Positive Solutions to Fractional Boundary Value Problems with Nonlinear Boundary Conditions

Abstract

We consider a system of boundary value problems for fractional differential equation given byD0+βϕp(D0+αu)(t)=λ1a1(t)f1(u(t),v(t)),t∈(0,1),D0+βϕp(D0+αv)(t)=λ2a2(t)f2(u(t),v(t)),t∈(0,1), where1<α,β≤2,2<α+β≤4,λ1,λ2are eigenvalues, subject either to the boundary conditionsD0+αu(0)=D0+αu(1)=0,u(0)=0,D0+β1u(1)-Σi=1m-2a1i D0+β1u(ξ1i)=0,D0+αv(0)=D0+αv(1)=0,v(0)=0,D0+β1v(1)-Σi=1m-2a2i D0+β1v(ξ2i)=0orD0+αu(0)=D0+αu(1)=0,u(0)=0,D0+β1u(1)-Σi=1m-2a1i D0+β1u(ξ1i)=ψ1(u),D0+αv(0)=D0+αv(1)=0,v(0)=0,D0+β1v(1)-Σi=1m-2a2i D0+β1v(ξ2i)=ψ2(v), where0<β1<1,α-β1-1≥0andψ1,ψ2:C([0,1])→[0,∞)are continuous functions. The Krasnoselskiis fixed point theorem is applied to prove the existence of at least one positive solution for both fractional boundary value problems. As an application, an example is given to demonstrate some of main results.