# Volume 60, № 10, 2008

### Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds

Antoniouk A. Val., Antoniouk A. Vict.

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1299–1316

We study the dependence on initial data for solutions of diffusion equations with globally non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may not be everywhere twice differentiable, we show that, under certain monotonicity conditions on the coefficients and curvature of the manifold, there are estimates exponential in time for the continuity of a diffusion process with respect to initial data. These estimates are combined with methods of the theory of absolutely continuous functions to achieve the first-order regularity of solutions with respect to initial data. The suggested approach neither appeals to the local stopping time arguments, nor applies the exponential mappings on the tangent space, nor uses imbeddings of a manifold to linear spaces of higher dimensions.

### On equiasymptotic stability of solutions of doubly-periodic impulsive systems

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1317–1325

A system of ordinary differential equations with impulse effects at fixed moments of time is considered. This system admits the zero solution. Sufficient conditions of the equiasymptotic stability of the zero solution are obtained.

### Differential equations with set-valued solutions

Komleva T. A., Plotnikov A. V., Skripnik N. V.

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1326–1337

Some special space of convex compact sets is considered and notions of a derivative and an integral for multivalued mapping different from already known ones are introduced. The differential equation with multivalued right-hand side satisfying the Caratheodory conditions is also considered and the theorems on the existence and uniqueness of its solutions are proved. In contrast to O. Kaleva's approach, the given approach enables one to consider fuzzy differential equations as usual differential equations with multivalued solutions.

### Inequalities for derivatives of functions in the spaces *L*_{p}

_{p}

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1338 – 1349

The following sharp inequality for local norms of functions $x \in L^{r}_{\infty,\infty}(\textbf{R})$ is proved: $$\frac1{b-a}\int\limits_a^b|x'(t)|^qdt \leq \frac1{\pi}\int\limits_0^{\pi}|\varphi_{r-1}(t)|^q dt \left(\frac{||x||_{L_{\infty}(\textbf{R})}}{||\varphi_r||_{\infty}}\right)^{\frac{r-1}rq}||x^{(r)}||^q_{\infty}r,\quad r \in \textbf{N},$$ where $\varphi_r$ is the perfect Euler spline, takes place on intervals $[a, b]$ of monotonicity of the function $x$ for $q \geq 1$ or for any $q > 0$ in the cases of $r = 2$ and $r = 3.$ As a corollary, well-known A. A. Ligun's inequality for functions $x \in L^{r}_{\infty}$ of the form $$||x^{(k)}||_q \leq \frac{||\varphi_{r-k}||_q}{||\varphi_r||_{\infty}^{1-k/r}} ||x||^{1-k/r}_{\infty}||x^{(r)}||^{k/r}_{\infty},\quad k,r \in \textbf{N},\quad k < r, \quad 1 \leq q < \infty,$$ is proved for $q \in [0,1)$ in the cases of $r = 2$ and $r = 3.$

### Lattice of normal subgroups of a group of local isometries of the boundary of a spherically homogeneous tree

Lavrenyuk Ya. V., Sushchanskii V. I.

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1350–1356

We describe the structure of the lattice of normal subgroups of the group of local isometries of the boundary of a spherically homogeneous tree LIsom ∂T. It is proved that every normal subgroup of this group contains its commutant. We characterize the quotient group of the group LIsom ∂T by its commutant.

### On conjugacy in groups of finite-state automorphisms of rooted trees

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1357–1366

We show that the conjugacy of elements of finite order in the group of finite-state automorphisms of a rooted tree is equivalent to their conjugacy in the group of all automorphisms of the rooted tree. We establish a criterion for conjugacy between a finite-state automorphism and the adding machine in the group of finite-state automorphisms of a rooted tree of valency 2.

### Finite absolute continuity of Gaussian measures on infinite-dimensional spaces

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1367–1377

We study the notion of finite absolute continuity for measures on infinite-dimensional spaces. For Gaussian product measures on \(\mathbb{R}^{\infty}\) and Gaussian measures on a Hilbert space, we establish criteria for finite absolute continuity. We consider cases where the condition of finite absolute continuity of Gaussian measures is equivalent to the condition of their equivalence.

### Local behavior of Q-homeomorphisms in Loewner spaces

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1378–1388

We study the problem of the elimination of isolated singularities for so-called *Q*-homeomorphisms in Loewner spaces. We formulate several conditions for a function *Q*(*x*) under which every *Q*-homeomorphism admits a continuous extension to an isolated singular point. We also consider the problem of the homeomorphicity of the extension obtained. The results are applied to Riemannian manifolds and Carnot groups.

### On the normality of families of space mappings with branching

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1389–1400

We study space mappings with branching that satisfy modulus inequalities. For classes of these mappings, we obtain several sufficient conditions for the normality of families.

### Global exponential stability of a class of neural networks with unbounded delays

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1401–1413

In this paper, the global exponential stability of a class of neural networks is investigated. The neural networks contain variable and unbounded delays. By constructing a suitable Lyapunov function and using the technique of matrix analysis, we obtain some new sufficient conditions for global exponential stability.

### Continuum cardinality of the set of solutions of one class of equations that contain the function of frequency of ternary digits of a number

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1414–1421

We study the equation *v*_{1 }(*x*) = *x*, where *v*_{1 }(*x*) is the function of frequency of the digit 1 in ternary expansion of *x*.
We prove that this equation has a unique rational solution and a continuum set of irrational solutions.
An algorithm for the construction of solutions is proposed. We also describe the topological and metric properties of the set of all solutions.
Some additional facts about equations *v _{i }*(

*x*),

*i*= 0,2, are also given.

### Problem with pulse action for a linear stochastic parabolic equation of higher order

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1422–1426

We prove a theorem on the well-posedness of the Cauchy problem for a linear higher-order stochastic equation of parabolic type with time-dependent coefficients and continuous perturbations whose solutions are subjected to pulse action at fixed times.

### Solutions of the Kirkwood–Salsburg equation for a lattice classical system of one-dimensional oscillators with positive finite-range many-body interaction potentials

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1427–1433

For a system of classical one-dimensional oscillators on the *d*-dimensional hypercubic lattice interacting via pair superstable and many-body positive
finite-range potentials, the (lattice) Kirkwood–Salsburg equation is proposed for the first time and is solved.

### On the hill stability of motion in the three-body problem

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1434–1440

We consider the special case of the three-body problem where the mass of one of the bodies is considerably smaller than the masses of the other two bodies and investigate the relationship between the Lagrange stability of a pair of massive bodies and the Hill stability of the system of three bodies. We prove a theorem on the existence of Hill stable motions in the case considered. We draw an analogy with the restricted three-body problem. The theorem obtained allows one to conclude that there exist Hill stable motions for the elliptic restricted three-body problem.