# Volume 59, № 3, 2007

### Oscillation of certain fourth-order functional differential equations

Agarwal R. P., Grace S. R., O’Regan D.

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 291–313

Some new criteria for the oscillation of fourth-order nonlinear functional differential equations of the form $$\frac{d^2}{dt^2} \left(a(t) \left(\frac{d^2x(t)}{dt^2}\right)^{α} \right) + q(t)f(x[g(t)])=0, \quad α>0,$$ are established.

### Integrable superconductivity and Richardson equations

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 314–326

For the integrable generalized model of superconductivity, the solution of the Richardson equations is studied for a model spectrum. For the case of a narrow band, the solution is presented in terms of generalized Laguerre or Jacobi polynomials. In the asymptotic limit, when the Richardson equations are transformed into a singular integral equation, the properties of the integration contour are discussed and the spectral density is calculated. The conditions of appearance of gaps in the spectrum are investigated.

### Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds

Bogolyubov N. N., Prykarpatsky A. K.

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 327–344

We study the differential-geometric aspects of generalized de Rham-Hodge complexes naturally related to integrable multidimensional differential systems of the M. Gromov type, as well as the geometric structure of the Chern characteristic classes. Special differential invariants of the Chern type are constructed, their importance for the integrability of multidimensional nonlinear differential systems on Riemannian manifolds is discussed. An example of the three-dimensional Davey-Stewartson-type nonlinear integrable differential system is considered, its Cartan type connection mapping, and related Chern-type differential invariants are analyzed.

### Whitney’s jets for Sobolev functions

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 345–358

We present two fundamental facts of the jet theory for Sobolev spaces $W^{m, p}$. One of them is that the formal differentiation of $k$-jets theory is compatible with the pointwise definition of Sobolev $(m - 1)$-jet spaces on regular subsets of Euclidean spaces $R^n$. The second result describes the Sobolev embedding operator of Sobolev jet spaces increasing the order of integrability of Sobolev functions up to the critical Sobolev exponent.

### On some periodic solutions of singularly perturbed parabolic equations

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 359–369

We present results from the theory of singular perturbations and, in particular, from a new branch of this theory (contrast alternating-type structures).

### Nonlocal boundary-value problem for linear partial differential equations unsolved with respect to the higher time derivative

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 370–381

We study the well-posedness of the problem with general nonlocal boundary conditions in the time variable and conditions of periodicity in the space coordinates for partial differential equations unsolved with respect to the higher time derivative. We establish the conditions of existence and uniqueness of the solution of the considered problem. In the proof of existence of the solution, we use the method of divided differences. We also prove metric statements on the lower bounds of small denominators appearing in constructing the solution of the problem.

### Stability analysis of large-scale functional differential systems

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 382–394

The present paper is focused on a new method for analysis of stability of solutions of a large-scale functional differential system via matrix-valued Lyapunov-Krasovskii functionals. The stability conditions are based on information about the dynamical behavior of subsystems of the general system and properties of the functions of interconnection between them.

### Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 395–411

We complete the group classification of systems of two coupled nonlinear reaction-diffusion equations with general diffusion matrix begun in author’s previous works. Namely, all nonequivalent equations with triangular diffusion matrix are classified. In addition, we describe symmetries of diffusion systems with nilpotent diffusion matrix and additional terms with first-order derivatives.

### Averaging of initial-value and multipoint problems for oscillation systems with slowly varying frequencies and deviated argument

Danylyuk I. M., Petryshyn R. I., Samoilenko A. M.

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 412–430

We prove new theorems on the substantiation of the method of averaging over all fast variables on a segment and a semiaxis for multifrequency systems with deviated argument in slow and fast variables. An algorithm for the solution of a multipoint problem with parameters is studied, and an estimate for the difference of solutions of the original problem and the averaged problem is established.

### International Conference "Mathematical Analysis and Differential Equations and Applications"

Samoilenko A. M., Savchuk V. V., Sokolenko I. V., Stepanets O. I.

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 431