Convergence theorems for ψ-expansive and accretive mappings

Let E be a real Banach space, and let A : D (A) ⊆ E → E be a Lipschitz, ψ-expansive and accretive mapping such that over(c o, -) (D (A)) ⊆ ∩_{λ > 0} R (I + λ A). Suppose that there exists x₀ ∈ D (A), where one of the following holds: (i) There exists R > 0 such that ψ (R) > 2 {norm of matrix} A (x₀) {norm of matrix}; or (ii) There exists a bounded neighborhood U of x₀ such that t (x - x₀) ∉ A x 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.