# Volume 55, № 11, 2003

### Monotonicity of Topological Entropy for One-Parameter Families of Unimodal Mappings

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1443-1449

For a special class of one-parameter families of unimodal mappings of the form *f* _{t}(*x*): [0, 1] → [0, 1], *f* _{t} = *atx*/(*x* + *t*), 0 ≤ *x* ≤ 1/2, we establish that, for *t* ε [0, 1/(*a* − 2)], *a* > 2, the topological entropy *h*(*f* _{t}) is a function monotonically increasing in the parameter. We prove that there exists a class of one-parameter families of unimodal mappings *f* _{t} that contains the family indicated above and establish conditions under which the topological entropy *h*(*f* _{t}) is a function monotonically increasing in the parameter.

### On Stabilization of Programmed Motion

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1450-1458

We obtain new results on stabilization of programmed motions.

### Subharmonics of a Nonconvex Noncoercive Hamiltonian System

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1459-1466

We study the problem of the existence of multiple periodic solutions of the Hamiltonian system $$J\dot x + u\nabla G\left( {t,u\left( x \right)} \right) = e\left( t \right),$$ where *u* is a linear mapping, *G* is a *C* ^{1}-function, and *e* is a continuous function.

### Global Attractor for a Nonautonomous Inclusion with Discontinuous Right-Hand Side

Kapustyan O. V., Kasyanov P. O.

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1467-1475

We consider a nonautonomous inclusion the upper and lower selectors of whose right-hand side are determined by functions with discontinuities of the first kind. We prove that this problem generates a family of multivalued semiprocesses for which there exists a global attractor compact in the phase space.

### Asymptotic Behavior of Solutions of Linear Singularly Perturbed General Separated Boundary-Value Problems with Initial Jump

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1476-1488

We obtain asymptotic estimates for solutions of singularly perturbed boundary-value problems with initial jumps.

### On the Growth of Meromorphic Solutions of an Algebraic Differential Equation in a Neighborhood of a Logarithmic Singular Point

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1489-1502

We prove that if an analytic function *f* with an isolated singular point at ∞ is a solution of the differential equation *P*(*z*ln*z*, *f*, *f*′) = 0, where *P* is a polynomial in all variables, then *f* has finite order. We study the asymptotic properties of a meromorphic solution with logarithmic singularity.

### Multidimensional Lagrange–Yen-Type Interpolation Via Kotel'nikov–Shannon Sampling Formulas

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1503-1520

Direct finite interpolation formulas are developed for the Paley–Wiener function spaces \(L_\diamondsuit ^2\) and \(L_{[-\pi, \pi]^d}^2\) , where \(L_\diamondsuit ^2\) contains all bivariate entire functions whose Fourier spectrum is supported by the set ♦ = Cl{(*u*, *v*) ∣ |*u*| + |*v*| < π], while in \(L_{[-\pi, \pi]^d}^2\) the Fourier spectrum support set of its *d*-variate entire elements is [−π, π]^{ d }. The multidimensional Kotel'nikov–Shannon sampling formula remains valid when only finitely many sampling knots are deviated from the uniform spacing. By using this interpolation procedure, we truncate a sampling sum to its irregularly sampled part. Upper bounds of the truncation error are obtained in both cases.

According to the Sun–Zhou extension of the Kadets \(\frac{1}{4}\) -theorem, the magnitudes of deviations are limited coordinatewise to \(\frac{1}{4}\) . To avoid this inconvenience, we introduce weighted Kotel'nikov–Shannon sampling sums. For \(L_{[-\pi, \pi]^d}^2\) , Lagrange-type direct finite interpolation formulas are given. Finally, convergence-rate questions are discussed.

### Cauchy Problem for Nonuniformly Parabolic Equations with Degeneracy

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1520-1529

In spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of the Cauchy problem for nonuniformly parabolic equations without restrictions on the power order of degeneracy of the coefficients. We obtain an estimate for the solution of the problem in the corresponding spaces.

### Estimation of a *K*-Functional of Higher Order in Terms of a *K*-Functional of Lower Order

Radzievskaya E. I., Radzievskii G. V.

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1530-1540

Let *U* _{j} be a finite system of functionals of the form \(U_j (g):= \int _0^1 g^(k_j) ( \tau ) d \sigma _j ( \tau )+ \sum_{l < k_j} c_{j,l} g^(l) (0)\) , and let \(W_{p,U}^r\) be the subspace of the Sobolev space \(W_p^r [0;1]\) , 1 ≤ *p* ≤ +∞, that consists only of functions *g* such that *U* _{j}(*g*) = 0 for *k* _{j} < *r*. It is assumed that there exists at least one jump τ_{ j } for every function σ_{ j }, and if τ_{ j } = τ_{ s } for *j* ≠ *s*, then *k* _{j} ≠ *k* _{s}. For the *K*-functional $$K(\delta, f; L_p ,W_{p,U}^r) := \inf \limits_{g \in W_{p,U}^r} {|| f-g ||_p + \delta (|| g ||_p + || g^(r) ||_p)},$$ we establish the inequality \(K(\delta^n , f;L_p ,W_{p,U}^r) \leqslant cK(\delta^r ,f; L_p ,W_{p,U}^r)\) , where the constant *c* > 0 does not depend on δ ε (0; 1], the functions *f* belong to *L* _{p}, and *r* = 1, ¨, *n*. On the basis of this inequality, we also obtain estimates for the *K*-functional in terms of the modulus of smoothness of a function *f*.

### Exact Solvability Conditions for the Cauchy Problem for Systems of First-Order Linear Functional-Differential Equations Determined by $(σ_1, σ_2, ... , σ_n; τ)$-Positive Operators

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1541-1568

We obtain new sufficient conditions under which the Cauchy problem for a system of linear functional-differential equations is uniquely solvable for arbitrary forcing terms. The conditions established are unimprovable in a certain sense.

### Criterion for the Uniqueness of a Solution of the Darboux–Protter Problem for Degenerate Multidimensional Hyperbolic Equations

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1569-1575

We obtain a criterion for the uniqueness of a regular solution of the Darboux–Protter problem for degenerate multidimensional hyperbolic equations.

### Lebesgue–Cech Dimensionality and Baire Classification of Vector-Valued Separately Continuous Mappings

Kalancha A. K., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1576-1579

For a metrizable space *X* with finite Lebesgue–Cech dimensionality, a topological space *Y*, and a topological vector space *Z*, we consider mappings *f*: *X* × *Y* → *Z* continuous in the first variable and belonging to the Baire class α in the second variable for all values of the first variable from a certain set everywhere dense in *X*. We prove that every mapping of this type belongs to the Baire class α + 1.

### On One Inequality in Approximation Theory

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1580-1585

We investigate one inequality in approximation theory and obtain necessary and sufficient conditions for the validity of this inequality. We present several examples demonstrating that the results obtained are unimprovable.