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Nonoscillation theory for second order half-linear differential equations in the framework of regular variation. (English) Zbl 1047.34034
The authors study regularity properties implying nonoscillation of the solutions of the half-linear equation \[ (| y'|^{\alpha- 1}y')'+ q(t)| y|^{\alpha- 1}y= 0 \] with \(\alpha> 0\) and \(q\) positive and continuous on the half-axis \(t\geq 0\). Some necessary and sufficient conditions for the existence of such solutions are presented. The results generalize corresponding ones for the linear equation (when \(\alpha= 1\)) as proved in [V. Marić, Regular variation and differential equations. Lecture Notes in Mathematics 1726. Berlin: Springer-Verlag (2000; Zbl 0946.34001)].

MSC:
34C11 Growth and boundedness of solutions to ordinary differential equations
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
34D05 Asymptotic properties of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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[1] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications 27, Cambridge Univ. Press, 1987. · Zbl 0617.26001
[2] O. Došly, Oscillation criteria for half-linear second order differential equations, Hiroshima Math. J. 28 (1998), 507–521. · Zbl 0920.34042
[3] O. Došly, Methods of oscillation theory of half-linear second order differential equations, Czechoslovak. Math. J. 50(125) (2000), 657–671. · Zbl 1079.34512
[4] Á. Elbert, A half-linear second order differential equation, Colloquia Math. Soc. Janos Bolyai 30: Qualitative Theory of Differential Equations, Szeged, pp. 153–180 (1979).
[5] Á. Elbert and A. Schneider, Perturbations of the half-linear Euler differential equation, Result. Math. 37 (2000), 56–83. · Zbl 0958.34029
[6] H. C. Howard and V. Marić, Regularity and nonoscillation of solutions of second order linear differential equations, Bull. T. CXIV de Acad. Serbe Sci. et Arts, Classe Sci. mat. nat. Sci. math. 22 (1997), 85–98. · Zbl 0947.34015
[7] H. C. Howard, V. Marić and Z. Radašin, Asymptotics of nonoscillatory solutions of second order linear differential equations, Zbornik Rad. Prirod.-Mat. Fak. Univ. Novi Sad, Ser. Mat. 20 (1990), 107–116.
[8] H. B. Hsu and C. C. Yeh, Nonoscillation criteria for second-order half-linear differential equations, Appl. Math. Lett. 8 (1995), 63–70. · Zbl 0844.34028
[9] H. B. Hsu and C. C. Yeh, Oscillation theorems for second order half-linear differential equations, Appl. Math. Lett. 9 (1996), 71–77. · Zbl 0877.34027
[10] J. Jaros and T. Kusano, A Picone-type identity for second order half-linear differential equations, Acta Math. Univ. Comenian. (NS) 68 (1999), 137–151. · Zbl 0926.34023
[11] M. Kitano and T. Kusano, On a class of second order quasilinear ordinary differential equations, Hiroshima Math. J. 25 (1995), 321–355. · Zbl 0835.34034
[12] T. Kusano, Y. Naito and A. Ogata, Strong oscillation and nonoscillation of quasilinear differential equations of second order, Differential Equations and Dynamical Systems 2 (1994), 1–10. · Zbl 0869.34031
[13] T. Kusano and Y. Naito, Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hungar. 76 (1997), 81–99. · Zbl 0906.34024
[14] H. J. Li and C. C. Yeh, Sturmian comparison theorem for half-linear second-order differential equations, Proc. Royal Soc. Edinburgh 125A (1995), 1193–1204. · Zbl 0873.34020
[15] H. C. Li and C. C. Yeh, Oscillation of half-linear second order differential equations, Hiroshima Math. J. 25 (1995), 585–594. · Zbl 0872.34019
[16] J. V. Manojlović, Oscillation criteria for second-order half-linear differential equations, Math. Comput. Modelling 30 (1999), 109–119. · Zbl 1042.34532
[17] V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics 1726, Springer-Verlag, Berlin-Heidelberg-New York, 2000. · Zbl 0946.34001
[18] V. Marić and M. Tomić, A trichotomy of solutions of second order linear differential equations, Zbornik Rad. Prirod.-Mat. Fak. Univ. Novi Sad, Ser. Mat. 14 (1984), 1–11. · Zbl 0597.34060
[19] V. Marić and M. Tomić, A classification of solutions of second order linear differential equations by means of regularly varying functions, Publ. Inst. Math. (Beograd) 48(62) (1990), 199–207.
[20] V. Marić and M. Tomić, Slowly varying solutions of second order linear differential equations, Publ. Inst. Math. (Beograd) 58(72) (1995), 129–136. · Zbl 0965.34044
[21] D. D. Mirzov, On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems, J. Math. Anal. Appl. 53 (1976), 418–425. · Zbl 0327.34027
[22] E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics 508, Springer-Verlag, Berlin-Heidelberg-New-York, 1976. · Zbl 0324.26002
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