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Vector Delay Differential Equations

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Book cover Nonoscillation Theory of Functional Differential Equations with Applications

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University—Kingsville, Kingsville, USA

    Ravi P. Agarwal

  2. Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Leonid Berezansky

  3. Department of Mathematics, University of Calgary, Calgary, Canada

    Elena Braverman

  4. Department of Computer Sciences and Mathematics, Ariel University Center of Samaria, Ariel, Israel

    Alexander Domoshnitsky

Authors
  1. Ravi P. Agarwal
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  2. Leonid Berezansky
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  3. Elena Braverman
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  4. Alexander Domoshnitsky
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Agarwal, R.P., Berezansky, L., Braverman, E., Domoshnitsky, A. (2012). Vector Delay Differential Equations. In: Nonoscillation Theory of Functional Differential Equations with Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3455-9_9

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