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Existence of positive solutions of higher-order quasilinear ordinary differential equations. (English) Zbl 1232.34054

Summary: The even-order quasilinear ordinary differential equation \[ D(\alpha _{n})D(\alpha _{n-1})\dots D(\alpha _{1})x+p(t)\left| x\right|^{\beta -1}x=0 \tag{1} \] is considered under the hypotheses that \(n\) is even, \(D(\alpha _{ i })x = (|x|\alpha i - 1 x)'\), \(\alpha _{ i } > 0(i = 1,2,\dots,n)\), \(\beta > 0\), and \(p(t)\) is a continuous, nonnegative, and eventually nontrivial function on an infinite interval \([a,\infty)\), \(a > 0\). The existence of positive solutions of \((1)\) is discussed, and basic results to the classical equation \(x^{(n)} + p(t)|x|^{b-1}x=0\) are extended to the more general equation \((1)\). In particular, necessary and sufficient integral conditions for the existence of positive solutions of \((1)\) are established in the case \(\alpha_{1}\alpha _{2}\dots \alpha _{ n } \neq \beta \).

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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