Generalized dichotomy for ordinary differential equations in a Banach space
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.
scientific article; zbMATH DE number 6049643