Random fixed points of uniformly Lipschitzian mappings

Let (Omega,Sigma) be a measurable space, X a Banach space, C a weekly convex subset of X and T: Omega x C --> C a random operator. We prove the random version of a deterministic fixed point theorem when T is an asymptotically nonexpensive mapping and the characteristic of convexity epsilon(0)(X) is less than 1. Let N(X) be the normal structure coefficient of X and kappa(0)(X) its Liftschitz constant. If T is kappa(omega)-uniformity Lipschitzian and there exists a constant c such that kappa(omega) less than or equal to c < 1 + root1 + 4N (X)(kappa(0)(X) - 1)/2, we prove that T has a random fixed point. (C) 2004 Elsevier Ltd. All rights reserved.