# Volume 66, № 4, 2014

### On the Existence of Mild Solutions of the Initial-Boundary-Value Problems for the Petrovskii-Type Semilinear Parabolic Systems with Variable Exponents of Nonlinearity

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 435–444

We study the initial-boundary-value problem with general homogeneous boundary conditions for the Petrovskii-type semilinear parabolic systems with variable exponents of nonlinearity in a cylindrical domain. The existence of mild solutions of this problem is proved.

### Stable Quasiorderings on Some Permutable Inverse Monoids

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 445–457

Let *G* be an arbitrary group of bijections on a finite set. By *I*(*G*), we denote the set of all injections each of which is included in a bijection from *G*. The set *I*(*G*) forms an inverse monoid with respect to the ordinary operation of composition of binary relations. We study different properties of the semi-group *I*(*G*). In particular, we establish necessary and sufficient conditions for the inverse monoid *I*(*G*) to be permutable (i.e., *ξ* ○ *φ* = *φ* ○ *ξ* for any pair of congruences on *I*(*G*)). In this case, we describe the structure of each congruence on *I*(*G*). We also describe the stable orderings on *I*(*A* _{ n }), where *A* _{ n } is an alternating group.

### Atoms in the *p*-localization of Stable Homotopy Category

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 458–472

We study *p*-localizations, where *p* is an odd prime, of the full subcategories \( {\mathcal{S}}^n \) of stable homotopy category formed by CW-complexes with cells in *n* successive dimensions. Using the technique of triangulated categories and matrix problems, we classify the atoms (indecomposable objects) in \( {\mathcal{S}}_p^n \) for *n* ≤ 4(*p* − 1) and show that, for *n* > 4(*p* − 1), this classification is wild in a sense of the representation theory.

### Logarithmic Derivative and the Angular Density of Zeros for a Zero-Order Entire Function

Mostova M. R., Zabolotskii N. V.

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 473–481

For an entire function of zero order, we establish the relationship between the angular density of zeros, the asymptotics of logarithmic derivative, and the regular growth of its Fourier coefficients.

### On Some Zero-Filiform Algebras

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 482–492

We present the description of algebras with the maximum nilpotency index given by certain special identities.

### Approximating Characteristics of the Analogs of Besov Classes with Logarithmic Smoothness

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 493–499

We obtain the exact-order estimates of some approximating characteristics for the analogs of Besov classes of periodic functions of several variables (with logarithmic smoothness).

### Remainders of Semitopological Groups or Paratopological Groups

Lin Fucai, Liu Chuan, Xie Li-Hong

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 500–509

We mainly discuss the remainders of Hausdorff compactifications of paratopological groups or semitopological groups. Thus, we show that if a nonlocally compact semitopological group *G* has a compactification *bG* such that the remainder *Y* = *bG \ G* possesses a locally countable network, then *G* has a countable *π* -character and is also first-countable, that if *G* is a nonlocally compact semitopological group with locally metrizable remainder, then *G* and *bG* are separable and metrizable, that if a nonlocally compact paratopological group has a remainder with sharp base, then *G* and *bG* are separable and metrizable, and that if a nonlocally compact ℝ_{1}-factorizable paratopological group has a remainder which is a *k* -semistratifiable space, then *G* and *bG* are separable and metrizable. These results improve some results obtained by C. Liu (Topology Appl., **159**, 1415–1420 (2012)) and A.V. Arhangel’skїǐ and M. M. Choban (Topology Proc., **37**, 33–60 (2011)). Moreover, some open questions are formulated.

### On Countable Almost Invariant Partitions of *G*-Spaces

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 510–517

For any *σ* -finite *G*-quasiinvariant measure *μ* given in a *G*-space, which is *G*-ergodic and possesses the Steinhaus property, it is shown that every nontrivial countable *μ*-almost *G*-invariant partition of the *G*-space has a *μ*-nonmeasurable member.

### Hyperbolic Variational Inequality of the Third Order with Variable Exponent of Nonlinearity

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 518–530

In Sobolev spaces with variable exponent, we consider the problem for a semilinear hyperbolic variational inequality of the third order. We establish conditions for the existence of a solution *u* of this problem such that *u* ∈ *L* ^{∞}((0, *T*); *V* _{1,0}(*Ω*)), *u* _{ t } ∈ *L* ^{∞}((0, *T*); *V* _{1,0}(*Ω*)) ∩ *L* ^{ p(x)}(*Q* _{ T }), and *u* _{ tt } ∈ *L* ^{∞}((0, *T*); *L* ^{2}(*Ω*)), where *V* _{1,0}(*Ω*) ⊂ *H* ^{1}(*Ω*).

### Common Fixed-Point Theorems for Nonlinear Weakly Contractive Mappings

Abbas M., Chandok S., Khan M. S.

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 531–537

Some common fixed-point results for mappings satisfying a nonlinear weak contraction condition within the framework of ordered metric spaces are obtained. The accumulated results generalize and extend several comparable results well-known from the literature.

### Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 538–550

Let \( {\mu}_{\varOmega, \overrightarrow{b}} \) be a multilinear commutator generalized by the *n*-dimensional Marcinkiewicz integral with bounded kernel *μ* _{Ώ} and let \( {b}_j\ \in Os{c_{\exp}}_{L^{r_j}} \) , 1 ≤ *j* ≤ *m*. We prove the following weighted inequalities for *ω* ∈ *A* _{∞} and 0 < *p* < ∞: $$ {\begin{array}{cc}\hfill {\left\Vert {\mu}_{\varOmega }(f)\right\Vert}_{L^p\left(\omega \right)}\le C{\left\Vert M(f)\right\Vert}_{L^p\left(\omega \right)},\hfill & \hfill \left\Vert {\mu}_{\varOmega, \overrightarrow{b}}(f)\right\Vert \hfill \end{array}}_{L^p\left(\omega \right)}\le C{\left\Vert {M}_{L{\left( \log L\right)}^{1/r}}(f)\right\Vert}_{L^p\left(\omega \right)}. $$

The weighted weak *L*(log *L*)^{1/r } -type estimate is also established for *p* =1 and *ω* ∈ *A* _{1}.

### Relationship Between the Green and Lyapunov Functions in Linear Extensions of Dynamical Systems

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 551–557

We study systems of linear extensions for dynamical systems. As a result, we establish the relationship between the design matrices in the structure of Green functions and alternating Lyapunov functions.

### Asymptotic Stability of Implicit Differential Systems in the Vicinity of Program Manifold

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 558–565

Sufficient conditions for the asymptotic and uniform asymptotic stability of implicit differential systems in a neighborhood of the program manifold are established. Sufficient conditions of stability are also obtained for the known first integrals. A class of implicit systems for which it is possible to find the derivative of the Lyapunov function is selected.

### On the Solvability of a System of Linear Equations Over the Domain Of Principal Ideals

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 566–570

We propose new necessary and sufficient conditions for the solvability of a system of linear equations over the domain of principal ideals and an algorithm for the solution of this system.

### Special Space Curves Characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 571-576

By using the facts that the condition$\det(α^{(1)}, α^{(2)}, α^{(3)}) = 0$ characterizes a plane curve and the condition $\det(α^{(2)}, α^{(3)}, α^{(4)}) = 0$ characterizes a curve of constant slope, we present special space curves characterized by the condition $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$, in different approaches. It is shown that the space curve is Salkowski if and only if $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$. The approach used in our investigation can be useful in understanding the role of the curves characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ in differential geometry.