A modified Mann algorithm for the general split problem of demicontractive operators

Öz

This work proposes a novel method for solving the general split common fixed point problem of demicontractive operators in the framework of real Hilbert spaces. Our proposed technique is principally based on the Mann algorithm. The proof of the weak convergence theorem is additionally established under some particular conditions. We subsequently verify the convergence of our algorithm via numerical examples.

Anahtar Kelimeler

• general split common fixed point problem
• demicontractive operator
• Mann algorithm

Kaynakça

1. [1] H.H. Bauschke, J.M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev. 38 (1996) 367-426.
2. [2] H.H. Bauschke, P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics, Springer, New York (2011).
3. [3] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl. 18 (2002) 441-453.
4. [4] Y. Censor, A. Segal, The split common ?xed point problem for directed operators, J. Convex Anal. 16(2) (2009) 587-600.
5. [5] Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms. 8(2) (1994) 221-239.
6. [6] W. Chaolamjiak, S.A. Khan, H.A. Hammad, H. Dutta, Weak and strong convergence results for the modified Noor iteration of three quasi-nonexpansive multivalued mappings in Hilbert spaces, Filomat. 34(8) (2020) 2495-2510.
7. [7] T.L. Hicks, J.D. Kubicek, On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl. 59 (1977) 498-504.
8. [8] A. Kangtunyakarn, Iterative scheme for finding solutions of the general split feasibility problem and the general constrained minimization problems, Filomat. 33(1) (2019) 233-243.

9. [9] S. Kesornporm, N. Pholasa, P. Cholamjiak, On the convergenece analysis of the gradient-CQ algorithms for the split feasibility problem, Numer. Algorithms. 84(3) (2020) 997-1017.
10. [10] S. Maruster, The solution by iteration of nonlinear equations in Hilbert spaces, Proc. Am. Math. Soc. 63 (1977) 69-73.
11. [11] A. Moudafi, The split common fixed-point problem for demicontractive mappings, Inverse Probl. 26(5) (2010) 055007.
12. [12] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Am. Math. Soc. 73 (1967) 595-597.
13. [13] S. Suantai, W. Cholamjiak, P. Cholamjiak, An implicit iteration process for a finite family of multi-valued mappings in Banach spaces, Appl. Math. Lett. 25 (2012) 1656-?1660.
14. [14] S. Suantai, K. Kankam, P. Cholamjiak, W. Cholamjiak, A parallel monotone hybrid algorithm for a finite family of G−nonexpansive mappings in Hilbert spaces endowed with a graph applicable in signal recovery, Comp. Appl. Math. 40 (2021) 145.
15. [15] R. Suparatulatorn, S. Suantai, N. Phudolsitthiphat, Reckoning solution of split common fixed point problems by using inertial self-adaptive algorithms, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113 (2019) 3101-3114.
16. [16] X.X. Zheng, Y. Yao, Y.C. Liou, L.M. Leng, Fixed point algorithms for the split problem of demicontractive operators, J. Nonlinear Sci. Appl. 10 (2017) 1263-1269.