# Vol. 23 (2017), no. 2

### On behavior at infinity of solutions of elliptic differential equations in a Banach space

M. L. Gorbachuk, V. M. Gorbachuk

MFAT **23** (2017), no. 2, 108-122

108-122

For a differential equation of the form $y''(t) - By(t) = 0, \ t \in (0, \infty)$, where $B$ is a weakly positive linear operator in a Banach space $\mathfrak{B}$, the conditions on the operator $B$, under which this equation is uniformly or uniformly exponentially stable are given. As distinguished from earlier works dealing only with continuous at 0 solutions, in this paper no conditions on behavior of a solution near 0 are imposed.

### Evolution of correlation operators of large particle quantum systems

MFAT **23** (2017), no. 2, 123-132

123-132

The paper deals with the problem of a rigorous description of the evolution of states of large particle quantum systems in terms of correlation operators. A nonperturbative solution to a Cauchy problem of a hierarchy of nonlinear evolution equations for a sequence of marginal correlation operators is constructed. Moreover, in the case where the initial states are specified by a one-particle density operator, the mean field scaling asymptotic behavior of the constructed marginal correlation operators is considered.

### On universal coordinate transform in kinematic changeable sets

MFAT **23** (2017), no. 2, 133-154

133-154

This work is devoted to a study of abstract coordinate transforms in kinematic changeable sets. Investigations in this direction may be interesting for astrophysics, because there exists a hypothesis that, in a large scale of the Universe, physical laws (in particular, the laws of kinematics) may be different from the laws acting in a neighborhood of our solar System.

### Asymptotic properties of the $p$-adic fractional integration operator

Anatoly N. Kochubei, Daniel S. Soskin

MFAT **23** (2017), no. 2, 155-163

155-163

We study asymptotic properties of the $p$-adic version of a fractional integration operator introduced in the paper by A. N. Kochubei, Radial solutions of non-Archimedean pseudo-differential equations, Pacif. J. Math. 269 (2014), 355-369.

### Fixed points of complex systems with attractive interaction

MFAT **23** (2017), no. 2, 164-176

164-176

We study the behavior of complex dynamical systems describing an attractive interaction between two opponents. We use the stochastic interpretation and describe states of systems in terms of probability distributions (measures) and their densities. For the time evolution we derive specific non-linear difference equations which generalize the well-known Lotka-Volterra equations. Our results state the existence of fixed points (equilibrium states) for various kinds of attractive interactions. Besides, we present an explicit description of the limiting distributions and illustrate abstract results by several examples.

### Initial-boundary value problems for two-dimensional parabolic equations in Hörmander spaces

MFAT **23** (2017), no. 2, 177-191

177-191

We investigate a general nonhomogeneous initial-boundary value problem for a two-dimensional parabolic equation in some anisotropic Hörmander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate Hörmander spaces.

### On well-behaved representations of $\lambda$-deformed CCR

D. P. Proskurin, L. B. Turowska, R. Y. Yakymiv

MFAT **23** (2017), no. 2, 192-205

192-205

We study well-behaved ∗-representations of a λ-deformation of Wick analog of CCR algebra. Homogeneous Wick ideals of degrees two and three are described. Well-behaved irreducible ∗-representations of quotients by these ideals are classified up to unitary equivalence.