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ON OPIAL INEQUALITIES INVOLVING HIGHER ORDER DERIVATIVES

  • Received : 2011.02.19
  • Published : 2012.11.30
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Abstract

In the present paper we establish some new Opial-type inequalities involving higher order partial derivatives. The results in special cases yield some of the recent results on Opial's inequality and provide new estimates on inequalities of this type.

Keywords

References

  1. R. P. Agarwal, Sharp Opial-type inequalities involving r-derivatives and their applications, Tohoku Math. J. 47 (1995), no. 4, 567-593. https://doi.org/10.2748/tmj/1178225462
  2. R. P. Agarwal and V. Lakshmikantham, Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, World Scientific, Singapore, 1993.
  3. R. P. Agarwal and P. Y. H. Pang, Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Academic Publishers, Dordrecht, 1995.
  4. R. P. Agarwal and P. Y. H. Pang, Sharp Opial-type inequalities in two variables, Appl. Anal. 56 (1995), no. 3-4, 227-242. https://doi.org/10.1080/00036819508840324
  5. R. P. Agarwal and E. Thandapani, On some new integro-differential inequalities, Anal. sti. Univ. "Al. I. Cuza" din Iasi 28 (1982), no. 1, 123-126.
  6. H. Alzer, An Opial-type inequality involving higher-order derivatives of two functions, Appl. Math. Lett. 10 (1997), no. 4, 123-128. https://doi.org/10.1016/S0893-9659(97)00071-2
  7. D. Bainov and P. Simeonov, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, 1992.
  8. P. R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc. 104 (1962), 470-475. https://doi.org/10.1090/S0002-9947-1962-0139706-1
  9. W. S. Cheung, On Opial-type inequalities in two variables, Aequationes Math. 38 (1989), no. 2-3, 236-244. https://doi.org/10.1007/BF01840008
  10. W. S. Cheung, Some new Opial-type inequalities, Mathematika 37 (1990), no. 1, 136-142. https://doi.org/10.1112/S0025579300012869
  11. W. S. Cheung, Some generalized Opial-type inequalities, J. Math. Anal. Appl. 162 (1991), no. 2, 317-321. https://doi.org/10.1016/0022-247X(91)90152-P
  12. W. S. Cheung, Opial-type inequalities with m functions in n variables, Mathematika 39 (1992), no. 2, 319-326. https://doi.org/10.1112/S0025579300015047
  13. W. S. Cheung, D. D. Zhao, and J. E. Pecaric, Opial-type inequalities for Differential Operators, to appear in Nonlinear Anal.
  14. K. M. Das, An inequality similar to Opial's inequality, Proc. Amer. Math. Soc. 22 (1969), 258-261.
  15. E. K. Godunova and V. I. Levin, An inequality of Maroni, Mat. Zametki 2 (1967), 221-224.
  16. L. K. Hua, On an inequality of Opial, Sci. Sinica 14 (1965), 789-790.
  17. B. Karpuz, B. Kaymakcalan, and U. M. Ozkan, Some multi-dimenstonal Opial-type inequalities on time scales, J. Math. Inequal. 4 (2010), no. 2, 207-216.
  18. J. D. Li, Opial-type integral inequalities involving several higher order derivatives, J. Math. Anal. Appl. 167 (1992), no. 1, 98-100. https://doi.org/10.1016/0022-247X(92)90238-9
  19. D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, Berlin, New York, 1970.
  20. D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Inequalities involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.
  21. Z. Opial, Sur une inegalite, Ann. Polon. Math. 8 (1960), 29-32. https://doi.org/10.4064/ap-8-1-29-32
  22. B. G. Pachpatte, On integral inequalities similar to Opial's inequality, Demonstratio Math. 22 (1989), no. 1, 21-27.
  23. B. G. Pachpatte, On Opial-type integral inequalities, J. Math. Anal. Appl. 120 (1986), no. 2, 547-556. https://doi.org/10.1016/0022-247X(86)90176-9
  24. B. G. Pachpatte, On some new generalizations of Opial inequality, Demonstratio Math. 19 (1986), no. 2, 281-291.
  25. B. G. Pachpatte, A note on generalized Opial type inequalities, Tamkang J. Math. 24 (1993), no. 2, 229-235.
  26. J. E. Pecaric, An integral inequality, Analysis, geometry and groups: a Riemann legacy volume, 471-478, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1993.
  27. J. E. Pecaric and I. Brnetic, Note on generalization of Godunova-Levin-Opial inequality, Demonstratio Math. 30 (1997), no. 3, 545-549.
  28. J. E. Pecaric and I. Brnetic, Note on the generalization of the Godunova-Levin-Opial inequality in several independent variables, J. Math. Anal. Appl. 215 (1997), no. 1, 274-282. https://doi.org/10.1006/jmaa.1997.5529
  29. G. I. Rozanova, Integral inequalities with derivatives and with arbitrary convex functions, Moskov. Gos. Ped. Inst. Vcen. Zap. 460 (1972), 58-65.
  30. D. Willett, The existence-uniqueness theorem for an n-th order linear ordinary differential equation, Amer. Math. Monthly 75 (1968), 174-178. https://doi.org/10.2307/2315901
  31. G. S. Yang, Inequality of Opial-type in two variables, Tamkang J. Math. 13 (1982), no. 2, 255-259.
  32. G. S. Yang, On a certain result of Z. Opial, Proc. Japan Acad. 42 (1966), 78-83. https://doi.org/10.3792/pja/1195522120
  33. C. J. Zhao and W. S. Cheung, Sharp integral inequalities involving high-order partial derivatives, J. Ineq. Appl. 2008 (2008), Article ID 571417, 10 pages. https://doi.org/10.1155/2008/571417

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