ON A NONLINEAR INTEGRAL EQUATION WITHOUT COMPACTNESS

Abstract

Abstract. The purpose of this paper is to obtain an existence result for the integral equation u (t) = ϕ (t, u (t)) + ∫ b a ψ (t, s, u (s)) ds, t ∈ [a, b] where ϕ: [a, b]×R → R and ψ: [a, b] × [a, b]×R → R are continuous functions which satisfy some special growth conditions. The main idea is to transform the integral equation into a fixed point problem for a condensing map T: C [a, b] → C [a, b]. The “a priori estimate method ” (which is a consequence of the invariance under homotopy of the degree defined for α-condensing perturbations of the identity) is used in order to prove the existence of fixed points for T. Note that the assump-tions on functions ϕ and ψ do not generally assure the compactness of operator T, therefore the Leray-Schauder degree cannot be used (see K. Deimling [2, Example 9.1, p. 69]). 1.