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Neutral Differential Equations

In: Nonoscillation Theory of Functional Differential Equations with Applications

Author

Listed:
  • 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)

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.

Suggested Citation

  • Ravi P. Agarwal & Leonid Berezansky & Elena Braverman & Alexander Domoshnitsky, 2012. "Neutral Differential Equations," Springer Books, in: Nonoscillation Theory of Functional Differential Equations with Applications, edition 127, chapter 0, pages 149-170, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-3455-9_6
    DOI: 10.1007/978-1-4614-3455-9_6
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