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Abstract  

The aim of this paper is to investigate sufficient conditions (Theorem 1) for the nonexistence of nontrivial periodic solutions of equation (1.1) withp ≡ 0 and (Theorem 2) for the existence of periodic solutions of equation (1.1).

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In [1] (p. 215), the authors Andronov, Leontovich-Andronova, Gordon, and Maier, consider the following equation:
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ \begin{gathered} \tfrac{{dx}}{{dt}} = y, \hfill \\ \tfrac{{dy}}{{dt}} = x + x^2 - \left( {\varepsilon _1 + \varepsilon _2 x} \right)y, \hfill \\ \end{gathered} \right.$$ \end{document}
wheree 1 ande 2 are real constants ande 1 ande 2 are not both zero. They proved that there are no non-trivial periodic solutions except possibly for the case . They left that case as an open problem. In this note we prove that there are indeed no non-trivial periodic solutions in the case either. Our method of proof consists essentially of constructing a Dulac function (see [6] and [9]) and using the conception of Duff's rotated vector field (see [4], [7], [8], [10], and [11]).
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The paper applies a numerical-analytical method for finding periodic solutions of the system of integro-differential equations
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\begin{gathered} \dot x = f(t,x,\mathop \smallint \limits_0^t \varphi (t,s,x(s))ds), t \ne t_i (x), \hfill \\ \Delta x|_{t = t_i (x)} = I_i (x). \hfill \\ \end{gathered}$$ \end{document}
Two theorems for existence of periodic solutions are proved for the cases whent = t i andt = t i(x).
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