Sturm–Picone theorem for fractional nonlocal equations

We establish a generalization of Sturm–Picone comparison theorem for pair fractional nonlocal equations: $$\begin{aligned} (-div. (A_1(x)\nabla ))^{s} u= & {} C_{1}(x) u \,\,\,\text { in }\,\,\Omega ,\\ 0 \,\,\,\,\text on }\,\,\,\,\,\partial \Omega , \end{aligned}$$ and (A_2(x)\nabla v= C_{2}(x) v in}\,\,\Omega on}\,\,\,\,\,\partial where $$\Omega \subset \mathbb {R}^n$$ is an open bounded subset with smooth boundary, $$0<s<1,\,\,A_1,\,A_2$$ are smooth, real symmetric positive definite matrices $${\overline{\Omega }}$$ $$C_{1}, C_{2}\in C^{\alpha }({\overline{\Omega }}).$$