Some Implicit Methods for Solving Harmonic Variational Inequalities

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Muhammad Aslam Noor
Khalida Inayat Noor


In this paper, we use the auxiliary principle technique to suggest an implicit method for solving the harmonic variational inequalities. It is shown that the convergence of the proposed method only needs pseudo monotonicity of the operator, which is a weaker condition than monotonicity.

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  1. G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl., 335(2007), 1294-1308.
  2. C. Baiocchi and A. Capelo, Variational and Quasi Variational Inequalities, John Wiley, New York, 1984.
  3. G. Cristescu and L. Lupsa, Non-connected Convexities and Applications, Kluwer Academic Publisher, Dordrechet, Holland, (2002).
  4. R. Glowinski, J. L. Lions and R. Tremolieres, Numerical Analysis of variational Inequalities, North-Holland, Ams- terdam, Holland, (1981).
  5. F. Giannessi and A. Maugeri, Variational Inequalities and Network equilibrium Problems, Plenum Press, New York, (1995).
  6. I. Iscan, Hermite-Hadamard type inequalities for harmonically convex functions. Hacettepe, J. Math. Stats., 43(6)(2014), 935-942.
  7. J. L. Lionns and Stampacchi, Variational inequalities, Commun. Pure Appl. Math. 20(1967), 491-512.
  8. C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer-Verlag, New York, (2006).
  9. M. A. Noor, General variational inequalities, Appl. Math. Letters, 1(1988), 119-121.
  10. M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251(2000), 217-229.
  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.
  13. M. A. Noor, Variational Inequalitie and Applications, Lecture Notes, COMSATS Institute of Information Technol- ogy, Islamabad, Pakistan, 2008-2016.
  14. M. A. Noor and K. I. Noor, Harmonic variational inequalities, Appl. Math. Inform. Sci. in press.
  15. M. A. Noor, K. I. Noor and S. Iftikhar, Integral inequalities for differentiable relative harmonic preinvex functions (survey), TWMS J. Pure Appl. Math. 7(1)(2016), 3-19.
  16. M. A. Noor, K. I. Noor and S. Iftikhar, Strongly generalized harmonic convex functions and integral inequalities, J. Math. Anal. in press.
  17. 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.
  18. G. Stampacchia, Formes bilineaires coercivities sur les sensembles convexes, C. R. Acad. Sci. Paris, 258(1964), 4413-4416.