# zbMATH — the first resource for mathematics

Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses. (English) Zbl 0877.34054
The delay differential equation with nonlinear impulses $\dot x(t)+ \sum^n_{i=1} p_i(t)x(t- \tau_i)=0,\quad t\neq t_j,\quad t\geq t_0,$ $x(t^+_j)- x(t_j)= I_j(x(t_j)),\quad j=1,2,\dots$ is considered, where $$p_i\in C([t_0,\infty),\mathbb{R}^+)$$, $$\tau_i\geq 0$$, $$i=1,2,\dots, n$$, $$I_j\in C(\mathbb{R},\mathbb{R})$$, $$j=1,2,\dots$$. A criterion for the existence of positive solutions for an equation without impulses is given.
A comparison of asymptotic behavior of its solutions with solutions of equations with impulses is established. Conditions, under which every nonoscillatory solution of the considered problem tends to zero as $$t\to\infty$$, are indicated.
Reviewer: J.Diblík (Brno)

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 34A37 Ordinary differential equations with impulses 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
Full Text:
##### References:
 [1] Arino, O.; Ladas, G.; Sficas, Y.G., On oscillations of some retarded differential equations, SIAM J. math. anal., 18, 64-73, (1987) · Zbl 0566.34053 [2] Chen, M.P.; Yu, J.S.; Shen, J.H., The persistence of nonoscillatory solutions of delay differential equations under impulsive perturbations, Computer math. anal., 27, 1-6, (1994) · Zbl 0806.34060 [3] Gopalsamy, K.; Zhang, B.G., On delay differential equations with impulses, J. math. anal. appl., 139, 110-122, (1989) · Zbl 0687.34065 [4] Györi, I.; Ladas, G., Oscillation theory of delay differential equations with applications, (1991), Clarendon Press Oxford · Zbl 0780.34048 [5] Kulenovic, M.R.S.; Ladas, G.; Meimaridou, A., Stability of solutions of linear delay differential equations, Proc. amer. math. soc., 100, 433-441, (1987) · Zbl 0645.34061 [6] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002 [7] Shen, J.H.; Wang, Z.C., Oscillation and asymptotic behavior of solutions of delay differential equations with impulses, Ann. differential equations, 10, 61-68, (1994) · Zbl 0801.34072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.