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Asymptotic behavior of solutions of nonautonomous half-linear differential systems. (English) Zbl 1174.34042
Summary: This paper is concerned with the asymptotic behavior of solutions of the class of second-order half-linear differential equations \[ (\varphi_p (\dot X ))+a (t) \varphi_ p (\dot X ) + b ( t ) \varphi_ p ( x ) = 0. \] The main purpose of this paper is to answer the question of how every solution approaches zero, under the assumption that the zero solution is globally asymptotically stable. Sufficient conditions are also given for the zero solution to be globally asymptotically stable. Moreover, an autonomous case is investigated in full detail and a geometrical classification is made based on the asymptotic behavior of solutions. The method used here is mainly phase plane analysis for a system equivalent to the half-linear differential equations. Some suitable examples are included to illustrate the main results. Global phase portraits are also attached for a deeper understanding.

MSC:
34D05 Asymptotic properties of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
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