Convergence Theorems of Iterations for a Pair of Nonexpansive Mappings in a Banach Space
Takayuki Tamura · 2000
Let £ be a real Banach space and let C be a nonempty closed convex subset of E. Then a mapping T of C into itself is called nonexpansive if \\Tx - Ty\\ 1, where {<*„} and {pn} are sequences in [0, 11. In the case of 5 = 7\ such an iteration scheme was considered by Ishikawa [4]. Das and Debata [2] studied the strong convergence of the iterates {*„} defined by (1) in the case when E is strictly convex and 5, T are quasi-nonexpansive mappings; see also Rhoades [7]. Takahashi and Tamura [11] proved the following: Let C be a nonempty closed convex subset of a uniformly convex Banach space E which satisfies Opial’s condition or whose norm is Frechet differentiable and let 5 and T be nonexpansive mappings of C into itself. Then for any initial data x\ in C , the iterates {*„} defined by (1), where or,, e [a, b] and e [a,b] for some a ,b e R with 0 < a < b < 1, converge weakly to a common fixed point of 5 and T . Further, they obtained the following : Let C be a nonempty closed convex subset of a strictly convex Banach space E and let 5 and T be nonexpansive mappings of C into itself such that 5(C) U T (C) is contained in a compact subset of C. Then for any initial data x\ in C, the iterates {*„} defined by (1) where an € [a, b] and /3n e [a, b] for some a, b e R with 0 < a < b < 1, converge strongly to a common fixed point of 5 and T .