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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 by Matsushita 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, ] in the context of a non-semigroup family of non-self mappings.

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Ž. Let E be Banach space with property U, m, m q 1, , g R, m g N, and a uniformly Gateaux differentiable norm; J: E ª E* a duality mapping; D a nonempty closed convex bounded subset of E; and T : D ª D a uniformly Ž . 1r 2 L-Lipschitzian asymptotically hemicontractive mapping with L -N E where Ž . 5 n 5 2