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Scalar Delay Differential Equations on Semiaxis with Positive and Negative Coefficients

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

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.

Keywords

Scalar Delay Differential Equation Nonoscillation Conditions Nonoscillatory Solution Oscillatory Coefficients Nonoscillation Criteria 
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.

References

  1. 67.
    Berezansky, L., Braverman, E.: On exponential stability of a linear delay differential equation with an oscillating coefficient. Appl. Math. Lett. 22, 1833–1837 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 78.
    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) MathSciNetGoogle Scholar
  3. 81.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 92.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 94.
    Chuanxi, Q., Ladas, G.: Oscillation in differential equations with positive and negative coefficients. Can. Math. Bull. 33, 442–451 (1990) CrossRefGoogle Scholar
  6. 143.
    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) MathSciNetzbMATHGoogle Scholar
  7. 154.
    Erbe, L.H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995) Google Scholar
  8. 155.
    Farell, K., Grove, E.A., Ladas, G.: Neutral delay differential equations with positive and negative coefficients. Appl. Anal. 27, 181–197 (1988) MathSciNetCrossRefGoogle Scholar
  9. 179.
    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) MathSciNetzbMATHGoogle Scholar
  10. 235.
    Kreith, K., Ladas, G.: Allowable delays for positive diffusion processes. Hiroshima Math. J. 15, 437–443 (1985) MathSciNetzbMATHGoogle Scholar
  11. 246.
    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) Google Scholar
  12. 259.
    Li, W., Quan, H.S.: Oscillation of higher order neutral differential equations with positive and negative coefficients. Ann. Differ. Equ. 11, 70–76 (1995) MathSciNetzbMATHGoogle Scholar
  13. 260.
    Li, W.-T., Quan, H., Wu, J.: Oscillation of first order neutral differential equations with variable coefficients. Commun. Appl. Anal. 3, 1–13 (1999) MathSciNetzbMATHGoogle Scholar
  14. 261.
    Li, W.-T., Jan, J.: Oscillation of first order neutral differential equations with positive and negative coefficients. Collect. Math. 50, 199–209 (1999) MathSciNetzbMATHGoogle Scholar
  15. 342.
    Yu, J.S.: Neutral differential equations with positive and negative coefficients. Acta Math. Sin. 34, 517–523 (1991) zbMATHGoogle Scholar
  16. 346.
    Yu, J.S., Yan, J.R.: Oscillation in first order differential equations with “integral smaller” coefficients. J. Math. Anal. Appl. 187, 371–383 (1994) MathSciNetCrossRefGoogle Scholar
  17. 357.
    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) MathSciNetzbMATHCrossRefGoogle Scholar

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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