Solvability and positive solution of a system of second-order boundary value problem with integral boundary conditions

Rochdi Jebari,Abderrahman Boukricha

doi:10.1186/s13661-014-0262-8

Abstract

The main purpose of this paper is to establish the existence, uniqueness and positive solution of a system of second-order boundary value problem with integral conditions. Using Banach’s fixed point theorem and the Leray-Schauder nonlinear alternative, we discuss the existence and uniqueness solution of this problem, and we apply Guo-Krasnoselskii’s fixed point theorem in cone to study the existence of positive solution. We also give some examples to illustrate our results.

Highlights

1 Introduction The systems of second-order ordinary differential equations arise from many fields in physics, biology and chemistry; for example, in the theory of nonlinear diffusion generated by nonlinear sources, in thermal ignition of gases, and in concentration in chemical or biological problems
The main purpose of the present paper is to investigate sufficient conditions for the existence, uniqueness and positive solution of the following problem: ui (t) + fi t, u (t), . . . , un(t) =, < t
Various types of boundary value problems involving integral condition have been studied by many authors using fixed point theorems on cones, fixed point index theory, the generalized quasilinearization method, the Leray-Schauder nonlinear alternative and the Leggett-Williams fixed point theorem [ – ]

Summary

Introduction

The systems of second-order ordinary differential equations arise from many fields in physics, biology and chemistry; for example, in the theory of nonlinear diffusion generated by nonlinear sources, in thermal ignition of gases, and in concentration in chemical or biological problems (see [ – ] and the references therein). The main purpose of the present paper is to investigate sufficient conditions for the existence, uniqueness and positive solution of the following problem: ui (t) + fi t, u (t), . Various types of boundary value problems involving integral condition have been studied by many authors using fixed point theorems on cones, fixed point index theory, the generalized quasilinearization method, the Leray-Schauder nonlinear alternative and the Leggett-Williams fixed point theorem [ – ].

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Conclusion