Positive Solutions for a System of Nonlinear Semipositone Boundary Value Problems with Riemann-Liouville Fractional Derivatives

Xiaowei Qiu,Donal O’Regan,Yujun Cui,Jiafa Xu

doi:10.1155/2018/7351653

Abstract

We study the existence of positive solutions for the system of nonlinear semipositone boundary value problems with Riemann-Liouville fractional derivatives D0+αD0+αu=f1t,u,u′,v,v′, 0<t<1, D0+αD0+αv=f2(t,u,u′,v,v′), 0<t<1, u0=u′0=u′(1)=D0+αu(0)=D0+α+1u(0)=D0+α+1u(1)=0, and v(0)=v′(0)=v′(1)=D0+αv(0)=D0+α+1v(0)=D0+α+1v(1)=0, where α∈(2,3] is a real number and D0+α is the standard Riemann-Liouville fractional derivative of order α. Under some appropriate conditions for semipositone nonlinearities, we use the fixed point index to establish two existence theorems. Moreover, nonnegative concave and convex functions are used to depict the coupling behavior of our nonlinearities.

Highlights

In this paper, we investigate the existence of positive solutions for the system of nonlinear semipositone boundary value problems with Riemann-Liouville fractional derivatives α α u = f1 (t, u, u󸀠, V, V󸀠 ), 0 < t < 1, α αV = f2 (t, u, u󸀠, V, V󸀠 ), 0 < t < 1,α α+1 u (0) = D0+ u (0)u (0) = u󸀠 (0) = u󸀠 (1) = D0+ α+1 u (1) = 0, = D0+ (1) α α+1 V (0) = D0+ V (0)
We study the existence of positive solutions for the system of nonlinear semipositone boundary value problems with Riemannα α α α
We investigate the existence of positive solutions for the system of nonlinear semipositone boundary value problems with Riemann-Liouville fractional derivatives α α

Summary

Semipositone Boundary Value Problems with

State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China. Α a real number and D0+ is the standard Riemann-Liouville fractional derivative of order α. Under some appropriate conditions for semipositone nonlinearities, we use the fixed point index to establish two existence theorems. Nonnegative concave and convex functions are used to depict the coupling behavior of our nonlinearities

Introduction

Then we have

The research is supported by the National Natural Science