Using Leray-Schauder degree

Abstract

Although the Leray-Schauder theory of topological degree [9, 7, 2, ll] is a powerful tool for work in nonlinear analysis, there are obstacles to its application. Two of these are: first, the degree theory provides only existence information, not information about the actual number of solutions of the corresponding equation; second, it is generally difficult to compute the Leray-Schauder degree of a given map. Our purpose here is to describe some methods for overcoming these obstacles. We show in Section 2 that if the map is differentiable, the degree is (with certain exceptions) a lower bound for the actual number of solutions of the equation, and with this result we prove an in-the-large implicit function theorem. In Section 3 we show that the degree of a corresponding map in a complex linear space is (again with certain exceptions) an upper bound for the number of solutions of the given equation. The results in Sections 2 and 3 are known for the Brouwer degree of maps in Euclidean space [2, 31 and we obtain the analogous results for the Leray-Schauder degree by using Smale’s [13] infinite-dimensional version of Sard’s Theorem. The crucial statement in Section 2 is Lemma 2 which is a special case of Smale’s theorem. (Actually this special case follows from earlier work. See [4].) Lemma 3 in Section 2 is also closely related to Smale’s work. See [13, p. 8661. In Section 4 we study nonlinear integral equations and reduce the problem of computing the Leray-Schauder degree to a finite-dimensional problem by using Bernstein polynomials. Then the results of Section 3 are applied to obtain an upper bound for the number of solutions of certain nonlinear integral equations (Theorem 3).