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Positive periodic solutions of impulsive delay differential equations with sign-changing coefficient. (English) Zbl 1064.34050

The authors consider a nonautonomous delay differential equation with impulses \[ \begin{cases} x^\prime(t)=-a(t)x(t)+\lambda h(t)f(x(t-\tau(t))), & t\neq t_k, \;\;k\in\mathbb Z, \\ x(t_k)=(1+b_k)x(t_k^-), & k\in\mathbb Z, \end{cases} \] where \(a\) and \(h\) change sign, and prove results on the existence and nonexistence of positive periodic solutions. By means of an auxiliary exponential function, the equation is transformed into an integral equation which encompasses the impulses. Then the authors prove the existence of at least one positive periodic solution for a positive \(\lambda\) by the Leray-Schauder degree theory and show the nonexistence of such a solution for all \(\lambda\) larger than a convenient number.

MSC:

34K13 Periodic solutions to functional-differential equations
34K45 Functional-differential equations with impulses
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