SYSTEMS OPTIMIZATION LABORATORY DEPARTMENT OF OPERATIONS RESEARCH STANFORD UNIVERSITY STANFORD, CALIFORNIA
|
|
- Arline Taylor
- 5 years ago
- Views:
Transcription
1 SYSTEMS OPTIMIZATION LABORATORY DEPARTMENT OF OPERATIONS RESEARCH STANFORD UNIVERSITY STANFORD, CALIFORNIA NIN DTIC A Monotone Complementarity Problem in Hilbert Space by Jen-Chih Yao fl ELECTE MI TECHNICAL REPORT SOL 90-7 APR April 1990 Research and reproduction of this report were partially supported by the National Science Foundation grant DMS ; Office of Naval Research grant N J-1659, and the Department of Energy grant DE-FG03-87ER Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do NOT necessarily reflect the views of the above sponsors. Reproduction in whole or in part is permitted for any purposes of the United States Government. This document has been approved for public release and sale; its distribution is unlimited. oq oo o-3
2 A Monotone Complementarity Problem.>-. in Hilbert Space Jen-Chih Yao AcSloft For April 1990 NTIS CRA & DTJC TAB 0 Department of Operations Research U0; ij f n o c ed Stanford University Stanford, CA By D..triti,, I Abstract Dist, I An existence theorem for a complementarity problem involving a weakly coercive monotone mapping over an arbitrary closed convex cone in a real Hilbert space is established Introduction Let H be a real Hilbert space with inner product (.,.) and norm ". Let K be a nonempty subset of H and f be a mapping from K into H. f is said to be weakly coercive if f is said to be monotone if (x,f(x)) --- o as Vxjl --- oo and x E K. (x - y,f(x) - f(y)) 0 for all x,y E K.
3 f is said to be strictly monotone if the above inequality is strict whenever x and y are distinct. f is said to be strongly monotone if there exists a positive number c such that (x - y,f(x) - f(y)) _ cllax - yll 2 for all x,y E K. A subset K of a real Hilbert space H is said to be a cone if Ax E K for all x E K and all A > 0. Let K be a dosed convex cone in H and dual cone K*, that is, K* ={uehi(ua' )0,Va'EK}. The complementarity problem (CP) is to find x E K such that f(x) E K" and (x,f(x)) = 0. (1) Problem (1) was formulated by Karamardian [7] and has been extensively studied in the literature. See, e.g., [2, 4, 5, 6, 7, 8] and the references therein. The purpose of this paper is to prove an existence theorem for a complementarity problem involving a weakly coercive monotone mapping over an arbitrary closed convex cone in a real Hilbert space. The main result extends some existing results; the method of the proof of this main result is to consider the family of finite-dimensional subspaces by using the known results for finite-dimensional spaces and to show that a certain net of solutions from such subspaces converges to a solution to CP. 2. The Main Result Now we prove the main result. Theorem 2.1. Let K be a closed convex cone in the real Hilbert space H. Let f be a weakly coercive monotone mapping from K into H which is continuous on K fl U for any 2
4 7 finite-dimensional subspace U of H. Then there exists x E K such that f(x) E K* and (x, f(x)) = 0. Proof. Let U be any finite-dimensional subspace of H with K fn U #. and let Pu be the orthogonal projection of H onto U. Let fu = Puf be the composition of Pu and f. Since the adjoint P of Pu is itself, we have limlul-o.oo, ueknu (u,fu(u)) = 00. Therefore fu is weakly coercive on K nl U. By 1, Corollary 2.3), there exists xu E K nl U such that (u-xu, f(xu)) O for allueknu. (2) Then by [7, Lemma 3.1], we have fu(xu) E (K n U)" and (xu,f(xu)) = 0. (3) Let A be the family of all finite-dimensional subspaces U of H with K nl U # 0 and Ku = {xv I U C V E A}. Since f is weakly coercive, it follows from (3) that there exists a constant r > 0 so that Ku C 3,. for all U E A where D, is the closure of the ball with center at 0 and radius r. For U E A, let /u' be the weak closure of Ku. Then the family I u I U E A} has the finite intersection property. Indeed, for U, V E A, let W E A be such that U U V C W. Then Ku fl Kv D Kw # 0. Since F, is weakly compact and Ku" C 3, for all U E A, it follows that nuea Ku/' # 0. Let x E n UEA U'. Suppose u E K is arbitrary and let U E A contain u. Since Ku is bounded and x E IRuW, there exists a sequence {xn} C Ku which converges to x weakly. Since f is monotone, by (2) we have (u - xz,f(u)) > 0 for all n. 3
5 Since (u -., f(u)) is wealdy continuous, we have (u -x,f(u)) > 0 for all u E K. For any u E K and any 0 < t < 1, let ut = tu + (1 - t)x. By substituting ut into (2), we have (ut- X,f(ut)) _ 0 for allo<t< 1. (4) Letting t approach 0 in (4), we get By (5) and [7, Lemma 3.1], it follows that (u -X, f(w)) _ 0 for all u E K. (5) f(x) E K* and (x,f(z)) = 0. The next corollary follows from Theorem 2.1 directly. Corollary 2.2. Let K be a closed convex cone in the real Hilbert space H. Let f be a mapping from K into H which is continuous on K fl U for any finite-dimensional subspace U of H. Then there exists a unique x E K such that f(z) E K* and (z,f(x)) = 0 under each of the following conditions: I. f is strictly monotone and weakly coercive, 2. f is strongly monotone. We note that Corollary extends a result of Nanda and Nanda [8, Theorem] where f is assumed to be strongly monotone and Lipschitzian. 4
6 References 1. M. Aganagi6, "Variational inequalities and generalized complementarity problems," Technical Report SOL 78-11, System Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, California, [See also "Contributions to Complementarity Theory," Ph.D. thesis, Department of Operations Research, Stanford University, Stanford, California, July 1978.] 2. M. S. Bazaraa, J. J. Goode and M. Z. Nashed, A nonlinear complementarity problem in mathematical programming in Banach space, Proc. Amer. Math. Soc. 35 (1972), K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, G. Isac and M. Th6ra, Complementarity problem and the existence of the post-critical equilibrium state of a thin elastic plate, J. Optim. Theory Appl. 58 (1988), G. Isac, Nonlinear complementarity problem and Galerkin method, J. Math. Anal. Appl. 108 (1985), S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl. 18 (1976), S. Karamardian, Generalized complementarity problem, J. Optim. Theory Appl. 8 (1971), S. Nanda and S. Nanda, A nonlinear complementarity problem in mathematical programming in Hilbert space, Bull. Austral. Math. Soc. 20 (1979),
7 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Man Da0nIW4 REPORT DOCMkENTA.TION PAGE RFR OPZDGFR I. REORT NMBERGOVT ACCESSION NO4. RECIPIENT'lS CATALOG NUMBER SOL r*-s. TYPEf OF REPORT G PERIOD COVERED A Monotone Complementarity Problem in Hilbert Space Technical Rteport 6. PERPO@ORG 1. REPORT NummaR 7. AUTHORWa 0. CONTRACT Oft GRANT NUMUEW@') Jen-Chih Yao NOOOl PERFORMING ORGANIZATION NAKE AND ADDRESS I. PROGRAM ELEMENT PROJECT, TASK AREAS 6ORK UNIT NUMBER$ Department of Operations Research - SOL Stanford University 1111 MA Stanford, CA It. CONTROLLING OFFICE 1NMEANO ADDRESS $11- REPORT DATE Office of Naval Research - Dept. of the Navy April N. Quincy Street IS. NUMBER OFPAGEts Arlington, VA Up IS. SECURITY CLASS. (of enie repen) UNCLASSIFIED I"a DECJ.ASMJICATIOM/DOWNGRACING 10. DISTRIBUTION STATEMENT (of ilif Rep.i) This document has been approved for public release and sale; its distribution is unlimited. 17. DISTRIBUTION STATEMENT (0900 abel 00m6di 81. "*., HAD@ df* S RSmt) IS. SUPPLEMENTARY NOTES IS. KEY WORDS (CmI*.u. as etvrn. aid. 110"00 a..m me Of 6F*S 410-*--&-)~ Hilbert space; complementarity problem; monotone mapping; weakly coercive mapping. 20. LUST RACT (Cendvee uowe side 41 aeee0ewr -and~& 6 W-* -0n000 An existence theorem for a complementarity problem involving a weakly coercive monotone mapping over an arbitrary closed convex cone in a real Hilbert space is established. DD AN EDITION or i Novsoi *mg-jg nes SECURITY CLASSIICATON OP "WIS PAGE (Vb WS ~
WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES
Fixed Point Theory, 12(2011), No. 2, 309-320 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES S. DHOMPONGSA,
More informationSTRONG 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 informationGeneralized 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 informationShih-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 informationSTRONG 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 informationViscosity 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 informationCONVERGENCE 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 informationSolution 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 informationOn 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 informationIterative algorithms based on the hybrid steepest descent method for the split feasibility problem
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (206), 424 4225 Research Article Iterative algorithms based on the hybrid steepest descent method for the split feasibility problem Jong Soo
More informationOn 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 informationCONVERGENCE 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 informationSTRONG CONVERGENCE OF A MODIFIED ISHIKAWA ITERATIVE ALGORITHM FOR LIPSCHITZ PSEUDOCONTRACTIVE MAPPINGS
J. Appl. Math. & Informatics Vol. 3(203), No. 3-4, pp. 565-575 Website: http://www.kcam.biz STRONG CONVERGENCE OF A MODIFIED ISHIKAWA ITERATIVE ALGORITHM FOR LIPSCHITZ PSEUDOCONTRACTIVE MAPPINGS M.O. OSILIKE,
More informationThe 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 informationGENERAL 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 informationON 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 informationExistence of Solutions to Split Variational Inequality Problems and Split Minimization Problems in Banach Spaces
Existence of Solutions to Split Variational Inequality Problems and Split Minimization Problems in Banach Spaces Jinlu Li Department of Mathematical Sciences Shawnee State University Portsmouth, Ohio 45662
More informationWEAK 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 informationBEST 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 informationSTRONG 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 informationPROJECTIONS 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 informationE.7 Alaoglu s Theorem
E.7 Alaoglu s Theorem 359 E.7 Alaoglu s Theorem If X is a normed space, then the closed unit ball in X or X is compact if and only if X is finite-dimensional (Problem A.25). Even so, Alaoglu s Theorem
More informationON SOME RESULTS ON LINEAR ORTHOGONALITY SPACES
ASIAN JOURNAL OF MATHEMATICS AND APPLICATIONS Volume 2014, Article ID ama0155, 11 pages ISSN 2307-7743 http://scienceasia.asia ON SOME RESULTS ON LINEAR ORTHOGONALITY SPACES KANU, RICHMOND U. AND RAUF,
More information1 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 informationON 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 informationViscosity Approximative Methods for Nonexpansive Nonself-Mappings without Boundary Conditions in Banach Spaces
Applied Mathematical Sciences, Vol. 2, 2008, no. 22, 1053-1062 Viscosity Approximative Methods for Nonexpansive Nonself-Mappings without Boundary Conditions in Banach Spaces Rabian Wangkeeree and Pramote
More informationFUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES. Yılmaz Yılmaz
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 33, 2009, 335 353 FUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES Yılmaz Yılmaz Abstract. Our main interest in this
More informationOn 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 informationSCALARIZATION 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("-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 informationYET 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 informationHAIYUN 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 informationIterative Convex Optimization Algorithms; Part One: Using the Baillon Haddad Theorem
Iterative Convex Optimization Algorithms; Part One: Using the Baillon Haddad Theorem Charles Byrne (Charles Byrne@uml.edu) http://faculty.uml.edu/cbyrne/cbyrne.html Department of Mathematical Sciences
More informationSOME GENERALIZATION OF MINTY S LEMMA. Doo-Young Jung
J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 6(1999), no 1. 33 37 SOME GENERALIZATION OF MINTY S LEMMA Doo-Young Jung Abstract. We obtain a generalization of Behera and Panda s result on nonlinear
More informationOn 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 informationAvailable online at J. Nonlinear Sci. Appl., 10 (2017), Research Article
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 2719 2726 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa An affirmative answer to
More informationThe Split Hierarchical Monotone Variational Inclusions Problems and Fixed Point Problems for Nonexpansive Semigroup
International Mathematical Forum, Vol. 11, 2016, no. 8, 395-408 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.6220 The Split Hierarchical Monotone Variational Inclusions Problems and
More informationA Relaxed Explicit Extragradient-Like Method for Solving Generalized Mixed Equilibria, Variational Inequalities and Constrained Convex Minimization
, March 16-18, 2016, Hong Kong A Relaxed Explicit Extragradient-Like Method for Solving Generalized Mixed Equilibria, Variational Inequalities and Constrained Convex Minimization Yung-Yih Lur, Lu-Chuan
More informationApproximating Fixed Points of Asymptotically Quasi-Nonexpansive Mappings by the Iterative Sequences with Errors
5 10 July 2004, Antalya, Turkey Dynamical Systems and Applications, Proceedings, pp. 262 272 Approximating Fixed Points of Asymptotically Quasi-Nonexpansive Mappings by the Iterative Sequences with Errors
More informationCommon fixed points of two generalized asymptotically quasi-nonexpansive mappings
An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) Tomul LXIII, 2017, f. 2 Common fixed points of two generalized asymptotically quasi-nonexpansive mappings Safeer Hussain Khan Isa Yildirim Received: 5.VIII.2013
More informationITERATIVE 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 informationAN ASYMPTOTIC MINTY S TYPE VARIATIONAL INEQUALITY WITH PSEUDOMONOTONE OPERATORS. G. Isac
Nonlinear Analysis Forum 8(1), pp. 55 64, 2003 AN ASYMPTOTIC MINTY S TYPE VARIATIONAL INEQUALITY WITH PSEUDOMONOTONE OPERATORS G. Isac Department of Mathematics Royal Military College of Canada P.O. Box
More informationAn iterative method for fixed point problems and variational inequality problems
Mathematical Communications 12(2007), 121-132 121 An iterative method for fixed point problems and variational inequality problems Muhammad Aslam Noor, Yonghong Yao, Rudong Chen and Yeong-Cheng Liou Abstract.
More informationCONVERGENCE 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 informationScalar 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 informationMonotone variational inequalities, generalized equilibrium problems and fixed point methods
Wang Fixed Point Theory and Applications 2014, 2014:236 R E S E A R C H Open Access Monotone variational inequalities, generalized equilibrium problems and fixed point methods Shenghua Wang * * Correspondence:
More informationStrong Convergence Theorems for Nonself I-Asymptotically Quasi-Nonexpansive Mappings 1
Applied Mathematical Sciences, Vol. 2, 2008, no. 19, 919-928 Strong Convergence Theorems for Nonself I-Asymptotically Quasi-Nonexpansive Mappings 1 Si-Sheng Yao Department of Mathematics, Kunming Teachers
More informationA convergence result for an Outer Approximation Scheme
A convergence result for an Outer Approximation Scheme R. S. Burachik Engenharia de Sistemas e Computação, COPPE-UFRJ, CP 68511, Rio de Janeiro, RJ, CEP 21941-972, Brazil regi@cos.ufrj.br J. O. Lopes Departamento
More informationOn 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 informationA Note of the Strong Convergence of the Mann Iteration for Demicontractive Mappings
Applied Mathematical Sciences, Vol. 10, 2016, no. 6, 255-261 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.511700 A Note of the Strong Convergence of the Mann Iteration for Demicontractive
More informationContents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping.
Minimization Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. 1 Minimization A Topological Result. Let S be a topological
More informationFixed 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 informationBrøndsted-Rockafellar property of subdifferentials of prox-bounded functions. Marc Lassonde Université des Antilles et de la Guyane
Conference ADGO 2013 October 16, 2013 Brøndsted-Rockafellar property of subdifferentials of prox-bounded functions Marc Lassonde Université des Antilles et de la Guyane Playa Blanca, Tongoy, Chile SUBDIFFERENTIAL
More informationIterative 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 informationSecond Order Elliptic PDE
Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic
More informationFIXED POINT THEOREM FOR NONEXPANSrVE SEMIGROUPS ON BANACH SPACE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 122, Number 4, December 1994 FIXED POINT THEOREM FOR NONEXPANSrVE SEMIGROUPS ON BANACH SPACE WATARU TAKAHASHI AND DOO HOAN JEONG (Communicated by
More informationSynchronal 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(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 informationVariational inequalities for set-valued vector fields on Riemannian manifolds
Variational inequalities for set-valued vector fields on Riemannian manifolds Chong LI Department of Mathematics Zhejiang University Joint with Jen-Chih YAO Chong LI (Zhejiang University) VI on RM 1 /
More informationCONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES
International Journal of Analysis and Applications ISSN 2291-8639 Volume 8, Number 1 2015), 69-78 http://www.etamaths.com CONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES
More informationStrong Convergence of the Mann Iteration for Demicontractive Mappings
Applied Mathematical Sciences, Vol. 9, 015, no. 4, 061-068 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.015.5166 Strong Convergence of the Mann Iteration for Demicontractive Mappings Ştefan
More informationA NOTE ON LINEAR FUNCTIONAL NORMS
A NOTE ON LINEAR FUNCTIONAL NORMS YIFEI PAN AND MEI WANG Abstract. For a vector u in a normed linear space, Hahn-Banach Theorem provides the existence of a linear functional f, f(u) = u such that f = 1.
More informationarxiv: 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 informationAnalysis Finite and Infinite Sets The Real Numbers The Cantor Set
Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered
More informationSHRINKING 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 informationThe equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators
Mathematical Communications 10(2005), 81-88 81 The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators Ştefan M. Şoltuz Abstract. We show that the Ishikawa iteration,
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics ON SOME APPROXIMATE FUNCTIONAL RELATIONS STEMMING FROM ORTHOGONALITY PRESERVING PROPERTY JACEK CHMIELIŃSKI Instytut Matematyki, Akademia Pedagogiczna
More informationSome unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces
An. Şt. Univ. Ovidius Constanţa Vol. 19(1), 211, 331 346 Some unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces Yonghong Yao, Yeong-Cheng Liou Abstract
More informationSteepest descent approximations in Banach space 1
General Mathematics Vol. 16, No. 3 (2008), 133 143 Steepest descent approximations in Banach space 1 Arif Rafiq, Ana Maria Acu, Mugur Acu Abstract Let E be a real Banach space and let A : E E be a Lipschitzian
More informationReceived 8 June 2003 Submitted by Z.-J. Ruan
J. Math. Anal. Appl. 289 2004) 266 278 www.elsevier.com/locate/jmaa The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense
More informationWeak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings
Mathematica Moravica Vol. 20:1 (2016), 125 144 Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings G.S. Saluja Abstract. The aim of
More informationVariational Inequalities. Anna Nagurney Isenberg School of Management University of Massachusetts Amherst, MA 01003
Variational Inequalities Anna Nagurney Isenberg School of Management University of Massachusetts Amherst, MA 01003 c 2002 Background Equilibrium is a central concept in numerous disciplines including economics,
More informationPolarization constant K(n, X) = 1 for entire functions of exponential type
Int. J. Nonlinear Anal. Appl. 6 (2015) No. 2, 35-45 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.252 Polarization constant K(n, X) = 1 for entire functions of exponential type A.
More informationStability of Adjointable Mappings in Hilbert
Stability of Adjointable Mappings in Hilbert arxiv:math/0501139v2 [math.fa] 1 Aug 2005 C -Modules M. S. Moslehian Abstract The generalized Hyers Ulam Rassias stability of adjointable mappings on Hilbert
More informationOn Semicontinuity of Convex-valued Multifunctions and Cesari s Property (Q)
On Semicontinuity of Convex-valued Multifunctions and Cesari s Property (Q) Andreas Löhne May 2, 2005 (last update: November 22, 2005) Abstract We investigate two types of semicontinuity for set-valued
More informationSTRONG CONVERGENCE OF AN IMPLICIT ITERATION PROCESS FOR ASYMPTOTICALLY NONEXPANSIVE IN THE INTERMEDIATE SENSE MAPPINGS IN BANACH SPACES
Kragujevac Journal of Mathematics Volume 36 Number 2 (2012), Pages 237 249. STRONG CONVERGENCE OF AN IMPLICIT ITERATION PROCESS FOR ASYMPTOTICALLY NONEXPANSIVE IN THE INTERMEDIATE SENSE MAPPINGS IN BANACH
More informationConvergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces
Mathematica Moravica Vol. 19-1 2015, 33 48 Convergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces Gurucharan Singh Saluja Abstract.
More informationThe structure of ideals, point derivations, amenability and weak amenability of extended Lipschitz algebras
Int. J. Nonlinear Anal. Appl. 8 (2017) No. 1, 389-404 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2016.493 The structure of ideals, point derivations, amenability and weak amenability
More informationA New Modified Gradient-Projection Algorithm for Solution of Constrained Convex Minimization Problem in Hilbert Spaces
A New Modified Gradient-Projection Algorithm for Solution of Constrained Convex Minimization Problem in Hilbert Spaces Cyril Dennis Enyi and Mukiawa Edwin Soh Abstract In this paper, we present a new iterative
More informationSum of two maximal monotone operators in a general Banach space is maximal
arxiv:1505.04879v1 [math.fa] 19 May 2015 Sum of two maximal monotone operators in a general Banach space is maximal S R Pattanaik, D K Pradhan and S Pradhan May 20, 2015 Abstract In a real Banach space,
More informationProblem Set 6: Solutions Math 201A: Fall a n x n,
Problem Set 6: Solutions Math 201A: Fall 2016 Problem 1. Is (x n ) n=0 a Schauder basis of C([0, 1])? No. If f(x) = a n x n, n=0 where the series converges uniformly on [0, 1], then f has a power series
More informationChapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems
Chapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems Elliptic boundary value problems often occur as the Euler equations of variational problems the latter
More informationFixed Point Theorem in Cone B-Metric Spaces Using Contractive Mappings
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 7 (2017), pp. 2997-3004 Research India Publications http://www.ripublication.com Fixed Point Theorem in Cone B-Metric Spaces
More informationSOME 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 informationTHE DUAL FORM OF THE APPROXIMATION PROPERTY FOR A BANACH SPACE AND A SUBSPACE. In memory of A. Pe lczyński
THE DUAL FORM OF THE APPROXIMATION PROPERTY FOR A BANACH SPACE AND A SUBSPACE T. FIGIEL AND W. B. JOHNSON Abstract. Given a Banach space X and a subspace Y, the pair (X, Y ) is said to have the approximation
More informationA regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces
Bin Dehaish et al. Journal of Inequalities and Applications (2015) 2015:51 DOI 10.1186/s13660-014-0541-z R E S E A R C H Open Access A regularization projection algorithm for various problems with nonlinear
More informationDirection of Movement of the Element of Minimal Norm in a Moving Convex Set
Journal of Convex Analysis Volume 14 (2007), No. 3, 455 463 Direction of Movement of the Element of Minimal Norm in a Moving Convex Set Renu Choudhary Department of Mathematics, University of Auckl, Private
More informationAnother Low-Technology Estimate in Convex Geometry
Convex Geometric Analysis MSRI Publications Volume 34, 1998 Another Low-Technology Estimate in Convex Geometry GREG KUPERBERG Abstract. We give a short argument that for some C > 0, every n- dimensional
More informationViscosity approximation methods for nonexpansive nonself-mappings
J. Math. Anal. Appl. 321 (2006) 316 326 www.elsevier.com/locate/jmaa Viscosity approximation methods for nonexpansive nonself-mappings Yisheng Song, Rudong Chen Department of Mathematics, Tianjin Polytechnic
More informationThe Nearest Doubly Stochastic Matrix to a Real Matrix with the same First Moment
he Nearest Doubly Stochastic Matrix to a Real Matrix with the same First Moment William Glunt 1, homas L. Hayden 2 and Robert Reams 2 1 Department of Mathematics and Computer Science, Austin Peay State
More informationOn the Weak Convergence of the Extragradient Method for Solving Pseudo-Monotone Variational Inequalities
J Optim Theory Appl 208) 76:399 409 https://doi.org/0.007/s0957-07-24-0 On the Weak Convergence of the Extragradient Method for Solving Pseudo-Monotone Variational Inequalities Phan Tu Vuong Received:
More informationA general iterative algorithm for equilibrium problems and strict pseudo-contractions in Hilbert spaces
A general iterative algorithm for equilibrium problems and strict pseudo-contractions in Hilbert spaces MING TIAN College of Science Civil Aviation University of China Tianjin 300300, China P. R. CHINA
More informationA Solution Method for Semidefinite Variational Inequality with Coupled Constraints
Communications in Mathematics and Applications Volume 4 (2013), Number 1, pp. 39 48 RGN Publications http://www.rgnpublications.com A Solution Method for Semidefinite Variational Inequality with Coupled
More informationThe 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 informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationRelaxed Quasimonotone Operators and Relaxed Quasiconvex Functions
J Optim Theory Appl (2008) 138: 329 339 DOI 10.1007/s10957-008-9382-6 Relaxed Quasimonotone Operators and Relaxed Quasiconvex Functions M.R. Bai N. Hadjisavvas Published online: 12 April 2008 Springer
More informationFURTHER STUDIES OF STRONGLY AMENABLE -REPRESENTATIONS OF LAU -ALGEBRAS
FURTHER STUDIES OF STRONGLY AMENABLE -REPRESENTATIONS OF LAU -ALGEBRAS FATEMEH AKHTARI and RASOUL NASR-ISFAHANI Communicated by Dan Timotin The new notion of strong amenability for a -representation of
More informationAffine iterations on nonnegative vectors
Affine iterations on nonnegative vectors V. Blondel L. Ninove P. Van Dooren CESAME Université catholique de Louvain Av. G. Lemaître 4 B-348 Louvain-la-Neuve Belgium Introduction In this paper we consider
More informationConvergence of Ishikawa Iterative Sequances for Lipschitzian Strongly Pseudocontractive Operator
Australian Journal of Basic Applied Sciences, 5(11): 602-606, 2011 ISSN 1991-8178 Convergence of Ishikawa Iterative Sequances for Lipschitzian Strongly Pseudocontractive Operator D. Behmardi, L. Shirazi
More informationStrong convergence theorems for asymptotically nonexpansive nonself-mappings with applications
Guo et al. Fixed Point Theory and Applications (2015) 2015:212 DOI 10.1186/s13663-015-0463-6 R E S E A R C H Open Access Strong convergence theorems for asymptotically nonexpansive nonself-mappings with
More information