Positive solutions for singular coupled integral boundary value problems of nonlinear higher-order fractional q-difference equations

Qingbo Zhao,Wengui Yang

doi:10.1186/s13662-015-0615-3

Abstract

This paper investigates the positive solutions for the singular coupled integral boundary value problem of nonlinear higher-order fractional q-difference equations. By applying a mixed monotone method and Guo-Krasnoselskii fixed point theorem, sufficient conditions for the existence and uniqueness results of the problem are established. An interesting example is presented to illustrate the main results.

Highlights

Due to the intensive development of the theory of fractional calculus itself and its varied applications in many fields of science and engineering, the fractional differential equation has gained considerable popularity and importance for the last several decades
There have been some papers dealing with the existence and multiplicity of solutions or positive solutions for boundary value problems involving nonlinear fractional differential equations; see [ – ] and references cited therein
We notice that boundary value problems for a coupled system of nonlinear fractional differential equations have been addressed by several researchers

Summary

Introduction

Due to the intensive development of the theory of fractional calculus itself and its varied applications in many fields of science and engineering, the fractional differential equation has gained considerable popularity and importance for the last several decades. Yuan et al [ ] and Jiang et al [ ] considered the positive solutions to the four-point coupled boundary value problems for systems of nonlinear semipositone fractional differential equations, respectively. By applying the generalized Banach contraction principle, the monotone iterative method, and Krasnoselskii’s fixed point theorem Zhao et al [ ] showed some existence results of positive solutions to nonlocal q-integral boundary value problem of nonlinear fractional q-derivatives equation. In [ ], the author considered the following coupled integral boundary value problem for systems of nonlinear semipositone fractional q-difference equations: Dαq u (t) + λf t, u(t), v(t) = , Dβq v (t) + λg t, u(t), v(t) = , Djqu ( ) = Djqv ( ) = , ≤ j ≤ n – , u( ) = μ v(s) dqs, v( ) = ν u(s) dqs,.

Then we have

This guarantees that