Global Uniqueness Theorem Research Articles

Local and Global Uniqueness Theorems of the N-th Order Partial Differential Equations

In this paper, we consider inequalities in which the function is an element of n-th partially order space. Local and Global uniqueness theorem of solutions of the n-the order Partial differential equation Obtained which are applications of Gronwall's inequalities.

On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating

Abstract This article surveys some results in the study of Iannelli [Su un modello di Oberbeck-Boussinesq relativo al moto di un fluido viscoso soggetto a riscaldamento, Fisica Matematica, Istituto Lombardo (rend. Sc.) A 121 (1987), 145–191], in which the motion of a viscous, compressible fluid in a two-dimensional domain, subject to heating at the walls, is studied. A global existence and uniqueness theorem for the time-dependent problem is given, and also, under more stringent assumptions, an existence and uniqueness theorem in the stationary case is given. A theorem on the asymptotic behavior for t → ∞ t\to \infty of the time-dependent solutions is proved.

Inverse problem for the system of viscoelasticity in anisotropic media with tetragonal form of elasticity modulus

For the reduced canonical system of integro-differential equations of viscoelasticity, direct and inverse problems of determining the velocity field of elastic waves and the relaxation matrix are considered. The problems are replaced by a closed system of Volterra integral equations of the second kind with respect to the Fourier transform in the variables $x_{1}$ and $x_{2}$ for the solution of the direct problem and unknowns of the inverse problem. Further, the method of contraction mappings in the space of continuous functions with a weighted norm is applied to this system. Thus, we prove global existence and uniqueness theorems for solutions of the problems.

Inverse Problem for Viscoelastic System in a Vertically Layered Medium

In this paper, we consider a three-dimensional system of first-order viscoelasticity equations written with respect to displacement and stress tensor. This system contains convolution integrals of relaxation kernels with the solution of the direct problem. The direct problem is an initial-boundary value problem for the given system of integro-differential equations. In the inverse problem, it is required to determine the relaxation kernels if some components of the Fourier transform with respect to the variables $x_1$ and $x_2$ of the solution of the direct problem on the lateral boundaries of the region under consideration are given. At the beginning, the method of reduction to integral equations and the subsequent application of the method of successive approximations are used to study the properties of the solution of the direct problem. To ensure a continuous solution, conditions for smoothness and consistency of initial and boundary data at the corner points of the domain are obtained. To solve the inverse problem by the method of characteristics, it is reduced to an equivalent closed system of integral equations of the Volterra type of the second kind with respect to the Fourier transform in the first two spatial variables $x_1$, $x_2$, for solution to direct problem and the unknowns of inverse problem. Further, to this system, written in the form of an operator equation, the method of contraction mappings in the space of continuous functions with a weighted exponential norm is applied. It is shown that with an appropriate choice of the parameter in the exponent, this operator is contractive in some ball, which is a subset of the class of continuous functions. Thus, we prove the global existence and uniqueness theorem for the solution of the stated problem.

Fractional crossover delay differential equations of Mittag-Leffler kernel: Existence, uniqueness, and numerical solutions using the Galerkin algorithm based on shifted Legendre polynomials

In the present work, we consider a class of fractional delay differential equations of order ρ with Atangana-Baleanu fractional derivatives in the Caputo sense. We convert our fractional delay problem to Volterra integral equation and used them to establish the local existence theorem using the Arzela-Ascoli theorem and Schauder’s fixed point theorem. After that, the contraction mapping theorem was used to prove the global existence and uniqueness theorem. For a numerical solution, we develop the Galerkin algorithm based on shifted Legendre polynomials to solve the utilized fractional delay problem. This algorithm is based on reducing such problems to those of solving a system of algebraic equations, also we state and prove the convergence and error estimate theorems. Four numerical examples, two linear and two nonlinear are presented to test the efficiency and accuracy of the proposed algorithm with some tables and figures to compare our results with the exact solutions. Several, conclusions, recommendations, inductions, and highlights were formulated in the last chapter to extinguish the perfection of the work.

Nonlinear Integral Equations with Potential-Type Kernels in the Nonperiodic Case

We find conditions under which a generalized potential-type operator acts continuously from a Lebesgue space with a general weight to its dual space and possesses the positivity property. Based on these conditions, we prove the global existence and uniqueness theorems for various classes of nonlinear integral equations of convolution type in real weighted Lebesgue spaces using the method of monotonic (in the Browder–Minty sense) operators. Also we obtain estimates of the norms of solutions, which imply that the corresponding homogeneous equations have only a trivial solution.

Simultaneous recovery of a locally rough interface and the embedded obstacle with its surrounding medium

Consider the problem of scattering of time-harmonic point sources by an infinite locally rough interface with bounded obstacles embedded in the lower half-space. The model problem is first reduced to an equivalent integral equation formulation defined in a bounded domain, where the well-posedness is obtained in L p by the classical Fredholm theory. Then a global uniqueness theorem is proved for the inverse problem of recovering the locally rough interface, the embedded obstacles and the wave number in the lower-half space by means of near-field measurements above the interface.

Mixed Neutral Caputo Fractional Stochastic Evolution Equations with Infinite Delay: Existence, Uniqueness and Averaging Principle

The aim of this article is to consider a class of neutral Caputo fractional stochastic evolution equations with infinite delay (INFSEEs) driven by fractional Brownian motion (fBm) and Poisson jumps in Hilbert space. First, we establish the local and global existence and uniqueness theorems of mild solutions for the aforementioned neutral fractional stochastic system under local and global Carathéodory conditions by using the successive approximations, stochastic analysis, fractional calculus, and stopping time techniques. The obtained existence result in this article is new in the sense that it generalizes some of the existing results in the literature. Furthermore, we discuss the averaging principle for the proposed neutral fractional stochastic system in view of the convergence in mean square between the solution of the standard INFSEEs and that of the simplified equation. Finally, the obtained averaging theory is validated with an example.

Задача определения ядер в двумерной системе уравнений вязкоупругости

For a two-dimensional system of integro-differential equations of viscoelasticity in an isotropic medium, the direct and inverse problems of determining the stress vector and particle velocity, as well as the diagonal hereditarity matrix, are studied. First, the system of two-dimensional viscoelasticity equations was transformed into a system of first-order linear equations. The thus composed system of first-order integrodifferential equations with the help of its special matrix was reduced to a normal form with respect to time and one of the spatial variables. Then, using the Fourier transform with respect to another spatial variable and integrating over the characteristics of the equations based on the initial and boundary conditions, it was replaced by a system of Volterra integral equations of the second kind, equivalent to the original problem. An existence and uniqueness theorem for the solution of the direct problem is given. To solve the inverse problem using the integral equations of the direct problem and additional conditions, a closed system of integral equations for unknown functions and some of their linear combinations is constructed. Further, the contraction mapping method (Banach principle) is applied to this system in the class of continuous functions with an exponential weighted norm. Thus, we prove the global existence and uniqueness theorem for the solutions of the stated problems. The proof of the theorems is constructive, i.e. with the help of the obtained integral equations, for example, by the method of successive approximations, a solution to the problems can be constructed.

Extrinsic Black Hole Uniqueness in Pure Lovelock Gravity

We define a notion of extrinsic black hole in pure Lovelock gravity of degree $k$ which captures the essential features of the so-called Lovelock-Schwarzschild solutions, viewed as rotationally invariant hypersurfaces with null $2k$-mean curvature in Euclidean space $\mathbb R^{n+1}$, $2\leq 2k\leq n-1$. We then combine a regularity argument with a rigidity result by Ara\'ujo-Leite to prove, under a natural ellipticity condition, a global uniqueness theorem for this class of black holes. As a consequence we obtain, in the context of the corresponding Penrose inequality for graphs established by Ge-Wang-Wu, a local rigidity result for the Lovelock-Schwarzschild solutions.

Basic theorem and global exponential stability of differential–algebraic neural networks with delay

A differential–algebraic neural network (DANN) with delay (DDANN) is proposed. Firstly, the global existence and uniqueness theorems are established for a DDANN, respectively. Next, a new differential–algebraic inequality is established. Then, a theorem on global exponential stability of DDANN is shown by using this inequality. As an application of DDANN, a very concise criterion on global exponential stability for a neutral-type neural network is given by using DDANNs. Finally, two examples are given to illustrate the theoretical results.

Applications of the Bielecki renorming technique

The renorming technique allows one to apply the Banach Contraction Principle for maps which are not contractions with respect to the original metric. This method was invented by Bielecki and manifested in an extremely elegant proof of the Global Existence and Uniqueness Theorem for ODEs. The present paper provides further extensions and applications of Bielecki’s method to problems stemming from functional analysis and from the theory of functional equations.

Memory kernel reconstruction problems in the integro‐differential equation of rigid heat conductor

The inverse problems of determining the energy–temperature relation α(t) and the heat conduction relation k(t) functions in the one‐dimensional integro‐differential heat equation are investigated. The direct problem is the initial‐boundary problem for this equation. The integral terms have the time convolution form of unknown kernels and direct problem solution. As additional information for solving inverse problems, the solution of the direct problem for x = x0 is given. At the beginning, an auxiliary problem, which is equivalent to the original problem, is introduced. Then the auxiliary problem is reduced to an equivalent closed system of Volterra‐type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the continuous class of functions with weighted norms, we prove the main result of the article, which is a global existence and uniqueness theorem of inverse problem solutions.

Recovering a potential in damped wave equation from Neumann-to-Dirichlet operator

The inverse coefficient problem of recovering the potential q(x) in the damped wave equation , (x, t) ∈ Ω T ≔ (0, ℓ) × (0, T) subject to the boundary conditions r(0)u x (0, t) = f(t), u(ℓ, t) = 0, from the Dirichlet boundary measured output ν(t) ≔ u(0, t), t ∈ (0, T] is studied. A detailed microlocal analysis of regularity of the direct problem solution in the subdomains defined by the characteristics as well as along these characteristics is provided. Based on this analysis, necessary regularity results and energy estimates are derived. It is proved that the Dirichlet boundary measured output uniquely determines the potential q(x) in the interval [0, h(T/2)] and this solution belongs to C(0, h(T/2)) with T < T*, where h(z) is the root of the equation , . Moreover, the global uniqueness theorem is proved. Compactness, invertibility and Lipschitz continuity of the Neumann-to-Dirichlet operator , Φ f [q](t) ≔ u(0, t; q) is proved. This allows us to prove an existence of a quasi-solution of the inverse problem defined as a minimum of the Tikhonov functional as well as its Fréchet differentiability. An explicit formula for the Fréchet gradient is derived by making use of the unique solution to corresponding adjoint problem. The proposed approach is leads to very effective gradient based computational identification algorithm.

Existence and Uniqueness of Chern Connection in the Klein-Grifone Approach

The Klein-Grifone approach to global Finsler geometry is adopted. A global existence and uniqueness theorem for Chern connection is formulated and proved. The torsion and curvature tensors of Chern connection are derived. Some properties and the Bianchi identities for this connection are investigated. A concise comparison between Berwald, Cartan and Chern connections is presented.

Iснування l-го моменту розв’язку стохастичних динамiчних систем Iто-Скорохода випадкової структури з зовнiшнiми збуреннями та нескiнченною пiслядiєю

In this article the definition of strong solution of Ito-Skorokhod stochastic dynamic systems of random structure with external disturbances and all prehistory is presented, important inequalities, which are used to prove Existence and Uniqueness theorems are proved. Global Existence and Uniqueness theorem is proved.

Global solvability of massless Dirac–Maxwell systems

We consider the Cauchy problem for massless Dirac–Maxwell equations on an asymptotically flat background and give a global existence and uniqueness theorem for initial values small in an appropriate weighted Sobolev space. The result can be extended via analogous methods to Dirac–Higgs–Yang–Mills theories.

Isotropic two-dimensional pseudo-Riemannian metrics uniquely constructed by a given curvature

Abstract We prove a global uniqueness theorem of reconstruction of a two-dimensional pseudo-metric by a given Gaussian curvature.

Global solvability of the initial boundary value problem for a model system of one-dimensional equations of polytropic flows of viscous compressible fluid mixtures

We consider the initial boundary value problem for a model system of one-dimensional equations which describe unsteady polytropic motions of a mixture of viscous compressible fluids. We prove the global existence and uniqueness theorem for the strong solution without restrictions on the structure of the viscosity matrix except standard properties of symmetry and positiveness.

The Cauchy problem of a fluid‐particle interaction model with external forces

In this paper, we consider the Cauchy problem of a fluid‐particle interaction model with external forces. We first construct the asymptotic profile of the system. The global existence and uniqueness theorem for the solution near the profile is given. Finally, optimal decay rate of the solution to the background profile is obtained by combining the decay rate analysis of a linearized equation with energy estimates for the nonlinear terms. The main method used in this paper is the energy method combining with the macro‐micro decomposition.