D. D. Bainov

On the asymptotic stability of systems with impulses by the direct method of Lyapunov

Abstract In the present paper the asymptotic and globally asymptotic stability of the zero solution of systems with impulses is investigated. For this purpose piecewise continuous auxiliary functions are used which are an analogue of Lyapunovs functions. The theorem of Marachkov on the asymptotic stability of systems without impulses is generalized. The results obtained are formulated in six theorems.

Continuous dependence of the solution of a system of differential equations with impulses on the impulse hypersurfaces

Abstract In the present paper the initial value problem is considered for systems of ordinary differential equations with impulses where the impulses are realized in the moments when the integral curve of the system meets some of previously given hypersurfaces called impulse hypersurfaces. Sufficient conditions for the continuous dependence of the solution on the impulse hypersurfaces are found.

Characteristic exponents of impulsive differential equations in a Banach space

The notion of general exponent of impulsive homogeneous differential equations is defined. A formula for the solution of impulsive nonhomogeneous differential equations is obtained and is used to establish a dependence between the existence of bounded solutions of such equations and the general exponent of the respective homogeneous equation.

Exponential dichotomy of linear impulsive differential equations in a Banach space

The dichotomy of linear impulsive equations in a Banach space is investigated.

Exponential stability of the solutions of singularly perturbed systems with impulse effect

Abstract In the present paper the exponential stability of the solutions of singularly perturbed systems with impulse effect is investigated. In order to obtain the main results the comparison method and piecewise continuous auxiliary functions which are analogues of Lyapunovs functions are used.

Stability of the General Exponent of Nonlinear Impulsive Differential Equations in a Banach Space

A theorem is proved which gurantees stability under small perturbations of the general exponent of impulsive nonlinear systems in a Banach space.

Lp-Equivalence of impulsive equations

By means of a modification of Schauders theorem, sufficient conditions for theLp-equivalence of impulsive nonlinear differential equations are found.

On the asymptotic equivalence of systems with impulse effect

Abstract In the present paper sufficient conditions for asymptotic equivalence and restricted asymptotic equivalence of a linear system of differential equations with impulse effect and its respective perturbed non-linear system are found.

Topological Equivalence and Exponential Dichotomy of Linear Impulsive Equations

The notions of kinematical similarity and topological equivalence for abstract impulsive differential equations are introduced and some of their properties related to the notion of exponential dichotomy are investigated.

Continuous dependence on a parameter of the solutions of impulsive differential equations in a Banach space

We prove that the solutions of an impulsive differential equation depend continuously on a small parameter under the assumption that the right-hand side of the equation and the impulse operators satisfy conditions of Lipschitz type.