Non-compact Operators Research Articles

A pseudo-monotonicity adapted to doubly nonlinear elliptic–parabolic equations

The case where B is an unbounded linear operator was considered 1rst by Bardos and Br7 ezis [4]. In the nonlinear case Raviart [20], Grange and Mignot [13], DiBenedetto and Showalter [11] proved existence results assuming that A and B are at least monotone operators, and B is compact. Alt and Luckhaus [1] investigated the case of a non-compact operator B, assuming A is strongly monotone; their work was extended more recently by Ka? cur [14], Filo and Ka? cur [12] or Zadrzy7 nska and ZajB aczkowski [26]. Berm7 udez et al. [7] devoted their work to the case where B is compact and strongly monotone and A is pseudo-monotone. We are interested in the same case excepted that B is no longer assumed to be strongly monotone, thus the equation may

Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$

We discuss estimates of the Hausdorff and fractal dimension of a global attractor for the semilinear wave equation $u_{t t} +\delta u_t -\phi (x)\Delta u + \lambda f(u) = \eta (x), x \in \mathbb R^N, t \geq 0,$ with the initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x),$ where $N \geq 3$, $\delta >0$ and $(\phi (x))^{-1}:=g(x)$ lies in $L^{N/2}(\mathbb R^N)\cap L^\infty (\mathbb R^N)$. The energy space $\mathcal X_0=\mathcal D^{1,2}(\mathbb R^N) \times L_g^2(\mathbb R^N)$ is introduced, to overcome the difficulties related with the non-compactness of operators, which arise in unbounded domains. The estimates on the Hausdorff dimension are in terms of given parameters, due to an asymptotic estimate for the eigenvalues $\mu$ of the eigenvalue problem $-\phi(x)\Delta u=\mu u, x \in \mathbb R^N$.

Spectral Analysis and Singular Value Computations of the Noncompact Frequency Response and Compression Operators in Sampled-Data Systems

This paper is motivated by the problem of computing the frequency response gain of general sampled-data systems with noncompact frequency response operators. We first show that, with the J-unitary transformation, the computation in the noncompact operator case can be reduced, in principle, to that in the compact operator case, to which an existing efficient and reliable bisection method can be applied. At the same time, however, we point out that there arise some critical problems in this reduction to the compact case which could be serious enough to invalidate the apparent success in the reduction. Through some spectral analysis of operators involving or related to the frequency response operators, we eventually prove that these critical problems can be circumvented after all, and we give an explicit result that shows how to compute the frequency response gain with a bisection method dealing only with finite-dimensional matrices. Extending the arguments, we also give a bisection method to compute the singular values of the frequency response operators and the associated compression operators.

Numerical approximation of the spectra of non-compact operators arising in buckling problems

The finite element method approximates the spectrum of an operator S by computing the spectra of a sequence of operators SN defined in terms of the finite element spaces. For the case that S is compact, convergence of the approximate spectra follows from the convergence of SN to S in the operator norm. We consider the case that S is non-compact, in which case such operator norm convergence cannot take place, and the approximations may be polluted by spurious eigenvalues. Pollution-free convergence of the eigenvalues can, however, be guaranteed outside the essential numerical range of S, which is related to the essential spectrum of S. We present results for estimating this essential numerical range and apply them to an algorithm for the buckling of three-dimensional bodies (that gives rise to a non-compact S). Our results show, for instance, that for the example of a circular disc, the algorithm will be free of spurious eigenvalues provided the body is thin enough. The case that singularities in the stresses can lead to non-physical spectral values being approximated is also investigated.

STRICTLY SINGULAR, NON-COMPACT OPERATORS EXIST ON THE SPACE OF GOWERS AND MAUREY

A strictly singular non-compact operator on the Gowers and Maurey space GM is constructed.

Measure of weak noncompactness under complex interpolation

Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón's complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if $T:A_{0}\rightarrow B_{0}$ or $T:A

The space of compact operators contains c 0 when a noncompact operator is suitably factorized

The space of compact operators contains c 0 when a noncompact operator is suitably factorized

Finite Element Analysis of a Quadratic Eigenvalue Problem Arising in Dissipative Acoustics

A quadratic eigenvalue problem arising in the determination of the vibration modes of an acoustic fluid contained in a cavity with absorbing walls is considered. The problem is shown to be equivalent to the spectral problem for a noncompact operator and a thorough spectral characterization is given. A numerical discretization based on Raviart--Thomas finite elements is analyzed. The method is proved to be free of spurious modes and to converge with optimal order. Implementation issues and numerical experiments confirming the theoretical results are reported.

Existence of a Global Attractor for Semilinear Dissipative Wave Equations on [formula omitted

We consider the semilinear hyperbolic problem utt+δut−φ(x)Δu+λf(u)=η(x), x∈RN, t>0, with the initial conditions u(x, 0)=u0(x) and ut(x, 0)=u1(x) in the case where N⩾3 and (φ(x))−1:=g(x) lies in LN/2(RN). The energy space X0=D1, 2(RN)×L2g(RN) is introduced, to overcome the difficulties related with the noncompactness of operators which arise in unbounded domains. We derive various estimates to show local existence of solutions and existence of a global attractor in X0. The compactness of the embedding D1, 2(RN)⊂L2g(RN) is widely applied.

Stability Rates for Linear Ill-Posed Problems with Compact and Non-Compact Operators

In this paper we deal with the ’strength’ of ill-posedness for ill-posed linear operator equations Ax = y in Hilbert spaces, where we distinguish according to M. Z. .Nashed the ill-posedness of type I if A is not compact, but we have R(A) \neq \bar{R(A)} for the range R(A) of A , and the ill-posedness of type II for compact operators A . From our considerations it seems to follow that-the problems with non-compact operators A are not ingeneral ’less’ ill-posed than the problems with compact operators. We motivate this statement by comparing the approximation and stability behaviour of discrete least-squares solutions and the growth rate of Galerkin matrices in both cases. Ill-posedness measures for compact operators A as discussed by B. Hofmann and U. Tautenhahn are derived from the decay rate of the non-increasing sequence of singular values of A . Since singular values do not exist for non-compact operators A , we introduce stability rates in order to have a common measure for the compact and non-compact cases. Properties of these rates are illustrated by means of convolution equations in the compact case and by means of equations with multiplication operators in thenon-compact case. Moreover, using increasing rearrangements of multiplier functions specific measures of ill-posedness called ill-posedness rates are considered for multiplication* operators. In this context, the character of sufficient conditions providing convergence rates of Tikhonov regularization are compared-for compact operators and multiplication. operators.

A degree theoretic approach to N-species periodic competition systems on the whole ℝN

Previous fixed-point index calculation results (see [4] and [7]) exploited in the study of population systems on bounded domains are no longer applicable to systems on the whole ℝn, due mainly to the lack of compactness. In this paper, we develop fixed-point index calculation techniques for non-compact operators so that they are applicable to systems on the whole ℝn. We illustrate the use of our fixed-point index calculation results by a simple representative model, namely, the Lotka–Volterra N-species periodic competition system on the whole ℝn.

The Berezin transform and operators on spaces of analytic functions

In this article we will illustrate how the Berezin transform (or symbol) can be used to study classes of operators on certain spaces of analytic functions, such as the Hardy space, the Bergman space and the Fock space. The article is organized according to the following outline. 1. Spaces of analytic functions 2. Definition and properties Berezin transform 3. Berezin transform and non-compact operators 4. Commutativity of Toeplitz operators 5. Berezin transform and Hankel or Toeplitz operators 6. Sarason’s example 1. Spaces of analytic functions. In this section we will introduce the spaces of analytic functions on which we will be working. We start with the following general definition. Definition 1.1. A reproducing functional Hilbert space on an open subset Ω of C is a Hilbert space H of functions on Ω such that for every w ∈ Ω the linear functional f 7→ f(w) is bounded on H. If H is a reproducing functional Hilbert space on set Ω, then by the Riesz Representation Theorem for every w ∈ Ω there is a unique element Kw ∈ H for which f(w) = 〈f,Kw〉, for all f ∈ H. We call the function Kw the reproducing kernel at w. Before we turn to a few examples we will prove some simple results about these reproducing kernels. The following proposition gives a way to compute the reproducing kernels. 1991 Mathematics Subject Classification: Primary 47B07, 47B35; Secondary 30C40, 31A05. Research of the author supported by summer grants from the University of Montana and the Montana University system. The paper is in final form and no version of it will be published elsewhere. [361]

Analysis of the Domain Integral Operator for Anisotropic Dielectric Waveguides

The domain integral equation for guided electromagnetic waves in anisotropic inhomogeneous guides is formulated. The spectral properties of the noncompact integral operator are analysed and the existence of guided modes is proved for lossless nonmagnetic media.

The order of convergence of eigenfrequencies in finite element approximations of fluid-structure interaction problems

In this paper we prove a double order for the convergence of eigenfrequencies in fluid-structure vibration problems. We improve estimates given recently for compressible and incompressible fluids. To do this, we extend classical results on finite element spectral approximation to nonconforming methods for noncompact operators.

Optimal feedback control for a semilinear evolution equation

In this paper, we consider a minimization problem of a cost functional associated to a nonlinear evolution feedback control system with a given boundary condition which includes the periodic one as a particular case. Specifically, by using an existence result for a system of inclusions involving noncompact operators (see Ref. 1), we first prove that the solution set of our problem is nonempty. Then, from the topological properties of this set, we derive the existence of a solution of the minimization problem under consideration.

Finitely-generated solutions of certain integral equations

Recent work has shown that the solutions of the second-kind integral equation arising from a difference kernel can be expressed in terms of two particular solutions of the equation. This paper establishes analogous results for a wider class of integral operators, which includes the special case of those arising from difference kernels, where the solution of the general case is generated by a finite number of particular cases. The generalisation is achieved by reducing the problem to one of finite rank. Certain non-compact operators, including those arising from Cauchy singular kernels, are amenable to this approach.

On the Solvability of Systems of Inclusions Involving Noncompact Operators

We consider the solvability of a system \[ \left \{ {\begin {array}{*{20}{c}} {y \in \bar F(x,y),} \\ {x \in \bar G(x,y)} \\ \end {array} } \right .\] of set-valued maps in two different cases. In the first one, the map $(x,y) - \circ \bar F(x,y)$ is supposed to be closed graph with convex values and condensing in the second variable and $(x,y) - \circ \bar G(x,y)$ is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case $\bar F$ is as before with compact, not necessarily convex, values and $\bar G$ is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set $x - \circ S(x)$ of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.

On the solvability of systems of inclusions involving noncompact operators

We consider the solvability of a system \[ { y ∈ F ¯ ( x , y ) , x ∈ G ¯ ( x , y ) \left \{ {\begin {array}{*{20}{c}} {y \in \bar F(x,y),} \\ {x \in \bar G(x,y)} \\ \end {array} } \right . \] of set-valued maps in two different cases. In the first one, the map ( x , y ) − ∘ F ¯ ( x , y ) (x,y) - \circ \bar F(x,y) is supposed to be closed graph with convex values and condensing in the second variable and ( x , y ) − ∘ G ¯ ( x , y ) (x,y) - \circ \bar G(x,y) is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case F ¯ \bar F is as before with compact, not necessarily convex, values and G ¯ \bar G is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set x − ∘ S ( x ) x - \circ S(x) of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.

Numerical analysis of boundary integral solution of the Helmholtz equation in domains with non-smooth boundaries

In the paper we carry out a complete analysis of several efficient numerical methods for the solution of boundary integral equations defined on a non-smooth boundary. In particular the solution of the Helmholtz equation in the exterior of a closed wedge is studied. The analytical behaviour of the solution of the resulting boundary integral equation (with a non-compact operator) near the wedge is investigated. Numerical analysis of the collocation and iterated collocation method for the problem is presented. Graded meshes are used to reflect the «singular» behaviour of the analytical solution, as well as the degree of the polynomial approximant, in order to yield results with «optimal convergence rates»

MAGNETIC GROUP ON THE PSEUDOSPHERE

The quantum mechanics of a particle moving on a pseudosphere under the action of a constant magnetic field is studied from an algebraic point of view. The magnetic group on the pseudosphere is SU(1, 1). The Hilbert space for the discrete part of the spectrum is investigated. The eigenstates of the non-compact operators (the hyperbolic magnetic translators) are constructed and shown to be expressible as continuous superpositions of coherent states. The planar limit of both the algebra and the eigenstates is analyzed. Some possible applications are briefly outlined.