Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On Opial Type Inequalities with Nonlocal Conditions and Applications

  • Bougoffa, Lazhar (Department of Mathematics, Faculty of Science, Al-imam University) ;
  • Daoud, Jamal Ibrahim (Department of Science in Engineering, International Islamic University)
  • Received : 2010.10.26
  • Accepted : 2010.12.09
  • Published : 2011.06.30
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Abstract

The purpose of this note is to give Opial type inequalities with nonlocal conditions. Also, a reverse of the original inequality with y(a) = y(b) = 0 is derived. We apply these inequalities to second-order differential equations with nonlocal conditions to derive several necessary conditions for the existence of solutions.

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References

  1. P. R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. math. Soc., 104(1962).
  2. R. C. Brown and M. Plum, An Opial-type inequality with an integral boundary con- dition, Proc. R. Soc. Lond. Ser. A, 461(2005), 2635-2651. https://doi.org/10.1098/rspa.2005.1449
  3. R. C. Brown and D. B. Hinton, Opial's inequality and oscillation of 2nd order equa- tions, Proc. Amer. math. Soc., 125(4)(1997), 1123-1129. https://doi.org/10.1090/S0002-9939-97-03907-5
  4. J. Calvert, Some generalizations of Opial's inequality, Proc. Amer. math. Soc., 18(1967), 72-75.
  5. K. M. Das, An inequality similar to Opials inequality, Proc. Amer. math. Soc., 22(1969), 258-261.
  6. B. J. Harris and Q. Kong, On the oscillation of di erential equations with an oscilla- tory coefficient, Trans. Amer. math. Soc., 347(1995), 1831-1839. https://doi.org/10.2307/2154978
  7. M. K. Kwong, On an Opial inequality with a boundary condition, J. Ineq. Pure Appl. Math., 8(1)(2007), Art. 4.
  8. D. S. Mitrinovic, J. E. Pecaric and A. M. Fink , Inequalities involving functions and their derivatives, Kluwer Academic Publishers.
  9. C. Olech, A simple proof of a certain result of Z. Opial, Ann. Polon. Math., 8(1960), 61-63. https://doi.org/10.4064/ap-8-1-61-63
  10. Z. Opail, Sur une inegalite, Ann. Polon. Math., 8(1960), 29-32. https://doi.org/10.4064/ap-8-1-29-32
  11. G. S. Yang, On a certain result of Z. Opial, Proc. Japan Acad., 42(1966), 78-83. https://doi.org/10.3792/pja/1195522120