Abstract
The purpose of this paper is to present the notion of weak relatively nonexpansive multi-valued mapping and to prove the strong convergence theorems of fixed point for weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. In this paper, the notion of multivalued resolvent of maximal monotone operator has been also presented which is a weak relatively nonexpansive multivalued mapping and can be used to find the zero point of maximal monotone operator.
Highlights
Introduction and PreliminariesIterative methods for approximating fixed points of multivalued mappings in Banach spaces have been studied by some authors, see for instance 1–4
For a given x1 ∈ C, let {xn} be the iterative sequence defined by xn 1 ΠCJ−1 αnJxn 1 − αn Jzn, zn ∈ T xn
The purpose of this paper is to present the notion of weak relatively nonexpansive multivalued mapping and to prove the strong convergence theorems for the weak relatively nonexpansive multivalued mappings in Banach spaces
Summary
Iterative methods for approximating fixed points of multivalued mappings in Banach spaces have been studied by some authors, see for instance 1–4. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized metric projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. Let C be a nonempty closed convex subset of a smooth Banach space E and T : C → C a multivalued mapping such that T x is nonempty for all x ∈ C. Let E be a uniformly convex and smooth Banach space, let C be a closed convex subset of E, and let T : C → C be a weak relatively nonexpansive multivalued mapping.
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