Strong Convergence Theorems for Mixed Equilibrium Problem and AsymptoticallyI-Nonexpansive Mapping in Banach Spaces

Abstract

This paper aims to use a hybrid algorithm for finding a common element of a fixed point problem for a finite family of asymptotically nonexpansive mappings and the set solutions of mixed equilibrium problem in uniformly smooth and uniformly convex Banach space. Then, we prove some strong convergence theorems of the proposed hybrid algorithm to a common element of the above two sets under some suitable conditions.

Generalized mixed equilibrium problems and quasi- ϕ-asymptotically nonexpansive multivalued mappings in Banach spaces

“…In 2012, Chang et al [9] considered the class of uniformly quasi-φ-asymptotically nonexpansive nonself mappings and studied in a uniformly convex and uniformly smooth real Banach space. In 2014, Deng et al [10] proved strong convergence theorems of the hybrid algorithm for common fixed point problem of finite family of asymptotically nonexpansive mappings and the set of solution of mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. In 2016, Ezeora [11] proved strong convergence theorems for a common element of the set of solution of generalized mixed equilibrium problem and the set of common fixed points of a finite family of multivalued strictly pseudocontractive mappings in real Hilbert spaces.…”

Strong Convergence Theorems for Split Common Fixed Point Problem of Bregman Generalized Asymptotically Nonexpansive Mappings in Banach Spaces

“…As stated in [4], let E be a Banach space with norm • . Let C be a nonempty closed convex subset of E and E * denote the dual space of E. Let B : C −→ E * be a nonlinear mapping and F : C × C −→ be a bifunction.…”

“…Application to the mixed equilibrium problem Lemma 4.1 [4]. Let E be a reflexive, strictly convex and smooth Banach space, and let C be a nonempty closed convex subset of E. Let f, g : C × C −→ be two bifunctions which satisfy the conditions (A 1 ) − (A 4 ), (B 1 ) − (B 3 )and(C), in (6), then for every x ∈ E and r > 0, there exists a unique point z ∈ C such thatf (z, y) + g(z, y) + 1 r y − z, jz − jx ≥ 0∀y ∈ C}In Reich, S, Sabach, S (2010)[18], when f (x) = 1 p x p then we have the following Lemma.…”

“…14 x, x ∈ (0, ∞). Now we check whether or not the mappings U and T are uniformly continuous Bregman generalized asymptotically nonexpansive.…”