Nonlinear Impulsive Integro-differential Equations Research Articles

Existence and Stability Results of Nonlinear Random Impulsive Integro-Differential Evolution Equations with Time-Varying Delays

This study examines the existence, uniqueness, and stability of nonlinear random impulsive integrodifferential equations with time-varying delays under sufficient conditions. Our study is based on the Leray-Schauder alternative fixed point theorem, Pachpatte’s inequality, and the Banach contraction principle. Besides, we generalize, extend, and develop some results for Kasinathan et al.

Study on the Multi-Point Boundary Value Problem for Second-Order Nonlinear Impulsive Integro-Differential Equation

In the field of biological control, there are a large number of systems that gradually evolve over a certain period of time. However, due to some natural or human intervention behavior, the system state will be subjected to some relatively short time interference, so that the system state changes in an instant. This sudden change of state makes the system not simply described by continuous or discrete dynamical systems, but by means of impulse dynamical systems. In this paper, the multi-point boundary value problem of a class of second-order nonlinear impulsive integral differential equations is studied on the basis of impulsive differential equation theory. The main results of this kind of equation are obtained by using the fixed point theorem of strict set contraction operator. Under certain assumptions, the existence of the solution of the equation is proved by constructing the operator on the special cone. Finally, combined with the practical application, the theory was applied to the stability prediction of biological ecosystem, and the correctness of the conclusion was verified.

On the Nonlinear Impulsive Volterra-Fredholm Integrodifferential Equations

In this paper, we investigate existence and uniqueness of solutions of nonlinear Volterra-Fredholm impulsive integrodifferential equations. Utilizing theory of Picard operators we examine data dependence of solutions on initial conditions and on nonlinear functions involved in integrodifferential equations. Further, we extend the integral inequality for piece-wise continuous functions to mixed case and apply it to investigate the dependence of solution on initial data through $epsilon$-approximate solutions. It is seen that the uniqueness and dependency results got by means of integral inequity requires less restrictions on the functions involved in the equations than that required through Picard operators theory.

Existence of solutions of an impulsive integro-differential equation with a general boundary value condition

<abstract><p>In this paper, we discuss the existence of solutions for a first-order nonlinear impulsive integro-differential equation with a general boundary value condition. New comparison principles are developed, and existence results for extremal solutions are obtained using the established principles and the monotone iterative technique. The results are more general than those of the periodic boundary problems, which may be widely applied in this field.</p></abstract>

Existence and uniqueness results for the first-order non-linear impulsive integro-differential equations with two-point boundary conditions

The article discusses the existence and uniqueness of solutions for a system of nonlinear integro-differential equations of the first order with two-point boundary conditions. The Green function is constructed, and the problem under consideration is reduced to equivalent integral equation. Existence and uniqueness of a solution to this problem is analyzed using the Banach contraction mapping principle. Schaefer’s fixed point theorem is used to prove the existence of solutions.

Mild Solution of Second-Order Impulsive Integro-Differential Evolution Equations of Volterra Type in Banach Spaces

In this paper, by using the cosine family theory, measure of non-compactness and the generalization of Darbo-fixed point theorem, we establish the existence of mild solutions of nonlinear second-order impulsive integro-differential evolution equations of Volterra type in Banach spaces. The result obtained herein generalizes and improves some known results.

Extremal Solutions for Nonlinear First-Order Impulsive Integro-Differential Dynamic Equations

This paper is concerned with the initial-value problem for nonlinear first-order impulsive integro-differential equations on time scales $$\mathbb{T}$$ . We establish certain existence criteria by using a fixed-point theorem for operator on cones, under which such problems have aminimal and amaximal solution lying in a corresponding region bounded by their lower and upper solutions.

Impulsive integral BVP for nonlinear integro-differential equation with monotone homomorphism in Banach spaces

In this paper, we consider a class of integral boundary value problems for nonlinear third-order impulsive integro-differential equation with a monotone homomorphism in real Banach space. By employing fixed point index theory, some sufficient criteria are obtained to ensure the existence of positive solutions. An example is given to demonstrate the application of our main results.

Multiple solutions of nonlinear fractional impulsive integro-differential equations with nonlinear boundary conditions

AbstractIn this paper, we study the existence of multiple solutions of fractional impulsive integrodifferential equations with nonlinear boundary conditions. By applying the Amann theorem and the method of upper and lower solutions, we obtain some new results on the multiple solutions.

Maximum Principles for Second-Order Impulsive Integro-Differential Equations with Integral Jump Conditions

We are concerned with two new maximum principles for second-order impulsive integro-differential equations with integral jump conditions. The impulse effects or jump conditions of this paper are involved in terms of integral of past states. As an application, we introduce a new definition of lower and upper solutions which leads to the development of the monotone iterative technique for a periodic boundary value problem related to a nonlinear second-order impulsive functional integro-differential equation with integral jump conditions.

On the asymptotic stability of impulsive neutral stochastic partial integrodifferential equations with variable delays and Poisson jumps

In this work, we study the existence and asymptotic stability in the \(p\)-th moment of mild solutions of nonlinear impulsive stochastic partial functional integrodifferential equations with delays. We suppose that the linear part possesses a resolvent operator in the sense given in Grimmer (Trans. Am. Math. Soc. 273(1), 333–349, 1982) and the nonlinear terms are assumed to be Lipschitz continuous. A fixed point approach is employed for achieving the required result. An example is provided to illustrate the results of this work.

Perturbation technique for a class of nonlinear implicit semilinear impulsive integro-differential equations of mixed type with noncompactness measure

By using the Arzela-Ascoli theorem, the Bellman inequality, and a monotone perturbation iterative technique in the presence of lower and upper solutions, we discuss the existence of mild solutions for a class of nonlinear first-order implicit semilinear impulsive integro-differential equations in Banach spaces. Under wide monotone conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique mild solution between lower and upper solutions.

STIELTJES DERIVATIVE METHOD FOR INTEGRAL INEQUALITIES WITH IMPULSES

The purpose of this paper is to obtain some integral inequalities with impulses by using the method of Stieltjes derivatives, and we use our results in the study of Lyapunov stability of solutions of a certain nonlinear impulsive integro-differential equation.

Asymptotic stability of impulsive stochastic partial integrodifferential equations with delays

In this paper, we study the existence and asymptotic stability in the p-th moment of mild solutions of nonlinear impulsive stochastic partial functional integrodifferential equations with delays. We suppose that the linear part possesses a resolvent operator in the sense given in Grimmer [R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc. 273(1) (1982), 333–349] and the nonlinear terms are assumed to be Lipschitz continuous. A fixed point approach is employed for achieving the required result. An example is provided to illustrate the results of this work.

A class of BVPS for first order impulsive functional integro-differential equations with a parameter

This paper is concerned with a class of boundary value problems for the nonlinear impulsive functional integro-differential equations with a parameter by establishing new comparison principles and using the method of upper and lower solutions together with monotone iterative technique. Sufficient conditions are established for the existence of extremal system of solutions for the given problem. Finally, we give an example that illustrates our results.

Positive solutions of Sturm-Liouville boundary value problems for singular nonlinear second-order impulsive integro-differential equation in Banach spaces

In this work, we investigate the existence of positive solutions of Sturm-Liouville boundary value problems for singular nonlinear second-order impulsive integro differential equation in a real Banach space. Some new existence results of positive solutions are established by applying fixed-point index theory together with comparison theorem. Some discussions and an example are given to demonstrate the applications of our main results.

A class of boundary value problems for first-order impulsive integro-differential equations with deviating arguments

This paper is concerned with a class of boundary value problems for nonlinear mixed impulsive integro-differential equations with deviating arguments. We establish a new comparison principle and use the method of upper and lower solutions together with the monotone iterative technique. Under suitable conditions, we obtain the existence results of extremal solutions for the problems. An example is also given to illustrate our results.

Impulsive anti-periodic boundary value problem of first-order integro-differential equations

This paper is concerned with the anti-periodic boundary value problem of first-order nonlinear impulsive integro-differential equations. We first establish a new comparison principle, and then obtain the existence of extremal solutions by upper–lower solution and monotone iterative techniques. Some examples are presented to illustrate the main results.

Second-Order Nonlinear Impulsive Integro-Differential Equations of Mixed Type and Optimal Controls in Fractional Power Spaces

A class of second-order nonlinear impulsive integro-differential equations of mixed type whose principal part is given by time-varying generating operators in fractional power spaces is considered. We introduce the reasonable PC-α-mild solution of second-order nonlinear impulsive integro-differential equations of mixed type and prove its existence. The existence of optimal controls for a Lagrange problem of systems governed by second-order nonlinear impulsive integro-equations of mixed type is also presented. An example is given for demonstration.

Necessary conditions of optimality for second-order nonlinear impulsive integro-differential equations on Banach spaces

We discuss a class of optimal control problems of systems governed by the second-order nonlinear impulsive integro-differential equations with time-varying generating operators in fractional power spaces. Introducing the reasonable P C – α -mild solution of the second-order nonlinear impulsive integro-differential equations we prove the existence of P C – α -mild solutions. Existence of optimal controls for a Lagrange problem of systems governed by the second-order nonlinear impulsive integro-equations is presented. We apply a direct approach to derive the maximum principle for the problem at hand.