Existence of a positive solution to systems of differential equations of fractional order

In this paper, we consider a system of (continuous) fractional boundary value problems given by {−D0+ν1y1(t)=λ1a1(t)f(y1(t),y2(t)),−D0+ν2y2(t)=λ2a2(t)g(y1(t),y2(t)), where ν1ν1, ν2∈(n−1,n]ν2∈(n−1,n] for n>3n>3 and n∈Nn∈N, subject either to the boundary conditions y1(i)(0)=0=y2(i)(0), for 0≤i≤n−20≤i≤n−2, and [D0+αy1(t)]t=1=0=[D0+αy2(t)]t=1, for 1≤α≤n−21≤α≤n−2, or y1(i)(0)=0=y2(i)(0), for 0≤i≤n−20≤i≤n−2, and [D0+αy1(t)]t=1=ϕ1(y), for 1≤α≤n−21≤α≤n−2, and [D0+αy2(t)]t=1=ϕ2(y), for 1≤α≤n−21≤α≤n−2. In the latter case, the continuous functionals ϕ1ϕ1, ϕ2:C([0,1])→Rϕ2:C([0,1])→R represent nonlocal boundary conditions. We provide conditions on the nonlinearities ff and gg, the nonlocal functionals ϕ1ϕ1 and ϕ2ϕ2, and the eigenvalues λ1λ1 and λ2λ2 such that the system exhibits at least one positive solution. Our results here generalize some recent results on both scalar fractional boundary value problems and systems of fractional boundary value problems, and we provide two explicit numerical examples to illustrate the generalizations that our results afford.