The solution by iteration of nonlinear equations of Hammerstein type

Abstract

Let E be a real uniformly smooth Banach space. Suppose N is a set-valued (locally) accretive map with open domain D(N) C E , K is a single-valued map with domain D(K) C E such that Im(N) C D(K) and K~ exists and is (locally) strongly accretive on Im(K) . If the equation f e x + KNx has a solution x* E D(N) , the strong convergence of iteration methods of the Ishikawa and Mann types to this solution is established. Related theorems deal with the iterative solution of nonlinear equations of the form f G x + Tx for the case where T is accretive. Explicit convergence rates are established. MIRAMARE TRIESTE November 1996 Permanent address: Department of Mathematics and Computer Science, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State, Nigeria.