Periodic solutions of the third order functional differential equations. (English) Zbl 1057.34098

The authors deal with the existence of \(2\pi\)-periodic solutions of third-order differential equations of the type \[ x'''(t)+ a(x'')^{2k-1}(t)+ b(x')^{2k-1}(t)+ cx^{2k-1}(t)+ g(t,x(t-\tau_1), x'(t-\tau_2))= p(t), \tag{1} \] where \(p(t+2\pi)= p(t)\), \(a\), \(b\), \(c\), \(\tau_1\) and \(\tau_2\) are real numbers, \(k\) is a positive integer, \(g: \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) is continuous and \(2\pi\)-periodic with respect to the first variable \(t\). The authors derive sufficient conditions for the existence of periodic solutions of (1). To this end, the authors use the continuation theorem from the theory of the coincidence degree, and a priori estimates.


34K13 Periodic solutions to functional-differential equations
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