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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

$\left(|{y}^{\text{'}}{|}^{\alpha -1}{y}^{\text{'}}{\right)}^{\text{'}}+q\left(t\right){|y|}^{\alpha -1}y=0$

with $\alpha >0$ and $q$ positive and continuous on the half-axis $t\ge 0$. 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 Qualitative theory of solutions of ODE: growth, boundedness 26A12 Rate of growth of functions of one real variable, orders of infinity, slowly varying functions 34D05 Asymptotic stability of ODE 34C15 Nonlinear oscillations, coupled oscillators (ODE)
##### References:
 [1] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications 27, Cambridge Univ. Press, 1987. [2] O. Došly, Oscillation criteria for half-linear second order differential equations, Hiroshima Math. J. 28 (1998), 507–521. [3] O. Došly, Methods of oscillation theory of half-linear second order differential equations, Czechoslovak. Math. J. 50(125) (2000), 657–671. · doi:10.1023/A:1022854131381 [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. · doi:10.1007/BF03322512 [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. [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. [9] H. B. Hsu and C. C. Yeh, Oscillation theorems for second order half-linear differential equations, Appl. Math. Lett. 9 (1996), 71–77. · doi:10.1016/0893-9659(96)00097-3 [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. [11] M. Kitano and T. Kusano, On a class of second order quasilinear ordinary differential equations, Hiroshima Math. J. 25 (1995), 321–355. [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. [13] T. Kusano and Y. Naito, Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hungar. 76 (1997), 81–99. · doi:10.1007/BF02907054 [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. · doi:10.1017/S0308210500030468 [15] H. C. Li and C. C. Yeh, Oscillation of half-linear second order differential equations, Hiroshima Math. J. 25 (1995), 585–594. [16] J. V. Manojlović, Oscillation criteria for second-order half-linear differential equations, Math. Comput. Modelling 30 (1999), 109–119. · doi:10.1016/S0895-7177(99)00151-X [17] V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics 1726, Springer-Verlag, Berlin-Heidelberg-New York, 2000. [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. [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. [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. · doi:10.1016/0022-247X(76)90120-7 [22] E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics 508, Springer-Verlag, Berlin-Heidelberg-New-York, 1976.