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Positive periodic solutions for a class of second-order differential equations with state-dependent delays. (English) Zbl 1459.34157

The following class of second-order differential equations with iterative state-dependent delays is considered: \[ x''(t)+p(t)x'(t)+q(t)x(t) = \frac{d}{dt}\left[f(t, x(t), x^{[2]}(t), \ldots, x^{[n]}(t))\right] +\sum^n_{j=1}c_j(t)x^{[j]}(t), \] where \(p\) and \(q\) are positive continuous real-valued functions, \(f: \mathbb{R}^{n+1} \to \mathbb{R}\) is continuous, \(x^{[0]}(t)=t, x^{[1]}(t) = x(t), x^{[2]}(t)=x(x(t)), \ldots, x^{[n]}(t)=x(x^{[n-1]}(t))\). By virtue of a Krasnoselskii fixed point theorem and some useful properties of a Green’s function, sufficient conditions are obtained for existence of positive periodic solutions. This is inspired by similar results available in literature.

MSC:

34K13 Periodic solutions to functional-differential equations
34K43 Functional-differential equations with state-dependent arguments
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