On a fractional Cauchy problem with singular initial data

Abstract

This article is dedicated to establishing the existence and uniqueness of solutions for the following problem: D α x ( t ) = F ( t , x ( t ) ) x ( 0 ) = x 0 , \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 x 0 {x}_{0} is the singular generalized function and F satisfies L ∞ {L}^{\infty } logarithmic type, D α {D}^{\alpha } is the Caputo derivative of order m − 1 < α < m m-1\lt \alpha \lt m with m ∈ N * m\in {{\mathbb{N}}}^{* } , which we will confirm to be present in Colombeau algebra. The Gronwall lemma is used in Colombeau’s algebra to establish the main results. To illustrate our theoretical analysis, we ended our work with an example.