Convergence of paths for pseudo-contractive mappings in Banach spaces

Abstract:

Let $X$ be a real Banach space, let $K$ be a closed convex subset of $X$, and let $T$, from $K$ into $X$, be a pseudo-contractive mapping (i.e. $(\lambda -1)$ $\|u-v\|\le \|(\lambda I-T)(u)-(\lambda I-T)(v)\|$ for all $u,v\in K$ and $\lambda >1)$. Suppose the space $X$ has a uniformly Gâteaux differentiable norm, such that every closed bounded convex subset of $K$ enjoys the Fixed Point Property for nonexpansive self-mappings. Then the path $t\to x_t\in K$, $t\in [0,1)$, defined by the equation $x_t=tTx_t+(1-t)x_0$ is continuous and strongly converges to a fixed point of $T$ as $t\to 1^-$, provided that $T$ satisfies the weakly inward condition.