Abstract

We present new quasi-nonexpansive mappings involving pseudomonotone bifunctions defined on convex sets in real Hilbert space. We investigate some properties concerning their fixed point sets. Some applications to equilibrium problems are discussed.

Author information

Contact details are reproduced from the original publication and may be historical.

Le Dung Muu

TIMAS, Thang Long University, Hanoi, Vietnam
and: Institute of Mathematics, VAST, Hanoi

ldmuu@math.ac.vn

Keywords

  • Convexity
  • pseudomonotonicity
  • quasi-nonexpansive mapping
  • fixed point
  • equilibria

Mathematics Subject Classification

47J25, 47N10, 90C25

References

  1. R. P. Agarwal, D. Oregan, D. R. Sahu: Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer (2000).Google Scholar
  2. G. Bigi, M. Castellani, M. Pappalardo, M. Passacantando: Existence and solution methods for equilibria, European J. Oper. Res. 227 (2013) 1–11.https://doi.org/10.1016/j.ejor.2012.11.037Google Scholar
  3. E. Blum, W. Oettli: From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994) 123–145.Google Scholar
  4. A. Cegielski: Iterative Methods for Fixed Point Problems in Hilbert Spaces, Springer, Berlin et al. (2012).https://doi.org/10.1007/978-3-642-30901-4Google Scholar
  5. P. L. Combettes, S. A. Hirstoaga: Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Analysis 6 (2005) 117–136.Google Scholar
  6. M. K. Ghosh, L. Debnath: Convergence of Ishikawa iterates of quasi-nonexpansive mapppings, J. Math. Anal. Appl. 207 (1997) 96–103.https://doi.org/10.1006/jmaa.1997.5268Google Scholar
  7. A. D. Ioffe, V. M. Tihomirov: Theory of Extremal Problems, North-Holland, Amsterdam et al. (1979).Google Scholar
  8. S. Ishikawa: Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974) 147–150.https://doi.org/10.1090/S0002-9939-1974-0336469-5Google Scholar
  9. A. N. Iusem, B. F. Svaiter: A variant of Korpelevich’s method for variational inequalities with a new search strategy, Optimization 42 (1992) 309–321.https://doi.org/10.1080/02331939708844365Google Scholar
  10. E. N. Khobotov: Modification of the extragradient method for solving variaional inequalities and certain optimization problem, USSR Comput. Math. Math. Phy. 27 (1987) 120–127.https://doi.org/10.1016/0041-5553(87)90058-9Google Scholar
  11. G. M. Korpelevich: The extragradient method for finding saddle points and other problems, Ekon. Mat. Metody 12 (1976) 747–756.Google Scholar
  12. G. Mastroeni: Gap functions for equilibrium problems, J. Glob. Optim 27 (2004) 411–426.https://doi.org/10.1023/A:1026050425030Google Scholar
  13. C. Moore: Iterative aproximation of fixed points of demicontractive maps, The Abdus Salam. Intern. Centre for Theoretical Physics, Trieste, Italy, Scientific Report No. IC/98/214 (1998).Google Scholar
  14. L. D. Muu, W. Oettli: Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Analysis 18 (1992) 1159–1166.https://doi.org/10.1016/0362-546X(92)90159-CGoogle Scholar
  15. L. D. Muu, T. D. Quoc: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model, J. Optim. Theory Appl. 142 (2009) 185–204.https://doi.org/10.1007/s10957-009-9529-0Google Scholar
  16. W. V. Petryshyn, T. E. Williamson: Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl. 43 (1973) 459–497.https://doi.org/10.1016/0022-247X(73)90087-5Google Scholar
  17. D. Q. Tran, L. D. Muu, V. H. Nguyen: Extragradient algorithms extended to equilibrium problems, Optimization 57 (2008) 749–776.https://doi.org/10.1080/02331930601122876Google Scholar
  18. T. R. Rockafellar: Convex Analysis, Princeton Univesity Press (1970).https://doi.org/10.1515/9781400873173Google Scholar
  19. P. T. Vuong, J. J. Strodiot, V. H. Nguyen: Extragradient methods and line-search algorithms for solving Ky Fan inequalities and fixed point problems, J. Optim. Theory Appl. 155 (2012) 605–627.https://doi.org/10.1007/s10957-012-0085-7Google Scholar
  20. Y. J. Wang, N. H. Xiu, J. Z. Zhang: Modified extragradient method for variational inequalities and verification of solution existence, J. Optim. Theory Appl. 119 (2003) 167–183.https://doi.org/10.1023/B:JOTA.0000005047.30026.b8Google Scholar

Suggested citation

T. V. Anh, L. D. Muu. “Quasi-Nonexpansive Mappings Involving Pseudomonotone Bifunctions on Convex Sets.” Journal of Convex Analysis 25 (2018), No. 4, 1105–1119. https://doi.org/10.68381/jca25070

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2018. Rights now held by Banach Press.