The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED VECTOR MINTY'S LEMMA

  • Received : 2012.04.30
  • Accepted : 2012.08.12
  • Published : 2012.08.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, the author defines a new generalized ${\eta}$, ${\delta}$, ${\alpha}$)-pseudomonotone mapping and considers the equivalence of Stampacchia-type vector variational-like inequality problems and Minty-type vector variational-like inequality problems for generalized (${\eta}$, ${\delta}$, ${\alpha}$)-pseudomonotone mappings in Banach spaces, called the generalized vector Minty's lemma.

Keywords

References

  1. M.R. Bai, S.Z. Zhoua & G.Y. Nib: Variational-like inequalities with relaxed $\eta-\alpha$ pseudomonotone mappings in Banach spaces Appl. Math. Lett. 19 (2006), 547-554. https://doi.org/10.1016/j.aml.2005.07.010
  2. C. Baiocchi & A. Capelo: Variational and Quasivariational Inequalities, Applications to Free Boundary Problems. John Wiley and Sons Ltd., New York (1984).
  3. A. Barbagallo: Regularity results for evolutionary nonlinear variational and quasi- variational inequalities with applicaions to dynamic equilibrium problems. J. Glob. Optim. 40 (2008), 29-39: DOI/O.1007/s 10898-007-9194-5. https://doi.org/10.1007/s10898-007-9194-5
  4. G.Y. Chen & X.Q. Yang: The vector complementarity problem and its equivalence with the minimal element. J. Math. Anal. Appl. 153 (1990), 136-158. https://doi.org/10.1016/0022-247X(90)90270-P
  5. M. Chinaie, T. Jabarootian, M. Rezaie & J. Zafarani: Minty's lemma and vector variational-like inequalities. J. Glob. Optim. 40 (2008), 463-473: DOI 10.1007/s 10898-007-9177-6.
  6. F. Giannessi: On connections among separation, penalization and regularization for variational inequalities with point-to-set-operators. Rendiconti del Circolo Matematico di Palermo, Series II, Suppl. 48 (1997), 11-18.
  7. F. Giannessi:On Minty variational principle, In New Trends in Mathematical Programming. Kluwer Academic Publishes, Dordrecht (1997).
  8. P. Hartman & G. Stampacchia: On some nonlinear elliptic differential functional equations. Acta Math. 115 (1996), 271-310.
  9. K.R. Kazmi & S.A. Khan: Existence of solutions to a generalized system. J. Optim. Theory Appl. 142 (2009), 355-361, DOI 10. 1007/s 10957-009-9530-7. https://doi.org/10.1007/s10957-009-9530-7
  10. D. Kinderlehrer & G. Stampacchia. An introduction to Variational Inequalities. Academic Press, New York (1980).
  11. B.-S. Lee & G.M. Lee: A vector version of Minty's lemma and application. Appl. Math. Lett. 12 (1999), 43-50.
  12. B.-S. Lee, G.M. Lee & S.-J. Lee: Variational-type inequalities for ($\eta,\theta,\delta$)-pseudo-monotone-type set-valued mappings in nonreflexive Banach spaces. Appl. Math. Lett. 15 (2002), 109-114. https://doi.org/10.1016/S0893-9659(01)00101-X
  13. B.-S. Lee & S.-J. Lee: A vector extension of Bechera and Panda's generalization of Minty's lemma. Indian. J. Pure Appl. Math. 31 (2000), no. 11, 1483-1489.
  14. G.M. Lee, D.S. Kim & H. Kuk: Existence of solutions for vector optimization problems. J. Math. Anal. Appl. 220 (1998), 90-98. https://doi.org/10.1006/jmaa.1997.5821
  15. G.M. Lee, D.S. Kim, B.S. Lee & N.D. Yen: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. Th. Meth. Appl. 34 (1998), 745-765. https://doi.org/10.1016/S0362-546X(97)00578-6
  16. L.-J. Lin & Z.-T. Yu: On some equilibrium problems for multimaps. J. Comp. Appl. Math. 129 (2001), 171-183. https://doi.org/10.1016/S0377-0427(00)00548-3
  17. S.B. Nadler Jr.: Multi-valued contraction mappings. Pacific J. Math. 30 (1969), no. 2, 475-488. https://doi.org/10.2140/pjm.1969.30.475
  18. G. Stampacchia: Variational Inequalities. In : A. Ghizzetti (ed.), Theory and Applications of Monotone Operators. PP. 101-192. Proc. NATO ADV. Study Institute, Venice (1968), Oderisi Gubbio (1969).