Vector Delay Differential Equations

  • Ravi P. Agarwal
  • Leonid Berezansky
  • Elena Braverman
  • Alexander Domoshnitsky


Chapter 9 is devoted to nonoscillation of systems of delay differential equations. Wazewski’s result claims that a solution of the vector differential equation is not less than a solution of the differential inequality if and only if the off-diagonal entries of the matrix are nonpositive. This property is discussed for vector delay equations; a sufficient condition is nonnegativity of the fundamental matrix. It is also demonstrated that nonpositivity of the off-diagonal entries, generally, is not necessary for the Wazewski property for linear systems with delays. The chapter includes sufficient positivity conditions for the fundamental matrix, comparison theorems, nonoscillation is also investigated for higher-order scalar delay differential equations. Positivity results are applied to construct estimates for the fundamental matrix and solutions, and to study stability; in particular, positivity of the fundamental matrix implies exponential stability of the vector delay differential equation, under some quite natural restrictions. Most of the results of the chapter are extended to systems with a distributed delay.


Vector Differential Delay Equations Fundamental Matrix Wazewski Exponential Stability Non-oscillatory Property 
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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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