Feasible iterative algorithms and strong convergence theorems for bi-level fixed point problems

Chang, Shih-Sen¹ ; Quan, Jing² ; Liu, Jingai³

¹ Center for General Education, China Medical University, Taichung, 40402, Taiwan
² Department of Mathematics, Yibin University, Yibin, Sichuan, 644007, China
³ Department of Mathematics and Physics, Beijing Institute of Petro-Chemical Technology, Beijing, 102617, China

Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 4, p. 1515-1528.

Résumé

The purpose of this paper is to introduce and study the bi-level split fixed point problems in the setting of infinite-dimensional Hilbert spaces. For solving this kind problems, some new simultaneous iterative algorithms are proposed. Under suitable conditions, some strong convergence theorems for the sequences generated by the proposed algorithm are proved. As applications, we shall utilize the results presented in the paper to study bi-level split equilibrium problem, bi-level split optimization problems and the bi-level split variational inequality problems. The results presented in the paper are new which also extend and improve many recent results.

Détail

DOI : 10.22436/jnsa.009.04.10

Classification : 47J25, 47H09, 65K10
Keywords: Bi-level split fixed point problem, bi-level split equilibrium problem, bi-evel split optimization problem, bi-level split variational inequality problem, split feasibility problem.

Bibliographie

[1] Ansari, Q. H.; Rehan, A. Split feasibility and fixed point problems, Nonlinear anal., Volume 2014 (2014), pp. 282-322

[2] Blum, E.; W. Oettli From optimization and variational inequalities to equilibrium problems, Math. Student, Volume 63 (1994), pp. 123-145

[3] Byrne, C. Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Problems, Volume 18 (2002), pp. 441-453

[4] Censor, Y.; Bortfeld, T.; Martin, B.; Trofimov, A. A unified approach for inversion problem in intensity-modulated radiation therapy, Phys. Med. Biol., Volume 51 (2006), pp. 2353-2365

[5] Censor, Y.; Elfving, T. A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, Volume 8 (1994), pp. 221-239

[6] Censor, Y.; Elfving, T.; Kopf, N.; Bortfeld, T. The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, Volume 21 (2005), pp. 2071-2084

[7] Censor, Y.; Motova, A.; A. Segal Perturbed projections ans subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl, Volume 327 (2007), pp. 1244-1256

[8] Censor, Y.; Segal, A. The split common fixed point problem for directed operators, J. Convex Anal., Volume 16 (2009), pp. 587-600

[9]

[10] Chang, S. S.; R. P. Agarwal Strong convergence theorems of general split equality problems for quasi-nonexpansive mappings, J. Inequal. Appl., Volume 2014 (2014), pp. 1-14

[11] Chang, S. S.; Cho, Y. J.; Kim, J. K.; Zhang, W. B.; Yang, L. Multiple-set split feasibility problems for asymptotically strict pseudocontractions, Abstr. Appl. Anal., Volume 2012 (2012), pp. 1-12

[12] Chang, S. S.; Lee, H. W. J.; Chan, C. K.; Zhang, W. B. A modified Halpern-type iterative algorithm for totally quasi-\(\phi\)-asymptotically nonexpansive mappings with applications, Appl. Math. Comput., Volume 218 (2012), pp. 6489-6497

[13] Chang, S. S.; Wang, L. Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces, Fixed Point Theory Appl., Volume 2014 (2014), pp. 1-14

[14] Chang, S. S.; Wang, L.; Tang, Y. K.; Wang, G. Moudafi's open question and simultaneous iterative algorithm for general split equality variational inclusion problems and general split equality optimization problems, Fixed Point Theory Appl., Volume 2014 (2014), pp. 1-17

[15] Chang, S. S.; Wang, L.; Tang, Y. K.; Yang, L. The split common fixed point problem for total asymptotically strictly pseudocontractive mappings, J. Appl. Math., Volume 2012 (2012), pp. 1-13

[16] Chang, S. S.; Wang, L.; Wang, X. R.; Wang, G. General Split Equality Equilibrium Problems with Application to Split Optimization Problems, J. Optim Theory Appl., Volume 166 (2015), pp. 377-390

[17] Chen, R.; Wang, J.; Zhang, H. General split equality problems in Hilbert spaces , Fixed Point Theory Appl., Volume 2014 (2014), pp. 1-8

[18] C. S. Chuang Strong convergence theorems for the split variational inclusion problem in Hilbert spaces, Fixed Point Theory Appl., Volume 2013 (2013), pp. 1-20

[19] Eslamian, M.; Latif, A. General split feasibility problems in Hilbert spaces, Abstr. Appl. Anal., Volume 2013 (2013), pp. 1-6

[20] Goebel, K.; Kirk, W. A. Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990

[21] He, Z.; Du, W. S. New feasible iterative algorithms and strong convergence theorems for bilevel split equilibrium problems, Fixed Point Theory Appl., Volume 2014 (2014), pp. 1-17

[22] Maingé, P. E. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., Volume 16 (2008), pp. 899-912

[23] Moudafi, A. Split monotone variational inclusions , J. Optim. Theory Appl., Volume 150 (2011), pp. 275-283

[24] A. Moudafi A relaxed alternating CQ algorithm for convex feasibility problems, Nonlinear Anal., Volume 79 (2013), pp. 117-121

[25] Moudafi, A.; Eman, A. S. Simultaneous iterative methods for split equality problem , Trans. Math. Program. Appl., Volume 1 (2013), pp. 1-11

[26] E. Naraghirad On an open question of Moudafi for convex feasibility problem in Hilbert spaces, Taiwanese J. Math., Volume 18 (2014), pp. 371-408

[27] Xu, H. K. A variable Krasnosel'skii-Mann algorithm and the multiple-sets split feasibility problem , Inverse Problems, Volume 22 (2006), pp. 2021-2034

[28] Yang, Q. The relaxed CQ algorithm for solving the split feasibility problem, Inverse Problems, Volume 20 (2004), pp. 1261-1266

[29] Zhao, J.; Yang, Q. Several solution methods for the split feasibility problem, Inverse Problems, Volume 21 (2005), pp. 1791-1799