Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in Banach spac
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Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in Banach spaces Godwin C Ugwunnadi1,2 , Bashir Ali3* , Ibrahim Idris3 and Maaruf S Minjibir3 * Correspondence: [email protected] 3 Department of Mathematical Sciences, Bayero University Kano, P.M.B. 3011, Kano, Nigeria Full list of author information is available at the end of the article
Abstract In this paper, we introduce a new iterative scheme by a hybrid method and prove a strong convergence theorem of a common element in the set of fixed points of a finite family of closed quasi-Bregman strictly pseudocontractive mappings and common solutions to a system of equilibrium problems in reflexive Banach space. Our results extend important recent results announced by many authors. MSC: 47H09; 47J25 Keywords: Bregman distance; quasi-Bregman strictly pseudocontractive map; fixed point
1 Introduction Let E be a real Banach space and C a nonempty closed convex subset of E. The normalized ∗ duality map from E to E (E∗ is the dual space of E) denoted by J is defined by J(x) = f ∈ E∗ : x, f = x = f . Let T : C → C be a map, a point x ∈ C is called a fixed point of T if Tx = x, and the set of all fixed points of T is denoted by F(T). The mapping T is called L-Lipschitzian or simply Lipschitz if there exists L > , such that Tx – Ty ≤ Lx – y, ∀x, y ∈ C and if L = , then the map T is called nonexpansive. Let g : C × C → R be a bifunction. The equilibrium problem with respect to g is to find z ∈ C such that g(z, y) ≥ ,
∀y ∈ C.
The set of solution of equilibrium problem is denoted by EP(g). Thus EP(g) := z ∈ C : g(z, y) ≥ , ∀y ∈ C . Numerous problems in physics, optimization and economics reduce to finding a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium ©2014 Ugwunnadi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Ugwunnadi et al. Fixed Point Theory and Applications 2014, 2014:231 http://www.fixedpointtheoryandapplications.com/content/2014/1/231
problem in Hilbert spaces; see for example Blum and Oettli [], Combettes and Hirstoaga []. Recently, Tada and Takahashi [, ] and Takahashi and Takahashi [] obtain weak and strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and set of fixed points of a nonexpansive mapping in Hilbert space. In particular, Takahashi and Zembayashi [] established a strong convergence theorem for finding a common element of the two sets by using the hybrid method introduced in Nakajo and Takahashi []. They also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space. Reich and Sabach [] and Kassay et al. [] proved some convergence theorems for the solution of
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