The main aim of this manuscript is to work on the split equilibrium problem with the combined results of the fixed point problem and split variational inequality problem. This paper is an extension of the recent work of Lohawech et al. We proposed a sequence that converges weakly to the common solution of all the three problems mentioned earlier. In the end, we supply some direct consequences of the main result, as the paper is an extension of various existing results.
Citation: Savita Rathee, Monika Swami. Algorithm for split variational inequality, split equilibrium problem and split common fixed point problem[J]. AIMS Mathematics, 2022, 7(5): 9325-9338. doi: 10.3934/math.2022517
The main aim of this manuscript is to work on the split equilibrium problem with the combined results of the fixed point problem and split variational inequality problem. This paper is an extension of the recent work of Lohawech et al. We proposed a sequence that converges weakly to the common solution of all the three problems mentioned earlier. In the end, we supply some direct consequences of the main result, as the paper is an extension of various existing results.
| [1] | A. Kangtunyakarn, A new iterative algorithm for the set of fixed-point problems of nonexpansive mappings and the set of equilibrium problem and variational inequality problem, Abstr. Appl. Anal., 2011 (2011), 1–24. |
| [2] |
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123–145. Corpus ID: 117484413. https://doi.org/10.2307/3977996 doi: 10.2307/3977996
|
| [3] |
G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonomika i Matematicheskie Metody, 12 (1976), 747–756. https://doi.org/10.3103/S0278641910030039 doi: 10.3103/S0278641910030039
|
| [4] |
G. Stampacchia, Formes bilineaires coercivites sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413–4416. https://doi.org/10.1186/s13660-018-1942-1 doi: 10.1186/s13660-018-1942-1
|
| [5] | H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, (2011), New York. https://doi.org/10.1007/978-1-4419-9467-7 |
| [6] |
H. K. Xu, Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl., 105 (2011), 360–378. https://doi.org/10.1007/s10957-011-9837-z doi: 10.1007/s10957-011-9837-z
|
| [7] |
H. K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl., 26 (2010), 17 pages. https://doi.org/10.1088/0266-5611/26/10/105018 doi: 10.1088/0266-5611/26/10/105018
|
| [8] |
H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Optim. Theory Appl., 298 (2004), 279–291. https://doi.org/10.1016/j.jmaa.2004.04.059 doi: 10.1016/j.jmaa.2004.04.059
|
| [9] |
L. C. Ceng, Q. H. Ansari, J. C. Yao, Some iterative methods for finding fixed points and for solving constrained convex minimization problems, Nonlinear Anal., 74 (2011), 5286–5302. https://doi.org/10.1016/j.na.2011.05.005 doi: 10.1016/j.na.2011.05.005
|
| [10] |
M. Suwannaprapa, N. Petrot, S. Suantai, Weak convergence theorems for split feasibility problems on zeros of the sum of monotone operators and fixed point sets in Hilbert spaces, Fixed Point Theory Appl., (2017), 6 pages. https://doi.org/10.1186/s13663-017-0599-7 doi: 10.1186/s13663-017-0599-7
|
| [11] |
M. Tian, B. N. Jiang, Weak convergence theorem for a class of split variational inequality problems and applications in a Hilbert space, J. Inequal. Appl., 123 (2017), 17 pages. https://doi.org/10.1186/s13660-017-1397-9 doi: 10.1186/s13660-017-1397-9
|
| [12] |
P. Lohawech, A. Kaewcharoen, A. Farajzadeh, Algorithm for the common solution of the split variational inequality problems and fixed point problems with applications, J. Inequal. Appl., (2018), 27 pages. https://doi.org/10.1186/s13660-018-1942-1 doi: 10.1186/s13660-018-1942-1
|
| [13] |
W. Takahashi, M. Toyoda, Weak convergence theorem for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417–423. https://doi.org/10.1023/A:1025407607560 doi: 10.1023/A:1025407607560
|
| [14] |
W. Takahashi, N. Nadezhkina, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191–201. https://doi.org/10.1007/s10957-005-7564-z doi: 10.1007/s10957-005-7564-z
|
| [15] |
W. Takahashi, H. K. Xu, J. C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, St-Valued Var. Anal., 23 (2015), 205–221. https://doi.org/10.1007/s11228-014-0285-4 doi: 10.1007/s11228-014-0285-4
|
| [16] | X. Qin, J. C. Yao, Projection splitting algorithms for nonself operators, J. Nonlinear Convex A., 18 (2017), 925–935. |
| [17] |
Y. Censor, A. Gibali, S. Reich, Algorithm for the split variational inequality problem, Numer. Algorithms, 58 (2012), 117–136. https://doi.org/10.1007/s11075-011-9490-5 doi: 10.1007/s11075-011-9490-5
|
| [18] |
Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587–600. https://doi.org/10.1088/0266-5611/26/5/055007 doi: 10.1088/0266-5611/26/5/055007
|
| [19] | Y. Censor, T. Elfving, A multiprojection algorithms using Bragman projection in a product space, 8 (1994), 221–239. https://doi.org/10.1007/BF02142692 |
| [20] |
Z. He, The split Equilibrium problem and its convergence algorithms, J. Inequal. Appl., 162 (2012), 15 pages. https://doi.org/10.1186/1029-242X-2012-162 doi: 10.1186/1029-242X-2012-162
|
Savita Rathee, Monika Swami. Algorithm for split variational inequality, split equilibrium problem and split common fixed point problem[J]. AIMS Mathematics, 2022, 7(5): 9325-9338. doi: 10.3934/math.2022517
DownLoad: