Strong convergence theorems for a semigroup of asymptotically nonexpansive mappings

Abstract

Let K be a nonempty closed convex subset of a real Banach space E . Let T : = { T ( t ) : t ≥ 0 } be a strongly continuous semigroup of asymptotically nonexpansive mappings from K into K with a sequence { L t } ⊂ [ 1 , ∞ ) . Suppose F ( T ) ≠ 0̸ . Then, for a given u ∈ K there exists a sequence { u n } ⊂ K such that u n = ( 1 − α n ) 1 t n ∫ 0 t n T ( s ) u n d s + α n u , for n ∈ N , where t n ∈ R + , { α n } ⊂ ( 0 , 1 ) and { L t } satisfy certain conditions. Suppose, in addition, that E is reflexive strictly convex with a Gâteaux differentiable norm. Then, the sequence { u n } converges strongly to a point of F ( T ) . Furthermore, an explicit sequence { x n } which converges strongly to a fixed point of T is proved.