| Non-Autonomous Systems: Asymptotic Behaviour and Weak Invariance Principles (2003) | |||||||||||||||
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| INTRODUCTION The initial-value problem for a non-autonomous ordinary dierential equations of the form _ x = f(t; x), x(0) = is considered, where f is of Caratheodory class (and so the initial-value problem has a solution and every solution can be maximally extended). Motivated by [18], one of the basic questions addressed in the paper is the following: if x is a global (forward-time) solution of the initial-value problem and g x 2 L for some function g, then what can one deduce about the asymptotic behaviour of x ? In the autonomous case _ x = f(x) with locally Lipschitz right-hand side and denoting, by ', the generated ow, the following observation is a consequence of results in [9]: let : [0; 1) ! [0; 1) be a continuous function with (s) = 0 , s = 0 and lim inf s!1 (s) > 0; if (k'(; )k) 2 L , then x(t) = '(t; ) ! 0 as t !1 (and so, in particular, for 1 p < 1, L trajectories of ows converge to zero as t !1). By contrast, if x is a global solution of the non-a | |||||||||||||||
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