scholarly journals Weak Convergence Theorems for Strictly Pseudocontractive Mappings and Generalized Mixed Equilibrium Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Jong Soo Jung
Keyword(s):  
Monotone Mapping ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Strictly Pseudocontractive Mapping ◽  
Mixed Equilibrium Problems ◽  
Inverse Strongly Monotone ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium ◽  
Strictly Pseudocontractive Mappings ◽  
New Iterative Method

We introduce a new iterative method for finding a common element of the set of fixed points of a strictly pseudocontractive mapping, the set of solutions of a generalized mixed equilibrium problem, and the set of solutions of a variational inequality problem for an inverse-strongly-monotone mapping in Hilbert spaces and then show that the sequence generated by the proposed iterative scheme converges weakly to a common element of the above three sets under suitable control conditions. The results in this paper substantially improve, develop, and complement the previous well-known results in this area.

scholarly journals Hybrid Algorithms for Minimization Problems over the Solutions of Generalized Mixed Equilibrium and Variational Inclusion Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
Thanyarat Jitpeera ◽  
Poom Kumam
Keyword(s):  
Real Hilbert Space ◽  
Monotone Mapping ◽  
Variational Inclusion ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Inclusion Problems ◽  
Minimization Problems ◽  
Inverse Strongly Monotone ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new general hybrid iterative algorithm for finding a common element of the set of solution of fixed point for a nonexpansive mapping, the set of solution of generalized mixed equilibrium problem, and the set of solution of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (2006), Yao and Liou (2010), Tan and Chang (2011), and other authors.

scholarly journals A General Iterative Algorithm for Generalized Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
Thanyarat Jitpeera ◽  
Poom Kumam
Keyword(s):  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Common Element ◽  
Generalized Mixed Equilibrium Problems ◽  
Mixed Equilibrium Problems ◽  
Inverse Strongly Monotone ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new general iterative method for finding a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (2006), Su et al. (2008), Klin-eam and Suantai (2009), Tan and Chang (2011), and some other authors.

scholarly journals Generalized mixed equilibrium problems and quasi- ϕ-asymptotically nonexpansive multivalued mappings in Banach spaces

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Bashir Ali ◽  
Lawal Umar ◽  
M. H. Harbau
Keyword(s):  
Iterative Algorithms ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Multivalued Mappings ◽  
Strong And Weak Convergence ◽  
Asymptotically Nonexpansive ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium ◽  
Nonexpansive Multivalued Mapping

Abstract In this paper, we introduce two iterative algorithms for finding a common element of the set of fixed points of a quasi- ϕ-asymptotically nonexpansive multivalued mapping and the sets of solutions of generalized mixed equilibrium problem in Banach space. Then, we prove strong and weak convergence of the sequences to element in the mentioned set. Our results generalize and improve recent results announced by many authors.

scholarly journals Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Tanom Chamnarnpan ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
Common Element ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

scholarly journals A New Hybrid Algorithm for a Pair of Quasi--Asymptotically Nonexpansive Mappings and Generalized Mixed Equilibrium Problems in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-22
Author(s):  
Jinhua Zhu ◽  
Shih-Sen Chang
Keyword(s):  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

The purpose of this paper is, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of quasi--asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

scholarly journals The Modified Block Iterative Algorithms for Asymptotically Relatively Nonexpansive Mappings and the System of Generalized Mixed Equilibrium Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Kriengsak Wattanawitoon ◽  
Poom Kumam
Keyword(s):  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Generalized Mixed Equilibrium Problems ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium ◽  
Modified Block

The propose of this paper is to present a modified block iterative algorithm for finding a common element between the set of solutions of the fixed points of two countable families of asymptotically relatively nonexpansive mappings and the set of solution of the system of generalized mixed equilibrium problems in a uniformly smooth and uniformly convex Banach space. Our results extend many known recent results in the literature.

scholarly journals A new Halpern-type algorithm for a generalized mixed equilibrium problem and a countable family of generalized nonexpansive-type maps

2018 ◽  
Vol 34 (2) ◽  
pp. 191-198
Author(s):  
C. E. CHIDUME ◽  
◽  
M. O. NNAKWE ◽  
Keyword(s):  
Equilibrium Problem ◽  
Hilbert Spaces ◽  
Common Fixed Points ◽  
Countable Family ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Mixed Equilibrium Problem ◽  
Uniformly Smooth ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

Let K be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space with dual space E∗. In this paper, a new iterative algorithm of Halpern-type is constructed and used to approximate a common element of a generalized mixed equilibrium problem and a common fixed points for a countable family of generalized nonexpansive-type maps. Application of our theorem, in the case of real Hilbert spaces, complements, extends and improves several important recent results. Finally, we give numerical experiments to illustrate the convergence of our sequence.

scholarly journals A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-39 ◽  
Author(s):  
Pongsakorn Sunthrayuth ◽  
Poom Kumam
Keyword(s):  
Iterative Algorithm ◽  
Equilibrium Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Generalized Mixed Equilibrium Problems ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Furthermore, we prove under some mild conditions that the proposed iterative algorithm converges strongly to a common element of the above four sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results presented in the paper improve and extend the recent ones announced by many others.

scholarly journals Generalized Mixed Equilibrium Problems and Fixed Point Problem for a Countable Family of Total Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Jinhua Zhu ◽  
Shih-Sen Chang ◽  
Min Liu
Keyword(s):  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Countable Family ◽  
Common Element ◽  
Generalized Mixed Equilibrium Problems ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

The purpose of this paper is first to introduce the concept oftotal quasi-ϕ-asymptotically nonexpansive mappingwhich contains many kinds of mappings as its special cases and then to use a hybrid algorithm to introduce a new iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for a countable family of total quasi-ϕ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in an uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

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