On the convergence of Ishikawa iterative process for a pair of single-valued and multi-valued nonexpansive mappings

Abstract

Let K be a nonempty compact convex subset of a complete CAT(0) space X and let t: K → K and T: K → CB(K) be a single-valued nonexpansive mapping and a multi-valued nonexpansive mapping, respectively. Assume, in addition, that F(t) ∩ F(T) ≠ ∅ and Tp = {p} for all p ∈ F(t) ∩ F(T). We prove that the sequence of modified Ishikawa iteration method generated from an arbitrary x₀ ∈ K by y_n = (1 - β_n) x_n ⊕ β_n z_n, x_n+1 = (1-α_n) x_n ⊕ α_nty_n, where z_n ∈ Tx_n and {α_n}, {β_n} are sequences of positive numbers satisfying 0< a ≤ α_n, β_n ≤ b < 1 converges strongly to a common fixed point of t and T; that is, there exists x ∈ K such that x = tx ∈ Tx. Our results extends and generalizes some theorems of T. Puttasontiphot [24].