## 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

#### 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
Accepted: 5 October 2016
First available in Project Euclid: 23 May 2017

https://projecteuclid.org/euclid.bjma/1495505201

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

Mathematical Reviews number (MathSciNet)
MR3679900

Zentralblatt MATH identifier
06754307

#### 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

#### 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.