Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space

Consider a linear impulsive equation in a Banach space $$\dot{x}(t)+A(t)x(t) = f(t), ~t \geq 0,$$ $$x(\tau_i +0)= B_i x(\tau_i -0) + \alpha_i,$$ with $\lim_{i \rightarrow \infty} \tau_i = \infty$. Suppose each solution of the corresponding semi-homogeneous equation $$\dot{x}(t)+A(t)x(t) = 0,$$ (2) is bounded for any bounded sequence $\{ \alpha_i \}$. The conditions are determined ensuring (a) the solution of the corresponding homogeneous equation has an exponential estimate; (b) each solution of (1),(2) is bounded on the half-line for any bounded $f$ and bounded sequence $\{ \alpha_i \}$ ; (c) $\lim_{t \rightarrow \infty}x(t)=0$ for any $f, \alpha_i$ tending to zero; (d) exponential estimate of $f$ implies a similar estimate for $x$.Comment: 19 pages, LaTex-fil

On oscillation of a Volterra integral equation with delay

AbstractThe purpose of this paper is to obtain sufficient conditions for oscillation of all solutions of the equation x(t) = f(t) + ∝at K(t, s, x(s), x(g(s))) ds to study the behaviour of its oscillatory solutions in a dependence on the distance between their consecutive zeros and to establish a theorem for localization of the zeros of its solutions

Asymptotic stability of the solutions of a linear singularly perturbed system with unbounded delay

AbstractSufficient conditions for asymptotic stability of the solutions of a linear singularly perturbed system of differential equations with unbounded delay have been found. Under the same conditions it is proved that for a locally Lipschitz initial function the initial value problem for the system degenerates regularly

Second method of Lyapunov and comparison principle for systems with impulse effect

AbstractIn the present paper questions of stability and boundedness of the solutions of systems with impulse effect at fixed moments with respect to a manifold are considered. The investigations are carried out by means of piecewise continuous vector-valued functions which are analogues of Lyapunov's functions. By means of a vector comparison equation and differential inequalities for piecewise continuous functions, theorems of stability and boundedness of the solutions of systems with impulses with respect to a manifold have been obtained

Estimates of solutions of impulsive parabolic equations and applications to the population dynamics

A theorem on estimates of solutions of impulsive parabolic equations by means of solutions of impulsive ordinary differential equations is proved. An application to the population dynamics is given

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

AbstractIn 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 Lyapunov's functions are used

Exponential stability of the solutions of the initial-value problem for systems with impulse effect

AbstractIn the present paper conditions have been found under which the exponential stability of a given solution of a system with impulse effect follows from the exponential stability of the respective system in variations

Oscillation of Neutral Différential Equations with "Maxima"

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