Abstract
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.
Highlights
Let E be a real Banach space and C a nonempty closed convex subset of E
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)
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 [ ]
Summary
Let E be a real Banach space and C a nonempty closed convex subset of E. 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. 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 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 some equilibrium and variational inequality problems in the setting of reflexive Banach spaces.
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