Neutral Differential Equations
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Chapter 6 deals with nonoscillation and oscillation properties of scalar linear neutral differential equations. There are two kinds of neutral equations, one of them can be integrated leading to a term with a concentrated delay and an integral term; the second type which is considered in this chapter has a derivative involved both without a delay and with one or several delays. The study of these equations is based on the functional properties of the linear operator of inner superposition (composition operator).
The main result of the chapter is the equivalence of the nonoscillation of the equation and the existence of a positive solution for a specially constructed nonlinear operator inequality. This result is applied here to obtain explicit nonoscillation conditions and prove comparison theorems. The second auxiliary result is the equivalence of oscillation properties of the neutral equation and a specially constructed equation with an infinite number of delays, such equations were considered in Chap. 4. This method allows to deduce sufficient oscillation conditions for neutral equations. This chapter also presents nonoscillation conditions and comparison results for neutral equations with positive and negative coefficients and results on existence of slowly oscillating solutions.
KeywordsNeutrality Equation Oscillatory Properties Nonoscillation Conditions Prove Comparison Theorems Contralateral Delay
- 24.Azbelev, N.V., Berezansky, L.M., Simonov, P.M., Chistykov, A.V.: Stability of linear systems with time-lag. Differ. Equ. 29, 153–160 (1993) Google Scholar
- 154.Erbe, L.H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995) Google Scholar
- 167.Gopalsamy, K.: Stability and Oscillation in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht, Boston, London (1992) Google Scholar
- 192.Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, New York (1991) Google Scholar
- 304.Philos, Ch.G., Purnaras, I.K.: Asymptotic properties, nonoscillation, and stability for scalar first order linear autonomous neutral delay differential equations. Electron. J. Differ. Equ. 2004(3) (2004), 17 pp. Google Scholar
- 345.Yu, J.S., Wang, Y.C.: Nonoscillation of a neutral delay differential equation. Rad. Mat. 8, 127–133 (1992/1996) Google Scholar