Symmetric positive solutions for a fractional singular integro-differential boundary value problem in presence of Caputo-Fabrizio fractional derivative

Abstract

We introduce the notion of Caputo-Fabrizio left and right derivatives. We present sufficient conditions for the existence of symmetric positive solutions for the following Caputo-Fabrizio fractional singular integro-differential boundary value problem \begin{align*} (2-\mu)\,{}^{CF}D_{0}^{\,\mu}x(t)+f(t,x(t))&=\left(\frac{\mu-1}{2-\mu}\right)^{2} \begin{cases} \int_{t}^{0}e^{-\frac{\mu-1}{2-\mu}(\tau-t)}\,x(\tau)d\tau,\hspace{0.4cm}&t\in(-1,0],\\ \int_{0}^{t}e^{-\frac{\mu-1}{2-\mu}(t-\tau)}\,x(\tau)d\tau,\hspace{0.4cm}&t\in[0,1), \end{cases}\\ x(\pm1)=x'(0^{\pm})&=0,\hspace{5.55cm}\mu\in(1,2), \end{align*} where the nonlinearity $f:(-1,\,1)\times(0,\infty)\rightarrow\mathbb{R}$ is continuous and singular at $t=-1$, $t=1$ and $x=0$.