Matrix operator equations

This article is devoted to the theory and applications of matrix operator equations in normed spaces. We describe in detail the general properties of matrix operators and their representing matrices. As the indexing set we take an arbitrary countable set. This is related to stochastic applications in which it is difficult to find an enumeration that fits well with the content. Sample applications are the Markov matrices and the operators they define. Comparisons with operators of this type arise in some stochastic problems of integral geometry and tomography.     

Generalized inverses and operator equations

In this paper we obtain conditions under which the operator equations of the types AX = C and $\text{AXA}{}^1\text{= C}$ have hermitian and nonnegative definite solutions; here A is assumed to be relatively invertible. In addition we obtain some properties of generalized inverses of operators. Lastly we pose some conjectures; one of them is that the set of all nonzero relatively invertible operators is not connected.     

On holomorphic operator function equations

Let A be an operator function which is holomorphic on a region G ^N and which has a global meromorphic relative inverse. It was proved by B. Gramsch [5],[6] that the operator equation A () = () (A () = ()), holomorphic on G, has a global meromorphic solution if it admits a pointwise solution on some nonempty open subset of G. In this paper we prove similar results for operator functions which do not have global relative inverses. The class of functions for which the result remains valid includes the semi-Fredholm functions. As in the case of functions which have global relative inverses, the theorems are much sharper for functions of one variable. In particular, for one variable many special properties of the function can be transferred to the solution, if desired.The research for this paper was done while the second author was employed at the Vrije Universiteit in Amsterdam.     

Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations

We develop index theories for linear selfadjoint operator equations and investigate multiple solutions for asymptotically linear operator equations. The operator equations consist of two kinds: the first has finite Morse index and can be used to investigate second order Hamiltonian systems and elliptic partial differential equations; the second may have infinite Morse index and can be used to investigate first order Hamiltonian systems.     

Composite wavelet bases for operator equations

This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit -cube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems, although this study is primarily motivated by our previous analysis of wavelet methods for pseudo-differential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales, as well as appropriate moment conditions.

     

Solvability of mixed type operator equations

In this paper we prove the existence of monotonic solution of the mixed type operator equations. We discuss also the existence of solutions of nonlinear integral equations of fractional orders. The technique rely on the concept of measure of noncompactness and its associated Darbo fixed point theorem.     

Approximation methods for nonlinear operator equations

Let be a real normed linear space and be a uniformly quasi-accretive map. For arbitrary define the sequence by where is a positve real sequence satisfying the following conditions: (i) ; (ii) . For , assume that 0$"> and that , where (the set of all positive integers): and is a strictly increasing function with . It is proved that a Mann-type iteration process converges strongly to . Furthermore if, in addition, is a uniformly continuous map, it is proved, without the condition on , that the Mann-type iteration process converges strongly to . As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.

     

Hermitian operator methods for reaction-diffusion equations

A variety of time-linearization, quasilinearization, operator-splitting, and implicit techniques which use compact or Hermitian operators has been developed for and applied to one-dimensional reaction-diffusion equations. Compact operators are compared with second-order accurate spatial approximations in order to assess the accuracy and efficiency of Hermitian techniques. It is shown that time-linearization, quasilinearization, and implicit techniques which use compact operators are less accurate than second-order accurate spatial discretizations if first-order approximations are employed to evaluate the time derivatives. This is attributed to first-order accurati temporal truncation errors. Compact operator techniques which use second-order temporal approximations are found to be more accurate and efficient than second-order accurate, in both space and time, algorithms. Quasilinearization methods are found to be more accurate than time-linearization schemes. However, quasilinearization techniques are less efficient because they require the inversion of block tridiagonal matrices at each iteration. Some improvements in accuracy can be obtained by using partial quasilinearization and linearizing each equation with respect to the variable whose equation is being solved. Operator-splitting methods which use compact differences to evaluate the diffusion operator were found to be less accurate than operator-splitting procedures employ second-order accurate spatial approximations. Comparisons among the methods presented in this paper are shown in terms of the L²-norm errors and computed wave speeds for a variety of time steps and grid spacings: The numerical efficiency is assessed in terms of the CPU time required to achieve the same accuracy.     

Shift-invariant spaces and linear operator equations

In this paper we investigate the structure of finitely generated shift-invariant spaces and solvability of linear operator equations. Fourier trans-forms and semi-convolutions are used to characterize shift-invariant spaces. Criteria are provided for solvability of linear operator equations, including linear partial difference equations and discrete convolution equations. The results are then applied to the study of local shift-invariant spaces. Moreover, the approximation order of a local shift-invariant space is characterized under some mild conditions on the generators. Supported in part by NSERC Canada under Grant OGP 121336.     

Unsymmetric meshless methods for operator equations

A general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well-posed linear operator equations is presented. The results provide optimal convergence rates, if the test and trial spaces satisfy a stability condition. Operators need not be elliptic, and the problems can be posed in weak or strong form without changing the theory. Non-stationary kernel-based trial and test spaces are shown to fit into the framework, disregarding the operator equation. As a special case, unsymmetric meshless kernel-based methods solving weakly posed problems with distributional data are treated in some detail. This provides a foundation of certain variations of the “Meshless Local Petrov-Galerkin” technique of S.N. Atluri and collaborators.     

On analytic solutions of operator differential equations

We find conditions on a closed operator A in a Banach space that are necessary and sufficient for the existence of solutions of a differential equation y′(t) = Ay(t), t ∈[0,∞),in the classes of entire vector functions with given order of growth and type. We present criteria for the denseness of classes of this sort in the set of all solutions. These criteria enable one to prove the existence of a solution of the Cauchy problem for the equation under consideration in the class of analytic vector functions and to justify the convergence of the approximate method of power series. In the special case where A is a differential operator, the problem of applicability of this method was first formulated by Weierstrass. Conditions under which this method is applicable were found by Kovalevskaya.     

Positive solutions of operator equations on half-line

This paper is concerned with an operator equation on half-line. Some theorems for the existence of its positive solutions are obtained by using the Krasnosel'skii-Guo theorem on cone expansion and compression in a special function space.