TY  - JOUR
ID  - Lyons2017
LB  - Lyons2017
AU  - Lyons, Jeffrey W.
AU  - Neugebauer, Jeffrey T.
TI  - Positive solutions of a singular fractional boundary value problem with a fractional boundary condition
ST  - Positive solutions of a singular fractional boundary value problem with a fractional boundary condition
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2017
VL  - 37
IS  - 3
SP  - 421
EP  - 434
DO  - http://dx.doi.org/10.7494/OpMath.2017.37.3.421
UR  - http://dx.doi.org/10.7494/OpMath.2017.37.3.421
AB  - 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.
KW  - fractional differential equation
KW  - singular problem
KW  - fixed point
SN  - 1232-9274
ER  -