SOME GENERALIZATION OF MINTY S LEMMA. Doo-Young Jung

Size: px
Start display at page:

Download "SOME GENERALIZATION OF MINTY S LEMMA. Doo-Young Jung"

Transcription

1 J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 6(1999), no SOME GENERALIZATION OF MINTY S LEMMA Doo-Young Jung Abstract. We obtain a generalization of Behera and Panda s result on nonlinear scalar case to the vector version. 1. Introduction and preliminaries In the recent decades, there have been a great deal of developments in the theory of optimization techniques. The study of variational inequalities and complementarity problems is also a part of this development because optimization problems can often be reduced to the solution of variational inequalities and complementarity problems. One of the important results of variational inequality theory is Minty s Lemma, which has interesting applications in the study of obstacles problems, confined plasmas, filtration phenomena, free-boundary problems, plasticity and viscoplasticity phenomena, elasticity problems, stochastic optimal control problems and others. Let, : X X R be the duality pairing. The classical Minty s Lemma (cf. [2] and [3]) is stated as follows: Theorem 1.1. Let X be a reflexive real Banach space, K a nonempty closed convex subset of X and X be the dual of X. Let T : K X be a monotone operator Received by the editors October 1, 1998, and in revised form December 22, Mathematics Subject Classification. 49J27 (47H05, 47J20). Key words and phrases. Vector variational inequality, Minty s lemma, finite dimensional continuous, η-hemicontinuous, convex cone. 33

2 34 DOO-YOUNG JUNG which is continuous on finite dimensional subspaces (or at least hemicontinuous). Then the followings are equivalent. (a) There exists an x 0 K such that T (x0 ), y x 0 0 for all y K. (b) There exists an x 0 K such that T (y), y x0 0 for all y K. Recently, Theorem 1.1 was generalized by Behera and Panda [1] to the nonlinear case. In this paper, we obtain a generalization of Behera and Panda s result on nonlinear scalar case to the vector version. Let X and Y be two topological vector spaces, K a nonempty convex subset of X, and C be a set-valued mapping from K into 2 Y such that for every x K, C(x) is a closed convex solid cone of Y, i.e., its interior is nonempty. Let T be an operator of X into the space L(X, Y ) of all continuous linear operators, and η : K K X be an operator. We define an order relation C(x) in Y by the convex cone C(x) as follows: for y 1, y 2 Y, y 1 C(x) y 2 y 2 y 1 C(x). Definition 1.1. Let T : K L(X, Y ) be an operator. T is said to be η-hemicontinuous in the sense of Stampacchia [5] if, for any x, y K, α [0, 1], the mapping α T (αy + (1 α)x), η(y, x) is continuous, where, : L(X, Y ) X Y is the duality pairing. Definition 1.2 [3]. An operator G: K Y is said to be continuous on finite dimensional subspaces if for every finite dimensional subspace M of X, the operator G: K M Y is weakly continuous. Definition 1.3 [4]. An operator G: X Y is said to be C(x)-convex if for every x 1, x 2 X and λ (0, 1), G(λx 1 + (1 λ)x 2 ) C(x) λg(x 1 ) + (1 λ)g(x 2 ),

3 SOME GENERALIZATION OF MINTY S LEMMA 35 i.e., λg(x 1 ) + (1 λ)g(x 2 ) G(λx 1 + (1 λ)x 2 ) C(x). 2. Main Results Now we consider the following generalized vector version of Minty s Lemma. Theorem 2.1. Let X and Y be topological vector spaces, K a nonempty convex subset of X, and {C(x): x K} a family of closed convex solid cones of Y. Suppose that T : K L(X, Y ) and η: K K X be operators such that (i) T (y), η(y, y) / Int C(x) for all x, y K, (ii) the operator x T (x), η(y, x) of K into Y is continuous on finite dimensional subspaces (or at least η- hemicontinuous) for each y K, (iii) the operator y T (x), η(y, x) of K into Y is convex for each x K, (iv) T (x), η(y, x) + T (y), η(x, y) / Int C(x) for all x, y K. Then the followings are equivalent: (a) there exists an x 0 K such that for all y K. (b) there exists an x 0 K such that T (y), η(x0, y) / Int C(x 0 ) for all y K. Proof. Suppose that there exists an x 0 K satisfying

4 36 DOO-YOUNG JUNG for all y K. Then it is easily shown that for such an x 0 we have T (y), η(x 0, y) / Int C(x 0 ) for all y K from (iv) by the fact that Int C(x) + Int C(x) = Int C(x). Conversely, suppose that there exists an x 0 K such that T (y), η(x0, y) / Int C(x 0 ) for all y K. For arbitrary x K, letting y λ = λx + (1 λ)x 0, 0 < λ < 1, we have a y λ K by the convexity of K. Hence T (yλ ), η(x 0, y λ ) / Int C(x 0 ). On the other hand, by the convexity of the operator y T (x), η(y, x), we have λ T (y λ ), η(x, y λ ) + (1 λ) T (y λ ), η(x 0, y λ ) T (y λ ), η(y λ, y λ ) / Int C(x 0 ). Since T (yλ ), η(x 0, y λ ) / Int C(x 0 ), we have T (yλ ), η(x, y λ ) / Int C(x 0 ) (2.1) from (i). Since the operator x T (x), η(y, x) of K into Y is continuous on finite dimensional subspaces (or at least η-hemicontinuous), taking the limit as λ approaches to 0 + in (2.1) we get for all y K by the closedness of Y \( Int C(x 0 )).

5 SOME GENERALIZATION OF MINTY S LEMMA 37 Corollary 2.2 [1]. Let X be a nonempty closed convex subset of a reflexive real Banach space X and let X be the dual of X. Let T : K X and θ : K K X be two maps such that (i) T (y), θ(y, y) = 0 for all y K, (ii) the map x T (x), θ(y, x) of K into R is continuous on finite dimensional subspaces (or at least hemicontinuous), for each y K, (iii) the map y T (x), θ(y, x) of K into R is convex for each x K, (iv) T (x), θ(y, x) + T (y), θ(x, y) 0 for all x, y K. Then the followings are equivalent: (a) x 0 K, T (x 0 ), θ(y, x 0 ) 0 for all y K. (b) x 0 K, T (y), θ(x 0, y) 0 for all y K. Proof. If we put C(x) = [0, ) in Theorem 2.1, it can be easily shown. References 1. A. Behera and G. K. Panda, A generalization of Minty s lemma, Indian J. Pure Appl. Math. 28 (1997), no. 7, MR 98j: F. E. Browder, Existence and approximation of solutions of nonlinear variational inequalities, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), MR 34# M. Chipot, Variational inequalities and Flow in Porous Media, Applied Mathematical Sciences, vol. 52, Springer-Verlag, New York, MR 85m: D. Kuroiwa, Convexity for set-valued maps, Appl. Math. Lett. 9 (1996), no. 2, MR 96m: G. Stampacchia, Variational inequalities, In: Theory and applications of monotone operators, Proceedings of the NATO Advanced Study Institute, Venice, Italy, 1968 (pp ), Edizioni Oderisi, Gubbio, MR 54# Department of Mathematics, College of Science, Kyungsung University, Daeyeondong, Nam-gu, Pusan , Korea

SCALARIZATION APPROACHES FOR GENERALIZED VECTOR VARIATIONAL INEQUALITIES

SCALARIZATION APPROACHES FOR GENERALIZED VECTOR VARIATIONAL INEQUALITIES Nonlinear Analysis Forum 12(1), pp. 119 124, 2007 SCALARIZATION APPROACHES FOR GENERALIZED VECTOR VARIATIONAL INEQUALITIES Zhi-bin Liu, Nan-jing Huang and Byung-Soo Lee Department of Applied Mathematics

More information

REMARKS ON SOME VARIATIONAL INEQUALITIES

REMARKS ON SOME VARIATIONAL INEQUALITIES Bull. Korean Math. Soc. 28 (1991), No. 2, pp. 163 174 REMARKS ON SOME VARIATIONAL INEQUALITIES SEHIE PARK 1. Introduction and Preliminaries This is a continuation of the author s previous work [17]. In

More information

ON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE. Sangho Kum and Gue Myung Lee. 1. Introduction

ON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE. Sangho Kum and Gue Myung Lee. 1. Introduction J. Korean Math. Soc. 38 (2001), No. 3, pp. 683 695 ON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE Sangho Kum and Gue Myung Lee Abstract. In this paper we are concerned with theoretical properties

More information

WEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE

WEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE Fixed Point Theory, Volume 6, No. 1, 2005, 59-69 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.htm WEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE YASUNORI KIMURA Department

More information

Applied Mathematics Letters

Applied Mathematics Letters Applied Mathematics Letters 25 (2012) 974 979 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On dual vector equilibrium problems

More information

Scalar Asymptotic Contractivity and Fixed Points for Nonexpansive Mappings on Unbounded Sets

Scalar Asymptotic Contractivity and Fixed Points for Nonexpansive Mappings on Unbounded Sets Scalar Asymptotic Contractivity and Fixed Points for Nonexpansive Mappings on Unbounded Sets George Isac Department of Mathematics Royal Military College of Canada, STN Forces Kingston, Ontario, Canada

More information

On pseudomonotone variational inequalities

On pseudomonotone variational inequalities An. Şt. Univ. Ovidius Constanţa Vol. 14(1), 2006, 83 90 On pseudomonotone variational inequalities Silvia Fulina Abstract Abstract. There are mainly two definitions of pseudomonotone mappings. First, introduced

More information

A CHARACTERIZATION OF REFLEXIVITY OF NORMED ALMOST LINEAR SPACES

A CHARACTERIZATION OF REFLEXIVITY OF NORMED ALMOST LINEAR SPACES Comm. Korean Math. Soc. 12 (1997), No. 2, pp. 211 219 A CHARACTERIZATION OF REFLEXIVITY OF NORMED ALMOST LINEAR SPACES SUNG MO IM AND SANG HAN LEE ABSTRACT. In [6] we proved that if a nals X is reflexive,

More information

Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis

Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis B. T. Kien, M.-M. Wong, N. C. Wong and J. C. Yao Communicated by F. Giannessi This research was partially

More information

Generalized Monotonicities and Its Applications to the System of General Variational Inequalities

Generalized Monotonicities and Its Applications to the System of General Variational Inequalities Generalized Monotonicities and Its Applications to the System of General Variational Inequalities Khushbu 1, Zubair Khan 2 Research Scholar, Department of Mathematics, Integral University, Lucknow, Uttar

More information

STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS

STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS ARCHIVUM MATHEMATICUM (BRNO) Tomus 45 (2009), 147 158 STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS Xiaolong Qin 1, Shin Min Kang 1, Yongfu Su 2,

More information

On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings

On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings Int. J. Nonlinear Anal. Appl. 7 (2016) No. 1, 295-300 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.341 On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous

More information

Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou

Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou J. Korean Math. Soc. 38 (2001), No. 6, pp. 1245 1260 DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE PROBLEMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou Abstract.

More information

The aim of this paper is to obtain a theorem on the existence and uniqueness of increasing and convex solutions ϕ of the Schröder equation

The aim of this paper is to obtain a theorem on the existence and uniqueness of increasing and convex solutions ϕ of the Schröder equation PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 1, January 1997, Pages 153 158 S 0002-9939(97)03640-X CONVEX SOLUTIONS OF THE SCHRÖDER EQUATION IN BANACH SPACES JANUSZ WALORSKI (Communicated

More information

HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM

HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM

More information

CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES. Jong Soo Jung. 1. Introduction

CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES. Jong Soo Jung. 1. Introduction Korean J. Math. 16 (2008), No. 2, pp. 215 231 CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES Jong Soo Jung Abstract. Let E be a uniformly convex Banach space

More information

Sequential Pareto Subdifferential Sum Rule And Sequential Effi ciency

Sequential Pareto Subdifferential Sum Rule And Sequential Effi ciency Applied Mathematics E-Notes, 16(2016), 133-143 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Sequential Pareto Subdifferential Sum Rule And Sequential Effi ciency

More information

ITERATIVE SCHEMES FOR APPROXIMATING SOLUTIONS OF ACCRETIVE OPERATORS IN BANACH SPACES SHOJI KAMIMURA AND WATARU TAKAHASHI. Received December 14, 1999

ITERATIVE SCHEMES FOR APPROXIMATING SOLUTIONS OF ACCRETIVE OPERATORS IN BANACH SPACES SHOJI KAMIMURA AND WATARU TAKAHASHI. Received December 14, 1999 Scientiae Mathematicae Vol. 3, No. 1(2000), 107 115 107 ITERATIVE SCHEMES FOR APPROXIMATING SOLUTIONS OF ACCRETIVE OPERATORS IN BANACH SPACES SHOJI KAMIMURA AND WATARU TAKAHASHI Received December 14, 1999

More information

Convergence Theorems of Approximate Proximal Point Algorithm for Zeroes of Maximal Monotone Operators in Hilbert Spaces 1

Convergence Theorems of Approximate Proximal Point Algorithm for Zeroes of Maximal Monotone Operators in Hilbert Spaces 1 Int. Journal of Math. Analysis, Vol. 1, 2007, no. 4, 175-186 Convergence Theorems of Approximate Proximal Point Algorithm for Zeroes of Maximal Monotone Operators in Hilbert Spaces 1 Haiyun Zhou Institute

More information

Abstract. The connectivity of the efficient point set and of some proper efficient point sets in locally convex spaces is investigated.

Abstract. The connectivity of the efficient point set and of some proper efficient point sets in locally convex spaces is investigated. APPLICATIONES MATHEMATICAE 25,1 (1998), pp. 121 127 W. SONG (Harbin and Warszawa) ON THE CONNECTIVITY OF EFFICIENT POINT SETS Abstract. The connectivity of the efficient point set and of some proper efficient

More information

ON THE RANGE OF THE SUM OF MONOTONE OPERATORS IN GENERAL BANACH SPACES

ON THE RANGE OF THE SUM OF MONOTONE OPERATORS IN GENERAL BANACH SPACES PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 11, November 1996 ON THE RANGE OF THE SUM OF MONOTONE OPERATORS IN GENERAL BANACH SPACES HASSAN RIAHI (Communicated by Palle E. T. Jorgensen)

More information

PROJECTIONS ONTO CONES IN BANACH SPACES

PROJECTIONS ONTO CONES IN BANACH SPACES Fixed Point Theory, 19(2018), No. 1,...-... DOI: http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html PROJECTIONS ONTO CONES IN BANACH SPACES A. DOMOKOS AND M.M. MARSH Department of Mathematics and Statistics

More information

SOME RESULTS ON 2-INNER PRODUCT SPACES

SOME RESULTS ON 2-INNER PRODUCT SPACES Novi Sad J. Math. Vol. 37, No. 2, 2007, 35-40 SOME RESULTS ON 2-INNER PRODUCT SPACES H. Mazaheri 1, R. Kazemi 1 Abstract. We onsider Riesz Theorem in the 2-inner product spaces and give some results in

More information

The sum of two maximal monotone operator is of type FPV

The sum of two maximal monotone operator is of type FPV CJMS. 5(1)(2016), 17-21 Caspian Journal of Mathematical Sciences (CJMS) University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-0611 The sum of two maximal monotone operator is of type

More information

On Gap Functions for Equilibrium Problems via Fenchel Duality

On Gap Functions for Equilibrium Problems via Fenchel Duality On Gap Functions for Equilibrium Problems via Fenchel Duality Lkhamsuren Altangerel 1 Radu Ioan Boţ 2 Gert Wanka 3 Abstract: In this paper we deal with the construction of gap functions for equilibrium

More information

Two-Step Iteration Scheme for Nonexpansive Mappings in Banach Space

Two-Step Iteration Scheme for Nonexpansive Mappings in Banach Space Mathematica Moravica Vol. 19-1 (2015), 95 105 Two-Step Iteration Scheme for Nonexpansive Mappings in Banach Space M.R. Yadav Abstract. In this paper, we introduce a new two-step iteration process to approximate

More information

Lecture 9 Monotone VIs/CPs Properties of cones and some existence results. October 6, 2008

Lecture 9 Monotone VIs/CPs Properties of cones and some existence results. October 6, 2008 Lecture 9 Monotone VIs/CPs Properties of cones and some existence results October 6, 2008 Outline Properties of cones Existence results for monotone CPs/VIs Polyhedrality of solution sets Game theory:

More information

Generalized vector equilibrium problems on Hadamard manifolds

Generalized vector equilibrium problems on Hadamard manifolds Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 1402 1409 Research Article Generalized vector equilibrium problems on Hadamard manifolds Shreyasi Jana a, Chandal Nahak a, Cristiana

More information

ON SOLVABILITY OF GENERAL NONLINEAR VARIATIONAL-LIKE INEQUALITIES IN REFLEXIVE BANACH SPACES

ON SOLVABILITY OF GENERAL NONLINEAR VARIATIONAL-LIKE INEQUALITIES IN REFLEXIVE BANACH SPACES ON SOLVABILITY OF GENERAL NONLINEAR VARIATIONAL-LIKE INEQUALITIES IN REFLEXIVE BANACH SPACES ZEQING LIU, JUHE SUN, SOO HAK SHIM, AND SHIN MIN KANG Received 28 December 2004 We introduce and study a new

More information

YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS

YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 103, Number 3, July 1988 YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS JOHN RAINWATER (Communicated by William J. Davis) ABSTRACT. Generic

More information

1 Introduction The study of the existence of solutions of Variational Inequalities on unbounded domains usually involves the same sufficient assumptio

1 Introduction The study of the existence of solutions of Variational Inequalities on unbounded domains usually involves the same sufficient assumptio Coercivity Conditions and Variational Inequalities Aris Daniilidis Λ and Nicolas Hadjisavvas y Abstract Various coercivity conditions appear in the literature in order to guarantee solutions for the Variational

More information

CONVERGENCE OF THE STEEPEST DESCENT METHOD FOR ACCRETIVE OPERATORS

CONVERGENCE OF THE STEEPEST DESCENT METHOD FOR ACCRETIVE OPERATORS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 12, Pages 3677 3683 S 0002-9939(99)04975-8 Article electronically published on May 11, 1999 CONVERGENCE OF THE STEEPEST DESCENT METHOD

More information

NONLINEAR THREE POINT BOUNDARY VALUE PROBLEM

NONLINEAR THREE POINT BOUNDARY VALUE PROBLEM SARAJEVO JOURNAL OF MATHEMATICS Vol.8 (2) (212), 11 16 NONLINEAR THREE POINT BOUNDARY VALUE PROBLEM A. GUEZANE-LAKOUD AND A. FRIOUI Abstract. In this work, we establish sufficient conditions for the existence

More information

Remark on a Couple Coincidence Point in Cone Normed Spaces

Remark on a Couple Coincidence Point in Cone Normed Spaces International Journal of Mathematical Analysis Vol. 8, 2014, no. 50, 2461-2468 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.49293 Remark on a Couple Coincidence Point in Cone Normed

More information

APPROXIMATING SOLUTIONS FOR THE SYSTEM OF REFLEXIVE BANACH SPACE

APPROXIMATING SOLUTIONS FOR THE SYSTEM OF REFLEXIVE BANACH SPACE Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 2 Issue 3(2010), Pages 32-39. APPROXIMATING SOLUTIONS FOR THE SYSTEM OF φ-strongly ACCRETIVE OPERATOR

More information

ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES

ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 3, 2018 ISSN 1223-7027 ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES Vahid Dadashi 1 In this paper, we introduce a hybrid projection algorithm for a countable

More information

ON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS

ON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS Chin. Ann. Math.??B(?), 200?, 1 20 DOI: 10.1007/s11401-007-0001-x ON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS Michel CHIPOT Abstract We present a method allowing to obtain existence of a solution for

More information

On the simplest expression of the perturbed Moore Penrose metric generalized inverse

On the simplest expression of the perturbed Moore Penrose metric generalized inverse Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated

More information

Best proximity point results in set-valued analysis

Best proximity point results in set-valued analysis Nonlinear Analysis: Modelling and Control, Vol. 21, No. 3, 293 305 ISSN 1392-5113 http://dx.doi.org/10.15388/na.2016.3.1 Best proximity point results in set-valued analysis Binayak S. Choudhury a, Pranati

More information

ON µ-compact SETS IN µ-spaces

ON µ-compact SETS IN µ-spaces Questions and Answers in General Topology 31 (2013), pp. 49 57 ON µ-compact SETS IN µ-spaces MOHAMMAD S. SARSAK (Communicated by Yasunao Hattori) Abstract. The primary purpose of this paper is to introduce

More information

EQUILIBRIUM FOR PERTURBATIONS OF MULTIFUNCTIONS BY CONVEX PROCESSES

EQUILIBRIUM FOR PERTURBATIONS OF MULTIFUNCTIONS BY CONVEX PROCESSES GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 3, 1996, 201-215 EQUILIBRIUM FOR PERTURBATIONS OF MULTIFUNCTIONS BY CONVEX PROCESSES H. BEN-EL-MECHAIEKH AND W. KRYSZEWSKI Abstract. We present a general equilibrium

More information

ON THE REGULARITY OF ONE PARAMETER TRANSFORMATION GROUPS IN BARRELED LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES

ON THE REGULARITY OF ONE PARAMETER TRANSFORMATION GROUPS IN BARRELED LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES Communications on Stochastic Analysis Vol. 9, No. 3 (2015) 413-418 Serials Publications www.serialspublications.com ON THE REGULARITY OF ONE PARAMETER TRANSFORMATION GROUPS IN BARRELED LOCALLY CONVEX TOPOLOGICAL

More information

On nonexpansive and accretive operators in Banach spaces

On nonexpansive and accretive operators in Banach spaces Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 3437 3446 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa On nonexpansive and accretive

More information

THE SEMI ORLICZ SPACE cs d 1

THE SEMI ORLICZ SPACE cs d 1 Kragujevac Journal of Mathematics Volume 36 Number 2 (2012), Pages 269 276. THE SEMI ORLICZ SPACE cs d 1 N. SUBRAMANIAN 1, B. C. TRIPATHY 2, AND C. MURUGESAN 3 Abstract. Let Γ denote the space of all entire

More information

ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES

ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M. I. OSTROVSKII (Communicated by Dale Alspach) Abstract.

More information

NEW MAXIMUM THEOREMS WITH STRICT QUASI-CONCAVITY. Won Kyu Kim and Ju Han Yoon. 1. Introduction

NEW MAXIMUM THEOREMS WITH STRICT QUASI-CONCAVITY. Won Kyu Kim and Ju Han Yoon. 1. Introduction Bull. Korean Math. Soc. 38 (2001), No. 3, pp. 565 573 NEW MAXIMUM THEOREMS WITH STRICT QUASI-CONCAVITY Won Kyu Kim and Ju Han Yoon Abstract. In this paper, we first prove the strict quasi-concavity of

More information

CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS

CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS Abstract. The aim of this paper is to characterize in terms of classical (quasi)convexity of extended real-valued functions the set-valued maps which are

More information

On an iterative algorithm for variational inequalities in. Banach space

On an iterative algorithm for variational inequalities in. Banach space MATHEMATICAL COMMUNICATIONS 95 Math. Commun. 16(2011), 95 104. On an iterative algorithm for variational inequalities in Banach spaces Yonghong Yao 1, Muhammad Aslam Noor 2,, Khalida Inayat Noor 3 and

More information

CONVERGENCE THEOREMS FOR STRICTLY ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN HILBERT SPACES. Gurucharan Singh Saluja

CONVERGENCE THEOREMS FOR STRICTLY ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN HILBERT SPACES. Gurucharan Singh Saluja Opuscula Mathematica Vol 30 No 4 2010 http://dxdoiorg/107494/opmath2010304485 CONVERGENCE THEOREMS FOR STRICTLY ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN HILBERT SPACES Gurucharan Singh Saluja Abstract

More information

(U 2 ayxtyj < (2 ajjxjxj) (2 a^jj),

(U 2 ayxtyj < (2 ajjxjxj) (2 a^jj), PROCEEDINGS Ol THE AMERICAN MATHEMATICAL SOCIETY Volume 59, Number 1. August 1976 THE MEANING OF THE CAUCHY-SCHWARZ- BUNIAKOVSKY INEQUALITY eduardo h. zarantonello1 Abstract. It is proved that a mapping

More information

SHRINKING PROJECTION METHOD FOR A SEQUENCE OF RELATIVELY QUASI-NONEXPANSIVE MULTIVALUED MAPPINGS AND EQUILIBRIUM PROBLEM IN BANACH SPACES

SHRINKING PROJECTION METHOD FOR A SEQUENCE OF RELATIVELY QUASI-NONEXPANSIVE MULTIVALUED MAPPINGS AND EQUILIBRIUM PROBLEM IN BANACH SPACES U.P.B. Sci. Bull., Series A, Vol. 76, Iss. 2, 2014 ISSN 1223-7027 SHRINKING PROJECTION METHOD FOR A SEQUENCE OF RELATIVELY QUASI-NONEXPANSIVE MULTIVALUED MAPPINGS AND EQUILIBRIUM PROBLEM IN BANACH SPACES

More information

THE NEARLY ADDITIVE MAPS

THE NEARLY ADDITIVE MAPS Bull. Korean Math. Soc. 46 (009), No., pp. 199 07 DOI 10.4134/BKMS.009.46..199 THE NEARLY ADDITIVE MAPS Esmaeeil Ansari-Piri and Nasrin Eghbali Abstract. This note is a verification on the relations between

More information

A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE. Myung-Hyun Cho and Jun-Hui Kim. 1. Introduction

A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE. Myung-Hyun Cho and Jun-Hui Kim. 1. Introduction Comm. Korean Math. Soc. 16 (2001), No. 2, pp. 277 285 A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE Myung-Hyun Cho and Jun-Hui Kim Abstract. The purpose of this paper

More information

The theory of monotone operators with applications

The theory of monotone operators with applications FACULTY OF MATHEMATICS AND COMPUTER SCIENCE BABEŞ-BOLYAI UNIVERSITY CLUJ-NAPOCA, ROMÂNIA László Szilárd Csaba The theory of monotone operators with applications Ph.D. Thesis Summary Scientific Advisor:

More information

EXISTENCE RESULTS FOR SOME VARIATIONAL INEQUALITIES INVOLVING NON-NEGATIVE, NON-COERCITIVE BILINEAR FORMS. Georgi Chobanov

EXISTENCE RESULTS FOR SOME VARIATIONAL INEQUALITIES INVOLVING NON-NEGATIVE, NON-COERCITIVE BILINEAR FORMS. Georgi Chobanov Pliska Stud. Math. Bulgar. 23 (2014), 49 56 STUDIA MATHEMATICA BULGARICA EXISTENCE RESULTS FOR SOME VARIATIONAL INEQUALITIES INVOLVING NON-NEGATIVE, NON-COERCITIVE BILINEAR FORMS Georgi Chobanov Abstract.

More information

Iterative common solutions of fixed point and variational inequality problems

Iterative common solutions of fixed point and variational inequality problems Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 1882 1890 Research Article Iterative common solutions of fixed point and variational inequality problems Yunpeng Zhang a, Qing Yuan b,

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 2, Issue 1, Article 12, 2001 ON KY FAN S MINIMAX INEQUALITIES, MIXED EQUILIBRIUM PROBLEMS AND HEMIVARIATIONAL INEQUALITIES

More information

SOME NEW DOUBLE SEQUENCE SPACES IN n-normed SPACES DEFINED BY A SEQUENCE OF ORLICZ FUNCTION

SOME NEW DOUBLE SEQUENCE SPACES IN n-normed SPACES DEFINED BY A SEQUENCE OF ORLICZ FUNCTION Journal of Mathematical Analysis ISSN: 2217-3412, URL: http://www.ilirias.com Volume 3 Issue 12012, Pages 12-20. SOME NEW DOUBLE SEQUENCE SPACES IN n-normed SPACES DEFINED BY A SEQUENCE OF ORLICZ FUNCTION

More information

Local strong convexity and local Lipschitz continuity of the gradient of convex functions

Local strong convexity and local Lipschitz continuity of the gradient of convex functions Local strong convexity and local Lipschitz continuity of the gradient of convex functions R. Goebel and R.T. Rockafellar May 23, 2007 Abstract. Given a pair of convex conjugate functions f and f, we investigate

More information

(1) H* - y\\ < (1 + r)(x - y) - rt(tx - ra)

(1) H* - y\\ < (1 + r)(x - y) - rt(tx - ra) PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 99, Number 2, February 1987 ITERATIVE APPROXIMATION OF FIXED POINTS OF LIPSCHITZIAN STRICTLY PSEUDO-CONTRACTIVE MAPPINGS C. E. CHIDUME ABSTRACT.

More information

A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS

A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS Fixed Point Theory, (0), No., 4-46 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS A. ABKAR AND M. ESLAMIAN Department of Mathematics,

More information

Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces

Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces YUAN-HENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua

More information

("-1/' .. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS. R. T. Rockafellar. MICHIGAN MATHEMATICAL vol. 16 (1969) pp.

(-1/' .. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS. R. T. Rockafellar. MICHIGAN MATHEMATICAL vol. 16 (1969) pp. I l ("-1/'.. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS R. T. Rockafellar from the MICHIGAN MATHEMATICAL vol. 16 (1969) pp. 397-407 JOURNAL LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERATORS

More information

STRONG CONVERGENCE THEOREMS FOR COMMUTATIVE FAMILIES OF LINEAR CONTRACTIVE OPERATORS IN BANACH SPACES

STRONG CONVERGENCE THEOREMS FOR COMMUTATIVE FAMILIES OF LINEAR CONTRACTIVE OPERATORS IN BANACH SPACES STRONG CONVERGENCE THEOREMS FOR COMMUTATIVE FAMILIES OF LINEAR CONTRACTIVE OPERATORS IN BANACH SPACES WATARU TAKAHASHI, NGAI-CHING WONG, AND JEN-CHIH YAO Abstract. In this paper, we study nonlinear analytic

More information

Layered Compression-Expansion Fixed Point Theorem

Layered Compression-Expansion Fixed Point Theorem Res. Fixed Point Theory Appl. Volume 28, Article ID 2825, pages eissn 258-67 Results in Fixed Point Theory and Applications RESEARCH ARTICLE Layered Compression-Expansion Fixed Point Theorem Richard I.

More information

BEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction

BEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction Bull Korean Math Soc 43 (2006), No 2, pp 377 387 BEST APPROXIMATIONS AND ORTHOGONALITIES IN -INNER PRODUCT SPACES Seong Sik Kim* and Mircea Crâşmăreanu Abstract In this paper, some characterizations of

More information

Some Generalizations of Caristi s Fixed Point Theorem with Applications

Some Generalizations of Caristi s Fixed Point Theorem with Applications Int. Journal of Math. Analysis, Vol. 7, 2013, no. 12, 557-564 Some Generalizations of Caristi s Fixed Point Theorem with Applications Seong-Hoon Cho Department of Mathematics Hanseo University, Seosan

More information

The Journal of Nonlinear Science and Applications

The Journal of Nonlinear Science and Applications J. Nonlinear Sci. Appl. 2 (2009), no. 2, 78 91 The Journal of Nonlinear Science and Applications http://www.tjnsa.com STRONG CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS AND FIXED POINT PROBLEMS OF STRICT

More information

The best generalised inverse of the linear operator in normed linear space

The best generalised inverse of the linear operator in normed linear space Linear Algebra and its Applications 420 (2007) 9 19 www.elsevier.com/locate/laa The best generalised inverse of the linear operator in normed linear space Ping Liu, Yu-wen Wang School of Mathematics and

More information

Fixed point theorems of nondecreasing order-ćirić-lipschitz mappings in normed vector spaces without normalities of cones

Fixed point theorems of nondecreasing order-ćirić-lipschitz mappings in normed vector spaces without normalities of cones Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 18 26 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Fixed point theorems of nondecreasing

More information

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space 1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization

More information

Some Aspects of 2-Fuzzy 2-Normed Linear Spaces

Some Aspects of 2-Fuzzy 2-Normed Linear Spaces BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 32(2) (2009), 211 221 Some Aspects of 2-Fuzzy 2-Normed Linear Spaces 1 R. M. Somasundaram

More information

Math 5052 Measure Theory and Functional Analysis II Homework Assignment 7

Math 5052 Measure Theory and Functional Analysis II Homework Assignment 7 Math 5052 Measure Theory and Functional Analysis II Homework Assignment 7 Prof. Wickerhauser Due Friday, February 5th, 2016 Please do Exercises 3, 6, 14, 16*, 17, 18, 21*, 23*, 24, 27*. Exercises marked

More information

The local equicontinuity of a maximal monotone operator

The local equicontinuity of a maximal monotone operator arxiv:1410.3328v2 [math.fa] 3 Nov 2014 The local equicontinuity of a maximal monotone operator M.D. Voisei Abstract The local equicontinuity of an operator T : X X with proper Fitzpatrick function ϕ T

More information

EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE. Leszek Gasiński

EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE. Leszek Gasiński DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 409 418 EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE Leszek Gasiński Jagiellonian

More information

Some Results on b-orthogonality in 2-Normed Linear Spaces

Some Results on b-orthogonality in 2-Normed Linear Spaces Int. Journal of Math. Analysis, Vol. 1, 2007, no. 14, 681-687 Some Results on b-orthogonality in 2-Normed Linear Spaces H. Mazaheri and S. Golestani Nezhad Department of Mathematics Yazd University, Yazd,

More information

Synchronal Algorithm For a Countable Family of Strict Pseudocontractions in q-uniformly Smooth Banach Spaces

Synchronal Algorithm For a Countable Family of Strict Pseudocontractions in q-uniformly Smooth Banach Spaces Int. Journal of Math. Analysis, Vol. 8, 2014, no. 15, 727-745 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.212287 Synchronal Algorithm For a Countable Family of Strict Pseudocontractions

More information

SOME FIXED POINT THEOREMS FOR ORDERED REICH TYPE CONTRACTIONS IN CONE RECTANGULAR METRIC SPACES

SOME FIXED POINT THEOREMS FOR ORDERED REICH TYPE CONTRACTIONS IN CONE RECTANGULAR METRIC SPACES Acta Math. Univ. Comenianae Vol. LXXXII, 2 (2013), pp. 165 175 165 SOME FIXED POINT THEOREMS FOR ORDERED REICH TYPE CONTRACTIONS IN CONE RECTANGULAR METRIC SPACES S. K. MALHOTRA, S. SHUKLA and R. SEN Abstract.

More information

Well-posedness for generalized mixed vector variational-like inequality problems in Banach space

Well-posedness for generalized mixed vector variational-like inequality problems in Banach space MATHEMATICAL COMMUNICATIONS 287 Math. Commun. 22(2017), 287 302 Well-posedness for generalized mixed vector variational-like inequality problems in Banach space Anurag Jayswal and Shalini Jha Department

More information

Some Properties of the Augmented Lagrangian in Cone Constrained Optimization

Some Properties of the Augmented Lagrangian in Cone Constrained Optimization MATHEMATICS OF OPERATIONS RESEARCH Vol. 29, No. 3, August 2004, pp. 479 491 issn 0364-765X eissn 1526-5471 04 2903 0479 informs doi 10.1287/moor.1040.0103 2004 INFORMS Some Properties of the Augmented

More information

Bhavana Deshpande COMMON FIXED POINT RESULTS FOR SIX MAPS ON CONE METRIC SPACES WITH SOME WEAKER CONDITIONS. 1. Introduction and preliminaries

Bhavana Deshpande COMMON FIXED POINT RESULTS FOR SIX MAPS ON CONE METRIC SPACES WITH SOME WEAKER CONDITIONS. 1. Introduction and preliminaries F A S C I C U L I M A T H E M A T I C I Nr 43 2010 Bhavana Deshpande COMMON FIXED POINT RESULTS FOR SIX MAPS ON CONE METRIC SPACES WITH SOME WEAKER CONDITIONS Abstract. The existence of coincidence points

More information

GENERAL NONCONVEX SPLIT VARIATIONAL INEQUALITY PROBLEMS. Jong Kyu Kim, Salahuddin, and Won Hee Lim

GENERAL NONCONVEX SPLIT VARIATIONAL INEQUALITY PROBLEMS. Jong Kyu Kim, Salahuddin, and Won Hee Lim Korean J. Math. 25 (2017), No. 4, pp. 469 481 https://doi.org/10.11568/kjm.2017.25.4.469 GENERAL NONCONVEX SPLIT VARIATIONAL INEQUALITY PROBLEMS Jong Kyu Kim, Salahuddin, and Won Hee Lim Abstract. In this

More information

PERIODIC SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER DIFFERENTIAL SYSTEMS WITH (q, p)-laplacian

PERIODIC SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER DIFFERENTIAL SYSTEMS WITH (q, p)-laplacian Electronic Journal of Differential Euations, Vol. 24 (24), No. 64, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PERIODIC SOLUIONS FOR NON-AUONOMOUS

More information

ON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM

ON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM Journal of Applied Analysis Vol. 9, No. 1 23, pp. 139 147 ON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM M. GALEWSKI Received July 3, 21 and, in revised form, March 26, 22 Abstract. We provide

More information

On operator equilibrium problems

On operator equilibrium problems MATHEMATICAL COMMUNICATIONS 581 Math. Commun. 19(2014), 581 587 On operator equilibrium problems Júlia Salamon 1, 1 Department of Mathematics and Computer Science, Sapientia Hungarian University of Transylvania,

More information

KKM-Type Theorems for Best Proximal Points in Normed Linear Space

KKM-Type Theorems for Best Proximal Points in Normed Linear Space International Journal of Mathematical Analysis Vol. 12, 2018, no. 12, 603-609 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2018.81069 KKM-Type Theorems for Best Proximal Points in Normed

More information

arxiv: v1 [math.oc] 21 Mar 2015

arxiv: v1 [math.oc] 21 Mar 2015 Convex KKM maps, monotone operators and Minty variational inequalities arxiv:1503.06363v1 [math.oc] 21 Mar 2015 Marc Lassonde Université des Antilles, 97159 Pointe à Pitre, France E-mail: marc.lassonde@univ-ag.fr

More information

CHAPTER 5. Birkhoff Orthogonality and a Revisit to the Characterization of Best Approximations. 5.1 Introduction

CHAPTER 5. Birkhoff Orthogonality and a Revisit to the Characterization of Best Approximations. 5.1 Introduction CHAPTER 5 Birkhoff Orthogonality and a Revisit to the Characterization of Best Approximations 5.1 Introduction The notion of orthogonality in an arbitrary normed space, with the norm not necessarily coming

More information

STRONG CONVERGENCE THEOREMS BY A HYBRID STEEPEST DESCENT METHOD FOR COUNTABLE NONEXPANSIVE MAPPINGS IN HILBERT SPACES

STRONG CONVERGENCE THEOREMS BY A HYBRID STEEPEST DESCENT METHOD FOR COUNTABLE NONEXPANSIVE MAPPINGS IN HILBERT SPACES Scientiae Mathematicae Japonicae Online, e-2008, 557 570 557 STRONG CONVERGENCE THEOREMS BY A HYBRID STEEPEST DESCENT METHOD FOR COUNTABLE NONEXPANSIVE MAPPINGS IN HILBERT SPACES SHIGERU IEMOTO AND WATARU

More information

arxiv:math/ v2 [math.gn] 21 Jun 2002

arxiv:math/ v2 [math.gn] 21 Jun 2002 ON FINITE-DIMENSIONAL MAPS II arxiv:math/0202114v2 [math.gn] 21 Jun 2002 H. MURAT TUNCALI AND VESKO VALOV Abstract. Let f : X Y be a perfect n-dimensional surjective map of paracompact spaces and Y a C-space.

More information

FUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM

FUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM Electronic Journal of Differential Equations, Vol. 28(28), No. 22, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) FUNCTIONAL

More information

New Coupled Common Fixed Point for Four Mappings satisfying Rational Contractive Expression

New Coupled Common Fixed Point for Four Mappings satisfying Rational Contractive Expression New Coupled Common Fixed Point for Four Mappings satisfying Rational Contractive Expression H K Nashine 1, A Gupta 2 1 Department of Mathematics, Amity University, Manth Kharora, Raipur-(Chhattisgarh),

More information

Continuous Sets and Non-Attaining Functionals in Reflexive Banach Spaces

Continuous Sets and Non-Attaining Functionals in Reflexive Banach Spaces Laboratoire d Arithmétique, Calcul formel et d Optimisation UMR CNRS 6090 Continuous Sets and Non-Attaining Functionals in Reflexive Banach Spaces Emil Ernst Michel Théra Rapport de recherche n 2004-04

More information

THE CARATHÉODORY EXTENSION THEOREM FOR VECTOR VALUED MEASURES

THE CARATHÉODORY EXTENSION THEOREM FOR VECTOR VALUED MEASURES PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 72, Number 1, October 1978 THE CARATHÉODORY EXTENSION THEOREM FOR VECTOR VALUED MEASURES JOSEPH KUPKA Abstract. This paper comprises three advertisements

More information

Research Article Existence and Duality of Generalized ε-vector Equilibrium Problems

Research Article Existence and Duality of Generalized ε-vector Equilibrium Problems Applied Mathematics Volume 2012, Article ID 674512, 13 pages doi:10.1155/2012/674512 Research Article Existence and Duality of Generalized ε-vector Equilibrium Problems Hong-Yong Fu, Bin Dan, and Xiang-Yu

More information

AFFINE EXTENSIONS OF FUNCTIONS WITH A CLOSED GRAPH. Marek Wójtowicz and Waldemar Sieg

AFFINE EXTENSIONS OF FUNCTIONS WITH A CLOSED GRAPH. Marek Wójtowicz and Waldemar Sieg Opuscula Math. 35, no. 6 (2015), 973 978 http://dx.doi.org/10.7494/opmath.2015.35.6.973 Opuscula Mathematica AFFINE EXTENSIONS OF FUNCTIONS WITH A CLOSED GRAPH Marek Wójtowicz and Waldemar Sieg Communicated

More information

Some results on stability and on characterization of K-convexity of set-valued functions

Some results on stability and on characterization of K-convexity of set-valued functions ANNALES POLONICI MATHEMATICI LVIII. (1993) Some results on stability and on characterization of K-convexity of set-valued functions by Tiziana Cardinali (Perugia), Kazimierz Nikodem (Bielsko-Bia la) and

More information

Yuqing Chen, Yeol Je Cho, and Li Yang

Yuqing Chen, Yeol Je Cho, and Li Yang Bull. Korean Math. Soc. 39 (2002), No. 4, pp. 535 541 NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY Yuqing Chen, Yeol Je Cho, and Li Yang Abstract. In this paper, we introduce the concept of lower semicontinuous

More information

REMARKS ON THE SCHAUDER TYCHONOFF FIXED POINT THEOREM

REMARKS ON THE SCHAUDER TYCHONOFF FIXED POINT THEOREM Vietnam Journal of Mathematics 28 (2000) 127 132 REMARKS ON THE SCHAUDER TYCHONOFF FIXED POINT THEOREM Sehie Park 1 and Do Hong Tan 2 1. Seoul National University, Seoul, Korea 2.Institute of Mathematics,

More information