Generalized dichotomy for ordinary differential equations in a Banach space
type: Item
instance of: scholarly article
zbMATH Open document ID: 1257.34041
publication date: 2012-06-25T00:00:00Z
full work available at URL: index.html
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34D10
zbMATH DE Number: 6049643
zbMATH Keywords: ordinary differential equations
zbMATH Keywords: generalized dichotomy
author: A. Georgieva
author: H. Kiskinov
MaRDI profile type: Publication
title: Generalized dichotomy for ordinary differential equations in a Banach space
published in: International Journal of Pure and Applied Mathematics
review text: From the text: Let \(X\) be an arbitrary Banach space with norm \(|\cdot|\) and identity \(I\) and let be \(J = [c, \infty)\), where \(c \in \mathbb{R}\). Let \(L(X)\) be the space of all linear bounded operators acting on \(X\) with the norm \(\|\cdot\|\). We consider the linear equation NEWLINE\[NEWLINE\frac{dx}{dt} = A(t)x,NEWLINE\]NEWLINE where \(A(t) \in L(X)\), \(t \in J\). The notion of exponential and ordinary dichotomy is fundamental in the qualitative theory of ordinary differential equations. In the given paper we introduce a (\(M, N, R\)) dichotomy, which is a generalisation of all dichotomies known by the authors. The aim of this paper is to study this dichotomy and the connection between it and the existence of a special kind of solutions to the related non-homogeneous differential equation. A special kind of roughness of the (\(M, N, R\)) dichotomy is introduced and considered. An example for an equation which is (\(M, N, R\)) dichotomous but not classical dichotomous is given.
reviewed by: S. Kostadinov
description: scientific article; zbMATH DE number 6049643