Positive solution of a system of integral equations with applications to boundary value problems of differential equations

Chunfang Shen,Hui Zhou,Liu Yang

doi:10.1186/s13662-016-0953-9

Chunfang Shen, Hui Zhou + Show 1 more

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https://doi.org/10.1186/s13662-016-0953-9

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Journal: Advances in Difference Equations	Publication Date: Oct 18, 2016
Citations: 8	License type: CC BY 4.0

Affiliation: Hefei Normal University

Abstract

In this paper, by using the Guo-Krasnoselskii theorem, we investigate the existence and nonexistence of positive solutions of a system of integral equation with parameters which can be seen as an effective generalization of various types of systems of boundary value problems for differential equation on continuous interval and time scales or fractional differential equations. We give a general approach of positive solutions to cover various systems of boundary value problems in a unified way, which avoids treating these problems on a case-by-case basis. Under some growth conditions imposed on the nonlinear term, we obtain explicit ranges of values of parameters with which the problem has a positive solution and has no positive solution, respectively. By giving some examples, we will show how our results may be applied to consider existence of positive solutions to a variety of system of boundary value problems of differential equations, differential equations on time scales or fractional differential equations.

Highlights

1 Introduction We consider the existence of eigenvalues yielding positive solutions to the system of integral equations (Pλ,μ,ζ )
The aim of this paper is to give a general approach of positive solutions to cover various systems of boundary value problems for differential equation on continuous interval and time scales or fractional differential equations in a unified way, which avoids treating these problems on a case-by-case basis
We consider the existence and nonexistence of positive solutions of integral equation system (Pλ,μ,ζ ) under the conditions (H )-(H ) and so the results obtained in this paper may include some known results as a special cases and can be applied to unconsidered boundary value problems which can be formulated as a system of integral equations like (Pλ,μ,ζ )

Summary

Introduction

We consider the existence of eigenvalues yielding positive solutions to the system of integral equations (Pλ,μ,ζ ). The aim of this paper is to give a general approach of positive solutions to cover various systems of boundary value problems for differential equation on continuous interval and time scales or fractional differential equations in a unified way, which avoids treating these problems on a case-by-case basis. We consider the existence and nonexistence of positive solutions of integral equation system (Pλ,μ,ζ ) under the conditions (H )-(H ) and so the results obtained in this paper may include some known results as a special cases and can be applied to unconsidered boundary value problems which can be formulated as a system of integral equations like (Pλ,μ,ζ ). We will show how our results may be applied to obtain eigenvalues yielding the existence of positive solutions to a variety of system of boundary value problems of differential equations, differential equations on time scales or fractional differential equations. By condition (H ), there exists R > such that for t ∈ [ , ], u(t) ≥ , v(t) ≥ , w(t) ≥ and u(t) + v(t) + w(t) ≤ R , f t, u(t), v(t), w(t) ≤ f s + ε u(t) + v(t) + w(t) , g t, u(t), v(t), w(t) ≤ g s + ε u(t) + v(t) + w(t) , h t, u(t), v(t), w(t) ≤ hs + ε u(t) + v(t) + w(t)

We define the set

Define thepositive constants λ

We consider the case

Choose a