In this paper, we propose extensions of Leray-Schauder boundary condition for a sum of two operators T + S in the case when T is an expansive operator and I − S is a completely continuous operator. As their applications, we investigate a class of fourth-order nonlinear boundary value problems with integral boundary conditions. We give conditions for the parameters of the considered boundary value problem that ensure existence of at least two non trivial bounded nonnegative classical solutions of the considered boundary value problem. The results in the paper are provided with a suitable example.