Banach Journal of Mathematical Analysis

Convergence theorems based on the shrinking projection method for hemi-relatively nonexpansive mappings, variational inequalities and equilibrium problems

Yeol Je Cho, Mi Kwang Kang, and Zi-Ming Wang

Full-text: Open access

Abstract

In this paper, we introduce a new hybrid projection algorithm based on the shrinking projection methods for two hemi-relatively nonexpansive mappings. Using the new algorithm, we prove some strong convergence theorems for finding a common element in the fixed points set of two hemi-relatively nonexpansive mappings, the solutions set of a variational inequality and the solutions set of an equilibrium problem in a uniformly convex and uniformly smooth Banach space. Furthermore, we apply our results to finding zeros of maximal monotone operators. Our results extend and improve the recent ones announced by Li [J. Math. Anal. Appl. 295 (2004) 115--126], Fan [J. Math. Anal. Appl. 337 (2008) 1041--1047], Liu [J. Glob. Optim. 46 (2010) 319--329], Kamraksa and Wangkeeree [J. Appl. Math. Comput. DOI: 10.1007/s12190-010-0427-2] and many others.

Article information

Source
Banach J. Math. Anal., Volume 6, Number 1 (2012), 11-34.

Dates
First available in Project Euclid: 14 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1337014662

Digital Object Identifier
doi:10.15352/bjma/1337014662

Mathematical Reviews number (MathSciNet)
MR2862540

Zentralblatt MATH identifier
1318.47085

Subjects
Primary: 47H05: Monotone operators and generalizations
Secondary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Keywords
Variational inequalities equilibrium problem hemi-relatively nonexpansive mappings shrinking projection method Banach space

Citation

Wang, Zi-Ming; Kang, Mi Kwang; Cho, Yeol Je. Convergence theorems based on the shrinking projection method for hemi-relatively nonexpansive mappings, variational inequalities and equilibrium problems. Banach J. Math. Anal. 6 (2012), no. 1, 11--34. doi:10.15352/bjma/1337014662. https://projecteuclid.org/euclid.bjma/1337014662


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References

  • Ya. Alber, Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, A. (ed.) Theory and Applications of Nonlinear Operators of Monotonic and Accretive Type, Marcel Dekker, New York, 1996, 15–50.
  • Ya. Alber and S. GuerreDelabriere, On the projection methods for fixed point problems, Analysis (Munich) 21 (2001), no. 1, 17–39.
  • Ya. Alber and A. Notik, On some estimates for projection operator in Banach space, Comm. Appl. Nonlinear Anal. 2 (1995), no. 1, 47–56.
  • Ya. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J. 4 (1994), no. 2, 39–54.
  • E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), no. 1-4, 123–145.
  • D. Butnariu, S. Reich and A.J. Zaslavski, Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numer. Funct. Anal. Optim. 24 (2003), no. 5-6, 489–508.
  • S.S. Chang, On Chidumes open questions and approximate solutions of multivalued strongly accretive mapping in Banach spaces, J. Math. Anal. Appl. 216 (1997), no. 1, 94–111.
  • C.E. Chidume and J. Li, Projection methods for approximating fixed points of Lipschitz suppressive operators, Panamer. Math. J. 15 (2005), no. 1, 29–39.
  • I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic, Dordrecht (1990).
  • P.L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), no. 1, 117–136.
  • J. Fan, A Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces, J. Math. Anal. Appl. 337 (2008), no. 2, 1041–1047.
  • S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150.
  • S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2002), no. 3, 938–945.
  • U. Kamraksa and R. Wangkeeree, Convergence theorems based on the shrinking projection method for variational inequality and equilibrium problems, J. Appl. Math. Comput. DOI: 10.1007/s12190-010-0427-2.
  • F. Kohasaha and W. Takahashi, Strong convergence of an iterative sequence for maximal monotone operators in Banach spaces, Abstr. Appl. Anal. 2004 (2004), no. 3, 239–249.
  • J. Li, On the existence of solutions of variational inequalities in Banach spaces, J. Math. Anal. Appl. 295(2004), 115–126.
  • Y. Liu, Strong convergence theorems for variational inequalities and relatively weak nonexpansive mappings, J. Global Optim. 46 (2010), no. 3, 319–329.
  • S. Matsushita and W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2004 (2004), no. 1, 37–47.
  • S.Y. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005), no. 2, 257–266.
  • A. Moudafi, Second-order differential proximal methods for equilibrium problems, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, 1–7.
  • X. Qin, Y.J. Cho and S.M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225 (2009), no. 1, 20–30.
  • X. Qin, S.Y. Cho and S.M. Kang, Strong convergence of shrinking projection methods for quasi-$\phi$-nonexpansive mappings and equilibrium problems, J. Comput. Appl. Math. 234 (2010), no. 3, 750–760.
  • S. Reich, Review of Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems by loana Cioranescu, Kluwer Academic Publishers, Dordrecht, 1990; Bull. Amer. Math. Soc. 26 (1992), 367–370.
  • S. Reich, A weak convergence theorem for the alternating method with Bregman distance, in: A.G. Kartsatos (Ed.), Theory and Applicationsof Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York, 1996, pp. 313–318.
  • S. Reich, Constructive techniques for accretive and monotone operators, Applied Nonlinear Analysis, in: Proceedings of the Third International Conference University of Texas, Arlington, TX, 1978, Academic Press, New York, (1979), pp. 335–345.
  • R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75–88.
  • M.V. Solodov and B.F. Svaiter, Forcing strong convergence of proximal point iterations in Hilbert space, Math. Program. 87 (2000), no. 1, 189–202.
  • Y. Su, Z. Wang and H.K. Xu, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Anal. 71 (2009), no. 11, 5616–5628.
  • Y. Su, M. Li and H. Zhang, New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators, Appl. Math. Comput. 217 (2011), no. 12, 5458–5465.
  • W. Takahashi, Nonlinear Functional Analysis, Yokohama-Publishers, Yokohama, 2000.
  • W. Takahashi, Convex Analysis and Approximation Fixed points. Yokohama Publishers, Yokohama, 2000.
  • W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70(2009), no. 1, 45–57.