Strong convergence of approximating fixed points for nonexpansive nonself-mappings in Banach spaces

Abstract

Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of E, and T C→E a nonexpansive mapping satisfying the inwardness condition. Assume that every weakly compact convex subset of E has the fixed point property. For u∈C and t∈(0, 1), let x_t be a unique fixed point of a contraction G_t C→E, defined by G_tx=tTx+(1−t)u, x∈C. It is proved that if {x_t} is bounded, then the strong lim_t→1x_t exists and belongs to the fixed point set of T Furthermore, the strong convergence of other two schemes involving the sunny nonexpansive retraction is also given in a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm.