Implicit function theorems for m-accretive and locally accretive set-valued mappings

Abstract

In this paper we elaborate new methods to the theory of parametric generalized equations governed by accretive mappings. Two implicit function theorems will be proved, one for m-accretive and another for locally accretive and continuous set-valued mappings. Making use of the theorem for m-accretive mappings we will study parametric nonlinear evolution problems. It will be also shown that there is possible a unified approach to the theory of implicit function theorems, namely many results on the stability of the solutions of perturbed nonsmooth or monotone generalized equations, variational inequalities [?, ?, 1, 5, 6, ?], as well the standard implicit function theorem (Theorem 15.1, [?]) are consequences of the present theorem for locally-accretive and continuous set-valued mappings. In the following investigations the geometrical properties of Banach spaces, the extension problems for accretive mappings and the existence of zeros for accretive mappings play an important role. The two implicit function theorems are independent because of the above mentioned extension problems. The possibility of extending an accretive mapping to an m-accretive one was studied in many papers [?, ?, 4]. The most important result is that if C is a closed, convex subset of a reflexive, strictly-convex Banach space X, which is not a nonexpansive retract of X, then no accretive mappings A : X ; X, with co (Dom (A)) = C, can be extended to an m-accretive mapping B : X ; X, with co(Dom (B)) = C. The closed balls in a non-Hilbert Banach space are not nonexpansive retracts of the whole space. A consistency condition appears in the following theorems. Such conditions are frequently used in implicit function theorems, see [?, ?, ?, ?], because they give ”almost” solutions for each parameter close enough to a fixed parameter and replace the continuity with respect to parameters. Our condition is that from the papers [?, ?] and as will be proved it is a weaker property than the lower-semicontinuity or the pseudo-continuity. The consistency will be used toghether with results on the existence of zeros for accretive mappings, see for example [?, ?, ?, ?, ?, 3, ?]. In this paper we do not use compactness