Abstract

We make a comparison among various existence theorems for monotone and nonmonotone variational inequalities and give necessary conditions for the solvability of the variational inequality problem.

Author information

Contact details are reproduced from the original publication and may be historical.

Antonino Maugeri

Dip. di Matematica e Informatica, Università di Catania, Viale A. Doria 6-I, 95125 Catania, Italy

maugeri@dmi.unict.it

Fabio Raciti

Dip. di Matematica e Informatica, Università di Catania, Viale A. Doria 6-I, 95125 Catania, Italy

fraciti@dmi.unict.it

Keywords

  • Brezis-pseudomonotonicity
  • Karamardian-pseudomonotonicity
  • Fan-hemicontinuity
  • hemicontinuity along line segments
  • coercivity

Mathematics Subject Classification

47H05, 52A41,39B82

References

  1. H. Brezis: Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier 18 (1968) 115–175.https://doi.org/10.5802/aif.280Google Scholar
  2. H. Brezis: Problèmes unilateraux, J. Math. Pures. Appl. 51 (1972) 1–168.https://doi.org/10.1007/BF02764814Google Scholar
  3. H. Brezis: Inéquations variationelles associées à des opérateurs d’évolution, in: Theory and Applications of Monotone Operators (Venice, 1968), A. Ghizzetti (ed.), Oderisi, Gubbio (1980) 249–264.Google Scholar
  4. H. Brezis, L. Nirenberg, G. Stampacchia: A remark on Ky Fan’s minimax principle, Boll. Unione Mat. Ital., IV. Ser. 6 (1972) 293–300.Google Scholar
  5. P. Daniele, A. Maugeri, W. Oettli: Time-dependent traffic equilibria, J. Optimization Theory Appl. 103 (1999) 543–555.https://doi.org/10.1023/A:1021779823196Google Scholar
  6. P. Daniele, S. Giuffré, G. Idone, A. Maugeri: Infinite dimensional duality and applications, Math. Ann. (2007) 221–239.https://doi.org/10.1007/s00208-007-0118-yGoogle Scholar
  7. P. Daniele, A. Maugeri: Variational inequalities and discrete and continuum models of network equilibrium problems, Math. Comput. Modelling 35 (2002) 689–708.https://doi.org/10.1016/S0895-7177(02)80030-9Google Scholar
  8. P. Daniele, S. Giuffré: General infinite dimensional duality and applications to evolutionary network equilibrium problems, Optim. Lett. 1 (2007) 227–243.https://doi.org/10.1007/s11590-006-0028-zGoogle Scholar
  9. M. B. Donato, A. Maugeri, M. Milasi, C. Vitanza: Duality theory for a dynamic Walrasian pure exchange economy, Pacific J. Optim. 4 (2008) 537–547.Google Scholar
  10. E. Ernst, M. Théra: A converse to the Lions-Stampacchia theorem, ESAIM, Control Optim. Calc. Var., electronic publication (August 2008), DOI 10.1051/cocv:2008054.https://doi.org/10.1051/cocv:2008054Google Scholar
  11. A. Domokos, J. Kolumbán: Variational inequalities with operator solutions, J. Glob. Optim. 23 (2002) 99–110.https://doi.org/10.1023/A:1014096127736Google Scholar
  12. K. Fan: A minimax inequality and applications, in: Inequalities III, O. Shisha (ed.), Academic Press, New York (1972) 103–113.Google Scholar
  13. K. Fan: Application of a theorem concerning sets with convex sections, Math. Ann. 163 (1966) 189–303.https://doi.org/10.1007/BF02052284Google Scholar
  14. K. Fan: Some properties of convex sets related to fix point theorems, Math. Ann. 266 (1984) 519–537.https://doi.org/10.1007/BF01458545Google Scholar
  15. P. M. Fitzpatrik: Surjectivity results for non-linear mappings from a Banach space to its dual, Math. Ann. 204 (1973) 177–188.https://doi.org/10.1007/BF01351586Google Scholar
  16. M. Frasca, A. Villani: A property of infinite-dimensional Hilbert spaces, J. Math. Anal. Appl. 139 (1989) 352–361.https://doi.org/10.1016/0022-247X(89)90112-1Google Scholar
  17. F. Giannessi, A. Maugeri (eds.): Variational Inequalities and Network Equilibrium Problems (Erice, 1994), Plenum Press, New York (1995).https://doi.org/10.1007/978-1-4899-1358-6Google Scholar
  18. J. Gwinner, F. Raciti: On a class of random variational inequalities on random sets, Numer. Funct. Anal. Optimization 2 (2006) 618–636.https://doi.org/10.1080/01630560600790819Google Scholar
  19. P. Hartman, G. Stampacchia: On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966) 271–310.https://doi.org/10.1007/BF02392210Google Scholar
  20. G. Idone, A. Maugeri, C. Vitanza: Variational Inequalities and the elastic-plastic torsion problem, J. Optimization Theory Appl. 117 (2003) 489–501.https://doi.org/10.1023/A:1023941520452Google Scholar
  21. G. Idone, A.Maugeri: Variational inequalities and a transport planning for an elastic and continuum model, J. Ind. Manag. Optim. 1 (2005) 81–86.https://doi.org/10.3934/jimo.2005.1.81Google Scholar
  22. G. Idone, A. Maugeri, C. Vitanza: Topics on variational analysis and applications to equilibrium problems, J. Glob. Optim. 28 (2004) 339–346.https://doi.org/10.1023/B:JOGO.0000026453.62250.61Google Scholar
  23. S. Karamardian: Complementarity problems over cones with monotone and pseudomonotone maps, J. Optimization Theory Appl. 18 (1976) 445–454.https://doi.org/10.1007/BF00932654Google Scholar
  24. B. T. Kien, J. C. Yao, N. D. Yen: On the solution existence of pseudomonotone variational inequalities, J. Glob. Optim. 41 (2008) 135–145.https://doi.org/10.1007/s10898-007-9251-0Google Scholar
  25. J. L. Lions, G. Stampacchia: Variational inequalities, Commun. Pure Appl. Math. 20 (1967) 493–519.https://doi.org/10.1002/cpa.3160200302Google Scholar
  26. A. Maugeri, W. Oettli, D. Schläger: A flexible form of Wardrop’s principle for traffic equilibria with side constraints, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 48 (1997) 185–193.Google Scholar
  27. A. Maugeri, F. Raciti: On general infinite dimensional complementarity problems, Optim. Lett. 2 (2008) 71–90.https://doi.org/10.1007/s11590-007-0044-7Google Scholar
  28. B. Ricceri: Basic existence theorems for generalized variational and quasi variational inequalities, in: Variational Inequalities and Network Equilibrium Problems (Erice, 1994), F. Giannessi, A. Maugeri (eds.), Plenum Press, New York (1995) 251–255.https://doi.org/10.1007/978-1-4899-1358-6_19Google Scholar
  29. G. Stampacchia: Formes bilinéaires sur les ensembles convexes, C. R. Acad. Sci., Paris 258 (1964) 4413–4416.Google Scholar
  30. N. D. Yen: On a problem of B. Ricceri on Variational Inequalities, in: Fixed Point Theory and Applications. Vol. 5 (Masan/China, 2001), Y. J. Cho et al. (ed.), Nova Science, Hauppauge (2004) 163–173.Google Scholar
  31. E. Zeidler: Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators, Springer, New York (1990).https://doi.org/10.1007/978-1-4612-0985-0Google Scholar

Suggested citation

A. Maugeri, F. Raciti. “On Existence Theorems for Monotone and Nonmonotone Variational Inequalities.” Journal of Convex Analysis 16 (2009), No. 3&4, 899–911. https://doi.org/10.68381/jca16055

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.