Abstract
The focus of this paper is to employ a new topological condition, based on admissibility of the function space topology, to provide existence results for a generalized vector quasi-variational inequality problem and its stronger form as well. The inequality problems involve a set-valued function, and the existence results are proved without using any monotonicity or convexity assumptions on the functions. Further, the solution sets of the inequality problems are shown to be closed and compact.
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References
Ansari QH, Chan W, Yang X (2006) Weighted quasi-variational inequalities and constrained nash equilibrium problems. Taiwanese J. Math. 10:361–380. https://doi.org/10.11650/twjm/1500403830
Ansari Q, Farajzadeh A, Schaible S (2009) Existence of solutions of vector variational inequalities and vector complementarity problems. J. Glob. Opt. 45:297–307. https://doi.org/10.1007/s10898-008-9375-x
Arens R, Dugundji J (1951) Topologies for function spaces. Pac J. Math. 1:5–31
Barbagallo A (2008) Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems. J. Glob. Opt. 40:29–39. https://doi.org/10.1007/s10898-007-9194-5
Bensoussan A, Lions J (1973) Nouvelle formulation de problèmes de contrôle impulsionnel et applications. C R Acad. Sci. Ser. I Math. 276:1189–1192
Berge C (1963) Topological Spaces. Oliver and Boyd, Edinburgh
Chang S, Salahuddin Ahmad M et al (2015) Generalized vector variational like inequalities in fuzzy environment. Fuzzy Sets Syst. 265:110–120. https://doi.org/10.1016/j.fss.2014.04.004
Chang S, Salahuddin Wang L et al (2020) Error bounds for mixed set-valued vector inverse quasi-variational inequalities. J. Inequal. Appl. 2020:1–16. https://doi.org/10.1186/s13660-020-02424-7
Chen G, Cheng G (1987) Vector variational inequality and vector optimization problem. In: Sawaragi Y, Inoue K, Nakayama H (eds) Toward Interactive and Intelligent Decision Support Systems. Springer, Berlin, pp 408–416
Chen G, Li S (1996) Existence of solutions for a generalized vector quasi-variational inequality. J. Opt. Theory Appl. 90:321–334. https://doi.org/10.1007/BF02190001
De Luca M (1995) Generalized Quasi-Variational Inequalities and Traffic Equilibrium Problem. In: Giannessi F, Maugeri A (eds) Variational Inequalities and Network Equilibrium Problems. Springer, Boston, pp 45–54
Ding X (1999) The generalized vector quasi-variational-like inequalities. Comput. Math. Appl. 37(6):57–67. https://doi.org/10.1016/S0898-1221(99)00076-0
Fan K (1961) A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142:305–310. https://doi.org/10.1007/BF01353421
Fang Y, Huang N (2003) The Vector F-Complementary Problems with Demipseudomonotone Mappings in Banach spaces. Appl. Math. Lett. 16:1019–1024. https://doi.org/10.1016/S0893-9659(03)90089-9
Gang X, Liu S (2009) Existence of solutions for generalized vector quasi-variational-like inequalities without monotonicity. Comput. Math. Appl. 58(8):1550–1557. https://doi.org/10.1016/j.camwa.2009.05.021
Giannessi F (1980) Theorem of alternative, quadratic programmes and complementarity problems. In: Cottle RW, Gianessi F, Lions JL (eds) Variational inequalities and complementarity problems. Wiley, Chichester, pp 151–186
Göpfert A, Riahi H, Tammer C et al (2003) Variational Methods in Partially Ordered Spaces. Springer, New York
Gupta, S., Sarma, R.D.: A new approach for solving generalized vector quasi-variational-like inequality problem. In: 5th International Conference on Recent Advances in Mathematical Sciences and its Applications (2021)
Gupta A, Sarma R (2017) A study of function space topologies for multifunctions. Appl. Gen. Topol. 18:331–344. https://doi.org/10.4995/agt.2017.7149
Gupta A, Kumar S, Sarma R et al (2021) A note on the generalized nonlinear vector variational-like inequality problem. J Funct Spaces. https://doi.org/10.1155/2021/4488217
Hebestreit N, Khan A, Köbis E et al (2019) Existence theorems and regularization methods for non-coercive vector variational and vector quasi-variational inequalities. J. Nonlinear Convex Anal. 20:565–591
Hu B (1990) A quasi-variational inequality arising in elastohydrodynamics. SIAM J. Math. Anal. 21(1):18–36. https://doi.org/10.1137/0521002
Husain S, Gupta S (2012) Existence of solutions for generalized nonlinear vector quasi-variational-like inequalities with set-valued mappings. Filomat 26:909–916. https://doi.org/10.2298/FIL1205909H
Kharroubi I, Ma J, Pham H et al (2010) Backward Sdes with constrained jumps and quasi-variational inequalities. Ann Probab. https://doi.org/10.1214/09-AOP496
Kum S, Kim W (2006) An extension of generalized vector quasi-variational inequality. Commun. Korean Math. Soc. 21:273–285. https://doi.org/10.4134/CKMS.2006.21.2.273
Lee BS, (2017) Minty lemma for inverted vector variational inequalities. Optimization 66(3):351–359. https://doi.org/10.1080/02331934.2016.1271799
Lenzen F, Becker F, Lellmann J et al (2012) A class of quasi-variational inequalities for adaptive image denoising and decomposition. Comput. Opt. Appl. 54:1–28. https://doi.org/10.1007/s10589-012-9456-0
Lin K, Yang D, Yao J (1997) Generalized vector variational inequalities. J. Opt. Theory Appl. 92:117–125. https://doi.org/10.1023/A:1022640130410
Milasi M (2014) Existence theorem for a class of generalized quasi-variational inequalities. J. Glob. Opt. 60:679–688. https://doi.org/10.1007/s10898-013-0114-6
Porter K (1993) The open-open topology for function spaces. Int J Math Math Sci. https://doi.org/10.1155/S0161171293000134
Wang Z, Chen Z, Che Z (2019) Gap functions and error bounds for vector inverse mixed quasi-variational inequality problems. Fixed Point Theory Appl. 2019:1–14. https://doi.org/10.1186/s13663-019-0664-5
Wangkeeree R, Yimmuang P (2013) Generalized nonlinear vector mixed quasi-variational-like inequality governed by a multi-valued map. J. Inequal. Appl. 2013:1–11. https://doi.org/10.1186/1029-242X-2013-294
Yao J, Zheng X (2018) Existence of solutions and error bound for vector variational inequalities in Banach spaces. Optimization 67:1–12. https://doi.org/10.1080/02331934.2018.1429433
Yu S, Yao J (1996) On vector variational inequalities. J. Opt. Theory Appl. 89:749–769. https://doi.org/10.1007/BF02275358
Acknowledgements
The authors thank Professor C. S. Lalitha, Department of Mathematics, University of Delhi South Campus, New Delhi, India for her valuable suggestions which helped us in improving the paper. The authors also acknowledge the valuable comments of the referees.
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Sonia, Sarma, R.D. A topological approach for vector quasi-variational inequalities with set-valued functions. Comput Manag Sci 20, 21 (2023). https://doi.org/10.1007/s10287-023-00457-z
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DOI: https://doi.org/10.1007/s10287-023-00457-z
Keywords
- Generalized vector quasi-variational inequality
- Fan-KKM lemma
- Admissibility
- Set-valued mapping
- Topological vector space