Convergence theorems for semigroup-type families of non-self mappings

Abstract

Motivated by a paper Chidume and Zegeye [Strong convergence theorems for common fixed points of uniformly L-Lipschitzian pseudocontractive semi-groups, Applicable Analysis, 86 (2007), 353–366], we prove several strong convergence theorems for a family (not necessarily a semigroup) ℱ = {T(t): t ∈ G} of nonexpansive or pseudocontractive non-self mappings in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm, where G is an unbounded subset of ℝ+. Our results extend and improve the corresponding ones byMatsushita and Takahashi [Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions,Nonlinear Analysis, 68 (2008), 412–419],Morales and Jung [Convergence of paths for pseudo-contractive mappings in Banach spaces, Proceedings of American Mathematical Society, 128 (2000), 3411–3419], Song [Iterative approximation to common fixed points of a countable family of nonexpansive mappings, Applicable Analysis, 86 (2007), 1329–1337], Song and Xu [Strong convergence theorems for nonexpansive semigroup in Banach spaces, Journal of Mathematical Analysis and Applications, 338 (2008), 152–161], Wong, Sahu, and Yao [Solving variational inequalities involving nonexpansive type mappings, Nonlinear Analysis, (2007) doi:10.1016/j.na. 2007.11.025] in the context of a non-semigroup family of non-self mappings.