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  • Research Article
  • Open Access

Oscillation of Higher-Order Neutral-Type Periodic Differential Equations with Distributed Arguments

Journal of Inequalities and Applications20072007:098423

  • Received: 19 October 2006
  • Accepted: 15 May 2007
  • Published:


We derive oscillation criteria for general-type neutral differential equations , , where , , , , , and are real numbers, the functions and are nondecreasing in for each fixed , and is periodic and continuous with respect to for each fixed . In certain special cases, the results obtained generalize and improve some existing ones in the literature.


  • Differential Equation
  • Real Number
  • Oscillation Criterion
  • Neutral Differential Equation
  • Periodic Differential Equation


Authors’ Affiliations

Department of Mathematics, Iowa State University, Ames, IA 50010, USA
Department of Mathematics, Middle East Technical University, Ankara, 06531, Turkey


  1. Hale J: Theory of Functional Differential Equations. 2nd edition. Springer, New York, NY, USA; 1977:x+365.View ArticleMATHGoogle Scholar
  2. Agarwal RP, Grace SR, O'Regan D: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:viii+337.View ArticleMATHGoogle Scholar
  3. Agarwal RP, Grace SR: Oscillation theorems for certain neutral functional-differential equations. Computers & Mathematics with Applications 1999,38(11–12):1–11. 10.1016/S0898-1221(99)00280-1MathSciNetView ArticleMATHGoogle Scholar
  4. Baĭnov DD, Mishev DP: Oscillation Theory for Neutral Differential Equations with Delay. IOP, Bristol, UK; 1992.Google Scholar
  5. Candan T, Dahiya RS: Oscillation behavior ofth order neutral differential equations with continuous delay. Journal of Mathematical Analysis and Applications 2004,290(1):105–112. 10.1016/j.jmaa.2003.09.072MathSciNetView ArticleMATHGoogle Scholar
  6. Dahiya RS, Zafer A: Oscillation theorems of higher order neutral type differential equations. Dynamical systems and differential equations. Discrete and Continuous Dynamical Systems 1998, 203–219. Added Volume I Added Volume IGoogle Scholar
  7. Das P: Oscillation criteria for odd order neutral equations. Journal of Mathematical Analysis and Applications 1994,188(1):245–257. 10.1006/jmaa.1994.1425MathSciNetView ArticleMATHGoogle Scholar
  8. Erbe LH, Kong Q, Zhang BG: Oscillation Theory for Functional-Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics. Volume 190. Marcel Dekker, New York, NY, USA; 1995:viii+482.Google Scholar
  9. Gopalsamy K, Lalli BS, Zhang BG: Oscillation of odd order neutral differential equations. Czechoslovak Mathematical Journal 1992,42(117)(2):313–323.MathSciNetGoogle Scholar
  10. Grace SR, Lalli BS: Oscillation theorems for certain neutral differential equations. Czechoslovak Mathematical Journal 1988,38(113)(4):745–753.MathSciNetGoogle Scholar
  11. Grace SR: Oscillation criteria forth order neutral functional-differential equations. Journal of Mathematical Analysis and Applications 1994,184(1):44–55. 10.1006/jmaa.1994.1182MathSciNetView ArticleMATHGoogle Scholar
  12. Grace SR: On the oscillations of mixed neutral equations. Journal of Mathematical Analysis and Applications 1995,194(2):377–388. 10.1006/jmaa.1995.1306MathSciNetView ArticleMATHGoogle Scholar
  13. Graef JR, Qian C, Yang B: Positive solutions of a higher order neutral differential equation. Mathematische Nachrichten 2003,256(1):17–28. 10.1002/mana.200310067MathSciNetView ArticleMATHGoogle Scholar
  14. Kiguradze IT: On the oscillatory character of solutions of the equation. Matematicheskiĭ Sbornik 1964, 65 (107): 172–187.MathSciNetGoogle Scholar
  15. Kitamura Y: Oscillation of functional-differential equations with general deviating arguments. Hiroshima Mathematical Journal 1985,15(3):445–491.MathSciNetMATHGoogle Scholar
  16. Ladas G, Sficas YG: Oscillations of neutral delay differential equations. Canadian Mathematical Bulletin 1986,29(4):438–445. 10.4153/CMB-1986-069-2MathSciNetView ArticleMATHGoogle Scholar
  17. Ladas G, Sficas YG: Oscillations of higher-order neutral equations. Journal of the Australian Mathematical Society 1986,27(4):502–511. 10.1017/S0334270000005105MathSciNetView ArticleMATHGoogle Scholar
  18. Philos ChG, Purnaras IK, Sficas YG: Oscillations in higher-order neutral differential equations. Canadian Journal of Mathematics 1993,45(1):132–158. 10.4153/CJM-1993-008-6MathSciNetView ArticleMATHGoogle Scholar
  19. Ouyang Z, Tang Q: Oscillation criteria for a class of higher order neutral differential equations. Annals of Differential Equations 2003,19(1):65–70.MathSciNetMATHGoogle Scholar
  20. Saker SH: Oscillation of higher order neutral delay differential equations with variable coefficients. Dynamic Systems and Applications 2002,11(1):107–125.MathSciNetMATHGoogle Scholar
  21. Staïkos VA: Basic results on oscillation for differential equations with deviating arguments. Hiroshima Mathematical Journal 1980,10(3):495–516.MathSciNetMATHGoogle Scholar
  22. Tanaka S: Oscillation of solutions of even order neutral differential equations. Dynamic Systems and Applications 2000,9(3):353–360.MathSciNetMATHGoogle Scholar
  23. Wang P, Shi W: Oscillatory theorems of a class of even-order neutral equations. Applied Mathematics Letters 2003,16(7):1011–1018. 10.1016/S0893-9659(03)90088-7MathSciNetView ArticleMATHGoogle Scholar
  24. Wang P, Wang M: Oscillation of a class of higher order neutral differential equations. Archivum Mathematicum 2004,40(2):201–208.MathSciNetMATHGoogle Scholar
  25. Wang P: Oscillations ofth-order neutral equation with continuous distributed deviating arguments. Annals of Differential Equations 1998,14(3):570–575.MathSciNetMATHGoogle Scholar
  26. Wang Z: A necessary and sufficient condition for the oscillation of higher-order neutral equations. The Tohoku Mathematical Journal 1989,41(4):575–588. 10.2748/tmj/1178227728MathSciNetView ArticleMATHGoogle Scholar
  27. Yang B, Zhang BG: Oscillation of a class of higher order neutral differential equations. Mathematische Nachrichten 1998, 193: 243–253. 10.1002/mana.19981930116MathSciNetView ArticleMATHGoogle Scholar
  28. Zhang G: Eventually positive solutions of odd order neutral differential equations. Applied Mathematics Letters 2000,13(6):55–61. 10.1016/S0893-9659(00)00054-9MathSciNetView ArticleMATHGoogle Scholar