Hybrid Methods for Solving Simultaneously an Equilibrium Problem and Countably Many Fixed Point Problems in a Hilbert Space

This paper presents a framework of iterative methods for finding a common solution to an equilibrium problem and a countable number of fixed point problems defined in a Hilbert space. A general strong convergence theorem is established under mild conditions. Two hybrid methods are derived from the p...

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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 160; no. 3; pp. 809 - 831
Main Authors: Nguyen, Thi Thu Van, Strodiot, Jean Jacques, Nguyen, Van Hien
Format: Journal Article
Language:English
Published: Boston Springer US 01.03.2014
Springer Nature B.V
Subjects:
Algorithms
Analysis
Applications of Mathematics
Approximation
Banach spaces
Calculus of Variations and Optimal Control; Optimization
Convergence
Engineering
Equilibrium
Fixed points (mathematics)
Hilbert space
Iterative methods
Joining
Mathematics
Mathematics and Statistics
Methods
Operations Research/Decision Theory
Optimization
Strategy
Studies
Theorems
Theory of Computation
Hybrid extragradient methods
Proximal point method
Equilibrium problems
Fixed-point problems
Armijo backtracking linesearch
Variational inequalities
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary:This paper presents a framework of iterative methods for finding a common solution to an equilibrium problem and a countable number of fixed point problems defined in a Hilbert space. A general strong convergence theorem is established under mild conditions. Two hybrid methods are derived from the proposed framework in coupling the fixed point iterations with the iterations of the proximal point method or the extragradient method, which are well-known methods for solving equilibrium problems. The strategy is to obtain the strong convergence from the weak convergence of the iterates without additional assumptions on the problem data. To achieve this aim, the solution set of the problem is outer approximated by a sequence of polyhedral subsets.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-013-0400-y