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Vietnam Journal of Mathematics

, Volume 44, Issue 3, pp 541–555 | Cite as

Opial-Type Inequalities with Two Unknowns and Two Functions on Time Scales

  • Safi S. Rabie
  • Samir H. Saker
  • Donal O’Regan
  • Ravi P. AgarwalEmail author
Article
  • 164 Downloads

Abstract

In this paper, we prove some new Opial-type inequalities for two unknowns and general kernels on time scales. Also, we prove some new dynamic Opial-type inequalities of higher-order involving two different weight functions. The proofs use Hölder’s inequality, the reverse Hölder’s inequality, the chain rule, and a power rule of integration on time scales.

Keywords

Opial’s inequality Time scales Hölder’s inequality 

Mathematics Subject Classification (2010)

26A15 26D10 26D15 39A13 34A40 

References

  1. 1.
    Agarwal, R.P., O’Regan, D., Saker, S.H.: Dynamic inequalities on time scales. Springer, Switzerland (2014)CrossRefzbMATHGoogle Scholar
  2. 2.
    Agarwal, R.P., Bohner, M.: Basic calculus on time scales and some of it’s applications. Results Math. 35, 3–22 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anastassiou, G.: Time scales inequalities. Int. J. Differ. Equ. 5, 1–23 (2010)MathSciNetGoogle Scholar
  4. 4.
    Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bohner, M., Kaymakçalan, B.: Opial inequalities on time scales. Ann. Pol. Math. 77, 11–20 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deniz, A.: Measure theory on time scales. A Thesis submitted to the graduate School of Engineering and Sciences of İzmir Institute of Technology for the degree of master of science in mathematics (2007)Google Scholar
  8. 8.
    Higgins, R.J., Peterson, A.: Cauchy functions and Taylor’s formula for time scales \(\mathbb {T}\). In: Aulbach, B. (ed.) Proceedings of the 6th International Conference on Difference Equations, pp 299–308. CRC Press LLC, Boca Raton (2004)Google Scholar
  9. 9.
    Karpuz, B., Özkan, U.M.: Some generalizations for Opial’s inequality involving several functions and their derivatives of arbitrary order on arbitrary time scales. Math. Inequal. Appl. 14, 79–92 (2011)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Li, Q.L., Cheung, W.S.: An Opial-type inequality on time scales. Abstr. Appl. Anal. 2013, 534083 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Saker, S.H.: New inequalities of Opial’s type on time scales and some of their applications. Discret. Dyn. Nat. Soc. 2012, 362526 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Saker, S.H.: Some Opial dynamic inequalities involving higher order derivatives on time scales. Discret. Dyn. Nat. Soc. 2012, 157301 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Wong, F.H., Lin, W.C., Yu, S.L., Yeh, C.C.: Some generalizations of Opial’s inequalities on time scales. Taiwan. J. Math. 12, 463–471 (2008)zbMATHGoogle Scholar
  14. 14.
    Zhao, C.J., Cheung, W.S.: On Opial inequalities involving higher order derivatives. Bull. Korean Math. Soc. 49, 1263–1274 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  • Safi S. Rabie
    • 1
  • Samir H. Saker
    • 1
  • Donal O’Regan
    • 2
  • Ravi P. Agarwal
    • 3
    Email author
  1. 1.Department of Mathematics, Faculty of ScienceMansoura UniversityMansouraEgypt
  2. 2.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland
  3. 3.Department of MathematicsTexas A and M University-KingsvilleKingsvilleUSA

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