Advertisement

Ukrainian Mathematical Journal

, Volume 64, Issue 9, pp 1403–1420 | Cite as

Comparison theorems and necessary/sufficient conditions for the existence of nonoscillatory solutions of forced impulsive differential equations with delay

  • Shao Yuan Huang
  • Sui Sun Cheng
Article
In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation
$$ {x}^{\prime\prime}(t)+p(t){{\left| {x\left( {g(t)} \right)} \right|}^{\eta }}\operatorname{sgn}\left( {x\left( {g(t)} \right)} \right)=e(t), $$
where η > 0, p, and g are continuous functions on [0,∞) such that p(t) ≥ 0, g(t) ≤ t, g′(t) ≥ α > 0, and lim t→∞ g(t) =∞. It is important to note that the condition g′(t) ≥ α > 0 is required. In the paper, we remove this restriction under the superlinear assumption η > 1. In fact, we can do even better by considering impulsive differential equations with delay and obtain necessary and sufficient conditions for the existence of nonoscillatory solutions and also a comparison theorem that enables us to apply known oscillation results for impulsive equations without forcing terms to get oscillation criteria for the analyzed equations.

Keywords

Impulsive Equation Nonlinear Differential Equation Complete Lattice Comparison Theorem Impulsive Effect 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. H. Nasr, “Necessary and sufficient conditions for the oscillation of forced nonlinear second order differential equations with delayed argument,” J. Math. Anal. Appl., 212, 21–59 (1997).MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. S. W. Wong, “Second order nonlinear forced oscillations,” SIAM J. Math. Anal., 19, No. 3, 667–675 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Y. G. Sun, “Necessary and sufficient condition for the oscillation of forced nonlinear differential equation with delay,” Pure Appl. Math., 18, No. 2, 170–173 (2002).MathSciNetzbMATHGoogle Scholar
  4. 4.
    M. S. Peng and W. G. Ge, “Oscillation criteria for second order nonlinear differential equations with impulses,” Comput. Math. Appl., 39, 217–225 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    R. P. Agarwal, S. R. Grace, and D. O’Regan, Oscillation Theory for Second Order Dynamic Equations, Taylor & Francis (2003).Google Scholar
  6. 6.
    V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore (19891).Google Scholar
  7. 7.
    N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Differential Equations with Impulsive Effects: Multivalued Right-Hand Sides with Discontinuities, de Gruyter, Berlin (2011).CrossRefGoogle Scholar
  8. 8.
    A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore (1995).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Shao Yuan Huang
    • 1
  • Sui Sun Cheng
    • 1
  1. 1.Tsing Hua UniversityHsinchuTaiwan

Personalised recommendations