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Sharp oscillation criteria for a class of fourth order nonlinear differential equations. (English) Zbl 1232.34053
The authors study the fourth order differential equation \[ (p(t)|x''|^{\alpha-1}x'')''+q(t)|x|^{\beta-1}x=0,\tag{1} \] where \(\alpha>0\), \(\beta>0\) are constants and \(p,\;q:[a,\infty)\to(0,\infty)\) are continuous functions.
They consider the super-half-linear case \(\alpha\leq 1<\beta\) and sub-half-linear case \(\beta<1\leq \alpha\) and prove for each case a necessary and sufficient condition of integral type which ensures that all solutions of (1) are oscillatory, i.e. every solution of (1) has an infinite sequence of zeros tending to infinity.
The results are of similar nature as the corresponding results of F. V. Atkinson [Pac. J. Math. 5, 643–647 (1955; Zbl 0065.32001)] and S. Belohorec [Mat.-Fyz. Čas., Slovensk. Akad. Vied 11, 250–254 (1961; Zbl 0108.09103)] derived for second order Emden–Fowler differential equation.
The proofs are based on suitable classification of all nonoscillatory solutions into six classes. Nonexistence of eventually positive solution is proved for each class separately.

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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[1] F.V. Atkinson, On second order nonlinear oscillation , Pacific J. Math. 5 (1955), 643-647. · Zbl 0065.32001 · doi:10.2140/pjm.1955.5.643
[2] S. Belohorec, Oscillatory solution of certain nonlinear differential equations of second order , Mat. Fyz. Casopis Sloven Akad. Vied. 11 (1961), 250-255. · Zbl 0108.09103
[3] K.-I. Kamo and H. Usami, Oscillation theorems for fourth-order quasilinear ordinary differential equations , Stud. Sci. Math. Hung. 39 (2002), 385-406. · Zbl 1026.34054 · doi:10.1556/SScMath.39.2002.3-4.10
[4] T. Kusano and M. Naito, Nonlinear oscillation of fourth order differential equations , Canada. J. Math. 28 (1976), 840-852. · Zbl 0432.34022 · doi:10.4153/CJM-1976-081-0
[5] —, On fourth-order non-linear oscillation , J. London Math. Soc. 14 1976, 91-105. · Zbl 0385.34014 · doi:10.1112/jlms/s2-14.1.91
[6] T. Kusano and T. Tanigawa, On the structure of positive solutions of a class of fourth order nonlinear differential equations , Ann. Mat. Pura Appl. 85 (2006), 521-536. · Zbl 1232.34056 · doi:10.1007/s10231-005-0165-5
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