Fixed Point Theorems for Lipschitzian Pseudo-Contractive Mappings

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