In this paper, we prove the existence and uniqueness of solution to the IMPULSIVE fuzzy FUNCTIONAL DIFFERENTIAL EQUATIONS under generalized Hukuhara differentiability via the principle of contraction mappings. Some examples are provided to illustrate the result.

In this paper, we prove the existence and uniqueness results to the random fractional FUNCTIONAL DIFFERENTIAL EQUATIONS under assumptions more general than the Lipschitz type condition. Moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.

This paper deals with the existence and uniqueness of the mild solution of the fractional integro-DIFFERENTIAL EQUATIONS with non-instantaneous impulses and state-dependent delay. Our arguments are based on the fixed point theory. Finally, an example to confirm of the results  is provided.

In this paper we introduce some stability criteria for IMPULSIVE fuzzy system of DIFFERENTIAL EQUATIONS with finite delay in states. Firstly, a new comparison principle for fuzzy DIFFERENTIAL system compared to crisp ordinary DIFFERENTIAL equation, based on a notion of upper quasi-monotone nondecreasing, in N dimentional state space is presented. Furthermore, in order to analyze the stability of fuzzy dynamical systems, vector Lyapunov like functions are defined. Then, by using these vector Lyapunov-like functions together with the new comparison theorem which is presented before, we will get results for some concepts of stability (eventual stability, asymptotic stability, strong stability, uniform stability and their combinations) for IMPULSIVE fuzzy delayed system of DIFFERENTIAL EQUATIONS. Moreover, some theorems for practical stability in terms of two measures are introduced and are proved. Finally, an illustrating example for stability checking of a system of DIFFERENTIAL EQUATIONS with fuzziness and time delay in states is given.

In this article we consider, IMPULSIVE initial value problems for a class of implicit fractional DIFFERENTIAL EQUATIONS involving the Caputo fractional derivative of order β in (1, 2]. The solutions of this nonlinear equation are analyzed by establishing sufficient conditions for existence and uniqueness using Banach's contraction mapping principle and the Schaefer's fixed point theorem. In addition, using the Banach contraction principle, we establish uniqueness result. To demonstrate main results two examples are presented.

The aim of this paper is to use the Reproducing kernel Hilbert Space Method (RKHSM) to solve the linear and nonlinear fuzzy IMPULSIVE fractional DIFFERENTIAL EQUATIONS. Finding the numerical solutions of this class of EQUATIONS are a difficult topic to analyze. In this study, convergence analysis, estimations error and bounds errors are discussed in detail under some hypotheses which provide the theoretical basis of the proposed algorithm. Some numerical examples indicate that this method is an efficient one to solve the mentioned EQUATIONS.

In this paper we introduce some stability criteria of nonlinear hybrid systems with time delay described by IMPULSIVE hybrid fuzzy system of DIFFERENTIAL EQUATIONS. Firstly, a comparison principle for fuzzy DIFFERENTIAL system based on a notion of upper quasi-monotone nondecreasing is presented. Here, for stability analysis of fuzzy dynamical systems, vector Lyapunov-like functions are defined. Then, by using these functions together with the new comparison theorem, we will get results for some concepts of stability (eventual stability, asymptotic stability, strong stability and uniform stability) for IMPULSIVE hybrid fuzzy delay DIFFERENTIAL systems. Furthermore, theorems for practical stability in terms of two measures are introduced and are proved. Finally, an illustrating example for stability checking of a DIFFERENTIAL system with fuzziness and time delay is given. Then, by introducing an applied example in Pharmacokinetics, we bridge theoretical concepts to the application of research in real world.