Arithmetic spectral transitions: A competition between hyperbolicity and the arithmetics of small denominators

In the given article for the mixed-type equation with a singular coeﬃcient the ﬁrst boundary value problem is studied. On the basis of property of completeness of the system of own functions of one-dimensional spectral problem the criterion of uniqueness is established. The solution the problem is constructed as the sum of series of Fourier - Bessel. At justiﬁcation of convergence of a row there is a problem of small denominators. In connection with that the assessment about apartness of small denominator from zero with the corresponding asymptotic which allows to prove the convergence of the series constructed in a class of regular solutions under some restrictions is given.

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METRICAL ESTIMATIONS OF DETERMINANT OF THE INTEGRAL PROBLEM FOR EQUATION OF VIBRATING OF STRING

PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY Number ◽

10.31471/2304-7399-2018-2(46)-38-46 ◽

2018 ◽

pp. 38-46

Author(s):

О. M. Medvid ◽

M. M. Symotyuk ◽

I. R. Tymkiv

Keyword(s):

Spatial Variable ◽

Integral Conditions ◽

Small Denominators ◽

Fractal Measure ◽

Integral Problem

Metrical theorems are inprocess set about estimations from below small denominators that arose up at investigational existence of periodic at times decision of task with integral conditions as moments after a spatial variable for equalization of small vibrating of string. For leading to of metrical estimations the concept of fractal measure and dimension of Hausdorff is from below applied.

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A nonlocal boundary-value problem for a fourth-order mixed-type equation

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-1-2 ◽

2020 ◽

Vol 17 (1) ◽

pp. 30-40

Author(s):

Kudratillo Fayazov ◽

Ikrombek Khajiev

Keyword(s):

Fourth Order ◽

Mixed Type ◽

Nonlocal Boundary ◽

Type Equation ◽

Small Denominators ◽

Mixed Type Equation ◽

The Stability ◽

Criterion Of Uniqueness ◽

Uniqueness Of A Solution ◽

Stability Of The Solution

The criterion of uniqueness of a solution of the problem with periodicity and nonlocal and boundary conditions is established by the spectral analysis for a fourth-order mixed-type equation in a rectangular region. When constructing a solution in the form of the sum of a series, we use the completeness in the space L_2, the system of eigenfunctions of the corresponding problem orthogonally conjugate. When proving the convergence of a series, the problem of small denominators arises. Under some conditions imposed on the parameters of the data of the problem and given functions, the stability of the solution is proved.

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