Nguyen Van Mau

On a class of singular integral equations with the linear fractional Carleman shift and the degenerate kernel

This article deals with the solvability, the explicit solutions of a class of singular integral equations with a linear-fractional Carleman shift and the degenerate kernel on the unit circle by means of the Riemann boundary value problem and of a system of linear algebraic equations. All cases about index of the coefficients in the equations are considered in detail.

ALGEBRAIC PROPERTIES OF GENERALIZED RIGHT INVERTIBLE OPERATORS

0. Introduction The theory of right invertible operators was started with works of D. Przeworska-Rolewicz [8]—[12] and then has been developed by M. Tasche [13]—[14], H. von Trotha [15], Z. Binderman [3] and many others (see [12]). The algebraic theory of generalized invertible operators was studied by P. M. Anselone and M. Z. Nashed [1], A. Ben-Israel and T. N. E. Greville [2], S. G. Caradus [4], M. Z. Nashed [5] and others (see [2]). However, the set of all generalized invertible operators is so large that, if we admit the axiom of choice, then every linear operator is generalized invertible [5]. Whereas, the generalized invertible operators do not satisfy desirable algebraic properties which the right invertible operators do (see [12]). For example, if a linear operator V € L(X) is generalized invertible and W is a generalized inverse of V, then there is not any general algorithm for constructing generalized inverses neither of V, n € N, nor of algebraic polynomials induced by V. Hence, there is the lack of effective methods to solve equations induced by algebraic polynomials with a generalized invertible operator.

PROPERTIES OF GENERALIZED ALMOST INVERSES

1. Let X be a linear space over a field 9 of scalars (were ?=IR or 5=C) . Denote by L(X) the set of all linear operators with domains and ranges in X. Let L Q(X)={AeL(X): dom A=X}. Following [10], denote by R(X) the set of all right invertible operators belonging to L(X). For a D€R(X) we write: K D={R€L 0(X): DR=I}, 9 n={FeL (X):F =F, FX=kerD, 3 FR=0}. R 6 * D Definition 1. (i) An operator VeL(X) is said to be a generalized almost invertible operator if there is an operator W e LQ(X) such that

INTERPOLATION PROBLEMS INDUCED BY RIGHT AND LEFT INVERTIBLE OPERATORS AND ITS APPLICATIONS TO SINGULAR INTEGRAL EQUATIONS

The interpolat ion problems induced by right invert ible operators were investigated by Przeworska-Rolewicz [ l ] . In the present paper we obtain necessary and suf f i c ient condit ions for general interpolation problems induced by r ight and l e f t invert ible operators to have a unique solution. As applicat ions , we deal with interpolation problems from theory of singular integral operators.

On a nonlinear difference equation with variable delay

Abstract In this paper, some results on the equi-boundedness of solutions, the stability of the zero and the existence of positive periodic solutions of nonlinear difference equation with variable delay xn+1=λnxn+αnF(xn−mn),    n=0, 1,…

GENERALIZED c(R) PROPERTY AND INTERPOLATION PROBLEMS INDUCED BY RIGHT INVERTIBLE OPERATORS

x_{n + 1} = \lambda _n x_n + \alpha _n F(x_{n - m_n } ),\,\,\,\,n = 0,\,1, \ldots

Operational Properties of Two Integral Transforms of Fourier Type and their Convolutions

are obtained.

On solutions of integral equations with analytic kernels and rotations

This paper gives the criteria for the system of initial operators to possess the c(ii)-property and the generalized c(ii)-property.