# Volume 58, № 11, 2006

### Yaroslav Borisovich Lopatinsky (09.11.1906 - 03.10.1981)

Gorbachuk M. L., Lyantse V. É., Markovskii A. I., Mikhailets V. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1443-1445

### Invariant cones and stability of linear dynamical systems

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1446–1461

We present a method for the investigation of the stability and positivity of systems of linear differential equations of arbitrary order. Conditions for the invariance of classes of cones of circular and ellipsoidal types are established. We propose algebraic conditions for the exponential stability of linear positive systems based on the notion of maximal eigenpairs of a matrix polynomial.

### On the correct solvability of the Dirichlet problem for operator differential equations in a Banach space

Gorbachuk M. L., Gorbachuk V. M.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1462–1476

We investigate the structure of solutions of an equation $y″(t) = By(t)$, where $B$ is a weakly positive operator in a Banach space B, on the interval $(0, \infty)$ and establish the existence of their limit values as $t → 0$ in a broader locally convex space containing $B$ as a dense set. The analyticity of these solutions on $(0, \infty)$ is proved and their behavior at infinity is studied. We give conditions for the correct solvability of the Dirichlet problem for this equation and substantiate the applicability of power series to the determination of its approximate solutions.

### Bilateral estimates for the support of a solution of the Cauchy problem for an anisotropic quasilinear degenerate equation

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1477–1486

We establish exact-order bilateral estimates for the size of the support of a solution of the Cauchy problem for a doubly nonlinear parabolic equation with anisotropic degeneration in the case where the initial data are finite and have finite mass.

### Inverse problem for a parabolic equation with strong power degeneration

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1487–1500

We consider the inverse problem of determining the time-dependent coefficient of the leading derivative in a full parabolic equation under the assumption that this coefficient is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solution of the problem under consideration.

### Solonnikov parabolic systems with quasihomogeneous structure

Ivasyshen S. D., Ivasyuk H. P.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1501–1510

We consider a new class of systems of equations that combine the structures of Solonnikov and Éidel’man parabolic systems. We prove a theorem on the reduction of a general initial-value problem to a problem with zero initial data and a theorem on the correct solvability of an initial-value problem in a model case.

### On the improvement of summability of generalized solutions of the Dirichlet problem for nonlinear equations of the fourth order with strengthened ellipticity

Kovalevskii A. A., Voitovich M. V.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1511–1524

We consider the Dirichlet problem for a class of nonlinear divergent equations of the fourth order characterized by the condition of strengthened ellipticity imposed on their coefficients. The main result of the present paper shows how the summability of generalized solutions of the given problem improves, depending on the variation in the exponent of summability of the right-hand side of the equation beginning with a certain critical value. The exponent of summability that guarantees the boundedness of solutions is determined more exactly.

### Cauchy problem for parabolic systems with pulse action

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1525–1535

For linear parabolic systems with pulse action, we establish the well-posedness of the Cauchy problem in normed Dini spaces.

### Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1536–1555

We study a regular elliptic boundary-value problem for a homogeneous differential equation in a bounded domain. We prove that the operator of this problem is a Fredholm (Noether) operator in a two-sided improved scale of functional Hilbert spaces. The elements of this scale are Hörmander-Volevich-Paneyakh isotropic spaces. We establish an *a priori* estimate for a solution and investigate its regularity.

### On spectral theorems for families of linearly connected self-adjoint operators with given spectra associated with extended Dynkin graphs

Ostrovskii V. L., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1556–1570

We prove spectral theorems for families of linearly connected self-adjoint operators with given special spectra associated with extended Dynkin graphs. We establish that all irreducible families of linearly connected operators with arbitrary spectra associated with extended Dynkin graphs are finite-dimensional.

### On the theory of the Beltrami equation

Ryazanov V. I., Srebro U., Yakubov E.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1571–1583

We study ring homeomorphisms and, on this basis, obtain a series of theorems on the existence of the so-called ring solutions for degenerate Beltrami equations. A general statement on the existence of solutions for the Beltrami equations that extends earlier results is formulated. In particular, we give new existence criteria for homeomorphic solutions $f$ of the class $W^{1, 1}_{\text{loc}}$ with f −1 ∈ $f^{—1} \in W^{1, 2}_{\text{loc}}$ in terms of tangential dilatations and functions of finite mean oscillation. The ring solutions also satisfy additional capacity inequalities.

### Andrei Reuter (1937-2006)

Bondarenko V. M., Drozd Yu. A., Kirichenko V. V., Mitropolskiy Yu. A., Samoilenko A. M., Samoilenko Yu. S., Sharko V. V., Stepanets O. I.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1584-1585