The Existence of Positive Solution to Three-Point Singular Boundary Value Problem of Fractional Differential Equation

Abstract

We investigate the existence of positive solution to nonlinear fractional differential equation three-point singular boundary value problem:Dqu(t)+f(t,u(t))=0,0<t<1,u(0)=0,u(1)=αD(q−1)/2u(t)|t=ξ, where1<q≤2is a real number,ξ∈(0,1/2],α∈(0,+∞)andαΓ(q)ξ(q−1)/2<Γ((q+1)/2),Dqis the standard Riemann-Liouville fractional derivative, andf∈C((0,1]×[0,+∞),[0,+∞)),lim⁡t→+0f(t,⋅)=+∞(i.e.,fis singular att=0). By using the fixed-point index theory, the existence result of positive solutions is obtained.