On the Nonlinear Impulsive Ψ\Psi--Hilfer Fractional Differential Equations

In this paper, we consider the nonlinear $\Psi$-Hilfer impulsive fractional differential equation. Our main objective is to derive the formula for the solution and examine the existence and uniqueness of results. The acquired results are extended to the nonlocal $\Psi$-Hilfer impulsive fractional differential equation. We gave an applications to the outcomes we procured. Further, examples are provided in support of the results we got.Comment: 2

(R2066) New Results of Ulam Stabilities of Functional Differential Equations of First Order Including Multiple Retardations

In this study, we pay attention to a functional differential equation (FDE) of first order including N-variable delays. We construct new sufficient conditions in relation to the Hyers-Ulam stability (HUS) and the generalized Hyers-Ulam-Rassias stability (GHURS ) of the FDE of first order including N-variable delays. By using Banach contraction principle (BCP), Picard operator and Gronwall lemma, we confirm two new theorems in relation to the HUS and the GHURS. The results of this study are new and extend, improve some earlier results of the HUS and the GHURS

A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator

In this paper, we propose a generalized Gronwall inequality through the fractional integral with respect to another function. The Cauchy-type problem for a nonlinear differential equation involving the psi-Hilfer fractional derivative and the existence and uniqueness of solutions are discussed. Finally, through generalized Gronwall inequality, we prove the continuous dependence of data on the Cauchy-type problem.1118710

Existence and Stability Results for Impulsive Integro-DifferentialEquations

In this paper, we study a new class of impulsiveintegro-differential equations for which the impulses are notinstantaneous. By using fixed point approach and techniques ofanalysis, we present the existence and uniqueness theorem and derivean interesting stability result in the sense of generalizedUlam-Hyers-Rassias

Existence and stability results for nonlinear fractional delay differential equations

We establish existence and uniqueness results for fractional order delay differential equations. It is proved that successive approximation method can also be successfully applied to study Ulam--Hyers stability, generalized Ulam--Hyers stability, Ulam--Hyers--Rassias stability, generalized Ulam--Hyers--Rassias stability, $\mathbb{E}_{\alpha}$--Ulam--Hyers stability and generalized $\mathbb{E}_{\alpha}$--Ulam--Hyers stability of fractional order delay differential equations