Impulsive Differential Equations Research Articles

Ulam Stability for Boundary Value Problems of Differential Equations—Main Misunderstandings and How to Avoid Them

Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied.

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.

Existence of solutions for boundary value problems for first order impulsive difference equations

In this work, we establish sufficient conditions for the existence of nonnegative solutions for a class of first order impulsive difference equations with a family of nonlinear boundary conditions. To prove our main result we use a new topological approach on the fixed point index theory for the sum of two operators in Banach spaces. An example is given to illustrate the main result.

Relative controllability for conformable impulsive delay differential equations

Abstract In this paper, we mainly study a class of conformable impulsive delay differential equations (CIDDEs). We first define a conformable impulsive delayed matrix function, and construct an explicit solution for linear CIDDEs by virtue of variation of constants method. Subsequently, based on impulsive delayed Grammian matrix, we study the relative controllability for the addressed linear equations. Moreover, with the help of Krasnoselskii’s fixed point theorem, relative controllability for the considered semilinear equations is proposed. Finally, two examples with numerical simulations are given to illustrate the main results.

Properties of the solutions to periodic conformable non-autonomous non-instantaneous impulsive differential equations

In this article, we study properties of the solutions to periodic non-autonomous conformable non-instantaneous impulsive differential equations. We use a conformable Cauchy matrix and obtain some basic properties of the periodic solution to the homogeneous and non=homogeneous problems. We consider the periodicity of solutions to nonlinear problem via a fixed theorem. For more information see https://ejde.math.txstate.edu/Volumes/2024/30/abstr.html

Analysis on nonlinear differential equation with a deviating argument via Faedo–Galerkin method

This article focuses on the impulsive fractional differential equation (FDE) of Sobolev type with a nonlocal condition. Existence and uniqueness of the approximations are determined via analytic semigroup and fixed point method. Convergence’s approximation is demonstrated by the idea of fractional power of a closed linear operator. Using an approximation procedure, a novel approach is reached. An illustration is used to clarify our key findings.

Almost sure exponential stability and stochastic stabilization of impulsive stochastic differential delay equations

In this paper, we mainly study the almost sure exponential stability of impulsive stochastic differential delay equations (ISDDEs) with bounded variable delays. The main technique is to compare ISDDEs with corresponding impulsive stochastic differential equations (ISDEs) without delay, to obtain the upper bound τ∗ of delays that ISDDEs can maintain stability by accurate calculation. The results show that if the corresponding ISDEs are almost surely exponentially stable, then the ISDDEs are also almost surely exponentially stable as long as these delays are less than τ∗. In addition, the stability theory established in this paper can be applied to noise stabilization based on sampled-data observations of a class of unstable impulsive systems. Taking impulsive Lurie systems (ILSs) as an example, we discuss the design of noise stabilization strategies.

An analysis of hybrid impulsive prey-predator-mutualist system on nonuniform time domains

Abstract In this work, we propose a hybrid impulsive prey-predator-mutualist model on nonuniform time domains. We have investigated the permanence/persistence results for the proposed model using the comparison theorems of impulsive differential equations and some dynamic inequality on the nonuniform time domains. In addition, we have established certain necessary requirements for the uniform asymptotic stability of the almost periodic solution and global attractivity of the proposed model. Furthermore, we provide several numerical examples on nonuniform time domains with computer simulation to demonstrate the viability of the results of the acquired analytical work.

IMPULSIVE DIFFERENTIAL EQUATION MODEL IN HIV-1 INHIBITION: ADVANCES IN DUAL INHIBITORS OF HIV-1 RT AND IN FOR THE PREVENTION OF HIV-1 REPLICATION

Reverse transcriptase (RT) and integrase (IN) are two pivotal enzymes in HIV-1 replication. RT converts the single-stranded viral RNA genome into double-stranded DNA and IN catalyzes the integration of viral double-stranded DNA into host DNA. Currently, dual inhibitors of HIV-1 RT and IN have become a hotspot in new anti-HIV drug research and development. A dual inhibitor of HIV-1 RT/IN does the same thing as the two independent drugs would do. In this paper, we develop a mathematical model comprising a system of nonlinear differential equations describing HIV-1 RT/IN catalyzed biochemical reactions based on Michaelis–Menten enzyme kinetic reaction. In the formulated model we incorporate HIV-1 RT/IN dual inhibitor which simultaneously works as a non-nucleoside RT inhibitor and IN inhibitor. To examine the efficacy of HIV-1 RT/IN dual inhibitor in the treatment of HIV-1 infection, we have introduced a one-dimensional impulsive differential equation model and determined an effective dosing regimen for applying the inhibitor numerically. Furthermore, the exact closed form solution of the impulsive differential equation model is carried out by using the Lambert W function and the local stability of the periodic solution is also obtained analytically. The results obtained from analytical as well as numerical studies provide a basic idea to investigate the minimum dose with the highest efficacy for administering HIV-1 RT/IN dual inhibitors to prevent HIV-1 infection.

A rumor propagation model in multilingual environment with time and state dependent impulsive control

In this paper we are interested in the problem of modeling, analysis and control of rumor spreading on online social networks. As the diversity of languages and the difference of user nodes in the network, a heterogeneous rumor spreading model with two language spreaders is proposed, and the impulsive isolation control strategy is considered. Firstly, the model with periodic impulsive isolation control and time-delay recovery rate is discussed. Based on the stability theory of impulsive differential equations, some conditions regarding the impulsive period and control gains are given for the rumor asymptotically disappearance and persistent existence. Secondly, a new type of event-triggered impulsive isolation control method is developed to address the tradeoff between restraining rumor and minimizing the cost of control, and some conditions are given to make the rumor disappear asymptotically. Furthermore, an improved event-triggered mechanism is established based on the fixed-time stability theory, which provides the conditions for the rumor to disappear within a fixed-time. Finally, some numerical simulations and an actual example are given to verify the rationality of the model and the effectiveness of the control methods.

Controllability of semilinear noninstantaneous impulsive neutral stochastic differential equations via Atangana-Baleanu Caputo fractional derivative

This study mainly concerns the controllability of semilinear noninstantaneous impulsive neutral stochastic differential equations via the Atangana-Baleanu (AB) Caputo fractional derivative (FD). The essential findings are created using methods and concepts from semigroup theory, stochastic theory, fractional calculus, K-set contraction, and measure of noncompactness. Finally, an example is provided to demonstrate the applications of the key findings.

Exponential stability results for second-order impulsive neutral stochastic differential equations

This paper is aimed to discuss the existence and exponential stability of mild solutions for second-order impulsive neutral stochastic differential equations (INSDEs). Firstly, we derive the existence of mild solutions for INSDEs by using the Banach fixed point theorem. Then, the exponential stability in the pth moment of mild solution for the considered INSDEs is proved by means of the generalized impulsive-integral inequality, which can effectively improve some known existing ones. Finally, as an application, an example is given to illustrate the efficiency of the obtained theoretical results.

Pth Moment Exponential Stability of Impulsive Stochastic Differential Equations with Unbounded Delays

Stochastic impulsive differential equations with Markovian switching have been applied extensively in various areas including ecology systems, neural networks, and control systems, and stability analysis is one fundamental premise of their applications. For two categories of Markovian switched impulsive stochastic differential functional equations with unbounded delays, this paper investigates the pth moment exponential stability by adopting stochastic Lyapunov stability theory and stochastic analysis approach. Several criteria on pth moment exponential stability have been acquired. In the proposed model, the time-varying coefficients and the hybrid impulsive effects are considered simultaneously. It can be seen that the criteria derived in this paper are more concise and the conditions are easier to verify compared with those existing results based on Razumikhin theory. Finally, two examples are illustrated to show the effectiveness of the theoretical findings.

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.

Neural Network Methods Based on Efficient Optimization Algorithms for Solving Impulsive Differential Equations

In view of the fact that impulsive differential equations have the discreteness due to the impulse phenomenon, this paper proposes a single hidden layer neural network method based extreme learning machine and a physicsinformed neural network method combined with learning rate attenuation strategy to solve linear impulsive differential equations and nonlinear impulsive differential equations respectively. For the linear impulsive differential equations, firstly, the interval is segmented according to the impulse points, and a single hidden layer neural network model is constructed, the weight parameters of hidden layer are randomly set, the optimal output parameters and solution of the first segment are obtained by the extreme learning machine algorithm, then we calculate the initial value of the second segment according to the jumping equation, and the remaining segments are solved in turn in the same way. Although the single hidden layer neural network method proposed can solve linear equations with high accuracy, it is not suitable for solving nonlinear equations. Therefore, we propose the physics-informed neural network combined with learning rate attenuation strategy to solve the nonlinear impulsive differential equations, then the Adam algorithm and L-BFGS algorithm are combined to find the optimal approximate solution of each segment. Numerical examples show that the single hidden layer neural network method with Legendre polynomials as the activation function and the physics-informed neural network method combined with learning rate attenuation strategy can solve linear and nonlinear impulsive differential equations with higher accuracy.

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.

Existence of solution sets for Φ-Laplacian for random impulsive differential equations

In this article, we provide results on the existence and compactness of the set of solutions for a class of initial-value problem representing a first-order random impulsive differential equation. The key tool to achieve our goal is based on an alternative nonlinear Leary-Schauder type theorem in the settings of generalized Banach spaces.

Existence Results of Impulsive Hybrid Fractional Differential Equations with Initial and Boundary Hybrid Conditions

In this paper, we establish sufficient conditions for the existence and uniqueness of solution of impulsive hybrid fractional differential equations with initial and boundary hybrid conditions. The proof of the main result is based on the classical fixed point theorems such as Banach fixed point theorem and Leray-Schauder alternative fixed point theorem. Two examples are included to show the applicability of our results.

Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response

<abstract><p>A good pulse control strategy should depend on the numbers of pests and natural enemies as determined via an integrated pest control strategy. Taking this into consideration, here, a nonlinear impulsive predator-prey model with improved Leslie-Gower and Beddington-DeAngelis functional response terms is qualitatively analyzed. The existence of a periodic solution for pest eradication has been obtained and the critical condition of global asymptotic stability has been established by using the impulsive differential equation Floquet theory. Furthermore, the conditions for the lasting survival of the system has been proved by applying a comparison theorem for differential equations. Additionally, a stable positive periodic solution has been obtained by applying bifurcation theory. To understand how nonlinear pulses affect the dynamic behavior of a system, MATLAB was used to conduct numerical simulations to show that the model has very complex dynamical behavior.</p></abstract>