Abstract
An optimal existing method for the approximation of common fixed points of countable families of nonlinear operators is introduced, by which a relaxed hybrid shrinking iterative algorithm is developed for the class of totally quasi-ϕ-asymptotically nonexpansive mappings, and a strong convergence theorem for solving generalized mixed equilibrium problems is established in the framework of Banach spaces. Since there is no need to impose the uniformity assumption on the involved countable family of mappings and no need to compute a complex series at each step in the iteration process, the result is more widely applicable than those of other authors with related interests.
Highlights
1 Introduction Throughout this paper we assume that E is a real Banach space with its dual E∗, C is a nonempty closed convex subset of E and J : E → E∗ is the normalized duality mapping defined by
Where φ : E × E → R+ ∪ { } denotes the Lyapunov functional defined by φ(x, y) = x – x, Jy + y, ∀x, y ∈ E
In, Qin et al [ ] proposed the following shrinking projection method to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of quasi-φ-nonexpansive mappings in the framework of Banach spaces:
Summary
[ ] ( ) A mapping T : C → C is said to be totally quasi-φ-asymptotically nonexpansive, if F(T) = ∅ and there exist nonnegative real sequences {νn}, {μn} with νn, μn → (as n → ∞) and a strictly increasing continuous function ζ : R+ ∪ { } → R+ ∪ { } with ζ ( ) = such that φ p, Tnx ≤ φ(p, x) + νnζ φ(p, x) + μn, ∀n ≥ , x ∈ C, p ∈ F(T), In , Qin et al [ ] proposed the following shrinking projection method to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of quasi-φ-nonexpansive mappings in the framework of Banach spaces:
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