Skip to main content
SpringerLink
Log in

Well-posedness of generalized vector variational inequality problem via topological approach

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

In this paper, we discuss well-posedness for a generalized vector variational inequality problem (GVVIP, in short) in the framework of topological vector spaces. Unlike in the available literature, we have adopted a topological approach using admissibility and convergence of nets, instead of monotonicity and convexity etc of the function involved. We provide necessary and sufficient conditions for a GVVIP to be well-posed in generalized sense. We give a characterization for GVVIP to be well-posed in generalized sense in terms of the upper semi-continuity of the approximate solution set map. Also, we provide some necessary conditions for a GVVIP to be well-posed in generalized sense in terms of Painlevé–Kuratowski convergence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ansari, Q.H.: On generalized vector variational-like inequalities. Ann. Sci. Math. Québec 19, 131–137 (1995)

    MathSciNet  Google Scholar 

  2. Arens, R., Dugundji, J.: Topologies for function spaces. Pacific J. Math. 1, 5–31 (1951)

    Article  MathSciNet  Google Scholar 

  3. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Systems & Control: Foundations & Applications. Birkhäuser Boston Inc, Boston, MA (1990)

    Google Scholar 

  4. Ceng, L.C., Hadjisavvas, N., Schaible, S., Yao, J.C.: Well-posedness for mixed quasivariational-like inequalities. J. Optim. Theory Appl. 139, 109–125 (2008)

    Article  MathSciNet  Google Scholar 

  5. Chang, S.S., Wang, L., Wang, X.R., Zhao, L.C.: Well-posedness for generalized (\(\eta \)-\(g\)-\(\varphi \))-mixed vector variational-type inequality and optimization problems. J. Inequal. Appl. 238, 16 (2019)

    MathSciNet  Google Scholar 

  6. Chen, G.-Y., Craven, B.D.: A vector variational inequality and optimization over an efficient set. Z. Oper. Res. 34, 1–12 (1990)

    MathSciNet  Google Scholar 

  7. Crespi, G.P., Guerraggio, A., Rocca, M.: Well-posedness in vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 132, 213–226 (2007)

    Article  MathSciNet  Google Scholar 

  8. Fang, Y.P., Hu, R., Huang, N.J.: Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints. Comput. Math. Appl. 55, 89–100 (2008)

    Article  MathSciNet  Google Scholar 

  9. Fang, Y.P., Huang, N.J.: Well-posedness for vector variational inequality and constrained vector optimization problem. Taiwan. J. Math. 11, 1287–1300 (2007)

    Article  Google Scholar 

  10. Fang, Y.P., Huang, N.J., Yao, J.C.: Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems. J. Glob. Optim. 41, 117–133 (2008)

    Article  MathSciNet  Google Scholar 

  11. Fang, Z.M., Li, S.J.: Painlevé–Kuratowski convergences of the solution sets to perturbed generalized systems. Acta Math. Appl. Sin. (English Series) 28(2), (2012)

  12. Giannessi, F.: Theorem of alternative, quadratic programs and complimentary problems. Variational inequalities and complementarity problems (Proc. Internat. School, Erice, 1978), pp. 151–186. Wiley, Chichester, (1980)

  13. Gupta, A., Kumar, S., Sarma, R.D., Garg, P.K., George, R.: A note on the generalized non-linear vector variational-like inequality problem. J. Func. Spaces 4488217, 7 (2021)

  14. Gupta, A., Sarma, R.D.: A study of function space topologies for multifunctions. App. Gen. Topol. 18, 331–344 (2017)

    Article  MathSciNet  Google Scholar 

  15. Huang, X.X., Yang, X.Q.: Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints. J. Ind. Manag. Optim 3, 671–684 (2007)

    Article  MathSciNet  Google Scholar 

  16. Jayswal, A., Jha, S.: Well-posedness for generalized mixed vector variational-like inequality problems in Banach space. Math. Commun. 22, 287–302 (2017)

    MathSciNet  Google Scholar 

  17. Jha, S., Das, P., Bandhyopadhyay, S., Treanţă, S.: Well-posedness for multi-time variational inequality problems via generalized monotonicity and for variational problems with multi-time variational inequality constraints. J. Comput. Appl. Math. 407(114033), 13 (2022)

    MathSciNet  Google Scholar 

  18. Kimura, K., Liou, Y.-C., Wu, S.-Y., Yao, J.-C.: Well-posedness for parametric vector equilibrium problems with applications. J. Ind. Manag. Optim 4, 313–327 (2008)

    Article  MathSciNet  Google Scholar 

  19. Kumar, S., Gupta, A., Garg, P.K., Sarma, R.D.: Topological solutions of \(\eta \)-generalized vector variational-like inequality problems. Math. Appl. (Brno) 10, 115–123 (2021)

    MathSciNet  Google Scholar 

  20. Kumar, S., Gupta, A., Garg, P.K.: Topological variants of vector variational inequality problem. La Math. (2022). https://doi.org/10.1007/s44007-022-00035-w

    Article  Google Scholar 

  21. Lalitha, C.S., Bhatia, G.: Well-posedness for parametric quasivariational inequality problems and for optimization problems with quasivariational inequality constraints. Optimization 59, 997–1011 (2010)

    Article  MathSciNet  Google Scholar 

  22. Lignola, M.B., Morgan, J.: Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution. J. Glob. Optim. 16, 57–67 (2000)

    Article  MathSciNet  Google Scholar 

  23. Munkres, J.R.: Topology: a first course. Prentice-Hall Inc, Englewood Cliffs, N.J. (1975)

    Google Scholar 

  24. Ruiz-Garzon, G., Osuna-Gomez, R., Rufian-Lizan, A.: Relationships between vector variational-like inequality and optimization problems. Eur. J. Oper. Res. 157, 113–119 (2004)

    Article  MathSciNet  Google Scholar 

  25. Salahuddin, S.: General set-valued vector variational inequality problems. Commun. Optim. Theory 26, 1–16 (2017)

    Google Scholar 

  26. Stampacchia, G.: Formes bilinéaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964)

    MathSciNet  Google Scholar 

  27. Tykhonov, A.N.: On the stability of the functional optimization. USSR Comput. Math. Math. Phys. 6, 26–33 (1966)

    Google Scholar 

  28. Usman, F., Khan, S.A.: A generalized mixed vector variational-like inequality problem. Non-linear Anal. 71, 5354–5362 (2009)

    Article  MathSciNet  Google Scholar 

  29. Virmani, G., Srivastava, M.: Various types of well-posedness for mixed vector quasivariational-like inequality using bifunctions. J. Appl. Math. Inform. 32, 427–439 (2014)

    Article  MathSciNet  Google Scholar 

  30. Ward, D.E., Lee, G.M.: On relations between vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 113, 583–596 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Delhi, Delhi, 110007, India

    Satish Kumar

  2. Department of Mathematics, Bharati College (University of Delhi), Delhi, 110058, India

    Ankit Gupta

Authors
  1. Satish Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ankit Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Satish Kumar.

Ethics declarations

Conflict of interest

The authors have no conflict of interest to declare.

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, S., Gupta, A. Well-posedness of generalized vector variational inequality problem via topological approach. Rend. Circ. Mat. Palermo, II. Ser 73, 161–169 (2024). https://doi.org/10.1007/s12215-023-00897-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-023-00897-1

Keywords

Mathematics Subject Classification

Search

Navigation