On a Nonlinear Integral Equation without Compactness

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 assumptions 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]).