# Volume 48, № 1, 1996

### Оценка рассчетных воздействий в параболических системах. $L_2$-подход

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 3-12

By using observations of solutions of the first initial boundary-value problem for a parabolic quasilinear equation with fast random oscillations, we estimate the nonlinear term of the equation. In the metric of the space $L_2$, we study large deviations of a nonparametric estimate of nonlinear influence.

### On the solvability of the dirichlet problem for elliptic nondivergent equations of the second order in a domain with conical point

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 13-24

We study the problem of solvability of the Dirichlet problem for second-order linear and quasilinear uniformly elliptic equations in a bounded domain whose boundary contains a conical point. We prove new theorems on the unique solvability of a linear problem under minimal smoothness conditions for the coefficients, right-hand sides, and the boundary of the domain. We find classes of solvability of the problem for quasilinear equations under natural conditions.

### General páley problem

Kondratyuk A. A., Kondratyuk Ya. V., Tarasyuk S. I.

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 25-34

In the class of functions *u* of finite lower order subharmonic in ℝ^{ p+2},*p* ∈ ℕ we establish an exact upper bound for $$\mathop {\lim }\limits_{r \to \infty } \inf \frac{{m_q (r,u^ + )}}{{T(r,u)}}, 1< q \le \infty ,$$ where*T(r, u)* is a Nevanlinna characteristic of the function *u* and*m* _{ q }(*r, u* ^{+}) is the integral*q*-mean of the function*u* ^{+},*u* ^{+} = max(*u*,0), on the sphere of radius*r*.

### Asymptotic representation of the perron root of a matrix-valued stochastic evolution

Shurenkov V. M., Yeleyko Ya. I.

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 35-43

We study an asymptotic representation of the Perron root of a matrix-valued stochastic evolution given by the transport equation.

### Random permanents of mixed sample matrices

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 44-49

We study the central limit theorem for mixed random permanents.

### Asymptotics of locally constrained control in optimal parabolic problems

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 50-56

We construct and justify asymptotic solutions of optimal parabolic problems with locally constrained control which depends only on time.

### The Hille-Yosida theorem for resolvent operators of multiparameter semigroups

Lavrent'ev A. S., Mishura Yu. S.

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 57-65

We consider multiparameter semigroups of two types (multiplicative and coordinatewise) and resolvent operators associated with such semigroups. We prove an alternative version of the Hille-Yosida theorem in terms of resolvent operators. For simplicity of presentation, we give statements and proofs for two-parameter semigroups.

### Multipoint problem for typeless factorized differential operators

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 66-79

We analyze the well-posedness of a problem with multipoint conditions in the time variable and periodic conditions in the spatial coordinates for differential operators decomposable into operators of the first order with complex coefficients. We establish conditions for the existence and uniqueness of the classical solution of the problem under consideration and prove metric theorems for the lower estimates of small denominators appearing in the process of construction of the solution.

### Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 80-89

In the space*L* _{q}, 1<*q*<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variables*B* _{1,θ} ^{ r } and*B* _{ p,α} ^{ r } by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross.

### Some remarks concerning the convergence of the numerical-analytic method of successive approximations

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 90-95

We establish new improved estimates necessary for the justification of the numerical-analytic method for the investigation of the existence and construction of approximate solutions of nonlinear boundary-value problems for ordinary differential equations.

### Asymptotics of the system of solutions of a general differential equation with parameter

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 96-108

We consider an*n*th-order differential equation $$a_0 (x)y^{(n)} (x) + a_1 (x)y^{(n - 1)} (x) + ... + a_n (x)y(x) = \lambda y(x)$$ with parameter λ ∈ ℂ on a finite interval [*a,b*]. Under the conditions that \(j = \overline {1,n} \) and*a* _{0} *(x)* is an absolutely continuous function which does not turn into zero on the interval [*a, b*], we establish asymptotic formulas of exponential type for the fundamental system of solutions of this equation provided that |λ| is sufficiently large.

### On linear systems with quasiperiodic coefficients and bounded solutions

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 109-115

For a discrete dynamical system ω_{ n }=ω_{0}+α*n*, where a is a constant vector with rationally independent coordinates, on the*s*-dimensional torus Ω we consider the set*L* of its linear unitary extensions*x* _{ n+1}=*A*(ω_{0}+α*n*)*x* _{ n }, where*A* (Ω) is a continuous function on the torus Ω with values in the space of*m*-dimensional unitary matrices. It is proved that linear extensions whose solutions are not almost periodic form a set of the second category in*L* (representable as an intersection of countably many everywhere dense open subsets). A similar assertion is true for systems of linear differential equations with quasiperiodic skew-symmetric matrices.

### On the eigenvalues of the fredholm operator

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 116-123

We prove that if ω(*t, x, K* _{2} ^{(m)} )⩽*c*(*x*)ω(*t*) for all*x*ε[*a, b*] and*x* ε [0,*b*-*a*] where*c* ∈*L* ^{1}(*a, b*) and ω is a modulus of continuity, then λ_{ n }=*O*(*n* ^{−m-1/2}ω(1/*n*)) and this estimate is unimprovable.

### On the uniqueness of a solution of the fourier problem for a system of sobolev-gal’pern type

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 124-128

We establish conditions for the uniqueness of a solution of the problem for a system of equations unresolved with respect to the time derivative without initial conditions in a noncylindrical domain. The system considered, in particular, contains pseudoparabolic equations.

### Separation of variables in linear extensions of dynamical systems on tori

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 129-132

We study the problem of separation of variables in linear extensions of dynamical systems on tori.

### Exact values of widths for certain functional classes

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 133-135

For classes of functions analytic in a unit circle in a Hardy-Banach space, we obtain exact values of Kolmogorov widths in the case where the metrics of the class and the space do not coincide.

### On the smoothness of bounded invariant manifolds of linear inhomogeneous extensions of dynamical systems

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 136-139

We establish sufficient conditions for the existence of smooth bounded invariant manifolds of dynamical systems.

### On periodic solutions of difference equations with continuous argument

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 140-145

We establish conditions for the existence of periodic solutions of systems of nonlinear difference equations with continuous argument.