Strong convergence theorems for a semigroup of nonexpansive mappings in Banach spaces

Abstract

Let E be a real reflexive Banach space, which admits a weakly sequentially continuous duality mapping of E into E*, and C be a nonempty closed convex subset of E. Let {T(t):t≥0} be a semigroup of nonexpansive self-mappings on C such that F:=∩ t≥0Fix(T(t))≠∅, where Fix(T(t))={x∈C: x=T(t)x}, and let f: C→C be a fixed contractive mapping. If {α n }, {β n }, {t n } satisfy some appropriate conditions, then a iterative process {x n } in C, defined by converges strongly to q∈F, and q is the unique solution in F to the following variational inequality: ⟨ (I−f)q, j(q−u)⟩≤0 for all u∈F. Our results extend and improve corresponding ones of Suzuki [T. Suzuki, On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proc. Amer. Math. Soc. 131 (2002), pp. 2133–2136.], Xu [H.K. Xu, A strong convergence theorem for contraction semigroups in Banach spaces, Bull. Aust. Math. Soc. 72 (2005), pp. 371–379.] and Chen and He [R.D. Chen and H. He, Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space, Appl. Math. Lett. 20 (2007), pp. 751–757.].