Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

RELATIVE RECIPROCAL VARIATIONAL INEQUALITIES

  • Received : 2017.02.27
  • Accepted : 2018.04.03
  • Published : 2018.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce a new class of reciprocal convex set which is called as relative reciprocal convex set. We establish a necessary and sufficient condition for the minimum of the differentiable relative reciprocal convex function. This condition can be viewed as a new class of variational inequality which is called relative reciprocal variational inequality. Using the auxiliary principle technique we discuss the existence criteria for the solution of relative reciprocal variational inequality. Some special cases which are naturally included in our main results are also discussed.

Keywords

References

  1. I. Ahmad, V. N. Mishra, R. Ahmad and M. Rahaman, An iterative algorithm for a system of generalized implicit variational inclusions SpringerPlus. 5 (2016), 16 pages. https://doi.org/10.1186/s40064-015-1619-x
  2. F. Al-Azemi and O. Calin, Asian options with harmonic average, Appl. Math. Infor. Sci. 9 (2015), 1-9.
  3. G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl. 335 (2007), 1294-1308. https://doi.org/10.1016/j.jmaa.2007.02.016
  4. C. Baiocchi and A. Capelo, Variational and Quasi Variational Inequalities, John Wiley, New York (1984).
  5. R. Glowinski, J. L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, Holland (1981).
  6. F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York (1995).
  7. I. Iscan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe. J. Math. Stats. 43(6) (2014), 935-942.
  8. J. L. Lions and G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math. 20, 491-512 (1967).
  9. M. A. Noor, General variational inequalities, Appl. Math. Letters. 1 (1988), 119-121. https://doi.org/10.1016/0893-9659(88)90054-7
  10. M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217-229. https://doi.org/10.1006/jmaa.2000.7042
  11. M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152 (2004), 199-277.
  12. M. A. Noor, Extended general variational inequalities, Appl. Math. Letters. 22 (2009), 182-186. https://doi.org/10.1016/j.aml.2008.03.007
  13. M. A. Noor, Variational Inequalities and Applications, Lecture Notes, COMSATS Institute of Information Technology, Islamabad, Pakistan, 2008-2016.
  14. M. A. Noor, K. I. Noor, M. U. Awan and S. Costache, Some integral inequalities for harmonically h-convex functions, U. P. B . Sci. Bull. Series A. 77(1) (2015), 5-16.
  15. M. A. Noor, K. I. Noor and S. Iftikhar, Nonconvex functions and integral inequalities, Punj. Uni. J. Math. 47(2) (2015), 19-27.
  16. M. A. Noor and K. I. Noor, Harmonic variational inequalities, Appl. Math. Infor. Sci. 10 (2016), 1811-1814. https://doi.org/10.18576/amis/100522
  17. M. A. Noor and K. I. Noor, Some implicit methods for solving harmonic variational inequalities, Inter. J. Anal. Appl. 12(1) (2016), 10-14.
  18. M. A. Noor, A. G. Khan, A. Pervez and K. I. Noor, Solution of harmonic variational inequalities by two-step iterative scheme, Turkish J. Ineq. 1(1) (2017), 46-52.
  19. G. Stampacchia, Formes bilineaires coercivities sur les ensembles convexes, C. R. Acad. Sci. Paris. 258 (1964), 4413-4416.