Positive solutions of boundary value problems for systems of second-order differential equations with integral boundary condition on the half-line

Shouliang Xi,Huipeng Ji,Mei Jia

doi:10.14232/ejqtde.2009.1.31

Abstract

In this paper, we study the existence of positive solutions of boundary value problems for systems of second-order differential equations with integral boundary condition on the half-line. By using the fixed-point theorem in cones, we show the existence of at least one positive solution with suitable growth conditions imposed on the nonlinear terms. Moreover, the associated integral kernels for the boundary value problems are given.

Highlights

In this paper we consider the existence of positive solutions for second-order boundary value problems (BVPs) for systems of differential equations with integral boundary condition on the half-line: u1 (t) + f1(t, u1(t), u2(t)) = 0, t ∈ (0, ∞), u2 (t) + f2(t, u1(t), u2(t)) = 0, t ∈ (0, ∞), u1(0) = u2(0) = 0
The existence of positive solutions was studied by Ma in 1999 for a type of three-point boundary value problem [2]
Many authors have studied the existence of positive solutions for multi-point boundary value problems, and obtained many sufficient conditions for the existence of positive solutions

Summary

Introduction

In this paper we consider the existence of positive solutions for second-order boundary value problems (BVPs) for systems of differential equations with integral boundary condition on the half-line: u1 (t) + f1(t, u1(t), u2(t)) = 0, t ∈ (0, ∞), u2 (t) + f2(t, u1(t), u2(t)) = 0, t ∈ (0, ∞), u1(0) = u2(0) = 0,. Boundary value problems with Riemann-Stieltjes integral boundary conditions (BCs) are being studied since they include BVPs with two-point, multipoint and integral BCs as special cases. Thereafter, Tian and Ge [27] studied the existence of positive solutions for the multi-point boundary value problem on the half-line. Motivated by the papers mentioned above, in this paper we investigate the existence of positive solutions of boundary value problems for systems of secondorder differential equations (1.1) with integral boundary condition on the halfline

Preliminaries

The main result

Illustration