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

Commun. Korean Math. Soc. 2002 Vol. 17, No. 1, 37-51 Printed March 1, 2002

Jong Soo Jung Dong-A University

Abstract : Let $X$ be a reflexive Banach space with a uniformly G\^ateaux differentiable norm, $C$ a nonempty bounded open subset of $X$, and $T$ a continuous mapping from the closure of $C$ into $X$ which is locally pseudo-contractive mapping on $C$. We show that if the closed unit ball of $X$ has the fixed point property for nonexpansive self-mappings and $T$ satisfies the following condition: there exists $z \in C$ such that $\Vert z - T(z)\Vert < \Vert x - T(x)\Vert$ for all $x$ on the boundary of $C$, 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 continuous and strongly converges to a fixed point of $T$ as $t \to 1^-$.