This paper is devoted to study a fractional equation involving Caputo-Katugampola derivative with nonlocal initial condition. Unlike previous papers, in this paper, the source function of problem is assumed having a singularity. We propose some reasonable conditions such that the problem has at least one mild solution or has a unique mild solution. The desired results are proved by using the Banach, Leray-Schauder and Krasnoselskii fixed point theorems. Some examples are given to confirm our theoretical findings.
Keywords: Caputo-Katugampola fractional derivative; Nonlinear integral equations; existence
2010 MSC: 26A33; 35A01; 35A02; 35R11