Order Linear Impulsive Differential Equations Research Articles

On the oscillation of second order linear impulsive differential equations

(Communicated by J. Yan) Abstract. For the second order linear impulsive differential equation with oscillatory coefficient ⎧ ⎩ (r(t)x � (t)) � +h(t)x(t )= 0, ttk, tk t0, k = 1,2,···, x(t + k )= akx(tk), x � (t + k )= bkx � (tk), k = 1,2,···, x(t + 0 )= x0, x � (t + 0 )= x� , (E) where h can be changed sign on (t0,∞), by using the equivalence transformation, we establish an associated impulsive differential equation with damping and give oscillation criteria for the equation. As applications, we obtain oscillation theorems for Eq.(E). Moveover, an example is also given to illustrate the relevance of the results.

Picone type formula for non-selfadjoint impulsive differential equations with discontinuous solutions

As the impulsive differential equations are useful in modelling many real processes observed in physics, chemistry, biology, engineering, etc., see [1, 11, 13, 20, 21, 22, 25, 26, 27], there has been an increasing interest in studying such equations from the point of view of stability, asymptotic behavior, existence of periodic solutions, and oscillation of solutions. The classical theory can be found in the monographs [9, 18]. Recently, the oscillation theory of impulsive differential equations has also received considerable attention, see [2, 14] for the Sturmian theory of impulsive differential equations, and [15] for a Picone type formula and its applications. Due to difficulties caused by the impulsive perturbations the solutions are usually assumed to be continuous in most works in the literature. In this paper, we consider second order non-selfadjoint linear impulsive differential equations with discontinuous solutions. Our aim is to derive a Picone type identity for such impulsive differential equations, and hence extend and generalize several results in the literature. We consider second order linear impulsive differential equations of the form

Stability criterion for second order linear impulsive differential equations with periodic coefficients

AbstractIn this paper we obtain instability and stability criteria for second order linear impulsive differential equations with periodic coefficients. Further, a Lyapunov type inequality is also established. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Boundary value problems for higher order linear impulsive differential equations

In this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied. Green's formula is defined for piecewise differentiable functions. Properties of Green's functions for higher order impulsive boundary value problems are introduced. An appropriate example of the Green's function for a boundary value problem is provided. Furthermore, eigenvalue problems and basic properties of eigensolutions are considered.

On the oscillation properties of first-order impulsive differential equations with a deviating argument

For first order linear impulsive differential equations with a deviating argument a number of oscillation theorems are proved (of the type of the Sturmian Theorem). Various criteria for oscillation or non-oscillation of the solutions of these equations are found. The oscillatory properties of some concrete equations of the type considered are investigated.

Oscillation and Nonoscillation for Second Order Linear Impulsive Differential Equations

We study the oscillation and nonoscillation for the second order linear impulsive differential equationu″=−p(t)u, wherep(t) is an impulsive function defined byp(t)=∑n=1∞anδ(t−tn), and we establish a necessary and sufficient condition for oscillation (or nonoscillation) of the equationu″=−p(t)u. We give some sufficient conditions for the oscillation and nonoscillation in two particular cases.