Advertisement

Well-Posedness for Mixed Quasivariational-Like Inequalities

  • L. C. Ceng
  • N. Hadjisavvas
  • S. Schaible
  • J. C. Yao
Article

Abstract

In this paper, we introduce concepts of well-posedness, and well-posedness in the generalized sense, for mixed quasivariational-like inequalities where the underlying map is multivalued. We give necessary and sufficient conditions for the various kinds of well-posedness to occur. Our results generalize and strengthen previously found results for variational and quasivariational inequalities.

Keywords

Mixed quasivariational-like inequalities Well-posedness Well-posedness in the generalized sense Multivalued maps Measure of noncompactness 

References

  1. 1.
    Ansari, Q.H., Yao, J.C.: Iterative schemes for solving mixed variational-like inequalities. J. Optim. Theory Appl. 108, 527–541 (2001) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984) MATHGoogle Scholar
  3. 3.
    Banas, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Marcel Dekker, New York (1980) MATHGoogle Scholar
  4. 4.
    Chan, D., Pang, J.S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 7, 211–222 (1982) MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Del Prete, I., Lignola, M.B., Morgan, J.: New concepts of well-posedness for optimization problems with variational inequality constraints. J. Inequal. Pure Appl. Math. 4, 26–43 (2003) MathSciNetGoogle Scholar
  6. 6.
    Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993) MATHGoogle Scholar
  7. 7.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research. Springer, Berlin (2003). Vols. 32 I and 32 II Google Scholar
  8. 8.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, vol. I. Kluwer Academic, Dordrecht (1997) MATHGoogle Scholar
  9. 9.
    Lechicki, A.: On bounded and subcontinuous multifunctions. Pac. J. Math. 75, 191–197 (1978) MATHMathSciNetGoogle Scholar
  10. 10.
    Lignola, M.B.: Well-posedness and L-well-posedness for quasivariational inequalities. J. Optim. Theory Appl. 128, 119–138 (2006) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lignola, M.B., Morgan, J.: Semicontinuity and episemicontinuity: equivalence and applications. Boll. Unione Mat. Ital. 8B, 1–6 (1994) MathSciNetGoogle Scholar
  12. 12.
    Lignola, M.B., Morgan, J.: Well-posedness for optimization problems with constraints defined by a variational inequality having a unique solution. J. Glob. Optim. 16, 57–67 (2000) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lignola, M.B., Morgan, J.: Vector quasivariational inequalities: approximate solutions and well-posedness. J. Convex Anal. 13, 373–384 (2006) MATHMathSciNetGoogle Scholar
  14. 14.
    Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Mosco, U.: Implicit variational problems and quasivariational inequalities. In: Summer School, Nonlinear Operators and the Calculus of Variations, Bruxelles, Belgium, 1975. Lecture Notes in Mathematics, vol. 543, pp. 83–156. Springer, Berlin (1976) CrossRefGoogle Scholar
  16. 16.
    Rockafellar, T.: Convex Analysis. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  17. 17.
    Schaible, S., Yao, J.C., Zeng, L.C.: Iterative method for set-valued mixed quasivariational inequalities in a Banach space. J. Optim. Theory Appl. 129, 425–436 (2006) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958) MATHMathSciNetGoogle Scholar
  19. 19.
    Tykhonov, A.N.: On the stability of the functional optimization problem. USSR J. Comput. Math. Math. Phys. 6, 631–634 (1966) Google Scholar
  20. 20.
    Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory. In: Zarantonello, E.H. (eds.) Contributions to Nonlinear Functional Analysis, pp. 237–424. Academic Press, New York (1971) Google Scholar
  21. 21.
    Zeng, L.C.: Iterative algorithm for finding approximate solutions of a class of mixed variational-like inequalities. Acta Math. Appl. Sinica 20, 477–486 (2004). English Series MATHCrossRefGoogle Scholar
  22. 22.
    Zeng, L.C.: Perturbed proximal point algorithm for generalized nonlinear set-valued mixed quasi-variational inclusions. Acta Math. Sinica 47, 11–18 (2004). Chinese Series MATHGoogle Scholar
  23. 23.
    Zeng, L.C., Yao, J.C.: Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces. J. Glob. Optim. 36, 483–496 (2006) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • L. C. Ceng
    • 1
  • N. Hadjisavvas
    • 2
  • S. Schaible
    • 3
  • J. C. Yao
    • 4
  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina
  2. 2.Department of Product and Systems Design EngineeringUniversity of the AegeanSyrosGreece
  3. 3.A.G. Anderson Graduate School of ManagementUniversity of CaliforniaRiversideUSA
  4. 4.Department of Applied MathematicsNational Sun-Yat-Sen UniversityKaohsiungTaiwan

Personalised recommendations