Uniqueness Theorem Research Articles

Uniqueness for an Inverse Quantum-Dirac Problem with Given Weyl Function

Abstract In this work, we consider a boundary value problem for a q-Dirac equation. We prove orthogonality of the eigenfunctions, realness of the eigenvalues, and we study asymptotic formulas of the eigenfunctions. We show that the eigenfunctions form a complete system, we obtain the expansion formula with respect to the eigenfunctions, and we derive Parseval’s equality. We construct the Weyl solution and the Weyl function. We prove a uniqueness theorem for the solution of the inverse problem with respect to the Weyl function.

Investigation and Methods for Solving Problems of Scattering of Electromagnetic Waves on Three-Dimensional Anisotropic Dielectric Structures

The problems of scattering of electromagnetic waves by anisotropic dielectric structures are studied on the basis of volumetric singular integral equations. Existence and uniqueness theorems for solutions are given for a wide class of media, including lossless media. An efficient method for solving integral equations based on the collocation method and fast discrete Fourier transform algorithms is described.

About the Existence and Uniqueness Theorem of Fuzzy Random Ordinary Differential Equations

Ordinary differential equations that includes stochastic processes in their vector fields are called random ordinary differential equations, that are considered through this work as a Weiner process or also called Brownian motion. In this paper, fuzzy random ordinary differential equations are considered, in which the fuzziness appears in the initial conditions in terms of triangular fuzzy numbers. Such equations are crucial in the theory of random dynamical systems and/or modern control theory and therefore the existence of a unique solution of such equations is of great importance. The statement and the proof of the existence and uniqueness theorem of fuzzy random ordinary differential equations is the main objective of this paper, which is proved using Banach contraction mapping theorem

Inverse Sturm-Liouville problem with conformable derivative and transmission conditions

In this paper, we study the inverse problem for Sturm-Liouville problem with conformable fractional differential operators of order $\alpha$, $0.5 < \alpha\leq 1$ and finite number of interior discontinuous conditions. For this aim first, the asymptotic formulas for solutions, eigenvalues and eigenfunctions of the problem are calculated. Then some uniqueness theorems for proposed inverse eigenvalue problem are proved. Finally, the Hald's theorem for conformable Sturm-Liouville problem is developed.

Strong uniqueness and alternation theorems for relative Chebyshev centers

In this paper, we give a strong uniqueness characterization theorem for the Chebyshev center of a set of infinitely many functions relative to a finite-dimensional linear space on a compact Hausdorff space. Additionally, we derive an alternation theorem for Chebyshev centers relative to a weak Chebyshev space on any compact set of the real line. Furthermore, we show an intrinsic characterization of those linear spaces where an alternation theorem holds.

Reticulation of an integral complete l-groupoid: An axiomatic approach

The reticulation of a unital ring R is a bounded distributive lattice L(A) whose main property is that the Zariski prime spectrum of R is homeomorphic with the Stone prime spectrum of L(A). In this paper we develop an axiomatic theory of reticulation in the abstract framework offered by the integral complete l-groupoids (= icl-groupoids). For an algebraic icl-groupoid A we define axiomatically three basic notions: pre-reticulation, semi-reticulation and reticulation. We prove a uniqueness theorem for the semi-reticulations of A and we characterize the algebraic icl-groupoids that admit a reticulation. Given an algebraic icl-groupoid A, we define the spectral closure of the m-prime spectrum SpecZ(A) of A, an abstraction of the spectral closure of the prime spectrum of a ring. When A is the icl-groupoid Id(R) of ideals in a ring R we obtain some of the Belluce results on the reticulation of R and the spectral closure of the prime spectrum of R.

The local Borg–Marchenko uniqueness theorem of matrix‐valued Dirac‐type operators for coefficients locally smooth at the right endpoint

AbstractWe provide an expression of the Weyl–Titchmarsh matrix associated with a self‐adjoint matrix‐valued Dirac‐type operator defined on [0, b) for . As a concrete application of it, we establish asymptotics of the difference of two Weyl–Titchmarsh matrices corresponding to two Dirac‐type operators , with a.e. on [0, a], in with , when the are sufficiently smooth in a right neighborhood of the point a and their right derivatives at a coincide up to a certain order. In addition to this, we also provide new proofs of the local Borg–Marchenko theorem and asymptotic high‐energy expansion of Weyl–Titchmarsh matrices associated with Dirac‐type operators.

The Inverse Problem of Determining the Coefficients Elliptic Equation

The paper considers the inverse problem in determining unknown coefficients in a linear elliptic equation. Theorems of existence, uniqueness and stability of the solution of inverse problems for a linear equation of elliptic type are proved. Using the method of sequential measurements, a regularizing algorithm is constructed to determine several coefficients.

On recovery of an unbounded bi-periodic interface for the inverse fluid-solid interaction scattering problem

Abstract This paper is concerned with the inverse scattering of acoustic waves by an unbounded periodic elastic medium in the three-dimensional case. A novel uniqueness theorem is proved for the inverse problem of recovering a bi-periodic interface between acoustic and elastic waves using the near-field data measured only from the acoustic side of the interface, corresponding to a countably infinite number of quasi-periodic incident acoustic waves. The proposed method depends only on a fundamental a priori estimate established for the acoustic and elastic wave fields and a new mixed-reciprocity relation established in this paper for the solutions of the fluid-solid interaction scattering problem.

Common best proximity points for a pair of mappings with certain dominating property

Abstract This article introduces a type of dominating property, partially inherited from L. Chen’s, and proves an existence and uniqueness theorem concerning common best proximity points. A certain kind of boundary value problem involving the so-called Caputo derivative can be formulated so that our result applies.

Reconstruction of Two Functions in the Model of Vibrations of a String One End of Which Is Placed in a Moving Medium

The paper considers an inverse problem of determining the coefficients in the model of small transverse vibrations of a homogeneous finite string one end of which is placed in a moving medium and the other is free. The vibrations are simulated by a hyperbolic equation on an interval. One boundary condition has a nonclassical form. Additional data for solving the inverse problem are the values of the solution of the forward problem with a known fixed value of the spatial argument. In the inverse problem, it is required to determine the function in the nonclassical boundary condition and a functional factor on the right-hand side of the equation. Uniqueness and existence theorems for the inverse problem are proved. For the forward problem, conditions for unique solvability are established in a form that simplifies the analysis of the inverse problem. For the numerical solution of the inverse problem, an algorithm is proposed for the stage-by-stage separate reconstruction of the sought-for functions using the method of successive approximations for integral equations.

Controlled polyhedral sweeping processes: Existence, stability, and optimality conditions

This paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis of truncated Hausdorff distances between moving polyhedra, we derive new existence and uniqueness theorems for sweeping trajectories corresponding to various classes of control functions acting in moving sets. Then we establish quantitative stability results, which provide efficient estimates on the sweeping trajectory dependence on controls and initial values. Our final topic, accomplished in finite-dimensional state spaces, is deriving new necessary optimality and suboptimality conditions for sweeping control systems with endpoint constrains by using constructive discrete approximations.

On Some Boundary Value Problems with a Shift for a Mixed Type Equation

Важным этапом в становлении теории краевых задач стали предложенные А.М. Нахушевым в 1969 году нелокальные задачи нового типа, впоследствии названные у нас краевыми задачами со смещением, а за рубежом — задачами (проблемами) Нахушева. Они являются обобщением задачи Трикоми, а так же содержат широкий класс корректных самосопряженных задач. Эти задачи сразу вызвали большой интерес многих авторов. За последние годы исследования задач со смещением для уравнений смешанного типа ведутся особенно интенсивно. Но в этих работах краевые условия, как правило, содержат классические операторы, в то время как нелокальным краевым задачам, содержащим операторы более сложной структуры и операторы дробного интегро-дифференцирования. Настоящая статья посвящена исследованию вопроса однозначной разрешимости краевых задач со смещением для уравнения смешанного эллиптико-гиперболического типа. Сформулированы корректные краевые задачи со смещением для уравнения смешанного типа. В данной работе исследованы вопросы однозначной разрешимости задач со смешением для уравнения смешанного эллиптико-гиперболического типа в смешанной области Ω, ограниченной в полуплоскости y>0 гладкой кривой Жордана, а в полуплоскости y<0 характеристиками уравнения (1). При ограничениях неравенственного типа на известные функции и различных порядках операторов дробного дифференцирования в краевом условии доказаны теоремы единственности. Существование решения задач доказывается путем редукции задач к уравнениям Фредгольма второго рода, безусловная разрешимость которых следует из единственности решения задач. An important stage in the development of the theory of boundary value problems was the proposed by A.M. Nakhushev in 1969, non-local problems of a new type, which were later called in our country boundary value problems with a shift, and abroad – Nakhushev problems (problems). They are a generalization of the Tricomi problem, and also contain a wide class of well-posed self-adjoint problems. These problems immediately aroused great interest of many authors. In recent years, studies of problems with a shift for equations of mixed type have been carried out especially intensively. But in these works, the boundary conditions, as a rule, contain classical operators, while non-local boundary value problems contain operators of a more complex structure and operators of fractional integro-differentiation. In this paper, we study the unique solvability of problems with mixing for an equation of mixed elliptic-hyperbolic type. Under constraints of unequal type on known functions and different orders of fractional differentiation operators in the boundary condition, uniqueness theorems are proved. The existence of a solution to the problems is proved by reducing the problems to Fredholm equations of the second kind, the unconditional solvability of which follows from the uniqueness of the solution to the problems.

On the Adjoint Problem in a Domain with Deviation Out from the Characteristic for the Mixed Parabolic-Hyperbolic Equation with the Fractional Order Operator

В настоящей статье доказана классическая, сильная разрешимость и вольтерровость сопряженной задачи c отходом от характеристики для уравнения смешанного параболо-гиперболического типа с оператором дробного порядка в смысле Герасимова-Капуто. Целью исследования является решение сопряженной задачи для уравнения смешанного параболо-гиперболического типа дробного порядка. Учитывая, свойств операторов дробного порядка найдены сопряженный оператор и применены постановки сопряженной задачи. Для исследования поставленной задачи в параболической частью смешанной области решается первой краевой задачи для уравнения параболического типа дробного порядка в смысле Герасимова-Капуто. Используя, свойств функции Райта получено функциональное соотношение на линии перехода. Точно также решая, задачи Коши гиперболической частью смешанной области находим функциональное соотношение. Следовательно, поставленная задача эквивалентным образом сводится к интегральному уравнению Вольтерра второго рода со слабой особенностью. Согласно теории интегральных уравнений Вольтерра второго рода доказывается однозначной разрешимость полученного уравнения. Кроме того, используя методы операторов интегро — дифференцирования дробного порядка, теории специальных функций, априорных оценок, теория интегральных уравнений доказываются теоремы единственности, существования и вольтерровость сопряженной задачи в области с отходом от характеристики для уравнения смешанного типа дробного порядка. Полученные результаты новые и отличаются от результатов М.А. Садыбекова и А.С. Бердышева. In this article, it was proved the classical, strong solvability and Volterra property of the adjoint problem with departure from the characteristic for an equation of mixed parabolic-hyperbolic type with a fractional order operator in the sense of Gerasimov-Caputo. The aim of the research is to solve the conjugate problem for the equation of a mixed parabolic-hyperbolic type of fractional order. Taking into account the properties of fractional order operators, the adjoint operator is found and the statements of the adjoint problem are applied. To study the formulated problem in the parabolic part of the mixed domain, the first boundary value problem for a parabolic type equation of fractional order in the sense of Gerasimov-Caputo is solved. Using the properties of the Wright function, a functional relation is obtained on the transition line. In the same way, solving the Cauchy problem with the hyperbolic part of the mixed domain, we find a functional relation. Consequently, the problem posed reduces in an equivalent way to a Volterra integral equation of the second kind with a weak singularity. According to the theory of Volterra integral equations of the second kind, the unique solvability of the resulting equation is proved. In addition, using the methods of integro-differentiation operators of fractional order, the theory of special functions, a priori estimates, the theory of integral equations, uniqueness, existence and Volterra theorems for the adjoint problem in a domain with deviation out of the characteristic for a mixed-type equation of fractional order are proved. The results obtained are new and differ from the results of M. A. Sadybekov and A. S. Berdyshev.

Leavitt path algebras of labelled graphs

A Leavitt labelled path algebra over a commutative unital ring is associated with a labelled space, generalizing Leavitt path algebras associated with graphs and ultragraphs as well as torsion-free commutative algebras generated by idempotents. We show that Leavitt labelled path algebras can be realized as partial skew group rings, Steinberg algebras, and Cuntz-Pimsner algebras. Via these realizations we obtain generalized uniqueness theorems, a description of diagonal preserving isomorphisms and we characterize simplicity of Leavitt labelled path algebras. In addition, we prove that a large class of partial skew group rings can be realized as Leavitt labelled path algebras.

Choice of Parameters for Solving Linear Nonclassical Equations of the First Kind

In this paper, we have chosen a regularization parameter for solving the linear Volterra integral equation of the first kind. Various questions for integral equations have been investigated in many papers. Even when the Volterra equation of the first type is an integral equation with an exact output, non-classical equations integrable by the limit are linear, and nonlinear integral equations are linear, and this is due to the need to develop new methods for the uniqueness of their solutions. But in this paper, we have obtained fundamental results for Fredholm integral equations of the first kind, where regularizing operators are constructed according to M. M. Lavrentiev for solving linear integral Fredholm equations. On the basis of the concept of the introduced derivative of a function with respect to an increasing function, linear Volterra integral equations of the first kind were studied. The aim of the study is to construct a regularizing operator and choose a regularization parameter. In the study, we have applied the concept of a derivative with respect to an increasing function, the regularization method according to M. M. Lavrentiev, methods of functional analysis, methods of transformation of equations, methods of integral and differential equations. The parameter for regularization is selected. Regularizing operator according to M. M. Lavrentiev is constructed and a uniqueness theorem is proved. The proposed methods can be used for the study of integral, integral-differential equations such as the Volterra integral equation of the first kind, as well as for the qualitative study of some applied processes in physics, ecology, medicine, geophysics, and the theory of control of complex systems. In connection with the application of integral equations, new areas are developing, for example, in the economic sciences, in some sections of biology, etc. They can be used in the further development of the theory of Volterra integral equations of the first kind. And also, when solving specific applied problems that lead to equations of the first kind.

Inverse Problem of Determining the Unknown Coefficient in the Beam Vibration Equation in an Infinite Domain

For the equation of transverse vibrations of a homogeneous beam, we consider the direct initial value problem in an infinite domain and study the inverse problem of determining the time-dependent beam stiffness coefficient. The solution of the direct problem is given with the help of fundamental solutions, and the existence and uniqueness of this solution are proved. Stability estimates are obtained for the solution of the inverse problem. Existence and uniqueness theorems for the solution of the inverse problem are proved using Banach’s contraction mapping principle.

A Uniqueness Theorem for Orthonormal Spline Series

In this paper we obtain recovery formulas for coefficients of orthonormal spline series by means of its sum, if the subsequence of partial sums of an orthonormal spline series converge in measure and the majorant of the subsequence of partial sums satisfies some necessary condition, provided that the spline system corresponds to a «regular» sequence. Additionally, it is proved that the regularity of the sequence is essential. Ստացել ենք, որ օրթոնորմալ սպլայն շարքերի գործակիցները վերականգնվում են շարքի գումարի միջոցով, եթե շարքի մասնակի գումարների ենթահաջորդականությունը ըստ չափի զուգամետ է, և այդ մասնակի գումարների ենթահաջորդականու- թյան մաժորանտը բավարարում է ինչ-որ անհրաժեշտ պայմանի, և որ այդ սպլայն համակարգին համապատասխանող հաջորդականությունը «ռեգուլյար» է: Ավելին, ապացուցված է, որ հաջորդականության ռեգուլյարությունը կարևոր է։ Получены формулы восстановления коэффициентов ортонормированных сплайн рядов с помощью их суммы, если подпоследовательность частичных сумм ортонормированного сплайн ряда сходится по мере, а мажоранта этой подпоследовательности частичных сумм удовлетворяет некоторому необходимому условию, в том случае, если сплайн система соответствует «регулярной» последовательности. Кроме того, доказывается, что регулярность последовательности является существенной.

Detachment waves in frictional contact: I — A two-mass system

This work studies a mathematical model for the motion of a two mass–spring–damper system with friction, and has a two-fold purpose. The first is to start a mathematical study of the way detachment or slip waves, when the system transits from stick to slip motion, propagate in discrete and continuous systems. The introduction of friction changes the problems into systems of differential set-valued inclusions, which are mathematically interesting, but rather complex. The second is to use such a ‘simple’ system as an example of a very general existence and uniqueness theorem for systems described by set-valued pseudomonotone operators, Kuttler and Shillor (1999) and Andrews et al. (2020).

Forward and Inverse Source Reconstruction Problems for the Equations of Vibrations of a Rectangular Plate

For the equation of vibrations of a rectangular plate, the initial-boundary value and inverse problems of finding the right-hand side (the source of vibrations) are studied. Solutions of the problems are constructed explicitly as sums of series, and the corresponding uniqueness and existence theorems are proved. When substantiating the existence of a solution to the inverse problem of determining the factor on the right-hand side, which depends on spatial coordinates, the problem of small denominators of two natural variables arises, for which estimates of the separation from zero with the corresponding asymptotics are established. These estimates made it possible to prove the existence theorem for this problem in the class of regular solutions by imposing certain smoothness conditions on the given boundary functions.