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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