Convergence analysis for combination of equilibrium problems and k-nonspreading set-valued mappings
DOI:
https://doi.org/10.22199/issn.0717-6279-2020-03-0037Keywords:
Shrinking projection hybrid method, Fixed point problem, k-nonspreading set-valued mappingsAbstract
We find a common solution of generalized equilibrium problems and the set of fixed points of a k-nonspreading setvalued mapping by using shrinking projection hybrid method. Finally, we compare the shrinking solution set after randomization by giving numerical example which justifies our main result.
References
A. Bnouhachem, “A hybrid iterative method for a combination of equilibria problem, a combination of variational inequality problems and a hierarchical fixed point problem”, Fixed point theory and applications, vol. 2014, no. 1, Art ID. 163, Jul. 2014, doi: 10.1186/1687-1812-2014-163
E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems”, The mathematics student, vol. 63, no. 1-4, pp. 123–145, 1994.
W. Cholamjiak and S. Suantai, “A new hybrid algorithm for a countable family of quasi-nonexpansive mappings and equilibrium problems”, Journal of nonlinear and convex analysis, vol. 12, no. 2, pp. 381-398, Aug. 2011. [On line]. Available: https://bit.ly/2ZElDfL
P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces”, Journal of nonlinear and convex analysis, vol. 6, pp. 117-136, 2005. [On line]- Available: https://bit.ly/3grPBti
S. Iemoto and W. Takahashi, “Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space”, Nonlinear analysis: theory, methods & applications, vol. 71, no. 12, pp. e2082–e2089, Dec. 2009, doi: 10.1016/j.na.2009.03.064
K. R. Kazmi, R. Ali, and S. H. Rizvi, “Common solution of a combination of split general variational-like inequality problems and a family of generalized asymptotically nonexpansive mappings”, Journal of advanced mathematical studies, vol. 10, no. 2, pp. 200-215, 2017.
K. R. Kazmi and S. H. Rizvi, “Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem”, Journal of the Egyptian Mathematical Society, vol. 21, no. 1, pp. 44–51, Apr. 2013, doi: 10.1016/j.joems.2012.10.009
K. R. Kazmi and S. H. Rizvi, “An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping”, Optimization letters, vol. 8, no. 3, pp. 1113–1124, Mar. 2013, doi: 10.1007/s11590-013-0629-2
K. R. Kazmi and R. Ali, “Common solution to an equilibrium problem and a fixed point problem for an asymptotically quasi- ϕ -nonexpansive mapping in intermediate sense”, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, vol. 111, no. 3, pp. 877–889, Sep. 2016, doi: 10.1007/s13398-016-0331-1
S. A. Khan and J.-W. Chen, “Gap functions and error bounds for generalized mixed vector equilibrium problems”, Journal of optimization theory and applications, vol. 166, no. 3, pp. 767–776, Nov. 2014, doi: 10.1007/s10957-014-0683-7
F. Kohsaka and W. Takahashi, “Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces”, SIAM journal on optimization, vol. 19, no. 2, pp. 824–835, 2008, doi: 10.1137/070688717
H. B. Liu, “Convergence theorems for a finite family of nonspreading and nonexpansive set-valued mappings and nonexpansive setvalued mappings and equilibrium problems with application”, Theoretical mathematics & applications, vol. 3, no. 3, pp. 49-61, 2013. [On line]. Available: https://bit.ly/2zAH72p
W. R. Mann, “Mean value methods in iteration”, Proceedings of the American Mathematical Society, vol. 4, no. 3, pp. 506–506, Mar. 1953, doi: 10.1090/S0002-9939-1953-0054846-3
K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups”, Journal of mathematical analysis and applications, vol. 279, no. 2, pp. 372–379, Mar. 2003, doi: 10.1016/S0022-247X(02)00458-4
S. Suantai, P. Cholamjiak, Y. J. Cho, and W. Cholamjiak, “On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces”, Fixed point theory and applications, vol. 2016, no. 1, Mar. 2016, doi: 10.1186/s13663-016-0509-4
R. Suparatulatorn, S. Suantai, and W. Cholamjiak, “Hybrid methods for a finite family of G-nonexpansive mappings in Hilbert spaces endowed with graphs”, AKCE international journal of graphs and combinatorics, vol. 14, no. 2, pp. 101–111, Aug. 2017, doi: 10.1016/j.akcej.2017.01.001
S. Suwannaut and A. Kangtunyakarn, “The combination of the set of solutions of equilibrium problem for convergence theorem of the set of fixed points of strictly pseudo-contractive mappings and variational inequalities problem”, Fixed point theory and applications, vol. 2013, no. 1, Nov. 2013, doi: 10.1186/1687-1812-2013-291
S. Suwannaut and A. Kangtunyakarn, “Convergence analysis for the equilibrium problems with numerical results”, Fixed point theory and applications, vol. 2014, no. 1, Art ID. 167, Aug. 2014, doi: 10.1186/1687-1812-2014-167
A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem”, Journal of optimization theory and applications, vol. 133, no. 3, pp. 359–370, May 2007, doi: 10.1007/s10957-007-9187-z
F. Usman and S. A. Khan, “A generalized mixed vector variational-like inequality problem”, Nonlinear analysis: theory, methods & applications, vol. 71, no. 11, pp. 5354–5362, Dec. 2009, doi: 10.1016/j.na.2009.04.023
Published
How to Cite
Issue
Section
Copyright (c) 2020 Suhel Ahmad Khan Khan, Kaleem Raza Kazmi, Watcharaporn Cholamjiak, Hemen Dutta

This work is licensed under a Creative Commons Attribution 4.0 International License.
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.