second-order differential equations three point boundary value

34

Elif Segah Oztas, Irfan Siap Reversible Codes over GF(16) and DNA Codes

35

Ebtisam EJAMAL Constraint Coefficient Problems for Subclasses of Univalent Functions

36

Ellaggoune Fateh, Zenkoufi Housseyn Local solution of the system of equations describing the motion of water in the desert areas

37

Ilias Elmouki, Smahane SAADI Optimal control of BCG immunotherapy in a mathematical model of superficial bladder cancer

38

Abdullah Said Erdogan, Allaberen Ashyralyev, Ali Ugur Sazaklıoglu Computational flow analysis on a two phase model with an unknown pressure function

39

viii

Ferchichi Mohamed Réda, Djerrai Asma Systèmes Dynamiques Discrets Bidimensionnels Relation entre Point Fixe et Point Focal

40

Frioui Assia, Assia Guezane Lakoud, R. Khaldi Solvability of a third order boundary value problem at resonance

41

Soumaya Ghnimi, Imed Ghanmi He’s method for solving some linear heat equation

42

AYŞE NUR GÜNCAN, Ulas Yamanci and Mehmet Gürdal Lacunary Statistical Limit Points in Random 2-Normed Spaces

43

Mehmet GÜRDAL, Mualla Birg ul Huban On I-convergence of Double Sequences in the Topology Induced by Random 2-Norms

44

Murat Güvercin, Feyzi Başar Some topological and geometric properties of the domain of the triple band matrix B(r,s,t) in the sequence space ℓ(p)

45

Hamidane Nacira, Assia Guezane Lakoud, R. Khaldi Existence and uniqueness of solution for a second order boundary value problem

46

Houas Mohamed, Abbas Moncef Solution de Meilleur Compromis pour le Problème du Plus Court Chemin Multicritère

47

Mahmut IŞIK, YAVUZ ALTIN and HIFSI ALTINOK THE PROPERTIES OF SOME SEQUENCE SPACES ON SEMINORMED SPACES

49

Abdennaceur Jarray, S. ABIDI Identification of the Diffusion in a Semi-Linear Problem

51

Rochdi Jebari, Ghanmi Imed and Abderrahman Boukricha Adomian Decomposition Method for solving nonlinear diffusion equation with convection term

52

Ali Karaisa, Feyzi Başar ON THE FINE SPECTRUM OF THE GENERALIZED DIFFERENCE OPERATOR DEFINED BY A DOUBLE SEQUENTIAL BAND MATRIX OVER THE SEQUENCE SPACE

53

Mohamed Kecies General approach of the root of a p-adic number

54

Mohamed Amine Kerker Hypergeometric solutions of a forth order Fuchsian partial differential equation

56

MOHAMMA SAEED KHAN SOME COUPLED FIXED POINT THEOREMS FOR MAPPINGS SATISFYING A RATIONAL EXPRESSION

57

ix

Amel Boudiaf, Drabla Saleh General boundary stabilization of memory type in thermoelasticity

58

Murat Kirisci, Feyzi Başar ON THE SPACES OF EULER ALMOST CONVERGENT AND EULER ALMOST NULL SEQUENCES

59

KOUACHI Samia, ZITOUNI Homogeneous Spaces of Elliptic Curves and associated groups

61

LEHBAB FATIMA, Z. Derriche et M. Bouhent Contribution a la dépollution des eaux colorées par les argiles anioniques

62

Lemdani Rachid, Moncef Abbas A broadcast chromatic number of a tree

63

Sara Litimein, M. Benchohra Abstract Fractional Integro-differential Equations With state-Dependent Delay

64

Belgacem Maher, Mahdjoubi Kouroch & Boukricha Abderrahmane On Mathematical Simulation of Cloaking

65

Ibrahima Mbaye Lagrange interpolation arising from a steady fluid structure interaction problem

66

Mebarek-Oudina Fateh, R. Bessaih Numerical Modeling of swirling flow in cylindrical configuration under the constant magnetic field

67

Hamza Medekhel, Mohamed Said Said Sur Quelque Problèmes aux Limites gouvernée par operateur de Laplace à poids dans un domaine polygône

69

Merabet Yahet, AHLEM LABDAOUI Statistical Bayesian Analysis of Experimental Data

71

Merabet Ismail, SERGE NICAISE AND DJAMEL AHMED CHACHA On the asymptotic behaviour of transmission thin shell problems

72

Ahcene MERAD, Abdelfatah BOUZIANI A Computational Method for integrodifferential hyperbolic equation with integral conditions.

73

MESSAOUDENE Hadia Comparison between classes of Joel Anderson and finite operators

75

Mikail ET, Murat KARAKAS and Muhammed ÇINAR On Statistical Convergence of Order α of Generalized Difference Sequences

76

x

Husnia Mohamed Eldanfour Modified Newton's methods with fifth or sixth –order convergence and multiple roots

77

Havva Nergiz, Feyzi Başar On The New Sequence Spaces Including The Spaces of All Convergent and Null Sequences

78

Fatih NURAY Statistical Derivative

79

Nawel Outili, M. Amroune and S. Hammoudi Operating Conditions Optimization of Catalytic Fixed Bed Reactor

81

Celal Çakan AN INEQUALITY RELATED TO THE RIESZ CORE OF DOUBLE SEQUENCES

82

Ahmet Faruk Çakmak, Feyzi Başar SPACE OF CONTINUOUS FUNCTIONS OVER THE FIELD OF NON-NEWTONIAN REAL NUMBERS

84

Raed Batahan On Generalized Gegenbauer Matrix Polynomials

86

Mourad Rahmani, Hacene Belbachir An inequality for n-convex functions

87

Samir RAHMANI, Abdelnasser DAHMANI Exponential inequalities for the Robbins Monro’s algorithm with associated Variables

88

RAHMOUNE Fazia, S. Ziani, L. Madi and M.S. Radjef Game Theory Approach to the Numerical Analysis of the M2 /M/1

89

Ahmet Sahiner Cone convergency for multiple sequences

90

Saltan Suna, M.T.KARAEV, M.GÜRDAL Basisity Problem and Weighted Shift Operators

91

Boukemara Ibtissem, Djellit Ilhem & Ferchichi Med Reda About a New Class of Bifurcations Generated by Piecewise Linear Maps

92

Djamila Seba, M. Benchohra Nonlinear Fractional Differential Inclusions in Banach Spaces

94

Hafidha Sebbagh, Mohammed Derhab Exact number of positive solutions for quasilinear boundary value problems with p-convex nonlinearities

95

xi

Seda Akbiyik, Irfan Siap MacWilliams Identity for Codes Over Forests

96

Neyaz Sheikh Convergence of wavelet expansions

97

Omar Slimani On Scaling Ill Conditionned Matrices

98

Yaser Sozen On a Combinatorial Laplacian and Homology of Compact Manifolds

99

SUDIP KUMAR PAL, Pratulananda Das & S. K. Ghosal SOME FURTHER REMARKS ON IDEAL SUMMABILITY IN 2-NORMED SPACES

100

Yassamina Tabet zatla, Naima Merzagui Existence of multiple positive solutions for a nonlocal boundary value problem with sign changing nonlinearities.

101

Özer Talo, Feyzi Başar Necessary and sufficient Tauberian conditions for the A^r method of summability

102

Sebiha Tekin, Feyzi Başar On the new sequence spaces including the spaces of absolutely p−summable and bounded sequences

104

Nouressadat Touafek On the solutions of some fractional systems of difference Equations

106

Touil Imene, Benterki Djamel A Theoretical and Numerical Performance of Central Trajectory Methods for Semi definite Programming

107

Houria Triki Generalized nonlinear Schrödinger equation with soliton solutions

108

YAHI Zahra, Bouroubi Sadek Some problems related to the average rank partition lattice a set

109

ULAŞ YAMANCI, M.T. Karaev, M. Gurdal On the Berezin symbols method, Abel convergence and related questions

110

Medine Yeilkayagil, Feyzi Başar Composited dual summability methods of the new sort

111

Serhat Yilmaz, Allaberen Ashyralyev On the Numerical Solution of Ultra Parabolic Equations with the Neumann Condition

112

Samia Youcefi, Johnny Henderson and Abdelghani Ouahab Existence and Solutions Set for ϕ-Laplacian Impulsive Differential Equations

113

xii

Anis Younes, A. Jarray and Y.Oual The Navier-Stokes problem in velocity-pressure formulation : convergence and Optimal Control

114

Zafer Cakir SPACE OF CONTINUOUS AND BOUNDED FUNCTIONS OVER THE FIELD OF NON-NEWTONIAN COMPLEX NUMBERS

115

ZERDANI Ouiza, Mustapha Moulai Optimizing a Nonlinear Function over the Integer Efficient Set

116

Mališa Žižovič, Nada Damljanovic, Jasmina Janjic Multiplicative method for multi criteria analysis

117

Bouhassoun Abdelkader Piecewise Decomposition method for solving fractional differential equation

118

BOUMAZA Nouri, A. Guezane-Lakoud Faedo Galerkin’s Method for non-local boundary value problem

119

Candan Murat Domain of the double sequential band matrix in the classical sequence spaces

121

Chaoui Abderrazek, Assia Guezane Lakoud Rothe-Galerkin’s method for a doubly nonlinear integrodifferential equations

122

CHEBBAH Mohammed, OUANES Mohand Production et Déploiment de logiciels distribuables p our Résolution de Problèmes Min-Max en Contrôle optimal

124

Nada Damljanovic, Mališa Žižovič and Jasmina Janjic MCD-method of minimal suitable values

126

Ugur Kadak, Hakan EFE & Feyzi BAŞAR Some sequence and function spaces by using the partial metric

127

Madad Khan & Nasir Khan Characterizations of Regular Abel-Grassmann's Groupoids

128

Saima Anis Coset diagram for the action of Picard Group on Q(i,√3)

131

Mahammed-Salah Abdelouahab & Nasr-eddine Hamri Hyper chaos in a fractional Chua system and its chaos-control and synchronization

132

Mücahit Akbiyik & Salim YÜCE Euler Savary’s Formula on Galilean Plane

133

Laboratory of Advanced Materials, Badji Mokhtar Annaba University Fatih University, Istanbul, Turkey

The Algerian-Turkish International days on Mathematics AITM 2

012

Survey on the domain of triangles in the sequence spaces

Feyzi Basar

Department of Mathematics,Fatih University,

Hadımkoy Campus, Buyukcekmece,34500 - Istanbul, Turkey

E-mail: [email protected], [email protected]

Abstract By ω, we denote the space of all real valued sequences. Any vector subspace of ω iscalled a sequence space. The domain λA of an infinite matrix A in a sequence space λ is defined by

λA =x = (xk) ∈ ω : Ax ∈ λ

,

which is a sequence space, where Ax = (Ax)n with (Ax)n =∑∞

k=0 ankxk, n ∈ N = 0, 1, 2, . . .. If Ais triangle, then one can easily observe that the sequence spaces λA and λ are linearly isomorphic, i.e.,λA

∼= λ.Although in most cases the new sequence space λA generated by the triangle matrix A from a sequencespace λ is the expansion or the contraction of the original space λ, it may be observed in some casesthat those spaces overlap. Define the S− and ∆−transform of x = (xk) ∈ ω by (Sx)n =

∑nk=0 xk and

(∆x)n = xn − xn−1, (x−1 ≡ 0), for all n ∈ N. Then, one can easily see that the inclusion λS ⊂ λstrictly holds for λ ∈ ℓ∞, c, c0. Further, one can deduce that the inclusion λ ⊂ λ∆ also strictly holdsfor λ ∈ ℓ∞, c, c0, ℓp, where 0 < p < ∞. However, if we define λ = c0 ⊕ spanz with z = (−1)k, i.e.,x ∈ λ if and only if x = s+αz for some s ∈ c0 and some α ∈ C, and consider the matrix A with the rowsAn defined by An = (−1)ne(n) for all n ∈ N, we have Ae = z ∈ λ but Az = e /∈ λ which lead us to theconsequences that z ∈ λ \ λA and e ∈ λA \ λ. That is to say that the sequence spaces λA and λ overlapbut neither contains the other. The approach constructing a new sequence space by means of the matrixdomain of a triangle matrix has recently been employed by number of researchers.

In this study, following Basar [Summability Theory and its Applications, Bentham Science Publishers,

e-books, Monographs, pp. xi+405, Istanbul-2012, ISBN: 978-1-60805-252-3], we summarize the literature

on the normed and paranormed sequence spaces derived by the domain of some triangle matrices.

2000 Mathematics Subject Classification: Primary 46A45; Secondary 40C05.Keywords: Normed and paranormed sequence spaces, matrix domain, α-, β- and γ-duals, triangle matrices, matrixtransformations.

1



012

Stability to Vector Lienard Equation with Constant Deviating Argument

Cemil Tunc

Department of Mathematics, Faculty of Sciences, Yuzuncu Yıl University, 65080, Van-TurkeyE-mail: [email protected]

Abstract In applied sciences, some practical problems concerning mechanics, the engineering tech-nique fields, economy, control theory, physics, chemistry, biology, medicine, atomic energy, informationtheory, etc. are associated with Lienard or modified Lienard equation. By this time, the qualitative prop-erties of solutions of scalar Lienard or modified Lienard equation with and without a deviating argumenthave been intensively discussed and are still being investigated in the literature. We refer the reader tothe papers or books of Ahmad and Rama Mohana Rao [1], Barnett [2], Burton ([3], [4]), Burton andZhang [5], Caldeira-Saraiva [6], Cantarelli [7], El’sgol’ts [8], El’sgol’ts and. Norkin [9], Gao and Zhao[10], Hale [11], Hara and Yoneyama ([12], [13]), Heidel ([14], [15]), Huang and Yu [16], Jitsuro and Yusuke[17], Kato ([18], [19]), Kolmanovskii and Myshkis [20], Krasovskiı [21], Li [22], Liu and Huang ([23], [24]),Liu and Xu [25], Liu [26],Long and Zhang [27], Luk [28], Malyseva [30], Muresan [31], Napoles Valdes[32], Sugie [33], Sugie and Amano [34], Sugie et al. [35], Tunc [36-39, 40-42], C. Tunc and E. Tunc [43],Yang [44], Ye et al. [45], Yoshizawa [46], Zhang ([47], [48]), Zhang and Yan [49], Zhou and Jiang [50],Zhou and Liu [51], Zhou and Xiang [52], Wei and Huang [53], Wiandt [54] and the references thereof.However, to the best of our knowledge from the literature, the stability and boundedness of solutions forvector Lienard equation with a deviating argument has not been discussed in the literature, yet.In this paper, we consider the vector Lienard equation with the multiple constant deviating arguments,τi > 0 :

X ′′(t) + F (X(t), X ′(t))X ′(t) +G(X(t)) +

n∑i=1

Hi(X(t− τi)) = P (t).

Some new results for the stability of solutions of this equation are obtained. By the work, we improve

some results in the literature.

References

[1] Ahmad, S. and Rama Mohana Rao, M., Theory of Ordinary differential Equations. WithApplications in Biology and engineering. Affiliated East-West Press Pvt. Ltd., New Delhi,1999.

[2] Barnett, S. A., new formulation of the Lienard-Chipart stability criterion. Proc. CambridgePhilos. Soc. 70 (1971), 269-274.

[3] Burton, T. A., On the equation x′′ + f(x)h(x′)x′ + g(x) = e(t). Ann. Mat. Pura Appl. (4)85 1970, 277-285.

2000 Mathematics Subject Classification:Keywords:

2


[4] Burton, T. A., Stability and Periodic solutions of Ordinary and Functional DifferentialEquations, Academic Press, Orlando, 1985.

[5] Burton, T. A. and Zhang, B., Boundedness, Periodicity, and Convergence of Solutions in aRetarded Lienard Equation. Ann. Mat. Pura Appl. (4) 165 (1993), 351-368.

[6] Caldeira-Saraiva, F., The boundedness of solutions of a Lienard equation arising in thetheory of ship rolling. IMA J. Appl. Math. 36 (1986), no. 2, 129-139.

[7] Cantarelli, G., On the stability of the origin of a non autonomous Lienard equation. Boll.Un. Mat. Ital. A (7) 10 (1996), 563-573.

[8] El’sgol’ts, L. E., Introduction to the Theory of Differential Equations with DeviatingArguments. Translated from the Russian by Robert J. McLaughlin Holden-Day, Inc., SanFrancisco, Calif.-London-Amsterdam, 1966.

[9] El’sgol’ts, L. E. and Norkin, S.B., Introduction to the Theory and Application of Dif-ferential Equations with Deviating Arguments. Translated from the Russian by John L.Casti. Mathematics in Science and Engineering, Vol. 105. Academic Press [A Subsidiary ofHarcourt Brace Jovanovich, Publishers], New York-London, 1973.

[10] Gao, S. Z. and Zhao, L. Q., Global asymptotic stability of generalized Lienard equation.

Chinese Sci. Bull. 40 (1995), no. 2, 105-109.

[11] Hale, J., Sufficient conditions for stability and instability of autonomous functional-differential equations. J . Differential Equations 1 (1965), 452-482.

[12] Hara, T. and Yoneyama, T., On the global center of generalized Lienard equation and itsapplication to stability problems. Funkcial. Ekvac. 28 (1985), no. 2, 171-192.

[13] Hara, T. and Yoneyama, T., On the global center of generalized Lienard equation and itsapplication to stability problems. Funkcial. Ekvac. 31 (1988), no. 2, 221-225.

[14] Heidel, J. W., Global asymptotic stability of a generalized Lienard equation. SIAM Journalon Applied Mathematics, 19 (1970), no. 3, 629-636.

[15] Heidel, J. W., A Liapunov function for a generalized Lienard equation. J . Math. Anal. Appl.39 (1972), 192-197.

[16] Huang, L. H. and Yu, J. S., On boundedness of solutions of generalized Lienard’s systemand its application. Ann. Differential Equations 9 (1993), no. 3, 311-318.

[17] Jitsuro, S. and Yusuke, A., Global asymptotic stability of non-autonomous systems ofLienard type. J . Math. Anal. Appl., 289 (2004), no.2, 673-690.

[18] Kato, J., On a boundedness condition for solutions of a generalized Lienard equation. J .Differential Equations 65 (1986), no. 2, 269-286.

[19] Kato, J., A simple boundedness theorem for a Lienard equation with damping. Ann. Polon.Math. 51 (1990), 183-188.

[20] Kolmanovskii, V. and Myshkis, A., Introduction to the Theory and Applications of Func-tional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.

[21] Krasovskiı, N. N., Stability of Motion. Applications of Lyapunov’s Second Method toDifferential Systems and Equations with Delay, Stanford, Calif.: Stanford University Press1963.

3


[22] Li, H. Q., Necessary and sufficient conditions for complete stability of the zero solution ofthe Lienard equation. Acta Math. Sinica 31 (1988), no. 2, 209-214.

[23] Liu, B. and Huang, L., Boundedness of solutions for a class of retarded Lienard equation.J . Math. Anal. Appl. 286 (2003), no. 2, 422-434.

[24] Liu, B. and Huang, L., Boundedness of solutions for a class of Lienard equations with adeviating argument. Appl. Math. Lett. 21 (2008), no. 2, 109-112.

[25] Liu, C. J. and Xu, S. L., Boundedness of solutions of Lienard equations. J . Qingdao Univ.Nat. Sci. Ed. 11 (1998), no. 3, 12-16.

[26] Liu, Z. R., Conditions for the global stability of the Lienard equation. Acta Math. Sinica38 (1995), no. 5, 614-620.

[27] Long, Wei; Zhang, Hong-Xia, Boundedness of solutions to a retarded Lienard equation.Electron. J . Qual. Theory Differ. Equ. 2010, No. 24, 9 pp,

[28] Luk, W. S., Some results concerning the boundedness of solutions of Lienard equations withdelay. SIAM J . Appl. Math. 30 (1976), no. 4, 768-774.

[29] Lyapunov, A.M., Stability of Motion, Academic Press, London, 1966.

[30] Malyseva, I. A., Boundedness of solutions of a Lienard differential equation. Differetial’niyeUravneniya 15 (1979), no. 8, 1420-1426.

[31] Muresan, M., Boundedness of solutions for Lienard type equations. Mathematica 40 (63)(1998), no. 2, 243-257.

[32] Napoles Valdes, J. E., Boundedness and global asymptotic stability of the forced Lienardequation. Rev. Un. Mat. Argentina 41 (2000), no. 4, 47-59 (2001).

[33] Sugie, J., On the boundedness of solutions of the generalized Lienard equation without thesignum condition. Nonlinear Anal. 11 (1987), no. 12, 1391-1397.

[34] Sugie, J. and Amano, Y., Global asymptotic stability of non-autonomous systems of Lienardtype. J . Math. Anal. Appl. 289 (2004), no. 2, 673-690.

[35] Sugie, J., Chen, D. L. and Matsunaga, H., On global asymptotic stability of systems ofLienard type. J . Math. Anal. Appl. 219 (1998), no. 1, 140-164.

[36] Tunc, C., Some new stability and boundedness results of solutions of Lienard type equationswith deviating argument. Nonlinear Anal. Hybrid Syst. 4 (2010), no. 1, 85-91.

[37] Tunc, C., A note on boundedness of solutions to a class of non-autonomous differentialequations of second order. Appl. Anal. Discrete Math. 4 (2010), no. 2, 361-372.

[38] Tunc, C., New stability and boundedness results of Lienard type equations with multipledeviating arguments. Izv. Nats. Akad. Nauk Armenii Mat. 45 (2010), no. 4, 47-56.

[39] Tunc, C., Boundedness results for solutions of certain nonlinear differential equations ofsecond order. J . Indones. Math. Soc. 16 (2010), no.2, 115-128.

[40] Tunc, C., Stability and boundedness of solutions of non-autonomous differential equationsof second order. J . Comput. Anal. Appl. 13 (2011), no. 6, 1067-1074.

4


[41] Tunc, C., On the stability and boundedness of solutions of a class of Lienard equations withmultiple deviating arguments. Vietnam J . Math. 39 (2011), no. 2, 177-190.

[42] Tunc, C., Uniformly stability and boundedness of solutions of second order nonlinear delaydifferential equations. Appl. Comput. Math. 10 (2011), no. 3, 449-462.

[43] Tunc, C. and Tunc, E., On the asymptotic behavior of solutions of certain second-orderdifferential equations. J . Franklin Inst. 344 (2007), no. 5, 391-398.

[44] Yang, Q. G., Boundedness and global asymptotic behavior of solutions to the Lienardequation. J . Systems Sci. Math. Sci. 19 (1999), no. 2, 211-216.

[45] Ye, Guo-Rong; Ding, Hui-Sheng; Wu, Xi-Lang, Uniform boundedness of solutions for aclass of Lienard equations. Electron. J . Differential Equations 2009, No. 97, 5 pp,

[46] Yoshizawa, T., Stability Theory by Liapunov’s Second Method. Publications of the Math-ematical Society of Japan, no. 9. The Mathematical Society of Japan, Tokyo 1966

[47] Zhang, B., On the retarded Lienard equation. Proc. Amer. Math. Soc. 115 (1992), no. 3,779-785.

[48] Zhang, B., Boundedness and stability of solutions of the retarded Lienard equation withnegative damping. Nonlinear Anal. 20 (1993), no. 3, 303-313.

[49] Zhang, X. S. and Yan, W. P., Boundedness and asymptotic stability for a delay Lienardequation. Math. Practice Theory 30 (2000), no. 4, 453-458.

[50] Zhou, X. and Jiang, W., Stability and boundedness of retarded Lienard-type equation.Chinese Quart. J . Math. 18 (2003), no. 1, 7-12.

[51] Zhou, J. and Liu, Z. R., The global asymptotic behavior of solutions for a nonautonomousgeneralized Lienard system. J . Math. Res. Exposition 21 (2001), no. 3, 410-414.

[52] Zhou, J. and Xiang, L., On the stability and boundedness of solutions for the retardedLienard-type equation. Ann. Differential Equations 15 (1999), no. 4, 460-465.

[53] Wei, J. and Huang, Q., Global existence of periodic solutions of Lienard equations withfinite delay. Dynam. Contin. Discrete Impuls. Systems 6 (1999), no. 4, 603-614.

[54] Wiandt, T., On the boundedness of solutions of the vector Lienard equation. Dynam. Sys-tems Appl. 7 (1998), no. 1, 141-143.

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012

Statistical convergence and its some generalizations

Rifat COLAK

Department of Mathematics, Firat University, 23119, Elazig − TURKIYEE-mail: [email protected]; [email protected]

Abstract A number sequence (xk) is said to be statistically convergent if there is a number L suchthat

limn→∞

1

n|k ≤ n : |xk − L| ≥ ε| = 0

for every ε > 0. We write S to denote the set of all statistically convergent sequences.Let λ = (λn) be a non-decreasing sequence of positive real numbers tending to ∞ such that λn+1 ≤ λn+1,λ1 = 1. The set of all such sequences will be denoted by Λ. Let λ = (λn) ∈ Λ. A sequence x = (xk) issaid to be λ−statistically convergent if for every ε > 0

limn→∞

1

λn|k ∈ In : |xk − L| ≥ ε| = 0

where In = [n− λn + 1, n]. Sλ will denote the set of all λ−statistically convergent sequences. In caseλ = (λn) = (n) , Sλ becomes S.Let 0 < α ≤ 1 be a fixed number. The sequence (xk) is said to be statistically convergent of order α if

limn→∞

1

nα|k ≤ n : |xk − L| ≥ ε| = 0

for every ε > 0. We use Sα to denote the set of all statistically convergent sequences of order α. In caseα = 1, Sα becomes S. We also define the sets Sα

λ and Sαλ in the similar manner.

The set of all strongly p−Cesaro summable sequences of order α will be denoted by wαp , i.e. x ∈ wα

p iff

limn→∞

1

nα

n∑k=1

|xk − ℓ|p = 0

for some ℓ (p > 0).We write [C, 1] and [V, λ] for the sets of sequences x = (xk) which are strongly Cesaro summable andstrongly (V, λ)- summable

We discusse the relations between the sequence sets S, Sα, Sλ , Sαλ , w

αp , [V, λ] , [V, λ, p] and so on for

different α’s and λ’s. Furthermore the similar definitions and discussions will be given for the double

sequences.


6


References

[1] Bhunia, S., Das, P., Pal, S. K., Restricting Statistical Convergence, Acta Math. Hungar ,DOI:

10.1007/s10474-011-0122-2

[2] Colak, R., Statistical convergence of order α, Modern Methods in Analysis and Its Applications, Anamaya

Pub., New Delhi, India 121-129, (2010)

[3] Colak, R., On Lambda Statistical convergence, Conference on Summability and Applications 2011, Istanbul

Commerce Univ. May 12-13, 2011, Istanbul

[4] R. Colak, Statistical Convergence of double sequences of order (α,β), The IV. Congress of the TWMS, July

01-03, 2011 Baku, Azarbaijan

[5] Colak, R., Bektas, C. A., λ−Statistical convergence of order α, Acta Math. Sci. 2011, 31B(3):953–959

[6] Colak, R.; Bektas, C. A. Errata to: ”λ-statistical convergence of order α, Acta Math.Sci. 2011, 31B(3):

953–959” [ MR2830535]. Acta Math. Sci. Ser. B Engl. Ed. 31 (2011), no. 5, 2099–2100

[7] Connor, J.S., The Statistical and Strong p-Cesaro Convergence of Sequences, Analysis 8 (1988) 47-63

[8] Duman O, Khan M K, Orhan C. A-statistical Convergence of Approximating Operators. Math Inequal Appl,

2003, 6(4): 689–699

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

[10] Fridy, J., On statistical convergence, Analysis 5 (1985) 301-313.

[11] Fridy, J. and Orhan, C., Lacunary statistical convergence, Pacific J. Math. 160 (1993) 43-51.

[12] Gadjiev, A. D. and Orhan, C., Some approximation theorems via statistical convergence, Rocky Mountain

J. Math. 32(1) (2002), 129-138.

[13] Kolk E., The statistical convergence in Banach spaces, Acta Comment. Univ. Tartu 928 (1991), 41-52.

[14] Maddox, I. J., Spaces of Strongly Summable Sequences, Quart. J. Math. Oxford (2), 18 (1967), 345-355

[15] Moricz, F., Statistical convergence of multiple sequences, Arch. Math. 81 (2003) 82-89

[16] Moricz, F., Tauberian theorems for Cesaro summable double sequences, Studia Math. 110 (1994) 83–96.

[17] Miller, H. I. and Orhan, C., On almost convergent and statistically convergent subsequences. Acta Math.

Hungar. 93(1-2) (2001), 135–151.

[18] Mursaleen, M., λ−statistical convergence, Math. Slovaca, 50(1) (2000), 111 -115

[19] Mursaleen, Osama H.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003)

223–231

[20] Pringsheim, A., Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900) 289–321.

[21] Rath, D. and Tripathy, B.C., On statistically convergent and statistically Cauchy sequences, Indian J. Pure.

Appl. Math., 25(4) (1994) 381-386

[22] Salat, T., On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139-150.

[23] Savas, E., Strong almost convergence and almost λ−statistical convergence. Hokkaido Math. J. 29(3) (2000),

531–536.

7


[24] Schoenberg, I. J., The integrability of certain functions and related summability methods, Amer. Math.

Monthly 66 (1959), 361-375.

[25] Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique, Colloquium Mathematicum,

vol.2, pp. 73-74, 1951.

[26] Zygmund, A., Trigonometric Series, Cambridge University Press, Cambridge, UK, 1979.

8



012

Characterisation of compact operators between certain BK spaces

EBERHARD MALKOWSKY

Department of Mathematics, Faculty of Sciences and Arts, Fatih University, 34500Buyukcekmece, Istanbul, Turkey

Department of Mathemnatics, Justus-Liebig University Giessen, D-35392 Giessen, GermanyE-mail: Eberhard.Malkowsky @math.uni-giessen.deema @Bankerinter.net

Abstract The sets c0(Λ), c(Λ) and c∞(Λ) of sequences that are Λ−strongly convergent to 0,Λ−strongly convergent and Λ−strongly bounded were first introduced and studied by Moricz [3]; the setc(Λ) generalises the concept of strong convergenceof Hyslop [1], and Kuttner and Thorpe [2]. We studysome important topological and geometric properties of the spaces c0(Λ), c(Λ) and c∞(Λ) and their dualspaces.

Furthermore, we present the complete lists of characterisations of the classes of matrix transformations

from those spaces into the spaces of bounded, convergent and null sequences, and of their subclasses of

compact matrix operators. These results are mainly achieved by the theory of BK spaces, and by the

means of the Hausdorff measure of noncompactness. Finally, we apply our own software to the graphical

representations of neighbourhoods and weak neighbourhoods in the topologies of our spaces.

References

[1] J. M. Hyslop, Note on the strong summability of series, Proc. Glasgow Math. Assoc. 1(1951/53),16-20

[2] B. Kuttner, B. Thorpe, Strong convergence, J. Reine Angew. Math. 311/312 (1979), 42-55

[3] F. M oricz, On strong convergence of numerical sequences and Fourier series, Acta Math.Hung. 54 (3-4) (1989) 319-327

2000 Mathematics Subject Classification: Primary: 46H05; Secondary: 40H05.Keywords: FK and BK spaces, dual spaces, matrix transformations, measure of noncompactness, compact opera-tors.

9



012

Double I-lacunary statistical convergence using Ideal

Ekrem SAVAS

Istanbul Commerce University, Department of Mathematics, Uskudar-Istanbul/TurkeyE-mail: [email protected], [email protected]

Abstract In this paper, we intend to introduce the concept of I- double statistical convergenceand I- double lacunary statistical convergence which naturally extends the notions of double statisticalconvergence and double lacunary statistical convergence. We mainly try to establish the relation betweenthese two summability notions.The double sequence θr,s = (kr, ls) is called double lacunary if there exist two increasing of integerssuch that

k0 = 0, hr = kr − kk−1 → ∞ as r → ∞

andl0 = 0, hs = ls − ls−1 → ∞ as s → ∞.

Notations: kr,s = krls, hr,s = hrhs, θr,s is determine by Ir,s = (k, l) : kr−1 < k ≤ kr&ls−1 < l ≤ ls,qr = kr

kr−1, qs = ls

ls−1, and qr,s = qr qs. We will denote the set of all double lacunary sequences by Nθr,s .

We now have the following definitions.Definition 1. A sequence x = xk,l is said to be I-lacunary statistically convergent to L or SI

θ2-

convergent to L if for any ϵ > 0 and δ > 0

(r, s) ∈ N :1

hr,s|(k, l) ∈ Ir,s : |xk,l − L| ≥ ϵ| ≥ δ ∈ I.

In this case we write xk,l → L(SIθ2). The class of all I- double lacunary statistically convergent sequences

will be denoted by SIθ2.

For I = Ifin, SIθ2- convergence again coincides with Sθ2 statistical convergence.

References

[1] P. Kostyrko, T. Salat, W.Wilczynki, I-convergence, Real Anal. Exchange, 26 (2)(2000/2001), 669-685.

[2] E. Savas and R. F. Patterson, Lacunary statistical convergence of multiple sequences, Appl.Math. Lett. 19 (2006), no. 6, 527534.

[3] R. F. Patterson and Ekrem Savas, Lacunary statistical convergence of double sequences ,Math. Commun. 10(2000), 55-61.


10


[4] E. Savas, R. F. Patterson, Some double lacunary sequence spaces defined by Orliczfunctions. Southeast Asian Bull. Math. 35 (2011), no.1, 103-110.

[5] Pratulananda Das, E. Savas, S. K. Ghosal, On generalizations of certain summability meth-ods using ideals, Appl. Math. Letters, 24 (2011), 1509 - 1514.

11



012

Elements of fixed point theory: Old and New

Smail Djebali

Laboratoire ”Theorie du point Fixe et Applications”E.N.S., B.P. 92, 16050 Kouba. Algiers, Algeria.

E-mail: [email protected]

Abstract In this talk, we review some classical results and present recent development of fixed

point theory. In particular, we put the stress on the fundamental properties of the sets, the topology

of the spaces and some boundary conditions satisfied by maps involved in the theorems. Both metric

and topological fixed point theory will be surveyed though the stress will be put on the three celebrated

fixed point theorems: The Banach, Brouwer and Schauder theorems. Examples and counter examples

illustrate the results presented in this survey. The main references for this talk are [1, 19, 22, 26, 30, 34,

35].

References

[1] R.A. Agarwal, M. Meehan and D. O’Regan, Fixed Point Theory and Applications, Cam-bridge Tracts in Mathematics, 141, Cambridge University Press, 2001.

[2] R.A. Agarwal, M. Meehan and D.R. Shu, Fixed Point Theory for Lipschitzian-type Map-pings with Applications, Fixed Point Theory and its Applications, 6, Springer, 2009.

[3] A.G. Aksoy and M.A. Khamsi, Nonstandard Methods in Fixed point Theory, Springer, NewYork, Berlin 1990.

[4] J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, In Lecture Notesin Pure and Applied Mathematics, vol. 60, Marcel Dekker, New York, 1980.

[5] K. Borsuk, Theory of Retracts, Polish Scienti¯c Publishers, Warszawa, 1967.

[6] C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, L.N.M.1965, Springer Verlag, 2009.

[7] I. Cioranescu, Geometry of Banach spaces, Duality Mappings and NonlinearProblems,Kluwer Academic, 1990.

[8] K. Deimling, Geometry of Banach Spaces, L.N.M. 485, Springer Verlag, 1975.

2000 Mathematics Subject Classification: 47H10, 54C15, 54C20, 54C55, 55M15.Keywords: Fixed point theorem; convex set; retract; homeomorphism; contraction; compact map; boundary condi-tion.

12


[9] K. Deimling, Nonlinear Functional Analysis, Springer, 1985.

[10] S. Djebali and K. Hammache, Furi-Pera fixed point theorems in Banach algebras withapplications, Acta. Univ. Palacki. Olomuc, Fac. rer. nat., Mathematica (47) (2008) 55-75.

[11] S. Djebali and K. Hammache, Fixed point theorems for nonexpansive maps in Banachspaces, Nonlinear Analysis 73 (2010) 3440-3449.

[12] S. Djebali and K. Hammache, Furi-Pera ¯xed point theorems for nonexpansive maps inBanach spaces, Fixed Point Theory, 13(2), 2012, to appear.

[13] S. Djebali, K. Hammache, and Z. Sahnoun, Fixed point theorems with the Furi-Pera interiorcondition on strictly star-shaped sets, submitted.

[14] J. Dugundji, Topoloy, 8th ed. Allyn and Bacon, Boston, 1973.

[15] M. Edelstein, Fixed point theorems in uniformly convex Banach spaces, Proc.Amer. Math.Soc., 44(2), (1974) 369374.

[16] R. Edwards, Functional Analysis. Theory and Applications, Holt-Rinehart-Winston, NewYork, 1965.

[17] M. Furi and P. Pera, A continuation method on locally convex spaces and Applications toode on noncompact intervals, Annales Polonici Mathematici, XLVII, (1987), 331-346.

[18] L. Gasinski and N.S. Papageorgiou, Nonlinear Analysis, Taylor and Francis vol. 9 (2005)

[19] K. Goebel, Concise Course on Fixed Point Theorems, Yokohama Publishers, 2002

[20] K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge UniversityPress, Cambridge, 1990

[21] D. GAohde, Zum prinzip der kontraktiven abbildung, Math. Nachr. 30 (1965) 251-258.

[22] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.

[23] N.M. Gulevich, ¯xed points of nonexpansive mappings, J of Math. Sc., 79(1)(1996) 755-815.

[24] S.T. Hu, Homotopy Theory, Academic Press, 1959.

[25] S.T. Hu, Theory of Retracts, Wayne State University Press, 1965.

[26] V.I. Istratescu, Fixed point Theory, An Introduction, D. Reidel Publishing Company, 1981.

[27] A.A. Ivanov, Fixed points of mappings of metric spaces, Translated from Zapiski NauchnykhSeminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova ANSSSR, 66 (1976) 5-102.

[28] A. Jimenez-Melado and C. H. Morales, Fixed point theorems under the interior condition,Proceeding of the A.M.S. 134(2) (2005) 501-507.

[29] A. Kaewcharoen and W.A. Kirk, Nonexpansive mapping defined on unbounded domains,Fixed Point Theory and Applications, 82080 (2006) 1-13.

[30] M.A. Khamsi and W.A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory,John Willey & Sons, 2001

13


[31] W.A. Kirk, A fixed point theorem for mappings which do not increase distance, Amer.Math Month. 72 (1965) 1002-1004.

[32] M.A. Krasnozels’ki, Positive Solutions of Operator Equations, Noordho, Groningen, TheNetherlands, 1964.

[33] N.S. Papageorgiou and S.Th. Kyrtsi-Yiallourou, Handbook of Applied Analysis, Springervol. 19 (2009)

[34] D.R. Smart, Fixed Point Theorems, Cambridge University Press, 1974

[35] E. Zeidler, Nonlinear Functional Analysis and its Applications. I : Fixed PointTheorems,Springer, 1985

14



012

Extrapolation method and some Nondensely defined impulsive semilinearneutral partial functional differential inclusions

Nadjet ABADA, R.P.AGARWAL, M.BENCHOHRA and H.HAMMOUCHE

Ecole Nationale Superieure,Constantine, ALGERIEE-mail:

Abstract In this paper, we use the extrapolation method combined with a fixed point theorem

for the sum of completely continuous and contraction operators, to etablish sufficient conditions for the

existence of mild solutions and extremal mild solutions for some classes of non-densely defined impulsive

semilinear neutral functional differential inclusions in separable Banach with infinite delay.

References

[1] N. Abada, M. Benchohra and H. Hammouche, Existence and controllability results forimpulsive partial functional differential inclusions, Nonlinear Anal. 69 (2008), 2892-2909.

[2] N. Abada, M. Benchohra and H. Hammouche, Existence and Controllability Results forNondensely Defined Impulsive Semi-linear Functional Differential Inclusions,Journal of Dif-ferential Equations, Vol. 246 (2009), 3834-3863. )

[3] N. Abada, M. Benchohra and H. Hammouche , Existence Results for Semilinear DifferentialEvolution Equations with Impulses and Delay ,CUBO A Mathematical Journal , volume12, No.2, (2010), 1-17.

[4] T.A. Burton and C. Kirk, A fixed point theorem of Krasnoselskiii-Schaefer type, Math.Nachr.

2000 Mathematics Subject Classification: 34A37, 34A60, 34G25, 34K30, 34K45Keywords:

15



012

Computing Index Of Graded Filiform and Quasi Filiform Lie Algebras

Adimi Hadjer

Bordj Bou-Arreridj UniversityE-mail:

Abstract The filiform and the quasi-filiform Lie algebras form a special class of nilpotent Lie

algebras. The aim of this paper is to compute the index and provide regular vectors of this two class of

nilpotent Lie algebras. we consider the graded filiform Lie algebras Ln, Qn, the n dimensionnal filiform

Lie algebras for n ⟨8, also the graded quasi-filiform Lie algebras and finally a Lie algebras whose nilradical

is Q2n.

1 Introduction

The filiform Lie algebras were introduced by M. Vergne (see [?]), she classified them up to di-mension 6 and also characterized the graded filiform Lie algebras. The classification of naturallygraded quasi-filiform Lie algebras is known; they have the characteristic sequence (n− 2, 1, 1)where n is the dimension of the algebra. The aim of this work is to give an extended versionof our paper [?] and to focus on filiform Lie algebras. We compute the index and provide theregular vectors of n-dimensional filiform Lie algebras for n < 8 and quasi-filiform Lie algebras.In the first , we summarize the index theory of Lie algebras. It is known that any n-dimensionalfiliform Lie algebra may be obtained by deformation of the one of the filiform Lie algebras Ln,so we consider the classification up to dimension 8 and compute for each Filiform Lie algebraits index and the set of all regular vectors. We compute olso the index of graded quasi-filiformLie algebras, and give regular vectors corresponding. In the last we compute the index of Liealgebras whose nilradical is Q2n.

2 Lie Algebras Index

Definition 2.1. A Lie algebras G over K is a pair consisting of a vector space V = G and askew-symmetric bilinear map [ , ] : G × G → G (x, y) → [x, y] satisfying the Jacobi identity

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 ∀x, y, z ∈ G.

Let V be a finite-dimensional vector space over K provided with the Zariski topology, G be aLie algebra and G∗ its dual. Then G actes on G∗ as follows:

G × G∗ → G∗

(X, f) 7→ X.f


16


where ∀ ∈ G : (X.f) (Y )= f ([X,Y ]) .Let f ∈ G∗ and Φf be a skew-symmetric bilinear form defined by

Φf : G × G → K(X,Y ) 7→ Φf (X,Y ) = f ([X,Y ])

We denote the kernel of the map Φf by Gf ,

Gf = x ∈ G : f([x, y]) = 0 ∀y ∈ G. (2.1)

Definition 2.2. The index of a Lie algebra G is the integer

χG = infdim Gf ; f ∈ G∗

.

A linear functional f ∈ G∗ is called regular if dim Gf = χG . The set of all regular linearfunctionals is denoted by G∗

r .

Remark 2.1. The set G∗r of all regular linear functionals is a nonempty Zariski open set.

Remark 2.2. In practice we search the minors of ordre n − χ (G) of non-zero determinanat ofthe matrix M such that M is the matrix corresponding of the multiplication table.

17



012

New Cebysev type inequalities for fractional integrals

Assia GUEZANE-LAKOUD

Laboratory of Advanced Materials, Badji Mokhtar University, B.P. 12, 23000, Annaba, AlgeriaE-mail: [email protected]

Aissaoui Fatima

Laboratory of Applied Mathematics and Modeling, 8 mai 1945 University, B.P 401, Guelma,Algeria


Abstract The aim of this paper is to establish new Cebysev type inequalities, for fractional

integrals.

References

[1] D.S.MITRONOVIV, J.E. PECARIC AND A.M FINK, Inequalities for Functions and theirIntegrals and Derivatives,Klumer Academic Publishers, Dordrecht, 1994

[2] G.ANASTASSIOU, M.R. HOOSHMANDASL, A. GHASSEMI, AND F. MOF-TAKHARZADEH Montgomery Identities For Fractional Integrals And Related Fractionalinequalities volume 10 (2009)

[3] G.A. Anastassiou, Ostrowski type inequalities, Proc. AMS 123 (1995), 3775-3781.

[4] G.A. Anastassiou, Fractional Differentiation Inequalities, Research Monograph, Springer,New York, 2009.

[5] G.A. Anastassiou, On Right Fractional Calculus, Chaos, Solitons and Fractals, 42 (2009),365-376.

[6] G.A. Anastassiou, Univariate right fractional Ostrowski inequalities, CUBO, accepted,2011.

[7] A.M.A. El-Sayed, M. Gaber, On the finite Caputo and finite Riesz derivatives, ElectronicJournal of Theoretical Physics, Vol. 3, No. 12 (2006), 81-95.

[8] G.S. Frederico, D.F.M. Torres, Fractional Optimal Control in the sense of Caputo and thefractional Noether’s theorem, International Mathematical Forum, Vol. 3, No. 10 (2008),479-493.


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012

STABILITY IN NONLINEAR NEUTRAL INTEGRO-DIFFERENTIALEQUATIONS WITH VARIABLE DELAY

ABDELOUAHEB ARDJOUNI

Department of Mathematics and Informatics, C.U. Souk-Ahras, Souk-Ahras 41000, AlgeriaE-mail: abd [email protected]

AHCENE DJOUDI

Laboratory of Applied Mathematics, Department of Mathematics, University of Annaba,P.O.Box 12, Annaba 23000, Algeria


Abstract In this paper we use the contraction mapping theorem to obtain asymptotic stability

results of a nonlinear neutral integro-differential equation with variable delay. An asymptotic stability

theorem with a necessary and sufficient condition is proved, which improves and generalizes some previous

results due to Burton [3], Becker and Burton [2] and Jin and Luo [4]. In the end we provide an example

to illustrate our claim.

References

[1] A. Ardjouni, A. Djoudi, Fixed points and stability in linear neutral differential equationswith variable delays. Nonlinear Analysis 74 (2011) 2062-2070.

[2] L. C. Becker, T. A. Burton, Stability, fixed points and inverse of delays, Proc. Roy. Soc.Edinburgh 136A (2006) 245-275.

[3] T. A. Burton, Fixed points and stability of a nonconvolution equation, Proceedings of theAmerican Mathematical Society 132 (2004) 3679–3687.

[4] C. H. Jin, J. W. Luo, Stability of an integro-differential equation, Computers and Mathe-matics with Applications 57 (2009) 1080-1088.

2000 Mathematics Subject Classification: 34K20, 34K30, 34K40.Keywords: Fixed points, Stability, Integro-differential equation, Variable delay.

19



012

Solving the traffic assignment problem using a new version of theFrank-Wolfe algorithm

Arrache Saida

Mathematics Department, University Saad Dahleb, BlidaE-mail: [email protected]

Ouafi Rachid

Mathematics Faculty, USTHBE-mail: rachid [email protected]

Abstract The Frank-Wolfe method was first introduced in quadratic programming at once itproved very effective for the resolution of large scale fiood problems, the Frank- Wolfe method, is famousfor its advantages: it is easy to implement and it performs well far from the optimal solution. However, ithas a property that makes it less favourable to use without modifications, namely that it shows very slowasymptotic convergence due to that the feasible solutions tend to zig-zag towards the optimal solution.To improve its performance, different modifications of the Frank-Wolfe method have been suggested,starting with the first I∠. J. Leblanc works until recent works of Ziyou Gao & Al.On our behalf, in this paper we propose a new improvement of the FW method (FWF) for solving thetraffic assignment problem, this modification consist to combine Fukushima direction (FWF), with awidened line search technique (FWλ). we choose the best possible combination which can join togetherthe maximum effectiveness to FW algorithm and preserve its convergence.The main proponent of this attempt is to avoid the zig-zagging in the path described by the solutionpoints of the pure FW method.

We also present preliminary computational studies in a C++ Builder5, in these we apply (FW), (FWλ),

(FWF), and (FWFλ) methods to some Traffic assignment problems. The computational results indicate

that the proposed algorithm yield satisfactory results within reasonable computational time comparing

to the other methods.

2000 Mathematics Subject Classification:Keywords: Algorithms, Convergence, Descent direction, Frank-Wolfe method, Line search, Traffic assignment.

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012

Local and extremal solutions of some fractional integrodifferential equationwith impulses

RAHIMA ATMANIA

LMA, Department of Mathematics, University of Annaba,P.O.Box 12, Annaba 23000, Algeria


Abstract Concepts of fractional analysis like differentiation and integration can be considered asa generalization of ordinary ones with integer order. However, much remains to be done before assum-ing that this generalization is really established. Fractional differential equations have been extensivelyapplied in many fields, for example in probability, viscoelasticity and electrical circuits. Different theo-retical studies about the subject were done by many mathematicians. For more details, we refer to thebook of K.S. Miller and B. Ross [2].

On the other side impulsive effects that appear in the modeling of phenomena subject to considerable

short-term changes may be part of studies of fractional differential problems. This topic was awake the

curiosity of many researchers in recent years to include [1]. In this presentation we study existence of local

and extremal solutions for some integrodifferential fractional equation with impulses by using fixed-point

theory and fractional analysis under suitable assumptions.

References

[1] M. Benchohra, B.A. Slimani, Existence and uniqueness of solutions to impulsive fractionaldifferential equations, E. J. D. E., 2009, No. 10, pp. 1–11.

[2] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional DifferentialEquations, John Wiley and Sons, 1993.

2000 Mathematics Subject Classification: 26A33, 34A12, 34A37.Keywords: local existence, extremal solution, integrodifferential equation, Caputo fractional derivative, impulsiveconditions, fixed point theory.

21



012

A Neighbourhood System of Fuzzy Numbers and its Topology

Salih Aytar

Suleyman Demirel University, Department of Mathematics32260 Isparta, TURKEY


Abstract It is shown that the neighbourhood system obtained by the neighbourhoods (whose radii

are positive fuzzy numbers) in a fuzzy number-valued metric space is a basis of a bonafide topology for

the set of all fuzzy numbers, and then the convergence with respect to this topology is introduced and

its basic properties are studied.

2000 Mathematics Subject Classification: Primary 40A05, Secondary 03E72;26E50Keywords:

22



012

Construction of Self-Dual and Isodual Cyclic Codes over Finite Chain Rings

Aicha Batoul and Kenza Guenda

Faculty of Mathematics USTHB, University of Science and Technology of Algiers, AlgeriaE-mail: [email protected], [email protected]

T. Aaron Gulliver

Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC,Canada


Abstract Codes over finite chain rings are a generalization of codes over fields. They have attractedsignificant attention from the scientific community because of their connection to non-linear codes overfields. Lattices can also be construction from self-dual codes over rings. This work first considers theconstruction of cyclic isodual codes over finite fields. Then the construction is generalized to finite chainrings. The codes are derived from duadic codes over Finite fields. In particular, we prove the followingresult.Theorem 1 Let fi, 1 ≤ i ≤ 2 be two monic polynomials which generate a pair of odd duadic codes oflength m over Fq. Let g(x) = (x− 1)fi(x)fj(−x). Then the cyclic code of length 2m over Fq generatedby g(x) is:i) a self-dual code if q is even;ii) an isodual code if q is odd.Let gi be the Hensel lift of the polynomial fi over a ffinite chain ring R with residual ffield Fq. Then thecyclic code of length 2m generated by G(x) = (x− 1)gi(x)gj(−x), is an isodual code over R.Let gi be the polynomial defined above. The free duadic codes Fi are defined as the cyclic codes generatedby gi. When the nilpotency index e is even we define the following pair of non free duadic codesE1 = ⟨(x− 1)g1(x), γ

e2 g1(x)g2(x)⟩ and E2 = ⟨(x− 1)g2(x), γ

e2 g1(x)g2(x)⟩

Let µ−1 be the permutation of Sm defined by i 7→ −i mod m. Then we have the following result.

Theorem Let Ei and Fi be the codes given above. If the splitting is given by µ−1 then Ei are self-dual

codes and Fi are isodual. If the cyclotomic classes modulo m are left invariant by µ−1, then the Fi are

isodual and the Ei are duals of each other.


23



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Asymptotic analysis of linearly elastostatic Signorini problem with Coulombfriction of shallow shell

A. Bensayah

Laboratoire des mathematiques appliquees, Universite Kasdi Merbah, B.P511, Ouargla 30000,Algerie

E-mail: bensayahabd@ gmail.com

S. Nicaise

Universite de Valenciennes et du Hainaut Cambresis, LAMAV, FR CNRS 2956, Institut desSciences et Techniques of Valenciennes, F-59313 -Valenciennes Cedex 9 France


A. Ghezal and D. A. Chacha

Laboratoire des mathematiques appliquees, Universite Kasdi Merbah, B.P511, Ouargla 30000,Algerie


Abstract In a recent work, Paumier [1, 2], studied the Signorini problem with friction in the linear

Kirchhoff-Love theory plates using the convergence method, then Leger and Miara [3] generalized this

study to the case of linearized shallow shell but without friction. The purpose of this paper is to extend

these results to the case of linearized shallow shell with a local Coulomb friction law.

References

[1] J.C. Paumier, Modelisation asymptotique d‘un probleme de plaque mince en contact uni-lateral avec frottement contre un obstacle rigide. Rapport Technique LMC-IMAG. (2002).

[2] J.C. Paumier, Le probleme de Signorini dans la theorie des plaques minces de Kirchhoff-Love. C. R. Acad. Sci. Paris. Ser 1. 567-570. 335(2002).

[3] A. Leger, B. Miara, Mathematical justification of the obstacle problem in the case of ashallow shell. J. Elasticity. 241-257. 90(2008).

[4] D.A. Chacha, A. Bensayah, Asymptotic modeling of a Coulomb fric- tional Signorini prob-lem for the von Karman plates. Compte Rendu de Mecanique, ScienceDirect. 846-850.336(2008).


24



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Positive solutions for a fractional boundary value problem with fractionalderivative condition

Assia Guezane LakoudLaboratory of Advanced Materials, Badji Mokhtar Annaba University,

B.P. 12, 23000, Annaba. AlgeriaE-mail: a [email protected]

S. BensebaaLaboratory of Advanced Materials, Badji Mokhtar Annaba University,

B.P. 12, 23000, Annaba. AlgeriaE-mail: salima [email protected]

Abstract In this paper, we consider the nonlinear fractional differential equation boundary-valueproblem with fractional derivative condition:

cDq0+u(t) = f(t, u(t),c Dq

0+u(t)), 0 ≺ t ≺ 1

u(0) = u//(0) = 0, u/(1) =c Dσ0+u(1),

where f : [0, 1] × R × R → R is given function , 2 < q < 3 and 0 < σ < 1. By means of a fixed-point

theorem on cones, existence, uniqueness and positivity results of solutions are established.

References

[1] R.L. Bagley, A theoretical basis for the application of fractional calculus to viscoelasticity,Journal of Rheeology 27(3) (1983) 201-210.

[2] N. Engheta, On fractional calculus and fractional multipolesin electromagnetism, IEEETrans. 44(4)(1996) 554-566.

[3] R. Hilfer, Application of Fractional Calculus in Phisics, Word Scientific, Singapore, 200,699-707.

[4] F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, NewYork, 1997.

[5] A. Guezane-Lakoud, R. Khaldi, Positive solution to a higher order fractional boundaryvalue problem with fractional integral condition, Romanian journal of mathematics andcomputer science, 2012, volume 2, p41-54.

2000 Mathematics Subject Classification:Keywords: Positive solution, Fractional Caputo derivative, Banach Contraction principle, Leray Schauder non-linearalternative, GuoKrasnoselskii Theorem.

25



012

Impulsive differential equations with variable times

A. Lakmeche

Djillali Liabes University, Sidi Bel Abbes, AlgeriaE-mail: [email protected]

F. Berrabah

Djillali Liabes University, Sidi Bel Abbes, AlgeriaE-mail: berrabah [email protected]

Abstract In this paper, Schauder-Tychonoff’s fixed point theorem and the notion of upper and

lower solutions are used to investigate the existence of solutions for first order impulsive equations.

References

[1] J. Dugundji and A. Granas, Fixed point Theory, Springer-Verlag, New York, 2003

[2] V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of impulsive DifferentialEquations, World Scientific, Singapore, 1989

2000 Mathematics Subject Classification:Keywords: Impulsive equations; upper and lower solutions; fixed point

26



012

The solutions of the octonionic equations in the formα (xβ) = ρ, (αx) β = ρ ve αxα = ρ

Cennet BOLATMustafa Kemal University, Faculty of Art and Science,

Department of Mathematics, Tayfur Sokmen Campus, Hatay, TurkeyE-mail: [email protected]

Ahmet IPEKMustafa Kemal University, Faculty of Art and Science,

Department of Mathematics, Tayfur Sokmen Campus, Hatay, TurkeyE-mail: [email protected]

Abstract We in this study give the methods to find the solutions of the octonionic equations with

one unknown such that α (xβ) = ρ, (αx)β = ρ and αxα = ρ. Two-sided equations like these equations

cannot be solved by any elementary methods, because the non-commutativity and non-associative of

octonion multiplication makes it difficult to simplify the equations any further. The methods given

in this note allow to reduce simply the equations in these forms to a real system of eight equations.

Furthermore, we present examples to illustrate our results in this study.

References

[1] C. Bolat, A. Ipek, On The Solutions of Linear Matrix Quaternionic Equations and TheirSystems, Submitted.

[2] C. Flaut, Some equation in algebras obtained by Cayley–Dickson process, An. St. Univ.Ovidius Constanta, 9(2) (2001), 45–68.

[3] E. Corrigan, C. Devchand, D.B. Fairlie and J. Nuyts, First-order equations for gauge fieldsin spaces of dimension greater than four, Nucl. Phys. B, 214(3) (1983), 452–464.

[4] J. M. Evans, Supersymmetric Yang-Mills theories and division algebras, Nucl. Phys. B, 298(1988), 92-108.

[5] R. E. Johnson, On the equation xα = x + β over an algebraic division ring, Bull. of theAmer. Math. Soc., 50 (1944), 202-207.

[6] R. M. Porter, Quaternionic linear and quadratic equations, J. Nat. Geom., 11(2) (1997),101–106.

[7] S. V. Shpakivskyi, Linear quaternionic equations and their systems, Adv. Appl. Cliff. Alg.,21 (2011), 637-645.

2000 Mathematics Subject Classification:Keywords: The systems of real equations; The octonionic equations.

27



012

STABILITY OF NONLINEAR DIFFERENTIAL EQUATION WITHDELAY VIA SCHAUDER’S AND KRASNOSELSKII’S THEOREMS

Ahleme Bouakkaz

Department of mathematics, 20 august 1955 University, P.O.Box 20, 21000, Skikda, AlgeriaE-mail: [email protected]

Ahcen Djoudi

Department of mathematics, Badji Mokhtar University, P.O.Box 12, 23000, Annaba, AlgeriaE-mail: [email protected]

Abstract This work is devoted to the study of the stability of the following nonlinear differentialequation with delay

x′(t) = −α(t)x3(t) + b(t)x3(t − r(t)).

by using Schauder ’s and Krasnoselskii ’s theorems.

The method used here “fixed-point technique” is one of the most efficient techniques for studying this

type of equations.

References

[1] Burton, T. A., Stability and periodic solutions of ordinary functional differential equations,Academic Press. NY, 1985.

[2] Burton,T. A. and Furumochi, T., Fixed points and problems in stability theory for ordinaryand functional differential equations. Dynamic Systems and Appl. 10 (2001), 89- 116.

[3] Burton, T. A., Liapunov functional, fixed points and stability by Kras- noseskiis theorem.Nonlinear studies 9 (2002), 181-190.

[4] Djoudi, A., and Khemis, R., Fixed point techniques and stability for neutral nonlineardifferential equations with unbounded delays, Georgian Math. J. Vo113 No 1 (2006), 25-34.

2000 Mathematics Subject Classification: 34K20, 47Hl0.Keywords: Nonlinear neutral differential equation, Contraction mapping, Sta- bility, Krasnosselskis theorem

28



012

Multivalued mixed variational inequalities with locally cocoercivemultivalued mappings

Boualem Alleche

Laboratoire de Mecanique, Physique et Modelisation Mathematique.Universite de Medea. Cite Ain Dheb. 26000 Medea. Algerie

E-mail: [email protected] - [email protected]

Abstract Let C be a nonempty closed convex subset of Rn and let F : Rn → 2Rn

be a multivaluedmapping such that F (x) is nonempty closed subset, for every x ∈ C. Suppose further that ϕ : C → R isa convex subdifferentiable function. We consider the following multivalued mixed variational inequality:

Find x∗ ∈ C such that

∃w∗ ∈ F (x∗) , 〈w∗, y − x∗〉+ ϕ (y)− ϕ (x∗) ≥ 0 ∀y ∈ C.

A large variety of problems arising in elasticity, fluid flow, economics, oceanography, transportation,optimization, pure and applied sciences can be seen as special cases of this problem. One usually callsF the cost operator and C the set of constraints. Recall that a multivalued mapping F is said to becocoercieve with a constant γ or briefly (γ-cocoercieve) on M if

∀x, x′ ∈ M, ∀w ∈ F (x) , ∀w′ ∈ F (x′)

γd2H (F (x) , F (x′)) ≤ 〈w − w′, x− x′〉.

We make use of a retraction mapping and some sequential approximation techniques of fixed point theoryto solve the multivalued mixed variational inequalities involving locally cocoercive multivalued mappings.We construct by using the Banach contraction principle converging sequences to the solutions and showhow to choose regularization parameters to compute these solutions.

2000 Mathematics Subject Classification: 65K10; 90C25; 47H10Keywords: Multivalued mixed variational inequality; cocoerciveness; fixed point; multivalued mapping; Hausdorffmetric.

29



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Periodic solutions for neutral nonlinear system of differential equations withtwo functionals delays

Deham HafsiaDepartment of mathematics, Badji Mokhtar University, B.P. 12, 23000, Annaba, Algeria

E-mail:

Djoudi AhceneDepartment of mathematics, Badji Mokhtar University, B.P. 12, 23000, Annaba, Algeria


Abstract In this paper, we use Krasnoselski’s fixed point theorem to show that neutral nonlinearsystem of differential equations with two functionals delays

d

dtx(t) = A(t)x(t) +

d

dtQ(t, x(t− g1(t)), x(t− g2(t)))

+f(t, x(t− g1(t)), x(t− g2(t)))

has a periodic solution, in the process we use the fondamental matrix solution of

y′ = A(t)y.

We also use the contraction mapping principle to show the existence of unique periodic solution of the

equation.

References

[1] Burton, T. A., A fixed point theorem of Krasnoselskii, Appl. Math. Lett. 11 (1998 ) 85-88.

[2] Burton, T. A., Krasnoselskiis inversion principle and fixed points, Nonlinear Anal.,30 (1997 ), 3975-3986.

[3] Dhage, B. C., On a fixed point of Krasnoselski-Schaeffer type, Electronic Journal of Qual-itative Theory of Differential Equations, No.6 (2002 ), 1-9.

[4] Dib, Y. M., Maroun, M. R., and Y.N. Raffoul, Periodicity and stability in neutral nonlineardifferential equations with functional delay, Electronic Journal of Differential Equations,Vol. 2005 (2005 ), No. 142, pp. 1-11.

[5] Hale, J., Theory of functional differential equations, Springer Verlag, NY, 1977.

[6] Raffoul, Y. N., Periodic solutions for neutral nonlinear differential equations with functionaldelays, Electron. J. Differential Equ. Vol. 2003 (2003 ), No. 102, 1 -7.

[7] Smart, D. R., Fixed Point Theorems, Cambridge Univ. Press, Cambridge, 1980.

2000 Mathematics Subject Classification: 34K20, 45J05, 45D05;Keywords: Nonlinear neutral differential equation, Periodic solution, Contraction mapping, Integral equation.

30



012

Some pecular dynamical properties of Three Dimensional Maps

Asma Djerrai

Laboratory of Mathematics, Dynamics & Modelization , University of Annaba , AlgeriaE-mail: [email protected]

Ilhem Djellit

Laboratory of Mathematics, Dynamics & Modelization , University of Annaba , AlgeriaE-mail: [email protected]

Abstract In this work we seek to characterize the reccurent dynamics of the three map having thefollowing structure

T (x, y, z) :

xn+1 = f(yn)yn+1 = g(zn)zn+1 = h(xn)

Where f : Y → X , g : Z → Y and h : X → Z are endomorphism maps. We present ourresults of T system on basis of the classical oligopoly model [3] and of the two -dimensionalcase (xn+1 = f(yn), yn+1 = g(xn)), see in [2] ,

we give some properties and characteristics, since this class of three-dimensionaldynamics is associated with the properties of one-dimensional maps. There is an in-teresting passage from the one-dimensional endomorphisms to the three-dimensionalendomorphisms.

References

[1] F.Argoul, A. Arneodo, ”From quasiperiodicity to chaos: an unstable sce-nario via period-doubling bifurcations tori. J. Mecanique Theor.Appl.,n0special,(1984),241-288.

[2] G. Bischi,C. Mammana.,L. Gardini,”Multistability and cyclic attractors induopoly games”: Chaos, Solutions and Fractals 11(2000), 543-564.

[3] A.Cournot,”Reseatches into the principles of the theory of wealth”. Engl. trans.,chapter VII, Irwin paper Back Classics in Economics(1963).

[4] D. Fournier-Prunaret, R. Lopez-Ruiz, T. Abdel Kaddous,”Route to Chaos inthree-Dimensional maps of logistic type”.

2000 Mathematics Subject Classification:Keywords: three-dimensional maps; Bifurcations; Fixed point, Cycles..

31


[5] M. Kopel, Simple and complex adjustment dynamics in cournot duopoly mod-els,Chaos solitons & Fractal 7 (12) (1996) 2031-204.

[6] I.Gumowski and C. Mira, Dynamique chaotiques (Cepadues Edi-tions,Toulouse,1980) .

[7] C. Mira, L.Gardini, A. Barugora , J. C.Cathala, ”Chaotic dynamics in two-dimensional noninvertible maps”, World Scientific, Series A,Vol.20, 1996.

[8] D.Rand,Exotic phenomena in games and duopoly models,J.Math.Econ.5(1978)173-184.

32



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On the Fine Spectrum of the Upper Triangle Double Band Matrix ∆+ on theSequence Space c0

Erdinc Dundar

Fethi Gemuhluoglu, Anadolu Ogretmen LisesiMalatya, Turkey


Feyzi Basar

Department of Mathematics, Fatih University, Hadımkoy Campus,Buyukcekmece, 34500 - Istanbul, Turkey


Abstract Define the upper triangle double band matrix ∆+ = (dnk)∞n,k=0 by

dnk =

1 , n = k,−1 , n+ 1 = k,0 , k > n+ 1 or 0 ≤ k < n

for all n, k ∈ N.In this study, after summarizing the related literature on the spectrum and fine spectrum of certain matrix

operators on some sequence spaces, we determine the fine spectrum of the matrix operator ∆+ defined

by an upper triangle double band matrix acting on the sequence space c0 with respect to the Goldberg’s

classification. As a new development, we give the approximate point spectrum, defect spectrum and

compression spectrum of the matrix operator ∆+ on the space c0. Although the corresponding results

of the present work coincides with those results derived by Basar et al. in [Subdivisions of the spectra

for difference operator over certain sequence spaces, under communication], the adjoint operators are

different.

2000 Mathematics Subject Classification: Primary 47A10, Secondary: 47B37.Keywords: Spectrum, fine spectrum, Goldberg’s classification, approximate point spectrum, defect spectrum, com-pression spectrum, upper triangle double band matrix.

33



012

TRIPLE POSITIVE SOLUTIONS FOR SYSTEM OF NONLINEARSECOND-ORDER DIFFERENTIAL EQUATIONS THREE POINT

BOUNDARY VALUE

Amir Elhaffaf


Mostepha Naceri


Abstract In this work,we apply the Legget-Williams fixed point theorems to obtain sufficientcondition for existence at least three positive solutions of boundary value problems for systems of second-order ordinary differential equations of the form

−u′′(t) + k2u(t) = f(t, u(t), v(t)), 0 < t < 1

−v′′(t) + ω2v(t) = g(t, u(t), v(t)), 0 < t < 1

u(0) = v(0) = 0u(1) = αu(η), v(1) = λv(β)

where f : (0, 1) × [0, +∞) × [0, +∞) → [0, +∞); g : [0, 1] × [0, +∞) × [0, +∞) → [0, +∞) and k, ω

are positives constants. 0 < η < 1, 0 < β < 1, 0 < α < α0, 0 < λ < λ0 .

2000 Mathematics Subject Classification: 34B10, 34B15, 34B14Keywords: Nonlinear second-order differential systems,Positive solutions,Legget- Williams fixed point theo-rems,boundary condition.

34



012

Reversible codes over GF (16) and Dna codes

Elif Segah Oztas

Department of Mathematics, Yıldız Technical University,Istanbul, TURKEY


Irfan Siap

Department of Mathematics, Yıldız Technical University,Istanbul, TURKEY


Mehmet Ozen

Department of Mathematics, Sakarya University,Sakarya, TURKEY


Abstract The error correction capability of DNA strands which is similar to the goal of error

correction in coding theory has attracted researchers from coding theory recently. A main goal has been

studying the structure of error correcting codes by imposing some restrictions so that they resemble DNA

structure. In this study, 16 element field (GF(16)) is used to code DNA as pairs. Reversible DNA codes of

even and odd length are built in the field GF(16) by a special family of polynomials. Further, Hamming

minimum distances are also studied.

2000 Mathematics Subject Classification:Keywords: Reversible codes, DNA codes

35



012

Constraint Coefficient Problems for Subclasses of Univalent Functions

Ebtisam. A. Eljamal

School of Mathematical Sciences, Faculty of Science and Technology,Universiti Kebangsaan Malaysia, Bangi, Malaysia

E-mail: n [email protected]

Abstract In the present paper, we introduce generalized differential operator of functions which

are analytic in the unit disk. Also we use this operator to introduce new classes Cγ and S∗γ of suitably

normalized close-to-convex and starlike univalent functions with positive coefficients, respectively. These

are two subclasses of the class SR of equally normalized univalent functions with real coefficients, the

class of positive real part functions with real coefficients. Using some lemmas on the extreme points of

closed convex classes, we solve the constraint problems of the first coefficients of Cγ and S∗γ for a fixed

second coefficient that is close to two. For these classes we also derive the radii of close-to-convexity,

starlikeness and convexity. Further, an application involving fractional calculus for functions in Cγ and

S∗γ are given. All of our results are sharp.

2000 Mathematics Subject Classification:Keywords: differential operator, real coefficients, starlikeness, convexity, close-to-convexity

36



012

Local solution of the system of equations describing the motion of water inthe desert areas

Fateh ELLAGGOUNE



Housseyn ZENKOUFI



Abstract Today the issue of water resource and desertification has become very important.

In this work, we propose a mathematical model of the flow of water in the desert regions expressed by a

system of equations describing the change in the amount of the steam in the atmosphere and that of the

quantity of water. Then we prove the existence and uniqueness of the local solution of this system.

References

[1] S. N. Antontsev, A; V. Kazhikhov, V. N. Monokov: Boundary value problems in mechanicsof non homogeneous fluids, Elsevier, 1990.

[2] P.-L. Lions: Mathematical topics in fluid mechanics, vol. 2, Clarendon, 1998.

[3] E. Pardoux: Integrales stochastiques hilbertiennes. Univ. Paris - Dauphine, 1976.

[4] E. Tornatore, H. Fujita Yashima: Equation stochastique monodimentonnelle d’un gazvisqueux barotropique (en italien), Ricerche di Matematica, vol. 46 pp. 255-283, 1997.

2000 Mathematics Subject Classification:Keywords: desertification, evaporation, condensation, saturated steam.

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012

Optimal control of BCG immunotherapy in a mathematical model ofsuperficial bladder cancer

Ilias ELMOUKI

Department of Mathematics and Computer sciences Faculty of sciences Ben M’Sik CasablancaMOROCCO

E-mail:

Smahane SAADI

Mathematical biology unit, Laboratory of Analysis, Modeling and Simulation,Department of Mathematics and Computer sciences, Faculty of sciences BenM ′Sik.

University of Hassan II Mohammedia –Casablanca MOROCCOAvenue Commandant Driss ELHARTI B. P : 7955-Ben M’Sik 20800 Casablanca


AbstractIn this talk, we aim to discuss the application of the Pontryagin’s maximum principle after introducingan optimal control in a mathematical model of BCG immunotherapy in superfficial bladder cancer.Interactions between tumor cells and immune responses represented by effector cells; describe a nonlinearsystem of four ordinary differential equations. Several studies have shown that we still don’t know whatis the optimal dose of BCG, main- tenance schedule, duration of treatment or is the optimal treatmentdifferent for every patient.For this reason, our ffirst goal is to ffind treatment regimens that minimize the cancer cell count.

We include numerical simulations based on a fourth-order iterative Runge-Kutta scheme which is used to

solve the optimality system of a boundary two-point value problem.

References

[1] SVETLANA BuNIMOVICH-MENDRAZITSKYA, ELIEZER SHOCHAT, LEWI STONE(2007), Mathematical Model of BCG Immunotherapy in Superficial Bladder Cancer; Bul-letin of Mathematical Biology, DOI 10.1007/sll538-007-9l95-z.

[2] ADRIAN P. VAN DER MEIJDEN, RICHARD J. SYLVESTER (2003), BCG Immunother-apy for Superficial Bladder Cancer: An Overview of the Past, the Present and the Future;EAU Update Series 1, 80-86.

[3] KIRSCHNER, D., PANETTA, J., (1998), Modelling immunotherapy of the tumor-immuneinteraction; J. Math. Biol., 37, 3, 235-252.

2000 Mathematics Subject Classification:Keywords: Superfficial bladder cancer; BCG immunotherapy; Pontryagin’s maximum principle; Optimal control.

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012

Computational flow analysis on a two phase model with an unknownpressure function

Allaberen Ashyralyev, Abdullah Said Erdogan and Ali Ugur Sazaklıoglu

Department of Mathematics, Fatih University,Buyukcekmece, Istanbul, TurkeyE-mail: [email protected]

Abstract Blood flow inside capillary can be modeled as a two-phase model which is a right hand

side identification problem with an unknown pressure function p (x) (see [1]-[5]). In this presentation, the

stability analysis of the problem is obtained. For obtaining approximate results, first and second orders

of accuracy difference schemes are presented. The Matlab implementation of these differences schemes

are generated.

References

[1] P. Guo, A.M. Weinstein, S. Weinbaum, A hydrodynamic mechanosensory hypothesis forbrush border microvilli, The American Journal of Physiology - Renal Physiology 279(4)(2000) 698-712.

[2] M. Sharan, A. S. Popel, A two-phase model for flow of blood in narrow tubes with increasedeffective viscosity near the wall, Biorheology 38 (2001) 415–428.

[3] A. Ashyralyev, A.S. Erdogan, N. Arslan, On the numerical solution of the diffusion equationwith variable space operator, Applied Mathematics and Computation 189(1) (2007) 682-689.

[4] N. Arslan, A.S. Erdogan, A. Ashyralyev, Computational fluid flow solution over endothelialcells inside the capillary vascular system, International Journal for Numerical Methods inEngineering 74(11) (2008) 1679-1689.

[5] A. Ashyralyev, A.S. Erdogan, N. Arslan, On the modified Crank–Nicholson differenceschemes for parabolic equation with non-smooth data arising in biomechanics, InternationalJournal for Numerical Methods in Biomedical Engineering 26(5) (2010) 501-510.


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Systemes Dynamiques Discrets Bidimensionnels. Relation entre Point Fixeet Point Focal

M. R. Ferchichi

Laboratoire de Mathematiques, Dynamique et ModelisationUniversite Badji Mokhtar - Annaba - Algerie


A. Djerrai

Laboratoire de Mathematiques, Dynamique et ModelisationUniversite Badji Mokhtar - Annaba - Algerie


Abstract Il existe des singularites specifiques aux systemes dynamiques discrets definis par desapplications bidimensionnelles ayant au moins une composante fractionnelle; ce sont les points focaux etles courbes prefocales [1]. Dans cette etude on s’interesse aux relations qui peuvent exister entre un pointfixe de T (application definissant le systeme dynamique) et un point focal d’une determination inverseT−1, si elle existe, de T [2].1 - La premiere relation est une localisation, dans le plan de phases, du point focal de T−1quand celui-ciest un point fixe de T .2 - La deuxieme relation est une equivalence entre le fait que le point focal Q soit un point fixe de T etque l’intersection des courbes prefocales de T−1 et de T−2 associees a Q soit non vide.3 - La troisieme relation est une condition necessaire ”une valeur propre nulle de la matrice Jacobiennede T au point Q” pour qu’un point focal de T−1 soit un point fixe de T .

Cette etude sera completee par deux exemples. Le premier est une application polynomiale ayant une

determination inverse fractionnelle admettant un seul point focal; le second est aussi une application

polynomiale ayant une determination inverse fractionnelle mais admettant deux points focaux.

References

[1] Bischi G.I., Gardini L. et Mira C., “Maps with Denominator. Part 1: Some Generic Prop-erties”, International Journal of Bifurcation and Chaos, Vol.9, N 1 (1999) p.119-153.

[2] Ferchichi M.R. et Djellit I., “On Some Properties of Focal Points”, Discrete Dynamics inNature and Society, Vol. 2009, Article ID 646258.

2000 Mathematics Subject Classification:Keywords: Systemes dynamiques discrets, Points focaux, Courbes prefocales

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012

Solvability of a third oder boundary value problem at resonance

Assia GUEZANE-LAKOUDLaboratory of Advanced Materials. Faculty of Sciences

University Badji Mokhtar, B.P. 12, 23000, Annaba, AlgeriaE-mail: [email protected]

Assia FRIOUILaboratory of Applied Mathematics and ModelingUniversity 8 mai 1945, B.P 401, Guelma, Algeria


Rabah KHALDILaboratory of Advanced Materials. Faculty of Sciences

University Badji Mokhtar, B.P. 12, 23000, Annaba, AlgeriaE-mail: [email protected]

Abstract This paper deals with a class of third oder boundary value problem at resonance case.

Some existence results are obtained by using the coincidence degree theory of Mawhin.

References

[1] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, in: NSF-CBMS Regional Conference Series in Math, No. 40, Amer. Math. Soc., Providence, RI,1979.

[2] W. Feng and J. R. L. Webb, Solvability of three-point boundary value problems at reso-nance, Nonlinear Anal. Theory, Methods and Appl. 30 (1997), 3227-3238.

[3] B. Liu, Solvability of multi-point boundary value problems at resonance (IV), Appl. Math.Comput. 143 (2003)275–299.

[4] C. P. Gupta, On third-order boundary value problem at resonance, Differential IntegralEquation 2 (1989) 1-12.

[5] R. P. Agarwal and D. O’Regan, Singular Differential and Integral Equations with Applica-tions (Kluwer, Dordrecht,2003).

[6] Z. Du, Solvability of functional differential equations with multi-point boundary value prob-lem at resonance, Comput.Math. Appl. 55, 2653–2661 (2008).

[7] B. Liu and Z. Zhao, A note on multi-point boundary value problems, Nonlinear Anal. 67,2680–2689 (2007).


41



012

He’s method for Solving some linear heat equation

Soumaya Ghnimi

Faculte des Sciences de Tunis,El Manar ,Tunis, Tunisie.E-mail: soumaya−[email protected]

Imed Ghanmi

Faculte des Sciences de Tunis,El Manar ,Tunis, Tunisie.E-mail: [email protected]

Abstract In this paper, a numerical algorithm, based on the homotopy perturbation method,

is applied for solving heat equation with an initial condition and non local boundary conditions. The

analytic solution of the linear heat equation is calculated in the form of a series with easily computable

components. We use three examples where the numerical application shows that the obtained solution

coincides with the exact one.

References

[1] J H He. Homotopy perturbation Method; a New nonlinear analytical technique. App MathComput,135(2005): 73-79.

[2] J H He. App Math. Comput 156 (2004), 527.

[3] J H He. Application of homotopy perturbation method to nonlinear wave equaton. ChaosSoliton Fract 26 (2005): 695-700.

[4] M A Rehman and M S A Taj. Fourth-order Method for nonhomogeneous Heat equationwith nonlocal Boundary conditions. Appl Math Sciences (2009). Vol 3, n37: 1811-1821.

[5] A Cheniguel and A Ayadi. Solving Heat Equation by the Adomian Decomposition Method.Proceedings of the World Congress on Engineering (2011). Vol I WCE, London U K.

2000 Mathematics Subject Classification: 35J05-35Q68-35J25-41A58-14F35Keywords: Homotopy perturbation method; Heat equation; Numerical methods.

42



012

Lacunary Statistical Limit Points in Random 2-Normed Spaces

Ayse Nur Guncan

Department of Mathematics, University of Suleyman Demirel, 32260, Isparta, TurkeyE-mail: [email protected]

Ulas Yamancı

Department of Mathematics, University of Suleyman Demirel, 32260, Isparta, TurkeyE-mail: ulas 19 @hotmail.com

Mehmet Gurdal

Department of Mathematics, University of Suleyman Demirel, 32260, Isparta, TurkeyE-mail: [email protected]

Abstract In this article we introduce the notion θ-cluster points, and investigate the relationbetween θ-cluster points and limit points of sequences in the topology induced by random 2-normedspaces and prove some important results.

2000 Mathematics Subject Classification: 40A35, 46A70, 47B35.Keywords: t-norm, random 2-normed space, Lacunary statistical convergence.

43



012

On I-convergence of Double Sequencesin the Topology Induced by Random 2-Norms

Mehmet Gurdal

Department of Mathematics, University of Suleyman Demirel32260, Isparta, Turkey


Mualla Birgul Huban

Department of Mathematics, University of Suleyman Demirel32260, Isparta, Turkey


Abstract In this article we introduce the notion I-convergence and I-Cauchy of double sequences

in the topology induced by random 2-normed spaces and prove some important results.

2000 Mathematics Subject Classification: Primary 40A35; Secondary 46A70, 54E70.Keywords: t-norm, random 2-normed space, ideal convergence, ideal Cauchy sequences, F -topology.This work is supported by Suleyman Demirel University with Project 2947-YL-11.

44



012

Some topological and geometric properties of the domain of the triple bandmatrix B(r, s, t) in the sequence space ℓ(p)

Murat Gvercin

The Graduate School of Sciences and Engineering, Fatih University, Hadımkoy Campus,Buyukcekmece, 34500 - Istanbul, Turkey


Feyzi Basar



Abstract The sequence space ℓ(p) was introduced by Maddox [Spaces of strongly summable

sequences, Quart. J. Math. Oxford (2)18(1967), 345–355]. In the present paper, the sequence space

ℓ(B, p) of non-absolute type has been studied which is the domain of the triple band matrix B(r, s, t) in

the sequence space ℓ(p). Furthermore, the α-, β- and γ-duals of the space ℓ(B, p) have been determined,

and the Schauder basis has been given. The classes of matrix transformations from the space ℓ(B, p)

to the spaces ℓ∞, c and c0 have been characterized. Additionally, the characterizations of some other

matrix transformations from the space ℓ(B, p) to the Euler, Riesz, difference, etc., sequence spaces have

been obtained by means of a given lemma. The last two sections of the paper have been devoted to some

results about the rotundity of the space ℓ(B, p) and conclusion.

2000 Mathematics Subject Classification: Primary 46A45; Secondary 46B45, 46A35.Keywords: Paranormed sequence space, matrix domain of a sequence space, α−, β− and γ−duals, triple bandmatrix and matrix transformations.

45



012

Existence and uniqueness of solution for a second order boundary valueproblem

Assia Guezane Lakoud

Laboratory of Advanced Materials, University Badji Mokhtar, B.P. 12, 23000, Annaba. AlgeriaE-mail: a [email protected]

Nacira Hamidane

Laboratory of Advanced Materials, University Badji Mokhtar, B.P. 12, 23000, Annaba. AlgeriaE-mail: [email protected]

Rabah KHALDI

Laboratory of Advanced Materials, University Badji Mokhtar, B.P. 12, 23000, Annaba. AlgeriaE-mail: [email protected]

Abstract Our work deals with a second order boundary value problem. Our aim is to give new

conditions on the nonlinear term, then, using Banach contraction principale and Leray Schauder nonlinear

alternative, we establish the existence and uniqueness of nontrivial solutions of the considered problem.

2000 Mathematics Subject Classification: 34K20, 34K30, 34K40.Keywords: Fixed point theorems, Second order boundary value problem, Integrals conditions, Banach contractionprinciple, Leray Schauder nonlinear alternative.

46



012

SOLUTION DE MEILLEUR COMPROMIS POUR LE PROBLEME DUPLUS COURT CHEMIN MULTICRITERE

HOUAS Mohamed

Laboratoire LAID3, Facult de Mathmatiques, U.S.T.H BP 32, El-Alia16111,Bab-Ezzouar, Alger- AlgrieE-mail: [email protected]

ABBAS Moncef

Laboratoire LAID3, Facult de Mathmatiques, U.S.T.H BP 32, El-Alia16111,Bab-Ezzouar, Alger- AlgrieE-mail: [email protected]

Abstract Dans ce papier nous tudions le problme du plus court chemin entre deux sommets, source

et puits dans un rseau multicritre. L’objectif de ce travail est de caractriser l’existence des solutions

admissibles et des solutions efficaces. Nous tudions les conditions ncessaires et suffisantes d’existence de

solutions admissibles et efficaces pour le problme de la recherche d’un plus court chemin d’un sommet

source un sommet puits dans un rseau bicritre, puis dans le cas gnral du rseau multicritre.

Les problmes de cheminement constituent un thme classique en recherche oprationnelle dontles applications sont trs nombreuses, notamment en transport et en tlcommunication. Parmiles problmes de cheminement les plus anciens et les plus traits on trouve le problme du pluscourt chemin. Il consiste chercher le meilleur chemin entre deux points, source et puits, afinde minimiser un critre prcis, gnralement, le temps, la distance ou le cot. Avec le dveloppementfulgurant de l’aide multicritre la dcision, et, en particulier, suite la nouvelle dmarche dansla formulation des problmes concrets de dcision qui tient compte de tous les points de vue, leproblme de cheminement classique monocritre [1] ne caractrise pas compltement les problmespratiques de cheminement. En effet, dans un rseau de transport ou de tlcommunication, plusieursparamtres peuvent tre associs chaque arc comme le temps, la distance, le cot,, etc. Il est doncclair qu’on est amen un problme de dcision multicritre particulier, celui de la recherche d’unplus court chemin dans un rseau multicritre [2, 3, 4, 5].

References

[1] Dijksta, E. W. A note on two problems in connexion with graphs. Numerische, Mathematik,1: 269-271, 1959.

2000 Mathematics Subject Classification:Keywords: Plus court chemin, plus court chemin multicritre, rseau, chemin efficace.

47


[2] Hansen, P. Bicriterion path problems. In G. Fandel and T. Gal, editors, Multiple CriteriaDecision Making Theory and Applications, volume 177 of Lecture Notes in Economics andMathematical Systems, pages 109-127. Springer Verlag, Berlin, 1979.

[3] Martins, E.Q.V. On a multicriteria shortest path problem. European Journal of OperationalResearch, 16: 236-245, 1984.

[4] Martins, E.Q.V, and Santos J.L.E. The labeling algorithm for the multiobjective shortestpath problem, 1999.

[5] Vincke, P. Problme multicritres. Cahiers du Centre d’Etudes de Recherche Oprationnelle,16: 425-439, 1974.

48



012

THE PROPERTIES OF SOME SEQUENCE SPACES ON SEMINORMEDSPACES

MAHMUT ISIK

Firat University, Department of Statistics, 23119, Elazig-TURKEYE-mail: [email protected]

YAVUZ ALTIN and HIFSI ALTINOK

Firat University, Department of Statistics, 23119, Elazig-TURKEYE-mail: [email protected], [email protected]

Abstract In this paper we introduce the sequence spaces

c0 (p, f, ϕ, q, s) , c (p, f, ϕ, q, s) , m (p, f, ϕ, q, s) using a modulus function f. We give various prop-

erties and some inclusion relations on these spaces.

References

[1] Z.U. AHMAD ANDM. MURSALEEN, An application of Banach limits, Proc. Amer. Math.Soc. 103, (1988), 244-246.

[2] Y. ALTIN AND M. ISIK, On some new seminormed sequence spaces defined by modulusfunctions. Hokkaido Math. J. 35 (2006), no. 3, 565–572.

[3] S. BANACH, Theorie des operations, Warszawa, 1932.

[4] V.K. BHARDWAJ, A generalization of a sequence space of Ruckle, Bull. Calcutta Math.Soc. 95 (5) (2003), 411-420.

[5] T. BILGIN, The sequence space ℓ (p, f, q, s) on seminormed spaces. Bull. Calcutta Math.Soc. 86 (4) (1994), 295-304.

[6] J. BOSS, Classical and Modern Methods in Summability. Oxford Univ. Press, 2000.

[7] P. K. KAMPTAN AND M. GUPTA, Sequence Spaces and Series, Marcel Dekker Inc., NewYork, 1981.

[8] G.G. LORENTZ, A contribution the theory of divergent series, Acta Math. 80 (1948),167-190.

2000 Mathematics Subject Classification: 40A05, 40C05, 40D05Keywords: Modulus function, invariant mean, seminorm, sequence space

49


[9] I.J. MADDOX, Spaces of strongly summable sequences, Quart. J.Math. Oxford 2 (18)(1967), 345-355.

[10] I.J. MADDOX, Sequence spaces defined by a modulus, Math. Proc. Camb. Phil. Soc. 100(1986), 161-166.

[11] M. MURSALEEN, On some new invariant matrix methods of summability, Quart. J. Math.Oxford 34 (2), (1983), 77-86.

[12] M. MURSALEEN, Matrix transformations between some new sequence spaces, Houston J.Math. 9 , (1983), 505-509.

[13] H. NAKANO, Concave modulars, J. Math. Soc. Japan. 5 (1953), 29-49.

[14] R.A. RAMI, Invariant means and invariant matrix method of summability, Duke Math. J.30, (1963), 81-94.

[15] W.H. RUCKLE, FK spaces in which the sequence of coordinate vectors is bounded, Canad.J. Math. 25 (1973), 973-978.

[16] S. LI, C. LI, AND Y.C. LI, On σ−limit and Sσ−limit in Banach spaces. Taiwanese J. Math.9 (2005), no. 3, 359–371.

[17] P. SCHAEFER, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972),104-110.

[18] A. WILANSKY, Functional Analysis, Blaisdell Publishing Company, New York, 1964.

50



012

Identification of the Diffusion in a Semi-Linear Problem

S. ABIDI

Faculty of Sciences of Tunis, Department of Mathematics, 1060 Tunis, TunisiaE-mail: [email protected]

A. JARRAY

Faculty of Sciences of Tunis, Department of Mathematics, 1060 Tunis, TunisiaE-mail: [email protected]

Abstract We give in this paper a variational method to identify the diffusion in a semilinear

parabolic problem. We show that the identification of the diffusion amounts to solving a problem of

optimal control. Under assumptions on the given nonlinearity we construct a sequence of linear problems

whose solutions converge to the solution of original one, and we use finite-elements to compute the cost

functional. In the end we use the sensibility method [4] to approximate the gradient of the cost functional.

References

[1] P.A. RAVTART, D. J.M.THOMAS - Introduction a l’analyse numerique des equations auxderivees partielles.

[2] J. L.LIONS, E. MAGENES - Problemes aux limites non homogenes (1968).

[3] M. Bouchiba, S. Abidi - Identification of the Diffusion in a linear Parabolic problem, Inter-national Journal of Applied Mathematics volume 23 No. 3 2010, 491-501.

[4] M. Kern - Problemes Inverses.

[5] J. Emile. Rakotoson - Analyse fonctionnelle appliquee aux equations aux derivees partielles.

2000 Mathematics Subject Classification:Keywords: Parabolic Equation, Diffusion, Finite-elements, Optimization

51



012

Adomian Decomposition Method for solving nonlinear diffusion equationwith convection term

Jebari Rochdi

Faculte des Sciences de Tunis,El Manar ,Tunis, TunisieE-mail: [email protected]

Ghanmi Imed


Boukricha Abderrahman


Abstract In this paper, the Adomian decomposition method is applied to solve the nonlinear

diffusion equation with convection term. This method yields an analytical solution in terms of a rapidly

convergent series with easily computable terms. The numerical results obtained by this method have

been compared with the exact solution to show that the Adomian decomposition method is a powerful

method and easy to use.

References

[1] Roman M. Cherniha,New Ansatze and Exact Solutions for Nonlinear Reaction-DiffusionEquations Arising in Mathematical Biology.Symmetry in Nonlinear Mathematical Physics1997, V.1, 138-146

[2] Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations,Chapman a Hall/CRC, Boca Raton, 2004.

[3] Talha Achouri, Khaled Omrani. Numerical solutions for the damped generalized regularizedlong-wave equation with a variable coefficient by Adomian decomposition method. CommunNonlinear Sci Numer Simulat 14 (2009) 2025-2033.

[4] Wazwaz AM. A new algorithm for calculating Adomian polynomials for nonlinear operators.Appl Math Comput 2000;111:53-69.

[5] Salah M. El-Sayed, Dogan Kaya. The decomposition method for solving (2 + 1)-dimensionalBoussinesq equation and (3 + 1)-dimensional KP equation. Applied Mathematics and Com-putation 157 (2004) 523-534.

2000 Mathematics Subject Classification: 35J05-35Q68-35J25-41A58-14F35Keywords: Adomian Decomposition Method, Nonlinear diffusion equation with convection term

52



012

ON THE FINE SPECTRUM OF THE GENERALIZED DIFFERENCEOPERATOR DEFINED BY A DOUBLE SEQUENTIAL BAND MATRIX

OVER THE SEQUENCE SPACE ℓ1

Ali Karaisa

Department of Mathematics, Konya University, Karacigan Mahallesi, Ankara Caddesi 74,42060-Konya, Turkey


Feyzi Bagar

Department of Mathematics, Fatih University, Hadimkoy Campus, Buyukcekmece, 34500 -Istanbul, Turkey


Abstract Let λ = (λk) be a strictly increasing sequence of positive reals tending to infinity, thatis

0 < λ0 < λ1 < λ2 < · · · and limk→∞

λk = ∞.

Define the infinite matrix Λ = (λnk)∞n,k=0 by

λnk =

λk−λk−1

λn, 0 ≤ k ≤ n

0, k > n

for all n, k ∈ N.Following Furkan, Bilgia and Kayaduman [Hokkaido Math. J., On the ffine spectrum of the generalizeddifference operator B(r, s) over the sequence spaces ℓ1 and bv, 35(2006), 897-908], we determine the finespectrum with respect to the Goldberg’s classification of the operator defined by the triangle matrix Λover the sequence space ℓ1. Additionally, we give the approximate point spectrum, defect spectrum andcompression spectrum of the matrix operator Λ over the space ℓ1.

2000 Mathematics Subject Classification: Primary 47A10, Secondary 47B37.Keywords: Spectrum of an operator, Lambda matrix, spectral mapping theorem, the sequence space ℓp, Goldberg’sclassification.

53



012

General approach of the root of a p-adic number

Mohamed Kecies

Centre universitaire de MilaE-mail: [email protected]

Abstract In this work, we applied the classical numerical method of the newton in the p-adic case tocalculate the cubic root of a p-adic number a ∈ Q∗

p where p is a prime number, and through the calculationof the approximate solution of the equation x3 − a = 0. We also determined the rate of convergence ofthis method and evaluated the number of iterations obtained in each step of the approximation.Let p be a prime number. The field Qp of p-adic numbers is the completion of the field Q of rationalnumbers with respect to the p-adic norm |·|p defined by

∀x ∈ Qp : |x|p =

p−vp(x), if x = 0

0, if x = 0,

with vp is the p-adic valuation defined by vp(x) = max r ∈ Z : pr | x.The p-adic norm induces a metric dp given by

dp : Qp ×Qp −→ R+

(x, y) 7−→ dp(x, y) = |x− y|p ,

this metric is called the p-adic metric. The iterative formula of the secant method is

∀n ∈ N : xn+1 =1

3x2n

(a+ 2x3

n

)(0.1)

if xn0is the cubic root of a of order r. Then

1. If p = 3, then xn+n0 is the cubic root of a of order 2nr − 3m(2n − 1) and

∀n ∈ N :

xn+n0+1 − xn+n0≡ 0 mod pφn

φn = 2nr −m(3 · 2n − 1)

2. If p = 3, then xn+n0 is the cubic root of a of order 2nr − 3(m+ 1)(2n − 1) and

∀n ∈ N :

xn+n0+1 − xn+n0 ≡ 0 mod 3φ′n

φ′n = 2nr − (m(3 · 2n − 1) + (3 · 2n − 2))

2000 Mathematics Subject Classification: 11E95, 65H04Keywords: Newton method, Hensel’s lemma, rate of convergence

54


References

[1] C.J. Zarowski, H.C. Card, On Addition and Multiplication with Hensel Codes. IEEE trans-actions on computers 39(12):1417-1423, December 1990.

[2] F. B. Vej, P-adic Numbers. Aalborg University. Departement Of Mathematical Sciences.[Online] Available: http://www.control.auc.dk/ jjl/oldpro/oldstu/mat3.ps. 18-12-2000.

[3] M. Knapp, C. Xenophotos. Numerical analysis meets number theory: using rootfindingmethods to calculate inverses ( mod pn). Appl. Anal. Discrete Math, 23-31, 4. (2010).

[4] S. Katok, p-adic analysis compared with real. Student Mathematical Library Vol. 37, Amer-ican Mathematical Society, 2007.

[5] T. Zerzaihi, M. Kecies, M. Knapp, Hensel codes of square roots of p-adic numbers. Appl.Anal. Discrete Math. 32-44,4 (2010).

55



012

Hypergeometric solutions of a forth order Fuchsian partial differentialequation

Mohamed Amine Kerker

Laboratoire de Mathematiques, EA 4535, Reims, FranceE-mail: [email protected]

Abstract We give an explicit representation of the solutions of the Cauchy problem, in termsof series of hypergeometric functions, for the following class of forth order fuchsian partial differentialequations

(t∂2

t − ∂2x + α∂t

) (t∂2

t − ∂2x + β∂t

)u (t, x) = 0, α, β ∈ C, β = α− 1

u (0, x) = xp∞∑

n=0anx

n, p ∈ C

ut (0, x) = 0.

(P )

We show that the solutions are holomorphic, ramified around the characteristic surface K : 4t−x2 = 0.

References

[1] E. Aysegul (2006), Some recursion relations for solutions of a class of equations. Appl.Math. Lett. 19, no. 10, 1095.1099.

[2] A. Bentrad (1994). Probleme de Cauchy caracteristique a donnees singulieres. Unicite etnon unicite de la solution, Analysis, 14, 303.310.

[3] S. Fujiie (1993), Singular Cauchy problems of higher order with characteristic surface, J.Math. Kyoto Univ. 33, 1.27.

[4] A. Weinstein (1955). On a class of partial differential equations of even order. Ann. Mat.Pura. Appl., 39, 245.254.

2000 Mathematics Subject Classification: 35A20, 35C10, 35C05, 33C05, 33L10.Keywords: Fuchsian operator, Gauss hypergeometric function.

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012

Some coupled fixed point theorems for mappings satisfying a rationalexpression

M.S.Khan

Department of Mathematics and Statistics, College of Science, Sultan Qaboos University,PoBox 36,PCode 123, Al-Khod, MUSCAT, Sultanate of Oman

E-mail:

Abstract The purpose of this paper is to establish some coupled coincidence point theorems for

a pair of mappings having a mixed g-monotone property satisfying a contractive condition of rational

type in the framework of partially ordered metric spaces. Also, we present a result on the existence and

uniqueness of coupled common fixed points. The results presented in the paper generalize and extend

several well-known results in the literature. An example is also provided to support our claim.

2000 Mathematics Subject Classification: 47H10, 54H25Keywords: Coupled fixed point, mixed g-monotone property, ordered metric spaces.

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012

General boundary stabilization of memory type in thermoelasticity

Amel Boudiaf

Department of mathematics, University of M ’sila, AlgeriaE-mail: [email protected]

Abstract In this work we study a nonlinear system of thermoelasticity, where a viscoelasticdamping acting on a part of the boundary. we establish a general decay result, from which the usualexponential and polynomial decay are only special cases.We are concerned with the folllowing system

utt − au+ β∇θ + f(u) = 0 in Ω× R+

cθt − kθ + βdivut = 0 in Ω× R+

u(., 0) = u0, ut(., 0) = u, on Ω

θ(., 0) = θ0, θt(., 0) = θ1, on Ω

where a viscoelastic damping acting on a part of the boundary.a, c, k, β are positive constants, f(u) behaves like |u|pu, Ω is a bouned domain of Rn, with a smoothboundary ∂Ω, such that Γ0,Γ1 is a partition of ∂Ω, ν is the outward normal to ∂Ω, u = u(x, t) ∈ Rn isthe displacement vector, θ = θ(x, t) is the difference temperature, and the relaxation function consideredto be positive, nonicreasing and belongs to W 1,2(0, +∞).

The boundary condition on Γ1 is the nonlocal viscoelastic condition responsible for the memory effect.


58



012

On the spaces of Euler almost convergent and Euler almost null sequences

Murat Kirisci

Department of Mathematical Education, Istanbul University,Beyazıt Yerleskesi, Muskule Sok. No: 1, 34470 Vefa – Istanbul, Turkey


Feyzi Basar



Abstract Let Er denotes the Euler means of order r. The Euler sequence spaces erp, er0, e

rc and

er∞ consisting of all sequences whose Er-transforms are in the spaces ℓp, c0, c and ℓ∞ are introduced byAltay and Basar [1], Altay et al. [2], and Mursaleen et al. [5]. Recently, Polat and Basar have studiedthe Euler spaces of difference sequences of order m, in [6].Let λ, µ be any two sequence spaces and A = (ank) be an infinite matrix of real numbers ank, wheren, k ∈ N. Then, we write Ax =

((Ax)n

), the A-transform of x, if (Ax)n =

∑k ankxk converges for each

n ∈ N. If x ∈ λ implies that Ax ∈ µ then we say that A defines a matrix mapping from λ into µ anddenote it by A : λ → µ. By (λ : µ), we mean the class of all infinite matrices such that A : λ → µ. Thedomain λA of an infinite matrix A in a sequence space λ is defined by λA :=

x = (xk) ∈ ω : Ax ∈ λ

which is a sequence space. If A is triangle, then one can easily observe that the sequence spaces λA andλ are linearly isomorphic, i.e. λA

∼= λ.

The concept almost convergence of a bounded sequence introduced by G.G. Lorentz [4]. Quite recently,

Basar and Kirisci have worked the domain of the generalized difference matrix B(r, s) in the sequence

space f of almost convergent sequences, in [3]. In this paper, following Basar and Kirisci [3], we essentially

deal with the domains fEr and (f0)Er of the Euler means of order r in the spaces f and f0 of almost

convergent and almost null sequences, respectively.

References

[1] B. Altay, F. Basar, On some Euler sequence spaces of non-absolute type, Ukrainian Math.J. 57(1)(2005), 1–17.

[2] B. Altay, F. Basar, M. Mursaleen, On the Euler sequence spaces which include the spacesℓp and ℓ∞ I, Inform. Sci. 176(10)(2006), 1450–1462.

[3] F. Basar, M. Kirisci, Almost convergence and generalized difference matrix, Comput. Math.Appl. 61(3)(2011), 602–611.

2000 Mathematics Subject Classification: Primary 46A45; Secondary 40C05.Keywords: Almost convergence, matrix domain of a sequence space, β− and γ−duals and matrix transformations.

59


[4] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80(1948),167–190.

[5] M. Mursaleen, F. Basar, B. Altay, On the Euler sequence spaces which include the spacesℓp and ℓ∞ II, Nonlinear Anal. 65(3)(2006), 707–717.

[6] H. Polat, F. Basar, Some Euler spaces of difference sequences of order m, Acta Math. Sci.27B(2)(2007), 254–266.

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012

Homogeneous Spaces of Elliptic Curves and associated groups

SACI -KOUACHI Samia and ZITOUNI

Department of mathematics, University Badji Mokhtar Annaba, AlgeriaE-mail: [email protected]

Abstract We are interested in homogeneous spaces of elliptic curves, we begin by studying

Weierstrass Cubic: algebraic structure of groups of Mordell-Weil invariants discriminating. We get the

coordinates of points−P , P1+P2 andmP,m ≥ 2, and the torsion groups T (E). We describe the valuations

and discounts. We study isomorphisms and isogenies of elliptic curves. We describe the cohomology of

abelian groups. This allowed us to develop the theory of homogeneous spaces, groups of Chatelet-Weil

groups and Selmer groups Chafarevich - Tate.


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012

CONTRIBUTION A LA DEPOLLUTION DES EAUX COLOREES PARLES ARGILES ANIONIQUES

Fatima Lehbab, Z. Derriche et M. Bouhent

Laboratoire de Physico-Chimie des Materiaux ;Catalyse et Environnement , Universite desSciences et de la Technologie d’Oran, Bp 1505 Oran El M’naouer , Algerie

E-mail: draa [email protected], bouhent [email protected]

Abstract Le travail que nous avons realise s’inscrit dans une perspective de contribution dans ledomaine de l’environnement. Outre le fait de traiter de la pollution des eaux, nous nous sommes familiariseavec les techniques de synthese, de caracterisation des materiaux solides cristallins de type ′′ argilesanioniques′′ appeles souvent par le nom ′′ d’Hydroxydes Doubles Lamellaires (HDL) ′′ et particulierementleur interet dans les applications de depollution des rejets industriels provenant de l’industrie telle letextile. . . et autres. L’objectif que nous nous sommes fixe est l’elimination des colorants anioniques paradsorption sur ces hydroxydes doubles lamellaires carbonates, leurs produits calcines ainsi que la phaseZn-al-cl. En effet ces argiles anioniques sont caracterisees par un empilement de feuillets d’hydroxydes demetaux divalents et trivalents [MII1-x MIIIx(OH)2]x+[Ax/mm-, nH2O]x- (dans notre cas M+2=Zn+2et M+3=Al+3) separes par des domaines inter feuillets occupes par l’anion A echangeable .Notre matrice a ete preparee par la methode de coprecipitation en milieu basique (pH=9) et constantavec un rapport molaire (Zn/Al) ; R=2.Le produit de synthese obtenu de ce rapport a ete caracterise par plusieurs techniques comme la diffractiondes rayons X (DRX) et spectroscopie infrarouge (IRTF), ce qui nous a permis de confirmer que le materiauobtenu correspond bien aux hydroxydes doubles lamellaires recherches.De part leurs proprietes d’echange anionique tres elevee, leur structure modulable et controlee et leureffet memoire unique, les HDLs sont des materiaux potentiellement tres interessants pour l’adsorption,l’intercalation des molecules de colorants en vue d’une remediation environnementale. Ces proprietesd’echange anionique nous ont permis d’evaluer leur efficacite dans l’elimination des colorants donnant desresultats tres satisfaisants sur les phases calcinees.

L’aspect economique de l’utilisation des materiaux adsorbant, rend important la reutilisation des argiles

anioniques vue leur pouvoir a se regenerer, la reutilisation de ces materiaux pour l’adsorption des colorants

a ete effectuee apres 4 cycles de regeneration.

2000 Mathematics Subject Classification:Keywords: HDL, adsorption, model, A85.

62



012

A broadcast chromatic number of a tree

Rachid LemdaniUniversity of Medea, AlgeriaE-mail: [email protected]

Moncef AbbasUSTHB, Algiers, Algeria


Abstract Let G = (V, E) be a simple graph with no isolated vertices. For every vertex u ∈ V ,

the open neighborhood of a vertex u is denoted by NG(u) and its degree |NG(u)| by degG(u) is the set fv

∈ G|uv ∈ E. The closed neighborhood of u is NG[u] = NG(u) ∪ u. The distance d(u, v) between two

vertices u and v in G is the number of edges on a shortest path between u and v in G.The eccentricity

e(v) of a vertex v is the largest distance from v to any vertex of G. The radius of G(rαd(G)) is the

smallest eccentricity in G.The diameter of G(diαm(G)) is the largest eccentricity in G. The vertex cover

number of G and is denoted by α0(G) is the cardinality of a minimum subset S of the vertices in a

graph such that every edge in the graph has at least one endpoint in S. A broadcast is a function

f : V → 0, 1, 2, . . . , diαm(G) that if every vertex v ∈ V, f(v) ≤ e(v). A broadcast coloring of order k

in G is a function π : V → 1, . . . , k that is if π(u) = π(v) implies that the distance between u and v

is more than π(u). The minimum order of a broadcast coloring is called the broadcast chromatic number

of G and is denoted by χb(G). It was introduced by Wayne Goddard, Sandra M. Hedetniemi, Stephen

T. Hedetniemi and John M. Harris, Douglas F. Rall in [1] and study its properties. They showed that it

is NP-hard to determine if χb(G) ≤ 4 and they determined the maximum broadcast chromatic number

of a tree.The case of a tree of diameter 4 is more complicated, but an explicit formula was given. The

key to the formula is the numbers of large and small neighbors of the central vertex (note that the vertex

is large if it has degree 4 or more, and small otherwise). We use these results and to prolong them on

the tree of diameter 5 while taking the central vertex of large degree, if the two central vertices are the

same degree, we take that who has the greatest sum of degree of all its neighbors (Note that the tree of

diameter 5 has two central vertices)

References

[1] J. Dunbar, D. Erwin, T. W. Haynes, S. M. Hedetniemi, and S. T. Hedet- niemi. Submitted.

[2] J. Dabney, Brian C. Dean, Stephen T. Hedetniemi, A Linear-Time Algo- rithm for Broad-cast Domination in a Tree. School of Computing, Clemson University. June 12, 2007.

[3] Jean E. Dunbara, David J. Erwinb, Teresa W. Haynesc, Sandra M. Hedet- niemid, StephenT. Hedetniemid, Broadcasts in graphs, DiscreteApplied Mathematics 154 (2006) 59-75.

2000 Mathematics Subject Classification:Keywords: Graph theory, Broadcast coloring, Broadcast chromatic number.

63



012

Abstract Fractional Integro-differential EquationsWith State-DependentDelay

S. Litimein

Laboratoire de Mathematiques, Universite de Sidi Bel-AbbesBP 89, 22000 Sidi Bel-Abbes, Algerie

E-mail: sara [email protected]

M. Benchohra

Laboratoire de Mathematiques, Universite de Sidi Bel-AbbesBP 89, 22000 Sidi Bel-Abbes, Algerie


Abstract In this paper we investigate the existence and uniqueness of solutions on a compact

interval for non-linear fractional integro-differential equations with state-dependent delay. Our results

will be obtained using suitable fixed point theorems and the technique of measures of noncompactness.

Some applications of the main result have been included.

References

[1] N. Abada, R. P. Agarwal, M. Benchohra and H. Hammouche, Existence results for non-densely defined impulsive semilinear functional differential equations with state-dependentdelay, Asian-Eur. J. Math., 1 (4) (2008), 449-468.

[2] R. P. Agarwal, B. Andradec, G. Siracusa, On fractional integro-differential equations withstate-dependent delay, Comput. Math. Appl. 63 (3) (2011), 1142-1149.

[3] R. P. Agarwal, M. Meechan and D. O’Regan, Fixed Point Theory and Applications, Cam-bridge University Press, Cambridge, 2001.

2000 Mathematics Subject Classification: 26A33; 45J05; 45G05.Keywords: Integral resolvent family, mild solution, fixed points, statedependent delay, measure of noncompactness.

64



012

Mathematical Simulation of Cloaking Metamaterial Structures

Maher Belgacem

Departement de Mathematiques, Faculte des Sciences de Tunis, TunisieUniversit de Rennes 1


Mahdjoubi Kouroch

IETR Universite de Rennes 1, Rennes 35700, FranceE-mail:

Boukricha Abderrahmane

Departement de Mathematiques, Faculte des Sciences de Tunis, TunisieE-mail:

Abstract There is currently a great deal of interest in the theoretical and practical possibility of

cloaking objects from the observation by electromagnetic waves.

In this paper we present a rigorous derivation of the material parameters for both the cylinder and rectan-

gle cloaking structures. Numerical results using these material parameters are presented to demonstrate

the cloaking e ect..

References

[1] A. Greenleaf, Y. Kurylev, M. Lassas and G. Ulhmann, Full-wave invisibility of active devicesat all frequencies, Comm. Math. Phys. 275 749-789 (2007).

[2] Q. LIN AND J. LI, Superconvergence analysis for Maxwell.s equations in dispersive media,Math. Comput., 77 (2008), pp. 757.771.

[3] A. Nicolet, F. Zolla, and S. Guenneau, Finite-Element Analysis of Cylindrical InvisibilityCloaks of Elliptical Cross Section, IEEE Trans. Magn. vol. 44, pp. 1150-1153 (2008).

[4] R. Weder, The boundary conditions for point transformed electromagnetic invisibilitycloaks, J. Phys. A: Math. Theor., vol. 41, p. 415401 (2008).

[5] D.Schurig, J. B. Pendry, and D. R. Smith, Calculation of material properties and ray tracingin transformation media, Opt. Express, vol. 14, pp. 9794-9804, (2006).

2000 Mathematics Subject Classification:Keywords: Maxwell’s equations, metamaterial, finite element method, invisible cloak

65



012

Lagrange interpolation arising from a steady fluid structure interactionproblem

Ibrahima Mbaye

University of Thies (Senegal), Department of MathematicsE-mail: [email protected]

Abstract In this paper, we use Lagrange interpolation to solve fluid structure interaction problem.

This study comes with a view to extend our previous approximation techniques in the resolution of

coupled problems. A combination of Lagrange interpolation and BFGS method leads to computation of

the structure displacement, the fluid veloccity and the fluid pressure.


66



012

Numerical Modeling of swirling flow in cylindrical configuration under theconstant magnetic field

F. Mebarek-Oudina

Departement des Sciences de la Matiere, Universite 20 Aout 1955, Skikda 21000 AlgerieLaboratoire d’Energetique Appliquee et de Pollution, Universite Mentouri, Constantine, Algerie


R. Bessaih

Laboratoire d’Energetique Appliquee et de Pollution, Universite Mentouri, Constantine, AlgerieE-mail: [email protected]

Abstract A numerical modeling of the mixed convection in cylindrical configurations with a

magnetic field was considered. The finite volumes method has been used to resolve the equations of

continuity, momentum (Navier-Stokes), energy and electric potential. The equations of mathematical

model are a partials differential equations, nonlinear elliptic, complex and coupled. The SIMPLER and

TDMA algorithms [1] are used to solve this system and obtain a solution. The computer code developed

here is validated via comparisons with numerical and experimental data founded in the literature. Sta-

bility diagrams are established according to the numerical results of this investigation. These diagrams

highlight the dependence of the critical Reynolds numbers ReCr with the values of the Richardson Ri and

Hartmann Ha numbers. The effect of the rotating disk, magnetic field and the bottom wall conductivity

on the flow is also studied [2-8]. The results obtained in this study will possibly allow the researchers and

industrialists to know the oscillatory modes of a low Prandtl number fluid with magnetic field, in order

to improve the quality of the semiconductors obtained during the crystal growth.

References

[1] S. V. Patankar, 1980. “Numerical Heat Transfer and Fluid Flow”. McGraw-Hill.

[2] H. Ben Hadid & D. Henry, 1996 “Numerical simulation of convective three-dimensionalflows in a horizontal cylinder under the action of a constant magnetic field”. Journal ofCrystal Growth, vol. 166, issues 1-4, pp. 436-445.

[3] F, Mebarek-Oudina & R, Bessaıh, 2011, Proceeding for the 15th International Meeting onThermal Sciences, Tlemcen University, Algeria.

[4] F, Mebarek-Oudina & R, Bessaıh, 2011, Book of abstracts for the International Conferenceon Applied Analysis and Algebra, Istanbul 29 June - 2 July, TURKEY.

2000 Mathematics Subject Classification:Keywords: Numerical Modeling, The finite volumes method, Rotating flow, Magnetic field, Liquid metal.

67


[5] F, Mebarek-Oudina & R, Bessaıh, 2007 “Magnetohydrodynamic Stability of Natural Con-vection Flows in Czochralski Crystal Growth. World Journal of Engineering” vol. 4 (4), pp.5-22.

[6] F.Mebarek-Oudina & R. Bessaıh, 2010 “Oscillatory Mixed Convection Flow in a CylindricalContainer with Rotating Disk Under Axial Magnetic Field” I. Review of Physics, vol. 4(1),pp. 45-51.

[7] A. Kharicha, A. Alemany, D. Bornas, 2004 “Influence of the Magnetic Field and the Con-ductance Ratio on the Mass Transfer Rotating Lid Driven Flow, ”International Journal ofHeat and Mass Transfer, vol. 47, pp.1997-2014.

[8] F.Mebarek-Oudina, R. Bessaıh and Ph. Marty, 2007, “Numerical simulation of the Magnetohydrodynamic Stability for Thermo Convective Flows”, Proceeding for the First Interna-tional Seminar on Fluid Dynamics and Materials Processing. ALGIERS, Juin 2-5.

68



012

Sur les Problemes aux Limites Elliptiques gouvernee par l’operateur deLaplace a poids dans un domaine plan polygone

H. MEDEKHEL

Departement de Mathematiques, Universite d’EL -Oued B.P789, EL-Oued 39000, AlgerieE-mail: [email protected]

M. S. SAID

Departement de Mathematiques, Universite Kasdi Merbah B.P511, Ouargla 30000, AlgerieE-mail: [email protected]

Abstract L’etude de 1‘equation de Laplace dans un polygone ou un polyedre, et generalement1‘etude des problemes elliptiques dans des domaines non reguliers n’est entamee que depuis une daterelativement recente; d’une part Grisvard [3] montre que la formule de Green construite pour le Laplaciendans le cas classique c’est a dire dans des domaines reguliers est encore valable dans les domaines nonreguliers tels que les polygones ou polyedres, par exemple, et en utilisant l’alternative de Fredholm, ilretrouve des resultats analogues a ceux du cas classique, ces resultats ne sont pas encore generalisees ades operateurs elliptiques aux derivees partielles d’ordre plus eleve dans des domaines de Rn, et ceci acause de la complexite des calculs.Le but de notre travail est d’etudier le role que jouent les fonctions poids dans 1‘etude du problemegenerale suivant :

u = f dans Ω Bu = g sur Γ (1)

Ou Ω est un ouvert plan de frontiere polygonale notee Γ1, et B etant un operateur differentiel aux deriveespartielles, d’ordre 0 ou 1 defini sur la frontiere Γ, on ne considerera ici que les deux cas: la condition deDirichlet, et celle de Neumann. f donnee dans L2(Ω).La transformation de Mellin, est bien adaptee a la geometrie du domaine, le noyau de Green aussi bienadaptee a la resolution de 1‘equation differentielle explicite, et de en evidence d’inegalite a priori. (Lemmede Peetre).[l, 2].

Ces resultats etaient connus en norme L2 dans certains cas particuliers voir Kondaratiev [6] cette etude

conduit naturellement a des espace avec poids, toutefois l’auteur en deduit certains resultats dans Hs.[4]

2000 Mathematics Subject Classification: 35A05, 45J05Keywords: Probleme de Poisson, transformation de Mellin, Espace a poids, Polygone, Regularite

69


References

[1] M.Merigot, Etude du probleme de u = f dans un polygone plan, Inegalites a priori,Boll.Un.Mat.Italie (4).10, pp. 577-597,1974.

[2] Lions,J.L.et Magenes, Problemes aux limtes non homogenes et Applica- tions,Tomel,Dunod,Paris 1968.

[3] P.Grisvard, Alternative de Fredholm relative au probleme de Dirichlet dans un polygoneou un polyedre, Boll.Un.Mat.Italie (4).51972 p. 132-164.

[4] P.Grisvard,Elliptic problems in nonsmooth domains,Pitman,London,1985.

[5] P.Grisvard,Singularites in Boundary value Problems, Masson,1992.

[6] V.A.Koundratiev,Probleme aux limites pour les equations elliptiques dans domaine avecpoint conique ou anguleux,Trudy Moskov Mat.Obsc 16,1967.

[7] MS.Said,B.Merouani,Role des poids dans 1‘etude de quelques problemes aux limites gou-vernes par le systeme de Lame dans un polygone, Roum.Sci.Techn-Mec. Appl.Bucaest 2002

70



012

Statistical Bayesian Analysis of Experimental Data

HAYET MERABET et AHLEM LABDAOUI

Laboratoire de Mathematiques Appliquees et ModelisationDepartement de Mathematiques, Universite Mentouri-Constantine

Route de Ain-El Bey, 25000 Constantine, AlgerieE-mail: [email protected]

Abstract The Bayesian researcher should know the basic ideas underlying Bayesian methodology

(i.e., Bayesian theory) and the computational tools used in modern Bayesian econometrics (i.e., Bayesian

computation). Bayesian econometrics typically involves extensive use of posterior simulation. Some of

the most important methods of posterior simulation are Monte Carlo integration, importance sampling,

Gibbs sampling and the Metropolis–Hastings algorithm. The Bayesian should also be able to put the

theory and computational tools together in the context of substantive empirical problems. We focus

primarily on recent developments in Bayesian computation. Then we focus on particular models (usually

regression based). Inevitably, we combine theory and computation in the context of particular models.

Although we have tried to be reasonably complete in terms of covering the basic ideas of Bayesian theory

and the computational tools most commonly used by the Bayesian, there is no way we can cover all

the classes of models used in econometrics. We propose to the user of analysis of variance and linear

regression model, which is the workhorse of econometrics, a practical, realistic and constructive statistical

inference, which brings a fresh look at his data. We illustrate the desirability and feasibility of Bayesian

methods by providing simple and direct answers. Accordingly, we have selected a few popular classes of

models (e.g., regression models with extensions and panel data models) to illustrate how the Bayesian

paradigm works in practice.

References

[1] J-M Marin et C. P. Robert (2007) Bayesian Core: A Practical Approach to ComputationalBayesian Statistics, Springer, New-York.

[2] M. Kyung, J. Gill, M. Ghosh and G. Casella: Penalized Regression, Standard Errors, andBayesian Lassos, Bayesian Analysis (2010) 5, Number 2, pp. 369-412.

2000 Mathematics Subject Classification:Keywords: Bayesian analysis, Markov Chain Monte Carlo Algorithms, Linear regression model.

71



012

On the asymptotic behavior of transmission thin shell problems

MERABET ISMAIL, SERGE NICAISE AND DJAMEL AHMED CHACHA

University of KASDI MERBAH OUARGLA, AlgeriaE-mail:

Abstract In this paper we study the asymptotic behavior of two-dimensional transmission problems

for the linear Koiter’s model of an elastic multi-structure composed of two thin shells with the same

thickness ε << 1. The membrane approximation, i.e., for ε = 0, fails to give a convenient approximate

solution, as the limit problem is ill posed. An appropriate dilation leads to an equivalent problem, for

which we prove strong convergence in some usual space.

2000 Mathematics Subject Classification:Keywords: thin shells, Koiter’s model, junction, boundary layers, sensitivity.

72



012

A Computational Method for integrodifferential hyperbolic equation withintegral conditions

Ahcene MERAD

Departement of Mathematics, The Larbi Ben M’Hidi University, Oum El Bouaghi, 04000,Algeria

E-mail: merad [email protected]

Abdelfatah BOUZIANI

Departement of Mathematics, The Larbi Ben M’Hidi University, Oum El Bouaghi, 04000,Algeria


Abstract In this work, we discuss the one dimensional integrodifferential hyperbolic equation

subject to nonlocal conditions.We use the method of Laplace transforms. Finally, we obtain the solution

by using a numerical technique for inverting the Laplace transforms.

1 Introduction

In the rectangle Q = (0, 1)× (0, T ) ,we consider the equation

∂2u

∂t2− ∂2u

∂x2= f (x, t) +

∫ t

0a (t− x)u (x, s) ds, x ∈ (0, 1) , t ∈ (0, T ) (1.1)

We adhere to equation (1.1) the initial conditions

u(x, 0) = φ (x) ,∂u(x, 0)

∂t= ψ (x) 0 ≤ x ≤ 1, (1.2)

and the nonlocal conditions∫ 1

0u (x, t) dx = 0,

∫ 1

0xu (x, t) dx = 0, 0 ≤ t ≤ T. (1.3)

2000 Mathematics Subject Classification:Keywords: Integrodifferential hyperbolic equation, Nonlocal condition, Laplace transform method.

73


References

[1] Abramowitz, M., Stegun. I.A., Handbook of Mathematical Functions, Dover, New York,1972.

[2] Ang, W.T., A Method of Solution for the One-Dimentional Heat Equation Subject toNonlocal Conditions, Southeast Asian Bulletin of Mathematics (2002), 26: 185-191.

[3] Benouar, N. E., Yurchuk. N. I., Mixed problem with an integral condition for parabolicequation with the Bessel operator, Differentsial’nye 27(1991) , 2094− 2098.

[4] Beılin, S. A., Existence of solutions for one-dimentional wave nonlocal conditions, Electron.J. Differential Equations 2001 (2001) , no. 76,1-8

[5] Bouziani, A., Mixed problem with boundary integral conditions for a certain parabolicequation, J. Appl. Math. Stochastic Anal.9 (1996) ,no.3, 323-330.

[6] Bouziani, A., Solution forte d’un probleme mixte avec une condition non locale pour uneclasse d’equations hyperbolique, Acad. Roy. Belg. Bull. Cl. Sci. 8 (1997) , 53-70.

[7] Bouziani, A., Strong solution for a mixed problem with nonlocal condition for certainpluriparabolic equations, Hiroshima Math. J. 27 (1997) ,no. 3, 373-390.

[8] Bouziani, A., On the solvability of nonlocal pluriparabolic problems, Electron, J. DifferentialEquations 2001(2001) ,1-16.

[9] Bouziani, A., Initial-boundary value problem with nonlocal condition for a viscosity equa-tion, Int. J. Math. & Math. Sci.30 (2002) , no. 6, 327-338.

[10] Bouziani, A., On the solvabiliy of parabolic and hyperbolic problems with a boundaryintegral condition, Internat. J. Math. & Math. Sci. 31 (2002) , 435-447.

[11] Bouziani, A., On the solvability of a class of singular parabolic equations with nonlocalboundary conditions in nonclassical function spaces, Internat. J. Math. & Math. Sci. 30(2002) , 435-447.

[12] Bouziani, A., On a classe of nonclassical hyperbolic equations with nonlocal conditions. J.Appl. Math. Stochastic Anal. 15 (2002) ,no. 2, 136-153.

[13] Bouziani, A.; N. Benouar, Mixed problem with integral conditions for a third order parabolicequation, kobe J. Math. 15 (1998) ,no. 1, 47-58..

[14] Stehfest,H., Numerical Inversion of the Laplace Transforme, Comm. ACM 13, 47-49 (seealso p624) (1970).

74



012

Comparison between classes of Joel Anderson and finite operators

MESSAOUDENE HADIA

Department of Mathematics, Tebessa University, Algeria.E-mail:

Abstract Let H be a separable infinite dimensional complex Hilbert space, and L(H) denote thealgebra of all bounded linear operators on H. The derivation of the operator A is defined by:δA :L(H) → L(H)X 7→ AX −XAThe main objective of this work is to compare the classes of operators for which the distance of theidentity operator and the derivation range is minimal (class of Joel Anderson noted JA(H)), or maximal(class of finite operator noted F(H)).And to prove that the class JA(H) has no algebraic structure and to give a necessary and sufficientcondition for a bounded linear operator A to be in JA(H) and to obtain some results concerning theform of operators in JA(H).

We proved that F(H) is a field and we present some properties of F(H) and give some classes of operators

which are in F(H).


75



012

On Statistical Convergence of Order α of Generalized Difference Sequences

Mikail ET

Department of Mathematics, Firat University, 23119, Elazıg - TURKEYE-mail: [email protected]

Murat KARAKAS

Department of Mathematics, Bitlis Eren University, Bitlis-TurkeyE-mail: [email protected]

Muhammed CINAR

Department of Mathematics, Mus Alparslan University, Mus -TurkeyE-mail: [email protected]

Abstract The idea of difference sequence spaces was introduced by Kızmaz and was generalized

by Et and Colak. Later on difference sequence spaces have been studied by Altay and Basar, Et and

Basarır, Malkowsky and Parashar, Mursaleen and many others. In this study we introduce the concept

Sαλ (∆m)−statistical convergence of order α. Also some relations between Sα

λ (∆m)−statistical conver-

gence of order α and strong Sβp (∆m)−summability of order β are given. Furthermore some relations

between the spaces ωα(p) [λ,M ] and Sα

λ (∆m) are examined.


76



012

Modified Newton’s methods with fifth or sixth -order convergence andmultiple roots

Husnia Mohamed Eldanfour

Department of Mathematics- Faculty of Science - Misurata University, PO Box 996 - Misurata- Libya


Abstract In paper [5] a modifications of the Newton’s method which produces iterative methods

with the fifth or sixth of convergence have been proposed. Here we study the order of convergence of

the methods when we have multiple roots. We prove that the order of convergence of the mNm go down

to one but, when the multiplicity p is known, it may be raised up to two and six by using two different

types of correction, when p is unknown we show that the mNm have converge faster than the classical

Newton’s method.

2000 Mathematics Subject Classification:Keywords: Newton’s method, order convergence, Multiple roots.

77



012

ON THE NEW SEQUENCE SPACES INCLUDING THE SPACES OF ALLCONVERGENT AND NULL SEQUENCES

Havva Nergiz

The Graduate School of Sciences and Engineering, Fatih University, Hadımkoy Campus,Buyukcekmece, 34500 - Istanbul, Turkey


Feyzi Basar



Abstract Let −∞ < r < ∞ and ω denotes the space of all complex valued sequences. Define thesubsets cr and cr0 of the space ω by

cr ; = x = (xk) ∈ ω : ∃l ∈ C ∋ limk→∞

xk

kr= l,

cr0 ; = x = (xk) ∈ ω : limk→∞

xk

kr= 0.

Let λ ∈ cr, cr0. The main results of this study are, as follows:1 ) The sets cr and cr0 are linear spaces with respect to the coordinate-wise addition and scalar multipli-cation of sequences.2 )(λ, dr∞) is a complete metric spaces, where

dr∞(x, y) = supk∈N

|xk − ykkr

|;x = (xk), y = (yk) ∈ λ.

3 )(λ, ∥ · ∥r∞) is a Banach space, where

∥x∥r∞ = supk∈N

|xk

kr|;x = (xk) ∈ λ.

4 ) The inclusion relation µ ⊂ λ with r ≥ 0 strictly holds, where µ ∈ c, c0.5 ) The α−, β− and γ-duals of the spaces cr and cr0 are determined.

6 ) The classes (λ : ℓ∞), (λ : f), (λ : c) and (µ : λ) of infinite matrices are characterized, where f denotes

the space of almost convergent sequences and µ ∈ ℓ∞, c, c0.

2000 Mathematics Subject Classification: Primary 46A45; Secondary 40C05.Keywords: Matrix domain of a sequence space, α−, β− and γ-duals and matrix transformations.

78



012

STATISTICAL CONTINUITY AND STATISTICAL DERIVATIVE

FATIH NURAY

DEPARTMENT OF MATHEMATICS, AFYON KOCATEPE UNIVERSITY,AFYONKARAHISAR, TURKEY


Abstract A sequence x = (xk) is said to be statistically convergent to the number u if for everyϵ > 0,

limn→∞

1

n|k ≤ n : |xk − u| ≥ ϵ| = 0

where the vertical bars indicate the number of elements in the enclosed set. In this case we writest− limxn = uf : R → R is said to be statistically continuous at u ∈ R provided that whenever st-limxn = u thenst− limf(xn) = f(u).

The purpose of this study is to give alternative definition of statistically continuous function and define

notion of statistical derivative for real valued functions.

References

[1] A. Borichev, R. Deville and E. Matheron, Strongly sequentially Continuous functions,Quaes- tiones Mathematicae 24 (2001), 535-548.

[2] A. M. Bruckner, Differentiation of real functions, Lecture Notes in Math.,659, Springer-Verlag, 1978.

[3] A. M. Bruckner, R. J. OMalley and B. S. Thomson, Path derivatives: a unified view ofcertain generalized derivatives, Tran. Amer. Math. Soc.,283 (1984), 97125.

[4] J. S. Connor, The statistical and strong p-Cesaro convergence of sequences, Analysis8(1988)46- 63.

[5] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951)241− 244.

[6] J.A. Fridy, On statistical convergence, Analysis 5(1985)301− 313.

[7] J.A. Fridy, Statistical limit points, Proc. Amer. Math. Soc. 118 (1993)1187− 1192.

[8] J.A. Fridy and C. Orhan, Statistical limit superior and limit inferior, Proc. Amer. Math.Soc. 125 (1997)3625− 3631.

2000 Mathematics Subject Classification: Primary 40A05; Secondary 40C05, 40D05.Keywords: statistical convergence, statistical continuity, statistical derivative.

79


[9] P. Kostyrko, T. Salat and W. Wilczynski, I-convergence, Real Anal. Exchange26(2000/2001), 669686.

[10] M. Laczkovich and G. Petruska, Remarks on a problem of A. M. Bruckner, Acta Math.Acad. Sci. Hungar., 38 (1981), 205214.

[11] S. Saks, Theory of the integral, Dover Publications, Inc. (New York).

[12] H. Shi, A type of path derivative, Real Anal.Exchange 28(2) (2002/2003), 279-286.

[13] A. Zygmund, Trigonometric series, 2nd Ed. Cambridge Univ. Press 1979.

80



012

Operating Conditions Optimization of Catalytic Fixed Bed Reactor

N.Outili, M. Amroune and S. Hammoudi

Department of industrial chemistry, University Mentouri Constantine,route Ain el Bey, Constantine.25000. Algeria

Laboratoire de l’ingenierie des procedes de l’environnement LIPEE-mail: n [email protected]

Abstract The importance of the production of ethylene oxide as intermediate product is indis-putable in chemical industry, since the total request of ethylene oxide continues to increase because ofits importance like intermediary to manufacture antifreeze, fibers of polyester and other petrochemicalproducts.The objective of this present work is the study of the catalytic fixed bed reactor used for this productionby seeking the optimal operating conditions: it is a question of adapting the operating conditions byholding account physical limitations of the reactor and industrial constraints.In this work we applied the pseudo-homogeneous model which already proved its effectiveness in thiscase, a code in Matlab was developed in order to solve the generated system of differential equations.We studied the influence of certain operational parameters on the selectivity and the thermal stability ofthe reactor, which can depend on a great number of external parameters but without having analyticalmodels of them.In this context we used the method “experimental plan”, which is often used to minimize the number ofmeasurement points of a process to obtain the maximum of information and the most influential factors,except that we have replaced measurements by numerical executions of the elaborate program.Indeed, we made variations of certain operational parameters in precise intervals, for which we calculatedthe corresponding selectivity and temperature of the reactor. By applying a factorial experimental plan,we could quantify the influence of the parameters; we find a correlation connecting the parameters to thekey factors, namely selectivity and temperature in order to facilitate optimization under constraints.

The results obtained are in agreements with those of the literature and make it possible to see that

the selectivity without constraints would be higher but the optimal operating conditions must take into

account the limits imposed by industrial constraints and process safety.

2000 Mathematics Subject Classification:Keywords: ethylene oxide, fixed bed, catalytic reactor, pseudo-homogeneous model, optimization, experimental plan

81



012

AN INEQUALITY RELATED TO THE RIESZ CORE OF DOUBLESEQUENCES

CELAL CAKAN

Inonu University Faculty of Education, 44100-Malatya/TURKEYE-mail: celal. [email protected]

Abstract A double sequence x = [xjk]∞j,k=0 is said to be convergent in the Pringsheim sense or

P-convergent if for every ϵ > 0 there exists an N ∈ N such that |xjk − ℓ| < ϵ whenever j, k > N , [3]. Inthis case, we write P − limx = ℓ. A double sequence x is bounded if

||x|| = supj,k≥0

|xjk| < ∞.

By ℓ2∞ we denote the space of all bounded double sequences.Let A = [αmn

jk ]∞j,k=0 be a four-dimensional infinite matrix of real numbers for all m, n = 0, 1, . . .. Thesums

ymn =∞∑j=0

∞∑k=0

αmnjk xjk

are called the A- transforms of the double sequence x = [xjk]. We say that a sequence x = [xjk] isA-summable to the limit ℓ if the A- transform of x = [xjk] exists for all m, n = 0, 1, . . . and is convergentto ℓ in the Pringsheim sense, i.e.,

limp,q→∞

p∑j=0

q∑k=0

αmnjk xjk = ymn

and limm,n→∞

ymn = ℓ.

Let (qi), (pj) be sequences of non-negative numbers which are not all zero and Qm = q1+q2++qm, q1 >0, Pn = p1 + p2 + · · · + pn, p1 > 0. The Riesz convergence of a double sequence x = [xjk] has beendefined in [1] by the P − lim tqpmn(x), where

tqpmn(x) =1

Qm

1

Pn

m∑i=1

n∑j=1

qipjX ij .

If x = [xjk] is Riesz convergent to s, then we write PR− limx = s. The set of all Riesz convergent doublesequences is denoted by c2R.The aim of this study is to give the necessary and sufficient conditions for a four dimensional matrixA = [αmn

jk ] to satisfy P − lim sup tqprs(Ax) ≤ Cσ(x) for all x ∈ ℓ2∞, where

Cσ(x) = limm,n

sup sups,t

1

mn

m∑j=0

n∑k=0

xjk

which defined in [2].

2000 Mathematics Subject Classification: 40C05,40J05,46A45Keywords: Double sequences, core of a sequence and Riesz convergence

82


References

[1] A.M. Alotaibi, C. Qakan, The Riesz convergence and Riesz core of double sequences, Journalof Inequalities and Applications 2012, 2012:56.

[2] C. Qakan, B. Altay, M. Mursaleen, The σ convergence and σ-core of double sequences,Applied Mathematics Letters 19 (2006), 1122-1128.

[3] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53(1900),289-321

83



012

SPACE OF CONTINUOUS FUNCTIONS OVER THE FIELD OFNON-NEWTONIAN REAL NUMBERS

Ahmet Faruk Cakmak

Department of Mathematical Engineering, Yildiz Technical University, Davutpaga Campus,Esenler, 80750 - Istanbul, TurkeyE-mail: [email protected]

Feyzi Bagar

Department of Mathematics, Fatih University, Hadimkoy Campus, Buyukcekmece, 34500 -Istanbul, Turkey


Abstract As an alternative to the classical calculus, Grossman and Katz introduced the non-Newtonian calculus in [2] consisting of the branches of geometric, anageometric and bigeometric calculus.Recently, Bashirov et al. [1] have recently emphasized on the non-Newtonian calculus and gave the resultswith applications corresponding to the well-known properties of derivative and integral in the classicalcalcu- lus. Quite recently, Uzer [3] has extended the multiplicative calculus to the complex valuedfunctions and interested in the statements of some fundamental theorems and concepts of multiplicativecomplex calculus, and demonstrated some analogies between the multiplicative complex calculus andclassical calculus by theoretical and numerical examples.Following Grossman and Katz, we construct the field R(N) of non-Newtonian real numbers and theconcept of non-Newtonian metric. Also we define and give the basic important properties of convergenceand continuity. Later, we emphasize on the sets C(N)[a, b], as the space of non-Newtonian continuousfunctions.Our main results are:Theorem 1. (R(N), u, ×) is a complete ffield.Theorem 2. An N-monotone sequence of N-real numbers converges if and only if it is N-bounded.Theorem 3. Let (an) and bn be N-convergent sequences of N-real numbers. Then for each pair of N-realnumbers α and β, the sequence λ× an uµ ×bn is N-convergent and

N − limn→∞

[λ× an uµ ×bn] = λ× (N − limn→∞

an)uµ ×(N − limn→∞

bn)

Moreover ; if an ≤ bn for all n ∈ N, then N − limn→∞

an ≤ N − limn→∞

bn.

Theorem 4. CN [a, b] is a metric space with the metric dN , where dN (x, y) = maxt∈[a,b]

|x(t) − y(t)| with

x, y ∈ CN [a, b].Theorem 5. CN [a, b] is a vector space with the algebraic operations deffined in the usual way.

Theorem 6. CN [a, b] is a Banach space with norm given by ∥x∥N = maxt∈[a,b]

|x(t)|.

2000 Mathematics Subject Classification: Primary 26A06, 11U10; Secondary 08A05, 46A45Keywords: Non-Newtonian calculus, algebraic structures with respect to non-Newtonian calculus, non-Newtonianfunction space.

84


References

[1] A.E. Bashirov, E.M. KUΓPlnar, A. O zyaplCl, Multiplicative calculus and its applications,

J. Math. Anal. Appl. 337(2008), 36-48.

[2] M. Grossman, R. Katz, Non-Newtonian Calculus, Lowell Technological Institute, 1972.

[3] A. Uzer, Multiplicative type complex calculus as an alternative to the classical calculus,Comput. Math. Appl. 60(2010), 2725-2737.

85



012

On Generalized Gegenbauer Matrix Polynomials

Raed S. Batahan

Department of Mathematics, Faculty of Sciences,Hadhramout University of Science and Technology,

50511, Mukalla, Yemen.E-mail:

Abstract The purpose of this talk is to consider a new generalization of the Gegenbauer matrix

polynomials in two variables. The hypergeometric matrix representation of the generalized Gegenbauer

matrix polynomials is presented here. Moreover, matrix differential recurrence relations and partial

differential equation concerning to these matrix polynomials are established.

2000 Mathematics Subject Classification:Keywords: Gegenbauer matrix polynomials; Gamma matrix function; Hypergeometric matrix function; matrixdifferential equations.

86



012

An inequality for n-convex functions

Mourad Rahmani and Hacene Belbachir

USTHB, Faculty of Mathematics, P. O. Box 32, El Alia,16111, Algiers, AlgeriaE-mail:

Abstract We extend Lupas inequality for n-convex (n-concave) functions.Let f, g : [a, b] → R be integrable functions, consider the Cebysev functional

T (f, g) =1

b− a

∫ b

a

f(x)g(x)dx− 1

(b− a)2

∫ b

a

f(x)dx

∫ b

a

g(x)dx (1)

where the integrals involved exist.The problem of estimating the functional in (1) under convexity has been studied by several authors. In1971, Atkinson [1] showed that if f, g are convex functions which are twice differentiable on [a, b] and∫ b

a

(x− a+ b

2)g(x)dx = 0,

then T (f, g) ≥ 0. In 1972, Lupas> [3] proved the following inequality for convex functions

T (f, g) ≥ 12

(b− a)4

∫ b

a

(x− a+ b

2)f(x)dx

∫ b

a

(x− a+ b

2)g(x)dx, (2)

with equality when at least one of the functions f, g is an affine function on [a, b]. We extend Lupassinequality for n-convex (n-concave) functions.

References

[1] E. V. Atkinson, An Inequality, Univ Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., (1971),No. 357-380, 5-6

[2] H. Belbachir, M. Rahmani, An Integral Inequality for n-convex Functions, Math. Inequal.Appl., 15 (1), 117-126, (2012).

[3] A. Lupas, An Integral Inequality for Convex Functions. Univ Beograd. Publ. Elektrotehn.Fak. Ser. Mat. Fiz., No. 381-409, 17-19, (1972).

2000 Mathematics Subject Classification:Keywords: Cebysev functional, convex functions of higher order, Legendre polynomials.

87



012

Exponential inequalities for the Robbins Monro’s algorithm with associatedvariables

Samir RAHMANI

Department of Technology, University A. Mira, Bejaia 06000, AlgeriaE-mail: [email protected]

Abdelnasser DAHMANI

Department of Mathematics, University A. Mira, Bejaia 06000, AlgeriaE-mail: a [email protected]

Abstract In this work, we establish exponential inequalities for the Robbins-Monro’s algortihm

with associated variables, and to precise the almost complete convergence rate of this algorithm.

References

[1] H. Robbins, S. Monro, A stochastic approximation method. Ann. Math. Stat, 22, N1(1951) 400-407.

[2] D.A. Ioannides, G.G. Roussas, Exponential inequality for associated random variables.Statist Probab Lett, 42 (1999), 423-431.

[3] T. Birkel, Moment bounds for associated sequences. Ann Probab, 16 (1988), 1184-1193.

[4] M. Duflo, Méthodes récursives aléatoires. Masson, 1990.

2000 Mathematics Subject Classification: 60E15, 60F15, 62L08, 62L20.Keywords: Robbins-Monro’s algorithm, associated random variable, exponential inequality, rate of convergence.

88



012

Game Theory Approach to the Numerical Analysis of the M2/M/1 Queue

H.ZIANI(1), MADI(2), F.RAHMOUNE(3) and M.S.RADJEF(4)

Lamos Laboratory - University of Bejaia - Bejaia 06000 - AlgeriaE-mail: (3)[email protected], (4)[email protected]

Abstract The study of economic behavior of service providers in a competition environment

is an important and interesting research issue. This paper deals with a game theory approach to the

numerical analysis of the batch arrival queuing system. We consider a single server Markovian queue. We

assume that arriving customers decide whether to enter the system or balk based on a natural reward-

cost structure, which incorporates their desire for service as well as their unwillingness to wait. We

examine customer behavior under various levels of information regarding the system state. Specifically,

before making the decision, a customer may or may not know the state of the server and/or the number

of present customers. We derive equilibrium strategies for the customers under the various levels of

information and analyze the stationary behavior of the system under these strategies. We also illustrate

further effects of the information level on the equilibrium behavior via numerical experiments.

2000 Mathematics Subject Classification:Keywords: Markovian Queuing Systems - Game Theory - Equilibrium customer strategies- Nash Equilibrium-Competition, , Nash Equilibrium.

89



012

Cone convergency for multiple sequences

Ahmet SAHINER

Department of Mathematics, Suleyman Demirel University,32260, Isparta, TURKEY


Abstract The aim of this paper is to introduce a new type convergency which is useful when a

d−multiple sequence is not convergent in some usual senses.

2000 Mathematics Subject Classification: Primary 40A05, 40B05; Secondary 26A03xKeywords: Double sequence; multiple sequence; statistical convergence; multiple natural density; cone convergency.

90



012

Basisity problem and weighted shift operators

Mubariz.T. Karaev

University of Suleyman Demirel, Isparta Vocational School32260, Isparta, Turkey.


Mehmet Gurdal and Suna Saltan

Department of Mathematics, University of Suleyman Demirel32260, Isparta, Turkey.


Abstract We investigate a basisity problem in the space ℓpA (D) and in its invariant sub-spaces. Namely, let W denote a unilateral weighted shift operator acting in the space ℓpA (D) ,1 ≤ p < ∞, by Wzn = λnz

n+1, n ≥ 0, with respect to the standard basis znn≥0 . Applying theso-called ”discrete Duhamel product” technique, it is proven that for any integer k ≥ 1 the sequence(wi+nk)

−1 · (W | Ei)kn

fn≥0

is a basic sequence in Ei := spanzi+n : n ≥ 0

equivalent to the basis

zi+nn≥0

if and only if∧f (i) = 0. We also investigate a Banach algebra structure for the subspaces

Ei, i ≥ 0.

2000 Mathematics Subject Classification: 46B15; 47B37, 47B47Keywords: Basis, Basic sequence, Discrete Duhamel Product, Banach algebra, Weighted Shift Operator.

91



012

About A New Class of Bifurcations Generated by Piecewise Linear Maps

I. Boukemara, I. Djellit and M. Ferchichi

University of Annaba, Faculty of Sciences, Department of Mathematics, B.P.12- 23000Annaba-ALGERIA

E-mail:

Abstract The purpose of this paper is to study the occurrence of a new bifurcation phenomena

for piecewise smooth maps. These phenomena are part of a rich new class of bifurcations which we call

border-collision bifurcations. Border collision bifurcations have been defined for continuous piecewise

smooth maps depending on parameters [14]. In the simplest case of one dimensional maps, border

collision bifurcations occur, as a parameter is varied, when a fixed or periodic point of the map collides

with the set of points (called border) where the map is not differentiable. We are interested in a family of

two-dimensional piecewise map T : R2 −→ R2 depending on three parameters, given by two maps T1T2

de fined in the regions R1, R2 , respectively:

T : (x, y) 7→

T1(x, y), if (x, y) ∈ R1

T2(x, y), if (x, y) ∈ R2.

where

T1 :

(xy

)7→

(1− ax− by

x

), R1 = (x, y) ∈ R2x ≥ 0,

T2 :

(xy

)7→

(1 + ax− by + c

x

), R2 = (x, y) ∈ R2x < 0,

defined by linear functions, as we recall, a, b, and c are real parameters. We examine the specificbifurcation phenomena that result from the piecewise-linear structure of this map. We show howthe model displays abrupt period-doubling bifurcations and a variety of different border-collisionbifurcations. This results are illustrated by a numerical experiment.

References

[1] Zh. T. Zhusubaliyev, E. A.Soukhoter in , E. Mosekilde,Border-collision bi-furcations on atwo-dimensional toru s. Chaos, Solitons and Fractals 13 (2002), 1889-1915.

[2] J. Laugesen, E. Mosekilde,Border-collision bifurcations in a dynamic man-agement game.,Computers Operations Research 33 (2006), 464478

2000 Mathematics Subject Classification:Keywords: Piecewise linear maps, attractor, border collision bifurcations, Chaos.

92


[3] F. Dercole, S. Maggi,Detection and continuation of a border collision bifur-cation in a forestfire model, Applied Mathematics and Computation 168 (2005) 623635.

[4] H. E. Nusse, E. Ott, J. A. York,Bor der-collision bifurcations: An expla-nation for observedbifurcation phenomena. Physical review E, volume 49, number 2, (1994).

93



012

Nonlinear Fractional Differential Inclusions in Banach Spaces

Djamila Seba

Departement de Mathematiques, Universite M’hamed Bougara de Boumerdes,35000, Boumerdes, Algerie


Mouffak Benchohra

Laboratoire de Mathematiques, Universite de Sidi Bel-Abbes,B.P. 89, 22000, Sidi Bel-Abbes, Algerie


Abstract In recent years, fractional Calculus have been addressed by several researchers. Fractionalderivatives provide an excellent tool for the description of memory and hereditary properties of variousmaterials and processes. These characteristics of the fractional derivatives make the fractional ordermodels more realistic and practical than the classical integer-order models. As a matter of fact, fractionalCalculus arise in many engineering and scientific disciplines such as physics, chemistry, biology, economics,control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, and fittingof experimental data.In this work we investigate the existence of solutions for an initial value problem (IVP for short) for thenonlinear fractional differential inclusion of the form

cDry(t) ∈ F (t, y(t),c Dr−1y(t)), for a.e. t ∈ J := [0, T ], 1 < r < 2, (0.1)

y(0) = y0, y′(0) = y1, (0.2)

where cDr is the Caputo fractional derivative, F : J × E × E → P(E) is a given multivalued mapsatisfying some assumptions that will be specified later, y0, y1 ∈ E and E is a Banach space with norm|.|.We devote our attention here, to prove that the initial value problem for the nonlinear fractional dif-

ferential inclusion (0.1)–(0.2) has at least one solution by applying a fixed point theorem due to Monch

for multivalued operators, combined with the concept of measures of noncompactness. This concept has

proved to be a very useful tool for seeking solutions for fractional differential equations in Banach spaces.


94



012

Exact number of positive solutions for quasilinear boundary value problemswith p−convex nonlinearities

Mohammed Derhab

Department of Mathematics, University Abou-Bekr Belkaid Tlemcen B.P. 119, Tlemcen,13000, Algeria


Hafidha Sebbagh

Department of Mathematics, University Abou-Bekr Belkaid Tlemcen B.P. 119, Tlemcen,13000, Algeria


Abstract By using the quadrature method, we study the exact number of positive solutions of thefollowing quasilinear boundary value problem:

−(φp(u′))′ = λf(u) in (0, 1),

u(0) = u(1) = 0,

where p > 1, φp(y) = |y|p−2y, (φp(u′))′ is the one dimensional p−Laplacian and f is a p−convex function

and λ is a positive real parameter.

2000 Mathematics Subject Classification: 34B15, 34C10Keywords: p−Laplacian; positive solutions; quadrature method; p−convex function

95



012

MacWilliams Identity for Codes Over Forests

Seda Akbiyik

Department of Mathematics, Faculty of Science,Yildiz Technical University, Istanbul, Turkey


Irfan Siap



Abstract In this work, the MacWilliams Identity is established for binary linear codes over forests

which are a special family of posets. With this approach, this establishment avoids the use of the dual

poset. Some examples are also provoided.

2000 Mathematics Subject Classification:Keywords: Poset, P-code, P-weight, Poset metric, dual poset, MacWilliams Identity, weight enumerator,trees,forests.

96



012

Convergence of Wavelet Expansions

N. A. Sheikh

Department of Mathematics National Institute of Technology Hazratbal SrinagarKashmir-190006 India


Abstract Frame analysis has been a popular subject for over ten years. Frames were introducedby Duffin and Schaeffer in the context of non- harmonic Fourier series. A sequence xn in a Hilbertspace H is a frame if there exist two numbers A,B > 0 such that for all x ∈ H we have

A ∥ x ∥2≤∑n

| < x, xn > |2 ≤ B ∥ x ∥2 .

The numbers A,B are called the frame bounds.In recent years, frame theory attracted attention of many mathematicians to study theoretical and ap-plication aspects. Wavelets and frames in Sobolev spaces are motivated by applications of wavelets innumerical algorithms and image processing, since the solution spaces of many partial differential equa-tions and the classes of images are now modelled by various sobolev spaces. Quite, often, wavelets areconstructed in L2(Rd) and their norm equivalence is established for other Sobolev spaces.Expansions in a frame series have properties similar in many respects to those of expansions in orthogonalsystem, and they are used in both theoretical studies and applications for signal and image analysis, datacompression and pattern recognition.In this paper, we show convergence, pointwise convergence and uniform convergence of the wavelet seriesL∑

l=1

∑j∈Z

∑k∈Zd

< f, ψlj,k > ψl

j,k andL∑

l=1

∑j∈Z

∑k∈Zd

< f, ψlj,k > ψl

j,k for any f∈ L2(Rd)

Further, the convergence investigation is also made using oblique extension principle on Sobolev spacesHs(Rd) and H−s(Rd). The Parseval frame is obtained for Hs(Rd) and H−s(Rd). Here Hs(Rd) is aHilbert space under the inner product

< f, g >Hs(Rd)=1

(2π)d

∫Rd

f(ψ)g(ψ)T(1+ ∥ ψ ∥2)sdψ, f, g ∈ Hs(Rd).

Moreover, for each g ∈ H−s(Rd),

< f, g >=1

(2π)d

∫Rd

f(ψ)g(ψ)Tdψ, f ∈ Hs(Rd)

defines a functional value on Hs(Rd).

2000 Mathematics Subject Classification: 42 C15 42C40Keywords: Sobolev spaces, Parseval frame, Oblique extension principle

97



012

On Scaling Ill Conditionned Matrices

Omar Slimani

Department of Math. University Badji Mokhtar Annaba, AlgeriaE-mail:

Abstract One challenge when we compute eigenvalues of a matrix or solve a linear system A.x = b

and so on, is when the matrix A is ill conditioned (ie : κ2(A) = ∥A∥2 ·∥∥A−1

∥∥2) is very big. Many technics

to scale A exist; one of the most popular is the method of W.H. Wilkinson which consist to render the

entries of A between −1 and +1. The idea which ispresented here consist to bring the problem of scalling

A to a non linear programming problem and then use the tools of optimization to solve it.


98



012

On a Combinatorial Laplacian and Homology of Compact Manifolds

Yasar Sozen

Fatih University, Buyukcekmece, 34500 Istanbul / TurkeyE-mail: [email protected]

Abstract This article considers a fixed closed oriented smooth 2m−manifold (m odd) M, and dual

cell-decompositions K, K ′ of M. It proves that C∗(K;R)⊕C∗(K′;R) is a symplectic chain complex, also

presents the corresponding Laplacian and proves the properties.

References

[1] P. Griffiths, J. Harris, Principles of Algebraic Geometry, John Willey Library Edition(1994).

[2] C. Ozel, Y. Sozen, Reidemeister torsion of product manifolds and its applications to quantumentanglement, Balkan J. Geom. Appl. Vol.17 No.2 (2012) 66–76.

[3] Y. Sozen, Reidemeister torsion of a symplectic complex, Osaka J. Of Math., Osaka J. Math.Vol. 45 No. 1 (2008) 1–39.

[4] Y. Sozen, On Fubini-Study form and Reidemeister torsion,, Topology and Appl. Vol. 156No. (2009) 951–955.

[5] Y. Sozen, Reidemeister torsion and period matrix of Riemann surfaces, Math. Slovaca, Vol.61, No. 61, Feb. 2011, 29–38

[6] Y. Sozen, A note on symplectic chain complex, AIP Conf. Proc. 1309, ICMS InternationalConference on Mathematical Science, 2010, Vol. 1309, 870–883.

[7] Y. Sozen, Symplectic chain complex and Reidemeister torsion of compact manifolds, Math.Scand., accepted.

[8] Y. Sozen, On a volume element of Hitchin component, Fundamenta Mathematicae, ac-cepted.

[9] E. Witten, On quantum gauge theories in two dimension, Commun. in Math.Phys. 141(1991), 153-209.

2000 Mathematics Subject Classification: 57N99; 55U15; 18G35Keywords: Symplectic chain complex, Combinatorial Laplacian, Homology, Compact manifold

99



012

SOME FURTHER REMARKS ON IDEAL SUMMABILITY IN2-NORMED SPACES

PRATULANANDA DAS, SUDIP KUMAR PAL AND SANJOY KR GHOSHAL

E-mail:

Abstract Very recently ideals were used to study summability of sequences in 2-normed spaces

by Gurdal et al ([2], [3], [6]) who investigated the convergence and Cauchy condition ( namely I and

I*-convergence and Cauchy conditions ). In this paper we make some further investigations in this line

which provides answers to two important questions regarding I and I*-Cauchy sequences which were left

unanswered. We then introduce new concepts of I and I*-divergence in 2-normed spaces and study their

certain properties.

References

[1] K.Dems, On I-Cauchy sequences, Real Anal. Exchange, 30 (1) (2004-2005), 123-128.

[2] M.Gurdal and I.Acik, On I-Cauchy sequences in 2-normed spaces, Math. Ineq. Appl., 11(2) (2008), 349-354.

[3] M.Gurdal and S.Pehlivan, The statistical convergence in 2-Banach spaces, Thai J.Math.,2(1)(2004), 107-113.

[4] J.A.Fridy, On statistical convergence, Analysis, 5 (1985), 301-313.

[5] P.Kostyrko, T.Salat, W.Wilczynki, I-convergence, Real Anal. Exchange, 26 (2) (2000/2001),669-685.

[6] P.Kostyroko, M.Macaj and M.Sleziak, I-convergence and extremal ilimit points, Math. Slo-vaca,55 (2005), 443-464.

[7] A.Sahiner, M.Gurdal, S.Saltan and H.Gunawan, Ideal convergence in 2-normed spaces.Taiwanese J. Math., 11 (5) (2007), 1477-1484.

[8] T.Salat, On Statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980),139-150.

2000 Mathematics Subject Classification: 40A05, 46A70, 40A99, 46A99Keywords: 2-normed spaces, I-Cauchy, I*-Cauchy, I-divergence, I*-divergence, condition (AP)

100



012

Existence of multiple positive solutions for a nonlocal boundary valueproblem with sign changing nonlinearities

Yasmina Tabet Zatla And Naima Merzagui

Department of Mathematics,University of Abou Bekr Belkaid Tlemcen, AlgeriaE-mail: [email protected]

Abstract In this paper, we study the existence of multiple positive solutions of a boundary value

problem

··x (t) + q(t)f(x(t)) = 0 0 < t < 1,

x(0) = 0, x(1) =∫ βα x(s)dg(s),

where the nonlinear terme f is allowed to change sign.We impose growth conditions on f whichyield the existence of at least two positive solutions by using a fxed point theorem in doublecones.

References

[1] A. V. Bitsadze, On the the ory of nonlocal boundary value problems, Soviet Math. Dock,30 (1964), 8-10

[2] Z. Chen and F. Xu, Multiple p ositive solutions for nonlinear second order m-point bound-ary value problems with sign changing nonlinearities, Electronic Journal of DifferentialEquations, Vol. 2008(2008), 1-12.

[3] G.A. Anastassiou, Ostrowski type inequalities, Proc. AMS 123 (1995), 3775-3781.

[4] W. G . Ge and J. L. Ren, Fixed p oint theorems in double cones and their applicatiosto nonlinear boundary value problems, Chinese Annals of Mathematics 27(A)(2006), 155-168(in Chinese).

[5] L. Karakostas and P. Tsamatos, Existence of multiple positive solutions for a nonlocalboundary value problem, Juliusz Schauder Center, 19(2002), 109-121.

2000 Mathematics Subject Classification:Keywords: Nonlocal boundary value problems, multiple positive solutions, fxed point theorem in double cones.

101



012

Necessary and sufficient Tauberian conditions for the Ar method ofsummability

Ozer Talo

Department of Mathematics, Celal Bayar University,Muradiye Yagcılar Campus, 45140 - Manisa , Turkey


Feyzi Basar



Abstract Let 0 < r < 1. Then the class Ar = (arnk) of Toeplitz matrices, introduced by Basar in

[Fırat Univ. Fen & Muh. Bil. Dergisi 5(1)(1993), 113–117], is given by

arnk =

1+rk

n+1 , 0 ≤ k ≤ n,

0 , k > n,

for all k, n ∈ N. We should note here that a number of papers were published on the sequence spacesdefined by the domain of the Ar matrices in some normed and paranormed sequence spaces by theresearchers.Moricz and Rhoades determined the necessary and sufficient Tauberian conditions for certain weigh-ted mean methods of summability in [Acta. Math. Hungar. 102(4) (2004), 279–285]. In the presentpaper, we deal with the necessary and sufficient Tauberian conditions for the Ar method by the followingtheorems:

Theorem 0.1. Let (xk) be a sequence of real numbers which is summable Ar to a finite limit l. Then

limn→∞

xn = l

if and only if the following two conditions are satisfied:

supλ>1

lim infn→∞

1

λn − n

λn∑k=n+1

[(1 + rk)xk − xn

]≥ 0

and

sup0<λ<1

lim infn→∞

1

n− λn

n∑k=λn+1

[xn − (1 + rk)xk

]≥ 0.

2000 Mathematics Subject Classification: 40E05, 40G05.Keywords: Summability by Ar methods, one-sided and two-sided Tauberian conditions, slowly oscillating sequences.

102


Theorem 0.2. Let (xk) ∈ ω be summable Ar to a finite limit. Then (xk) converges to the same limit ifand only if one of the following two conditions is satisfied

infλ>1

lim supn→∞

∣∣∣∣∣ 1

λn − n

λn∑k=n+1

[(1 + rk)xk − xn

]∣∣∣∣∣ = 0

or

inf0<λ<1

lim supn→∞

∣∣∣∣∣ 1

n− λn

n∑k=λn+1

[xn − (1 + rk)xk

]∣∣∣∣∣ = 0.

103



012

On the new sequence spaces including the spaces of absolutely p−summableand bounded sequences

Sebiha TekinThe Graduate School of Sciences and Engineering, Fatih University, Hadımkoy Campus,

Buyukcekmece, 34500 - Istanbul, TurkeyE-mail: [email protected]

Feyzi BasarDepartment of Mathematics, Fatih University, Hadımkoy Campus,


Abstract Let 0 ≤ r < ∞ and ω denotes the space of all complex valued sequences. Define thesubsets ℓrp and ℓr∞ of the space ω by

ℓrp :=

x = (xk) ∈ ω :

∞∑k=1

∣∣∣xk

kr

∣∣∣p < ∞

, (0 < p < ∞),

ℓr∞ :=

x = (xk) ∈ ω : sup

k∈N

∣∣∣xk

kr

∣∣∣ < ∞,

where N denotes the set of positive integers.The main results of this study are, as follows:

1) The sets ℓrp and ℓr∞ form a linear space with respect to the coordinate-wise addition and scalarmultiplication of sequences.

2) (ℓrp, drp) and (ℓr∞, dr∞) are the complete metric spaces, where

drp(x, y) =

( ∞∑k=1

∣∣∣∣xk − ykkr

∣∣∣∣p)1/p

; x = (xk), y = (yk) ∈ ℓrp, (p ≥ 1),

dr∞(x, y) = supk∈N

∣∣∣∣xk − ykkr

∣∣∣∣ ; x = (xk), y = (yk) ∈ ℓr∞.

3) (i) Let p ≥ 1. Then, (ℓrp, ∥ · ∥rp) is a Banach space, where

∥x∥rp =

( ∞∑k=1

∣∣∣xk

kr

∣∣∣p)1/p

; x = (xk) ∈ ℓrp.

(ii) Let 0 < p < 1. Then, (ℓrp, ∥ · ∥rp′) is a complete p−normed space, where

∥x∥rp′ =

∞∑k=1

∣∣∣xk

kr

∣∣∣p ; x = (xk) ∈ ℓrp.

2000 Mathematics Subject Classification: Primary 46A45; Secondary 40C05.Keywords: Matrix domain of a sequence space, α−, β− and γ−duals and matrix transformations.

104


4) (ℓr∞, ∥ · ∥r∞) is a Banach space, where

∥x∥r∞ = supk∈N

∣∣∣xk

kr

∣∣∣ ; x = (xk) ∈ ℓr∞.

5) The inclusion relations ℓp ⊂ ℓrp with r > 1 and ℓ∞ ⊂ ℓr∞ and ℓ∞ ⊂ ℓrp strictly hold.

6) The α−, β− and γ−duals of the spaces ℓrp and ℓr∞ are determined.

7) The classes (ℓrp : ℓ∞), (ℓrp : c), (ℓ∞ : ℓr∞) and (ℓ∞ : ℓrp) of infinite matrices are characterized.

105



012

On the solutions of some fractional systems of difference equations

Nouressadat Touafek

LMAM Laboratory, Mathematics Department, Jijel University, AlgeriaE-mail: [email protected]

Abstract Studying properties of systems of rational difference equations allow to understand thebehavior of some real life phenomena in biology, control theory, economics, physics, sociology, etc, whichare modelled by such systems. Although systems of difference equations are very simple in form, it isextremely difficult to understand thoroughly the behaviors of their solutions. Recently many papers aredevoted to this subject, see for example [1-4] and references cited therein.In this paper we deal with the solutions of some systems of difference equations on a rational forms. We getthe form of the solutions and in order to illustrate our results and to support our theoretical discussions,we consider several interesting numerical examples which represent different types of qualitative behaviorof the solutions.

References

[1] B. D. Iricanin, N. Touafek, On a second-order max-type system of difference equations, accepted.

[2] A. Y. Ozban, On the positive solutions of the system of rational difference equations, xn+1 =1/yn−k, yn+1 = yn/xn−myn−m−k, J. Math. Anal. Appl., 323, (2006), p. 26-32.

[3] N. Touafek, E. M. Elsayed, On the solutions of systems of rational difference equations, Math.Comput. Modelling., 55, (2012), p. 1987-1997.

[4] I. Yalcınkaya, On the global asymptotic behavior of a system of two nonlinear difference equations,Ars Comb., 95, (2010), p. 151-159.

2000 Mathematics Subject Classification: Primary 39A10, Secondary 40A05.Keywords: Behavior of solutions, periodic solutions, system of difference equations.

106



012

A Theoretical and Numerical Performance of Central Trajectory Methodsfor Semidefinite Programming

Touil ImeneUniversite de Jijel

Laboratoire de mathematiques appliqueesE-mail: i [email protected]

Benterki DjamelUniversite De Setif

E-mail: dj [email protected]

Abstract In this work, we present a feasible primal algorithm for linear semidefinite programmingdefinite by

(SDP )

min[⟨C,X⟩ = tr(CX) =n∑

i,j=1

CijXij ]

AX = b ,X ∈ Sn+.

Where b ∈ Rm, Sn+ designates the cone of the positive semidefinite matrix on the linear space

of (n × n) symmetrical matrix Sn. A is a linear operator of Sn in Rm defined byAX =(⟨A1, X⟩, ⟨A2, X⟩, ..., ⟨Am, X⟩)t . The matrices C and (Ai)i=1,m are in Sn. Sn

++, designate the set ofthe positive definite matrices of Sn. The problem (SDP) based on the direction of Alizadeh, Haeberlyand Overton (AHO). To study (SDP ), we replace it by the perturbed equivalent problem

(SDP )µ

min[fµ (X) = ⟨C,X⟩+ µg(X) + nµ lnµ] , µ > 0AX = b,

g(X) =

− ln (detX) if X ∈ Sn

++

+∞ otherwise.

We establish the existence and uniqueness of the optimal solution of (SDP)µ and its convergence to the

optimal solution of (SDP). We present some numerical simulations which show the effectiveness.

References

[1] D. Benterki, J. P. Crouzix and B. Merikhi, A numerical feasible interior point method forlinear semidefinite programs, RAIRO- Operations Research, 41(1) (2007) 49-59.

[2] J. P. Crouzix and B. Merikhi, A logarithm barrier method for semidefinite programming,RAIRO - Operations Research 42 (2) (2008)123-139.

[3] R. D. C. Monteiro and P. R. Zanjacomo, A note on the existence of Alizadeh-Haeberly-Overton direction for semidefinite programming, Mathematical Programming 78 (1997),393-396.

2000 Mathematics Subject Classification: 90C22, 90C51Keywords: Linear Semidefinite Programming, Primal-Dual Interior Point Methods, Central Trajectory Methods

107



012

Generalized nonlinear Schrodinger equation with soliton solutions

Houria Triki

Laboratoire de Physique des Rayonnements, Departement de Physique,Universite Badji Mokhtar, BP 12, Annaba 23000, Algerie


Abstract The propagation of soliton pulses in an homogeneous optical fiber that is described

by the nonlinear Schrodinger equation including higher-order nonlinear and dispersion effects is studied.

By applying the solitary wave ansatz method, we derive various forms of exact soliton solutions for the

considered model. All the physical parameters in the solitary wave solutions are obtained as functions of

the dependent model coefficients. Such solutions may be useful to explain the dynamics of wave propa-

gation in nonlinear optical fiber systems with non-Kerr terms. Parametric conditions for the formation

of soliton pulses are determined.

2000 Mathematics Subject Classification:Keywords: Nonlinear Schrodinger equation, soliton solution, Solitary wave ansatz, optical fiber.

108



012

Some problems related to the average rank partition lattice a set

Yahi Zahra

LAID3, Faculty of Mathematics, USTHB, Bab Ezzouar, Algiers, AlgeriaE-mail: [email protected]

Bouroubi Sadek

Universite Bjaia, Faculty of Economic, Management and Trade Sciences, Algiers, AlgeriaE-mail: [email protected]

Abstract The problems related to the lattice of partitions of a set are several. Some problemsrelated to the maximum size of an antichain, the minimum size of a cut set and the processing imagetheory which is one of preoccupation of computer scientists.

Our study focus on some parameters related directly to the lattice of partitions set witch was the study

objectives of more researchers, K. Engel, R. Canfiled and Sadek Bouroubi. One of this is the Bell numbers

representing the numbers of all partitions of set, which is a sum of Sterling numbers of the second space

that we show an important property using a random variable.We have also studied another important

parameter called the average number of blocks in this lattice which is a quotient of two Bell numbers less

then one. We will present some numerical results using Maple software related to this parameter.

References

[1] S. Bouroubi, Bell Numbers and Engel’s conjecture, Rostock. Math. Kolloq., 62, 61-70(2007).

[2] E. R. Canfield and H. Harper, A large antichains in the partition lattice, Random structuresAlgorithms 6, 89-104, 1995. Numer.

[3] K. Engel, On the average rank of an element in a filter of the partition lattice , J Com-bin.Theory Ser, A 64, 67-78, 1994.

2000 Mathematics Subject Classification:Keywords: partition, lattice, average, optimality

109



012

On the Berezin symbols method,Abel convergence and related questions

Mubariz T. Karaev

University of Suleyman Demirel, Isparta Vocational School32260, Isparta, Turkey.


Mehmet Gurdal and Ulas Yamancı

Department of Mathematics, University of Suleyman Demirel32260, Isparta, Turkey.

E-mail: [email protected], ulas 19 @hotmail.com

Abstract We investigate some problems related with Berezin symbols of operators on Hardy and

Bergman spaces and their applications in summability theory and in solution of Beurling problem.We

also discuss in terms of Berezin symbols the solutions of the Beurling problem.

2000 Mathematics Subject Classification: 47B35.Keywords: Reproducing kernel, Berezin symbol, Toeplitz operator, Hardy space, Bergman space point.This work is supported by the Scientific and Technological Research Council of Turkey (TUBITAK) with Project109T590. Also, this work is supported by Suleyman Demirel University with Project 3115-YL-12.

110



012

Composited dual summability methods of the new sort

Medine YesilkayagilDepartment of Mathematics, Usak University,

1 Eylul Campus, 64200 - Us.ak, TurkeyE-mail: medine.yesilkayagil

Feyzi BasarDepartment of Mathematics, Fatih University, Hadımkoy Campus,


Abstract Following Altay and Basar [Some paranormed Riesz sequence spaces of non-absolutetype, Southeast Asian Bull. Math. 30(5)(2006), 591–608], we define the duality relation between a pairof infinite matrices. Our focus is the dual summability methods of the new sort. Let us suppose thatthe infinite matrices A = (ank) and B = (bnk) transform the sequences x = (xk) and y = (yk) which areconnected with the relation y = Rtx to the sequences u = (un) and v = (vn), respectively, i.e.,

un =(Ax

)n=

∑k

ankxk for each n ∈ N, (0.1)

vn =(By

)n=

∑k

bnkyk for each n ∈ N. (0.2)

It is clear here that the method B is applied to the Rt-transform of the sequence x = (xk) while themethod A is directly applied to the terms of the sequence x = (xk). We shall say in this situation thatthe methods A and B in (0.1), (0.2) are the dual of the new sort if un becomes vn (or vn becomes un)under the application of the formal summation by parts. This statement is equivalent to the relation

ank =

∞∑j=k

tkTj

bnj or bnk = ∆

(anktk

)Tk for all k, n ∈ N.

Suppose that C = (cnk) is a strongly regular triangle matrix. Define the matrices D = (dnk) andE = (enk) by the usual matrix product as D = CA and E = CB which are called as composited matriceswhile the matrices A and B are called as original matrices.Our main results are:

Theorem 0.1. The composited matrices are dual of the new sort if and only if the original matrices aredual of the new sort.

Theorem 0.2. Every A limitable sequence is limitable D. However, the converse of this fact does nothold, in general.

Theorem 0.3. Every B limitable sequence is limitable E. However, the converse of this fact does nothold, in general.

Theorem 0.4. The duality relation of the new sort isn’t preserved under the usual inverse operation.

2000 Mathematics Subject Classification: 40C05.Keywords: Dual summability methods, generalized dual summability methods, Riesz means, almost convergence.

111



012

A Note on the Modified Crank-Nicholson Difference Schemes for UltraParabolic Equations with the Neumann Condition

Allaberen Ashyralyev, Serhat Yilmaz

Department of Mathematics, Fatih University, Istanbul, TurkeyE-mail: [email protected]

Abstract In this paper, we focus on studying the stability of second order difference scheme forthe approximate solution of the initial boundary value problem for ultra parabolic equations

∂u(t,s)∂t + ∂u(t,s)

∂s +Au(t, s) = f(t, s), 0 < t, s < T,u(0, s) = ψ(s), 0 ≤ s ≤ T,u(t, 0) = φ(t), 0 ≤ t ≤ T

in an arbitrary Banach space E with a strongly positive operator A. For approximately solving thisproblem, r-modified Crank-Nicolson difference schemes of the second-order of accuracy

uk,m−uk−1,m−1

τ +Auk,m = fk,m, 1 ≤ k, m ≤ ruk,m−uk−1,m−1

τ + A2 (uk,m + uk−1,m−1) = fk,m, r + 1 ≤ k, m ≤ N,

fk,m = f(tk, sm), tk = kτ, sm = mτ, 1 ≤ k, m ≤ N, Nτ = 1,u0,m = ψm, ψm = ψ(sm), 0 ≤ m ≤ N,uk,0 = φk, φk = φ(tk), 0 ≤ k ≤ N

are presented. The stability estimates for the solution of these difference schemes is established. In

applications, the stability in maximum norm of difference shemes for multidimensional ultra parabolic

equations with Neu- mann condition is established. Applying the difference schemes, the numerical

methods are proposed for solving one dimensional ultra parabolic equations.


112



012

Existence and Solutions Set for ϕ-Laplacian Impulsive Differential Equations

Samia Youcefi and Abdelghani Ouahab

Laboratory of Mathematics, Sidi-Bel-Abbes University PoBox 89, 22000 Sidi-Bel-Abbes, AlgeriaE-mail: [email protected]; [email protected]

Johnny Henderson

Department of Mathematics, Baylor University Waco, Texas 76798-7328 USAE-mail: [email protected]

Abstract In this paper, we present a couple of results on existence and the topological structureof the solutions set for initial value problems for the following first-order impulsive differential equation,

(ϕ(y′))′ = f(t, y(t)), a.e. t ∈ [0, b],y(t+k )− y(t−k ) = Ik(y(t

−k )), k = 1, . . . , m,

y′(t+k )− y′(t−k ) = Ik(y′(t−k )), k = 1, . . . , m,

y(0) = A, y′(0) = B,

where f : [0, b]× R → R is a given function, 0 = t0 < t1 < . . . < tm < tm+1 = b, m ∈ N. The functionsIk, Ik ∈ C(R, R) characterize the jump in the solutions at impulse points tk, k = 1, . . . , m., ϕ: R → Ris a suitable monotone homeomorphism, and A, B ∈ R. For this setting, the proofs of the two resultspresented, while involving some cases, are quite straight for word.

For the final result of the paper, the hypotheses are modified so that the nonlinearity f depends on y′.

For this latter case the impulsive conditions and initial conditions remain the same. Because of the de-

pendency on y′, the proof of the result presented is somewhat more involved. Of course, the second case

also covers the first case when f is independent of y′.

2000 Mathematics Subject Classification: 34A37, 34K45.Keywords: ϕ-Laplacian, fixed point theorems, impulsive solution set, com- pactness.

113



012

The Navier-Stokes problem in velocity-pressure formulation:convergence andOptimal Control

A. Younes

Faculte des Sciences de Tunis, Tunisie.E-mail: [email protected]

A. Jarray


Y.Ouali


Abstract In this paper, we study the nonlinear Navier-Stokes problem in velocity-pressure formu-

lation. We construct a sequence of a Newton-linearized problems and we show that the sequence of weak

solutions converges towards the solution of the nonlinear one in a quadratic way. A control problem on

the homogeneous problem is considered.

References

[1] V.Girault, P-A Raviart . An analysis of upwind schemes for the Navier-Stokes equations.SIAM J. Numer.Anal. Vol. 19 n 2. po.312-333.(1982).

[2] E. Hopf On Non-Linear Partial Differential Equations. Lecture Series of the Symp. onpartial Diff. Equations Berkeley. (1955).

[3] A. Younes, S.Abidi: ”The Dirichlet Navier-Stokes Problem in the Velocity-Pressure For-mulation.” International Journal of Applied Mathematics Volume 24 No. 3 2011, 469-477.

[4] Reviart Ecland ”Formulation variationnelle et optimisation”, book.

[5] J.C.De Los Reyes and R. Griesse ”state-constrained optimal control of the stationaryNavier-Stokes equations ”J. Math. Anal. Appl. 343 (2008) 257-272.

2000 Mathematics Subject Classification: 35J20 ,49J96.Keywords: Navier-Stokes equations ,Newton’s algorithm , Approximation Convergence , Control.

114



012

Space of continuous and bounded functions over the field of non-NewtonianComplex numbers

Zafer Cakir

Department of Mathematical Engineering, Gumushane University,29100 - Gumushane, Turkey


Abstract As an alternative to the classical calculus, Grossman and Katz introduced the non-Newtonian calculus in [1] consisting of the branches of geometric, anageometric and bigeometric calculus.Bashirov et al. [2] have recently emphasized on the non-Newtonian calculus and gave the results withapplications corresponding to the well-known properties of derivative and integral in the classical calculus.Recently, Uzer [3] has extended the multiplicative calculus to the complex valued functions and inter-ested in the statements of some fundamental theorems and concepts of multiplicative complex calculus,and demonstrated some analogies between the multiplicative complex calculus and classical calculus bytheoretical and numerical examples.

In this study, using the field C(G) of non-Newtonian complex numbers, we respectively define the sets

B(A) non-Newtonian bounded functions space and C[a, b] non-Newtonian continuous functions space.

Later we investigate some properties of these function spaces and give the obtained results in theorems.

References

[1] M. Grossman, R. Katz, Non-Newtonian Calculus, Lowell Technological Institute, 1972.

[2] A.E. Bashirov, E.M. Kurpınar, A. Ozyapıcı, Multiplicative calculus and its applications, J.Math. Anal. Appl. 337(2008), 36–48.

[3] A. Uzer, Multiplicative type complex calculus as an alternative to the classical calculus,Comput. Math. Appl. 60(2010), 2725–2737.

2000 Mathematics Subject Classification: Primary 32C15, 32M25; Secondary 08A05, 30H99.Keywords: Algebraic structures with respect to non-Newtonian calculus, non-Newtonian bounded functions space,non-Newtonian continuous functions space.

115



012

Optimizing a Nonlinear Function over the Integer Efficient Set

Ouiza Zerdani

University of Tizi-Ouzou, Laboratory LAROMAD, Faculty of sciences, Tizi-Ouzou, AlgeriaE-mail: [email protected]

Mustapha Moulai

University of USTHB, Laboratory LAID3, Faculty of mathematics, AlgeriaE-mail: mustapha [email protected]

Abstract The problem of optimizing a real valued function over the efficient set of a multiple

objective linear program has some applications in multiple objective decisions making. This problem

of optimizing over the efficient set can be classified as a hard global optimization problem. The main

difficulty of this problem arises from the fact that its feasible domain, in general, is non-convex and not

given explicitly as a constrained set of an ordinary mathematical programming problem. In this work

an algorithm is developed that optimizes an arbitrary nonlinear function over an integer efficient set of

a vector affine fractional program without explicitly having to enumerate all the efficient solutions. The

proposed method is based on a cutting plane technique and on a simple selection technique that improves

the objective value at each iteration. A numerical illustration is included to explain the proposed method.

2000 Mathematics Subject Classification: 90C10, 90C20, 90C26, 90C29, 90C32Keywords: Integer programming; Optimization over the efficient set; Multiple objective linear fractional program-ming; Level sets; Global optimization

116



012

Multiplicative method for multi criteria analysis

Malisa Zizovic

University Singidunum, Danijelova 32, 11000 Belgrade, SerbiaE-mail: [email protected]

Abstract In this paper one new method for multi criteria analysis based on multiplicative evalu-

ations of alternatives by criteria is given.

References

[1] C. L. Hwang, K. Yoon, Multiple attribute decision making methods and applications,Spriner-Verlag, 1981.

[2] M. Radojicic, M. Zizovic, Applications of methods of multi criteria analysis in buissnisdecision making, Technical faculty in Cacak, Serbia 1998( Monograph in serbian).

[3] M. Zizovic, N. Damljanovic, V. Lazarevic, N. Deretic, New method for multicriteria analysis,U.P.B. Sci. Bull. Series A 73 (2) (2011) 13–22.


117



012

Piecewise Decomposition method for solving fractional differential equation

Abdelkader Bouhassoun

University of Oran Senia, Faculty of sciences,Mathematics department, B.P 1524, Oran 31000


Abstract In this work, a modification of the telescoping decomposition method applied to nonlinear

differential equations is presented. This method yields a series solution with accelerated convergence.

Some illustrative examples are given with a comparison withe the Adomian method.

References

[1] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal.Appl. 135 (1988) 501—544.

[2] K. Diethelm and J. F. Neville; Analysis of fractional differential equations, J. Math. Anal.Appl. 265 (2002), no. 2, 229–248.

[3] A. A. Kilbas and S. A. Marzan; Cauchy problem for differential equation with Caputoderivative. Fract. Calc. Appl. Anal. 7 (2004), no. 3, 297–321.

[4] S. Momani, Z. Odibat; Numerical comparison of methods for solving linear differentialequations of fractional order, Chaos Sol. Fract. 31 (5) (2007) 1248–1255.

[5] I. Podlubny; Fractional Differential equations, Mathematics in Science and Engineering,vol, 198, Academic Press, 1999.

[6] M. Al-Refai; Telescoping Decomposition Method for Solving First Order Nonlinear Dif-ferential Equations, Proceedings of the International Multi-Conference of Engineers andComputer Scientists 2008 Vol. II (IMECS 2008), 2008, Hong Kong.

[7] Z. Odibat, S. Momani and H. Xu; A reliable algorithm of homotopy analysis method for solv-ing nonlinear fractional differential equations, Applied Mathematical Modelling 34 (2010),593–600.


118



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Galerkin Method Applied for a Boussinesq Equation with NonlocalCondition



Nouri Boumaza

Department of Mathematics, Tebessa University, AlgeriaE-mail: [email protected]

Abstract This paper deals with the solvability and uniqueness of a higher dimention mixed non

local problem for a Boussinesq equation. The uniqueness and existence of a generalized solution is proved

with the help of an a priori estimate and the galerkin approximation method, respectively.

References

[1] G. Avalishvili, D. Gordiziani, On a class of spatial non-local problems for some hyperbolicequations, G. M. J. V7 (2000), N 3, 417–425.

[2] D. Bahuguna, S. Abbas, J. Dabas, Partial functional differential equation with an integralcondition and applications to population dynamics, Nonlinear Anal, TMA 69 (2008), 2623–2635.

[3] A. Beilin, On a Mixed nonlocal problem for a wave equation, Electron. J. Differential Equa-tions 103, (2006), 1–10.

[4] A. Bouziani, N. Benouar, Probleme mixte avec conditions integrales pour une equationshyperboliques, Bull. Belg. Math.Soc.3, (1996),137–145.

[5] J.R. Cannon, Y. Lin, A Galerkin procedure for diffusion equations subject to specificationof mass, SIAM J. Numer. Anal. 24 (1987) 499–515..

[6] A. Guezane-Lakoud, N. Boumaza, Galerkin method applied for a non local problem; IJA-MAS, 19, (2010), 72-89

[7] A. Guezane-Lakoud, Jaydev Dabas, and Dhirendra Bahuguna, “Existence and Uniquenessof Generalized Solutions to a Telegraph Equation with an Integral Boundary Condition viaGalerkin’s Method,” Int J M M S, vol. 2011, Article ID 451492, 14 pages, 2011.

2000 Mathematics Subject Classification:Keywords: Nonlocal condition, a Priori estimate, Galerkin’s method, Boussinesq equation.

119


[8] S. Mesloub, F. Mesloub, On the higher dimension Boussinesq equation with a non classicalcondition, Mathematical Methods in the Applied Sciences, Volume 34, Issue 5, pages 578–586, 30 March 2011

[9] L. S. Pulkina, Initial-Boundary Value Proble with Nonlocal Boundary Condition for aMultidimensional Hyperbolic Equation, Differential equations. (2008), Vol. 44, N 8, 1119-1125.

120



012

Domain of the double sequential band matrix in the classical sequence spaces

Murat Candan

Inonu University, Faculty of Arts Sciences, Department of Mathematics,Malatya-44280 /Turkey.

E-mail: [email protected], murat [email protected]

Abstract Let r = (rn)∞n=0 and s = (sn)

∞n=0 be given convergent sequences of positive real numbers.

Define the sequential generalized difference matrix B(r, s) = bnk(r, s) by

bnk(r, s) :=

rn , (k = n),sn , (k = n− 1),0 , (0 ≤ k < n− 1 or k > n),

for all k, n ∈ N, the set of natural numbers. Let λ denotes the any one of the classical spaces ℓ∞, c, c0and ℓp of bounded, convergent, null and absolutely p−summable sequences, respectively, and λ also be

the domain of the double sequential band matrix B(r, s) in the sequence space λ, where 1 ≤ p < ∞.

The present paper is devoted for studying on the sequence space λ. Furthermore, the β− and γ−duals

of the space λ are determined, and the Schauder bases for the spaces c, c0 and ℓp are given, and some

topological properties of the spaces c0, ℓ1 and ℓp are examined. Finally, the classes (λ1 : λ2) and (λ1 : λ2)

of infinite matrices are characterized, where λ1 ∈ ℓ∞, c, c0, ℓp, ℓ1 and λ2 ∈ ℓ∞, c, c0, ℓ1.

2000 Mathematics Subject Classification:Keywords: Matrix domain of a sequence space, β− and γ−duals, Schauder basis and matrix transformations.

121



012

Rothe-Galerkin’s method for a doubly nonlinear integrodifferential equations

A. Chaoui





Abstract In this paper we propose a new approximation scheme for solving doubly nonlinear

initial boundary value problem with Volterra operator. Existence, uniqueness of solution as well as some

regularity result are obtained via Rothe-Galerkin method.

References

[1] D. Bahuguna, Quasilinear integrodifferential equations in Banach spaces, Nonlinear analy-sis, Vol. 24, N.2(1995) 175-183.

[2] R. Eymard, M. Gutnic and D. Hilhorst, The finite volume method for an elliptic parabolicequation, Acta Math. Univ. Comenianae. vol. LXVII, 1(1998), pp. 181-195.

[3] Alt, H. W. and S. Luckhaus(1983). Quasilinear elliptic-parabolic differential equations.Mathematische Zeitschrift 183, 311-341.

[4] Brezis H., Analyse fonctionnelle, theorie et applications, Masson, Paris, 1983.

[5] A. Guezane-Lakoud, D. Belakroum, Rothe’s method for a telegraph equation with integralconditions, Nonlinear Anal. 70(2009) 3842-3853.

[6] A. Guezane-Lakoud, M. S. Jasmati, A. Chaoui, Rothe’s method for an integrodifferentialequation with integral conditions, Nonlinear Analysis. 72 (2010) 1522-1530. (2010) .

[7] J. Kacur and R. Van Keer, On the numerical solution of semilinear parabolic problems inmulticomponent structures with Volterra operators in the transmission conditions and inthe boundary conditions, Z. Angew. Math. Mech. 75(1995),no. 2,91-103.

2000 Mathematics Subject Classification: 34K20, 35k55, 35A35, 65M20Keywords: Rothe’s method, A priori estimate, Integrodifferential equation, Weak solution

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[8] Jager,W. and J. kacur(1995). Solution of doubly nonlinear and degenerate parabolic prob-lems by relaxation schems. Mathematical Modelling and numerical Analisis 29(5), 605-627.

[9] J. Kacur (1999). Solution to strongly nonlinear parabolic problems by a linear approxima-tion schema. IMA Journal of Numerical Analisis 19, 119-145.

[10] Mesloub, Nonlinear nonlocal mixed problem for a second order pseudoparabolic equation.Journal of Mathematical Analysis and ApplicationsVolume 316, Issue 1, 1 April 2006, Pages189-209.

[11] Showalter,R. E. monotone Operators in Banach Space and Nonlinear partial DifferentialEquations. Mathematical surveys and monographs.

[12] Marian Slodicka, An approximation scheme for a nonlinear degenerate parabolic equationwith a second-order differential Volterra operator, Journal of computational and appliedmathematics.168 (2004) 447-458.

[13] E. Rothe, Two-dimensional parabolic boundary value problems as a limiting case of one-dimensional boundary value problems. Math. Ann. 102(1930), 650-670.

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012

Production et Deploiment de logiciels distribuables pour Resolution deProblemes Min-Max en Controle Optimal

CHEBBAH Mohammed

Laboratoire LAROMAD Universite Tizi ouzouE-mail: [email protected]

OUANES Mohand

Laboratoire LAROMAD Universite Tizi ouzouE-mail:

Abstract Essentiellement les Methodes du Support et Adaptee Seront Implementees, leurs De-

ploiements a Travers un Logiciel type Visual Basic Executable nous permettera des Simulations au Cont-

role Optimal type discret et continu. Ces Methodes trouvent aussi leurs places lorsqu il sagit de resoudre

des problemes quadratiques ou des problemes non lineaires.

1 Introduction

Dans ce travail on va presenter un logiciel pour la resolution de programmes lineaires et par lasuite aux quadratiques avec des methodes.-Methode addaptee-Methode du support et autres. ;Avec contraintes bornees, simples et contraintes generalisees. Cela suppose 1‘implementation desMe- thodes de resolutions,avec une etudes comparative. De plus on s‘interessera aux problemesde Controle OptimalLes Problemes de depart : 1) Premier Probleme f(x) = min

k(c′kx+ αk) → max

x. d1 ≤ x ≤ d2

2 ) Deuxieme Probleme f(x) = mink

(c′kx+ αk) → maxx

. Ax = b, d1 ≤ x ≤ d2

La resolution de ces deux types de Problemes,avec la Methode Adaptee ou la Methode duSupportvont nous permettre de resoudre pratiquement tous les problemes Quadratiques et tous lesproblemes non lineaires moyennant des Algorithmes de corrections etablis pour cet effet.De plus Sont Importants pour la suite,seront utilises pour la Resolution des Problemes Min −

Max en Controle Optimal du Type par exemple:-le systeme dynamique:-dx(t)

dt= Ax(t) +

bu(t), x(O) = x0, T = [0;T1] Intervalle de Temps et par la suite les problemes Quadratiques.Nous allons faire cela pour 02 cas -CAS DISCERT (temps discret).-CAS CONTINU (temps continu).

2000 Mathematics Subject Classification:Keywords: Min-MAX Methodes Controle Optimal Discret continu Support Adaptee Deploiment Production Logicielsite http: //damoum.voila.net

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C ‘est a dire que en tout, nous allons resoudre et implementer (05) cinq problemes. Pourles Details de la theorie de Resolution ainsi que les Simulations Numeriques Issues de NotreRealisation Informatique ,on peut consulter le Site web http: //damoum.voila.netPresentation du Plan de Travail:Resolution et Implementation de la Methode adaptee et Support -Simulations et comparaisonavec Le Simplexe -Resolution et Implementation de la Methode adaptee et Support Min-Maxavec Contraintes bornees, Simples et Contraintes generalisees -Idem pour: le Probleme de Con-trole Optimal Min-Max avec la Methode adaptee et Support Cas Discret-Resolution et Imple-mentation du Probleme de Controle Optimal Min-Max avec la Methode adaptee et Support CasContinu-Production de Logiciels PerspectivesConclusion: Pour la suite:Methode du support pour Problemes Quadratiques convexes et nonconvexes -Methode duale du support pour Problemes Quadratiques convexes et non convexes -Resolution de problemes non lineaires convexes et non convexes vers une Globalisation.-Methodedu support pour Problemes Quadratiques en Controle Optimal-Methode duale du support pourProblemes Quadratiques en Controle Optimal.

References

[1] M. R. GABASSOV et F.M KIRILLOVA. . Me’thodes de Programmation Line’aire. EditionMinsk T2 1978.

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012

MCD-method of minimal suitable values

Malisa Zizovic

University Singidunum, Danijelova 32, 11000 Belgrade, SerbiaE-mail: [email protected]

Nada Damljanovic

University of Kragujevac, Technical faculty in Cacak, Cacak, SerbiaE-mail: [email protected]

Jasmina Janjic

University of Pristina, Faculty of Agriculture, Kopaonicka bb, 38228 Lesak, SerbiaE-mail: [email protected]

Abstract This paper deals with one of the classical multi criteria analysis problems: the alternatives

a1, a2, . . . , am to be ranked by criteria c1, c2, . . . , cn. Starting with given set of alternatives, we make

one finite partially ordered set of new weaker alternatives depended on given ones. In this set we include

a set of minimal suitable points P1, P2, . . . , Pk, and by weighted distances between alternatives aiand points Ps, we make the total order of alternatives by new method of multi criteria analysis.

References

[1] C. L. Hwang, K. Yoon, Multiple attribute decision making methodsand applications,Spriner-Verlag, 1981.

[2] M. Radojicic, M. Zizovic, Applications of methods of multi criteria analysis in buissnisdecision making, Technical faculty in Cacak, Serbia 1998( Monograph in serbian).

[3] M. Zizovic, N. Damljanovic, V. Lazarevic, N. Deretic, New method for multicriteria analysis,U.P.B. Sci. Bull. Series A 73 (2) (2011) 13-22.


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012

Some sequence and function spaces by using the partial metric

Ugur Kadak

Department of Mathematics, Gazi University, Teknikokullar,06500 - Ankara, Turkey


Hakan Efe

Department of Mathematics, Gazi University, Teknikokullar,06500 - Ankara, Turkey


Feyzi Basar



Abstract The concept of partial metric was introduced by Matthews in [Partial metric topology,in: S. Andima, et al. (Eds.), Proc. 8th Summer Conference on Topology and Its Applications, in: Ann.New York Acad. Sci. 728(1994), 183–197]. By ℓ∞, c and ℓq, we denote the classical sequence spaces ofall bounded, convergent and absolutely q-summable sequences, respectively, where 1 ≤ q < ∞. In thisstudy, we deal with the partial metric sequence spaces (ℓ∞(p), p∞), (c(p), p∞) and (ℓq(p), pq), where

p∞(x, y) := supk∈N

p(xk, yk) ; (x = (xk), y = (yk) ∈ ℓ∞(p) or x = (xk), y = (yk) ∈ c(p)),

pq(x, y) :=

[ ∞∑k=1

p(xk, yk)q

]1/q

; (x = (xk), y = (yk) ∈ ℓq(p)),

where N denotes the set of positive integers. We show that the partial metric classical sequence spaces(ℓ∞(p), p∞), (c(p), p∞) and (ℓq(p), pq) are complete. Additionally, we also examine the partial metricspaces (C[a, b], P∞) and (B[a, b], P∞) of non-negative real valued continuous and bounded functionsdefined on the closed interval [a, b], where

P∞(f, g) := supt∈[a,b]

p(f(t), g(t)) ; (f, g ∈ C[a, b] or f, g ∈ B[a, b]).

We also prove that the partial metric function spaces (C[a, b], P∞) and (B[a, b], P∞) are complete and

give some examples of partial metric spaces.

2000 Mathematics Subject Classification: Primary 46A45; Secondary 40C05.Keywords: Sequence space, space of bounded functions, space of continuous functions, partial metric.

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012

Characterizations of Regular Abel-Grassmann’s Groupoids

Madad Khan

Department of Mathematics, COMSATS Institute of Information TechnologyAbbottabad, K.P.K., Pakistan

Academic Visitor(May 19 to June 06, 2012)Mathematical Institute, University of Oxford, 24-29 St Giles, OX1 3LB, Oxford, UK


Nasir Khan

Department of Mathematics, COMSATS Institute of Information TechnologyAbbottabad, K.P.K., PakistanE-mail: [email protected]

Abstract In this paper, we introduce a new class of a non-associative algebraic structure namely

regular AG-groupoid and characterized it using its ideals.

1 Introduction

The idea of generalization of a commutative semigroup was first introduced by Kazim andNaseeruddin in 1972 (see [2]). They named it as a left almost semigroup (LA-semigroup). It isalso called an Abel-Grassmann’s groupoid (AG-groupoid) [11].An AG-groupoid is a groupoid S whose elements satisfy the left invertive law (ab)c = (cb)a, for alla, b, c ∈ S. In an AG-groupoid, the medial law [2] (ab)(cd) = (ac)(bd) holds for all a, b, c, d ∈ S.An AG-groupoid may or may not contains a left identity. If an AG-groupoid contains a leftidentity, then it is unique [5]. In an AG-groupoid S with left identity, the paramedial law(ab)(cd) = (db)(ca) holds for all a, b, c, d ∈ S. If an AG-groupoid contains a left identity, then itsatisfies the following law

a(bc) = b(ac), for all a, b, c ∈ S. (1)

An AG-groupoid is a non-associative and non-commutative algebraic structure mid way betweena groupoid and a commutative semigroup. This structure is closely related with a commutativesemigroup, because if an AG-groupoid contains a right identity, then it becomes a commutativesemigroup [5]. The connection of a commutative inverse semigroup with an AG-groupoid hasbeen given in [6] as: a commutative inverse semigroup (S, ) becomes an AG-groupoid (S, ·)under a·b = ba−1, for all a, b ∈ S. An AG-groupoid (S, .) with left identity becomes a semigroup(S, ), where ”” is defined as: for all x, y ∈ S, there exists a ∈ S such that x y = (xa)y [12].

2000 Mathematics Subject Classification: 20M10, 20N99Keywords: AG-groupoid, regular AG-groupoid, ideal, bi-ideal and quasi-ideal.

128


An AG-groupoid S is said to be locally associative if (aa)a = a(aa) for all a ∈ S. Mushtaqand Iqbal [7] proved that in a locally associative AG-groupoid S, (ab)n = anbn for all a, b ∈ S.Moreover it has been proved in [8] that if S is a locally associative AG∗∗-groupoid then anbm =bman for all a, b ∈ S and m,n ≥ 2. For basic notions and results one can see [3,12].An AG-groupoid has many characteristics similar to that of a commutative semigroup. Forinstance a2b2 = b2a2, for all a, b holds in a commutative semigroup, while this equation alsoholds for an AG-groupoid with left identity e, moreover ab = (ba)e for any subset a, b of anAG-groupoid. Now our aim is to discover some new investigations for an regular AG-groupoidusing the properties of its ideals.

2 Preliminary

Let S be an AG-groupoid. By an AG-subgroupoid of S, we means a non-empty subset A ofS such that A2 ⊆ A.A non-empty subset I of an AG-groupoid S is called a left (right) ideal of S if SI ⊆ I (IS ⊆ I)and it is called a two-sided ideal if it is both left and a right ideal of S. A non-empty subsetB of S is called a generalized bi-ideal of S if (BS)B ⊆ B and an AG-subgroupoid B of S iscalled a bi-ideal of S if (BS)B ⊆ B. A subset Q of S is called a quasi-ideal if QS ∩SQ ⊂ Q. Anideal I of S is called an idempotent if I2 = I.

Definition 2.1. An element a of an AG-groupoid S is called regular if there exist x ∈ S suchthat a = (ax)a and S is called regular, if every element of S is regular.

References

[1] P. Holgate, Groupoids satisfying a simple invertive law, The Math. Stud., 1− 4, 61(1992),101− 106.

[2] M. A. Kazim and M. Naseeruddin, On almost semigroups, The Alig. Bull. Math.,2(1972), 1− 7.

[3] Madad Khan and N. Ahmad, Characterizations of left almost semigroups by their ideals,Journal of Advanced Research in Pure Mathematics, 2(2010), 61− 73.

[4] Madad Khan and S. A . Khan. On decomposition of Abel-Grassmann’s groupoids, Acceptedin Semigroup Forum.

[5] Q. Mushtaq and S. M. Yousuf, On LA-semigroups, The Alig. Bull. Math., 8(1978), 65−70.

[6] Q. Mushtaq and S. M. Yusuf, On LA-semigroup defined by a commutative inverse semi-group, Math. Bech., 40(1988), 59− 62.

[7] Q. Mushtaq and Q. Iqbal, Decomposition of a locally associative LA-semigroup, SemigroupForum, 41(1990), 155− 164.

[8] Q. Mushtaq and Madad Khan, Semilattice decomposition of locally associative AG∗∗-groupoids, Algeb. Colloq., 16(2009)17− 22.

[9] Q. Mushtaq and Madad Khan, Topological structures on Abel-Grassmann’s groupoids,Accepted in Semigroup Forum.

[10] P. V. Protic and N. Stevanovic, On Abel-Grassmann’s groupoids, Proc. Math. Conf.Pristina, (1994), 31− 38.

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[11] P. V. Protic and N. Stevanovic, AG-test and some general properties of Abel-Grassmann’sgroupoids, PU. M. A., 4, 6(1995), 371− 383.

[12] N. Stevanovic and P. V. Protic, Composition of Abel-Grassmann’s 3-bands, Novi Sad, J.Math., 2, 34(2004), 175− 182.

130



012

Coset diagram for the action of Picard Group on Q(i,√3)

Saima Anis

COMSATS Institute of Information Technology Abbottabad, PakistanE-mail:

Abstract The Picard group Γ is PSL(2,Z[i]). We have defined coset diagram for the Picard

group. It has been observed that some elements of Q(i,√3)of the form a+b

√3

c and their conjugatesa−b

√3

c over Q (i) have different signs in the coset diagram for the action of Γ on the biquadratic field

Q(i,√3), these are called ambiguous numbers. We have noticed that ambiguous numbers in the coset

diagram for the action of Γ on Q(i,√3) form a unique pattern. It has been shown that there are finite

number of ambiguous numbers in an orbit Γα, where α is ambiguous, and they form a closed path and

it is the only closed path in the orbit Γα. We have devised a procedure to obtain ambiguous numbers of

the form a+k√3

c , where k is a positive integer.


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012

Hyper chaos in a fractional Chua system and its chaos-control andsynchronization

Mohammed-Salah Abdelouahab

Department of Mathematics and Informatics,Mila University Centre 43000, Algeria

E-mail:

Nasr-Eddine Hamri

Department of Mathematics and Informatics,Mila University Centre 43000, Algeria

E-mail:

Abstract The chaos control and synchronization of fractional-order chaotic systems has recentlyattracted increasing attention due to its potential applications in secure communication and controlprocessing.A hyperchaotic phenomenon is characterized by the existence of at last tow positive Lyapunov exponentswhich can increase the randomness and higher unpredictability of the corresponding system and thismakes message masking more effective by giving rise to more complex time series. So the hyperchaosmay be more useful in secure communication and encryption etc.

In this paper, a fractional hyperchaotic four-dimensional Chua’s circuit is introduced, where the capacitor

and the inductor of the original circuit are replaced by a fractional electric element called fractance.

The resulting circuit is described by four-dimensional fractional system. Based on the Routh-Hurwitz

conditions for the fractional-order systems, we obtain the sufficient conditions for realizing the control

of the fractional circuit and the synchronization between two hyperchaotic systems with different orders;

especially between the fractional system and its integer order counterpart. Numerical simulations show

the effectiveness of the theoretical analysis.


132



012

Euler Savary’s Formula on Galilean Plane

Mucahit Akbiyik



Salim Yuce



Abstract In this work, a canonical relative system for one-parameter Galilean planar motion

was defined. In addition, Euler-Savary formula, which gives the relationship between the curvature of

trajectory curves, was obtained with the help of this relative system.

2000 Mathematics Subject Classification:Keywords: Kinematics, Galilean Plane, Differential Geometry.

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