Impulsive BVPs with nonlinear boundary conditions for the second order differential equations without growth restrictions ✩

The paper deals with the impulsive nonlinear boundary value problem u (t ) = f t,u(t),u (t ) , g1(u(a),u(b)) = 0, g2(u (a),u (b)) = 0, u(tj+) = Ij (u(tj )), j = 1, . . . , p, u (tj+) =Mj (u (tj )), j = 1, . . . , p, where J = [a, b], f ∈ Car(J × R2), g1,g2 ∈ C(R2), Ij,Mj ∈ C(R).We prove the existence of a solution to this problem under the assumption that there exist lower and upper functions associated with the problem. Our proofs are based on the Schauder fixed point theorem and on the method of a priori estimates. No growth restrictions are imposed on f, g1,g2, Ij,Mj .  2004 Elsevier Inc. All rights reserved.