A spectral approach to local solvability of some nonlinear equations in normed spaces

Abstract

IN A RECENT paper [l] M. Furi and A. Vignoli outlined a nonlinear spectral theory which allowed them to get some surjectivity results for mapsfof the form I J-T, where I denotes the identity operator of a Banach space E, 1. is a real or complex parameter and T is a completely continuous and quasibounded operator acting on E. The author’s paper [2] was a generalization of such an approach to the case when facts between two normed spaces X and Y, and has the formf = L AT, where L is a linear Fredhohn operator of index zero and T satisfies suitable conditions. Some sufficient conditions for the surjectivity of L IT, or more generally for the global solvability of the equation L(x) AT(x) = y, were given there. The purpose of this paper, on the contrary, is to study the local solvability of this equation, in the sense that if L(x,) AT(x,) = yo, then there exist a neighbourhood U of x0 and a neighbourhood V of y,,, such that the equation L(x) AT(x) = p has a solution in U for all )’ in V. The underlying ideas are essentially two: the Mawhin’s coincidence degree theory of a couple of operators (L, T), (see Section l), and a notion of local characteristic value of the couple (L, T) (see Section 2). By joining these two tools we get in Section 3 some local solvability results for the equation L(x) AT(x) = y, and a surjectivity result for L AT, which completes the surjectivity results of [2]. Finally some examples of application of these results to boundary value problems for nonlinear second order differential equations are given in Section 4. It must be explicitly remarked that the present paper is also a generalization of some ideas of Furi and Vignoli [3], who gave local solvability results for the equation x AT(x) = y by making use of the Leray-Schauder topological degree and of a local spectrum argument.