Approximation of the zeros of m-accretive operators

A theorem which gives a necessary and sufficient condition for the strong convergence of the semigroup generated by an m-accretive operator defined on the whole Banach space E and of the steepest descent approximation process to a zero of a quasi-accretive operator A in Banach spaces is presented. The theorem states that the sequence generated by the steepest descent approximation process converges hyperstrongly to the element of the null space of A. This theorem is reformulated in the more general setting in which the operator is defined on a proper subset of E and takes values in E.