Generalized mixed equilibrium problem in Banach spaces

Abstract

This paper uses a hybrid algorithm to find a common element of the set of solutions to a generalized mixed equilibrium problem, the set of solutions to variational inequality problems, and the set of common fixed points for a finite family of quasi-φ-nonexpansive mappings in a uniformly smooth and strictly convex Banach space. As applications, we utilize our results to study the optimization problem. It shows that our results improve and extend the corresponding results announced by many others recently.

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

“…Then they proved that the sequence {x n } converges strongly to Π C x, where Π C is the generalized projection from E onto C. In this article, motivated and inspired by Kamimura et al [22], Li and Song [23], Iiduka and Takahashi [17], Zhang [24] and Inoue et al [25], we introduce the new hybrid algorithm (3.1) below. Under appropriate difference conditions, we will prove that the sequence {x n } generated by algorithms (3.1) converges strongly to the point…”

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

“…( [66]) It follows from Lemma 2.12 that the mapping K r : C C defined by (2.3) is a relatively nonexpansive mapping. Thus, it is quasi-j-nonexpansive.…”

“…( [66]) Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let B : C E* be a continuous and monotone mapping, : C ℝ be a lower semi-continuous and convex function, and f be a bifunction from C × C to ℝ satisfying (A1) -(A4). For r >0 and × E, then there exists u C such that…”

Hybrid shrinking iterative solutions to convex feasibility problems and a system of generalized mixed equilibrium problems

“…In [28], the author gives an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping.…”