Existence of a unique positive solution for a singular fractional boundary value problem

TL;DR

This paper proves the existence of a unique positive solution for a nonlinear fractional boundary value problem with singularity, using fixed point theorem, for equations of order between 3 and 4.

Contribution

It establishes the first known result on the existence and uniqueness of positive solutions for such singular fractional boundary value problems.

Findings

Proved existence of a unique positive solution.

Applied fixed point theorem to fractional differential equations.

Addressed singularity at boundary points.

Abstract

In the present work, we discuss the existence of a unique positive solution of a boundary value problem for nonlinear fractional order equation with singularity. Precisely, order of equation $D_{0+}^\alpha u(t)=f(t,u(t))$ belongs to $(3,4]$ and $f$ has a singularity at $t=0$ and as a boundary conditions we use $u(0)=u(1)=u'(0)=u'(1)=0$. Using fixed point theorem, we prove the existence of unique positive solution of the considered problem.