Editorial backlog

  1. Makhmudov O.I., Niyozov I.E. The Cauchy problem for the system of elasticity theory
    Status: reviewing
    Abstract.
    In this paper, we considered the problem of analytical continuation of the solution of the system equations of the moment theory of elasticity in spacious bounded domain from its values and values of its strains on part of the boundary of this domain, i.e., the Cauchy's problem. The condition of solvability of this problem is considered.
    Date of submission: 28 December 2022 г.


  2. EXISTENCE AND FINITE-TIME STABILITY OF SOLUTIONS FOR A CLASS OF NONLINEAR HILFER FUZZY FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH TIME-DELAYS
    Status: reviewing
    Abstract.
    In the current paper, we investigate a novel class of nonlinear Hil- fer fuzzy fractional stochastic differential equations with time-delays. Firstly, we convert the system under consideration into an analogous integral system. Secondly, using Schauder and Banach fixed point theorem, the existence and uniqueness results of solutions for nonlinear Hilfer fuzzy fractional stochastic differential equations are then established. Additionally, we explore the finite- time stability result of solution for the system under consideration. Lastly, an example is provided to visualize the theoretical results.
    Date of submission: 11 January 2023 г.


  3. Baidaulet A.T., Suleimenov K.M. On embedding in Lorentz spaces (a distant case)
    Status: reviewing
    Abstract.
    The paper studies the upper bound of a non-increasing non-negative function from the space $L^{p}(0,1)$ through the modulus of continuity of variable increment $\omega_{p,\alpha,\psi}(f,\delta)$. It is shown that for an increment of a function of the form $f(x)-f(x+hx^{\alpha}\psi(x))$ in the evaluation of the continuity module will take the form $\omega_{p,\alpha,\psi}\left(f,\frac{\delta}{\delta^{\alpha}\psi\left(\frac{1}{\delta}\right)}\right)$. The embedding of $\tilde H_{p,\alpha,\psi}^\omega\subset L(\mu,\nu)(\mu\not=p)$(distant case) is also studied.
    Date of submission: 29 Aprel 2023 г.


  4. ABSOLUTE SUMMABILITY OF FACTORED FOURIER INTEGRALS
    Status: reviewing
    Abstract.
    In this paper we study the absolute Norlund summability of factored Fourier Integrals. In some sense, our main result is a generalization of a special case of result of S. N. Lal and and A. k. Singh. Also, in particular case our result generalize the result of L. Boitsun. Also, as a corollary, we obtain a result concerning absolute Cesaro summability of factored Fourier integrals.
    Date of submission: 19 May 2023 г.


  5. Tukhliev D.K., Shabozov M.Sh. On the mean-square approximation of functions in the Bergman space and the value of the widths of some classes of functions
    Status: reviewing
    Abstract.
    In this paper studies extremal problems related to the best approximation of functions analytic in the unit circle and belonging to the Bergman space $B_2$. A number of exact theorems are obtained and the values of various n-widths of some classes of functions in $B_2$ are calculated.
    Date of submission: 16 June 2023 г.


  6. On density of polynomials in the algebra of holomorphic functions of exponential type on a linear lie group
    Status: reviewing
    Abstract.
    It is shown by the author in [J. Lie Theory 29:4, 1045–1070, 2019] that for every connected linear complex Lie group the algebra of polynomials (regular functions) is dense in the algebra of holomorphic functions of exponential type. However, the argument is quite involved. Here we present a short proof.
    Date of submission: 20 June 2023 г.


  7. Generalized composition operators on weighted Fock spaces
    Status: reviewing
    Abstract.
    Given analytic functions $g$ and $\Phi$ on the complex plane $\mathbb C$, we characterize bounded and compact properties of generalized composition operators $J_g^\Phi$ and $C^\Phi_g$, induced by $g$ and $\Phi$, on weighted Fock spaces $F^\Psi_g$ with weight function $\Psi$ satisfying some smoothness condition. Moreover, we investigate the Schatten $S_p(F^\Psi_2)$ class membership property of these operators.
    Date of submission: 27 June 2023 г.


  8. Bukusheva A.V., Galaev S.V. Geometry of sub-Riemannian manifolds equipped with a semimetric quarter-symmetric connection
    Status: reviewing
    Abstract.
    The semimetric quarter-symmetric connection on a sub-Riemannian manifold of contact type is introduced. This connection is given by an intrinsic metric connection and two structural endomorphisms, which save the distribution of a sub-Riemannian manifold. Conditions for the metricity of the introduced connection are found. We study structural endomorphisms of a semimetric connection consistent with a sub-Riemannian quasi-statistical structure. We study the properties of a semimetric quarter-symmetric connection defined on a nonholonomic Kenmotsu manifold and on an almost quasi-Sasakian manifold. Conditions are found when these manifolds are Einstein manifolds relative to a quarter-symmetric connection.
    Date of submission: 07 July 2023 г.


  9. Parfenov A.I. Inductive methods for the Hardy inequality on trees
    Status: reviewing
    Abstract.
    We study the two weight Hardy inequality on a rooted tree as well as its versions for trees with boundary and for the family of all dyadic cubes. In the general and diagonal cases, several new inductive criteria for the validity of the Hardy inequality are established. In the lower triangular case, we simplify two known proofs of the criterion due to Arcozzi, Rochberg and Sawyer (2002) which are based on the Marcinkiewicz interpolation theorem and the capacitary criterion, and also give new proofs based on induction, the inductive formula for capacity and the integration by parts formula. For the diagonal case, the last proof yields the optimal constant $p$ which coincides with Bennett's constant in the Hardy inequality for sequences.
    Date of submission: 17 July 2023 г.


  10. Rakhimova A.I. Hypercyclic and chaotic operators in the space of analytic functions in the band
    Status: reviewing
    Abstract.
    This article discusses the space $H(\Omega_r)$ of analytic functions in the band $\Omega_r$, endowed with the standard topology of uniform convergence on compacts of $\Omega_r$. It examines the issues of hypercyclicity, chaoticity and frequently hypercyclicity of differentiation and shift operators by definitions and using classical theorems. The main results of the article are given in Theorems 5, 10 and 11. In Theorem 5 it is proved that the linear continuous operator $T$ in $H(\Omega_r)$ commuting with the differential operator is hypercyclic. Theorem 10 shows that it is chaotic, and Theorem 11 is frequently hypercyclic in $H(\Omega_r)$.
    Date of submission: 10 Avgust 2023 г.


  11. Оценки жесткости кручения выпуклой области через новые геометрические характеристики области
    Status: reviewing
    Abstract.
    В статье введены новые геометрические характеристики выпуклой области с конечной длиной границы и приведен алгоритм их вычисления. Доказан ряд изопериметрических неравенств между новыми функционалами и известными интегральными характеристиками области. Отметим, что некоторые неравенства имеют широкий класс экстремальных областей. Рассмотрены приложения новых характеристик к задаче об оценке жесткости кручения выпуклой области.
    Date of submission: 15 Avgust 2023 г.


  12. Braichev G.G. О нулях и тейлоровских коэффициентах целой функции логарифмического роста
    Status: reviewing
    Abstract.
    В статье для важного класса целых функций нулевого порядка выявляются непосредственные, прямые связи между скоростью стремления к бесконечности последовательности нулей и скоростью стремления к нулю последовательности тейлоровских коэффициентов. Применяя коэффициентную характеризацию роста целых функций и некоторые тауберовы теоремы из выпуклого анализа, мы получаем точные асимптотические оценки, связывающие нули~$\lambda_n$ и спрямленные по Адамару тейлоровские коэффициенты~$\hat{f_n}$ для целых функций логарифмического роста. В ситуациях, когда функция обладает той или иной регулярностью поведения, упомянутые оценки переходят в точные асимптотические формулы. Например, если целая функция имеет регулярный по Борелю рост и точка $a=0$ не является ее борелевским исключительным значением, то при $n\to\infty$ справедливо асимптотическое равенство $\ln |\lambda_n|\sim \ln(\hat{f}_{n-1}/\hat{f_n})$. Результат верен и для функций совершенно регулярного логарифмического роста, причем в~последнем случае дополнительно можно утверждать, что $\ln|\lambda_1\lambda_2\,\ldots\,\lambda_n|\sim\ln\hat{f_n}^{-1}$ при $n\to\infty$.
    Date of submission: 15 Avgust 2023 г.


  13. Durdiev D.K. An undetermined coefficient problem for a mixed equation of parabolic-hyperbolic type with non-local boundary conditions on the characteristics
    Status: reviewing
    Abstract.
    For an equation of a mixed parabolic-hyperbolic type with a characteristic line of type change, we study the inverse problem associated with the search for an unknown coefficient at the lowest term of the parabolic equation. In the direct problem, we consider an analog of the Tricomi problem for this equation with a nonlocal condition on the characteristics in the hyperbolic part and Dirichlet’s conditions in the parabolic part of the domain. To determine undetermined coefficient, with respect to the solution, defined in the parabolic part of the domain, the integral overdetermination condition is specified. Global results on the unique solvability of the inverse problem in the sense of the classical solution are proved.
    Date of submission: 12 September 2023 г.


  14. Krause Mean Processes Generated by Cubic Stochastic Matrices with Positive Influences
    Status: reviewing
    Abstract.
    The Krause mean process serves as a comprehensive model for the dynamics of opinion exchange within multi-agent system wherein opinions are represented as vectors. In this paper, we propose a framework for opinion ex- change dynamics by means of the Krause mean process that is generated by a cubic doubly stochastic matrix with positive influences. The primary objective is to establish a consensus within the multi-agent system.
    Date of submission: 20 October 2023 г.


  15. Well-posedness and Stability result for a Timoshenko system with thermodiffusion effects and time-varying delay term
    Status: reviewing
    Abstract.
    n the actual article, we investigate a Timoshenko beam model with thermodiffusion effects and a time-dependent delay. We show that the problem is well-posed in the sense of C0-semigroup theory. We illustrate the general decay result of the problem’s solution using a suitable Liapunov function.
    Date of submission: 21 October 2023 г.


  16. Haliullin S.G. Extreme points for a total convex structure of generalized states
    Status: reviewing
    Abstract.
    It is well known that the set of states of a certain quantum mechanical system is closed from the point of view of the operational approach if we want to form mixtures or convex combinations. That is, if $s_1$ and $s_2$ are states, then so are $\lambda s_1+(1-\lambda) s_2$, where $0 < \lambda < 1$, must be states. We can define a convex combination of elements in a linear space, but unfortunately, in the general case, linear space is artificial for a set of states and has no physical meaning, but the operation of forming mixtures of states has a natural meaning. For this reason, an abstract definition of mixtures will be given, which does not depend on the concept of linearity. We will call this space a convex structure. The paper will consider state spaces, spaces of generalized states in which pure states, operations, effects associated with operations are distinguished. We will also consider ultraproducts of sequences of these structures, operations and effects.
    Date of submission: 01 November 2023 г.


  17. Shishkin K.A., Gumerov R.N., Lipacheva E.V. A categorical criterion for the existence of universal $C^*$-algebras
    Status: reviewing
    Abstract.
    The article deals with categories which determine universal $C^*$-algebras. These categories are called the compact $C^*$-relations. They were introduced by T.A.~Loring. For a given set $X$, the compact $C^*$-relation on $X$ is the category whose objects are functions from $X$ to $C^*$-algebras and morphisms are $\ast$-homomorphisms of $C^*$-algebras making the appropriate triangle diagrams commute. Moreover, these functions and $\ast$-homo\-mor\-phisms satisfy certain axioms. In this article, we prove that every compact $C^*$-relation is both complete and cocomplete. As an appli\-cation of the completeness of compact $C^*$-relations, we obtain the criterion for the existence of universal $C^*$-algebras.
    Date of submission: 03 November 2023 г.


  18. Кутлымуратов Б.Ж. О некоторых множествах, достаточных для голоморфного продолжения интегрируемых функций с граничным свойством Морера
    Status: reviewing
    Abstract.
    В данной статье рассматриваются интегрируемые функции, заданные на границе ограниченной области $D$ в ${{\mathbb{C}}^{n}}$, $n>1$, и обладающие обобщенным граничным свойством Морера. Исследуется вопрос о существовании голоморфного продолжения таких функций в область \(D\) для некоторых достаточных множеств \(\Gamma \) комплексных прямых.
    Date of submission: 18 December 2023 г.


  19. Podkletnova S.V. A series of boundary value problems for the Euler-Darboux equation with two lines of degeneracy
    Status: reviewing
    Abstract.
    This paper introduces a new partial differential equation which is an extension of the well-known Euler-Poisson-Darboux equation. Based on the proven properties of the solution of the introduced equation, general solutions are found explicitly for various values of parameters, existence and uniqueness theorems are proved. Based on the general solutions of the introduced equation, Cauchy problems and modified Cauchy problems are set and solved in the area of the upper right-angled triangle. Explicit solutions are derived. The existence and uniqueness theorems of the solution of all the tasks are proved.
    Date of submission: 23 December 2023 г.


  20. On the Level Sets of the Generalized Resolvent Norm of Operators Pencils
    Status: reviewing
    Abstract.
    This paper presents a proof demonstrating that the generalized resolvent operator, defined within a Hilbert space, cannot remain constant within any open subset of the resolvent set. We also study the same result over a complex uniformly convex Banach space under certain conditions. These findings extend existing results in the literature.
    Date of submission: 23 December 2023 г.


  21. Napalkov V.V., Nuyatov A.A. On the question of embedding Hilbert spaces with reproducing kernel
    Status: reviewing
    Abstract.
    In this paper, necessary and sufficient conditions for embedding one Hilbert space with a reproducing kernel (RKHS) into another RKHS are obtained.This article is a continuation of the authors' works, in which the problem of the coincidence or equivalence of two RKHS's was studied. An important role is played by the condition of matching two two complete systems of functions with some linear continuous operator, introduced by the authors earlier. The results obtained are illustrated with specific examples.
    Date of submission: 27 December 2023 г.


  22. Nazarov S.A. Different types of localization for eigenfunctions of scalar mixed boundary value problems in thin polyhedra
    Status: reviewing
    Abstract.
    Построена асимптотика собственных пар смешанной краевой задачи для оператора Лапласа в тонком многограннике с параллельными сближенными основаниями и скошенными узкими боковыми гранями. На основаниях назначены условия Дирихле, а на боковых гранях --- условия Дирихле или Неймана, распределение которых по граням, а также углы наклона последних оказывают существенное влияние на поведение собственных функций при истончении области. Обнаружены ситуации, в которых собственные функции распределены вдоль всего многогранника и локализованы около его боковых граней или вершин. Результаты основаны на анализе спектра (точка отсечки, изолированные собственные числа, пороговые резонансы и пр.) вспомогательных задач в полуполосе и четверти слоя со скошенными торцом и боковыми сторонами соответственно. Сформулированы открытые вопросы, относящиеся как к спектральному, так и асимптотическому анализу.
    Date of submission: 05 January 2024 г.


  23. Krivosheev A.S., Krivosheeva O.A. Interpolation and fundamental principle
    Status: reviewing
    Abstract.
    In this paper we study the spaces of analytic functions in convex domains of the complex plane. We consider subspaces of such spaces. These subspaces are invariant with respect to the differentiation operator. The problem of the fundamental principle for the invariant subspace is investigated, i.e. the problem of representation all its elements using a series the eigenfunctions and associated functions of the differentiation operator in this subspace --- exponents and exponential monomials. We give a complete description of the space of sequences of the coefficients of series by which functions from the invariant subspace are represented. The problem of multiple interpolation in spaces of entire functions of exponential type is also studied. The duality of problems of interpolation and fundamental principle is considered. The problem of this duality is completely solved. The duality of the problems of fundamental principle and interpolation for an arbitrary convex domain without any restrictions is established.
    Date of submission: 07 January 2024 г.


  24. Existence of gradient Gibbs measures for the \emph{HC}-Blume-Capel model with a countable number of states on a Cayley tree
    Status: reviewing
    Abstract.
    In this paper, we study the \emph{HC}-Blume-Capel model with a countable set of $\mathbb Z$ spin values on a Cayley tree. This model is determined by a countable set of parameters (i.e., the activity function $\theta>0$, $i\in \mathbb Z$). We solve the model exactly on the Cayley trees of order one and two. We show the co-existence of Gibbs measure and gradient Gibbs measures under the some values of parameters.
    Date of submission: 08 January 2024 г.


  25. Makhrova E.N. On homoclinic points of continuous maps on one-dimensional ramified continua
    Status: reviewing
    Abstract.
    Let $X$ be a dendroid, $f:X\to X$ be a continuous map, $p$ be a periodic point of $f$ in $X$ and let $x$ be a homoclinic point in $X$ to a periodic point $p$. We study the properties of a homoclinic point $x$ and the unstable manifold of a point $p$. We investigate the local structure of a dendroid $X$ under which the existence of a homoclinic point by definition of A.~Poincare implies the existence of a homoclinic point by definition of L.~Block. We also present differences in the properties of homoclinic points and the unstable manifols of periodic points for continuous maps defined on dendroids, dendrites and finite trees.
    Date of submission: 09 January 2024 г.


  26. Two shared sets problem in wider sense on $\mathbb{C}
    Status: reviewing
    Abstract.
    In this manuscript, in view of the introduced definition of weighted sharing of sets in wider sense, we nurture the relation between two meromorphic functions having multiple poles, sharing the zeros of two sets of polynomials, each characterized by distinct zeros. In the applications part of our paper we have further refined our results for a specific class of functions and supported by examples to enhance the coherence of the paper.
    Date of submission: 09 January 2024 г.


  27. On the Convergence of Random Fourier--Jacobi Series in $\mathrm{L}_{[-1,1]}^\mathrm{p}(d\mu_{\zeta,\eta})$ Space
    Status: reviewing
    Abstract.
    On the Convergence of Random Fourier--Jacobi Series in $\mathrm{L}_{[-1,1]}^\mathrm{p}(d\mu_{\zeta,\eta})$ Space
    Date of submission: 11 January 2024 г.


  28. BIHARMONIC HYPERSURFACES IN TERMS OF THE INDUCED METRIC OF TENSOR RICCI
    Status: reviewing
    Abstract.
    The manuscript presents an investigation into biharmonic hypersurfaces specifically focusing on their study within the Sasakian space form using the induced metric of tensor Ricci. We explore the existence of necessary and sufficient condition for biharmonic hypersurface in the context. Furthermore, we demonstrate biharmonic Hopf hypersurface is minimal when the gradient of the mean curvature aligns with the structural vector fields.
    Date of submission: 31 January 2024 г.


  29. A time-dependent inverse source problem for an integro-differential pseudoparabolic equation
    Status: reviewing
    Abstract.
    In this paper, we consider two inverse source problems for a pseudo-parabolic equation with memory. PDEs with a memory, in par- ticular, pseudo-parabolic equations have many applications in the various field of science and technology, since many of mathematical models of them express by such type equations. However, the appearance of such memory term in equation, along with this important physical meaning, in mathe- matical point of view, this causes some mathematical difficulties in both of analytical and numerical studies. The studying here inverse problems consist of finding a time dependent right-hand side coefficient by one of two type integral overdetermination conditions. Under suitable conditions on the data, we establish the existence and uniqueness of strong solutions. Along these analytical results, we also investigate the numerical solutions of these inverse problems. The created numerical algorithms tested by examples.
    Date of submission: 02 February 2024 г.


  30. $(N,\varepsilon)$-PSEUDOSPECTRA OF CLOSED LINEAR OPERATORS ON ULTRAMETRIC BANACH SPACES
    Status: reviewing
    Abstract.
    In this paper, we establish that the essential pseudospectrum of closed linear operator pencils is invariant under perturbation of completely continuous linear operators on ultrametric Banach spaces over a spherically complete field K and we obtain a characterization of the essential pseudospec- trum of a closed linear operator pencils by means of the spectra of any per- turbed completely continuous operators. Furthermore, we introduce and study the concept of (n,ε)-pseudospectrum of closed linear operators and the notion of (n,ε)-pseudospectrum of closed linear operator pencils on ultrametric Ba- nach spaces. We prove some results about them. Finally, several illustrative examples are provided.
    Date of submission: 02 February 2024 г.


  31. Kabanko M.V., Malyutin K.G. Interpolation sets in spaces of functions of finite order in the half-plane
    Status: reviewing
    Abstract.
    Examples of interpolation sets in the space of functions of finite order that are analytic in the upper half-plane are given. These examples are similar to interpolation sets in the space of analytic bounded functions in the upper half-plane.
    Date of submission: 03 February 2024 г.


  32. Reconstruction of the Potential Function of Discontinuous Sturm-Liouville Operator from Spectral Data
    Status: reviewing
    Abstract.
    This paper deals with the inverse spectral problem of the discontinuous Sturm-Liouville operator that is indicated in the way: to determine the potential $q(x)$ and the boundary constant $h$ according to spectral data. Finally, the reconstruction algorithm of the potential $q(x)$ from the spectral data is given.
    Date of submission: 07 February 2024 г.


  33. BOUND FOR CERTAIN HANKEL DETERMINANTS AND THE ZALCMAN CONJECTURE ASSOCIATED WITH MULTIVALENT BOUNDED TURNING FUNCTIONS OF ORDER ALPHA
    Status: reviewing
    Abstract.
    In this paper, we investigate for a sharp upper bound to certain generalized second Hankel determinant, the Zalcman conjecture and an upper bound to the third, fourth Hankel determinants for the class of multivalent analytic bounded turning functions of order α, for α ∈ [0,1). Further, we estimate an upper bound for third and fourth Hankel determinants with respect to two-fold and three-fold symmetric functions belongs to the same class. The practical tools applied in the derivation of our main results are the coefficient inequalities of the Carathéodory class P.
    Date of submission: 09 February 2024 г.


  34. Kudasheva E.G., Menshikova E.B., Khabibullin B.N. Dual construction and the existence of (pluri)subharmonic minorant
    Status: reviewing
    Abstract.
    Рассматривается проблемы существования и построения субгармонической или плюрисубгармонической функции, огибающей снизу функцию на подмножестве в конечномерном вещественном или комплексном пространстве. Такие проблемы естественным образом возникали в теориях равномерных алгебр, потенциала и комплексного потенциала, что нашло отражение в работах Д.~А.~ Эдвардса, Т.~В.~Гамелина, Е.~А.~Полецкого, С.~Бу и В.~Шахермайера, Б. Коула и Т.~Рансфорда, Ф. Ларуссона и Р. Сигурдссона и многих других. В наших работах 1990-х гг. и последних лет было показано, что эти проблемы играют ключевую роль при исследовании нетривиальности весовых пространств голоморфных функций, при описании нулевых множеств и подмножеств функций из таких пространств, в вопросах представления мероморфных функций в виде отношения голоморфных функций с ограничениями на их рост, при изучении аппроксимации экспоненциальными системами в функциональных пространствах и пр. Основные результаты статьи о существовании субгармонической или плюрисубгармонической функции-миноранты выводятся из нашей общей теоретико-функциональной схемы, которая позволяет дать двойственное определение нижней огибающей относительно выпуклого конуса в проективном пределе векторных решёток. Эта схема разрабатывалась нами в последние годы и основана на развитии абстрактной формы выметания. Идеология абстрактного выметания восходит к А. Пуанкаре и М.~В.~Келдышу в рамках выметания мер и субгармонических функций в теории потенциала. Она широко используется в теории вероятности, например, в известной монографии П.~Мейера, а также отражена, зачастую неявно, в монографиях Г.~П.~Акилова, С.~С.~Кутателадзе, А.~М.~Рубинова и др., связанных с теорией упорядоченных векторных пространств и решёток. В нашей статье разработанная нами схема адаптируется для выпуклых подконусов конуса всех субгармонических или плюрисубгармонических функций. Это позволяет получить новые критерии существования субгармонической или плюрисубгармонической миноранты для функций на области.
    Date of submission: 11 February 2024 г.


  35. Conservative numerical scheme for solving the Cahn-Hillard-Navier-Stokes system of equations
    Status: reviewing
    Abstract.
    This article presents a conservative numerical algorithm for solving the Cahn-Hillard-Navier-Stokes system of equations, which is used to simulate multiphase flows. For the Cahn-Hillard equation a numerical scheme has been constructed taking into account the fluid flow velocity. For the Navier-Stokes equation, the control volume method is used, which is modified to take into account the addition of surface forces and a phase field variable to the equation. The implementation of the proposed numerical algorithm is described in detail. The conservativeness of the proposed discrete scheme is shown through numerical simulation. Numerical results indicate the potential usefulness of the proposed method for calculating the dynamics of dispersed systems.
    Date of submission: 20 February 2024 г.


  36. Gladkov A.L. Global and blow-up solutions for a parabolic equation with nonlinear memory under nonlinear nonlocal boundary condition
    Status: reviewing
    Abstract.
    In this paper we consider parabolic equation with nonlinear memory and absorption \begin{equation*} u_t= \Delta u + a \int_0^t u^q (x,\tau) \, d\tau - b u^m, \;x \in \Omega,\;t>0, \end{equation*} under nonlinear nonlocal boundary condition \begin{equation*} u(x,t) = \int_{\Omega}{k(x,y,t)u^l(y,t)}\,dy, \; x\in\partial\Omega, \; t > 0, \end{equation*} and nonnegative continuous initial datum. Here $ a, b,\,q, \,m,\,l $ are positive numbers, $\Omega$ is a bounded domain in $\mathbb{R}^N$ for $N\geq1$ with smooth boundary $\partial\Omega,$ $k(x,y,t)$ is a nonnegative continuous function defined for $x \in \partial \Omega$, $y \in \overline\Omega$ and $ t \ge 0.$ We prove that every solution of the problem is global if $\max (q,l) \leq 1$ or $\max (q,l) > 1$ and $ l < (m + 1)/2, \, q \leq m.$ For $l>\max\{1, (p+1)/2\}$ and positive for small values of $ t$ function $k(x,y,t)$ solutions blow up in finite time for large enough initial data. The obtained results improve previously established conditions for the existence and the absence of global solutions.
    Date of submission: 20 February 2024 г.


  37. Petrosyan G.G. On systems of semilinear fractional differential inclusions with non-densely defined operators in Banach spaces.
    Status: reviewing
    Abstract.
    В настоящей работе изучаются системы полулинейных дифференциальных включений дробных порядков. Предполагается, что линейные части включений представлены операторами Хилле-Иосида в банаховых пространствах. Нелинейные части включений являются многозначными отображениями типа Каратеодори, зависящими от времени и конечного набора функций. Для исследования задачи существования решений такой системы используется теория дробного математического анализа, теория обобщенных метрических пространств, а также теория топологической степени для многозначных уплотняющих отображений. Идея решения поставленной задачи состоит в следующем. Мы представляем разрешающий многозначный оператор для данной системы и описываем его свойства. Показано, в частности, что этот мультиоператор является уплотняющим относительно специальной векторной меры некомпактности. Это дает возможность, применяя некоторые теоремы о неподвижной точке для указанных мультиоператоров, доказать локальную и глобальную теоремы существования интегральных решений данной системы. В последнем случае обосновывается также компактность множества таких решений и полунепрерывная сверху зависимость множества решений от начальных данных.
    Date of submission: 28 February 2024 г.