A generalization of Opial’s inequality and applications to second-order dynamic equations

  • Başak KarpuzEmail author
  • Billûr Kaymakçalan
  • Özkan Öcalan
Original Research


In this paper, we extend to arbitrary time scales some results of [Proc. Amer. Math. Soc., vol. 125, no. 4, pp. 1123–1129, (1997)], where R. C. Brown and D. B. Hinton investigate oscillation of a second-order differential equation. We also provide some examples on nontrivial time scales to illustrate the applicability of the results.


Disconjugate Disfocal Opial inequality Second-order dynamic equations Time scale 

Mathematics Subject Classification (2000)

34A40 39A13 26D15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwal R. P. and Pang P. Y. H., Opial inequalities with applications in differential and difference equations, Kluwer, Dordrecht, (1995)zbMATHGoogle Scholar
  2. 2.
    Beesack P. R. and Das K. M., Extensions of Opial’s inequality, Pasific J. Math., 26, 215–232, (1968)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bohner M. and Peterson A., Dynamic equations on time scales: An introduction with applications, Boston, MA: Birkhäuser Boston Inc., (2001)zbMATHGoogle Scholar
  4. 4.
    Bohner M. and Kaymakçalan B., Opial inequalities on time scales, Ann. Polon. Math., 77(1), 11–20, (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Brown R. C. and Hinton D. B., Opial’s inequality and oscillation of 2nd order equations, Proc. Amer. Math. Soc., 125(4), 1123–1129, (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Eloe P., Positive solutions of boundary-value problems for disfocal ordinary differential equations, J. Comput. Appl. Math., 88(1), 71–78, (1998)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Guseinov S. H. and Kaymakçalan B., On a disconjugacy criterion for second order dynamic equations on time scales, J. Comput. Math. Appl., 141(1–2), 187–196, (2002)zbMATHCrossRefGoogle Scholar
  8. 8.
    Karpuz B. and Özkan U. M., Some generalizations for Opial’s inequality involving several functions and their derivatives of arbitrary order on arbitrary time scales, (submitted)Google Scholar
  9. 9.
    Nehari Z., Disconjugate linear differential operators, Trans. Amer. Math. Soc., 129, 500–516, (1967)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Opial Z., Sur une inégalité, Ann. Polon. Math., 8, 29–32, (1960)zbMATHMathSciNetGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2010

Authors and Affiliations

  • Başak Karpuz
    • 1
    Email author
  • Billûr Kaymakçalan
    • 2
  • Özkan Öcalan
    • 3
  1. 1.Faculty of Science and Arts, Department of Mathematics, ANS CampusAfyon Kocatepe UniversityAfyonkarahisarTurkey
  2. 2.Department of Mathematical SciencesGeorgia Southern UniversityStatesboroUSA
  3. 3.Faculty of Science and Arts, Department of Mathematics, ANS CampusAfyon Kocatepe UniversityAfyonkarahisarTurkey

Personalised recommendations