AbstractLetEbe a realq‐uniformly smooth Banach space. SupposeTis a strongly pseudo‐contractive map with open domainD(T) inE.Suppose further that T has a fixed point inD(T).Under various continuity assumptions onTit is proved that each of the Mann iteration process or the Ishikawa iteration method converges strongly to the unique fixed point of T. Related results deal with iterative solutions of nonlinear operator equations involving strongly accretive maps. Explicit error estimates are also provi