First extremal point comparison for a fractional boundary value problem with a fractional boundary condition

Abstract:

Let $n \geq 2$ be a natural number, and let $n-1<\alpha \le n$ and $0<\gamma \le \alpha -1$ be real numbers. Let $\beta >0$ and $b\in (0,\beta ]$. We compare first extremal points of the differential equations $D_{0+}^\alpha u+p(t)u=0$, $D_{0+}^\alpha u+q(t)u=0$, $t\in (0,\beta )$, each of which satisfies the boundary conditions $u^{(i)}(0)=0$, $i=0,1,\dots ,n-2$, $\quad D_{0^+}^\gamma u(b)=0$. While it is assumed that $q$ is nonnegative, no sign restrictions are put on $p$. The fact that the associated Green’s function $G(b;t,s)$ is nonnegative and increasing with respect to $b$ plays an important role in the analysis.