Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces

Abstract

The purpose of this paper is to introduce a new hybrid projection method for finding a common element of the set of common fixed points of two relatively quasi-nonexpansive mappings, the set of the variational inequality for anα-inverse-strongly monotone, and the set of solutions of the generalized equilibrium problem in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Base on this result, we also get some new and interesting results. The results in this paper generalize, extend, and unify some well-known strong convergence results in the literature.

Modified Proximal point algorithms for finding a zero point of maximal monotone operators, generalized mixed equilibrium problems and variational inequalities

“…These results were extended to more general Banach spaces see [20,21]. In 2003, Kohsaka and Takahashi [21] introduced the following iterative sequence for a maximal monotone operator A in a smooth and uniformly convex Banach space: x 1 = x E and 13) where J is the duality mapping from E into E* and J r = (I + rA) -1 J.…”

The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings

“…Many authors developed the shrinking projection method for solving (mixed) equilibrium problems and fixed point problems in Hilbert and Banch spaces; see, [12,15,16,[47][48][49][50][51][52][53][54][55][56][57] and references therein.…”

A modified Mann iterative scheme by generalized f-projection for a countable family of relatively quasi-nonexpansive mappings and a system of generalized mixed equilibrium problems

“…The asymptotic behavior of a relatively nonexpansive mapping was studied in [32][33][34]. The class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [11,[32][33][34][35] which requires the strong restriction: F(T) = F(T) . In order to explain this better, we give the following example [36] of relatively quasi-nonexpansive mappings which is not relatively nonexpansive mapping.…”