Impulsive Functional Differential Equations Research Articles

A note on generalized exponential stability of impulsive stochastic functional differential equations

This brief considers the generalized exponential stability of impulsive stochastic functional differential equations (ISFDEs). Initially, An extension of Halanay inequality is derived. Subsequently, by employing this generalized Halanay inequality, we establish the generalized exponential stability result for the given systems.

Existence of a Class of Doubly Perturbed Stochastic Functional Differential Equations with Poisson Jumps

In this paper, we use successive approximations and Picard iterative method to establish the existence and uniqueness of mild solution for a class of doubly perturbed impulsive neutral stochastic functional differential equations with Poisson jumps in Hilbert spaces. An example of a doubly perturbed stochastic differential equation with delays is given to illustrate our main results.

Stability of impulsive stochastic functional differential equations with delays

In this paper, we consider the global asymptotical stability of stochastic functional differential equations with impulsive effects. First, by constructing the Lyapunov function, some stability criteria of impulsive stochastic functional differential equations are established. Second, we propose an application to investigate the effectiveness of the obtained results.

Approximate controllability of impulsive evolution stochastic functional differential equations driven by a fractional Brownian motion

In this paper, we study the approximate controllability of certain class of impulsive evolution stochastic functional differential equations, with variable delays, driven by a fractional Brownian motion in a separable real Hilbert space. We derive a new set of sufficient conditions for approximate controllability using a stochastic analysis of fractional Brownian motion with Hurst parameter H ∈ (1/2,1) and a Schaefer's fixed point theorem. An example is considered at the end of the paper to illustrate the obtained abstract results.

A System of Coupled Impulsive Neutral Functional Differential Equations: New Existence Results Driven by Fractional Brownian Motion and the Wiener Process

Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk’s fixed point theorem in extended Banach spaces. This paper aims to extend current results to a differential-inclusions scenario. The motivation of this paper for impulsive neutral differential equations is to investigate the existence of solutions for impulsive neutral differential equations with fractional Brownian motion and a Wiener process (topics that have not been considered and are the main focus of this paper).

Asymptotic behaviour of the non-oscillating solutions of first order nonlinear impulsive functional-differential equations of neutral type

This paper deals with nonlinear first-order impulsive functional-differential equations of neutral type with variable coefficients. The asymptotic behaviour of the non-oscillating solutions of such equations is investigated as a function of the values of the variable coefficient in the neutral term to serve as a basis for further studies of the oscillation of the solutions of such equations.

Study of Non-Linear Impulsive Neutral Fuzzy Delay Differential Equations with Non-Local Conditions

This manuscript aims to investigate the existence and uniqueness of fuzzy mild solutions for non-local impulsive neutral functional differential equations of both first and second order, incorporating finite delay. Furthermore, the study explores the properties of fuzzy set-valued mappings of a real variable, where these mappings exhibit characteristics such as normality, convexity, upper semi-continuity, and compact support. The application of the Banach fixed-point theorem is employed to derive the results. The research extensively employs fundamental concepts from fuzzy set theory, functional analysis, and the Hausdorff metric. Additionally, an illustrative example is provided to exemplify the practical implementation of the proposed concept.

Some criteria on exponential stability of impulsive stochastic functional differential equations

This paper is mainly concerned with th moment exponential stability of impulsive stochastic functional differential equations. By using the Lyapunov direct method and mathematical induction, we obtain a new generalized impulsive Halanay inequality and some new criteria, which contain two scenarios regarding influence of impulses. On the one hand, when the system itself is unstable, a suitable unidirectional impulse or sequence of impulses can be added to make it stable. On the other hand, we discuss how much impulse perturbation the system can withstand without destabilizing the original system. Finally, two examples are illustrated to show the validity and universality of our results.

Fractional neutral functional differential equations driven by the Rosenblatt process with an infinite delay

Abstract This paper concerns a class of fractional impulsive neutral functional differential equations with an infinite delay driven by the Rosenblatt process. A set of sufficient conditions are established for the existence of new mild solutions using fixed point theory. Finally, an illustrative example is provided to demonstrate the applicability of the theoretical result.

Stability of stochastic functional differential equations with impulses

The paper studies the stability of impulsive stochastic functional differential equations. At first, we obtained a new generalization of the Halanay inequality. Then, by applying the generalized Halanay inequality, the exponential stability results of impulsive stochastic functional differential equations are obtained.

Controllability of second order neutral impulsive fuzzy functional differential equations with Non-Local conditions

Controllability of second order neutral impulsive fuzzy functional differential equations with Non-Local conditions

Some sufficient conditions of existence and trajectory controllability for impulsive and initial value fractional order functional differential equations

For a class of impulsive fractional differential equation with fractional order r∈(1,2) by means of solution operator Sr(t), we develop the mild solution and deal with the existence results of mild solution for the considered equation with state-dependent delay through three fixed point theorems, namely Banach, Nonlinear Leray–Schauder Alternative and Krasnoselskii’s fixed point theorem. Next, we define the mild solution via an operator function Wr,β(τ) for abstract version of the considered equation. Further, we study the trajectory controllability (which is a stronger notion than other controllability) of the equation. In the last, we present some examples to illustrate the established results through partial fractional differential equations.

A NECESSARY CONDITION OF VIABILITY FOR IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION

Viability theory is a mathematical theory that offers mathematical metaphors of evolution of macrosystems arising in biology, economics, cognitive sciences, games, and similar areas, as well as in nonlinear systems of control theory. The viability problem has been studied by many authors by using various frameworks and techniques and is still one of the active directions of differential equations. The viability property in a stochastic framework was explored first by Aubin and Da Prato (Aubin & Da Prato, 1990). In this paper, we give a necessary condition for viability results of an impulsive stochastic functional differential equation driven by a fractional Brownian motion with Hurst parameter 1/2 < H <1.

APPROXIMATE CONTROLLABILITY OF IMPULSIVE STOCHASTIC SYSTEMS DRIVEN BY ROSENBLATT PROCESS AND BROWNIAN MOTION

In this paper we consider a class of impulsive stochastic functional differential equations driven simultaneously by a Rosenblatt process and standard Brownian motion in a Hilbert space. We prove an existence and uniqueness result and we establish some conditions ensuring the approximate controllability for the mild solution by means of the Banach fixed point principle. At the end we provide a practical example in order to illustrate the viability of our result.

Exponential Stability of Impulsive Neutral Stochastic Functional Differential Equations

This paper focuses on the problem of the pth moment and almost sure exponential stability of impulsive neutral stochastic functional differential equations (INSFDEs). Based on the Lyapunov function and average dwell time (ADT), two sufficient criteria for the exponential stability of INSFDEs are derived, which manifest that the result obtained in this paper is more convenient to be used than those Razumikhin conditions in former literature. Finally, two numerical examples and simulations are given to verify the validity of our result.

Existence of solutions for fractional impulsive neutral functional differential equations driven by fractional Brownian motion

Abstract In this paper, we consider a class of fractional impulsive neutral stochastic functional differential equations with infinite delay driven by a fractional Brownian motion in a real separable Hilbert space. We prove the existence of mild solutions by using stochastic analysis and a fixed-point strategy.

Wellposedness of impulsive functional abstract second-order differential equations with state-dependent delay

Abstract This study investigates the functional abstract second order impulsive differential equation with state-dependent delay. The major result of this study is that the abstract second-order impulsive differential equation with state-dependent delay system has at least one solution and is unique. After that, the wellposed condition is defined. Following that, we look at whether the proposed problem is wellposed. Finally, some illustrations of our findings are provided.

Exponential ultimate boundedness and stability of stochastic differential equations with impulese

AbstractThe paper mainly studies globally pth moment exponentially ultimate boundedness and pth moment exponential stability of impulsive stochastic functional differential equations. By using the Lyapunov direct method of Razumikhin‐type condition and the principle of comparison, some sufficient conditions for globally pth moment exponentially ultimate boundedness and globally pth moment exponential stability are presented. Theorems require the linear coefficients of the upper bound of Lyapunov differential operators are time‐varying functions; this generalizes the previous results. When the time delay is not considered in the system, a unified criterion is given to achieve boundedness and stability when the system is disturbed by stabilizing impulse and destabilizing impulse. It shows that the stochastic differential equation may be unbounded or unstable, and it can be bounded or stable by adding appropriate impulsive perturbation. Finally, we use two examples to illustrate the validity of our results.

Dynamical Behavior of Stochastic Markov Switching Hepatitis B Epidemic Model with Saturated Incidence Rate

The article researches a stochastic hepatitis B epidemic model with saturated incidence rate, which is perturbed by both white noise and colored noise. Firstly, we obtain a significant criterion R 0 S which relies on environmental noises. By means of Lyapunov function approach, we show that there is a stationary distribution if R 0 S > 1 . Its condition implies that when white noise is small, in the stochastic model, there exists a stochastic positive equilibrium state without changing the basic properties of its corresponding deterministic model. Secondly, we derive sufficient criteria for extinction of the disease. Finally, we propose a definition of the solution to an impulsive stochastic functional differential equation with Markovian switching (ISFDM).

A note on controllability of impulsive neutral stochastic functional differential equations driven by a Rosenblatt process in a Hilbert space

In this paper we study the controllability results of impulsive neutral stochastic functional differential equations with variable delays driven by Rosenblatt process in a Hilbert space. The controllability results are obtained by the Banach fixed point theorem. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.