Abstract
In this paper, the common solution problem (P1) of generalized equilibrium problems for a system of inverse-strongly monotone mappings and a system of bifunctions satisfying certain conditions, and the common fixed-point problem (P2) for a family of uniformly quasi-ϕ-asymptotically nonexpansive and locally uniformly Lipschitz continuous or uniformly Hölder continuous mappings are proposed. A new iterative sequence is constructed by using the generalized projection and hybrid method, and a strong convergence theorem is proved on approximating a common solution of (P1) and (P2) in Banach space.2000 MSC: 26B25, 40A05
Highlights
Common solution problems with their applications have been discussed
In 2010, Chang et al [10] discussed the common solution of a generalized equilibrium problem and a common fixed-point problem for two relatively nonexpansive mappings, and established a strong convergence theorem on the common solution problem
Chang et al [11] established a strong convergence theorem on solving the common fixed-point problem for a family of uniformly quasi-j-asymptotically nonexpansive and uniformly Lipschitz continuous mappings in a uniformly smooth and strictly convex Banach space with the KadecKlee property
Summary
Common solution problems (i.e., to find a common element of the set of solutions of equilibrium problems and/or the set of fixed points of mappings and/or the set of solutions of variational inequalities) with their applications have been discussed. F (Si) fixed points of Si. Motivated by the works in [8,9,10,11], in this paper we will produce a new iterative sequence approximating a common solution of (P1) and (P2) (i.e., some point belonging to F ∩ G), and show a strong convergence theorem in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, where {Si}∞ i=1 in (P2) is a family of uniformly quasi-j-asymptotically nonexpansive mappings and for each i ≥ 1, Si is locally uniformly Lipschitz continuous or uniformly Hölder continuous with order Θi. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, {xn} and{yn} be two sequences of E, and u ∈ E. Let C be a nonempty closed convex subset of a smooth and strictly convex reflexive Banach space E, and let A : C ® E* be a δ-inverse-strongly monotone mapping and f : C × C ® R be a bifunction satisfying the following conditions (B1) f(z, z) = 0, ∀z Î C;.
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