The solution by iteration of nonlinear equations involving Psi-strongly accretive operators

Abstract

Let E be an arbitrary real Banach space and let K C E be a proximal subset of E. Suppose T : K i —> E is a Lipschitz ψ-strongly accretive operator such that for any given / G E the equation Tx = f has a solution x* E D(T). It is proved that modified iteration processes of the Mann and Ishikawa types converge strongly to x*. Related results deal with the solution of the equation x + λTx = f, λ > 0 for a ψ-strongly accretive or ψ-strongly dissipative T and the iterative approximation of fixed points to a ψ-strongly pseudocontractive T. We also discuss the case where the operator T is only uniformly continuous. MIRAMARE TRIESTE December 1996 Permanent address: Department of Mathematics and Computer Science, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State, Nigeria.