Existence and Estimates of Positive Solutions for Some Singular Fractional Boundary Value Problems

Abstract

We establish the existence and uniqueness of a positive solution $u$ for the following fractional boundary value problem${\mathrm{:D}}^{\alpha }u(x)=-a(x)u^{\sigma }(x)$, $x\in (0,1)$ with the conditions $\mathrm{}{\lim }_{x\to 0^{+}}$$x^{2-\alpha }u(x)=0$, $u(1)=0$, where , $\sigma \in (-1,1)$, and $a$ is a nonnegative continuous function on $(0,1)$ that may be singular at $x=0$ or $x=1$. We also give the global behavior of such a solution.