Acta Mathematica Vietnamica

, Volume 42, Issue 2, pp 369–394 | Cite as

Some Inequalities for Partial Derivatives on Time Scales

  • Tran Dinh PhungEmail author


We prove some weighted inequalities for compositions of functions on time scales which are in turn applied to establish some new dynamic Opial-type inequalities in several variables. Some generalizations and applications to partial differential dynamic equations are also considered.


Opial’s inequality Time scale Partial differential dynamic equation 

Mathematics Subject Classification (2010)

26D15 26D10 26E70 



The author would like to express his deepest gratitude to Assoc. Prof. Dinh Thanh Duc, Prof. Vu Kim Tuan and Nguyen Du Vi Nhan for their comments and suggestions to improve this paper.

This work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2014.32.


  1. 1.
    Agarwal, R.P., Bohner, M., Peterson, A.: Inequalities on time scales: a survey. Math. Inequal. Appl. 4, 535–557 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Agarwal, R.P., O’regan, D., Saker, S.H.: Dynamic Inequalities on Time Scales. Springer International Publishing, Switzerland (2014)CrossRefzbMATHGoogle Scholar
  3. 3.
    Andrić, M., Barbir, A., Pec̆arić, J.: Generalizations of Opial-type inequalities in several independent variables. Demonstr. Math. 47, 839–847 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Beesack, P.R.: On an integral inequality of Z. Opial. Trans. Am. Math. Soc. 104, 470–475 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bohner, M., Peterson, A. (eds.): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)Google Scholar
  7. 7.
    Bohner, M., Kaymakçalan, B.: Opial inequalities on time scales. Ann. Polon. Math. 77, 11–20 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brnetić, I., Pec̆arić, J.: Note on the generalization of the Godunova-Levin-Opial type inequality in several independent variables. J. Math. Anal. Appl. 215, 545–549 (1997)Google Scholar
  9. 9.
    Cheung, W.S.: On Opial-type inequalities in two variables. Aequationes Math. 38, 236–244 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duc, D.T., Nhan, N.D.V., Xuan, N.T.: Inequalities for partial derivatives and their applications. Canad. Math. Bull. 58, 486–496 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Godunova, E.K., Levin, V.I.: On an inequality of Maroni. Mat. Zametki 2, 221–224 (1967)MathSciNetGoogle Scholar
  12. 12.
    Higgins, R.J., Peterson, A.: Cauchy functions and Taylor’s formula for time scales \(\mathbb {T}\). In: Proceedings of the 6th International Conference on Difference Equations, pp 299–308. CRC, Boca Raton (2004)Google Scholar
  13. 13.
    Hilger, S.: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Karpuz, B., Kaymakçalan, B., Öcalan, Ö.: A generalization of Opial’s inequality and applications to second-oder dynamic equations. Diff. Equ. Dyn. Syst. 18, 11–18 (2010)CrossRefzbMATHGoogle Scholar
  15. 15.
    Li, L., Han, M.: Some new dynamic Opial type inequalities and applications for second order integro-differential dynamic equations on time scales. Appl. Math. Comput. 232, 542–547 (2014)MathSciNetGoogle Scholar
  16. 16.
    Opial, Z.: Sur une inégalité. Ann. Polon. Math. 8, 29–32 (1960)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Pachpatte, B.G.: On two independent variable Opial-type integral inequalities. J. Math. Anal. Appl. 125, 47–57 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pachpatte, B.G.: Some inequalities similar to Opial inequality. Demonstr. Math. 26, 643–647 (1993)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Pečarić, J.: An Integral Inequality, pp 471–478. Hadronic Press, Palm Harbor (1993)zbMATHGoogle Scholar
  20. 20.
    Saker, S.H.: Some Opial-type inequalities on time scales. Abstr. Appl. Anal. 2011(265316), 19 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Saker, S.H.: Some Opial dynamic inequalities involving higher order derivatives on time scales. Discrete Dyn. Nature Soc. 2012(157310), 22 (2012)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Saker, S.H., Osman, M.M., O’regan, D., Agarwal, R.P.: Some new Opial dynamic inequalities with weight functions on time scales. Math. Inequal. Appl. 18, 1171–1187 (2015)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Srivastava, H.M., Tseng, K.L., Tseng, S.J., Lo, J.C.: Some weighted Opial-type inequalities on time scales. Taiwanese. J. Math. 14, 107–122 (2010)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Wong, F.H., Yeah, C.C., Lian, W.C.: An extension of Jensen’s inequality on time scales. Adv. Dyn. Syst. Appl. 1, 113–120 (2006)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Yin, L., Zhao, C.: Some new generalizations of Maroni inequality on time scales. Demonstr. Math. 46, 645–654 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Zhao, C.J., Cheung, W.S.: On some Opial-type inequalities. J. Inequal. Appl. 2011, 7 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Department of MathematicsQuy Nhon UniversityBinh DinhVietnam

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