CARPATHIAN J. MATH. 19 (2003), No. 1, 67-72 Positive solutions of nonlinear functional-integral equations ANDREI HORVAT-MARC Abstract. In this paper we study the conditions are required for existence of at least one positive solution of the functional-integral equation u (x)= g (x)+ h 0 k (x, s) F (u)(s)ds, x ∈ [0,h] where F : C [0,h] → C [0,h] is an operator. Our approach to the problem is based on the Krasnoselskii’s compression-expansion fixed point theorem. 1. Introduction In this paper, we consider the nonlinear integral equation (1.1) u (x)= g (x)+ h  0 k (x, s) F (u)(s) ds, x ∈ [0,h] where F : C [0,h] → C [0,h], g : [0,h] → R and k : [0,h] × [0,h] → R. In particular case, when F is the Nemitskii’s operator attached to the function f : [0,h] × R → R, i.e. for u ∈ C [0,h] F (u)(x)= N f u (x)= f (x, u (x)) , x ∈ [0,h], then equation (1.1) became (1.2) u (x)= g (x)+ h  0 k (x, s) f (s, u (s)) ds, x ∈ [0,h] . The existence of positive solutions for (1.2) was studied in several papers [1, 2, 5, 7, 8, 9] and reference therein. For example, in [5] are established existence results of positive solutions for (1.2) and their applications to the boundary-value problem with integral boundary conditions. In [1], the equation (1.2) is used to studie the solutions for the two-point boundary value problem. The idea of this paper was suggest in [7], where are presented existence results of multiple nonnegative continuous solutions of a nonlinear integral equation on both a compact interval and semi-infinite interval. Applications Received: 11.05.2003; In revised form: 08.02.2004 2000 Mathematics Subject Classification. 45G10, 45M20, 47G10. Key words and phrases. Krasnoselskii’s fixed point theorem, Banach space, nonlinear Fredholm integral equations. 67