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Global asymptotic stability of a class of nonautonomous integro-differential systems and applications. (English) Zbl 1058.45004

In Section 2 of this paper the authors establish some fundamental criteria for global asymptotic stability of the following \(n\)-dimensional nonautonomous integro-differential system \[ \dot x_{i}(t)= -b_{i}(t)x_{i}(t)+f_{i}(t,x_{1}(t),\ldots,x_{n}(t); x_{1}(t-\tau_{i1}(t)), \ldots, x_{n}(t-\tau_{in}(t)); \]
\[ \int_{-\infty}^{t}k_{i1}(t-s)x_{1}(s)\,ds,\ldots, \int_{-\infty}^{t}k_{in}(t-s)x_{n}(s)\,ds), \quad t\geq 0, \;i=1,\ldots, n. \] Section 3 is concerned with the existence and global asymptotic stability of an unique equilibrium of the autonomous case of the above cited system. The following section is devoted to the periodic case in which the coincidence degree theory is used. In the last section, some particular models arising from mathematical biology and neural networks are presented.

MSC:

45J05 Integro-ordinary differential equations
45G15 Systems of nonlinear integral equations
45M10 Stability theory for integral equations
92B20 Neural networks for/in biological studies, artificial life and related topics
92D25 Population dynamics (general)
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