UDC 517.95A survey of works of the authors and their disciples devoted to the investigation of problems with nonlocal conditions with respect to a selected variable in cylindrical domains is presented. These problems are considered for linear equations and systems of partial differential equations that, in general, are ill posed in the Hadamard sense and whose solvability in certain scales of functional spaces is established for almost all (with respect to Lebesgue measure) vectors composed of the coefficients of the problem and the parameters of the domain.

Асимптотичний метод дослідження резонАнсних режимів неліній них згинАльних коливАнь пружного вАлА Purpose. To develop a method for determining resonant modes of industrial equipment of elastic shaft type, which is widely used in the mining industry, through the study of mathematical model of nonlinear oscillations. Mathematical models of oscillatory systems previously were studied in the literature mainly based on the numerical and experimental approaches. This paper proposes using a combination of the wave theory of motion and asymptotic methods of nonlinear mechanics using special apparatus of periodic functions to investigate the vibrational dynamics of the system and conditions of resonance phenomena in it, as well as to describe the method for determining the resonance curves to increase the margin of safety of industrial equipment. Methodology. Methods for studying resonance amplitudes and frequencies, determining the strength characteristics of equipment are based on the use of asymptotic methods of nonlinear mechanics, wave motion theory and theory of special Ateb-functions. Findings. In this work the conditions of resonance amplitude and frequency depending on the system parameters were obtained analytically for these nonlinear vibrational systems of elastic shaft and the overall method for determining the resonance curves was described. Originality. For the first time a complete analysis of the impact of physical, mechanical and geometrical factors of the dynamic pricess on the resonant frequency and amplitude in systems such as elastic shaft was conducted on the UDC 534.1

Abstract. We establish the necessary and sufficient conditions of existence of nontrivial quasi-polynomial solutions of the problem in a layer for homogeneous partial differential equation with s + 1 variables of second order in time variable and generally infinite order in other s (spatial) variables with Dirichlet boundary conditions in time. We apply the differential-symbol method for constructing such quasi-polynomial solutions. We also give examples of problems for which we construct other solutions besides of quasi-polynomial ones.

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