The purpose of this paper is to obtain an existence result for the integral equation u(t) = j(t, u(t)) + b ò a y(t, s, u(s))ds, t Î[a, b]. where j : [ a, b ] ́ R ® R and y : [ 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 a -condensing perturbations of the identity) is used in order to prove the existence of fixed points for T . Note that the assumptions on functions j and y do not generally assure the compactness of operator T , therefore the Leray-Schauder degree cannot be used (see K. Deimling).