Convergence theorems for image-expansive and accretive mappings Original Research Article

Let EE be a real Banach space, and let A:D(A)⊆E→EA:D(A)⊆E→E be a Lipschitz, ψψ-expansive and accretive mapping such that View the MathML sourceco¯(D(A))⊆∩λ>0R(I+λA). Suppose that there exists x0∈D(A)x0∈D(A), where one of the following holds: (i) There exists R>0R>0 such that ψ(R)>2‖A(x0)‖ψ(R)>2‖A(x0)‖; or (ii) There exists a bounded neighborhood UU of x0x0 such that t(x−x0)∉Axt(x−x0)∉Ax for x∈∂U∩D(A)x∈∂U∩D(A) and t<0t<0. An iterative sequence {xn}{xn} is constructed to converge strongly to a zero of AA. Related results deal with the strong convergence of this iteration process to fixed points of ψψ-expansive and pseudocontractive mappings in real Banach spaces. The convergence results established in this paper are new for this more general class of ψψ-expansive and accretive or pseudocontractive mappings.