Ukrains’kyi Matematychnyi Zhurnal (Ukrainian Mathematical Journal) Editor-in-Chief: A. N. Timokha ISSN: 0041-6053, 1027-3190 Ukrainian Mathematical Journal is published by Springer Nature and Institute of Mathematics of the National Academy of Sciences of Ukraine (est. 1934). The journal is a comprehensive international platform whose aim consists of publishing original research and review articles across various branches of pure and applied mathematics. Submitted manuscripts can be written in either English or Ukrainian. Their single-blind peer review is handled by the Editorial Board. After being accepted, Ukrains’kyi Matematychnyi Zhurnal (est. 1949) publishes the full-scale Ukrainian-language articles and abstracts of English submissions, whereas Ukrainian Mathematical Journal collects and issues, in full, the accepted English-language articles together with the Ukrainian-language manuscript translations. The journal scopes cover but not restricted to the eight main heading: Algebra; ODE & Dynamical Systems; Geometry, Topology & Fractals; Functional Analysis, Operator Theory & PDEs; Theory of Functions; Mathematical Physics; Probability & Statistics; Numerical & Applied Mathematics The authors are obligated to mention one of these heading and the corresponding editor when submitting the manuscript. UMZh indexed in: MathSciNet, zbMATH, Scopus, Web of Science, Google Scholar.
Institute of Mathematics, NAS of Ukraine
Pseudo almost periodic dynamical behaviors of hematopoiesis model
UDC 517.929, 517.518 We present a hematopoiesis model with nonlinear harvesting terms. By using the fixed-point theorem and constructing suitable Lyapunov functionals, we establish new sufficient criteria required to guarantee the existence and uniqueness of positive pseudo almost periodic solutions for the considered model with Stepanov-like pseudo almost periodic coefficients. Furthermore, the exponential stability of the obtained solution is investigated. It is shown that every positive solution exponentially converges to a unique positive pseudo almost periodic solution. Finally, an illustrative example is provided to demonstrate the efficiency of our theoretical results.
Copyright (c) 2026 Mohammed Zarhouni, Mohamed Zitane
On solutions of some extreme problems in the classes of mappings defined in a ring
UDC 517.548 We consider classes of mappings defined in a domain of the Euclidean space. In terms of the modulus of the families of paths connecting the boundaries of a ring in this domain, we define a certain functional. The maximum of this functional is found under certain restrictions imposed on the analyzed classes.
Copyright (c) 2026 Руслан Салімов, Євген Севостьянов, Денис Ромаш
Weakly nonlinear boundary-value problem for a two-term sequential fractional differential equation
UDC 517.927, 517.23 We establish the solvability conditions and propose an algorithm for the construction of solution of a weakly nonlinear boundary-value problem for the two-term differential equation with sequential Caputo derivative. A special case in which the sequential derivative is a combination of the ordinary derivative and the Caputo fractional derivative is analyzed.
AMS Subject Classification: 26A33; 34A08; 34B15
On the proximate order and lower proximate order for meromorphic functions
For a function $\Gamma(r)=\exp\left\{\displaystyle\int\nolimits_1^r\dfrac{\gamma(t)}{t}dt\right\},$ $\gamma(r)$ is a proximate order, we deduce the expression $\Gamma(r)=r^{\gamma(r)}L(r),$ where $L(r)$ is a slowly varying function on $[1,+\infty),$ i.e., $rL'(r)/L(r)\to 0$ as $r\to+\infty.$ We define the notions of proximate order $\rho(r)$ and proximate lower order $\lambda(r)$ of a function $f$ meromorphic in $\mathbb{C}$ such that either $\underline{\Delta}(D)=\liminf_{r\to+\infty}T(r,f)/D(r)>0$ and $\overline{\Delta}(H)=\limsup_{r\to+\infty}T(r,f)/H(r)=+\infty$ or $\underline{\Delta}(D)=0$ and $\overline{\Delta}(H)<+\infty,$ where $D(r)=\exp\left\{\displaystyle\int\nolimits_1^r\dfrac{\rho(t)}{t}dt\right\}$ and $H(r)=\exp\left\{\displaystyle\int\nolimits_1^r\dfrac{\lambda(t)}{t}dt\right\}.$ The obtained results reveal the incorrectness of Lemma 1 and Theorem 1 from the paper by S. H. Dwivedi [S. H. Dwivedi, Compos. Math., 22, No. 1, 39–48 (1970)]. We also generalize the statement of Lemma 2 in the cited paper as follows: If a function $\phi(r)$ is such that $r\phi'(r)/\phi(r)\to\phi_0$ as $r \to +\infty,$ then $\displaystyle\int\nolimits_1^r\dfrac{\phi(t)}{t^{1+\alpha}}dt \sim \dfrac{\phi(r)}{(\phi_0-\alpha)r^\alpha}$ for $0 \le \alpha < \phi_0$ and $\displaystyle\int\nolimits_r^{+\infty}\dfrac{\phi(t)}{t^{1+\alpha}}dt \sim \dfrac{\phi(r)}{(\alpha-\phi_0)r^\alpha}$ for $\alpha > \phi_0$ as $r \to +\infty.$
Copyright (c) 2026 Микола Заболоцький, Тарас Заболоцький, Мар'яна Мостова
On weakly $(n-1)$-convex sets in low-dimensional Euclidean spaces
UDC 514.172 Wе study the properties of generalized convex sets in the real $n$-dimensional Euclidean space $\mathbb{R}^n,$ where $n=2,3,$ known as weakly $(n-1)$-convex sets. An open set in $\mathbb{R}^2$ (respectively, $\mathbb{R}^3$) is called weakly $1$-convex (respectively, weakly $2$-convex) if, for every boundary point of the set, there exists a straight line (respectively, a plane) that passes through this point but does not intersect the indicated set. A point in the complement of a set in $\mathbb{R}^2$ (respectively, $\mathbb{R}^3$) is called a $1$-nonconvexity (respectively, $2$-nonconvexity) point of the set if every straight line (respectively, every plane) that passes through this point intersects the set. It is shown that each connected component of the $1$-nonconvexity-point set corresponding to a planar weakly $1$-convex set is either the interior of a convex polygon or the interior of a convex generalized polygon. It is proved that the $2$-nonconvexity-point set corresponding to an open weakly $2$-convex set in $\mathbb{R}^3$ is itself an open and weakly $2$-convex set. We also study some convexity properties of $2$-nonconvexity-point sets corresponding to weakly $2$-convex sets in $\mathbb{R}^3$.
Evolutionary inequalities for functionals with fixed points in the complex plane
Dedicated to the memory of Professor Oleksandr Bakhtin UDC 517.54 We propose an approach to getting evolutionary-type estimates for the products of inner radii of mutually non-overlapping multiply connected domains with fixed points in the complex plane for all values of a certain parameter $\gamma\in(0, n].$
Normally solvable boundary-value problems with degenerate impulsive action
UDC 517.927 We establish solvability conditions for normally solvable boundary-value problems for a system of ordinary differential equations with degenerate impulsive actions at fixed times in the presence of infinitely many impulsive perturbations. The distinctions are established between the structure of solutions to a system of ordinary differential equations with nondegenerate impulsive actions at fixed times (in the presence of infinitely many impulsive perturbations) and the solutions to the system with finitely many degenerate impulsive perturbations.
Copyright (c) 2026 Петер Беннер, Сергій Чуйко, Віктор Чуйко
Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials
UDC 517.983, 517.984 We study the resolvents and scattering matrices for the following pairs of unbounded self-adjoint operators in $\mathbb{L}_{2}(\mathbf{R}_{3})$: the standard Laplace operator $A$ and its self-adjoint perturbation, which rigorously realizes the formal sum of $A$ with linear combinations of infinitely many zero-range potentials. By using Krein’s resolvent formula for the perturbed operator, we establish sufficient conditions for the resolvent difference of the analyzed pair of operators to be a nuclear operator. Under these conditions, we deduce and analyze an explicit formula for the corresponding scattering matrices.
unbounded operator, B-Weyl, Left and right B-Weyl operators, Left and right Drazin invertible operators, Left and right B-Weyl spectra, Left and right Drazin spectra.
Tauberian theorems for the statistical deferred Cesàro summability of sequences of fuzzy numbers
UDC 517.521 We introduce the concept of statistical deferred Cesàro summability for sequences of fuzzy numbers and examine its fundamental properties within the framework of fuzzy number spaces. It is shown that every bounded and statistically convergent sequence is statistically deferred Cesàro summable to the same fuzzy number provided that the deferred Cesàro mean is properly deferred, thus showing that this summability method agrees with statistical convergence. However, the converse of this statement is, in general, not true. This means that additional conditions are required for the statistical convergence to follow from the statistical deferred Cesàro summability. Furthermore, we establish necessary and sufficient Tauberian conditions under which the statistical convergence of a certain subsequence of a sequence of fuzzy numbers follows from its statistical deferred Cesàro summability. Unlike the existing Tauberian results for the Cesàro summability of fuzzy numbers, the present work deals with the statistically deferred framework, which requires the use of essentially different techniques.
Copyright (c) 2026 Ebrar Yıldırım, İbrahim Çanak
Generalized Fourier–Cesàro sums for functions of bounded variation
UDC 517.518, 517.521 We investigate the Fourier–Cesàro sums with respect to general orthonormal systems, when a function $f$ belongs to a certain differentiable class of functions.
General Fourier series, Fourier coefficients, Cesàro summability, functions of bounded variation, orthonormal systems, Banach spaces.
Copyright (c) 2026 Vakhtang Tsagareishvili, Giorgi Tutberidze, Giorgi Cagareishvili
Fibonacci-based generalizations of Bessel polynomials and their applications to Fibonacci-analogue Bessel filters
UDC 517.584, 511.176 In the theory of orthogonal polynomials, extending classical polynomial families to multivariate and deformed settings often leads to rich and challenging generalizations. Motivated by recent developments in Fibonacci calculus and Bessel polynomial theory, this paper introduces the $F$-Bessel (Fibonacci–Bessel) polynomials, the reverse $F$-Bessel polynomials, and their two-dimensional counterparts, the 2D $F$-Bessel polynomials. Several fundamental properties of these polynomial families are established, including generating functions, identities, differential recurrence relations, and determinant representations. The reverse $F$-Bessel polynomials are further employed to construct Fibonacci analogues of classical Bessel filters. Pole distributions for the third-, fourth-, fifth-, and sixth-order $F$-Bessel filters are analyzed, while the gain, phase, phase delay, and group delay characteristics of the third-order filter are numerically investigated and compared with those of the corresponding classical Bessel filter. The results demonstrate that the proposed $F$-Bessel filters preserve the desirable features of classical Bessel filters while exhibiting several enhanced performance characteristics.
$F$-Bessel polynomials, Reverse $F$-Bessel polynomials, Fibonomial coefficient, 2D $F$-Bessel polynomials, $F$-Bessel Filter
Surjective isometry in reflexive strongly facially symmetric spaces
UDC 517.982, 517.986 We study the isometry properties of strongly facially symmetric spaces. It is shown that a linear operator in a reflexive strongly facially symmetric space is a surjective isometry if and only if it maps the set of indecomposable geometric tripotents onto itself and preserves the orthogonality relations within this set.
Strongly facially symmetric space, norm exposed face, geometric tripotent, geometric Peirce projections, surjective isometry.
Copyright (c) 2026 Jumabek Seypullaev, Dilfuza Eshniyazova
Some properties of an extended Fourier convolution operator and its applications
UDC 517.444, 517.968, 517.982 We study some boundedness properties and the Banach algebra structure of the extended Fourier convolution operator. As applications of these results, we establish the solvability of some classes of Fredholm integral equations and differential equations.
Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials
UDC 517.587, 512.643 We consider a Sobolev-type inner product known from the literature. This inner product is a modification of the inner product of Meixner polynomials containing an additional term at a point $a\in\mathbb{R},$ involving the forward-difference operator $\Delta$ and a weight $\lambda>0.$ Here, the point $a$ is chosen to guarantee that the spectrum of Meixner polynomials does not intersect the interval $(a, a+1).$ This inner product and its generalization already exist in the literature for various other measures. We consider a specific measure. For the resulting orthonormal Sobolev-type polynomials, we obtain a five-term recurrence relation. The associated pentadiagonal matrix is analyzed in relation to the Jacobi pencil matrix. The behavior of zeros with respect to the parameters $a$ and $\lambda$ is investigated and the Plancherel–Rotach-type asymptotic behavior is obtained.
Copyright (c) 2026 N. Neha, A. Swaminathan
Radius type inequalities of certain harmonic univalent mappings and their partial sums
UDC 517.546, 517.57 We study a subclass of harmonic univalent mappings defined by a differential inequality and denoted by $\mathcal{R}_{H}^{0}(\gamma,\delta,\lambda).$ This subclass was introduced in the paper [S. Çakmak, E. Yaşar, S. Yalçin, Hacet. J. Math. Stat., 51, № 1, 172–186 (2022)]. We determine the Bohr and Bohr–Rogosinski radii for this family. Sufficient conditions for the invariance of the partial sums of the function class $\mathcal{R}_{H}^{0}(\gamma,\delta,\lambda)$ are presented. In addition, we compute the radius of convexity for the sections of members of the family.
Copyright (c) 2026 Akash Meher, Priyabrat Gochhayat
On $\mathsf{q}$-starlike functions related to $\mathsf{q}$-Bernoulli numbers
UDC 517.54 We define a class of $\mathsf{q}$-starlike functions related to $\mathsf{q}$-Bernoulli numbers and study the coefficient bounds, Hankel determinants, and logarithmic coefficient bounds for this class of functions. The sharpness of the obtained estimates is also discussed. Our results generalize results already known from the literature.
Copyright (c) 2026 Kashif Javed, Mohsan Raza
Multilinear polynomial identities under the action of generalized skew derivations
UDC 512.552 Suppose that $\mathcal{H}_1, \mathcal{H}_2$ and $\mathcal{H}_3$ are generalized skew derivations on prime ring $\mathcal{R}$ with ${\rm char}(\mathcal{R})\neq 2$ such that $\mathcal{H}_1\big(\mathcal{H}_2(\pi(q))\pi(q)\big)=\mathcal{H}_3(\pi(q)^2)$ for all $q=(q_1,\ldots,q_n) \in \mathcal{R}^n,$ where $\pi(q_1,\ldots,q_n)$ denotes the multilinear polynomial over $\mathcal{C},$ which is noncentral and nonidentity. We present all possible configurations of the maps $\mathcal{H}_1, \mathcal{H}_2,$ and $\mathcal{H}_3.$ This extends the results obtained by V. de Filippis [Comm. Algebra, 49, No. 7, 2987–3009 (2021)].
Copyright (c) 2026 Pallavee Gupta, S. K. Tiwari, Lovepreet Singh
Additivity of Heegaard genera under annulus sum
UDC 515.162 Let $M$ be a compact 3-manifold containing an essential annulus $A$ separating $M$ into two manifolds $M_1$ and $M_2.$ We prove that if, for each $i=1,2,$ $g(M_i) \geq 2$ and $M_i$ admits a Heegaard splitting with Heegaard distance not smaller than $2\bigl(g(M_1) + g(M_2)\bigr) - 1,$ then $g(M) = g(M_1) + g(M_2).$ As a corollary, we present a sufficient condition for the superadditivity of the tunnel number of knots under connected sum.
Heegaard genus, Heegaard distance, annulus sum,
Copyright (c) 2026 Qi-Long Guo, Qian-Hui Dai, Shu-Xin Wang
Geometric classification of totally umbilical quasi-hemislant submanifolds
UDC 514.76, 514.77 We study the geometric characteristics and classification of totally umbilical quasi-hemislant submanifolds in almost product Riemannian manifolds. We focus on three primary distributions: the slant distribution $D^\theta,$ the totally real distribution $D^\bot,$ and the invariant distribution $D,$ providing a detailed analysis of their integrability conditions and conditions of being totally geodesic. Through these investigations, we found that these submanifolds can be categorized as totally geodesic, CR submanifolds, or exhibits other interesting geometric properties, such as being extrinsic spheres or proper semislant submanifolds depending on specific geometric conditions. We establish necessary and sufficient conditions for each classification and describe possible structural configurations of these submanifolds. Our results contribute to a broader comprehension of quasi-hemislant submanifolds in differential geometry by providing new classifications and geometric perspective in almost product Riemannian structures.
Copyright (c) 2026 Tanveer Fatima, Sharief Deshmukh
UDC 512.553, 512.661 We study the structure of the right proper flat profile for a class of modules defined as the collection of all proper classes flatly generated by a given module class. Focusing on the classes of all right modules and finitely presented right modules, we characterize several types of rings based on the structure and cardinality of their proper flat profiles. For example, the class of all right modules has a single-element profile if and only if the ring is von-Neumann regular, and over a right Noetherian ring, it has exactly two elements under special conditions. We also show that, over two-sided Noetherian rings, this profile cannot have exactly three elements. In addition, we compare the proper flat profiles of various module classes and investigate their relationship with proper injective and proper projective profiles. For instance, a ring is of finite representation type precisely when certain proper flat and proper projective profiles coincide. Finally, we study the cases where the proper flat profile forms a chain and, for finitely presented right modules, describe the structure of left Noetherian, commutative Noetherian, and two-sided Artinian rings under this assumption.
Proper flat profile, flatly generated proper class, finitely presented module
ring and module theory, homological algebra
Copyright (c) 2026 Yılmaz Durǧun, Salahattin Özdemir, Zübeyir Türkoğlu
Existence results for $\varphi$-Caputo fractional semilinear evolution equations
UDC 517.937, 517.98, 517.23 We study the existence of mild solutions for a new class of nonlocal fractional semilinear integrodifferential evolution equations with a nondensely defined closed linear operator involving the generalized \( \varphi \)-Caputo fractional derivative. By applying two fixed-point theorems for condensing operators, in combination with the \( C_0 \)-semigroup theory and the concept of noncompactness measure, we derive sufficient conditions for the existence of solutions. An illustrative example is presented to highlight the applicability and robustness of our theoretical findings.
Copyright (c) 2026 Walid Benhadda, Ali El Mfadel, Abderrazak Kassidi, M'hamed Elomari
Hyper-commuting generalized derivations acting on Lie ideals in prime rings
UDC 512.552 Let $\mathscr{R}$ be a prime ring with its Utumi ring of quotients $\mathscr{U}$ and extended centroid $\mathcal{C}.$ Also let $n\geq 1$ be a fixed integer and let ${\rm char}(\mathscr{R})\neq 2.$ Suppose that $\mathscr{F}, \mathcal{G},$ and $\mathscr{H}$ are three generalized derivations of $\mathscr{R}$ and $\mathscr{L}$ is a noncentral Lie ideal of $\mathscr{R}.$ If $$\bigr[\mathscr{F}(\mathcal{G}(u)u)-u\mathscr{H}(u),u\bigr]_n=0$$ for all $u \in \mathscr{L},$ then we describe the structure of the maps $\mathscr{F}, \mathcal{G},$ and $\mathscr{H}.$
Copyright (c) 2026 B. Dhara, M. S. Tammam El-Sayiad
A new approach to signal denoising in the analytic univalent function space
UDC 517.54 We introduce a new Gegenbauer-polynomial-based subclass of analytic bi-univalent functions and establish sharp coefficients and functional bounds, which reveal a previously unexplored relationship between the geometric function theory and signal denoising. Unlike classical Fourier and wavelet approaches, the proposed framework enforces intrinsic analytic coherence across the coefficients yielding hierarchical stability on individual, pairwise, and higher-order levels. The analytic bounds supported by the graphical evidence demonstrate how geometric constraints naturally suppress noise-induced distortions and provide a mathematically rigorous and structurally novel approach to denoising based on the univalent function theory. This is achieved by developing a robust new class of analytic $F$-starlike functions with the help of normalized bivariate Gegenbauer polynomials, the sigmiod function, and starlike-Bazilevic functions denoted by $BS_q^\mu$ via the generalized discrete probability distribution, convolution, and subordination principles. The consequences of the obtained results are demonstrated in the corollaries and graphically. We present three new coefficient inequalities that extend and refine several standard topics in the geometric function theory, including the Bieberbach inequality, the Fekete–Szego problem, and the Zalcman or Hankel-type quadratic relation. Beyond their theoretical significance for complex analysis, we use a linear combination of functions to gain an insight into their application to denoising in signal processing. It is concluded that the method and the results are novel in their reinterpretation for signal processing, where the coefficient bounds support denoising by enforcing thresholding, structural consistency, and energy stability.
Analytic, univalent, polynomials, Quantum
Copyright (c) 2026 Adeniyi Abimbola Ayeni, Abiodun Tinuoye Oladipo
Families of spiral-like functions connected with the Rabotnov function
UDC 517.546, 517.58 We explore new families of analytic functions defined through the concepts of subordination and the Rabotnov function. We provide convolution properties and sharp coefficient bounds for functions within these families and examine their inclusion relations with other well-known analytic function families.
A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space
UDC 514.752, 514.748 By using the parallel transport frame field, we examine the geometric properties of a soliton surface $\Psi=\Psi(s,t)$ associated with the Betchov–Da Rios equation in four-dimensional Euclidean space. We obtain derivative formulas for the parallel transport frame field of a unit-speed $s$-parameter curve $\Psi=\Psi(s,t),$ for all $t.$ We deduce two basic geometric invariants of the soliton surface, $k$ and $h,$ and some other important invariants, such as Gaussian curvature, mean curvature vector, and Gaussian torsion. With the aid of these, we prove a set of theorems that describe the conditions under which the soliton surface is flat, minimal, semiumbilic, or Wintgen ideal (superconformal) by using these surface invariants. In addition, we present a theorem that characterizes the curvature ellipse of the Betchov–Da Rios soliton surface with respect to the parallel transport frame field in $E^{4}.$ Finally, we construct an example of a Betchov–Da Rios soliton surface, obtain its geometric invariants, and show its projections into the three-dimensional space to illustrate our theoretical results.
Copyright (c) 2026 Mustafa Altın, Ahmet Kazan
Hom-structures and bihom-structures of antiflexible and pre-antiflexible algebras
UDC 512.554, 512.628 We define and study the structures of Pre-Anti-Flexible algebras in the Hom- and BiHom-cases. More precisely, in the BiHom-case, the corresponding algebraic structure is defined by two products $\triangleleft,$ $\triangleright$ and two linear maps $f$ and $g$ on $A.$ This structure is denoted by $\big(A,\triangleleft,\triangleright,f,g\big)$ and if $f=g,$ then we get the Hom-version denoted by $\big(A,\triangleleft,\triangleright,f\big).$ Our main results can be desribed as follows: we define some algebraic structures and investigate some properties and relationships between the BiHom-Pre-Anti-Flexible algebras and BiHom-Anti-Flexible algebras. Furthermore, we prove that any BiHom-Anti-Flexible algebra equipped with a Rota–Baxter operator defines a BiHom-Pre-Anti-Flexible algebra. Finally, we introduce the notion of Nijenhuis operator in the BiHom setting and demonstrate that a Nijenhuis operator on a BiHom-Anti-Flexible algebra specifies a BiHom-Pre-Anti-Flexible algebra.
anti-flexible algebras, pre-anti-flexible algebras, Rota-Baxter operators, Nijenhuis operators
Copyright (c) 2026 Walid Aloulou, Mansour Jebli
UDC 519.17, 517.98, 519.8 Let $G$ be a simple finite graph and let $\mathcal U_G$ be the corresponding quantum graph. We study the game algebra $C(\mathrm{Qut}(\mathcal U_G))$ of the quantum automorphisms of $\mathcal U_G.$ It is shown that, for the complete graph $K_n,$ the algebra $C(\mathrm{Qut}(\mathcal U_{K_n}))$ is not commutative even for all $n\geq 3,$ unlike $C(\mathrm{Qut}(K_n))=C(S_n^+).$ Moreover, we prove that, for any graph $G$ with $|V(G)|\geq 3,$ the quantum graph $\mathcal U_G$ admits nonlocal symmetry, which means that there exists a perfect quantum no-signaling correlation for the quantum automorphism game for $\mathcal U_G,$ which is not local.
Copyright (c) 2026 Ольга Островська, Василь Островський, Людмила Туровська
Inverse scattering problem for a hyperbolic system of equations on the semiaxis
UDC 517.98, 517.95 We study а scattering problem for the hyperbolic systems of first-order differential equations on the semiaxis under the assumption that all solutions exhibit asymptotic behaviors at infinity corresponding either to incident or to scattered waves. The problem is reduced to finding the solution of the system in the case where all incident waves are known together with a certain homogeneous boundary condition. Under these assumptions, the solution exists and is unique. Therefore, we observe the appearance of a scattering operator on the semiaxis, which transforms all incident waves into scattered waves. For a special case in which the numbers of incident and scattered waves are equal, we establish the detailed properties of the scattering operator, including its factorization into Volterra factors of the opposite polarities. This factorization makes it possible to solve the inverse scattering problem on the semiaxis, i.e., to unambiguously reconstruct all potentials in the system according to the two known scattering operators corresponding to two different boundary conditions.
scattering problem, inverse problem, integral representations of solutions.
Copyright (c) 2026 Леонід Нижник, Нізамеддін Ш. Іскендеров
Even positive-definite hyperbolically convex functions in a Hilbert space. Representations of Hilbert spaces by self-adjoint operators
UDC 517.98 The paper consists of two parts. In the first part, we establish sufficient and necessary conditions for the integral representation of even positive-definite hyperbolically convex (h.c.) functions $k(x),$ $x\in H$. These functions are continuous in the $j$-topology. The positive definiteness of a function is understood in the pointwise sense. The analyzed theorem is a modified version of the Berezansky theorem presented in [Yu. M. Berezansky and I. M. Gali, Ukr. Mat. Zh., 24, № 4, 351–372 (1972)]. The integral representation for some other positive-definite kernels were considered in [O. V. Lopotko and I. I. Rudynskyi, Ukr. Mat. Zh., 34, № 3, 310–312 (1982)] and [O. V. Lopotko, Dop. Akad. Nauk Ukr., Ser. A, 8, 11–13 (1991)]. In the second part, we prove the integral representation for a family of commutative self-adjoint operators $\mathcal{A}_x$ connected by algebraic relations. For this purpose, we construct a rigging (chain) $H_{\kappa}=H_0\supset H_+=L_2\supset \mathcal{D}$ for $x\in H.$ Our proof is based on the integral representation of even positive-definite h.c. functions of infinitely many variables (see [Yu. M. Berezansky and Yu. G. Kondratyev, Spectral methods in infinite-dimensional analysis, Naukova Dumka, Kyiv (1988)] and [O. V. Lopotko, Bukov. Mat. Zh., 11, № 1, 26–38 (2023)]). Some other forms of generalizations of this kind were considered in [Yu. M. Berezansky and A. A. Kalyuzhny, Ukr. Mat. Zh., 36, № 4, 417–421 (1984); A. A. Kurepa, Canad. Math. J., 12, 45–50 (1960)]; and [Yu. S. Samoilenko, Spectral Theory of the Sets of Self-Adjoint Operators, Naukova Dumka, Kyiv (1984)].
integral representation, positively definite kernel, topology, rigging, operators
First boundary-value problem on the semiaxis for an isotropic superdiffusion equation of order greater than one
UDC 517.95, 517.983, 519.21 The Dirichlet problem is studied for a one-dimensional isotropic superdiffusion equation of order $\alpha\in(1;2)$ on the semiaxis. The analytic solutions of this problem are obtained by the mapping method in the classes of continuous Hўоlder functions, which may have an integrable singularity with respect to time. The probability interpretation of the solutions is clarified and the uniqueness of the regular solution is proved on an infinite time interval.
Fractional Laplacian, Riesz fractional derivative operator, random Levy processes, Dirichlet boundary value problem, isotropic superdiffusion equation