Strong convergence theorems for locally pseudo-contractive mappings in Banach spaces

Abstract

Let X be a reflexive Banach space with a uniformlyG^ateaux differentiable norm, C a nonempty bounded open subsetof X, and T a continuous mapping from the closure of C intoX which is locally pseudo-contractive mapping on C. We showthat if the closed unit ball of X has the fixed pointproperty for nonexpansive self-mappings and T satisfies thefollowing condition: there exists z in C such that Vert z -T(z)Vert < Vert x - T(x)Vert for all x on the boundary ofC, then the trajectory t longmapsto z_t in C, t in [0,1)defined by the equation z_t = tT(z_t) + (1 - t)z is continuousand strongly converges to a fixed point of T as t to 1^-.