Abstract
AbstractIn this paper, we prove strong and weak convergence theorems for a mapping defined on a bounded, closed and convex subset of a uniformly convex Banach space, satisfying the RCSC condition. This condition was introduced by Karapınar (Dynamical Systems and Methods, 2012). We first establish the demiclosed principle for the mapping satisfying the RCSC condition. Then, using this principle, we establish the weak and strong convergence theorems. Results in the paper extend and improve a number of important results in this literature such as Khan and Suzuki (Nonlinear Anal. 80:211-215, 2013) and Reich (J. Math. Anal. Appl. 67:274-276, 1979).
Highlights
Let C be a nonempty closed convex subset of a Banach space X
A mapping T : C → C is said to be nonexpansive if Tx – Ty ≤ x – y for all x, y ∈ C
Khan and Suzuki [ ] proved a weak convergence theorem for a mapping satisfying condition (C) in uniformly convex Banach spaces whose dual has the Kadec-Klee property
Summary
Let C be a nonempty closed convex subset of a Banach space X. Khan and Suzuki [ ] proved a weak convergence theorem for a mapping satisfying condition (C) in uniformly convex Banach spaces whose dual has the Kadec-Klee property. Motivated by the above mentioned works, in this paper, we prove some weak and strong convergence theorems for generalized nonexpansive ((RCSC)-condition) mappings in a uniformly convex Banach space, which has the Kadec-Klee property. Let C be a nonempty subset of a Banach space X and T : C → C be a mapping satisfying (RCSC)-condition. A Banach space X is said to have the Kadec-Klee property if, for every sequence {xn} in X which converges weakly to a point x ∈ X with xn converging to x , {xn} converges strongly to x. Let C be a nonempty bounded convex subset of a uniformly convex Banach space X, and let T : C → C be a mapping satisfying (RCSC)-condition.
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