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.
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 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.
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.
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