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Nonoscillatory half-linear differential equations and generalized Karamata functions. (English) Zbl 1103.34017

Summary: We introduce a natural generalization of the concept of regularly varying functions in the sense of Karamata, and show that the class of generalized Karamata functions is a well-suited framework for the study of the asymptotic behavior of nonoscillatory solutions of the half-linear differential equation \[ \bigl(p(t)|y'|^{\alpha-1}y'\bigr)'+q (t)|y|^{\alpha-1}y=0. \]

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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[1] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27 (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0617.26001
[2] Á. Elbert, A half-linear second order differential equation, Colloquia Math. Soc. Janos Bolyai 30; Qualitative Theory of Differential Equations, Szeged, 1979, pp. 153-180.; Á. Elbert, A half-linear second order differential equation, Colloquia Math. Soc. Janos Bolyai 30; Qualitative Theory of Differential Equations, Szeged, 1979, pp. 153-180.
[3] Howard, H. C.; Marić, V., 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, 85-98 (1997) · Zbl 0947.34015
[4] Jaroš, J.; Kusano, T., Remarks on the existence of regularly varying solutions for second order linear differential equations, Publ. Inst. Math. (Beograd) (N.S.), 72, 86, 113-118 (2002) · Zbl 1052.34042
[5] J. Jaroš, T. Kusano, Self-adjoint differential equations and generalized Karamata functions, Bull. T. CXXIX de Acad. Serbe Sci. et Arts, Classe Sci. Mat. Nat. Sci. Math. 29 (2004) 25-60.; J. Jaroš, T. Kusano, Self-adjoint differential equations and generalized Karamata functions, Bull. T. CXXIX de Acad. Serbe Sci. et Arts, Classe Sci. Mat. Nat. Sci. Math. 29 (2004) 25-60.
[6] Jaroš, J.; Kusano, T.; Tanigawa, T., Nonoscillation theory for second order half-linear differential equations in the framework of regular variation, Results Math., 43, 129-149 (2003) · Zbl 1047.34034
[7] Kusano, T.; Naito, Y., Oscillation and nonoscillation theorems for second order quasilinear differential equations, Acta. Math. Hungar., 76, 81-99 (1997) · Zbl 0906.34024
[8] Kusano, T.; Naito, Y.; Ogata, A., Strong oscillation and nonoscillation of quasilinear differential equations of second order, Differential Equations and Dynamical Systems, 2, 1-10 (1994) · Zbl 0869.34031
[9] V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics, vol. 1726, Springer, Berlin, Heidelberg, New York, 2000.; V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics, vol. 1726, Springer, Berlin, Heidelberg, New York, 2000. · Zbl 0946.34001
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