Uniqueness Theorem Research Articles

Some fixed point and stability results in $ b $-metric-like spaces with an application to integral equations on time scales

<abstract><p>This paper presents the stability theorem for the $ T $-Picard iteration scheme and establishes the existence and uniqueness theorem for fixed points concerning $ T $-mean nonexpansive mappings within $ b $-metric-like spaces. The outcome of our fixed point theorem substantiated the existence and uniqueness of solutions to the Fredholm-Hammerstein integral equations defined on time scales. Additionally, we provided two numerical examples from distinct time scales to support our findings empirically.</p></abstract>

Existence and Uniqueness Theorems for the Pfaff Equation with Continuous Coefficients

Existence and Uniqueness Theorems for the Pfaff Equation with Continuous Coefficients

Finite-time stability and uniqueness theorem of solutions of nabla fractional $ (q, h) $-difference equations with non-Lipschitz and nonlinear conditions

<abstract><p>In this paper, the discrete $ (q, h) $-fractional Bihari inequality is generalized. On the grounds of inequality, the finite-time stability and uniqueness theorem of solutions of $ (q, h) $-fractional difference equations with non-Lipschitz and nonlinear conditions is concluded. In addition, the validity of our conclusion is illustrated by a nonlinear example with a non-Lipschitz condition.</p></abstract>

Boltzmann-type kinetic equations and discrete models

The known non-linear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim-Uehling-Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables $F(x,y;v,w)$. The function $F$ is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the kinetic equations mentioned above correspond to different forms of the function (polynomial) $F$. Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas which are similar to those used for the construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similar to the Boltzmann $H$-function. The existence and uniqueness theorem for global in time solution of the Cauchy problem for these models is proved. Moreover, it is proved that the solution converges to the equilibrium solution when time goes to infinity. The properties of the equilibrium solution and the connection with solutions of the wave kinetic equation are discussed. The problem of the approximation of the Boltzmann-type equation by its discrete models is also discussed. The paper contains a concise introduction to the Boltzmann equation and its main properties. In principle, it allows one to read the paper without any preliminary knowledge in kinetic theory. Bibliography: 61 titles.

A uniqueness theorem for inverse problems in quasilinear anisotropic media II

We study the question of whether one can uniquely determine scalar quasilinear conductivity in an anisotropic medium by making voltage and current measurements at the boundary. We prove a global uniqueness in the $ C^{2,\alpha} $ category, by showing that the $ C^{2,\alpha} $ quasilinear conductivity in an anisotropic medium can be uniquely determined by the voltage and current measurements at the boundary, i.e., by the Dirichlet to Neumann map, assuming that an anisotropic linear conductivity can be identified by its Dirichlet to Neumann map up to a diffeomorphism that fixes the boundary.

Uncertainty distributions of solutions to nabla Caputo uncertain difference equations and application to a logistic model

<p>The nabla fractional-order uncertain difference equation with Caputo-type was analyzed in this article. To begin, the existence and uniqueness theorem of solutions for nabla Caputo uncertain difference equations with almost surely bounded uncertain variables was presented. Furthermore, the uncertainty distributions of the solutions for the proposed equations were obtained by establishing a connection between the solutions of equations and their $ \alpha $-paths based on new comparison theorems. Finally, an application of the uncertain difference equations in a logistic population model involving Allee effect was provided and examples were performed to demonstrate the validity of the theoretical results presented.</p>

On uniqueness for series in the general Franklin system

We prove some uniqueness theorems for series in general Franklin systems. In particular, for series in the classical Franklin system our result asserts that if the partial sums $S_{n_i}(x)=\sum_{k=0}^{n_i}a_kf_k(x)$ of a Franklin series $\sum_{k=0}^{\infty}a_kf_k(x)$ converge in measure to an integrable function $f$ and $\sup_i|S_{n_i}(x)|<\infty$, for $x\notin B$, where $B$ is some countable set and $\sup_i(n_i/n_{i-1})<\infty$, then this is the Fourier-Franklin series of $f$. Bibliography: 29 titles.

О решении смешанной задачи для уравнения колебаний движущегося вязкоупругого полотна

A model initial boundary value problem of small transverse oscillations of a viscoelastic moving web with a hinged condition of fastening is considered. The vibrations of such a canvas are described by a linear differential equation of the 5th order in a spatial variable with constant coefficients. It is worth noting that the equation includes mixed derivatives of the desired function both with respect to the spatial variable and with respect to time. The paper describes a technique for constructing a solution in the form of a functional series based on a system of basis functions. To solve the initial-boundary value problem under the additional condition of conservation of energy, a condition is obtained that ensures the uniqueness of the solution. A special class of functions for which the uniqueness theorem holds is explicitly described.

Boltzmann-type kinetic equations and discrete models

The known non-linear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim-Uehling-Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this end we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables $F(x,y;v,w)$. The function $F$ is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the kinetic equations mentioned above correspond to different forms of the function (polynomial) $F$. Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas which are similar to those used for the construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similar to the Boltzmann $H$-function. The existence and uniqueness theorem for global in time solution of the Cauchy problem for these models is proved. Moreover, it is proved that the solution converges to the equilibrium solution when time goes to infinity. The properties of the equilibrium solution and the connection with solutions of the wave kinetic equation are discussed. The problem of the approximation of the Boltzmann-type equation by its discrete models is also discussed. The paper contains a concise introduction to the Boltzmann equation and its main properties. In principle, it allows one to read the paper without any preliminary knowledge in kinetic theory. Bibliography: 61 titles.

A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons

A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons

Stability of the time-dependent identification problem for a fractional Telegraph equation with the Caputo derivative

The telegraph equation , in a Hilbert space is investigated. Here is a self-adjoint, positive operator, is the Caputo derivative. An inverse problem is considered in which, along with , also a time varying factor of the source function is unknown. To solve this inverse problem, we take the additional condition with an arbitrary bounded linear functional .Existence and uniqueness theorem for the solution to the problem under consideration is proved. Inequalities of stability are obtained.

FIXED POINT RESULT USING TWO DOMINATED MAPPING ON A CLOSED BALL

In the present work, we obtain the result of fixed-point theorem through quasi-contraction mapping and improve the work of A. Shoaib et.al (2015) in a left and right K-sequentially 0-complete ordered quasi-partial spaces respectively, where locally contractive condition satisfied on a closed set. We can use this result to solve the complication of computer algorithms and study it. In this paper, some fixed-point results of self-mapping which is defined on quasi partial metric spaces are given by using dominated mapping (A. Shoaib et.al, 2015) in a left and right K-sequentially 0-complete ordered quasi-partial spaces respectively. Where locally contractive condition satisfied on a closed set. And by taking advantage of these results, the necessary conditions for self-mappings on quasi partial metric spaces in quasi contraction are investigated and prove existence and uniqueness theorem of fixed point for contraction mapping. We can use this result to solve the complication of computer algorithms and study it.

KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY

This paper studies the inverse problem of determining a multidimensional kernel function of an integral term which depends on the time variable \(t\) and \((n-1)\)-dimensional space variable \(x'= \left(x_1,\ldots, x_ {n-1}\right)\) in the \(n\)-dimensional diffusion equation with a time-variable coefficient at the Laplacian of a direct problem solution. Given a known kernel function, a Cauchy problem is investigated as a direct problem. The integral term in the equation has convolution form: the kernel function is multiplied by a solution of the direct problem's elliptic operator. As an overdetermination condition, the result of the direct question on the hyperplane \(x_n = 0\) is used. An inverse question is replaced by an auxiliary one, which is more suitable for further investigation. After that, the last problem is reduced to an equivalent system of Volterra-type integral equations of the second order with respect to unknown functions. Applying the fixed point theorem to this system in Hölder spaces, we prove the main result of the paper, which is a local existence and uniqueness theorem.

What is a real black hole?

Abstract Four dimensional Einstein equation with non-vanishing energy momentum tensor is solved to give a solution where the event horizon of a celestial body is outside of the body, which corresponds to a black hole by definition. There is no singularity at the center of the object unlike the case of Schwarzschild solution. We do not have the gravitational collapse. We do not have the gravitational collapse thus avoiding the Penrose-Hawking singularity theorems. The main feature of our black hole solution is that it has the horizon inside the black hole in addition to the usual outside horizon. This inner horizon singularity is not removable by a coordinate transformation because the determinant of metric tensor is singular at this point but there is no singularity neither in the density nor in the pressure indicating there is no gravitational collapse.

Characterization of the null energy condition via displacement convexity of entropy

AbstractWe characterize the null energy condition for an ‐dimensional, time‐oriented Lorentzian manifold in terms of convexity of the relative ‐Renyi entropy along displacement interpolations on null hypersurfaces. More generally, we also consider Lorentzian manifolds with a smooth weight function and introduce the Bakry–Emery ‐null energy condition that we characterize in terms of null displacement convexity of the relative ‐Renyi entropy. As application we then revisit Hawking's area monotonicity theorem for a black hole horizon and the Penrose Singularity Theorem from the viewpoint of this characterization and in the context of weighted Lorentzian manifolds.

A generalization of uniqueness theorems of meromorphic mappings from complete Kähler manifolds

In this paper, we generalize and extend some recent unicity theorems for meromorphic mappings of a complete Kähler manifold into [Formula: see text] sharing few hyperplanes as well as hypersurfaces in subgeneral position without counting multiplicity, where all zeros with multiplicities more than a certain number do not need to be counted. Further, the assumption of nondegeneracy of the mappings is omitted in some of our results.

Vortex-type equations on compact Riemann surfaces

In this paper, we prove a priori estimates for some vortex-type equations on compact Riemann surfaces. As applications, we recover existing estimates for the vortex bundle Monge-Ampère equation, prove an existence and uniqueness theorem for the Calabi-Yang-Mills equations on vortex bundles and get estimates for J-vortex equation. We prove an existence and uniqueness result relating Gieseker stability and the existence of almost Hermitian Einstein metrics, i.e., a Kobayashi-Hitchin type correspondence. We also prove Kählerness of the negative of the symplectic form which arises in the moment map interpretation of the Calabi-Yang-Mills equations in [9].

On the packing/covering conjecture of infinite matroids

The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem of Edmonds and Fulkerson. A packing for a family (Mi:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({M_i}:i \\in \\Theta)$$\\end{document} of matroids on the common edge set E is a system (Si:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({S_i}:i \\in \\Theta)$$\\end{document} of pairwise disjoint subsets of E where Si is panning in Mi. Similarly, a covering is a system (Ii: i ∈ Θ) with ∪i∈ΘIi=E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\cup _{i \\in \\Theta}}{I_i} = E$$\\end{document} where Ii is independent in Mi. The conjecture states that for every matroid family on E there is a partition E=Ep⊔Ec\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E = {E_p} \\sqcup {E_c}$$\\end{document} such that (Mi↾Ep:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({M_i}\\upharpoonright{E_p}:i \\in \\Theta)$$\\end{document} admits a packing and (Mi.Ec:i∈Θ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({M_i}.{E_c}:i \\in \\Theta)$$\\end{document} admits a covering. We prove the case where E is countable and each Mi is either finitary or cofinitary. To do so, we give a common generalisation of the singular matroid intersection theorem of Ghaderi and the countable case of the Matroid Intersection Conjecture by Nash-Williams by showing that the conjecture holds for countable matroids having only finitary and cofinitary components.

“The theorem says…”: Engineering students making meaning of solutions to Ordinary Differential Equations

There is a need for further studies on students’ learning of Differential Equations (DEs), especially in advanced undergraduate and graduate courses. Research on the mathematical education of engineers shows a conflict between students’ demands for practical, contextualized pedagogies and the need for abstract reasoning and appropriate use of mathematical results. Few papers focus on engineering students’ interpretation of theorems and their use as tools in argumentation and problem-solving. This paper takes a sociocultural stance on learning and employs dialogical inquiry – a methodology rooted in Bakhtinian theory, newly developed for collaborative inquiry and qualitative data analysis – to investigate the meanings that senior engineering students made while working on a task designed to evaluate their understanding of Existence and Uniqueness Theorems (EUTs) of solutions of DEs. We identified two important epistemological disconnections that explain the difficulties that some of our students faced in making meaning of solutions of DEs and the EUT.

Regularity of the Pressure Function for Weak Solutions of the Nonstationary Navier–Stokes Equations

We study the nonstationary system of Navier–Stokes equations for an incompressible fluid. Based on a regularized problem that takes into account the relaxation of the velocity field into a solenoidal field, the existence of a pressure function almost everywhere in the domain under consideration for solutions in the Hopf class is substantiated. Using the proposed regularization, we prove the existence of more regular weak solutions of the original problem without smallness restrictions on the original data. A uniqueness theorem is proven in the two-dimensional case