Opuscula Math. 37, no. 3 (2017), 421-434
http://dx.doi.org/10.7494/OpMath.2017.37.3.421

Opuscula Mathematica

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

Jeffrey W. Lyons
Jeffrey T. Neugebauer

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.

Keywords: fractional differential equation, singular problem, fixed point.

Mathematics Subject Classification: 26A33, 34A08, 34B16.

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• Jeffrey W. Lyons
• Nova Southeastern University, Department of Mathematics, Fort Lauderdale, FL 33314 USA
• Jeffrey T. Neugebauer
• Eastern Kentucky University, Department of Mathematics and Statistics, Richmond, KY 40475 USA
• Communicated by Theodore A. Burton.
• Revised: 2016-10-29.
• Accepted: 2016-10-29.
• Published online: 2017-01-30.