Uniqueness Theorem Research Articles

Implications of Raychaudhuri equation and geodesic focusing in interacting two fluid systems

The present work analyses Raychaudhuri equation (RE) in Einstein gravity for two interacting fluids. Focusing theorem, a consequence of RE, is directly related to the singularity theorems of Penrose and Hawking. In the present work, the signature of the Raychaudhuri scalar or convergence scalar has been examined for two types of interaction forms namely, (i) Q=H(αρ1+βρ2) (linear) and (ii) Q=Hξρ1ρ2αρ1+βρ2 (non-linear). Finally, the phenomena of focusing of a congruence of geodesics and possible avoidance of singularity have been discussed based on the nature of the two fluids and the interaction term operating between them.

Criterion of the Existence of a Strongly Continuous Resolving Family for a Fractional Differential Equation with the Hilfer Derivative

In the qualitative theory of differential equations in Banach spaces, the resolving families of operators of such equations play an important role. We obtained necessary and sufficient conditions for the existence of strongly continuous resolving families of operators for a linear homogeneous equation resolved with respect to the Hilfer derivative. These conditions have the form of estimates on derivatives of the resolvent of a linear closed operator from the equation and generalize the Hille–Yosida conditions for infinitesimal generators of C0-semigroups of operators. Unique solvability theorems are proved for the corresponding inhomogeneous equations. Illustrative examples of the operators from the considered classes are constructed.

A New Method for the Exact Controllability of Linear Parabolic Equations

This work solves the exact controllability to zero in the final time for a linear parabolic problem when the control only acts in a part of the spatial domain. Specifically, it is proved, by compactness arguments, the existence of a partially distributed control. The lack of regularity in the problem prevents the use of standard techniques in this field, that is, Carleman’s inequalities. Controlling a parabolic equation when the diffusion is discontinuous and only acts in a part of the domain is interesting, for example, as in the spreading of a brain tumor. The proof is based on a new maximum principle in the final time; in a linear parabolic equation, with a right-hand side that changes sign in a certain way, and an initial datum of a constant sign, the solution at the final time has the same sign as the initial datum. As a consequence of the exact control result, we prove a unique continuation theorem when the data are not regular.

Common fixed-point theorems on complex partial b- metric space

In this paper, we establish some unique common fixed-point theorems for two pairs of weakly compatible mappings in complex partial b-metric space which generalize and improves several well- known common fixed point results in partial b-metric space.

Equivariant Infinite Loop Space Theory: The Space Level Story

We rework and generalize equivariant infinite loop space theory, which shows how to construct G G -spectra from G G -spaces with suitable structure. There is a classical version which gives classical Ω \Omega - G G -spectra for any topological group G G , but our focus is on the construction of genuine Ω \Omega - G G -spectra when G G is finite. We also show what is and is not true when G G is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical G G -spectra for general G G but fails for genuine G G -spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature.

Uniqueness theorems in the steady vibration problems of the Moore–Gibson–Thompson thermoporoelasticity

Abstract In this paper, the linear model of Moore–Gibson–Thompson thermoporoelasticity is considered and the governing equations of motion and steady vibrations are given. The basic system of equations of steady vibrations with respect to the displacement vector, the changes of temperature and fluid pressure are proposed. Then the radiation conditions are established and Green’s first identity is obtained. Finally, on the basis of this identity, the uniqueness theorems for classical solutions of the boundary value problems of steady vibrations in the theory of MGT thermoporoelaticity are proved.

Rigidity Aspects of Penrose’s Singularity Theorem

In this paper, we study rigidity aspects of Penrose’s singularity theorem. Specifically, we aim to answer the following question: if a spacetime satisfies the hypotheses of Penrose’s singularity theorem except with weakly trapped surfaces instead of trapped surfaces, then what can be said about the global spacetime structure if the spacetime is null geodesically complete? In this setting, we show that we obtain a foliation of MOTS which generate totally geodesic null hypersurfaces. Depending on our starting assumptions, we obtain either local or global rigidity results. We apply our arguments to cosmological spacetimes (i.e., spacetimes with compact Cauchy surfaces) and scenarios involving topological censorship.

A uniqueness theorem for Sturm-Liouville equations with a spectral parameter nonlinearly contained in the boundary condition

In this work, we consider the inverse scattering problem for Sturm-Liouville equation on the semi-infinite interval with a nonlinear spectral parameter in the boundary condition. The scattering data of the problem is defined and the properties of the scattering data are investigated. The fundamental equation is obtained and uniqueness of the algorithm to the potential with given scattering data is studied.

NAYMARK PROBLEM FOR AN ORDINARY DIFFERENTIAL EQUATION WITH A FRACTIONAL DISCRETE DISTRIBUTED DIFFERENTIATION OPERATOR

For an ordinary differential equation with a fractional discretely distributed differentiation operator, the Naimark problem is studied, where the boundary conditions are specified in the form of linear functionals. This allows us to cover a fairly wide class of linear local and nonlocal conditions. A necessary and sufficient condition for the unique solvability of the problem is obtained. A representation of the solution to the problem under study is found in terms of special functions. The theorem of existence and uniqueness of the solution is proven.

THE EXISTENCE AND UNIQUENESS THEOREMS FOR STOCHASTIC DIFFERENTIAL-DIFFERENCE HYBRID SYSTEMS

A stochastic differential-difference hybrid system is a system of interacting variables whose dynamics are described by stochastic differential equations for some of them and difference equations for others. Systems with two types of difference equations are examined: first, a difference equation in the form of a process involving the multiplicative Wiener process, and second, a difference equation with delay. The existence and uniqueness theorems for both systems have been proven. The basic conditions on the system’s parameters are local Lipschitz conditions and linear growth order.

Black Hole and Equipotential Photon Surface Uniqueness in Four-Dimensional Asymptotically Flat Electrostatic Electro-Vacuum Spacetimes

AbstractWe study four-dimensional asymptotically flat electrostatic electro-vacuum spacetimes with a connected black hole, photon sphere, or equipotential photon surface inner boundary. Our analysis, inspired by the potential theory approach by Agostiniani–Mazzieri, allows to give self-contained proofs of known uniqueness theorems of the sub-extremal, extremal, and super-extremal Reissner–Nordström spacetimes. We also obtain new results for connected photon spheres and equipotential photon surfaces in the extremal case. Finally, we provide, up to a restriction on the range of their radii, the uniqueness result for connected (both non-degenerate and degenerate) equipotential photon surfaces in the super-extremal case, not yet treated in the literature.

On the (k,φ)-Hilfer Langevin fractional coupled system having multi point boundary conditions and fractional integrals

In this manuscript, we investigate the (k,φ)-Hilfer Langevin fractional coupled system having multi point boundary conditions and fractional integrals. This study is crucial as it addresses the complexities of (k,φ)- Hilfer Langevin coupled system with multi point conditions, which have applications in many scientific and engineering fields. Unlike previous research, our work introduces novel techniques for analyzing these system, leading to more accurate and comprehensive results. The major findings include new existence and uniqueness theorems, along with improved stability conditions, which enhance the understanding and potential applications of these systems. We demonstrate the existence, uniqueness and different kinds of Ulam stability for our suggested model. We prove the uniqueness of the solution by utilizing Banach contraction mapping principle. The existence of solution is studied by using Krasnoselskii's fixed point theorem. We also explore the different types of Ulam stability under the specific conditions. We presented an example at the end to support our main results.

The Uniqueness Theorem for Nonlocal Hydrodynamic Media

The Uniqueness Theorem for Nonlocal Hydrodynamic Media

Inverse nodal problems for integro-differential Dirac operators with parameter-nonlocal integral boundary condition by using numerical methods

In this study, we apply numerical methods to solve inverse nodal problems for the integro-differential Dirac operator with parameter-nonlocal integral boundary conditions. We apply the Bernstein method and Legendre multi-wavelets method to obtain approximate solutions for the potentials, and also present numerical examples to demonstrate their effectiveness. We then employ the numerical method to investigate the uniqueness theorem with respect to the potentials. The proofs are constructive.

Tricomi Problem Analogue for a Second Order Mixed Type Equation

In this paper, we investigated an analogue of the Tricomi problem for one equation of parabolic-hyperbolic type of the second order with a heat conduction operator in the parabolicity region and with a degenerate hyperbolic operator of the first kind in the hyperbolicity region. The type change line is characteristic for a parabolic equation and non-characteristic for a hyperbolic one. The problem is studied when the values of the sought function are given on non-characteristic segments of the boundary of a parabolic parabolic equation, and a value is also given on the characteristic of the hyperbolic equation at. Previously, similar problems were the subject of research by many authors, such as, for example, the works of A.~M.~Nakhushev and Kh.~G.~Bzhikhatlova, M.~S.~Salakhitdinov and A.~S.~Berdysheva. In this paper, theorems on the uniqueness and existence of a regular solution to the problem under study are proven. When proving the uniqueness theorem for the solution of the problem, modern methods of the theory of fractional calculus are used, and the existence is proven using the method of integral equations. In the case when the coefficients of the equation under consideration are constant, the solution is found and written out explicitly. The theorems proved generalize the previously obtained results both in terms of sufficiency conditions for the uniqueness of a solution to an analogue of the Tricomi problem, and in terms of the existence theorem.

Kernel Determination Problem in the Third Order 1D Moore-Gibson-Thompson Equation with Memory

In this study, we address the inverse problem of determining the convolution kernel function in the third-order Moore--Gibson--Thompson (MGT) equation, which is commonly used to model fluid motion with memory effects. Specifically, we focus on the determination of the unknown kernel, which governs the memory term in the equation. Initially, we employ the Fourier spectral method to solve the direct initial-boundary-value problem for a non-homogeneous MGT equation that with the memory term. The Fourier spectral method allows us to leverage the problem's inherent linearity and spatial homogeneity, leading to an efficient and explicit construction of the solution. The direct problem is analyzed under appropriate initial and boundary conditions, which are carefully specified to ensure mathematical consistency. To solve the inverse problem, we introduce an additional condition-typically a form of observational data such as at certain points-which provides the necessary constraints for determining the kernel. We prove local existence and uniqueness theorems for solution of the problem.

Nonlocal boundary value problem for a nonhomogeneous parabolic type equation with two degenerate lines

This work is devoted to the study of the conditional correctness of the nonlocal boundary value problem for a nonhomogeneous parabolic type equation with two degenerate lines. In this article we investigated well-posed of the problem depending of coefficients in conditions for a problem. In the case of ill-posed, based on the idea of A.N. Tikhonov, the conditional correctness of the problem are proved, namely, the theorems of uniqueness and conditional stability on the set of correctness are proved. For all cases we get a priori estimate of the solution we used the logarithmic convexity method and the results of the spectral problem considered by S.G. Pyatkov

On one linear inverse problem with nonlocal boundary conditions for the second kind, second order mixed type equation in an unbounded parallelepiped

In this article the correctness of a linear inverse problem with nonlocal boundary conditions for the three-dimensional second kind, second order mixed type equation in an unbounded parallelepiped is investigated. The existence and uniqueness theorems for a generalized solution to a linear inverse problem with a nonlocal boundary condition are proved in a certain class of integrable functions. The \"-regularization", a priori estimates, approximation sequences, Fourier transform methods are applied.

Hawking-Type Singularity Theorems for Worldvolume Energy Inequalities

AbstractThe classical singularity theorems of R. Penrose and S. Hawking from the 1960s show that, given a pointwise energy condition (and some causality as well as initial assumptions), spacetimes cannot be geodesically complete. Despite their great success, the theorems leave room for physically relevant improvements, especially regarding the classical energy conditions as essentially any quantum field theory necessarily violates them. While singularity theorems with weakened energy conditions exist for worldline integral bounds, so-called worldvolume bounds are in some cases more applicable than the worldline ones, such as the case of some massive free fields. In this paper, we study integral Ricci curvature bounds based on worldvolume quantum strong energy inequalities. Under the additional assumption of a—potentially very negative—global timelike Ricci curvature bound, a Hawking-type singularity theorem is proved. Finally, we apply the theorem to a cosmological scenario proving past geodesic incompleteness in cases where the worldline theorem was inconclusive.

Mutual singularity of Riesz products on the unit sphere

We prove analogs of Peyrière’s mutual singularity theorem for standard and generalized Riesz products on the unit sphere of C n \mathbb {C}^n , n ≥ 2 n\ge 2 . As a corollary, we obtain an analog of Zygmund’s dichotomy for the Riesz products under consideration.