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Keywords:
oscillatory; nonoscillatory; Riccati differential equation; Sturm Comparison Theorem
Summary:
In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation \[ (r(t)\Phi (u^{\prime }(t)))^{\prime }+c(t)\Phi (u(t))=0, \] where (i) $r,c\in C([t_{0}, \infty )$, $\mathbb{R}:=(-\infty , \infty ))$ and $r(t)>0$ on $[t_{0},\infty )$ for some $t_{0}\ge 0$; (ii) $\Phi (u)=|u|^{p-2}u$ for some fixed number $p> 1$. We also generalize some results of Hille-Wintner, Leighton and Willet.
References:
[1] W. J.  Coles: A simple proof of a well-known oscillation theorem. Proc. Amer. Math. Soc. 19 (1968), 507. MR 0223644 | Zbl 0155.12802
[2] Á.  Elbert: A half-linear second order differential equation. Colloquia Math. Soc.  J. Bolyai 30: Qualitivative Theorem of Differential Equations, Szeged, 1979, pp. 153–180. MR 0680591
[3] A. M.  Fink and D. F. St.  Mary: A generalized Sturm comparison theorem and oscillatory coefficients. Monatsh. Math. 73 (1969), 207–212. DOI 10.1007/BF01300536 | MR 0244561
[4] B. J.  Harris: On the oscillation of solutions of linear differential equations. Mathematika 31 (1984), 214–226. DOI 10.1112/S0025579300012432 | MR 0804196 | Zbl 0574.34015
[5] E.  Hille: Non-oscillation theorems. Trans. Amer. Math. Soc. 64 (1948), 234–252. DOI 10.1090/S0002-9947-1948-0027925-7 | MR 0027925 | Zbl 0031.35402
[6] A.  Kneser: Untersuchungen über die reelen Nullstellen der Integrale linearer Differentialgleichungen. Math. Ann. 42 (1893), 409–435. DOI 10.1007/BF01444165 | MR 1510784
[7] M. K.  Kwong and A.  Zettl: Integral inequalities and second order linear oscillation. J.  Diff. Equations 45 (1982), 16–33. DOI 10.1016/0022-0396(82)90052-3 | MR 0662484
[8] W.  Leighton: The detection of the oscillation of solutions of a second order linear differential equation. Duke J.  Math. 17 (1950), 57–62. DOI 10.1215/S0012-7094-50-01707-8 | MR 0032065 | Zbl 0036.06101
[9] W.  Leighton: Comparison theorems for linear differential equations of second order. Proc. Amer. Math. Soc. 13 (1962), 603–610. DOI 10.1090/S0002-9939-1962-0140759-0 | MR 0140759 | Zbl 0118.08202
[10] H. J.  Li and C. C.  Yeh: Sturmian comparison theorem for half-linear second order differential equations. Proc. Roy. Soc. Edin. 125A (1995), 1193–1204. MR 1362999
[11] H.  J.  Li and C. C.  Yeh: On the nonoscillatory behavior of solutions of a second order linear differential equation. Math. Nachr. 182 (1996), 295–315. DOI 10.1002/mana.19961820113 | MR 1419898
[12] J. D.  Mirzov: On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems. J.  Math. Anal. Appl. 53 (1976), 418–425. DOI 10.1016/0022-247X(76)90120-7 | MR 0402184 | Zbl 0327.34027
[13] R. A.  Moore: The behavior of solutions of a linear differential equation of second order. Pacific J.  Math. 5 (1955), 125–145. DOI 10.2140/pjm.1955.5.125 | MR 0068690
[14] C.  Sturm: Sur les équations différentielles linéaires du second order. J.  Math. Pures Appl. 1 (1836), 106–186.
[15] C.  Swanson: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New York-London, 1968. MR 0463570 | Zbl 0191.09904
[16] C. T.  Taam: Nonoscillatory differential equations. Duke Math.  J. 19 (1952), 493–497. DOI 10.1215/S0012-7094-52-01951-0 | MR 0051994
[17] D.  Willett: On the oscillatory behavior of the solutions of second order linear differential equations. Ann. Polon. Math. 21 (1969), 175–194. MR 0249723 | Zbl 0174.13701
[18] A.  Wintner: On the comparison theorem of Kneser-Hille. Math. Scand. 5 (1957), 255–260. MR 0096867 | Zbl 0080.29801
[19] D.  Willett: Classification of second order linear differential equations with respect to oscillation. Adv. Math. 3 (1969), 594–623. DOI 10.1016/0001-8708(69)90011-5 | MR 0280800 | Zbl 0188.40101
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