scholarly journals A System of Mixed Equilibrium Problems, a General System of Variational Inequality Problems for Relaxed Cocoercive, and Fixed Point Problems for Nonexpansive Semigroup and Strictly Pseudocontractive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-35 ◽  
Author(s):  
Poom Kumam ◽  
Phayap Katchang
Keyword(s):  
Fixed Points ◽  
Iterative Algorithm ◽  
Equilibrium Problems ◽  
General System ◽  
Common Fixed Points ◽  
Iterative Sequence ◽  
Pseudocontractive Mappings ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium ◽  
Strictly Pseudocontractive Mappings

We introduce an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a general system of variational inequalities for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in Hilbert spaces. Furthermore, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems. Our results extend and improve the recent results of Chang et al. (2010) and many others.

scholarly journals Common Solutions of Generalized Mixed Equilibrium Problems, Variational Inclusions, and Common Fixed Points for Nonexpansive Semigroups and Strictly Pseudocontractive Mappings

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
Poom Kumam ◽  
Usa Hamphries ◽  
Phayap Katchang
Keyword(s):  
Fixed Points ◽  
Equilibrium Problems ◽  
Common Fixed Points ◽  
Common Element ◽  
Monotone Mappings ◽  
Generalized Mixed Equilibrium Problems ◽  
Pseudocontractive Mappings ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Strictly Pseudocontractive Mappings

We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez-Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors.

scholarly journals Convergence Theorems for BregmanK-Mappings and Mixed Equilibrium Problems in Reflexive Banach Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-18 ◽  
Author(s):  
Bashir Ali ◽  
M. H. Harbau
Keyword(s):  
Fixed Points ◽  
Equilibrium Problem ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Common Fixed Points ◽  
Finite Family ◽  
Iterative Sequence ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new mixed equilibrium problem with a relaxed monotone mapping in a reflexive Banach space and prove the existence of solution of the equilibrium problem. Using Bregman distance, we introduce the concept of BregmanK-mapping for a finite family of Bregman quasiasymptotically nonexpansive mappings and show the fixed point set of the BregmanK-mapping is the set of common fixed points of{Ti}i=1N. Using the BregmanK-mapping, we introduce an iterative sequence for finding a common point in the set of a common fixed points of the finite family of Bregman quasiasymptotically nonexpansive mappings and the set of solutions of some mixed equilibrium problems. Strong convergence of the iterative sequence is proved. Our results generalise and improve many recent results in the literature.

scholarly journals A New Hybrid Algorithm for Solving a System of Generalized Mixed Equilibrium Problems, Solving a Family of Quasi--Asymptotically Nonexpansive Mappings, and Obtaining Common Fixed Points in Banach Space

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
J. F. Tan ◽  
S. S. Chang
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Generalized Mixed Equilibrium Problems ◽  
Asymptotically Nonexpansive ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

The main purpose of this paper is to introduce a new hybrid iterative scheme for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, set of common fixed points of a family of quasi--asymptotically nonexpansive mappings, and null spaces of finite family of -inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. The results presented in the paper improve and extend the corresponding results announced by some authors.

scholarly journals An iterative algorithm for the system of split mixed equilibrium problem

2020 ◽  
Vol 53 (1) ◽  
pp. 309-324
Author(s):  
Ibrahim Karahan ◽  
Lateef Olakunle Jolaoso
Keyword(s):  
Variational Inequality ◽  
Iterative Algorithm ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Equilibrium System ◽  
Convergence Conditions ◽  
Mixed Equilibrium Problems ◽  
Split Variational Inequality ◽  
Mixed Equilibrium ◽  
Split Mixed Equilibrium Problem

AbstractIn this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.

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