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

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

Category: Vol 34/2018 no. 1 Tags: positive solution, Nonlinear fractional differential equations, singular boundary value problem

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In the present work, we discuss the existence of a unique positive solution of a boundary value problem for a 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 a fixed point theorem, we prove the existence of unique positive solution of the considered problem.