Existence and Global Asymptotic Behavior of Positive Solutions for Nonlinear Fractional Dirichlet Problems on the Half-Line

Abstract

We are interested in the following fractional boundary value problem: $D^{\alpha }u(t)+a\left(t\right)u^{\sigma }=0$, $t\in (0,\infty )$, ${\lim }_{t\to 0}{t}^{2-\alpha }u(t)=0$, ${\lim }_{t\to \infty }{t}^{1-\alpha }u(t)=0$, where , $\sigma \in (-\mathrm{1,1})$, $D^{\alpha }$ is the standard Riemann-Liouville fractional derivative, and $a$ is a nonnegative continuous function on $(0,\infty )$ satisfying some appropriate assumptions related to Karamata regular variation theory. Using the Schauder fixed point theorem, we prove the existence and the uniqueness of a positive solution. We also give a global behavior of such solution.