Abstract
We introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed quasi‐ϕ‐asymptotically nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality problem for a γ‐inverse strongly monotone mapping in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others.
Highlights
Introduction and PreliminaryLet E be a Banach space with the dual E∗
Let C be a nonempty closed convex subset of E and f : C × C → R a bifunction, where R is the set of real numbers
In this paper, inspired and motivated by the works mentioned above, we introduce an iterative process for finding a common element of the set of common fixed points of a finite family of closed quasi-φ-asymptotically nonexpansive mappings, the solution set of equilibrium problem, and the solution set of the variational inequality problem for a γ-inverse strongly monotone mapping in Banach spaces
Summary
Let E be a Banach space with the dual E∗. A mapping A : D A ⊂ E → E∗ is said to be monotone if, for each x, y ∈ D A , the following inequality holds: Ax − Ay, x − y ≥ 0. A is said to be γ-inverse strongly monotone if there exists a positive real number γ such that x − y, Ax − Ay ≥ γ Ax − Ay 2, ∀x, y ∈ D A
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