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We discuss two techniques useful in the investigation of periodic solutions of broad classes of non-linear non-autonomous ordinary differential equations, namely the trigonometric collocation and the method based upon periodic successive approximations.

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Numerical-analytical methods based on variation approach to the tasks of mechanics of deformable solid, liquid, gas, hydro- and thermodynamics, multilinked contact and on analytical decisions of loosely-coupled tasks are presented in this paper. The suggested numerical-analytical method allows investigating on high technological level the high-strength joints with guaranteed interference (pressed, hydropressed, thermal, multilayered, autofrettaged, multi-contact (polyjoints)) at various stages of life cycle, except for recycling. Method allow: to determine mode of deformation, joint loading capacity in view of technology factor influence; to calculate parameters of technological processes with the diverse contact effects proceeding in joints, assembled by various methods.

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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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