Abstract
We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.
Highlights
Let X be a real Banach space whose dual space is denoted by X∗
We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces
The authors [5] established an existence theorem for MVVI (15) in a smooth Banach space X and proved that the sequence generated by their iterative algorithm converges strongly to a solution of MVVI (16)
Summary
Let X be a real Banach space whose dual space is denoted by X∗. The normalized duality mapping J : X → 2X∗ is defined byJ (x) = {x∗ ∈ X∗ : ⟨x, x∗⟩ = ‖x‖2 = x∗2} , ∀x ∈ X, (1)where ⟨⋅, ⋅⟩ denotes the generalized duality pairing. The authors [5] established an existence theorem for MVVI (15) in a smooth Banach space X and proved that the sequence generated by their iterative algorithm converges strongly to a solution of MVVI (16).
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