Nguyen Minh Tuan

Generalized convolutions and the integral equations of the convolution type

This article gives six new generalized convolutions of the integral transforms of Fourier type, and investigates a class of integral equations of convolution type by using the constructed convolutions. Namely, the explicit solutions in L 1(ℝ d ) of a class of integral equations of convolution type are obtained.

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.

Applications of generalized convolutions associated with the Fourier and Hartley transforms

In this paper we present new generalized convolutions with weight-function associated with the Fourier and Hartley transforms, and consider applications. Namely, using the generalized convolutions, we construct normed rings on the space L(R), provide the sufficient and necessary condition for the solvability of a class of integral equations of convolution type, and receive the explicit solutions of those equations.

ON THE SOLVABILITY OF SYSTEMS OF LINEAR EQUATIONS ON COMMUTATIVE SEMIGROUP

are necessary for the system (1.1) to have a solution in S. However, in general, they do not provide sufficient conditions for system (1.1) to be solvable. In [4], Theorems 2.4 and 2.5 gave two conditions on the operators so that the compatibility conditions are sufficient for the solvability of (1.1). The purpose of this paper is to study some kinds of conditions on the operators lt so that the compatibility conditions will be sufficient for the system (1.1) to have a solution in S. Theorems 2.2 and 2.4 (Corollaries 2.1 and 2.2) consider li in a class of generalized invertible operators (in a class of one-side invertible operators) respectively. Each of Theorems 2.3, 2.6, 2.7 deals with conditions on li which are weaker than that of Theorems 2.4, 2.5 in [4], under which the system (1.1) to be solvable in S. Theorems 2.8, 2.9 and 2.10 provide another

On the Criteria for Initial Operators Possessing (c)-Property and Generalized (c)-Property

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