Dynamics and stability of impulsive hybrid setvalued integro-differential equations with delay

We study the dynamics and stability theory for impulsive hybrid set integro-differential equations with delay. Sufficient conditions for the stability of the null solution of impulsive hybrid set integro-differential equations with delay are presented.     

Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions

The existence, uniqueness and continuous dependence of a mild solution of an impulsive neutral functional differential evolution nonlocal Cauchy problem in general Banach spaces are studied, by using the fixed point technique and semigroup of operators.     

Almost periodic solutions for abstract impulsive differential equations

This paper studies the existence and uniqueness of exponentially stable almost periodic solutions for abstract impulsive differential equations in Banach space. The investigations are carried out by means of the fractional powers of operators. We construct an example to illustrate the feasibility of our results.     

Attractors for differential equations with unbounded delays

We prove the existence of attractors for some types of differential problems containing infinite delays. Applications and examples are provided to illustrate the theory, which is valid for both cases with and without explicit dependence of time, and with or without uniqueness of solutions, as well.     

Impulsive stability for systems of second order retarded differential equations

This paper is concerned with systems of impulsive second order delay differential equations. We prove that unstable systems can be stabilized by imposition of impulsive controls. The main tools used are Lyapunov functionals, stability theory and control by impulses.     

Lyapunov theorems for measure functional differential equations via Kurzweil‐equations

We consider measure functional differential equations (we write measure FDEs) of the form , where f is Perron–Stieltjes integrable, is given by , with , and and are the distributional derivatives in the sense of the distribution of L. Schwartz, with respect to functions and , , and we present new concepts of stability of the trivial solution, when it exists, of this equation. The new stability concepts generalize, for instance, the variational stability introduced by ?. Schwabik and M. Federson for FDEs and yet we are able to establish a Lyapunov‐type theorem for measure FDEs via theory of generalized ordinary differential equations (also known as Kurzweil equations).     

Autonomous and non-autonomous attractors for differential equations with delays

The asymptotic behaviour of some types of retarded differential equations, with both variable and distributed delays, is analysed. In fact, the existence of global attractors is established for different situations: with and without uniqueness, and for both autonomous and non-autonomous cases, using the classical notion of attractor and the recently new concept of pullback one, respectively.     

Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay

In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472].     

Monotone method for first order functional differential equations with retardation and anticipation

This paper discusses first order functional differential equations with retardation and anticipation. By using the method of upper and lower solutions coupled with monotone iterative technique, new existence results are obtained which improve known results in the literature.     

Functional differential inclusions generated by functional differential equations with discontinuities

Given a functional differential equation with a discontinuity, a construction of its extension in the shape of a functional differential inclusion is offered. This construction can be regarded as a generalization of the famous Filippov approach to study ordinary differential equations with discontinuities. Some basic properties of the solutions of the introduced functional differential inclusions are studied. The developed approach is applied to analysis of gene regulatory networks with general delays.     

Boundary value problems for first order differential equations of mixed type

We apply the monotone iterative method to formulate existence results for integro-differential equations with nonlinear boundary conditions. Some results for integro-differential inequalities are also given. Examples illustrate obtained results.     

Qualitative analysis and applications of a kind of state-dependent impulsive differential equations

This paper analyzes a certain type of impulsive differential equations (IDEs). Several useful theorems for its periodic solutions and their stabilities are given. The key idea is that a periodically time-dependent IDE can be transformed into the state-dependent IDE. As applications of our theory, the optimization problems in population dynamics are studied. That is, the maximum sustainable yields of single population models with periodically impulsive constant harvesting are discussed. Furthermore, we apply these results to the studies of the order-1 periodic solutions and their stability of a single population model with stage structure in which the mature is impulsively proportionally harvested while the immature is impulsively added with the constant.     

Positive solutions of p-type retarded functional differential equations

For systems of retarded functional differential equations with unbounded delay and with finite memory sufficient and necessary conditions of existence of positive solutions on an interval of the form [_t0,∞)$[t_0,\infty )$ are derived. A general criterion is given together with corresponding applications (including a linear case, too). Examples are inserted to illustrate the results.     

The monotone iterative technique for impulsive hybrid set valued integro-differential equations

We develop the monotone method for impulsive hybrid set integro-differential equations in all its generality. Some interesting observations are presented.     

Existence theorems for nonlinear functional differential equations of neutral type

Conditions are found upon satisfaction of which the differential equation

Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions

In this paper, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some nondensely defined impulsive semilinear functional differential inclusions in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these equations and the topological structure of the solutions set are considered too.     

A new approach to practical stability of impulsive functional differential equations in terms of two measures

In this work, we consider a new approach to the practical stability theory of impulsive functional differential equations. With Lyapunov functionals and Razumikhin technique, we use a new technique in the division of Lyapunov functions, given by Shunian Zhang, and obtain conditions sufficient for the uniform practical (asymptotical) stability of impulsive delay differential equations. An example is also discussed to illustrate the advantage of the proposed results.     

Boundedness and periodicity in impulsive ordinary and functional differential equations

In this paper, we study the boundedness and periodicity of solutions for impulsive ordinary differential equations (ODEs) and functional differential equations (FDEs). By using Horn’s fixed point theorem, Hale–Yoshizawa type criteria for the existence of T$T$-periodic solutions are established. To use these criteria, we also give two new boundedness theorems, and establish new existence results for T$T$-periodic solutions which show that the impulsive perturbations do contribute to yielding periodic solutions even when the underlying systems do not enjoy periodic solutions.