Nonautonomous Differential Equations in Banach Space and Nonrectifiable Attractivity in Two-Dimensional Linear Differential Systems

Abstract

We study the asymptotic behaviour on a finite interval of a class of linear nonautonomous singular differential equations in Banach space by the nonintegrability of the first derivative of its solutions. According to these results, the nonrectifiable attractivity on a finite interval of the zero solution of the two-dimensional linear integrable differential systems with singular matrix-elements is characterized.

Characterization for Rectifiable and Nonrectifiable Attractivity of Nonautonomous Systems of Linear Differential Equations

“…Then , ∈ 1 ([ 0 , ∞)). Let 0 be a constant matrix determined in (4). If 0 is a new matrix defined by 0 = − 0 , then from (4), (23), and = 1/ , we conclude…”

“…, be matrix elements of ( ). Also, let 0 , and 0 = Re 0 ± Im 0 be matrix elements and eigenvalues of 0 , respectively, where 0 appears in (4). Rewriting (4) in terms of matrix elements and eigenvalues, we get…”

“…In the next theorem, the nonrectifiable attractivity is obtained without supposing the a priori estimates (37) and (58), since they immediately follow from the asymptotic behaviour of ( ) near = 0 given in (4).…”

“…Finally, we point out that in some essential cases, the model system (7) allows the explicit form of all its solutions. Such systems are called the integrable systems; for details, see [4]. Hence, the rectifiable and nonrectifiable attractivity of the zero solution of (7) in such cases can be reproven in an explicit way.…”

“…The proofs of the main results are given in Sections 4, 5, and 6. Such a kind of topics has been considered for the first time in [4] but only in the case of the so-called integrable systems (systems that allow all solutions in explicit forms), where the asymptotic integration near = 0 is not required. About the asymptotic integration near = ∞ of differential equations and systems, we refer reader to [5][6][7][8][9][10][11][12].…”