%0 Journal Article
%F Lyons2017
%A Lyons, Jeffrey W.
%A Neugebauer, Jeffrey T.
%T Positive solutions of a singular fractional boundary value problem with a fractional boundary condition
%! Positive solutions of a singular fractional boundary value problem with a fractional boundary condition
%J Opuscula Mathematica
%O Opuscula Math.
%D 2017
%V 37
%N 3
%P 421-434
%R http://dx.doi.org/10.7494/OpMath.2017.37.3.421
%U http://dx.doi.org/10.7494/OpMath.2017.37.3.421
%X 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.
%K fractional differential equation
singular problem
fixed point
%@ 1232-9274