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Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses. (English) Zbl 0877.34054
The delay differential equation with nonlinear impulses $$\dot x(t)+ \sum^n_{i=1} p_i(t)x(t- \tau_i)=0,\quad t\ne t_j,\quad t\ge t_0,$$ $$x(t^+_j)- x(t_j)= I_j(x(t_j)),\quad j=1,2,\dots$$ is considered, where $p_i\in C([t_0,\infty),\bbfR^+)$, $\tau_i\ge 0$, $i=1,2,\dots, n$, $I_j\in C(\bbfR,\bbfR)$, $j=1,2,\dots$. 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)

34K25Asymptotic theory of functional-differential equations
34A37Differential equations with impulses
34K99Functional-differential equations
Full Text: DOI
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