Fractional differential equations are an important branch of mathematics and have been considered under many different fractional derivatives. Among them, differential equations with Riesz-Caputo fractional derivatives have also attracted the attention of many researchers. Studying differential equations that may have singularity coefficients is more difficult than usual because they require several complex techniques. In the present paper, we consider a nonlinear pantograph differential equation where the source function may have a temporal singularity. Using the contraction principle, we prove that the problem has a unique solution under some appropriate conditions. Furthermore, we define a new type of Ulam-Hyers stability and show the main equation of the problem is stable in the mentioned sense. To obtain the main results, a new inequality is proposed and proved. Some examples are constructed to confirm the validity and feasibility of the theoretical results.

We consider a boundary value problem involving a fractional differential equation with a g-Caputo fractional derivative. This paper establishes some new criteria for the existence of solutions to the problem, differing from those obtained by previous researchers. The method is based on the construction of a novel Green’s function and the application of the Schauder fixed point theorem. Examples are provided to illustrate the fundamental distinctions between our results and earlier work.

In this paper, we consider initial value problems for nonlinear fractional equations where source functions may discontinuous. We obtain the existence and uniqueness of maximal mild solutions of the problem. We also give some appropriate conditions such that mild solutions of the problem blow-up at a finite time. Furthermore, we discuss the continuous dependence of mild solutions of the problem with respect to fractional order.

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

In this paper we discus on a Lyapunov-type inequality for a fractional differential equation involving sequential generalized Caputo fractional derivatives with boundary conditions. The results presented in this paper is new to the corresponding results in the literature.