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Existence results of the system of generalized variational inequalities with multi-valued mappings

  • Mijanur RahamanEmail author
  • Prinkya Gupta
  • Javid Iqbal
  • Rais Ahmad
Original Paper
  • 93 Downloads

Abstract

In this article, we study a generalized variational inequality problem with multi-valued mappings over product sets and the system of generalized variational inequalities with multi-valued mappings which are equivalent problems. By developing the idea of generalized densely relatively pseudomonotone mappings, and by using well-known Fan-KKM theorem and fixed point theorem, we prove existence results of our problem. We construct an example of generalized Nash equilibrium problem in the context of our problem, and as an application of our results, we establish an existence of coincident point result.

Keywords

Dense-segment set Variational inequality problem System of variational inequalities Generalized densely relatively pseudomonotone 

Mathematics Subject Classification

49J40 47J20 47H10 

References

  1. 1.
    Ansari, Q.H., Khan, Z.: Densely relative pseudomonotone variational inequalities over product of sets. J. Nonlinear Convex Anal. 7, 179–166 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ansari, Q.H., Khan, Z.: Relatively \(B\)-pseudomonotone variational inequalities over product of sets. J. Inequal. Pure Appl. Math. 4(1), 1–8 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Chowdhury, M.S.R., Tan, K.K.: Generalization of Ky Fan minimax inequality with applications to generalized variational inequalities for pseudomonotone operators and fixed point theorems. J. Math. Anal. Appl. 204, 910–926 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fan, K.: A generalization of Tychnoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Khusbhu, Khan, Z.: Generalized monotonicities and its applications to the system of general variational inequalities. Int. J. Inno. Res. Sci. Engg. Tech 3, 13459–13464 (2014)Google Scholar
  6. 6.
    Luc, D.T.: Existence results for densely pseudomonotone variational inequalities. J. Math. Anal. Appl. 254, 291–308 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Pang, J.S.: Asymmetric variational inequalities over product of sets: applications and iterative methods. Math. Prog. 31, 206–219 (1985)CrossRefzbMATHGoogle Scholar
  8. 8.
    Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. C.R. Acad. Sci, Paris I 258, pp. 4413–4416 (1964)Google Scholar
  9. 9.
    Wu, K.Q., Huang, N.J.: Vector variational-like inequalities with relaxed \(\eta -\alpha \) pseudomonotone mappings in Banach spaces. J. Math. Inequal. 1(2), 281–290 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l. 2017

Authors and Affiliations

  • Mijanur Rahaman
    • 1
    Email author
  • Prinkya Gupta
    • 2
  • Javid Iqbal
    • 2
  • Rais Ahmad
    • 1
  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia
  2. 2.Department of Mathematical SciencesB.G.S.B. UniversityRajouriIndia

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