Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities

Journal of Optimization Theory and Applications

Volumn 111, Issue 2, 2000, Pages 299-323

  Chadli, O ^a   Chbani, Z ^a   Riahi, H ^a  

^a FACULTÉ DES SCIENCES SEMLALIA   (Morocco)

Author keywords

Complementarity problems; Equilibrium problems; Monotone bifunctions and operators; Variational inequalities; Viscosity principle

Indexed keywords

COERCIVE FORCE; VARIATIONAL TECHNIQUES;

COMPLEMENTARITY PROBLEMS; CONDITION; EQUILIBRIUM PROBLEM; EXISTENCE OF SOLUTIONS; MONOTONE BIFUNCTION; MONOTONE OPERATORS; NEW RESULTS; RESEARCH AREAS; VARIATIONAL INEQUALITIES; VISCOSITY PRINCIPLE;

VISCOSITY;

EID: 0035217295     PISSN: 00223239     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (42)

1. 1. BAIOCCHI, C., and CAPELO, A., Variational and Quasivariational Inequalities, Applications to Free Boundary Problems, John Wiley and Sons, New York, NY, 1984.
2. 2. BLUM, E., and OETTLI, W., From Optimization and Variational Inequalities to Equilibrium Problems, Mathematics Student, Vol. 63, pp. 123-145, 1994.
3. 3. BEN-EL-MECHAIEKH, H., DEGUIRE, P., and GRANAS, A., Une Alternative Non-Lineaire en Analyse Convexe et Applications, Comptes Rendus de l'Academie des Sciences de Paris, Vol. 295, pp. 257-259, 1982.
4. 4. BLUM, E., and OETTLI, W., Variational Principle for Equilibrium Problems, 11th International Meeting on Mathematical Programming, Matrafured, Hungary, 1992.
5. 5. BIANCHI, M., Una Classe di Funzioni Monotone Generalizzate, Rivista di Matematica per le Scienze Economiche e Sociali, Vol. 16, pp. 17-32, 1993.
6. 6. BIANCHI, M., and SCHAIBLE, S., Generalized Monotone Bifunctions and Equilibrium Problems, Journal of Optimization Theory and Applications, Vol. 90, pp. 31-43, 1996.
7. 7. SCHAIBLE, S., Generalized Monotonicity: A Survey, Proceeding of the Symposium on Generalized Convexity, Edited by S. Komlosi, T. Rapcsak, and S. Schaible, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 405, pp. 229-249, 1994.
8. 8. SCHAIBLE, S., Generalized Monotonicity: Concepts and Uses, Proceedings of the Meeting on Variational Inequalities and Network Equilibrium Problems, Edited by F. Giannessi and A. Maugeri, Plenum Publishing Corporation, New York, NY, pp. 289-299, 1995.
9. 9. DING, X. P., and TARAFDAR, E., Monotone Generalized Variational Inequalities and Generalized Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 88, pp. 107-122, 1996.
10. 10. COTTLE, R. W., and YAO, J. C., Pseudomonotone Complementarity Problems in Hilbert Spaces, Journal of Optimization Theory and Applications, Vol. 75, pp. 281-295, 1992.
11. 11. GIUSTI, E., Minimal Surfaces and Functions of Bounded Variations, Birkhäuser, Basel, Switzerland, 1984.
12. 12. ATTOUCH, H., Viscosity Solutions of Minimization Problems, SIAM Journal on Optimization, Vol. 6, pp. 769-806, 1996.
13. 13. COTTLE, R. W., and PANG, J. S., A Least Element Theory of Solving Linear Complementarity Problems as Linear Programs, Mathematics of Operations Research, Vol. 3, pp. 155-170, 1978.
14. 14. SCHAIBLE, S., and YAO, J. C., On the Equivalence of Nonlinear Complementarity Problems and Least-Element Problems, Mathematical Programming, Vol. 70, pp. 191-200, 1995.
15. 15. BRÉZIS, H., NIRENBERG, L., and STAMPACCHIA, G., A Remark on Ky Fan's Minimax Principle, Bollettino della Unione Matematica Italiana, Vol. 6, pp. 293-300, 1972.
16. 16. KARAMARDIAN, S., and SCHAIBLE, S., Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37-46, 1990.
17. 17. SAIGAL, R., Extension of the Generalized Complementarity Problem, Mathematics of Operations Research, Vol. 1, pp. 260-266, 1976.
18. 18. KARAMARDIAN, S., Complementarity Problems over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445-454, 1976.
19. 19. RIDDELL, R. C., Equivalence of Nonlinear Complementarity Problems and Least-Element Problems in Banach Lattices, Mathematics of Operations Research, Vol. 6, pp. 462-474, 1981.
20. e Cycle, Université Paul Sabatier, Toulouse, France, 1983.
21. 21. HASSOUNI, A., Opérateurs Quasimonotones: Application à Certains Problèmes Variationnels, Thèse d'Etat, Université Paul Sabatier, Toulouse, France, 1993.
22. 22. CORREA, R., JOFRÉ, A., and THIBAULT, L., Subdifferential Monotonicity as Characterization of Convex Functions, Numerical Functional Analysis and Optimization, Vol. 15, pp. 531-535, 1994.
23. 23. AUSSEL, D., CORVELLEC, J. N., and LASONDE, M., Subdifferential Characterization of Quasiconvexity and Convexity, Journal of Convex Analysis, Vol. 1, pp. 195-201, 1994.
24. 24. LUC, D. T., Characterizations of Quasiconvex Functions, Bulletin of the Australian Mathematical Society, Vol. 48, pp. 393-406, 1993.
25. 25. POLIQUIN, R. A., Subgradient Monotonicity and Convex Functions, Nonlinear Analysis, Vol. 14, pp. 305-317, 1990.
26. 26. HADJISAVVAS, N., and SCHAIBLE, S., Quasimonotone Variational Inequalities in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 90, pp. 55-111, 1996.
27. 27. DING, X. P., Browder-Hartman-Stampacchia Quasivariational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 173, pp. 577-587, 1993.
28. 28. KOMLOSI, S., Generalized Monotonicity and Generalized Convexity, Journal of Optimization Theory and Applications, Vol. 84, pp. 361-376, 1995.
29. 29. KARAMARDIAN, S., SCHAIBLE, S., and CROUZEIX, J. P., First Characterization of Generalized Monotone Maps, Journal of Optimization Theory and Applications, Vol. 76, pp. 399-413, 1993.
30. 30. OETTLI, W., and THÉRA, M., On Maximal Monotonicity of Perturbed Mappings, Bollettino della Unione Matematica Italiana, Vol. 7, pp. 47-55, 1995.
31. 31. OETTLI, W., and RIAHI, H., On Maximal α-Monotonicity of Sums of Operators, Communications in Applied Nonlinear Analysis, Vol. 5, pp. 1-17, 1998.
32. 32. FAN, K., A Generalization of Tychonoff's Fixed-Point Theorem, Mathematishe Annalen, Vol. 142, pp. 305-310, 1961.
33. 33. TAN, K. K., and YUAN, Z. Z., A Minimax Inequality with Applications to the Existence of Equilibrium Points, Bulletin of the Australian Mathematical Society, Vol. 47, pp. 483-503, 1993.
34. 34. BOURBAKI, N., Espaces Vectoriels Topologiques, Chapters 1-5, Masson, Paris, France.
35. 35. AUBIN, J. P., Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer Verlag, Berlin, Germany, 1993.
36. 36. BERGE, C., Espaces Topologiques: Fonctions Multivoques, Dunod, Paris, France, 1959.
37. 37. SION, M., On General Minimax Theorems, Pacific Journal of Mathematics, Vol. 8, pp. 171-176, 1958.
38. 38. SIMONS, S., You Cannot Generalize the Minimax Theorem Too Much, Rendiconti del Seminario Matematico e Fisico di Milano, Vol. 59, pp. 59-64, 1989.
39. 39. YAO, J. C., Multivalued Variational Inequalities with K-Monotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 391-403, 1994.
40. 40. ATTOUCH, H., and BRÉZIS, H., Duality for Sum of Functions in General Banach Spaces, As Respects of Mathematics and Its Applications, Edited by J. Barroso, North Holland, Amsterdam, Holland, pp. 125-133, 1986.
41. 41. PHELPS, R. R., Convex Functions, Monotone Operators, and Differentiability, 2nd Edition, Springer Verlag, New York, NY, 1993.
42. 42. RUDIN, W., Functional Analysis, McGraw-Hill Book Company, New York, NY, 1973.