Let E be a real Banach space, and let A:D(A)E→E be a Lipschitz, ψ-expansive and accretive mapping such that . Suppose that there exists x₀D(A), where one of the following holds: (i) There exists R>0 such that ψ(R)>2A(x₀); or (ii) There exists a bounded neighborhood U of x₀ such that t(x−x₀)Ax for x∂U∩D(A) and t<0. An iterative sequence {x_n} is constructed to converge strongly to a zero of A. Related results deal with the strong convergence of this iteration process to fixed points of ψ-expansive and pseudocontractive mappings in real Banach spaces. The convergence results established in this paper are new for this more general class of ψ-expansive and accretive or pseudocontractive mappings.