In this article, a two-point boundary value problem for an integro-differential equation in which the highestorder derivatives appear in the integral term is considered. The Dzhumabaev parametrization method is applied to solve the problem. The original problem is reduced to an equivalent problem for an integrodifferential equation with parameters. The resulting problem includes an integro-differential equation with parameters, an initial condition, and an additional relation. Conditions for the existence and uniqueness of a solution to the integro-differential equation with parameters are established in terms of the coefficients and kernels of the equation, as well as the boundary functions. An explicit representation of the solution in terms of the parameters is constructed. The unique solvability of the original two-point boundary value problem is established in terms of the initial data. A special case of the integro-differential equation with the highest-order derivative appearing in the integral term, subject to two-point boundary conditions, is also investigated. The Dzhumabaev parametrization method is used to solve the problem. An explicit form of the solution is obtained.
Mathematical modeling of various real-world processes frequently leads to boundary value problems (BVPs) for third-order partial differential equations of mixed and composite types, which have no classical analogues in mathematical physics. Foundational studies by A.V. Bitsadze and M.S. Salakhitdinov first addressed well-posed boundary value problems for degenerate equations of third-order mixed and mixed-composite types. A key approach in their work involved representing the general solution of a composite-type equation as a sum of functions, which proved essential for operators constructed as products of permuted differential operators. Following these foundational contributions, the study of third-order partial differential equations involving Lavrentiev-Bitsadze, Gellerstedt, heat conduction, and string-type operators has been further advanced by both international and Uzbek mathematicians. Despite these developments, boundary value problems for third-order equations of parabolic-hyperbolic and elliptic-hyperbolic types with singular coefficients remain largely unexplored. In this article, we formulate and investigate boundary value problems for a third-order elliptic equation with a singular coefficient. The existence and uniqueness of classical solutions are rigorously proved. A new extremum principle for third-order equations is developed and applied to establish uniqueness. The existence of a solution is reduced to a singular integral equation of normal type, which is regularized using the classical Carleman-Vekua method, leading to an equivalent Fredholm equation of the second kind. The analytical framework is complemented by a numerical scheme that verifies the theoretical results and illustrates the qualitative behavior of solutions near the degenerate boundary. Furthermore, a numerical illustration is provided to demonstrate the stability and smoothness of the obtained solutions even in the presence of singular coefficients. Finally, the potential biomedical relevance of the model is discussed through its application to steady-state diffusion processes in tumor tissues.
This article develops a formal framework for fractal structures within classical first-order model theory. The notion of fractality is reformulated in purely logical terms by replacing metric self-similarity with logical self-similarity induced by elementary endomorphisms of structures. A hierarchy of morphisms is introduced, including endomorphisms, elementary endomorphisms, and endiks, which preserve the truth of formulas in one or both directions. Based on these morphisms, fractal subsets and fractal models are defined via finite families of elementary self-maps. On the syntactic level, fractality is expressed through finite systems of T-elementary syntactic endomorphisms generating a stabilization process called the fractal corridor (a sequence of theories generated by iterated application of syntactic endomorphisms). A compatibility condition between syntactic and semantic fractality is formulated and proved. Using a Henkin-type construction, syntactic operators are lifted to semantic self-maps of a canonical model, yielding fractal completeness. A corresponding compactness theorem is also established. All constructions are carried out within standard first-order logic.
The well-known Hardy inequalities, formulated in both continuous and discrete cases, play an important role in mathematical analysis, differential equations and many other branches of mathematics. The original forms of these inequalities were subsequently extended and generalized in various directions, leading to the development of Hardy inequalities as an independent and significant area of research. A central problem in the theory of weighted inequalities is the characterization of conditions under which inequalities involving Hardy-type operators hold. Many cases of weighted estimates for linear integral Hardy-type operators have been considered, and there is a large number of books and scientific articles on this topic. More recently, considerable attention has been given to iterated Hardy-type operators due to their application in Morrey-type spaces. This paper analyzes a class of operators formed by iterating two operators, one of which involves a kernel satisfying conditions that generalize those considered previously. The study examines Hardy-type inequalities associated with these iterated operators and establishes necessary and sufficient conditions for their validity. The characterization of weighted Hardy inequalities involving iterated operators can now be applied to study of bilinear weighted Hardy-type inequalities.
In this paper, we provide a characterization of the boundedness of positive sublinear operators that are superposition of three operators: Copson, Hardy, and Tandori (supremal) operators defined on the halfaxis from a weighted Lebesgue space Lp(v) to another weighted Lebesgue space L1(w), where v and w are weight functions on the half-axis (0,∞), and 1 < p < ∞. Our characterization is entirely different from existing results in the literature. The motivation for investigating such inequalities stems from the problem of finding a minimal rearrangement invariant space that contains the cones of non-increasing rearrangement of the functions represented by generalized fractional maximal function acting on functions from weighted Lorentz function spaces. More specifically, by obtaining two-sided estimates for the best constant in the corresponding inequality, we derive a characterization of the associate space of minimal rearrangement invariant spaces containing cones of non-increasing rearrangement of generalized fractional maximal function. To achieve this goal, we are using discretization and anti-discretization methods. In particular, we extend existing discretization techniques to handle the operators formed by iterating the Copson, Hardy, and Tandori operators. We first establish a discrete characterization in Theorem 3. Then, applying anti-discretization techniques, we derive a continuous characterization in Theorem 1.
In this paper, we solve the Cauchy problem for a loaded fractional diffusion equation in an infinite strip. The loaded term is defined as the trace of the fractional derivative of the desired solution on a continuous curve lying inside the domain. We consider all three cases of possible distribution of the order of differentiation in the loaded term (µ) and the order of the time-fractional derivative in the principal differential part of the equation (α). In the first case considered (α>µ), the problem under study is reduced to an integral equation. In the second case (α =µ), we obtain a functional equation. In the third case (α <µ), we are dealing with a differential equation. We show that the condition α>µ ensures the unique solvability of the problem under consideration. In the case of an essentially loaded equation (α≤µ), the problem may lose both uniqueness and solvability. In particular, it is shown that if α<µ, then the problem under consideration ceases to be uniquely solvable, and the corresponding homogeneous problem has infinitely many nontrivial solutions. Moreover, in this case, the solvability requires additional conditions that narrow the set of admissible input data.
In this paper, we investigate the interpolation properties of discrete net spaces np,q(M) and examine their applications to the analysis of linear operators acting on these spaces. These spaces are characterized by the property that, for monotonically non-increasing sequences, the norm in np,q(M) coincides with the norm of the discrete Lorentz space lp,q(M). At the same time, unlike Lorentz spaces, these spaces np,q(M) may contain sequences that do not tend to zero, making them suitable for the study of more general function spaces and operator classes. The main result of this paper is an analogue of Marcinkiewicz-type interpolation theorem for discrete net spaces np,q(M), which offers a powerful tool to study the boundedness of linear operators within this framework. By extending classical interpolation techniques to discrete nets, the theorem enables researchers to derive strong-type estimates for operators based on weak-type estimates on local nets. Consequently, this approach provides a unified framework for obtaining boundedness results, demonstrating the utility of discrete net spaces in analyzing operators within harmonic analysis. These findings contribute significantly to understanding the structural properties of discrete net spaces. Furthermore, they introduce innovative tools for applications in harmonic analysis, operator theory, and related mathematical fields where such spaces naturally arise, ultimately paving the way for advanced theoretical developments and broader analytical applications.
We consider the qualitative properties of solutions to a coupled system of nonhomogeneous doubly nonlinear parabolic equations on the whole line with an exponentially varying density. The characteristic features of degeneracy at vanishing values and gradients are analyzed, and the need for weak solutions and reliable comparison estimates is identified and justified. Using a nonlinear splitting method, we construct explicit comparison functions and, on this basis, apply a comparison principle to obtain global existence of nonnegative solutions for sufficiently small initial data in the slow-diffusion regime. In addition, a self-similar reduction is performed via a nonlinear change of variables, which converts the problem into an auxiliary system for similarity profiles. An asymptotic representation of these self-similar solutions is derived, and the dependence of the solution behaviour on the governing parameters is clarified. It is shown how the parameters affect spatial localization and finite-speed propagation, and a Fujita-type criterion is obtained that provides conditions for the existence and nonexistence of global solutions. To support the analytical results, numerical simulations implemented in Python produce solution profiles and graphical illustrations of the nonlinear diffusion dynamics. The computations agree with the qualitative predictions and help visualize the transition between parameter regimes.
This article deals with the problems of constructing and analyzing a collective risk model for an insurance company when the time evolution is defined on a general time scale. The relevance of the study is determined by the need to describe premium accumulation and claim payments occurring at discrete or irregular time instants within a unified analytical framework. The characteristic features of the classical risk model and its extension to time scales are analyzed, and the need to investigate the behavior of the non-ruin probability under such a generalization is identified and justified. On the basis of the study, the authors construct an analogue of the classical model on time scales and derive a dynamic equation for the distribution of the number of claims. An integral equation on a time scale for the non-ruin probability is formulated. Conditions ensuring the correctness of the constructed model are established. It is proved that the nonruin probability defined on a family of time scales converges pointwise to the corresponding probability in the classical continuous-time risk model as the graininess function tends to zero. It is shown that the proposed approach provides a rigorous justification of the transition from discrete to continuous risk models.
We explore the existence and uniqueness criteria for solutions of a Liouville–Caputo fractional differential equation with the nonlinearity containing the unknown function as well as its lower order fractional derivative, and supplemented with a set of nonlocal fractional boundary data with respect to initial and final segments of the given domain. Integral boundary conditions offer an effective approach to model the flow and drag phenomena in arbitrary shaped vessels, heat conduction, biomedical computational fluid dynamics, engineering problems, etc. The notion of segmental type nonlocal fractional integral boundary conditions introduced in this paper is novel and specializes to periodic/anti-periodic boundary data under a suitable choice of the parameters involved in these conditions (see the second last paragraph of Introduction). We apply Krasnosel’ski˘i’s fixed point theorem and Leray-Schauder’s nonlinear alternative to prove two existence results for the problem at hand, while the uniqueness of its solutions is established via Banach’s contraction mapping principle. Examples are constructed for illustrating the obtained results. Our work is useful in the given configuration as it leads to a new direction for research on fractional boundary value problems. The paper concludes with some interesting observations.
In this article, we first establish an algebraic hyperstructure called a ternary Menger hyperalgebra of rank n, where n is a natural number. The algebraic hyperstructure can be regarded as a novel generalization of ternary semihypergroups. In particular, by setting the natural number n equal to 1, the algebraic hyperstructures of ternary Menger hyperalgebras of rank 1 and ternary semihypergroups are the same. And then, we extend some fundamental results on the ternary semihypergroup theory to study on ternary Menger hyperalgebras of rank n including subhyperalgebras and homomorphisms. Moreover, we investigate some interesting algebraic connections among Menger algebras of rank n, Menger hyperalgebras of rank n, ternary Menger algebras of rank n and ternary Menger hyperalgebras of rank n. In this section, we present that the algebraic hyperstructure of ternary Menger hyperalgebras of rank n can also be considered as an extension of the concepts Menger hyperalgebras of rank n and ternary Menger algebras of rank n. Finally, we use algebraic hyperstructures of ternary Menger hyperalgebras of rank n to construct the so-called diagonal ternary semihypergroups of the ternary Menger hyperalgebras of rank n.
In the age of digitalisation and rapid technological progress, it is easy to forget that every computer program, every algorithm, and the very structure of our thinking are founded on mathematical logic. In this abstract yet vitally important field, Kazakhstan has its own recognised authority. In May 2026, Aibat Rafhatovich Yeshkeyev marks his 70th birthday. He is a Doctor of Physical and Mathematical Sciences, Professor, and Professor-Researcher at the Karaganda National Research University named after àcademician Ye.A. Buketov, a man whose work in model theory has built a bridge between Kazakhstani science and the international academic community.
In this paper, we investigate several topological properties of n-inner product spaces with respect to the inner products and norms defined on the quotient spaces we constructed. The construction is carried out with respect to a set of n linearly independent vectors, ensuring a consistent analytical framework. This construction was performed in several ways, each resulting in multiple quotient spaces. On each of these quotient spaces, we defined an inner product, along with the corresponding induced norm. Quotient spaces with similar structures are grouped into equivalence classes, thereby yielding several classes of quotient spaces. Within this framework, several topological aspects, including weak convergence, strong convergence, Cauchy sequences, and completeness, are examined with respect to classes of quotient spaces. Consequently, for each aspect, multiple definitions are formulated relative to these classes. We showed that the various definitions associated with a given topological property are equivalent to one another, regardless of the class of quotient spaces we used. Finally, the minimal number of norms within a given class required for an effective investigation of these properties is determined, thereby contributing to a more efficient and non-redundant analytical framework.
This paper investigates a time-fractional parabolic equation with Zaremba-type boundary conditions. The main objective of the present work is to construct reliable and efficient numerical approximations for such problems. To this end, stable finite difference schemes are developed within a consistent analytical framework. A key result is obtaining a coercive stability estimate for the first-order scheme, which guarantees its consistency and supports its practical use in computations. In addition, both first- and second-order schemes are implemented in the one-dimensional case using a modified Gaussian elimination approach. This implementation simplifies the solution process and improves computational reliability when handling the resulting systems. The behavior of the proposed methods is examined through several numerical experiments designed to reflect different parameter choices and settings. The results demonstrate that the schemes achieve the expected levels of accuracy, consistency, and efficiency. An accompanying error analysis explains the observed outcomes and supports the theoretical findings. The numerical results, presented in tables, show strong agreement with the theoretical predictions, thereby confirming the validity and effectiveness of the proposed approach. These conclusions highlight the practical applicability of the proposed numerical schemes for solving this fractional parabolic problem with mixed boundary conditions.
MA-semirings form a proper subclass of inverse semirings that properly contains both the class of rings and the class of distributive lattices with the least element. In this paper, we study generalized derivations satisfying certain algebraic identities of MA-semirings with involution. The main objective of this research is to investigate identities involving three, two, one generalized derivation in MA-semirings with involution, ensuring commutativity. Hermitian and skew-Hermitian elements are primarily used to formulate the basic tools for the development of this paper and these notions are the fundamental units of the second kind involution. Involution of the second kind plays a key role not only for proving the main results (see Theorems 1, 3, 5) but also it enables us to observe more results from their proofs (see Theorems 2, 4, 6). Since every derivation is a generalized derivation, the results obtained naturally extend a variety of results on derivations. Moreover, several well-established results on derivations of MA-semirings and rings under the similar environment can be concluded as special cases.
The study of matrix operators acting between weighted sequence spaces lp,v and lq,u has become an important direction in functional analysis, particularly due to its close connection with Hardy-type inequalities and the general theory of linear operators on discrete structures. A key problem in this framework is determining when such operators are bounded and obtaining precise value or sharp estimates for their operator norms. Although considerable attention has been devoted to matrix operators whose entries satisfy the so-called Oinarov conditions, including several extensions to broader classes of kernels, the literature still lacks comprehensive norm estimates, especially in the case 1 < q < p < ∞. In this paper, we establish necessary and sufficient criteria for the boundedness of matrix operators with entries satisfying the Oinarov conditions. Furthermore, we provide both lower and upper estimates for their norms. These results not only refine previously known inequalities but also provide new tools for analyzing the structure and behavior of weighted sequence spaces. Applications of our findings include spectral characterization of matrix operators, investigation of oscillatory and non-oscillatory properties of solutions to higher-order difference equations, and the evaluation of sequences via their discrete derivatives.
The stability of differential equations with periodic and quasiperiodic coefficients is a central topic in modern stability theory, with important applications in mechanics, physics, and dynamical systems. A classical result in this area is the Lyapunov integral criterion, which provides stability conditions for linear second-order equations with periodic coefficients. In this paper, we extend this criterion to equations with quasiperiodic coefficients. Our analysis is based on the method of periodic characteristics, which has proven effective in the study of multiperiodic solutions for systems with a diagonal differentiation operator. Within this framework, the multiperiodicity condition is reduced to a functional equation, and a Floquet-type representation of the matricant of the associated system is derived. This representation shows that multiperiodicity of solutions follows from the purely imaginary nature of the characteristic multipliers and the periodicity of the helical characteristics. The obtained results confirm that the Lyapunov integral criterion remains valid for equations with quasiperiodic coefficients. More generally, they demonstrate the effectiveness of the characteristic method for analyzing stability in complex dynamical systems, thereby extending the scope of classical stability theory.
This article discusses the problems of constructing and classifying algebras generated by modular products of cycles. We demonstrate that algebras of binary isolating, used to analyze relationships between binary formulas of a theory, can be naturally interpreted in terms of metric properties of graphs. A characteristic feature of the modular product is that with sufficiently large cycle parameters (m,n > 4), the diameter of such a graph does not exceed three. This makes it possible to define an algebra of binary formulas using only four labels. For small cycle parameters, the presence of simplices is identified and justified. Based on the analysis, we propose a generalized scheme combining modular products of cycles and their extended versions. It is proved that for m,n > 4, the algebra of binary isolating formulas for the theory of Cm∇Cn is isomorphic to the algebra of simplices of corresponding diameter. Explicit Cayley tables are constructed for products involving small cycles (C3–C6), leading to general descriptions of algebras Mo (odd) and Me (even). The proposed approach provides new opportunities for classifying theories and establishing correspondences between algebras and graphs, underlining its relevance for modern model theory and structural combinatorics.
In this paper, a non-polynomial quartic spline technique with a fitting parameter is applied to solve a second-order singularly perturbed differential-difference equation (SPDDE) having small shifts. Taylor series expansion is employed for the delayed and advanced terms in the considered problem to produce a singularly perturbed differential equation (SPDE), and then a non-polynomial quartic spline technique is applied. To manage the layer structure, a fitting parameter is introduced in the proposed computational method; based on the step size, this parameter is evaluated using the theory of singular perturbation theory. Two model examples with left-end boundary layer behavior are considered to theory validate the theoretical finding. The convergence method is analyzed, and the solutions are reported in terms of maximum absolute error with quadratic convergence rate using the fitting parameter. For comparison, solutions without the fitting parameter are reported for test problems. The graphs depict the layer profile for the values of perturbation and shift parameters using the fitting factor and the oscillations without it. The proposed scheme gives uniformly convergent and valid results.
This article deals with the fundamental problems in the mathematical theory of fractional differential equations, specifically focusing on the analytical solvability of boundary value problems in time-dependent domains. The relevance of the study implies the necessity of developing methods for equations with nonlocal operators modeling anomalous diffusion. A one-dimensional diffusion equation containing a RiemannLiouville fractional derivative with respect to time is examined. The characteristic features of the problem, posed in a non-cylindrical domain bounded by a moving linear boundary and a fixed spatial coordinate, are analyzed. The need to handle inhomogeneous boundary data is identified, and the problem is initially reduced to one with homogeneous conditions. On the basis of the study, the author constructs the fundamental solution in a quarter-plane by means of the bilateral Laplace transform and obtains the Green function for the Dirichlet problem. It is shown that the solution can be expressed through an integral representation in terms of a specific boundary density. This density satisfies a Volterra-type integral equation with a weakly singular kernel. Using the contraction mapping principle, it is proved that this equation has a solution. Consequently, the existence of a regular solution to the original boundary value problem is established.
