Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Polá?ik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.     

Unconditional bases and the order structure

The paper deals with an analytic characterization of unconditional bases, a characterization of the unconditional character of a Schauder base for the associated cone and a number of counterexamples supporting various conditions appearing in the theorems.     

Generic convergence of infinite products of positive linear operators

We present several results concerning the asymptotic behavior of (random) infinite products of generic sequences of positive linear operators on an ordered Banach space. In addition to a weak ergodic theorem we also obtain convergence to an operator of the formf(·) wheref is a continuous linear functional and is a common fixed point.     

Relations between convolution type operators on intervals and on the half-line

This paper is devoted to the question to obtain (algebraic and topologic) equivalence (after extension) relations between convolution type operators on unions of intervals and convolution type operators on the half-line. These operators are supposed to act between Bessel potential spaces,H ^s,p , which are the appropriate spaces in several applications. The present approach is based upon special properties of convenient projectors, decompositions and extension operators and the construction of certain homeomorphisms between the kernels of the projectors. The main advantage of the method is that it provides explicit operator matrix identities between the mentioned operators where the relations are constructed only by bounded invertible operators. So they are stronger than the (algebraic) Kuijper-Spitkovsky relation and the Bastos-dos Santos-Duduchava relation with respect to the transfer of properties on the prize that the relations depend on the orders of the spaces and hold only for non-critical orders:S – 1/p . For instance, (generalized) inverses of the operators are explicitly represented in terms of operator matrix factorization. Some applications are presented.This research was supported by Junta Nacional de Investigação Científica e Tecnológica (Portugal) and the Bundesminister für Forschung und Technologie (Germany) within the projectSingular Operators-new features and applications, and by a PRAXIS XXI project under the titleFactorization of Operators and Applications to Mathematical Physics.     

On the behavior of attractors under finite difference approximation

We recall that the long-time behavior of the Kuramoto-Sivashinsky equation is the same as that of a certain finite system of ordinary differential equations. We show how a particular finite difference scheme approximating the Kuramoto-Sivashinsky may be viewed as a small C ¹ perturbation of this system for the grid spacing sufficiently small. As a consequence one may make deductions about how the global attractor and the flow on the attractor behaves under this approximation. For a sufficiently refined grid the long-time behavior of the solutions of the finite difference scheme is a function of the solutions at certain grid points, whose number and position remain fixed as the grid is refined. Though the results are worked out explicitly for the Kuramoto-Sivashinsky equation, the results extend to other infinite-dimensional dissipative systems.     

Approximation of approximation numbers by truncation

LetA be a bounded linear operator onsome infinite-dimensional separable Hilbert spaceH and letA _n be the orthogonal compression ofA to the span of the firstn elements of an orthonormal basis ofH. We show that, for eachk1, the approximation numberss _k(A_n) converge to the corresponding approximation numbers _k(A) asn. This observation implies almost at once some well known results on the spectral approximation of bounded selfadjoint operators. For example, it allows us to identify the limits of all upper and lower eigenvalues ofA _n in the case whereA is selfadjoint. These limits give us all points of the spectrum of a selfadjoint operator which lie outside the convex hull of the essential spectrum. Moreover, it follows that the spectrum of a selfadjoint operatorA with a connected essential spectrum can be completely recovered from the eigenvalues ofA _n asn goes to infinity.     

On a class of stably finite nuclear C*-algebras

In this note we show that a separable C^*-algebra is nuclear and has a quasidiagonal extension by (the ideal of compact operators on an infinite-dimensional separable Hilbert space) if and only if it is anuclear finite algebra (NF-algebra) in the sense of Blackadar and Kirchberg, and deduce that every nuclear C^*-subalgebra of aNF-algebra isNF. We show that strongNF-algebras satisfy a Følner type condition.     

A closed graph theorem for order bounded operators

The closed graph theorem is one of the cornerstones of linear functional analysis in Fréchet spaces, and the extension of this result to more general topological vector spaces is a di?cult problem comprising a great deal of technical difficulty. However, the theory of convergence vector spaces provides a natural framework for closed graph theorems. In this paper we use techniques from convergence vector space theory to prove a version of the closed graph theorem for order bounded operators on Archimedean vector lattices. This illustrates the usefulness of convergence spaces in dealing with problems in vector lattice theory, problems that may fail to be amenable to the usual Hausdorff-Kuratowski-Bourbaki concept of topology.     

On discrete inequalities of the Poincaré type

The aim of the present paper is to establish some new discrete inequalities of the Poincaré type involving functions ofn independent variables and their first order forward differences. The proofs given here are quite elementary and our results provide new estimates on this type of discrete inequalities.     

On the approximation of positive operators and the behaviour of the spectra of the approximants

LetT be a positive linear operator on the Banach latticeE and let (S _n ) be a sequence of bounded linear operators onE which converge strongly toT. Our main results are concerned with the question under which additional assumptions onS _n andT the peripheral spectra (S _n ) ofS _n converge to the peripheral spectrum (T) ofT. We are able to treat even the more general case of discretely convergent sequences of operators.     

Completely integrable systems of the KdV type related to isospectral periodic regular difference operators

Completely integrable KdV systems are described on coadjoint orbits, which are isospectral classes of periodic regular difference operators. The finite zone solutions for some field equations are then obtained if the equations are written on a jet bundle of maps with values in a Lie group and if the orbits are truncated invariantly with regard to the group action.     

The strongest intrinsic meaning of sequential-evaluation convergence

For classical Banach sequence spaces c₀(X), l_∞(X) and lp(X) (0<p<+∞) we have found the strongest intrinsical meanings of their β-duals, and two basic convergence results are established in the β-duals.     

Generic existence and uniqueness of positive eigenvalues and eigenvectors

We consider a closed cone of positive operators on an ordered Banach space and prove that a generic element of this cone has a unique positive eigenvalue and a unique (up to a positive multiple) positive eigenvector. Moreover, the normalized iterations of such a generic element converge to its unique eigenvector.     

On the spectral radius of positive operators on Banach sequence spaces

Let K₁,…,K_n be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,∞)×…×[0,∞)→[0,∞) of n variables, we define a non-negative matrix and consider the inequality

     

Compactness properties of certain integral operators related to fractional integration

Suppose 1≤p,q≤∞ and α > (1/p−1/q)₊. Then we investigate compactness properties of the integral operator when regarded as operator from L_p[0,1] into L_q[0,1]. We prove that its Kolmogorov numbers tend to zero faster than exp(−c_αn^1/2). This extends former results of Laptev in the case p=q=2 and of the authors for p=2 and q=∞. As application we investigate compactness properties of related integral operators as, for example, of the difference between the fractional integration operators of Riemann–Liouville and Weyl type. It is shown that both types of fractional integration operators possess the same degree of compactness. In some cases this allows to determine the strong asymptotic behavior of the Kolmogorov numbers of Riemann–Liouville operators. In memoria of Eduard (University of the West Indies) who passed away in October 2004.     

Banach algebras of structured matrix sequences

We discuss the asymptotic behavior of matrix sequences belonging to a special class of non-commutative Banach algebras and study, in particular, the stability, and more general the Fredholm property of such sequences. The abstract results are applied to finite sections of band-dominated operators, especially in the case l^p(Z), 1?p?∞.     

Monotone iterative sequences for nonlinear boundary value problems of fractional order

In this paper we extend the maximum principle and the method of upper and lower solutions to boundary value problems with the Caputo fractional derivative. We establish positivity and uniqueness results for the problem. We then introduce two well-defined monotone sequences of upper and lower solutions which converge uniformly to the actual solution of the problem. A numerical iterative scheme is introduced to obtain an accurate approximate solution for the problem. The accuracy and efficiency of the new approach are tested through two examples.     

Approximation of p-multiplier Operators via their Spectral Projections

Conditions for a p-multiplier $\psi: {\mathbb{Z}} \to {\mathbb{C}}Conditions for a p-multiplier are presented which ensure that the corresponding operator T_ψ, acting in , can be approximated by linear combinations of p-multiplier projections coming from the uniform operator closed, unital algebra of operators generated by T_ψ. Functions of bounded variation on play an important role, as do certain Λ (p)-sets. Dedicated to the memory of H. H. Schaefer Werner J. Ricker: Former Alexander von Humboldt Fellow at the Universit?t Tübingen, hosted by Prof. H.H. Schaefer from Sept. 1987 – Feb. 1988.     

On non-surjective coarse isometries between Banach spaces

Assume that X, Y are real Banach spaces, Y has uniform convexity of type p (≥ 1), and f: X → Y is a standard coarse isometry. In this paper, we show that if

then there is a linear isometry U : X → Y so that

where is defined by

Representation properties of coarse isometries in free ultrafilter limits on are also discussed.