Theory Of Linear Operators Research Articles

A dual-phase-lag porous-thermoelastic problem with microtemperatures

<abstract><p>In this work, we consider a multi-dimensional dual-phase-lag problem arising in porous-thermoelasticity with microtemperatures. An existence and uniqueness result is proved by applying the semigroup of linear operators theory. Then, by using the finite element method and the Euler scheme, a fully discrete approximation is numerically studied, proving a discrete stability property and a priori error estimates. Finally, we perform some numerical simulations to demonstrate the accuracy of the approximation and the behavior of the solution in one- and two-dimensional problems.</p></abstract>

Every Lateral Band is the Kernel of an Orthogonally Additive Operator

{In this paper we continue a study of relationships between the lateral partial order $\sqsubseteq$ in a vector lattice (the relation $x \sqsubseteq y$ means that $x$ is a fragment of $y$) and the theory of orthogonally additive operators on vector lattices. It was shown in~\cite{pMPP} that the concepts of lateral ideal and lateral band play the same important role in the theory of orthogonally additive operators as ideals and bands play in the theory for linear operators in vector lattices. We show that, for a vector lattice $E$ and a lateral band $G$ of~$E$, there exists a vector lattice~$F$ and a positive, disjointness preserving orthogonally additive operator $T \colon E \to F$ such that ${\rm ker} \, T = G$. As a consequence, we partially resolve the following open problem suggested in \cite{pMPP}: Are there a vector lattice~$E$ and a lateral ideal in $E$ which is not equal to the kernel of any positive orthogonally additive operator $T\colon E\to F$ for any vector lattice $F$?

Boundary conditions for hyperbolic relaxation systems with characteristic boundaries of type II

This paper is a continuation of our preceding work on hyperbolic relaxation systems with characteristic boundaries of type I. Here we focus on the characteristic boundaries of type II, where the boundary is characteristic for the equilibrium system and is non-characteristic for the relaxation system. For this kind of characteristic initial-boundary-value problems (IBVPs), we introduce a three-scale asymptotic expansion to analyze the boundary-layer behaviors of the general multi-dimensional linear relaxation systems. Moreover, we derive the reduced boundary condition under the Generalized Kreiss Condition by resorting to some subtle matrix transformations and the perturbation theory of linear operators. The reduced boundary condition is proved to satisfy the Uniform Kreiss Condition for characteristic IBVPs. Its validity is shown through an error estimate involving the Fourier-Laplace transformation and an energy method based on the structural stability condition.

Isometric embeddability of [formula omitted] into [formula omitted

In this paper, we study existence of isometric embedding of Sqm into Spn, where 1≤p≠q≤∞ and n≥m≥2. We show that for all n≥m≥2 if there exists a linear isometry from Sqm into Spn, where (q,p)∈(1,∞]×(1,∞)∪(1,∞)∖{3}×{1,∞} and p≠q, then we must have q=2. This mostly generalizes a classical result of Lyubich and Vaserstein. We also show that whenever Sq embeds isometrically into Sp for (q,p)∈(1,∞)×[2,∞)∪[4,∞)×{1}∪{∞}×(1,∞)∪[2,∞)×{∞} with p≠q, we must have q=2. Thus, our work complements work of Junge, Parcet, Xu and others on isometric and almost isometric embedding theory on non-commutative Lp-spaces. Our methods rely on several new ingredients related to perturbation theory of linear operators, namely Kato-Rellich theorem, theory of multiple operator integrals and Birkhoff-James orthogonality, followed by thorough and careful case by case analysis. The question whether for m≥2 and 1<q<2, Sqm embeds isometrically into S∞n, was left open in Bull. London Math. Soc. 52 (2020) 437-447.

Gaps between Operator Norm and Spectral and Numerical Radii of the Tensor Product of Operators

One of the fundamental problem of the Spectral Theory of Linear Operators is to determine of the geometric place of the spectrum of the given operator and calculate the spectral and numerical radii of this operator. Other important problem in this theory is to explained the situation the spectral (numerical) radius is equal or not to operator norm. The only known way to calculate the spectral radius to date is the classical Gelfand formula, which often presents great technical challenges. Also, there is not yet a method of calculating the numerical radius for an operator. It should be noted that the finding the numerical range and numerical radius means maximally localizing the spectrum of an operator. The main purpose of this paper is to determine the relations gaps between operator norm and spectral and numerical radii of the tensor product operators associated with the compatible gaps of coordinate operators.

Stability of singular waves for Dullin–Gottwald–Holm equation

In this paper, the asymptotic stability of the solutions near the explicit singular waves of Dullin–Gottwald–Holm equation is studied based on the commutator estimate, the semi-group theory of linear operator and the Banach fixed point theorem.

Analyticity of solutions to thermo‐elastic‐plastic flow problem with microtemperatures

AbstractIn this paper, we study some qualitative and numerical properties of new equations including the coupled effects of thermal elastic‐plastic theory with microtemperatures. We establish the necessary and sufficient conditions to guarantee that the model dissipates energy. The one‐dimensional case, which corresponds to isotropic hardening problem, is chosen in order to present some qualitative and numerical properties. With the help of the semigroup theory of linear operators, we prove the well‐posedness of the one‐dimensional problem corresponding to plastic flow. Then, we show that the associated semigroup is not analytical in general, except for a special case. The exponential stability of the solutions is kept in all cases. Finally, a numerical tool, based on the finite element method, is developed to validate the proposed model and to show its capability. Particular attention is paid to the consideration of the elastoplastic behavior in the development of this tool.

Exponential stabilization of a microbeam system with a boundary or distributed time delay

This paper addresses the stabilization problem of a microscale beam system subject to a delay. Several situations are considered depending whether the delay occurs as a boundary or interior/distributed term. In both cases, the microbeam system is shown to be well posed in the sense of semigroups theory of linear operators. More importantly, using the energy method, the exponential stability is established as long as the parameter of the delay term is small.

Global Perturbation of Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

Partial stability of linear systems with respect to a given component of the phase vector

A new geometric approach to the study of the partial stability of linear systems is proposed, which is based on the application of the geometric theory of linear operators. Using the theory of conjugate spaces and conjugate linear operators, bases are constructed in which the system under study takes the canonical form. A cyclic subspace with respect to the conjugate linear operator is considered. A basis is constructed for the dual space of a linear operator, in which its matrix takes the canonical form. This basis corresponds to the dual basis of the original linear space. Then, in a pair of bases of dual spaces the system under study takes the simplest form. The geometric properties of the system are realized using a non-singular linear transformation in the space of a part of the components of the system’s phase vector. This allows us to decompose the system under study in order to obtain necessary and sufficient conditions for the partial stability of the linear system. In an equivalent system, an independent subsystem is distinguished, whose nature of stability determines the behavior of the investigated component of the original system’s phase vector. The relationship between the partial stability of the system and the existence of an invariant subspace of a linear operator characterizing the dynamics of the system is established. The canonical form of the resulting subsystem makes it easy to exclude auxiliary variables and write an equation equivalent to this system. The application of the obtained results to the solution of the problem of partial stability for linear systems with constant coefficients belonging to the classes of ordinary differential equations, discrete systems and systems with deviating argument is shown. An example of a linear system of differential equations is given to illustrate the result obtained.

Spectral Problem and Initial Value Problem of a Nonlocal Sturm-Liouville Equation

In this paper, we considered the spectral problem and initial value problem of a nonlocal Sturm-Liouville equation with fractional integrals and fractional derivatives. We proved that the fractional operator associated to the nonlocal Sturm-Liouville equation is self-adjoint in Hilbert space. And then, we derived the corresponding spectral problem consists of countable number of real eigenvalues, and the algebraic multiplicity of each eigenvalue is simple. We also discussed the orthogonal completeness of the corresponding eigenfunction system in the Hilbert spaces. Furthermore, we obtained asymptotic properties of eigenvalues and the number of zeros of eigenfunctions by using the perturbation theory for linear operators. Finally, we studied the uniqueness of solutions for the nonlocal Sturm-Liouville equation under some initial value conditions.

Generalized Bernstein-Chlodowsky-Kantorovich type operators involving Gould-Hopper polynomials

In the present article, we establish a link between the theory of positive linear operators and the orthogonal polynomials by defining Bernstein-Chlodowsky-Kantorovich operators based on Gould-Hopper polynomials (orthogonal polynomials) and investigate the degree of convergence of these operators for unbounded continuous functions having a polynomial growth. In this connection, the moments of the operators are derived first, and then the approximation degree of the considered operators is established by means of the complete and the partial moduli of continuity. Next, we focus on the rate of convergence of these operators for functions in a weighted space. The associated Generalized Boolean Sum (GBS) operator of the operators under study is defined, and the degree of approximation is studied with the aid of the mixed modulus of smoothness and the Lipschitz class of Bögel continuous functions.

Existence, stability and controllability results of stochastic differential equations with non-instantaneous impulses

ABSTRACT This paper is devoted to the study of a new class of non-instantaneous impulsive stochastic differential equations driven by mixed fractional Brownian motion in separable Hilbert spaces. Based on the stochastic analysis theory, analytic semigroup theory of linear operators, fractional powers of operators, and a fixed point technique, a new set of sufficient conditions are derived to ensure the existence and uniqueness of mild solutions for the proposed stochastic system. Moreover, we also investigate the asymptotic behaviour of mild solutions and controllability results for the proposed stochastic system. Finally, an example is given to demonstrate the applicability of our main results.

Semigroup method for a deteriorating system with multiple vacations of a repairman

By using the method of functional analysis, especially $$C_0$$ -semigroup theory and spectral theory of linear operators we prove the well-posedness and the existence of a unique positive dynamic solution of a deteriorating system with multiple vacations of a repairman. Moreover, we show spectral properties of the system operator which imply that the system does not have a steady-state solution.

A New Subclass of Certain Analytic Univalent Functions Associated with Hypergeometric Functions

The main objective of the present paper is to give with using the linear operator theory and also hypergeometric representations of related functions a new special subclass $\mathcal{TS}_{p}(2^{-r},2^{-1}), r\in \mathbb{ Z }^{+}$ of uniformly convex functions and in addition a suitable subclass of starlike functions with negative Taylor coefficients. Furthermore, the provided trailblazer outcomes in presented study are generalized to certain functions classes with fixed finitely many Taylor coefficients.

FINITENESS OF THE NUMBER OF CRITICAL VALUES OF THE HARTREE-FOCK ENERGY FUNCTIONAL LESS THAN A CONSTANT SMALLER THAN THE FIRST ENERGY THRESHOLD

We study the Hartree-Fock equation and the Hartree-Fock energy functional universally used in many-electron problems. We prove that the set of all critical values of the Hartree-Fock energy functional less than a constant smaller than the first energy threshold is finite. Since the Hartree-Fock equation which is the corresponding Euler-Lagrange equation is a system of nonlinear eigenvalue problems, the spectral theory for linear operators is not applicable. The present result is obtained establishing the finiteness of the critical values associated with orbital energies less than a negative constant and combining the result with the Koopmans' well-known theorem. The main ingredients are the proof of convergence of the solutions and the analysis of the Fr\'echet second derivative of the functional at the limit point.

Asymptotic Stability of Singular Solution for Camassa-Holm Equation

The aim of this paper is to study singular dynamics of solutions of Camassa-Holm equation. Based on the semigroup theory of linear operators and Banach contraction mapping principle, we prove the asymptotic stability of the explicit singular solution of Camassa-Holm equation.

Frank Featherstone Bonsall. 31 March 1920—22 February 2011

Frank Bonsall played a significant role in the mathematical life of the United Kingdom in the decades following the Second World War. He had a particular impact in Scotland and the north of England, especially in research and graduate education. His research interests focused primarily on functional analysis, the area of mathematics that brings together various strands of analysis under a single abstract framework, and on the related theory of linear operators on Banach spaces. He influenced a generation of young mathematicians with the elegance of his written and oral expositions, both of his own research and that of others. The quality of his caring and thorough research supervision was reflected in his many PhD students who would continue in research and go on to successful academic careers in their own right, both in the United Kingdom and beyond.

Editorial: Biographical Memoirs, Volume 69

Editorial: Biographical Memoirs, Volume 69

Existence of solutions for subquadratic convex operator equations at resonance and applications to Hamiltonian systems

This paper investigates the existence of solutions to subquadratic operator equations with convex nonlinearities and resonance by means of the index theory for self-adjoint linear operators developed by Dong and dual least action principle developed by Clarke and Ekeland. Applying the results to subquadratic convex Hamiltonian systems satisfying several boundary value conditions including Bolza boundary value conditions, generalized periodic boundary value conditions and Sturm–Liouville boundary value conditions yield some new theorems concerning the existence of solutions or nontrivial solutions. In particular, some famous results about solutions to subquadratic convex Hamiltonian systems by Mawhin and Willem and Ekeland are special cases of the theorems.