Iterative approximation of solutions to nonlinear equations involving m-accretive operators in Banach spaces

Abstract

Let E be an arbitrary real Banach space and T :E→E be a Lipschitz continuous accretive operator. Under the lack of the assumption lim n→∞ α n =lim n→∞ β n =0, we prove that the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation x+ Tx= f. Moreover, this result provides a convergence rate estimate for some special cases of such a sequence. Utilizing this result, we imply that if T :E→E is a Lipschitz continuous strongly accretive operator then the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation Tx= f. Our results improve, generalize and unify the ones of Liu, Chidume and Osilike, and to some extent, of Reich.