Generalized variational-like inequalities for pseudo-monotone type III operators

Mohammad Chowdhury; Kok-Keong Tan

Open Mathematics (2008)

  • Volume: 6, Issue: 4, page 526-536
  • ISSN: 2391-5455

Abstract

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Our aim in this paper is mainly to prove some existence results for solutions of generalized variational-like inequalities with (η,h)-pseudo-monotone type III operators defined on non-compact sets in topological vector spaces.

How to cite

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Mohammad Chowdhury, and Kok-Keong Tan. "Generalized variational-like inequalities for pseudo-monotone type III operators." Open Mathematics 6.4 (2008): 526-536. <http://eudml.org/doc/269274>.

@article{MohammadChowdhury2008,
abstract = {Our aim in this paper is mainly to prove some existence results for solutions of generalized variational-like inequalities with (η,h)-pseudo-monotone type III operators defined on non-compact sets in topological vector spaces.},
author = {Mohammad Chowdhury, Kok-Keong Tan},
journal = {Open Mathematics},
keywords = {generalized variational-like inequalities; 0-diagonally concave relation; pseudo-monotone type III operators},
language = {eng},
number = {4},
pages = {526-536},
title = {Generalized variational-like inequalities for pseudo-monotone type III operators},
url = {http://eudml.org/doc/269274},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Mohammad Chowdhury
AU - Kok-Keong Tan
TI - Generalized variational-like inequalities for pseudo-monotone type III operators
JO - Open Mathematics
PY - 2008
VL - 6
IS - 4
SP - 526
EP - 536
AB - Our aim in this paper is mainly to prove some existence results for solutions of generalized variational-like inequalities with (η,h)-pseudo-monotone type III operators defined on non-compact sets in topological vector spaces.
LA - eng
KW - generalized variational-like inequalities; 0-diagonally concave relation; pseudo-monotone type III operators
UR - http://eudml.org/doc/269274
ER -

References

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  1. [1] Aliprantis C., Brown D., Equilibria in markets with a Riesz space of commodities, J. Math. Econom., 1983, 11, 189–207 http://dx.doi.org/10.1016/0304-4068(83)90036-8 Zbl0502.90006
  2. [2] Aubin J.P., Ekeland I., Applied nonlinear analysis, John Wiley & Sons, Inc., New York, 1984 Zbl0641.47066
  3. [3] Brézis H., Nirenberg L., Stampacchia G., A remark on Ky Fan’s minimax principle, Boll. Un. Mat. Ital., 1972, 6, 293–300 Zbl0264.49013
  4. [4] Browder F.E., Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc., 1965, 71, 780–785 http://dx.doi.org/10.1090/S0002-9904-1965-11391-X Zbl0138.39902
  5. [5] Chowdhury M.S.R., Generalized variational inequalities for upper hemicontinuous and demi operators with applications to fixed point theorems in Hilbert spaces, Serdica Math. J., 1998, 24, 163–178 Zbl0941.47054
  6. [6] Chowdhury M.S.R., The surjectivity of upper-hemi-continuous and pseudo-monotone type II operators in reflexive Banach Spaces, Ganit, 2000, 20, 45–53 Zbl1063.47503
  7. [7] Chowdhury M.S.R., Haque A.L., Thompson H.B., Quasem G.M.A., The surjectivity of pseudo-monotone type III operators in reflexive Banach Spaces, Nonlinear Funct. Anal. Appl., 2004, 9, 37–44 Zbl1069.47507
  8. [8] Chowdhury M.S.R., Tan K.-K., Generalization of Ky Fan’s minimax inequality with applications to generalized variational inequalities for pseudo-monotone operators and fixed point theorems, J. Math. Anal. Appl., 1996, 204, 910–929 http://dx.doi.org/10.1006/jmaa.1996.0476 
  9. [9] Chowdhury M.S.R., Tan K.-K., Generalized variational inequalities for quasi-monotone operators and applications, Bull. Polish Acad. Sci. Math., 1997, 45, 25–54 Zbl0885.47018
  10. [10] Chowdhury M.S.R., Tarafdar E., Hemi-continuous operators and some applications, Acta Math. Hungar., 1999, 83, 251–261 http://dx.doi.org/10.1023/A:1006725207966 Zbl0934.47038
  11. [11] Chowdhury M.S.R., Thompson H.B., Generalized variational-like inequalities for pseudomonotone type II operators, Nonlinear Analysis, 2005, 63, 321–330 http://dx.doi.org/10.1016/j.na.2005.01.060 Zbl1159.49300
  12. [12] Ding X.P., Lee C., Yu S.-J., Algorithm of solutions for a system of generalized mixed implicit quasi-variational inclusions involving (h-η)-maximal monotone mappings, Taiwanese J. Math., 2007, 11, 577–593 Zbl1138.49011
  13. [13] Ding X.P., Tarafdar E., Generalized nonlinear variational inequalities with nonmonotone set-valued mappings, Appl. Math. Lett., 1994, 7, 5–11 http://dx.doi.org/10.1016/0893-9659(94)90002-7 Zbl0808.49013
  14. [14] Ding X.P., Tarafdar E., Existence and uniqueness of solutions for a general nonlinear variational inequality, Appl. Math. Lett., 1995, 8, 31–36 http://dx.doi.org/10.1016/0893-9659(94)00106-M 
  15. [15] Ding X.P., Tarafdar E., Monotone generalized variational inequalities and generalized complementarity problems, J. Optim. Theory Appl., 1996, 88, 107–122 http://dx.doi.org/10.1007/BF02192024 Zbl0843.49004
  16. [16] Ding X.P., Tarafdar E., Generalized variational-like inequalities with pseudomonotone set-valued mappings, Arch. Math. (Basel), 2000, 74, 302–313 Zbl1013.47026
  17. [17] Fan K., A minimax inequality and applications, In: Shisha O. (Ed.), Inequalities III, Proc. Third Sympos. (1969 University of California, Los Angeles), Academic Press, New York, 1972, 103–113 
  18. [18] Fang S.C., Peterson E.L., Generalized variational inequalities, J. Optim. Theory Appl., 1982, 38, 363–383 http://dx.doi.org/10.1007/BF00935344 Zbl0471.49007
  19. [19] Harker P.T., Pang J.S., Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Programming, 1990, 48, 161–220 http://dx.doi.org/10.1007/BF01582255 Zbl0734.90098
  20. [20] Hartman P., Stampacchia G., On some non-linear elliptic differential-functional equations, Acta Math., 1966, 115, 271–310 http://dx.doi.org/10.1007/BF02392210 Zbl0142.38102
  21. [21] Jou C.R., Yao J.C., Extension of generalized multi-valued variational inequalities, Appl. Math. Lett., 1993, 6, 21–25 http://dx.doi.org/10.1016/0893-9659(93)90026-J 
  22. [22] Karamardian S., The complementarity problem, Math. Programming, 1972, 2, 107–129 http://dx.doi.org/10.1007/BF01584538 Zbl0247.90058
  23. [23] Kinderlehrer D., Stampacchia G., An introduction to variational inequalities and their applications, Academic Press, New York-London, 1980 Zbl0457.35001
  24. [24] Kneser H., Sur un théorème fondamental de la théorie des jeux, C. R. Acad. Sci. Paris, 1952, 234, 2418–2420 (in French) Zbl0046.12201
  25. [25] Mancino O., Stampacchia G., Convex programming and variational inequalities, J. Optimization Theory Appl., 1972, 9, 3–23 http://dx.doi.org/10.1007/BF00932801 Zbl0213.45202
  26. [26] Minty G.J., Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 1962, 29, 341–346 http://dx.doi.org/10.1215/S0012-7094-62-02933-2 Zbl0111.31202
  27. [27] Minty G.J., On a “monotonicity” method for the solution of nonlinear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A., 1963,50, 1038–1041 http://dx.doi.org/10.1073/pnas.50.6.1038 Zbl0124.07303
  28. [28] Moreau J.J., Principes extrémaux pour le problème de la naissance de la cavitation, J. Mécanique, 1966, 5, 439–470 (in French) Zbl0147.44504
  29. [29] Noor M.A., General variational inequalities, Appl. Math. Lett., 1988, 1, 119–122 http://dx.doi.org/10.1016/0893-9659(88)90054-7 
  30. [30] Noor M.A., Nonconvex functions and variational inequalities, J. Optim. Theory Appl., 1995, 87, 615–630 http://dx.doi.org/10.1007/BF02192137 Zbl0840.90107
  31. [31] Parida J., Sahoo M., Kumar A., A variational-like inequality problem, Bull. Austral. Math. Soc., 1989, 39, 225–231 http://dx.doi.org/10.1017/S0004972700002690 Zbl0649.49007
  32. [32] Pascali D., Sburlan S., Nonlinear mappings of monotone type, Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1978 Zbl0423.47021
  33. [33] Peng J.M., Equivalence of variational inequality problems to unconstrained optimization, Math. Programming, 1997, 78, 347–355 Zbl0887.90171
  34. [34] Rockafellar R.T., Convex analysis, Princeton University Press, Princeton, N.J., 1970 Zbl0193.18401
  35. [35] Rockafellar R.T., Wets R.J.-B., Variational analysis, Springer-Verlag, Springer-Verlag, Berlin, 1998 Zbl0888.49001
  36. [36] Stefanov S.M., A Lagrangian dual method for solving variational inequalities, Math. Inequal. Appl., 2002, 5, 597–608 Zbl1015.49007
  37. [37] Yamashita N., Taji K., Fukushima M., Unconstrained optimization reformulations of variational inequality problems, J. Optim. Theory Appl., 1997, 92, 439–456 http://dx.doi.org/10.1023/A:1022660704427 Zbl0879.90180
  38. [38] Zhou J.X., Chen G., Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl., 1988, 132, 213–225 http://dx.doi.org/10.1016/0022-247X(88)90054-6 

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