Approximating common fixed points of families of quasi‐nonexpansive mappings

Abstract

ABSTRACT. This paper deals with a family of quasi-nonexpansive mappings in a uniformly convex Banach space, and the convergence of iterates generated by this family. A fixed point theorem for two quasi-nonexpansive mappings is then proved. This theorem is then extended for a finite family of quasinonexpansive mappings. It is shown that Ishikawa's result follows as special cases of results proved in this paper.

Weak and Strong Convergence for Quasi-Nonexpansive Mappings in Banach Spaces

“…Next, we show the strong convergence of the Mann iteration process with errors when E is a real uniformly convex Banach space and T is a quasi-nonexpansive self-mapping satisfying Condition A, which generalizes the result due to Senter-Dotson [10]. Finally, we show that the iteration {xn} defined by+γnun converges strongly to a common fixed point of T and S when E is a real uniformly convex Banach space and T, S are two quasi-nonexpansive self-mappings satisfying Condition D, which generalizes the result due to Ghosh-Debnath [3]. …”

“…Finally, we prove that the iteration {x n } defined by (3) converges strongly to a common fixed point of T and S when E is a real uniformly convex Banach space and T, S are two quasi-nonexpansive self-mappings satisfying Condition D, which generalizes the result due to Ghosh-Debnath [3].…”

“…From the Condition D, (18) and (19), we obtain lim n→∞ d(x n , F) = 0. By using similar method in the proof of Theorem 6, {x n } converges strongly to a common fixed point of T and S. □ As a direct consequence, taking γ n = 0 for all n ≥ 1 in Theorem 8, we obtain the following result, which improves Theorem 1 of Ghosh-Debnath [3] under much less restriction on the iterative parameter {α n }.…”

“…For two mappings S, T of C into itself, we also consider a more general iterative scheme of the type (cf., Ghosh-Debnath [3], Xu [13]) emphasizing the randomness of errors as follows:…”

“…On the other hand, Senter-Dotson [10] proved that if E is a real uniformly convex Banach space, and C is a nonempty closed convex subset of E, and T : C → C is a quasi-nonexpansive mapping satisfying Condition A, then for any x 1 ∈ C, the sequence {x n } defined by x n+1 = (1 − α n )x n + α n T x n converges strongly to some fixed point of T under the assumption that {α n } in [0, 1] is chosen so that α n ∈ [a, b] for all n ≥ 1 and some a, b ∈ (0, 1). Ghosh-Debnath [3] proved that if E is a real uniformly convex Banach space and C is a nonempty closed convex subset of E and T, S : C → C are two quasi-nonexpansive mappings, and T, S satisfy Condition C with F (T ) ∩ F (S) ̸ = ∅, then for any x 1 in C, the sequence {x n } defined by…”