Measure compact,almost compact,and integral operators of the first,second, and third kind

Under study are the metric, algebraic, spectral, and characteristic properties of measure compact, almost compact, and integral operators of the first, second, and third kind in spaces of integrable functions.     

A third order,semi-explicit method in the numerical solution of first kind Volterra integral equations

We present a new numerical method for the solution of nonsingular Volterra integral equations of the first kind. It belongs to a new class of methods that are semi-explicit, provide self-starting algorithms and possess favourable stability properties. The third order convergence of the particular method exhibited is established, under suitable conditions, and numerical results are illustrated.     

Fredholm equations of the first kind

A method is described for solving the Fredholm integral equation of the 1st kind by passage to the moment L-problem. The problem is reduced to a linear programming problem. A bound of the method for a particular normed space is derived. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 34–38.     

Periodic solutions for a kind of second order neutral functional differential equations

In this paper, criteria are established for the existence of periodic solutions to a second order sublinear neutral differential equation. Our method is based on careful a priori estimation and continuation theorem, and our sublinear condition is an improvement of the boundedness condition in some recent results.     

On volterra equations of the first kind

The existence of a solution β of the equation $$\int_0^t {a(t - s)d\beta (s) = 1, t > 0} $$ is studied under fairly general assumptions on the function a. Sufficient conditions for the measure β to be absolutely continuous or satisfy some additional regularity properties are given. An extension to nonconvolution kernels is also considered.     

On the theory of convolution equations of the third kind, II

Existence results of Part I of the paper are generalized to two types of autoconvolution equations of the third kind having free terms with nonzero values at x=0 like the well-known Bernstein-Doetsch equation for the Jacobian theta zero functions. Also uniqueness results for the linear convolution equations in Part I of the paper are extended to more general function spaces. Further, a special class of integro-differential equations with autoconvolution integral and two classes of the linear singular Abel-Volterra equations are dealt with.     

Projection methods for equations of the second kind

A class of projection methods, differing from the classical projection methods, is studied for the equation y = f + Ky, where K is a compact linear operator in a Banach space E, and f?E, In these methods K is approximated by a finite-rank operator K_n, which is constructed with the aid of certain projection operators, and which satisfies K_nz = Kz for all z belonging to a chosen subspace U_n ? E. Under certain conditions, it is shown that the convergence of the approximate solution is faster than that of any classical projection method based on the subspace U_n. In an example, U_n is taken to consist of piecewise constant functions, and the projections are so chosen that the method becomes equivalent to a single iteration of a classical method, the collocation method; in this case the error (in the supremum norm) is $\text{O(}\text{1}\text{n}{}^2\text{)}$, compared with $\text{O(}\text{1}\text{n}\text{)}$ for the collocation method.     

Integral equations of the third kind with unbounded operators

We consider linear functional equations of the third kind in L ₂ with arbitrary measurable coefficients and unbounded integral operators with kernels satisfying broad conditions. We propose methods for reducing these equations by linear continuous invertible transformations either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. To the integral equations obtained after the reduction, one can apply various exact and approximate methods of solution; in particular, the two approximate methods developed in this article.     

Autoconvolution equations of the third kind with power-logarithmic coefficients

Existence and uniqueness theorems for a basic class of autoconvolution equations of the third kind with power-logarithmic coefficeint and free term are derived. Under suitable assumptions the existence of a solitary solution and of a one-parametric family of solutions is proved.     

On the theory of convolution equations of the third kind

The autoconvolution equation of the third kind with coefficient of general power type is dealt with by the method of weighted norms developed for equations with coefficients of linear and integer power type in recent joint work of the author with L. Berg, J. Janno, and B. Hofmann. For this equation two existence theorems and a uniqueness theorem are proved. Further, as an auxiliary equation a linear singular integral equation of Abel is treated anew and the existence of solutions to a related class of linear Volterra equations of the third kind is derived.     

Systems of convolution equations of the first and second kind on a finite interval and factorization of matrix-functions

Systems of n convolution equations of the first and second kind on a finite interval are reduced to a Riemann boundary value problem for a vector function of length 2n. We prove a theorem about the equivalence of the Riemann problem and the initial system. Sufficient conditions are obtained for the well-posedness of a system of the second kind. Also under study is the case of the periodic kernel of the integral operator of a system of the first and second kind.     

On the similarity of linear operators in L p to integral operators of the first and second kind

We construct an example of a compact operator of the third kind in L _p (p ≠ 2) not similar to any integral operator of the first or second kind. This example shows that not every linear equation of the third kind in L _p (p ≠ 2) can be reduced by an invertible continuous linear change to an equivalent integral equation of the first or second kind. The example also proves the impossibility of a characterization of integral and Carleman integral operators in L _p (p ≠ 2) in terms of the spectrum and its components.     

Linear differential equations and functional analysis,V

Mathematische Annalen -