Abstract
In this paper, we discuss the solution of generalized vector bifunction variational inequalities of Stampachhia and Minty types with the concept of generalized cone. Existence results have been established over compact and non-compact sets in Banach spaces without using convexity condition on a bifunction. Some relationship between generalized vector bifunction variational inequalities and vector optimization problems have been established.
Similar content being viewed by others
References
Al-Homidan, S., Ansari, Q..H.: Generalized Minty vector variational-like inequalities and vector optimization problems. J. Optim. Theory Appl. 144, 1–11 (2010)
Ansari, Q.H., Lee, G.M.: Nonsmooth vector optimization problems and Minty vector variational inequalities. J. Optim. Theory Appl. 146, 1–16 (2010)
Bianchi, M., Hadjisavvas, N., Schaible, S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92(3), 527–542 (1997)
Ceng, L.C., Yao, J.C.: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214(1), 186–201 (2008)
Fan, Ky.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142(3), 305–310 (1961)
Farajzadeh, A.P., Lee, B.S.: Vector Variational-like inequality problem and vector optimization problem. Appl. Math. Lett. 23(1), 48–52 (2010)
Giannessi, F.: Theorems of the alternative, quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. John Wiley and Sons, New York (1980)
Guu, S.M., Li, J.: Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets. Nonlinear Anal. Theory Methods Appl. 71, 2847–2855 (2009)
Hanson, M.A.: On sufficiency of the Kuhn–Tucker condition. J. Math. Anal. Appl. 80(2), 545–550 (1981)
Kazmi, K.R.: Existence of solutions for vector optimization. Appl. Math. Lett. 9(6), 19–22 (1996)
Komlósi, S.: On the Stampacchia and Minty variational inequalities. In: Generalized Convexity and Optimization for Economic and Financial Decisions, pp. 231–260 (1999)
Lalitha, C.S., Mehta, M.: Vector-variational Inequalities with Cone-pseudomonotone bifunction. J. Math. Program. Oper. Res. 54, 327–338 (2005)
Mishra, S.K., Noor, M.A.: On vector variational-like inequality problem. J. Math. Anal. Appl. 311(1), 69–75 (2005)
Mishra, S.K., Upadhyay, B.B.: Some relation between vector variational inequality problems and nonsmooth vector optimization problems using Quasi efficiency. Positivity 17(4), 1071–1083 (2013)
Ruiz-Garzón, G., Osuna-Gómez, R., Rufián-Lizana, A.: Relationship between vector variational-like inequality and optimization problems. Eur. J. Oper. Res 157(1), 113–119 (2004)
Sahu, B.K., Kumar, S., Pani, S.: An auxiliary problem principle for the solutions of mixed invex equilibrium problems in Banach spaces. OPSEARCH 1–16 (2023)
Sahu, B.K., Pani, S., Mohapatra, R.N.: Mixed invex equilibrium problems with generalized relaxed monotone and relaxed invariant pseudomonotone mappings. Math. Inequal. Appl. 23, 201–215 (2020)
Yu, S.J., Yao, J.C.: On vector variational inequalities. J. Optim. Theory Appl. 89(3), 749–769 (1996)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kumar, S., Pani, S. Solvability of generalized vector bifunction variational inequality problem. J. Appl. Math. Comput. 70, 1325–1338 (2024). https://doi.org/10.1007/s12190-024-01997-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-024-01997-6