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A theorem on average Liapunov functions. (English) Zbl 0542.34043
For a system of ordinary differential equations, J. Hofbauer [ibid. 91, 233-240 (1981; Zbl 0449.34039)] gave a criterion, framed in terms of an ’average’ Lyapunov function, for a compact set to be a repeller (in a certain strong sense). From the point of view of applications this criterion is extremely useful, as it is often considerably easier to find an average Lyapunov function than a Lyapunov function in the usual sense. The aim of this paper is to increase the range of applicability of Hofbauer’s result by removing certain unnecessary technical restrictions. Our result is illustrated by applying it to prove a rather strong coexistence criterion for an ecological system.

MSC:
34D20 Stability of solutions to ordinary differential equations
92D25 Population dynamics (general)
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