Sturm–Picone theorem for fractional nonlocal equations

We establish a generalization of Sturm–Picone comparison theorem for a pair of fractional nonlocal equations: (-div.(A1(x)∇))su=C1(x)uinΩ,u=0on∂Ω,and (-div.(A2(x)∇))sv=C2(x)vinΩ,v=0on∂Ω,where Ω ⊂ Rⁿ is an open bounded subset with smooth boundary, 0<s<1,A1,A2 are smooth, real symmetric and positive definite matrices on Ω ¯ and C<inf>1</inf>, C<inf>2</inf>∈ C^α(Ω ¯).