Classification of positive solutions of generalized functional-differential equations. (English) Zbl 1042.34562
Summary: In this paper, we give a classification of nonoscillatory solutions of the quasilinear differential equation \((*)\) \((\phi(y'(t)))'+f(t,y(g(t)))=0\), \(t\geq a\). Under suitable conditions on \(f\) and \(g\), we obtain a classification of nonoscillatory solutions of \((*)\) according to their asymptotic behavior as \(t\to \infty\). We also find some necessary and sufficient conditions to guarantee the existence of nonoscillatory and oscillatory solutions for the above equation.
MSC:
| 34K11 | Oscillation theory of functional-differential equations |
References:
| [1] | Agarwal, R. P.; Wong, P. J.Y., Oscillatory behavior of solutions of certain second order nonlinear differential equations, J. Math. Anal. Appl., 198, 337-354 (1996) · Zbl 0855.34039 |
| [2] | Agarwal, R. P.; Wong, P. J.Y., The oscillation and asymptotically monotone solutions of second-order quasilinear differential equations, Appl. Math. Comput., 79, 207-237 (1996) · Zbl 0889.39009 |
| [3] | Elbert, Á.; Kusano, T., Oscillation and nonoscillation theorems for a class of second order quasilinear differential equations, Acta Math. Hungarica, 56, 325-336 (1990) · Zbl 0732.34035 |
| [4] | Erbe, L., Oscillation criteria for second order nonlinear delay equations, Canda. Math. Bull., 16, 49-56 (1973) · Zbl 0272.34095 |
| [5] | Kusano, T.; Naito, M., Nonlinear oscillation of second order differential equations with deviating argument, Ann. Mat. Pura Appl., 106, 171-185 (1975) · Zbl 0316.34083 |
| [6] | Kusano, T.; Ogata, A.; Usami, H., Oscillation theory for a class of second order quasilinear ordinary differential equations with application to partial differential equations, Japan. J. Math., 19, 131-147 (1993) · Zbl 0792.34033 |
| [7] | Kusano, T.; Wang, J., Oscillation properties of half-linear functional differential equations of the second order, Hiroshima Math. J., 25, 371-385 (1995) · Zbl 0840.34079 |
| [8] | Wang, J., Oscillation and nonoscillation theorems for a class of second order quasilinear functional differential equations, Hiroshima Math. J., 27, 449-466 (1997) · Zbl 0893.34064 |
| [9] | Edwards, R. E., Functional Analysis: Theory and Applications (1965), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York · Zbl 0182.16101 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
