Conditions of solvability of the nonlocal boundary-value problem for a differential-operator equation with weak nonlinearity

We study the solvability of a nonlocal boundary-value problem for a differential equation with nonlinearity. The linear part of the equation has complex coefficients which together with the coefficient in nonlocal conditions are considered to be the parameters of the problem. The area of change of each parameter is limited by a complex circle with its center at the origin. The nonlinear part of the equation is given by a smooth function that satisfies, together with its derivatives, some conditions of growth in the Dirichlet--Fourier space scale. The proofs are based on the differentiable Nash--Moser iteration scheme, where the main difficulty is to get estimates of the interpolation type for the inverse linearized operators obtained at each step of the iteration. The estimation is connected with the problem of small denominators which is solved, by using a metric approach on the set of parameters of the problem.

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

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.

Counting algebraic numbers in short intervals with rational points

In 2012 it was proved that real algebraic numbers follow a non­uniform but regular distribution, where the respective definitions go back to H. Weyl (1916) and A. Baker and W. Schmidt (1970). The largest deviations from the uniform distribution occur in neighborhoods of rational numbers with small denominators. In this article the authors are first to specify a gene ral condition that guarantees the presence of a large quantity of real algebraic numbers in a small interval. Under this condition, the distribution of real algebraic numbers attains even stronger regularity properties, indicating that there is a chance of proving Wirsing’s conjecture on approximation of real numbers by algebraic numbers and algebraic integers.

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

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.

On nonlocal boundary value problem for the equation of motion of a homogeneous elastic beam with pinned-pinned ends

In the current paper, in the domain $D=\{(t,x): t\in(0,T), x\in(0,L)\}$ we investigate the boundary value problem for the equation of motion of a homogeneous elastic beam $$ u_{tt}(t,x)+a^{2}u_{xxxx}(t,x)+b u_{xx}(t,x)+c u(t,x)=0, $$ where $a,b,c \in \mathbb{R}$, $b^2<4a^2c$, with nonlocal two-point conditions $$u(0,x)-u(T, x)=\varphi(x), \quad u_{t}(0, x)-u_{t}(T, x)=\psi(x)$$ and local boundary conditions $$u(t, 0)=u(t, L)=u_{xx}(t, 0)=u_{xx}(t, L)=0.$$ Solvability of this problem is connected with the problem of small denominators, whose estimation from below is based on the application of the metric approach. For almost all (with respect to Lebesgue measure) parameters of the problem, we establish conditions for the solvability of the problem in the Sobolev space. In particular, if $\varphi\in\mathbf{H}_{q+\rho+2}$ and $\psi \in\mathbf{H}_{q+\rho}$, where $\rho>2$, then for almost all (with respect to Lebesgue measure in $\mathbb{R}$) numbers $a$ exists a unique solution $u\in\mathbf{C}^{\,2}([0,T];\mathbf{H}_{q})$ of the problem considered.

On the steady-state resonant acoustic–gravity waves

The steady-state interaction of acoustic–gravity waves in an ocean of uniform depth is researched theoretically by means of the homotopy analysis method (HAM), an analytic approximation method for nonlinear problems. Considering compressibility, a hydroacoustic wave can be produced by the interaction of two progressive gravity waves with the same wavelength travelling in opposite directions, which contains an infinite number of small denominators in the framework of the classical analytic approximation methods, like perturbation methods. Using the HAM, the infinite number of small denominators are avoided once and for all by means of choosing a proper auxiliary linear operator. Besides, by choosing a proper ‘convergence-control parameter’, convergent series solutions of the steady-state acoustic–gravity waves are obtained in cases of both non-resonance and exact resonance. It is found, for the first time, that the steady-state resonant acoustic–gravity waves widely exist. In addition, the two primary wave components and the resonant hydroacoustic wave component might occupy most of wave energy. It is found that the dynamic pressure on the sea bottom caused by the resonant hydroacoustic wave component is much larger than that in the case of non-resonance, which might even trigger microseisms of the ocean floor. All of these might deepen our understanding and enrich our knowledge of acoustic–gravity waves.

DIRICHLET PROBLEM FOR PULKIN’S EQUATION IN A RECTANGULAR DOMAIN

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.