An Iterative Method for a Common Solution of a Combination of the Split Equilibrium Problem, a Finite Family of Nonexpansive Mapping and a Combination of Variational Inequality Problem
Keywords:Split equilibrium problem; variational inequality problem; fixed point problem; projection method; nonexpansive mapping.
The present paper aims to deal with an iterative algorithm for finding common solution of the combination of the split equilibrium problem and a finite family of non-expansive mappings and the combination of variational inequality problem in the setting of real Hilbert spaces. Further, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution to these problems. A numerical example is presented to illustrate the proposed method and convergence result. The results and method presented in this paper generalize, extend and unify some known results in the literatures.
E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium prob- lems, The Mathematics Student, 63 (1994), 123-145.
A. Bnouhachem, An iterative algorithm for system of generalized equilibrium problems and fixed point problem, Fix. Point Th. Appl. 2014(235) (2014), 1-21.
A. Bnouhachem, A modified projection method for a common solution of a system of vari- ational inequalities, a split equilibrium problem and a hierarchical fixed-point problem, Fix. Point Th. Appl. 2014(22) (2014), 1-25.
A. Bnouhachem, S. Al-Homidan and Q. H. Ansari, An iterative method for common solutions of equilibrium problems and hierarchical fixed point problems, Fix. Point Th. Appl. 194(2014), 1-21.
A. Bnouhachem, A hybrid iterative method for a combination of equilibria problem, a com- bination of variational inequality problems and a hierarchical fixed point problem, Fix. Point Th. Appl. 163 (2014), 1-29.
A. Bnouhachem, Strong convergence algorithm for approximating the common solutions of a variational inequality, a mixed equilibrium problem and a hierarchical fixed-point problem, J. Inequal. Appl. 154(2014), 1-24.
L. C. Ceng and J. C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214(1) (2008), 186-201.
S. S. Chang, H. W. Joseph Lee and C. K. Chan, A new method for solving equilibrium prob- lem fixed point problem and variational inequality problem with application to optimization, Nonlinear Anal. 70 (2009), 3307-3319.
C. E. Chidume and M. O. Nnakwe, Iterative algorithms for split variational inequalities and generalized split feasibility problems with applications, J. Nonlinear Var. Anal. 3 (2019), 127- 140.
F. Cianciaruso, G. Marino, L. Muglia and Y. Yao, A hybrid projection algorithm for find- ing solutions of mixed equilibrium problem and variational inequality problem, Fix. Point Th. Appl. 2010 (2010), Article ID 383740, 19 pages.
P. L. Combettes and S. A. Hirstoaga, Equilibrium programming using proximal like algo- rithms, Math. Prog. 78 (1997), 29-41.
P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), 117-136.
J. Deepho, W. Kumam and P. Kumam, A new hybrid projection algorithm for solving the split generalized equilibrium problems and the system of variational inequality problems, J. Math. Model. Algor. 13(4) (2014), 405-423.
J. Deepho, J. M. Moreno and P. Kumam, A viscosity of Cesaro mean approximation method for split generalized equilibrium, variational inequality and fixed point problems, J. Nonlinear Sci. Appl. 9, 1475-1496 (2016).
M. Farid, The subgradient extragradient method for solving mixed equilibrium problems and fixed point problems in Hilbert spaces, J. Appl. Numer. Optim. 1 (2019), 335-345.
K. R. Kazmi and S. H. Rizvi, Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for nonexpansive semigroup, Math. Sci. 7(1) (2013), 1-10.
K. R. Kazmi and S. H. Rizvi, Iterative approximation of a common solution of a split equi- librium problem, a variational inequality problem and a fixed point problem, J. Egypt. Math. Society 21(1) (2013), 44-51.
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 equilibrium problem and fixed point problems, Fix. Point Th. Appl. 2012(186) (2012), 1-26.
C. Marino, L. Muglia and J. C. Yao, Viscosity methods for common solutions of equilib- rium and variational inequality problems via multi-step iterative algorithms and common fixed points. Nonlinear Anal. 75(4) (2012), 1787-1798.
W. Phuengrattana and K. Lerkchaiyaphum, On solving the split generalized equilibrium problem and the fixed point problem for a countable family of nonexpansive multivalued map- pings, Fix. Point Th. Appl. 2018(1) (2018), 1-17.
X. Qin, M. Shang and Y. Su, A general iterative method for equilibrium problem and fixed point problem in Hilbert spaces, Nonlinear Anal. 69(11) (2008), 3897-3909.
S. Reich and S. Sabach, Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces, optimization theory and related topics, Contemp. Math. 568 (2012), 225-240.
R. T. Rockafellar, On the maximality of sums nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75-88 (1970).
G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexs, C. R. Acad. Sci. Paris. 258 (1964), 4413-4416.
S. Suwannaut and A. Kangtunyakarn, The combination of the set of solutions of equilib- rium problem for convergence theorem of the set of fixed points of strictly pseudo-contractive mappings and variational inequalities problem, Fix. Point Th. Appl. 291 (2013), 1-26.
S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69(3) (2008), 1025- 1033.
H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66(1) (2002), 240-256.
J. Zhao, D. Hou and H. Zong, Several iterative algorithms for solving the multiple-set split common fixed-point problem of averaged operators, J. Nonlinear Funct. Anal. 2019 (2019), Article ID 39.
How to Cite
Copyright (c) 2021 Tamkang Journal of Mathematics
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.