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Positive periodic solutions of nonautonomous functional differential equations depending on a parameter. (English) Zbl 1007.34066

Here, the authors study the existence of positive periodic solutions for a first-order functional-differential equations of the form \[ y'(t)=-a(t)y(t)+\lambda h(t)f(y(t-\tau(t))),\tag{1} \] where \(a=a(t), h=h(t)\), and \(\tau=\tau(t)\) are continuous \(T\)-periodic functions, \(T>0, \lambda>0\) are constants, \(f=f(t)\) and \(h=h(t)\) are positive and \(\int_{0}^{T}a(t) dt>0\). They give additional conditions on the function \(f\) to show that there exists \(\lambda^*>0\) such that (1) has at least one positive \(T\)-periodic solution for \(\lambda\in(0,\lambda^*]\) and does not have any \(T\)-periodic positive solutions for \(\lambda>\lambda^*\). Their technique is based on the well-known upper and lower solutions method.

MSC:

34K13 Periodic solutions to functional-differential equations
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