Banach Journal of Mathematical Analysis

A hybrid shrinking projection method for common fixed points of a finite family of demicontractive mappings with variational inequality problems

Suthep Suantai and Withun Phuengrattana

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this article, we prove some properties of a demicontractive mapping defined on a nonempty closed convex subset of a Hilbert space. By using these properties, we obtain strong convergence theorems of a hybrid shrinking projection method for finding a common element of the set of common fixed points of a finite family of demicontractive mappings and the set of common solutions of a finite family of variational inequality problems in a Hilbert space. A numerical example is presented to illustrate the proposed method and convergence result. Our results improve and extend the corresponding results existing in the literature.

Article information

Source
Banach J. Math. Anal., Volume 11, Number 3 (2017), 661-675.

Dates
Received: 7 July 2016
Accepted: 5 October 2016
First available in Project Euclid: 23 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1495505201

Digital Object Identifier
doi:10.1215/17358787-2017-0010

Mathematical Reviews number (MathSciNet)
MR3679900

Zentralblatt MATH identifier
06754307

Subjects
Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.
Secondary: 46C05: Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Keywords
demicontractive mappings inverse strongly monotone mapping shrinking projection method variational inequality problems Hilbert spaces

Citation

Suantai, Suthep; Phuengrattana, Withun. A hybrid shrinking projection method for common fixed points of a finite family of demicontractive mappings with variational inequality problems. Banach J. Math. Anal. 11 (2017), no. 3, 661--675. doi:10.1215/17358787-2017-0010. https://projecteuclid.org/euclid.bjma/1495505201


Export citation

References

  • [1] Q. H. Ansari and J. C. Yao, Systems of generalized variational inequalities and their applications, Appl. Anal. 76 (2000), no. 3–4, 203–217.
  • [2] K. Aoyama and F. Kohsaka, Fixed point and mean convergence theorems for a family of $\lambda$-hybrid mappings, J. Nonlinear Anal. Optim. 2 (2011), no. 1, 87–95.
  • [3] C. E. Chidume and S. Măruşter, Iterative methods for the computation of fixed points of demicontractive mappings, J. Comp. Appl. Math. 234 (2010), no. 3, 861–882.
  • [4] T. L. Hicks and J. D. Kubicek, On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl. 59 (1977), no. 3, 498–504.
  • [5] H. Iiduka and W. Takahashi, Weak convergence theorems by Cesáro means for nonexpansive mappings and inverse-strongly-monotone mappings, J. Nonlinear Convex Anal. 7 (2006), no. 1, 105–113.
  • [6] A. Kangtunyakarn, Strong convergence of the hybrid method for a finite family of nonspreading mappings and variational inequality problems, Fixed Point Theory Appl. 2012, no. 188.
  • [7] Y. Kimura, Convergence of a sequence of sets in a Hadamard space and the shrinking projection method for a real Hilbert ball, Abstr. Appl. Anal. 2010, no. 582475.
  • [8] Y. Kimura, K. Nakajo, and W. Takahashi, Strongly convergent iterative schemes for a sequence of nonlinear mappings, J. Nonlinear Convex Anal. 9 (2008), no. 3, 407–416.
  • [9] F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel) 91 (2008), no. 2, 166–177.
  • [10] M. O. Osilike and F. O. Isiogugu, Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal. 74 (2011), no. 5, 1814–1822.
  • [11] A. Latif, L. C. Ceng, and Q. H. Ansari, Multi-step hybrid viscosity method for systems of variational inequalities defined over sets of solutions of an equilibrium problem and fixed point problems, Fixed Point Theory Appl. 2012, no. 186.
  • [12] G. Marino and H. K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), no. 1, 336–346.
  • [13] C. Martinez-Yanesa and H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), no. 11, 2400–2411.
  • [14] K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003), no. 2, 372–379.
  • [15] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. (N.S.) 73 (1967), 591–597.
  • [16] N. Petrot and R. Wangkeeree, A general iterative scheme for strict pseudononspreading mapping related to optimization problem in Hilbert spaces, J. Nonlinear Anal. Optim. 2 (2011), no. 2, 329–336.
  • [17] G. Stampacchia, Formes bilinéaries coercitives sur les ensembles convexes, C. R. Math. Acad. Sci. Paris 258 (1964), 4413–4416.
  • [18] W. Takahashi, Nonlinear Function Analysis: Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2000.
  • [19] W. Takahashi, Y. Takeuchi, and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 341 (2008), no. 1, 276–286.
  • [20] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), no. 2, 417–428.
  • [21] H. Zegeye and N. Shahzad, Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl. 62 (2011), no. 11, 4007–4014.