Abstract
In this paper, we consider a generalized set-valued mixed equilibrium problem (in short, GSMEP) in real Hilbert space. Related to GSMEP, we consider a generalized Wiener-Hopf equation problem (in short, GWHEP) and show an equivalence relation between them. Further, we give a fixed-point formulation of GWHEP and construct an iterative algorithm for GWHEP. Furthermore, we extend the notion of stability given by Harder and Hick [3] and prove the existence of a solution of GWHEP and discuss the convergence and stability analysis of the iterative algorithm. Our results can be viewed as a refinement and improvement of some known results in the literature.
Highlights
We give a fixed-point formulation of generalized Wiener-Hopf equation problem (GWHEP) and construct an iterative algorithm for GWHEP
We extend the notion of stability given by Harder and Hick [3] and prove the existence of a solution of GWHEP and discuss the convergence and stability analysis of the iterative algorithm
Various kinds of iterative schemes have been proposed for solving equilibrium problems and variational inequalities, see for example [1,2,3,4,5,6,7,8,9,11,12,13,15]
Summary
Equilibrium problems, as the important extension of variational inequalities, have been widely studied in recent years. Shi [15] initially used Wiener-Hopf equation to study the variational inequalities. In 2002, Moudafi [8] has studied the convergence analysis for a mixed equilibrium problem involving single-valued mappings. Many authors given in [2,4,5,6,11,12,15] used various generalizations of WienerHopf equations to develop the iterative algorithms for solving various classes of variational inequalities and mixed equilibrium problems involving single and set-valued mappings. Inspired by the works given in [2,4,5,6,8,11,12,15], in this paper, we consider a generalized set-valued mixed equilibrium problem (GSMEP) in real Hilbert space. By exploiting the technique of this paper, one can generalize and improve the results given in [1,2,3,4,5,6,7,8,9,11,12,13,15]
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