Ukrains’kyi Matematychnyi Zhurnal

Fixed-point theorem for an infinite Toeplitz matrix and its extension to general infinite matrices

2024-04-10T20:39:42+03:00

UDC 517.9

In [V. M. Abramov, Bull. Austral. Math. Soc., 104, 108–117 (2021)], the fixed-point equation was studied for an infinite nonnegative particular Toeplitz matrix. In the present paper, we provide an alternative proof for the existence of a positive solution in the general case. The presented proof is based on the application of a version of the M. A. Krasnosel'ski fixed-point theorem. The results are then extended to the equations with infinite matrices of a general type.

Leonardo and hyper-Leonardo numbers via Riordan arrays

2024-04-10T20:39:48+03:00

UDC 511

A generalization of the Leonardo numbers is defined and called the hyper-Leonardo numbers. Infinite lower triangular matrices, whose elements are Leonardo and hyper-Leonardo numbers are considered. Then the $A$- and $Z$-sequences of these matrices are obtained. Finally, the combinatorial identities between the hyper-Leonardo and Fibonacci numbers are obtained using the fundamental theorem of the Riordan arrays.

A note on the mapping theorem of essential pseudospectra on a Banach space

2024-04-10T20:39:40+03:00

UDC 517.98

The main goal of the paper is to determine some basic properties of the essential pseudospectrum of a bounded linear operator $A$ defined on a Banach space $X.$ We also prove two different versions of the essential pseudospectral mapping theorem.

Boundary-value problems for the Lyapunov equation. І

2024-04-10T20:39:34+03:00

UDC 517.9

We study boundary-value problems for the Lyapunov operator-differential equation. By using the theory of Moore–Penrose pseudoinverse operators and its development, we establish conditions for the existence of generalized solutions and propose algorithms for their construction.

Abelian model structures on comma categories

2024-04-10T20:40:02+03:00

UDC 512.64

Let $\mathsf{A}$ and $\mathsf{B}$ be bicomplete Abelian categories, which both have enough projectives and injectives and let $T\colon\mathsf{A}\rightarrow\mathsf{B}$ be a right exact functor. Under some mild conditions, we show that hereditary Abelian model structures on $\mathsf{A}$ and $\mathsf{B}$ can be amalgamated into a global hereditary Abelian model structure on the comma category $(T\downarrow\mathsf{B})$. As an application of this result, we give an explicit description of a subcategory that consists of all trivial objects of the Gorenstein flat model structure on the category of modules over a triangular matrix ring.

The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$

2024-04-10T20:52:47+03:00

UDC 517.9

Let $n\in \mathbb{N},$ $n\geq 2.$ An element $(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}(^n E)$ if\/ $\|x_1\| = \ldots = \|x_n\| = 1$ and $|T(x_1, \ldots, x_n)| = \|T\|, $ where ${\mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E.$ For $T\in {\mathcal L}(^n E),$ we define \begin{align*}{\rm Norm}(T) = \big\{(x_1, \ldots, x_n)\in E^n\colon (x_1, \ldots, x_n) \mbox{is a norming point of} T\big\}.\end{align*} The ${\rm Norm}(T)$ is called the {\em norming set} of $T.$ For $m\in \mathbb{N},$ $m\geq 2, $ we characterize ${\rm Norm}(T)$ for every $T\in {\mathcal L}\big({}^m l_1^n\big),$ where $l_1^n = \mathbb{R}^n$ with the $l_1$-norm. As applications, we classify ${\rm Norm}(T)$ for every $T\in {\mathcal L}\big({}^m l_{1}^n\big)$ with $n = 2, 3$ and $m = 2.$

Relationship between the Boyanov–Naydenov problem and Kolmogorov-type inequalities

2024-04-10T20:38:04+03:00

UDC 517.5

We prove that the Boyanov–Naidenov problem $\|x^{(k)}\|_{q,\, \delta} \to \sup,$ $k= 0,1, \ldots ,r-1,$ on the classes of functions $\Omega^r_p(A_0, A_r) := \{x\in L^r_{\infty}\colon \|x^{(r)}\|_{\infty}\le A_r,\ L(x)_p\le A_0 \},$ where $q \ge 1$ for $k\ge 1$ and $q \ge p$ for $k=0,$ is equivalent to the problem of finding the sharp constant $C = C(\lambda)$ in the Kolmogorov-type inequality \begin{gather}\|x^{(k)}\|_{q,\, \delta} \leq C L(x)_{p}^{\alpha} \big\|x^{(r)}\big\|_\infty^{1-\alpha}, \quad x\in \Omega^{r}_{p, \lambda}, \tag{1}\end{gather} where $\alpha=\dfrac{r-k+1/q}{r+1/p},$ $\|x\|_{p,\, \delta}:=\sup \{\|x\|_{L_p[a,\, b]}\!\colon a, b \in {\rm \bf R},\ 0< b-a \le \delta \},$ $\delta > 0,$ $\Omega^{\,r}_{p, \lambda}:= \bigcup \{\Omega^{\,r}_p(A_0, A_r)\colon A_0 = A_r L(\varphi_{\lambda, r})_p \},$ $\lambda > 0,$ $\varphi_{\lambda, r}$ is a contraction of the ideal Euler spline of order $r,$ and $L(x)_p:=\sup\big\{ \|x\|_{L_p[a,\, b]}\colon a, b \in {\rm \bf R},\ |x(t)|>0,\ t\in (a, b)\big\}.$

In particular, we obtain a sharp inequality of the form (1) on the classes $\Omega^{\,r}_{p, \lambda},$ $\lambda > 0.$ We also prove the theorems on relationships for the Boyanov–Naidenov problems on the spaces of trigonometric polynomials and splines and establish the relevant sharp Bernstein-type inequalities.

Sufficient conditions and radius problems for the Silverman class

2024-04-10T20:36:39+03:00

UDC 517.5

For $0<\alpha\leq1$ and $\lambda>0,$ let \begin{equation}\label{1} G_{\lambda,\alpha}=\left\{f\in\mathcal{A}\colon \left|\frac{1-\alpha+\alpha zf''(z)/f'(z)}{z f'(z)/f(z)}-(1-\alpha)\right|<\lambda,\ z\in\mathbb{D}\right\}.\tag{0.1}\end{equation} The general form of the Silverman class introduced by Tuneski and Irmak [Int. J. Math. and Math. Sci., {\bf 2006}, Article~ID 38089 (2006)]. Our differential inequality formulation lays out several sufficient conditions for this class. Further, we consider a class $\Omega$ given by \begin{equation}\label{omega}\Omega=\left\{f\in\mathcal{A}\colon |zf'(z)-f(z)|<\frac{1}{2},\ z\in\mathbb{D}\right\}.\tag{0.2}\end{equation} For these two classes, we establish inclusion relations involving some well-known subclasses of $\mathcal{S}^*$ and compute radius estimates featuring various pairings of these classes.

Laguerre–Cayley functions and related polynomials

2024-04-10T20:36:26+03:00

UDC 517.587

We investigate the main properties of Laguerre–Cayley functions and related polynomials, which can be regarded as an essential component of mathematical apparatus of the functional-discrete (FD-) method for solving the Cauchy problem for an abstract homogeneous evolutionary equation of fractional order.

Some new Cesàro sequence spaces of order $\alpha$

2024-04-10T20:36:23+03:00

UDC 517.98

We introduce the spaces $\ell_\infty(C_\alpha),$ $f(C_\alpha),$ and $f_0(C_\alpha)$ of Ces\`{a}ro bounded, Ces\`{a}ro almost convergent, and Ces\`{a}ro almost null sequences of order $\alpha,$ respectively. Moreover, we establish some inclusion relations for these spaces and determine the $\alpha$-, $\beta$- and $\gamma$-duals of the spaces $\ell_\infty(C_\alpha)$ and $f(C_\alpha).$ Finally, we characterize the classes of matrix transformations from the space $f(C_\alpha)$ to any sequence space $Y$ and from any sequence space $Y$ to the space $f(C_\alpha).$

Rotational interval exchange transformations

2024-04-24T14:20:06+03:00

UDC 517.5

We show the equivalence of two possible definitions of a rotational interval exchange transformation: by the first definition, it is the first return map for a circle rotation onto a union of finitely many circle arcs and, by the second definition, it is an interval exchange with a scheme (in the sense of interval rearrangement ensembles) whose dual is also an interval exchange scheme.

On continuous extension to the boundary of a domain of the Cauchy-type integral with parameter-dependent density

2024-04-10T20:35:51+03:00

UDC 517.54

We establish sufficient conditions for the continuous extension of a Cauchy-type integral whose density depends on the parameter to a nonsmooth integration line.