Abstract
In Chap. 3, nonoscillation of equations with several delays and coefficients of different signs is considered. Unlike equations with positive coefficients, the existence of a positive solution in this case does not imply positivity of the fundamental function, as the first example of the chapter demonstrates. Also, nonoscillatory solutions do not necessarily tend to zero. There have been many mistakes made when studying such equations, one of them made by the authors in their paper in the Journal of Mathematical Analysis and Applications published in 2002. A corrected result on the relation of nonoscillation and positivity of the fundamental function is included in this chapter. In addition, the chapter presents comparison results, explicit nonoscillation conditions, discussion on the asymptotic properties of nonoscillatory solutions and the analysis of the equation with one delay term and an oscillating coefficient. For such an equation, examples demonstrate that even if the positive part of the coefficient “prevails”, this still does not guarantee that nonoscillatory solutions tend to zero.
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References
Berezansky, L., Braverman, E.: On exponential stability of a linear delay differential equation with an oscillating coefficient. Appl. Math. Lett. 22, 1833–1837 (2009)
Berezansky, L., Domshlak, Y.: Differential equations with several deviating arguments: Sturmian comparison method in oscillation theory, I. Electron. J. Differ. Equ. 40, 1–19 (2001)
Berezansky, L., Domshlak, Y., Braverman, E.: On oscillation properties of delay differential equations with positive and negative coefficients. J. Math. Anal. Appl. 274, 81–101 (2002)
Cheng, S.S., Guan, X.P., Yang, J.: Positive solutions of a nonlinear equation with positive and negative coefficients. Acta Math. Hung. 86, 169–192 (2000)
Chuanxi, Q., Ladas, G.: Oscillation in differential equations with positive and negative coefficients. Can. Math. Bull. 33, 442–451 (1990)
Domshlak, Yu.I., Aliev, A.I.: On oscillatory properties of the first order differential equations with one or two arguments. Hiroshima Math. J. 18, 31–46 (1988)
Erbe, L.H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995)
Farell, K., Grove, E.A., Ladas, G.: Neutral delay differential equations with positive and negative coefficients. Appl. Anal. 27, 181–197 (1988)
Guo, S.J., Huang, L.H., Chen, A.P.: Existence of positive solutions and oscillatory solutions of differential equations with positive and negative coefficients. Math. Sci. Res. Hot-Line 5, 59–65 (2001)
Kreith, K., Ladas, G.: Allowable delays for positive diffusion processes. Hiroshima Math. J. 15, 437–443 (1985)
Ladas, G., Sficas, Y.G.: Oscillation of delay differential equations with positive and negative coefficients. In: Proceedings of the International Conference on Qualitative Theory of Differential Equations, University of Alberta, June 18–20, pp. 232–240 (1984)
Li, W., Quan, H.S.: Oscillation of higher order neutral differential equations with positive and negative coefficients. Ann. Differ. Equ. 11, 70–76 (1995)
Li, W.-T., Quan, H., Wu, J.: Oscillation of first order neutral differential equations with variable coefficients. Commun. Appl. Anal. 3, 1–13 (1999)
Li, W.-T., Jan, J.: Oscillation of first order neutral differential equations with positive and negative coefficients. Collect. Math. 50, 199–209 (1999)
Yu, J.S.: Neutral differential equations with positive and negative coefficients. Acta Math. Sin. 34, 517–523 (1991)
Yu, J.S., Yan, J.R.: Oscillation in first order differential equations with “integral smaller” coefficients. J. Math. Anal. Appl. 187, 371–383 (1994)
Zhang, X., Yan, J.R.: Oscillation criteria for first order neutral differential equations with positive and negative coefficients. J. Math. Anal. Appl. 253, 204–214 (2001)
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Agarwal, R.P., Berezansky, L., Braverman, E., Domoshnitsky, A. (2012). Scalar Delay Differential Equations on Semiaxis with Positive and Negative Coefficients. In: Nonoscillation Theory of Functional Differential Equations with Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3455-9_3
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