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Generalized Vector Equilibrium-Like Problems without Pseudomonotonicity in Banach Spaces
Journal of Inequalities and Applications volume 2007, Article number: 061794 (2007)
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.
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.
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_2
Chen G-Y, Goh CJ, Yang XQ: Existence of a solution for generalized vector variational inequalities. Optimization 2001,50(1–2):1–15. 10.1080/02331930108844550
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.2004.02.051
Khan MF, Salahuddin : On generalized vector variational-like inequalities. Nonlinear Analysis 2004,59(6):879–889.
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.2006.02.019
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.1997.5192
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.1999.6312
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:1022640130410
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:1004665830923
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:1017581009670
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)00055-5
Ansari QH, Konnov IV, Yao J-C: On generalized vector equilibrium problems. Nonlinear Analysis 2001,47(1):543–554. 10.1016/S0362-546X(01)00199-7
Ansari QH, Oettli W, Schläger D: A generalization of vectorial equilibria. Mathematical Methods of Operations Research 1997,46(2):147–152. 10.1007/BF01217687
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)00121-4
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)00277-4
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.2005.09.010
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.
Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.
Fan K: A generalization of Tychonoff's fixed point theorem. Mathematische Annalen 1961,142(3):305–310. 10.1007/BF01353421
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Ceng, LC., Guu, SM. & Yao, JC. Generalized Vector Equilibrium-Like Problems without Pseudomonotonicity in Banach Spaces. J Inequal Appl 2007, 061794 (2007). https://doi.org/10.1155/2007/61794
- Banach Space
- Convex Subset
- Existence Result
- Existence Theorem
- Generalize Vector