Open Access

Generalized Vector Equilibrium-Like Problems without Pseudomonotonicity in Banach Spaces

Journal of Inequalities and Applications20072007:061794

https://doi.org/10.1155/2007/61794

Received: 10 January 2007

Accepted: 21 March 2007

Published: 8 May 2007

Abstract

Let and be real Banach spaces, a nonempty closed convex subset of , and a multifunction such that for each is a proper, closed and convex cone with , where denotes the interior of . Given the mappings , , and , we study the generalized vector equilibrium-like problem: find such that for all for some . By using the KKM technique and the well-known Nadler result, we prove some existence theorems of solutions for this class of generalized vector equilibrium-like problems. Furthermore, these existence theorems can be applied to derive some existence results of solutions for the generalized vector variational-like inequalities. It is worth pointing out that there are no assumptions of pseudomonotonicity in our existence results.

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Authors’ Affiliations

(1)
Department of Mathematics, Shanghai Normal University
(2)
Department of Business Administration, College of Management, Yuan-Ze University
(3)
Department of Applied Mathematics, National Sun Yat-Sen University

References

  1. Giannessi F: Theorems of alternative, quadratic programs and complementarity problems. In Variational Inequalities and Complementarity Problems (Proc. Internat. School, Erice, 1978). Edited by: Cottle RW, Giannessi F, Lions J-L. John Wiley & Sons, Chichester, UK; 1980:151–186.Google Scholar
  2. Ansari QH, Siddiqi AH, Yao J-C: Generalized vector variational-like inequalities and their scalarizations. In Vector Variational Inequalities and Vector Equilibria, Nonconvex Optim. Appl.. Volume 38. Edited by: Giannessi F. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:17–37. 10.1007/978-1-4613-0299-5_2View ArticleGoogle Scholar
  3. Chen G-Y, Goh CJ, Yang XQ: Existence of a solution for generalized vector variational inequalities. Optimization 2001,50(1–2):1–15. 10.1080/02331930108844550MathSciNetView ArticleMATHGoogle Scholar
  4. Chadli O, Yang XQ, Yao J-C: On generalized vector pre-variational and pre-quasivariational inequalities. Journal of Mathematical Analysis and Applications 2004,295(2):392–403. 10.1016/j.jmaa.2004.02.051MathSciNetView ArticleMATHGoogle Scholar
  5. Khan MF, Salahuddin : On generalized vector variational-like inequalities. Nonlinear Analysis 2004,59(6):879–889.MathSciNetView ArticleMATHGoogle Scholar
  6. Ceng L-C, Yao J-C: Generalized Minty's lemma for generalized vector equilibrium problems. Applied Mathematics Letters 2007,20(1):32–37. 10.1016/j.aml.2006.02.019MathSciNetView ArticleGoogle Scholar
  7. Konnov IV, Yao J-C: On the generalized vector variational inequality problem. Journal of Mathematical Analysis and Applications 1997,206(1):42–58. 10.1006/jmaa.1997.5192MathSciNetView ArticleMATHGoogle Scholar
  8. Konnov IV, Yao J-C: Existence of solutions for generalized vector equilibrium problems. Journal of Mathematical Analysis and Applications 1999,233(1):328–335. 10.1006/jmaa.1999.6312MathSciNetView ArticleMATHGoogle Scholar
  9. Lin KL, Yang D-P, Yao J-C: Generalized vector variational inequalities. Journal of Optimization Theory and Applications 1997,92(1):117–125. 10.1023/A:1022640130410MathSciNetView ArticleMATHGoogle Scholar
  10. Konnov IV, Schaible S: Duality for equilibrium problems under generalized monotonicity. Journal of Optimization Theory and Applications 2000,104(2):395–408. 10.1023/A:1004665830923MathSciNetView ArticleMATHGoogle Scholar
  11. Ansari QH, Konnov IV, Yao J-C: Existence of a solution and variational principles for vector equilibrium problems. Journal of Optimization Theory and Applications 2001,110(3):481–492. 10.1023/A:1017581009670MathSciNetView ArticleMATHGoogle Scholar
  12. Lee B-S, Lee G-M: A vector version of Minty's lemma and application. Applied Mathematics Letters 1999,12(5):43–50. 10.1016/S0893-9659(99)00055-5MathSciNetView ArticleMATHGoogle Scholar
  13. Ansari QH, Konnov IV, Yao J-C: On generalized vector equilibrium problems. Nonlinear Analysis 2001,47(1):543–554. 10.1016/S0362-546X(01)00199-7MathSciNetView ArticleMATHGoogle Scholar
  14. Ansari QH, Oettli W, Schläger D: A generalization of vectorial equilibria. Mathematical Methods of Operations Research 1997,46(2):147–152. 10.1007/BF01217687MathSciNetView ArticleMATHGoogle Scholar
  15. Ansari QH, Yao J-C: An existence result for the generalized vector equilibrium problem. Applied Mathematics Letters 1999,12(8):53–56. 10.1016/S0893-9659(99)00121-4MathSciNetView ArticleMATHGoogle Scholar
  16. Li J, Huang N-J, Kim JK: On implicit vector equilibrium problems. Journal of Mathematical Analysis and Applications 2003,283(2):501–512. 10.1016/S0022-247X(03)00277-4MathSciNetView ArticleMATHGoogle Scholar
  17. Ceng L-C, Yao J-C: An existence result for generalized vector equilibrium problems without pseudomonotonicity. Applied Mathematics Letters 2006,19(12):1320–1326. 10.1016/j.aml.2005.09.010MathSciNetView ArticleGoogle Scholar
  18. Giannessi F: On Minty variational principle. In New Trends in Mathematical Programming, Appl. Optim.. Volume 13. Kluwer Academic Publishers, Boston, Mass, USA; 1998:93–99.View ArticleGoogle Scholar
  19. Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.MathSciNetView ArticleMATHGoogle Scholar
  20. Fan K: A generalization of Tychonoff's fixed point theorem. Mathematische Annalen 1961,142(3):305–310. 10.1007/BF01353421MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Lu-Chuan Ceng et al. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.