We study the Banach-Saks sets in Banach lattices. It also investigated some of the consequences derived from this study. Among other things, we establish that each $L$-weakly compact set is Banach-Saks. Also, we characterize Banach lattices in which the solid hull of a Banach-Saks set is also Banach-Saks. Next, we give a necessary and sufficient condition under which an order bounded almost Dunford-Pettis operator between two Banach lattices maps Banach-Saks sets into $L$-weakly compact sets. In addition, we establish a new characterization of Banach lattices with order continuous norm and order weakly compact operators using Banach-Saks set and the Cesaro convergence.
The paper investigates a nonlocal problem with homogeneous boundary conditions for a nonhomogeneous partial differential equation involving the generalized differentiation operator $z\frac{\partial}{\partial z}$, acting on a function of the complex variable $z$ with scalar coefficients. The uniqueness and existence theorems for the solution of the problem in the space $\mathbf{H}_{q}^n(\mathcal{D})$ are proved. The Hadamard well-posedness of the problem is established. This distinguishes the problem from Hadamard ill-posed nonlocal problems with a vector complex variable $z$.
Using the tails of the approximants of continued fractions and their multidimensional generalizations, branched continued fractions, canonical numerators and denominators have been defined, and the conditions for the degeneracy of approximants have been investigated. Certain types of figured approximants, specifically $\mathcal{A}$-, $\mathcal{B}$-, and $\mathcal{C}$-figured approximants and their applications, are considered. Various types of convergence for continued and branched continued fractions have been classified. By analogy with Leighton's example, a $\mathcal{C}$-figured-convergent two-dimensional branched continued fraction with independent variables has been constructed; in its structure, every fourth element in each constituent continued fraction is an arbitrary complex number different from $0$ and $-1/2$. Methods for investigating the convergence of branched continued fractions are analyzed.
At present, the approximation properties of harmonic Poisson integrals for the upper half-plane, which satisfy the Laplace equation in Cartesian coordinates, have not been sufficiently studied. This paper studies the approximation properties of conjugate harmonic Poisson integrals for the upper half-plane on classes of Hölder functions. An exact equality is found for the upper bound of the deviation of the conjugate harmonic Poisson integral from its boundary value in a uniform metric.
Let $\mathscr{F}$ be an $\omega$-closed family of the subsets of the set of all non-negative integers $\omega$ and $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ be the bicyclic extension generated by $\mathscr{F}$, which are defined in [Visnyk Lviv Univ. Ser. Mech.-Mat. 2020, 90, 5-19]. In [Visnyk Lviv Univ. Ser. Mech.-Mat. 2023, 95, 28-45] we describe injective monoid endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ with the family $\mathscr{F}^3=\{[0),[1),[2)\}$ and show that for every injective monoid endomorphism $\varepsilon$ of $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ there exists a positive integer $k$ such that $\varepsilon=\alpha_{[k]}$ and the mapping $\alpha_{[k]}\colon \boldsymbol{B}_{\omega}^{\mathscr{F}^3}\to\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ is defined by the formula \[(i,j,[p))\alpha_{[k]}=\begin{cases}(ki,kj,[p)), & \text{if}\; p\in\{0,1\},\\(k(i+1)-1,k(j+1)-1,[2)), & \text{if}\; p=2,\end{cases}\] for all $i,j\in\omega$. In this paper, we describe injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$. In particular, we construct injective endomorphisms $\varpi_3$ and $\lambda$ of $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ and describe the subsemigroup $\left\langle\lambda,\varpi_3\right\rangle$ of the semigroup $\boldsymbol{End}_{\textsf{inj}}(\boldsymbol{B}_{\omega}^{\mathscr{F}^3})$ of all injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$, generated by $\varpi_3$ and $\lambda$. We show that for every injective endomorphism $\varepsilon$ of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ there exists an injective endomorphism $\iota\in\left\langle\lambda,\varpi_3\right\rangle$ such that $\varepsilon=\alpha_{[k]}\circ\iota$ for some positive integer $k$, where $\alpha_{[k]}$ is an injective monoid endomorphism of $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$. Also, we prove that the submonoid of $\boldsymbol{End}_{\textsf{inj}}(\boldsymbol{B}_{\omega}^{\mathscr{F}^3})$, which is generated by the set $\big\{\alpha_{[k]}\colon k\in\mathbb{N}\big\}$ and by the endomorphism $\lambda$, is isomorphic to the semidirect product $(\mathbb{N},\cdot)\ltimes_{\mathfrak{h}}(\omega,+)$.
