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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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