Naslov
Nonautonomous differential equations in Banach space and nonrectifiable attractivity in two- dimensional linear differential systems

Autori
Miličić, Siniša ; Pašić, Mervan

Izvornik
Abstract and applied analysis (1085-3375) 2013 (2013); 935089-1

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
linear differential systems; nonautonomous; attractivity; rectifiability

Sažetak
We study the asymptotic behaviour near $t=0$ of all solutions $\mathbf{; ; ; x}; ; ; \in C^1((0, t_0] ; \mathbb{; ; ; X}; ; ; )$ of linear nonautonomous differential equation $(1.1)$: $\mathbf{; ; ; x}; ; ; '=A(t)\mathbf{; ; ; x}; ; ; $, $t\in (0, t_0]$, where $\mathbb{; ; ; X}; ; ; $ is an arbitrary Banach space and $A\colon(0, t_0]\to L(\mathbb{; ; ; X}; ; ; )$ is an operator-valued function which may be singular at $t=0$. In terms of asymptotic behaviour of the operator norm $\|A(t)\|$ near $t=0$, we characterize a kind of singular behaviour near $t=0$ of all solutions $x(t)$ of equation $(1.1)$ by the nonintegrability of $\|\mathbf{; ; ; x}; ; ; '\|_{; ; ; \mathbb{; ; ; X}; ; ; }; ; ; $ on the interval $(0, t_0]$. Next, according to previous results in particular when $\mathbb{; ; ; X}; ; ; =\mathbb{; ; ; R}; ; ; ^2$ and $(1.1)$ is a linear integrable system, we study the so-called nonrectifiable attractivity of the zero solution of $(1.1)$ as a geometric kind of singular behaviour of all solutions of $(1.1)$ near $t=0$: under some sufficient conditions, the convergence to zero of $\|\mathbf{; ; ; x}; ; ; \|_{; ; ; \mathbb{; ; ; X}; ; ; }; ; ; $ as $t$ goes to $0$ as well as the infiniteness of length of corresponding solution curve of $x(t)$ are characterized in terms of singular behaviours of matrix-elements of $A(t)$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika, Temeljne tehničke znanosti

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