On a fractional Cauchy problem with singular initial data

This article is dedicated to establishing the existence and uniqueness of solutions for the following problem: Dαx(t)=F(t,x(t))x(0)=x0,\left\{\begin{array}{l}{D}^{\alpha }x\left(t)=F\left(t,x\left(t))\hspace{1.0em}\\ x\left(0)={x}_{0},\hspace{1.0em}\end{array}\right. where x0{x}_{0} is the singular generalized function and F satisfies L∞{L}^{\infty } logarithmic type, Dα{D}^{\alpha } is the Caputo derivative of order m−1