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On the asymptotic behavior at infinity of solutions to quasi-linear differential equations. (English) Zbl 1224.34098
Summary: Sufficient conditions are formulated for the existence of non-oscillatory solutions to the equation \[ y^{(n)}+\sum _{j=0}^{n-1}a_{j}(x)y^{(j)}+p(x)| y| ^{k} \operatorname {sgn} y =0 \] with \( n\geq 1\), real (not necessarily natural) \(k>1\), and continuous functions \(p\) and \(a_{j}\) defined in a neighborhood of \(+\infty \). For this equation, with positive potential \(p\), a criterion is formulated for the existence of non-oscillatory solutions with non-zero limit at infinity. In the case of even order, a criterion is obtained for all solutions of this equation at infinity to be oscillatory. Sufficient conditions are obtained for the existence of a solution to this equation which is equivalent to a polynomial.

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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