Fixed point theorems for Lipschitzian pseudo-contractive mappings

Abstract:

Let X be a Banach space and $D \subset X$. A mapping $U:D \to X$ is said to be pseudo-contractive if, for all $u,v \in D$ and all $r > 0,\left \| {u - v} \right \| \leqq \left \| {(1 + r)(u - v) - r(U(u) - U(v))} \right \|$. A recent fixed point theorem of W. V. Petryshyn is used to prove: If G is an open bounded subset of X with $0 \in G$ and $U:\bar G \to X$ is a lipschitzian pseudo-contractive mapping satisfying (i) $U(x) \ne \lambda x$ for $x \in \partial G,\lambda > 1$ , and (ii) $(I - U)(\bar G)$ is closed, then U has a fixed point in $\bar G$. This result yields fixed point theorems for pseudo-contractive mappings in uniformly convex spaces and for “strongly” pseudo-contractive mappings in reflexive spaces.