Givi P Giorgadze’s scientific contributions

In the present paper we consider the following Satisfaction Problem of Consumers Demands (SPCD): The supplier must supply the measurable system of the measure mk to the k-th consumer at the moment tk for 1 � k � n. The measure of the supplied measurable system is changed under action of some dynamical system; What minimal measure of measurable system must take the supplier at the initial moment t = 0 to satisfy demands of all consumers ? In this paper we consider Satis- faction Problem of Consumers Demands measured by ordinary “Lebesgue measures” in R1 for various dynamical systems in R1. In order to solve this problem we use Liouville type theorems for them which describes the dependence between initial and resulting measures of the entire system.

By using an infinite-dimensional "Lebesgue measure" in an infinite-dimensional separable Banach space B with Schauder basis a solution of a heat equation with initial value problem on B is constructed. Properties of uniformly distributed real-valued sequences in an interval of the real axis are used for a construction of a certain algorithm which gives an approximation of corresponding solutions.

By using the technique of "Fourier differential operator" in R ∞ and Laplace transforms, a representation in a multiple trigonometric series of the solution of a certain generalized heat equation of many variables is obtained.

This article presents main results of investigations of the authors which were obtained during the last five years by the partially support on the Shota Rustaveli National Science Foundation (Grant no. 31–24). These results are Liouville-type theorems and describe the behavior of various phase motions in terms of ordinary and standard “Lebesgue measures” in R∞. In this context, the following three problems are discussed in this paper: Problem 1. An existence and uniqueness of partial analogs of the Lebesgue measure in various function spaces; Problem 2. A construction of various dynamical systems with domain in function spaces defined by various partial differential equations; Problem 3. To establish the validity of Liouville-type theorems for various dynamical systems with domains in function spaces in terms of partial analogs of the Lebesgue measure.

By using a structure of the "Fourier differential operator" in R ∞ , we describe a new and essentially different approach for a solution of the old functional problem posed by R. D. Carmichael in 1936. More precisely, under some natural restrictions, we express in an explicit form the general solution of the linear inhomogeneous differen-tial equation of infinite order with real constant coefficients. In addition, we construct an invariant measure for the corresponding differential equation. Also, we describe a certain approach for a solution of an initial value problem for a special class of linear inhomogeneous partial differential equations of infinite order with real constant coefficients and describe behaviors of corresponding dynamical systems in terms of partial analogs of the Lebesgue measure.

By using a technique developed in [1], we describe a new and essentially different approach for a solution of the old functional problem (a) posed by R. D. Carmichael in [2](cf. p. 199). More precisely, under some general restrictions, we express in an explicit form the general solution of the non-homogeneous ordinary differential equation of the infinite order with real constant coefficients. In addition, we construct an invariant measure for the corresponding differential equation. By using the structure of the "Fourier differential operator" in R ∞ , we give a certain approach for a solution of an initial value problem for a special class of linear and non-homogeneous partial differential equations (with constant coefficients) of infinite order in two variables. Also, we describe behaviors of corresponding dynamical systems in R ∞ in terms of ordinary and standard "Lebesgue measures" [3]. References [1] Pantsulaia G., Giorgadze G., On some applications of infinite-dimensional cellular matrices,