Ulam stability and data dependence for fractional differential equations with Caputo derivative

In this paper, Ulam stability and data dependence for fractional differential equations with Caputo fractional derivative of order $\alpha$ are studied. We present four types of Ulam stability results for the fractional differential equation in the case of $0<\alpha<1$ and $b=+\infty$ by virtue of the Henry-Gronwall inequality. Meanwhile, we give an interesting data dependence results for the fractional differential equation in the case of $1<\alpha<2$ and $b<+\infty$ by virtue of a generalized Henry-Gronwall inequality with mixed integral term. Finally, examples are given to illustrate our theory results

Ulam Stabilities for Partial Impulsive Fractional Differential Equations

summary:In this paper we investigate the existence of solutions for the initial value problems (IVP for short), for a class of implicit impulsive hyperbolic differential equations by using the lower and upper solutions method combined with Schauder’s fixed point theorem

Analytical study of ABC-fractional pantograph implicit differential equation with respect to another function

This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo ($\mathcal{ABC}$) fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers $(\mathbb{UH})$, generalized $\mathbb{UH}$, Ulam-Hyers-Rassias $(\mathbb{UHR})$ and generalized $\mathbb{UHR}$ are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature