Strong convergence theorems for accretive operators in Banach spaces

AbstractLet E be a reflexive Banach space with a uniformly Gâteaux differentiable norm and let A ⊂ E × E be m-accretive. Assume that S: cl(D(A)) → E is bounded, strongly accretive, and continuous. For x ϵ E and t > 0, let xt be the unique solution of the inclusion x ϵ Sxt + tAxt. It is proved that if there exists a nonexpansive retraction P of E onto cl(D(A)), then the strong limt → 0xt exists. Moreover, it is shown that if every weakly compact convex subset of E has the fixed point property for nonexpansive mappings and 0 ϵ R(A), then the strong limt → ∞xt exists and belongs to A−10. An interesting application to a convergence result for an implicit iterative scheme is also included