Author ORCID Identifier
Jeffrey T. Neugebauer 
https://orcid.org/0000-0003-2450-4169
Department
Mathematics and Statistics
Document Type
Article
Publication Date
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.
Recommended Citation
Lyons, J. W., & Neugebauer, J. T. (2017). Positive solutions of a singular fractional boundary value problem with a fractional boundary condition. Opuscula Mathematica, 37(3), 421. doi:10.7494/opmath.2017.37.3.421
Journal Title
Opuscula Mathematica
