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Keywords:
half-linear equation; Riccati technique; variational principle; reciprocity principle; principal solution; oscillation and nonoscillation criteria
Summary:
In this paper we investigate oscillatory properties of the second order half-linear equation \[ (r(t)\Phi (y^{\prime }))^{\prime }+c(t)\Phi (y)=0, \quad \Phi (s):= |s|^{p-2}s. \qquad \mathrm{{(*)}}\] Using the Riccati technique, the variational method and the reciprocity principle we establish new oscillation and nonoscillation criteria for (*). We also offer alternative methods of proofs of some recent oscillation results.
References:
[1] C. D. Ahlbrandt: Equivalent boundary value problems for self-adjoint differential systems. J. Differential Equations 9 (1971), 420–435. DOI 10.1016/0022-0396(71)90015-5 | MR 0284636 | Zbl 0218.34020
[2] C. D. Ahlbrandt: Principal and antiprincipal solutions of self-adjoint differential systems and their reciprocals. Rocky Mountain J. Math. 2 (1972), 169–189. DOI 10.1216/RMJ-1972-2-2-169 | MR 0296388
[3] M. Del Pino, M. Elquenta, R. Manachevich: Generalizing Hartman’s oscillation result for $(|x^{\prime }|^{p-2}x^{\prime })^{\prime }+c(t)|x|^{p-2}x=0$, $p>1$. Houston J. Math. 17 (1991), 63–70. MR 1107187
[4] O. Došlý: Oscillation criteria for half-linear second order differential equations. Hiroshima J. Math. 28 (1998), 507–521. MR 1657543
[5] O. Došlý: A remark on conjugacy of half-linear second order differential equations. Math. Slovaca 2000 (to appear). MR 1764346
[6] O. Došlý: Principal solutions and transformations of linear Hamiltonian systems. Arch. Math. 28 (1992), 113–120. MR 1201872
[7] O. Došlý, A. Lomtatidze: Oscillation and nonoscillation criteria for half-linear second order differential equations. (to appear). MR 2259737
[8] Á. Elbert: A half-linear second order differential equation. Colloq. Math. Soc. János Bolyai 30 (1979), 158–180.
[9] Á. Elbert: Oscillation and nonoscillation theorems for some non-linear ordinary differential equations. Lecture Notes in Math. 964 (1982), 187–212. DOI 10.1007/BFb0064999
[10] Á. Elbert: Asymptotic behaviour of autonomous half-linear differential systems on the plane. Studia Sci. Math. Hungar. 19 (1984), 447–464. MR 0874513 | Zbl 0629.34066
[11] Á. Elbert, T. Kusano: Principal solutions of nonoscillatory half-linear differential equations. Adv. Math. Sci. Appl. 18 (1998), 745–759.
[12] E. Hille: Nonoscillation theorems. Trans. Amer. Math. Soc. 64 (1948), 234–252. DOI 10.1090/S0002-9947-1948-0027925-7 | MR 0027925
[13] J. Jaroš, T. Kusano: A Picone type identity for half-linear differential equations. Acta Math. Univ. Comen. 68 (1999), 137–151. MR 1711081
[14] T. Kusano, Y. Naito: Oscillation and nonoscillation criteria for second order quasilinear differential equations. Acta Math. Hungar. 76 (1997), 81–99. DOI 10.1007/BF02907054 | MR 1459772
[15] T. Kusano, Y. Naito, A. Ogata: Strong oscillation and nonoscillation of quasilinear differential equations of second order. Differential Equations Dynam. Systems 2 (1994), 1–10. MR 1386034
[16] H. J. Li: Oscillation criteria for half-linear second order differential equations. Hiroshima Math. J. 25 (1995), 571–583. MR 1364075 | Zbl 0872.34018
[17] H. J. Li, C. C. Yeh: Sturmian comparison theorem for half-linear second order differential equations. Proc. Roy. Soc. Edinburgh 125A (1996), 1193–1204. MR 1362999
[18] A. Lomtatidze: Oscillation and nonoscillation of Emden-Fowler type equation of second order. Arch. Math. 32 (1996), 181–193. MR 1421855
[19] R. Mařík: Nonnegativity of functionals corresponding to the second order half-linear differential equation, submitted.
[20] 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
[21] J. D. Mirzov: Principal and nonprincipal solutions of a nonoscillatory system. Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100–117. MR 1001343
[22] A. Wintner: On the non-existence of conjugate points. Amer. J. Math. 73 (1951), 368–380. DOI 10.2307/2372182 | MR 0042005
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