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Lyapunov numbers for a countable system of ordinary differential equations. (English) Zbl 0731.34052
The paper is devoted to the investigation of the asymptotic properties of the system of differential equations \[ (1)\quad \dot x_ i=g_ i(t)x_ i+\sum^{\infty}_{j=1,\quad j\neq i}(g_{ij}(t)+\tilde g_{ij}(t))x_ j,\quad x_ i(t_ 0)=x_ i^ 0,\quad i=1,2,..., \] where \(g_ i(t)\), \(g_{ij}(t)\) and \(\tilde g_{ij}(t)\) are continuous functions (in general, complex-valued) of the real variable t for \(t_ 0\leq t<\infty\) and \(x^ 0=(x^ 0_ 1,x^ 0_ 2,...)\in C\), C being the space of convergent sequences. The system (1) can be written in the form \[ (2)\quad \dot x=A(t)x,\quad x(0)=x^ 0. \] Suppose that for each \(t\in [0,\infty)\), A(t) is the infinitesimal generator of a \(C^ 0\)- semigroup, A(t): \(D\subset C\to C\) is densely defined and assume the existence, uniqueness and continuous dependence of the solutions of (1) in \([0,\infty)\). Using the Ważewski topological method, the author shows that under some convenient conditions on \(g_ i(t)\), \(g_{ij}(t)\), \(\tilde g_{ij}(t)\), there exists a system of linearly independent solutions \((x_ i(t))^{\infty}_{i=1}\), \(x_ i(t)=(x_{1i},x_{2i},...)\) such that \[ \lim_{t\to \infty}(x_{ik}(t)/x_{kk}(t))=0 \] for every \(i\neq k\). The sufficient conditions for \[ \lambda (x_ i(t))=\limsup_{t\to \infty}\frac{1}{t}\int^{t}_{0}Re(g_ i(s))ds,\quad i=1,2,..., \] where \(\lambda (x_ i(t))\) is the Lyapunov number of the solution \(x_ i(t)\) are given too. The results generalize those of Z. Szmydt and O. Perron.
Reviewer: J.Kalas (Brno)
MSC:
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
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