Consider the first-order linear differential equation with several non-monotone retarded arguments
{x^{\prime}(t)+\sum_{i=1}^{m}p_{i}(t)x(\tau_{i}(t))=0}
,
{t\geq t_{0}}
, where the functions
{p_{i},\tau_{i}\in C([t_{0},\infty),\mathbb{R}^{+})}
, for every
{i=1,2,\ldots,m}
,
{\tau_{i}(t)\leq t}
for
{t\geq t_{0}}
and
{\lim_{t\to\infty}\tau_{i}(t)=\infty}
.
New oscillation criteria which essentially improve the known results in the literature are established.
An example illustrating the results is given.

We show the existence of the unique solution of impulsive differential equation x 0 .t / D a .t / .x .t / x .bt 1c// C f .t / ; t ¤ n 2 Z C D f1; 2; : : :g ; t 0; x .t / D c t x .t / C d t ; t D n 2 Z C ; with the initial conditions x. 1/ D x 1 ; x .0/ D x 0 ; where b:c denotes the floor integer function. Moreover, we obtain sufficient conditions for the asymptotic constancy of this equation and we compute, as t ! 1, the limits of the solutions of the impulsive equation with c n D 0 in terms of the initial conditions, a special solution of the corresponding adjoint equation and a solution of the corresponding difference equation.

Abstract. This paper concerns with the existence of the solutions of a second order impulsive delay differential equation with a piecewise constant argument. Moreover, oscillation, nonoscillation and periodicity of the solutions are investigated.

We prove the existence and uniqueness of the solutions of a second order mixed type impulsive differential equation with piecewise constant arguments. Moreover, we study oscillation, non-oscillation and periodicity of the solutions.

In this paper, two classes of techniques, based on multiple scales perturbation analysis and the averaged energy (or Lyapunov function) method, are employed to investigate interesting nonlinear dynamical regimes in a system of coupled Rayleigh-Van der Pol oscillators with time-delayed displacement and velocity feedback. Such systems are currently of topical interest in many applications. The multiple scales

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