# Volume 66, № 1, 2014

### Fixed-Point Theorems for Multivalued Generalized Nonlinear Contractive Maps in Partial Metric Spaces

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 3–16

We prove some fixed-point results for multivalued generalized nonlinear contractive mappings in partial metric spaces, which generalize and improve the corresponding recent fixed-point results due to Ćirić [L. B. Ćirić, “Multivalued nonlinear contraction mappings,” *Nonlin. Anal.*, **71**, 2716–2723 (2009)] and Klim and Wardowski [D. Klim and D. Wardowski, “Fixed-point theorems for set-valued contractions in complete metric spaces,” *J. Math. Anal. Appl.*, **334**, 132–139 (2007)].

### On the Boundary Behavior of One Class of Mappings in Metric Spaces

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 17–29

We study the problem of extension to the boundary of continually ring *Q*-homeomorphisms relative to a *p*-module between continual domains in metric spaces with measures and formulate the conditions for the function *Q* and the boundaries of domains under which every continually ring *Q*-homeomorphism admits a continuous or homeomorphic extension to the boundary. The accumulated results yield, in particular, important applications to fractals in ℝ^{ n } *, n ≥* 2*.*

### Queueing Systems with Resume Level

Bratiichuk N. S., Sliwinska D.

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 30–40

A new approach is proposed for the investigation of the characteristics of queueing systems of the *M/G/*1*/b*-type with finite waiting rooms and a resume level of the input flow. A convenient algorithm is proposed for the numerical evaluation of stationary parameters of the system. Its efficiency is demonstrated for a specific system.

### Classification of the Regular Components of Two-Dimensional Inner Maps

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 41–48

We propose a topological classification of dynamical systems generated by two-dimensional inner maps on the fully invariant regular components of a wandering set with special attracting boundary (to within the topological conjugacy).

### Estimates for the Approximations of the Classes of Analytic Functions by Interpolation Analogs of the De-La-Vallée–Poussin Sums

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 49–62

We establish two-sided estimates for the exact upper bounds of approximations by the interpolation analogs of the de-la-Vallée-Poussin sums on the classes of 2*π* -periodic functions *C* _{ β,s } ^{ ψ } specified by the sequences *ψ*(*k*) and shifts of the argument *β* , *β ∈* ℝ*,* under the condition that the sequences *ψ*(*k*) satisfy the d’Alembert *D* _{ q } *, q ∈* (0*,* 1)*,* condition. Similar estimates are obtained for the classes *C* _{ β } ^{ ψ } *H* _{ ω } generated by convex moduli of continuity *ω*(*t*)*.* Under the conditions *n − p → ∞* and *p → ∞,* the indicated estimates turn into asymptotic equalities.

### On Nearly *ℳ*-Supplemented Subgroups of Finite Groups

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 63–70

A subgroup *H* is called nearly *ℳ*-supplemented in a finite group *G* if there exists a normal subgroup *K* of *G* such that *HK* ⊴ *G* and *TK* < *HK* for every maximal subgroup *T* of *H.* We obtain some new results on supersoluble groups and their formation by using nearly *ℳ*-supplemented subgroups and study the structure of finite groups.

### Kolmogorov-Type Inequalities for Fractional Derivatives on the Half Line

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 71–78

We obtain new sharp Kolmogorov-type inequalities for the fractional derivatives of functions defined on the half line.

### Distribution of Random Variable Represented by a Binary Fraction with Three Identically Distributed Redundant Digits

Makarchuk O. P., Pratsiovytyi M. V.

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 79–88

We present the complete solution of the problem of pure Lebesgue type of the distribution of random variable *χ* represented by a binary fraction with three identically distributed redundant digits.

### Decay of the Solutions of Parabolic Equations with Double Nonlinearity and the Degenerate Absorption Potential

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 89–107

We study the behavior of solutions for the parabolic equation of nonstationary diffusion with double nonlinearity and a degenerate absorption term: $$ {\left({\left| u\right|}^{q-1} u\right)}_t-{\displaystyle \sum_{i=1}^N\frac{\partial }{\partial {x}_i}\left({\left|{\nabla}_x u\right|}^{q-1}\frac{\partial u}{\partial {x}_i}\right)+{a}_0(x){\left| u\right|}^{\lambda -1} u=0,} $$ where \( {a}_0(x)\ge {d}_0\; \exp \left(-\frac{\omega \left(\left| x\right|\right)}{{\left| x\right|}^{q+1}}\right) \) , *d* _{0} = const > 0, 0 ≤ *λ* < *q*, *ω*(⋅) ϵ *C*([0, + ∞)), *ω*(0) = 0, *ω*(*τ*) > 0 for *τ* > 0, and \( {\displaystyle {\int}_{0+}\frac{\omega \left(\tau \right)}{\tau} d\tau <\infty } \) . By the local energy method, we show that a Dini-type condition imposed on the function ω(·) guarantees the decay of an arbitrary solution for a finite period of time.

### On Two-Dimensional Model Representations of One Class of Commuting Operators

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 108–127

In the work by V. A. Zolotarev, *Dokl. Akad. Nauk Arm. SSR*, **63**, No. 3, 136–140 (1976), a triangular model is constructed for a system of twice-commuting linear bounded completely nonself-adjoint operators {*A* _{1}, *A* _{2}} ([*A* _{1}, *A* _{2}] = 0, [*A* _{1} ^{∗} , *A* _{2}] = 0) such that rank (*A* _{1})_{ I }(*A* _{2})_{ I } = 1 (2*i*(*A* _{ k })_{ I } = *A* _{ k } − *A* _{ k } ^{∗} , *k* = 1, 2) and the spectrum of each operator *A* _{ k }, *k* = 1, 2*,* is concentrated at zero. The indicated triangular model has the form of a system of operators of integration over the independent variable in *L* _{ Ω } ^{2} where the domain *Ω* = [0, *a*] × [0, *b*] is a compact set in ℝ^{2} bounded by the lines *x* = *a* and *y* = *b* and a decreasing smooth curve *L* connecting the points (0*, b*) and (*a,* 0)*.*

### On Generalized Regularized Trace of a Fourth-Order Differential Operator with Operator Coefficient

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 128–134

We deduce a formula for the trace of a boundary-value problem with unbounded operator coefficient and boundary conditions depending on the parameter.

### A Ring of Pythagorean Triples over Quadratic Fields

Harnchoowong A., Somboonkulavudi C.

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 135–139

Let *K* be a quadratic field and let *R* be the ring of integers of *K* such that *R* is a unique factorization domain. The set *P* of all Pythagorean triples in *R* is partitioned into *P* _{ η } *,* sets of triples 〈*α*, *β*, *γ*〉 in *P* where *η* = *γ − β.* We show the ring structures of each *P* _{ η } and *P* from the ring structure of *R.*

### On Kropina Change for *m*th Root Finsler Metrics

Peyghan E., Tabatabaeifar T., Tayebi A.

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 140–144

We study the Kropina change for *m*th root Finsler metrics and establish necessary and sufficient condition under which the Kropina change of an *m*th root Finsler metric is locally dually flat. Then we prove that the Kropina change of an *m*th root Finsler metric is locally projectively flat if and only if it is locally Minkowskian.