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On the positive solutions of a model system of nonlinear ordinary differential equations. (Russian. English summary) Zbl 1459.34057

Summary: This article investigates the properties of positive solutions of a model system of two nonlinear ordinary differential equations with variable coefficients. We found the new conditions on coefficients for which an arbitrary solution \((x(t), y(t))\) with positive initial values \(x(0)\) and \(y(0)\) is positive, nonlocally continued and bounded at \(t>0\). For this conditions we investigated the question of global stability of positive solutions via method of constructing the guiding function and the method of limit equations. Via the method of constructing the guide function we proved that if the system of equations has a positive constant solution \((x_*, y_*)\), then any positive solution \((x(t), y(t))\) at \(t\rightarrow +\infty\) approaches \((x_*, y_*)\). And in the case when the coefficients of the system of equations have finite limits at \(t\rightarrow +\infty\) and the limit system of equations has a positive constant solution \((x_{\infty}, y_{\infty})\), via method of limit equations we proved that any positive solution \((x(t), y(t))\) at \(t\rightarrow +\infty\) approaches \((x_{\infty}, y_{\infty})\). The results obtained can be generalized for the multidimensional analog of the investigated system of equations.

MSC:

34A34 Nonlinear ordinary differential equations and systems
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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References:

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