Neutral Differential Equations

Ravi P. Agarwal, Leonid Berezansky, Elena Braverman and Alexander Domoshnitsky
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Ravi P. Agarwal: Texas A&M University—Kingsville, Department of Mathematics
Leonid Berezansky: Ben-Gurion University of the Negev, Department of Mathematics
Elena Braverman: University of Calgary, Department of Mathematics
Alexander Domoshnitsky: Ariel University Center of Samaria, Department of Computer Sciences and Mathematics

Chapter Chapter 6 in Nonoscillation Theory of Functional Differential Equations with Applications, 2012, pp 149-170 from Springer

Abstract: Abstract 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.

Keywords: Neutrality Equation; Oscillatory Properties; Nonoscillation Conditions; Prove Comparison Theorems; Contralateral Delay (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/978-1-4614-3455-9_6

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