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Periodic solutions in general scalar non-autonomous models with delays. (English) Zbl 1292.47062

In this paper, existence of periodic solutions is considered for a broad class of scalar differential equations with delays, serving as population models. The authors first establish an existence theorem for the abstract functional differential equation \(x'(t) = \Phi(x)(t)\), where \(\Phi\) is a continuous mapping from the space of continuous \(T\)-periodic functions to itself. Then this theorem is applied to various models, so that concrete sufficient conditions can be easily derived for the existence of periodic solutions. Examples of models include \(x'(t) = -a(t)+h(t, x(t), x(t-\tau(t, x(t))))\), \(x'(t) = -a(t)x(t)+g(t, x(t), x(t-\tau(t, x(t))))\) and \(x'(t) = -f(t, x(t))+g(t, x(t), x(t-\tau(t, x(t))))\). It is also demonstrated that the technique can be used to expand on well-known results as well as to shorten existing proofs in the literature.

MSC:

47N20 Applications of operator theory to differential and integral equations
92D25 Population dynamics (general)
34K13 Periodic solutions to functional-differential equations
34K05 General theory of functional-differential equations
46B25 Classical Banach spaces in the general theory
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