Skip to main content

Advertisement

SpringerLink
Log in

Mixed projection methods for systems of variational inequalities

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

Let H be a real Hilbert space. Let \(F:D(F) \subseteq H \to H, K : D(K) \subseteq H \to H\) be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u *,v *]∈D(F) × D(K) such that \(\left\{\begin{array}{lll}&\langle Fu^* - v^*, x - u^*\rangle \geq 0,\quad x \in D(F),\\ &\langle Kv^* + u^*, y - v^*\rangle \geq 0,\quad y \in D(K)\end{array}\right.\) This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.

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. Brézis H. (1968). Equations et inequations nonlineaires dans les espaces vectoriels en dualite. Annales de l’Institut Fourier 18: 115–175

    Google Scholar 

  2. Chidume C.E. and Zegeye H. (2003). Approximation methods for nonlinear operator equations. Proc. Am. Math. Soc. 131: 2467–2478

    Article  Google Scholar 

  3. Chidume C.E. and Zegeye H. (2004). Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space. Proc. Am. Math. Soc. 133: 851–858

    Article  Google Scholar 

  4. Chidume C.E. and Zegeye H. (2003). Iterative approximation of solutions of nonlinear equations of Hammerstein type. Abst. Appl. Anal. 6: 353–365

    Article  Google Scholar 

  5. Chidume C.E., Zegeye H. and Aneke S.J. (2002). Approximation of fixed points of weakly contractive non-self maps in Banach spaces. J. Math. Anal. Appl. 270: 189–199

    Article  Google Scholar 

  6. Harker P.T. and Pang J.S. (1990). Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Prog. Series B. 48: 161–220

    Article  Google Scholar 

  7. Karamardian S. and Schaible S. (1990). Seven kinds of monotone maps. J. Optim. Theory Appl. 66: 37–47

    Article  Google Scholar 

  8. Karamardian S. (1976). Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18: 445–455

    Article  Google Scholar 

  9. Minty G.J. (1962). Monotone (nonlinear) operators in Hilbert spaces. Duke Math. J. 29: 341–346

    Article  Google Scholar 

  10. Morales C.H. (1985). Surjectivity theorems for multi-valued mappings of accretive type. Commentationes Mathematicae Universitatis Carolinae 26: 397–413

    Google Scholar 

  11. Osilike M.O., Aniagbosor S.C. and Akuchu B.G. (2002). Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces. PanAm. Math. J. 12: 77–88

    Google Scholar 

  12. Reich S. (1980). Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75: 287–292

    Article  Google Scholar 

  13. Schaible S. (1995). Generalized monotonicity: concepts and uses. In: Giannessi, F. and Maugeri, A. (eds) variational inequalities and network equilibrium problems, pp 289–299. Plenum Publishing Corporation, New York

    Google Scholar 

  14. Takahashi W. and Zhang P.J. (1988). The closedness property and the pseudo-A-properness of accretive operators. J. Math. Anal. Appl. 132: 548–557

    Article  Google Scholar 

  15. Weng X. (1991). Fixed point iteration for local strictly pseudo-contractive mappings. Proc. Am. Math. Soc. 113: 727–731

    Article  Google Scholar 

  16. Yao J.C. (1994). Variational inequalities with generalized monotone operators. Math. Operat. Res. 19: 691–705

    Article  Google Scholar 

  17. Zarantonello E.H. (1960). Solving functional equations by contractive averaging, Mathematics Research Center Report#160, Mathematics Research Center. University of Wisconsin, Madison

    Google Scholar 

  18. Zeng L.C. (1994). Iterative algorithms for finding approximate solutions for general strongly nonlinear variational inequalities. J. Math. Anal. Appl. 187: 352–360

    Article  Google Scholar 

  19. Zeng L.C. (1996). Iterative algorithms for finding approximate solutions to completely generalized strongly nonlinear quasivariational inequalities. J. Math. Anal. Appl. 201: 180–194

    Article  Google Scholar 

  20. Zeng L.C. (1998). On a general projection algorithm for variational inequalities. J. Optim. Theory Appl. 97: 229–235

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Lu-Chuan Zeng

  2. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 804, Taiwan

    Jen-Chih Yao

Authors
  1. Lu-Chuan Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jen-Chih Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jen-Chih Yao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zeng, LC., Yao, JC. Mixed projection methods for systems of variational inequalities. J Glob Optim 41, 465–478 (2008). https://doi.org/10.1007/s10898-007-9258-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-007-9258-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation