Taiwanese Journal of Mathematics

Absence of Real Roots of Characteristic Functions of Functional Differential Equations with Nine Real Parameters

Shao-Yuan Huang and Sui-Sun Cheng

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We consider the oscillation of a class of first order neutral differential equations with nine real parameters. This relatively difficult problem is completely solved by applying the Cheng-Lin envelope method to find the exact conditions for the absence of real roots of the associated characteristic function. Several specific examples are also included to illustrate these conditions.

Article information

Taiwanese J. Math., Volume 15, Number 1 (2011), 395-432.

First available in Project Euclid: 18 July 2017

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Primary: 34C10: Oscillation theory, zeros, disconjugacy and comparison theory

functional differential equations Cheng-Lin envelope method characteristic function oscillation criteria dual sets


Huang, Shao-Yuan; Cheng, Sui-Sun. Absence of Real Roots of Characteristic Functions of Functional Differential Equations with Nine Real Parameters. Taiwanese J. Math. 15 (2011), no. 1, 395--432. doi:10.11650/twjm/1500406182. https://projecteuclid.org/euclid.twjm/1500406182

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  • S. J. Bilchev, M. K. Grammatikopoulos and I. P. Stavroulakis, Oscillation criteria in higher order neutral equations, J. Math. Anal. Appl., 183 (1994), 1-24.
  • S. S. Cheng and Y. Z. Lin, Exact regions of oscillation for a neutral differential equation, Proc. Royal Soc. Edin., 130A (2000), 277-286.
  • S. S. Cheng and Y. Z. Lin, The exact region of oscillation for first order neutral differential equation with delays, Quarterly Appl. Math., 64(3) (2006), 433-445.
  • S. S. Cheng and Y. Z. Lin, Dual Sets of Envelopes and Characteristic Regions of Quasi-Polynomials, World Scientific, 2009.
  • I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991.
  • H. S. Ren and Z. X. Zheng, The algebraic criteria of oscillation of linear neutral differential equations with delays, J. Biomath., 13(1) (1998), 43-46, (in Chinese).
  • H. S. Ren, On the accurate distribution of characteristic roots and stability of linear delay differential systems, Northeastern Forestry University Press, Harbin, 1999, (in Chinese).