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 Strong Convergence Theorems for a Generalized Mixed Equilibrium Problem and a Family of Total Quasi--Asymptotically Nonexpansive Multivalued Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
J. F. Tan ◽  
S. S. Chang
Keyword(s):  
Equilibrium Problem ◽  
Infinite Family ◽  
Convex Banach Space ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Multivalued Mappings ◽  
Mixed Equilibrium Problem ◽  
Asymptotically Nonexpansive ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

The main purpose of this paper is by using a hybrid algorithm to find a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of common fixed points for a infinite family of total quasi--asymptotically nonexpansive multivalued mapping in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results announced by some authors.

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 Hybrid Extragradient Method with Regularization for Convex Minimization, Generalized Mixed Equilibrium, Variational Inequality and Fixed Point Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-27 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Juei-Ling Ho
Keyword(s):  
Real Hilbert Space ◽  
Extragradient Method ◽  
Iterative Algorithms ◽  
Common Element ◽  
Monotone Mappings ◽  
Strong And Weak Convergence ◽  
Generalized Mixed Equilibrium ◽  
Hybrid Extragradient Method ◽  
Mixed Equilibrium ◽  
Fréchet Differentiable

We introduce two iterative algorithms by the hybrid extragradient method with regularization for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings and the set of fixed points of an asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions.

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.

scholarly journals A New Hybrid Algorithm for a System of Generalized Mixed Equilibrium Problems and Fixed Point of Quasi--Asymptotically Nonexpansive Mappings

2011 ◽  
Vol 2011 ◽  
pp. 1-18
Author(s):  
Jinhua Zhu ◽  
Shih-Sen Chang
Keyword(s):  
Hybrid Algorithm ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
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 to use a new hybrid algorithm 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 an infinite family 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 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 Strong Convergence Theorems for Countable Families of Uniformly Quasi-ϕ-Asymptotically Nonexpansive Mappings and a System of Generalized Mixed Equilibrium Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-27 ◽  
Author(s):  
Siwaporn Saewan ◽  
Poom Kumam
Keyword(s):  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
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 to present a new hybrid block iterative scheme by the generalizedf- projection method for finding a common element of the fixed point set for a countable family of uniformly quasi-ϕ-asymptotically nonexpansive mappings and the set of solutions of the system of generalized mixed equilibrium problems in a strictly convex and uniformly smooth Banach space with the Kadec-Klee property. Furthermore, we prove that our new iterative scheme converges strongly to a common element of the aforementioned sets. The results presented in this paper improve and extend important recent results in the literature.

scholarly journals Existence and Strong Convergence Theorems for Generalized Mixed Equilibrium Problems of a Finite Family of Asymptotically Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Rabian Wangkeeree ◽  
Hossein Dehghan ◽  
Pakkapon Preechasilp
Keyword(s):  
Strong Convergence ◽  
Equilibrium Problem ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Finite Family ◽  
Generalized Mixed Equilibrium Problem ◽  
Mixed Equilibrium Problem ◽  
Asymptotically Nonexpansive ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We first prove the existence of solutions for a generalized mixed equilibrium problem under the new conditions imposed on the given bifunction and introduce the algorithm for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of finite family of asymptotically nonexpansive mappings. Next, the strong convergence theorems are obtained, under some appropriate conditions, in uniformly convex and smooth Banach spaces. The main results extend various results existing in the current literature.

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.

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