Strong convergence theorem of a hybrid projection algorithm for a family of quasi-φ-asymptotically nonexpansive mappings

Cis a bounded closed convex subset of a Hilbert spaceH,TandS:C→Care two asymptotically nonexpansive mappings such thatST=TS. We establish a strong convergence theorem forSandTin Hilbert space by hybrid method. The results generalize and unify many corresponding results.

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Strong Convergence Theorem for Solving Generalized Mixed Equilibrium Problems and Fixed Point Problems for Total Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

Journal of Applied Mathematics ◽

10.1155/2012/506210 ◽

2012 ◽

Vol 2012 ◽

pp. 1-21

Author(s):

Zhaoli Ma ◽

Lin Wang ◽

Yunhe Zhao

Keyword(s):

Strong Convergence ◽

Banach Spaces ◽

Convergence Theorem ◽

Equilibrium Problems ◽

Nonexpansive Mappings ◽

Strong Convergence Theorem ◽

Generalized Mixed Equilibrium Problems ◽

Asymptotically Nonexpansive ◽

Mixed Equilibrium Problems ◽

Generalized Mixed Equilibrium

We introduce an iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed points for countable families of total quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in an uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results.

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

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2011/956852 ◽

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.

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Modified Hybrid Block Iterative Algorithm for Convex Feasibility Problems and Generalized Equilibrium Problems for Uniformly Quasi-ϕ-Asymptotically Nonexpansive Mappings

Abstract and Applied Analysis ◽

10.1155/2010/357120 ◽

2010 ◽

Vol 2010 ◽

pp. 1-22 ◽

Cited By ~ 14

Author(s):

Siwaporn Saewan ◽

Poom Kumam

Keyword(s):

Equilibrium Problems ◽

Nonexpansive Mappings ◽

Infinite Family ◽

Projection Algorithm ◽

Strong Convergence Theorem ◽

Convex Feasibility Problems ◽

Asymptotically Nonexpansive Mappings ◽

Convex Feasibility ◽

Asymptotically Nonexpansive ◽

Generalized Equilibrium

We introduce a modified block hybrid projection algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results.

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