Open Access

Oscillation and nonoscillation theorems for a class of even-order quasilinear functional differential equations

Journal of Inequalities and Applications20062006:42120

https://doi.org/10.1155/JIA/2006/42120

Received: 13 November 2005

Accepted: 30 January 2006

Published: 16 July 2006

Abstract

We are concerned with the oscillatory and nonoscillatory behavior of solutions of even-order quasilinear functional differential equations of the type , where and are positive constants, and are positive continuous functions on , and is a continuously differentiable function such that , . We first give criteria for the existence of nonoscillatory solutions with specific asymptotic behavior, and then derive conditions (sufficient as well as necessary and sufficient) for all solutions to be oscillatory by comparing the above equation with the related differential equation without deviating argument.

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Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, Faculty of Science and Mathematics, University of Niš
(2)
Department of Mathematics, Faculty of Science Education, Joetsu University of Education

References

  1. Kiguradze IT: On the oscillatory character of solutions of the equation. Matematicheskii Sbornik. New Series 1964, 65 (107): 172–187.MathSciNetGoogle Scholar
  2. Kusano T, Lalli BS: On oscillation of half-linear functional-differential equations with deviating arguments. Hiroshima Mathematical Journal 1994,24(3):549–563.MathSciNetMATHGoogle Scholar
  3. Kusano T, Naito M: Comparison theorems for functional-differential equations with deviating arguments. Journal of the Mathematical Society of Japan 1981,33(3):509–532. 10.2969/jmsj/03330509MathSciNetView ArticleMATHGoogle Scholar
  4. Kusano T, Wang J: Oscillation properties of half-linear functional-differential equations of the second order. Hiroshima Mathematical Journal 1995,25(2):371–385.MathSciNetMATHGoogle Scholar
  5. Mahfoud WE: Comparison theorems for delay differential equations. Pacific Journal of Mathematics 1979,83(1):187–197.MathSciNetView ArticleMATHGoogle Scholar
  6. Tanigawa T: Oscillation and nonoscillation theorems for a class of fourth order quasilinear functional differential equations. Hiroshima Mathematical Journal 2003,33(3):297–316.MathSciNetMATHGoogle Scholar
  7. Tanigawa T: Oscillation criteria for a class of higher order nonlinear differential equations. Memoirs on Differential Equations and Mathematical Physics 2006, 37: 137–152.MathSciNetMATHGoogle Scholar
  8. Tanigawa T, Fentao W: On the existence of positive solutions for a class of even order quasilinear differential equations. Advances in Mathematical Sciences and Applications 2004,14(1):75–85.MathSciNetMATHGoogle Scholar
  9. Wang J: Oscillation and nonoscillation theorems for a class of second order quasilinear functional-differential equations. Hiroshima Mathematical Journal 1997,27(3):449–466.MathSciNetMATHGoogle Scholar

Copyright

© J. Manojlovi´c and T. Tanigawa. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.