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

Abstract

The paper deals with the impulsive nonlinear boundary value problemwhere J = [a, b], f ∈ Car(J × R 2 ), g 1 , g 2 ∈ C(R 2 ), I j , M j ∈ 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, g 1 , g 2 , I j , M j .

Positive Solutions of Higher-Order Boundary-Value Problems

“…[10] obtained sufficient conditions for the existence of solutions of fourthorder BVPs based on the existence of a pair of strong lower and upper solutions. BVPs with special nonlinear BCs have also been studied in the literature (see [8][9][10][11][22][23][24]. We remark that the BVPs in the general form (1.1), (1.2) are important because of their applications to physical, biological and chemical phenomena (see [2,7,21]).…”

Boundary value problems for higher order linear impulsive differential equations

“…As a matter of fact, we emphasize that the idea of considering higher order impulsive differential equations with discontinuity in all derivatives is not new [7,[10][11][12]18,22,27]. In addition, there are many studies on boundary value problems with discontinuity conditions including periodic boundary value problems and eigenvalue problems [4][5][6]9,13,15,17,19,21,26,28,29].…”

Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations

“…When f or I k are superlinear, there is no paper concerned the solvability of these problems. On the third hand, in the past ten years, fixed point theorems in cones in Banach spaces are extensively used to obtain one, two, or three positive solutions of two-point, or multi-point BVPs for second, or higher order differential equations with or without impulses effects, one may see [29][30][31][32] and references therein. However, there is no paper concerned with the existence of positive solutions of periodic BVPs for first order impulsive differential equations with nonlinear boundary conditions.…”