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Neutral Differential Equations

  • Ravi P. Agarwal
  • Leonid Berezansky
  • Elena Braverman
  • Alexander Domoshnitsky
Chapter
  • 925 Downloads

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Ravi P. Agarwal
    • 1
  • Leonid Berezansky
    • 2
  • Elena Braverman
    • 3
  • Alexander Domoshnitsky
    • 4
  1. 1.Department of MathematicsTexas A&M University—KingsvilleKingsvilleUSA
  2. 2.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  3. 3.Department of MathematicsUniversity of CalgaryCalgaryCanada
  4. 4.Department of Computer Sciences and MathematicsAriel University Center of SamariaArielIsrael

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