## 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
Full Text:

### References:

 [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 · 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. [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. [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. · Zbl 0946.34001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.