Viscosity Approximation Methods for Split Equilibrium Problem and Fixed Point Problem for Finite Family of Nonexpansive Mappings in Hilbert Spaces

Article Sidebar

PDF
Published: Apr 6, 2021
DOI: https://doi.org/10.14456/past.2021.8
Keywords:
Split equilibrium problem Variational inequality Fixed point Nonexpansive mapping

Main Article Content

Jitsupa Deepho
Faculty of Science, Energy and Environment, King Mongkut’s University of Technology North Bangkok, Rayong Campus (KMUTNB-Rayong)
Poom Kumam
King Mongkut's University of Technology Thonburi
Pakeeta Sukprasert
Rajamangala University of Technology Thanyaburi

Abstract

In this paper, we present a new iterative scheme bases on the hybrid viscosity approximation method and the hybrid steepest-descent method for finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the split equilibrium problem in Hilbert spaces.

Article Details

How to Cite
1.
Deepho J, Kumam P, Sukprasert P. Viscosity Approximation Methods for Split Equilibrium Problem and Fixed Point Problem for Finite Family of Nonexpansive Mappings in Hilbert Spaces. Prog Appl Sci Tech. [Internet]. 2021 Apr. 6 [cited 2024 Apr. 26];11(1):25-37. Available from: https://ph02.tci-thaijo.org/index.php/past/article/view/243325
Issue
Vol. 11 No. 1 (2021): January - April 2021
Section
Mathematics and Applied Statistics
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Crossref
Scopus
Google Scholar
Europe PMC

References

Halpern B., Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961.

Xu H.K., An iterative approach to quadratic optimization, J Optimiz Theory App, 116 (2003), 659-678.

Marino G., Xu H.K., A general iterative method for nonexpansive mappings in hilbert spaces, J Math Anal Appl, 318 (2006), 43–52.

Yamada I., Butnariu D., Censor Y., Reich S., The hybrid steepest descent method for the variational inequality problems over the intersection of fixed points sets of nonexpansive mappings, Inherently Parallel Algorithms in Feasibility and Optimization and Their Application, North-Holland, Amsterdam, 2001.

Tian M., A general iterative algorithm for Nonexpansive mappings in hilbert spaces, Nonlinear Anal-Theor., 73 (2010), 689–694.

Zhou H.Y., Wang P.A., A simpler explicit iterative algorithm for a class of variational inequalities in hilbert spaces, J Optimiz Theory App., 161 (2014), 716–727.

Zhang C., Yang C., A new explicit iterative algorithm for solving a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings, Fixed Point Theory A., 2014 (2014) 60.

He Z., The Split equilibrium problem and its convergence algorithms, J Inequal Appl., 2012, 2012:162.

Chang S.S., Lee H.W.J., Chan C.K., A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear Anal-Theor., 70 (2009), 3307–3319.

Katchang P., Kumam P., A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space, Appl Math Comput., 32 (2010), 19–38.

Qin X., Shang M., Su Y., A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Anal-Theor., 69 (2008), 3897-3909.

Plubtieng S., Punpaeng R., A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J Math Anal Appl., 336 (2007), 455–469.

Combettes P.L., Hirstoaga S.A., Equilibrium programming in Hilbert spaces, J Nonlinear Cconvex A., 6 (2005), 117-136.

Tada A., Takahashi W, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J Optimiz Theory App, 133 (2007), 359–370.

Takahashi S., Takahashi W., Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J Math Anal Appl., 331 (2007), 506–515.

Takahashi S., Takahashi W., Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal-Theor., 69 (2008), 1025–1033.

Byrne C., Censor Y., Gibali A., Reich S., The split common null point problem. J Nonlinear Cconvex A, 13(4), 759–775 (2012).

Blum E., Oettli W., From optimization and variational inequalities to equilibrium problems, Math Stud., 63 (1994), 123–145.

Lopez G., Martin V., Xu H.K., Iterative algorithm for the multiple-sets split feasibility problem, in: Biomedical Mathematics: Promising Direction in Imaging, Therapy Planning and Inverse Problems, 2009, 243-279.

Xu H.K., Kim T.H., Convergence of hybrid steepest-descent methods for variational inequalities, J Optimiz Theory App., 119 (2003), 659-678.

Moudafi, A, Split monotone variational inclusions. J Optimiz Theory App., 150, 275-283 (2011).

Kazmi K.R., Rizvi S.H., Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for nonexpansive semigroup, Mathematical Sciences, Vol. 7:1 (2013), doi:10.1186/2251–7456–7–1.

Cianciaruso F., Marino G., Muglia L., Yao Y., A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory A., 2010, 383740 (2010).

Suzuki T., Strong convergence theorems for an infinite family of nonexpansive mappings in general Banach spaces, Fixed Point Theory A., Vol. 1, 103–123 (2005).

Shimoji K., Takahashi W., Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwan J Math., Vol. 5 (2), 87–404 (2001).

Osilike M.O., Igbokwe D.I.,Weak and Strong Convergence Theorems for Fixed Points of Pseudocontractions and Solutions of Monotone Type Operator Equations, Comput Math Appl, Vol. 40, 559–567, (2000).

Suzuki T., Strong Convergence of Krasnoselskii and Mann’s Type Sequences for One-Parameter Nonexpansive Semigroups Without Bochner Integrals, J Math Anal Appl, Vol. 305, 227–239 (2005).