Iterative approximation of solutions of nonlinear equations of Hammerstein type

Suppose XX is a real qq-uniformly smooth Banach space and F,K:X→XF,K:X→X are Lipschitz ϕϕ-strongly accretive maps with D(K)=F(X)=XD(K)=F(X)=X. Let u∗u∗ denote the unique solution of the Hammerstein equation u+KFu=0u+KFu=0. An iteration process recently introduced by Chidume and Zegeye is shown to converge strongly to u∗u∗. No invertibility assumption is imposed on KK and the operators KK and FF need not be defined on compact subsets of XX. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.