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Kinetical systems. (English) Zbl 0903.34043

The system \[ \frac {dy}{dx} = AG(t,y), \quad (t,y) \in \mathbb{R} \times \mathbb{R}^n \tag \(*\) \] occuring in physical chemistry and biology is investigated. Here is \(A:=C-C'\) where \(C,C'\) are \(n\times m\) matrices whose elements \(c_{ij}\), \(c'_{ij}\) are nonnegative integers such that rank\((A)<m\) and \(G: = [G_j]\) is an \(m\)-vector \[ G_j (t,y) : = -r_j(t) \prod ^n_{k=1} y_k^{c_{kj}} + d_j (t) \prod ^n_{k=1} y_k^{c_{kj}'}. \] It is supposed that \(r_j\), \(d_j : \mathbb{R} \rightarrow [0,\infty)\) are continuous functions.
Using the method of inequalities, the elementary fixed point method and the invariance principle the author derives some basic properties of nonnegative solutions to the nonautonomous system \((*)\), proves the existence of a critical point in the autonomous case and derives some asymptotic properties of a special form of \((*)\).
Reviewer: M.Ráb (Brno)

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
92E20 Classical flows, reactions, etc. in chemistry
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References:

[1] J. K. Hale: Ordinary Differential Equations. R.E.Krieger Pub. Comp. INC. Malabar, Florida, 1980, 2nd · Zbl 0433.34003
[2] Ph. Hartman: Ordinary Differential Equations. John Wiley & Sons Inc., New York-London-Sydney, 1964. · Zbl 0125.32102
[3] N. Rouche, P. Habets, M. Laloy: Stability Theory by Liapunov’s Direct Method. Springer Verlag, New York-Berlin-Heidelberg, 1977. · Zbl 0364.34022
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