Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals

Abstract

In this paper, we prove Krasnoselskii and Mann's type convergence theorems for nonexpansive semigroups without using Bochner integral and without assuming the strict convexity of Banach spaces. One of our main results is the following: let C be a compact convex subset of a Banach space E and let { T ( t ) : t ⩾ 0 } be a one-parameter strongly continuous semigroup of nonexpansive mappings on C. Let { t n } be a sequence in [ 0 , ∞ ) satisfying lim inf n → ∞ t n < lim sup n → ∞ t n and lim n → ∞ ( t n + 1 − t n ) = 0 . Let λ ∈ ( 0 , 1 ) . Define a sequence { x n } in C by x 1 ∈ C and x n + 1 = λ T ( t n ) x n + ( 1 − λ ) x n for n ∈ N . Then { x n } converges strongly to a common fixed point of { T ( t ) : t ⩾ 0 } .