Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses.

*(English)*Zbl 0877.34054The 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.

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) |

##### Keywords:

delay differential equation; nonlinear impulses; positive solutions; asymptotic behavior; nonoscillatory solution
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\textit{J. Yu} and \textit{J. Yan}, J. Math. Anal. Appl. 207, No. 2, 388--396 (1997; Zbl 0877.34054)

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