# Volume 54, № 11, 2002

### On the Construction and Growth of Solutions of Degenerate Functional Differential Equations of Neutral Type

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1443-1451

We consider degenerate linear functional differential equations in Banach spaces and construct solutions of exponential and hyperexponential growth. We establish conditions for the unique solvability of an initial-value problem and describe the set of initial functions. The results are applied to partial differential equations with time delay

### Morrey Regularity of Solutions of Nonlinear Elliptic Systems of Arbitrary Order with Restrictions on the Modulus of Ellipticity

↓ Abstract

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1452-1466

We investigate the dependence of the regularity of generalized solutions of nonlinear elliptic systems on the modulus of ellipticity and regularity of the right-hand side. We establish Morrey regularity with limit exponent determined by the modulus of ellipticity in the case where the right-hand side belongs to a space with a norm stronger than the Dini function. These conditions are exact for second-order systems, namely, for any violation of the Dini condition, we construct a solution that does not belong to the Morrey space with limit exponent.

### Complete Solvability of the Cauchy Problem for Petrovskii Parabolic Equations in *S*-Type Spaces

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1467-1479

We establish the correct solvability (in both directions) of the Cauchy problem for Petrovskii parabolic equations with time-dependent coefficients in *S*-type spaces. We also prove that a solution of this problem stabilizes to zero in the sense of the topology of these spaces.

### A Differential Boundary Operator of the Sturm–Liouville Type on a Semiaxis with Two-Point Integral Boundary Conditions

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1480-1485

We prove that a differential boundary operator of the Sturm–Liouville type on a semiaxis with two-point integral boundary conditions that acts in the Hilbert space *L* _{2}(0, ∞) is closed and densely defined. The adjoint operator is constructed. We also establish criteria for the maximal dissipativity and maximal accretivity of this operator.

### Spectrum and States of the BCS Hamiltonian in a Finite Domain. III. BCS Hamiltonian with Mean-Field Interaction

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1486-1505

We investigate the spectrum of a model Hamiltonian with BCS and mean-field interaction in a finite domain under periodic boundary conditions. The model Hamiltonian is considered on the states of pairs and waves of density charges and their excitations. It is represented as the sum of three operators that describe noninteracting pairs, the interaction between pairs, and the interaction between pairs and waves of density charges. The last two operators tend to zero in the thermodynamic limit, and the spectrum of the model Hamiltonian coincides with the spectrum of noninteracting pairs with chemical potential shifted by mean-field interaction. It is shown that the model and approximating Hamiltonians coincide in the thermodynamic limit on their ground and excited states and both have two branches of eigenvalues and eigenvectors.

### On the Asymptotic Integration of a System of Linear Differential Equations with a Small Parameter in the Coefficients of a Part of Derivatives

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1505-1517

We propose an asymptotic method for the integration of one type of systems of linear differential equations with a small parameter in the coefficients of a part of derivatives.

### Solutions of Weakly-Perturbed Linear Systems Bounded on the Entire Axis

Boichuk А. A., Boichuk О. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1517-1530

We establish conditions under which solutions of weakly-perturbed systems of linear ordinary differential equations bounded on the entire axis *R* emerge from the point ε = 0 in the case where the corresponding unperturbed homogeneous linear differential system is exponentially dichotomous on the semiaxes *R* _{+} and *R* _{−}.

### Binary Transformations and (2 + 1)-Dimensional Integrable Systems

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1531-1550

A class of nonlinear nonlocal mappings that generalize the classical Darboux transformation is constructed in explicit form. Using as an example the well-known Davey–Stewartson (DS) nonlinear models and the Kadomtsev–Petviashvili matrix equation (MKP), we demonstrate the efficiency of the application of these mappings in the (2 + 1)-dimensional theory of solitons. We obtain explicit solutions of nonlinear evolution equations in the form of a nonlinear superposition of linear waves.

### Approximation of the Classes $B_{p,θ}^Ω$ of Periodic Functions of Many Variables in Uniform Metric

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1551-1559

We obtain order estimates for the best $М$-term trigonometric approximations and approximations by Fourier sums for the classes $B_{p,θ}^Ω$ of periodic functions of many variables in the uniform metric.

### On Solvable Groups with Proper Quotient Groups of Finite Rank

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1560-1568

We study solvable groups of infinite special rank all proper normal subgroups of which define quotient groups of finite special rank.

### Derivation of Moment Equations for Solutions of a System of Differential Equations Dependent on a Semi-Markov Process

Dzhalladova I. A., Valeyev K. G.

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1569-1573

We present a new method for the derivation of moment equations for solutions of a system of nonlinear differential equations dependent on a finite-valued semi-Markov process. For systems of linear equations, we compare the results obtained with known ones.

### Best Uniform Approximation of a Family of Functions Continuous on a Compact Set

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1574-1580

We investigate the problem of the best uniform approximation of a function continuous on a compact set. We generalize the principal results of this investigation to the problem of the best simultaneous uniform approximation of a family of functions continuous on a compact set.

### On the Suslin Number of Totally-Bounded Left-Topological Groups

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1581-1573

For every infinite cardinal α, we construct a zero-dimensional totally-bounded left-topological group with Suslin number α.