Advanced Impulsive Differential Equations With Piecewise Constant Arguments

Abstract

Abstract. We prove the existence and uniqueness of the solutions of a class of first order nonhomogeneous advanced impulsive differential equations with piecewise constant arguments. We also study the conditions of periodicity, oscillation, nonoscillation and global asymptotic stability for some special cases.

Oscillation of a nonlinear impulsive differential equation system with piecewise constant argument

“…On the other hand, in 1994, the case of studying discontinuous solutions of differential equations with piecewise continuous arguments has been proposed as an open problem by Wiener [10]. Due to this open problem, linear impulsive differential equations with piecewise constant arguments have been dealt with in [11][12][13]. Moreover, cellular neural networks with piecewise constant argument have been investigated in [14][15][16].…”

Oscillation and Periodicity of a Second Order Impulsive Delay Differential Equation with a Piecewise Constant Argument

“…and this equality yields us to the characteristic equation (4). On the other hand, if λ 1 , λ 2 and λ 3 are different roots of Eq.…”

“…There are also some papers of the authors: In [9], the oscillatory and periodic solutions of the first order linear scalar impulsive delay differential equation and in [4], existence and uniqueness and also oscillatory and periodic solutions of a class of first order nonhomogeneous advanced impulsive differential equations with piecewise constant arguments were studied. The asymptotic convergence of first order delay and advanced impulsive differential equations with piecewise constant arguments were also considered in [2], [3] and [14].…”

“…The asymptotic convergence of first order delay and advanced impulsive differential equations with piecewise constant arguments were also considered in [2], [3] and [14].…”

Convergence of solutions of an impulsive differential system with a piecewise constant argument

“…In [5] Bereketoglu and Karakoc obtained su¢ cient conditions for the asymptotic constancy of the solutions of a system of nonhomogeneous linear impulsive pantograph equations. In [16], [7], [8] and [6] authors considered the asymptotic constancy of di¤erent types of impulsive di¤erential equations with piecewise constant arguments and formulated the limit value of the solutions in terms of the initial condition and the solution of the integral equation for each type of equations.…”