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

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.