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 := boolean AND(t >= 0)Fix(T(t)) not equal empty set, where Fix(T(t)) = {x is an element of C: x = T(t)x}, and let f: C -> C be a fixed contractive mapping. If {alpha(n)}, {beta(n)}, {t(n)} satisfy some appropriate conditions, then a iterative process {x(n)} in C, defined by x(n) = alpha(n)y(n) + (1 - alpha(n))T(t(n))x(n), y(n) = beta(n)f(x(n-1)) + (1 - beta(n))x(n-1) converges strongly to q is an element of F, and q is the unique solution in F to the following variational inequality: LT (I - f)q, j(q - u)> LT = 0 for all u is an element of 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.].