Positive solutions of a singular fractional boundary value problem with a fractional boundary condition

Abstract

For $\alpha\in(1,2]$ the singular fractional boundary value problem $D^{\alpha}_{0^+}x+f\left(t,x,D^{\mu}_{0^+}x\right)=0,\quad 0\lt t\lt 1,$ satisfying the boundary conditions $x(0)=D^{\beta}_{0^+}x(1)=0$, where $\beta\in(0,\alpha-1]$, $\mu\in(0,\alpha-1]$, and $D^{\alpha}_{0^+}$, $D^{\beta}_{0^+}$ and $D^{\mu}_{0^+}$ are Riemann-Liouville derivatives of order $\alpha$, $\beta$, and $\mu$ respectively, is considered. Here $f$ satisfies a local Carathéodory condition, and $f(t, x, y)$ may be singular at the value 0 in its space variable $x$. Using regularization and sequential techniques and Krasnosel’skii’s fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given.