#### Department

Mathematics and Statistics

Article

2017

#### 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.

#### Journal Title

Opuscula Mathematica

37

3

421

434

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