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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.

##### MSC:
 34K13 Periodic solutions to functional-differential equations
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##### References:
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