Construction of a common element for the set of solutions of fixed point problems and generalized equilibrium problems in Hilbert spaces

Muhammad Aqeel Ahmad Khan

Abstract


In this paper, we propose and analyse an iterative algorithm for the approximation of a common solution for a finite family of k-strict pseudo-contractions and two finite families of generalized equilibrium problems in the setting of Hilbert spaces. Strong convergence results of the proposed iterative algorithm together with some applications to solve the variational inequality problems are established in such setting. Our results generalize and improve various existing results in the current literature.

Keywords


fixed point; strict pseudo-contraction; equilibrium problem; variational inequality problem; inverse strongly monotone mapping; shrinking projection method

Mathematics Subject Classification (2010)


Primary 47H05, 47H09; Secondary 49H05

References


Bauschke, Heinz H., and Jonathan M. Borwein. "On projection algorithms for solving convex feasibility problems." SIAM Rev. 38, no. 3 (1996): 367-426.

Blum, Eugen, and Werner Oettli. "From optimization and variational inequalities to equilibrium problems." Math. Student 63, no. 1-4 (1994): 123-145.

Browder, Felix E., and Wolodymyr V. Petryshyn. "Construction of fixed points of nonlinear mappings in Hilbert space." J. Math. Anal. Appl. 20 (1967): 197-228.

Cholamjiak, Prasit, and Suthep Suantai. "Convergence analysis for a system of generalized equilibrium problems and a countable family of strict pseudocontractions." Fixed Point Theory Appl. 2011 (2011): Art. ID 941090.

Combettes, Patrick L., and Sever A. Hirstoaga. "Equilibrium programming in Hilbert spaces." J. Nonlinear Convex Anal. 6, no. 1 (2005): 117-136.

Combettes, Patrick L. "Constrained image recovery in a product space." Vol. 2 of Proceedings of the IEEE International Conference on Image Processing , 2025-2028. Washington DC: IEEE Computer Society Press, 1995.

Fan, Ky. "A generalization of Tychonoff's fixed point theorem." Math. Ann. 142 (1961): 305-310.

Huang, Chunyan, and Xiaoyan Ma. "On generalized equilibrium problems and strictly pseudocontractive mappings in Hilbert spaces." Fixed Point Theory Appl. 2014, 2014:145.

Kangtunyakarn, Atid. "Hybrid algorithm for finding common elements of the set of generalized equilibrium problems and the set of fixed point problems of strictly pseudocontractive mapping." Fixed Point Theory Appl. 2011 (2011): Art. ID 274820.

Khan, Muhammad A.A., and Hafiz Fukhar-ud-din. "Strong convergence by the shrinking effect of two half-spaces and applications." Fixed Point Theory Appl. 2013, 2013:30.

Khan, Muhammad A.A., and Hafiz Fukhar-ud-din, and Abdul R. Khan. "Mosco convergence results for common fixed point problems and generalized equilibrium problems in Banach spaces." Fixed Point Theory Appl. 2014, 2014:59.

Khánh, Phan Quoc, and Le Minh Luu. "On the existence of solutions to vector quasivariational inequalities and quasicomplementarity problems with applications to traffic network equilibria." J. Optim. Theory Appl. 123, no. 3 (2004): 533-548.

Kim, Jong Kyu. "Convergence theorems of iterative sequences for generalized equilibrium problems involving strictly pseudocontractive mappings in Hilbert spaces." J. Comput. Anal. Appl. 18, no. 3 (2015): 454-471.

Kim, Jong Kyu, and Sun Young Cho, and Xiaolong Qin. "Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings." J. Inequal. Appl. 2010 (2010): Art. ID 312602.

Luu, Do Van. "On constraint qualifications and optimality conditions in locally Lipschitz multiobjective programming problems." Nonlinear Funct. Anal. Appl. 14, no. 1 (2009): 81-97.

Marino, Giuseppe, and Hong-Kun Xu. "Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces." J. Math. Anal. Appl. 329, no. 1 (2007): 336-346.

Moudafi, Abdellatif. "Second-order differential proximal methods for equilibrium problems." JIPAM. J. Inequal. Pure Appl. Math. 4, no. 1 (2003): Article 18.

Moudafi, Abdellatif, and Michel Théra. "Proximal and dynamical approaches to equilibrium problems." Vol. 477 of Lecture Notes in Econom. and Math. Systems, 187-201. Berlin: Springer, 1999.

Scherzer, Otmar. "Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems." J. Math. Anal. Appl. 194, no. 3 (1995): 911-933.

Su, Tran Van. "Second-order optimality conditions for vector equilibrium problems." J. Nonlinear Funct. Anal. 2015 (2015): Article ID 6.

Takahashi, Satoru, and Wataru Takahashi. "Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space." Nonlinear Anal. 69, no. 3 (2008): 1025-1033.

Wen, Dao-Jun, and Yi-An Chen. "General iterative methods for generalized equilibrium problems and fixed point problems of $k$-strict pseudo-contractions." Fixed Point Theory Appl. 2012, 2012:125.

Zhang, Lingmin, and Yan Hao. "Fixed point methods for solving solutions of a generalized equilibrium problem." J. Nonlinear Sci. Appl. 9, no. 1 (2016): 149-159.


Full Text: PDF

Download statistics: 1439

Licencja Creative Commons
This article by Muhammad Aqeel Ahmad Khan is governed by the Creative Commons Attribution-ShareAlike 4.0 International(CC BY-SA 4.0) licence.


e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej