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.
  • Received: 2016-08-04.
  • Revised: 2016-10-29.
  • Accepted: 2016-10-29.
  • Published online: 2017-01-30.
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Cite this article as:
Jeffrey W. Lyons, Jeffrey T. Neugebauer, Positive solutions of a singular fractional boundary value problem with a fractional boundary condition, Opuscula Math. 37, no. 3 (2017), 421-434, http://dx.doi.org/10.7494/OpMath.2017.37.3.421

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