Iterative solution of nonlinear equations involving set-valued uniformly accretive operators

Abstract

Let E be a real normed linear space and let A : E ↦ 2 E be a uniformly continuous and uniformly quasi-accretive multivalued map with nonempty closed values such that the range of ( I – A) is bounded and the inclusion f ϵ Ax has a solution x* ϵ E. It is proved that Ishikawa and Mann type iteration processes converge strongly to x*. Further, if T : E ↦ 2 E is a uniformly continuous and uniformly hemicontractive set-valued map with bounded range and a fixed point x* ϵ E, it is proved that both the Mann and Ishikawa type iteration processes converge strongly to x*. The strong convergence of these iteration processes with errors is also proved.