Iterative methods for solving the split common fixed point problem of demicontractive mappings in Hilbert spaces

2079
Downloads

4260
Views
Authors
Chunxiang Zong
 Department of Mathematics, Nanchang University, Nanchang 330031, P. R. China.
Yuchao Tang
 Department of Mathematics, Nanchang University, Nanchang 330031, P. R. China.
Abstract
The split common fixed point problem was proposed in recent years
which required to find a common fixed point of a family of mappings
in one space whose image under a linear transformation is a common
fixed point of another family of mappings in the image space. In
this paper, we study two iterative algorithms for solving this split
common fixed point problem for the class of demicontractive mappings
in Hilbert spaces. Under mild assumptions on the parameters, we
prove the convergence of both iterative algorithms. As a consequence, we obtain new convergence
theorems for solving the split
common fixed point problem for the class of directed mappings. We compare the performance of the proposed iterative
algorithms with the existing iterative algorithms and conclude from the numerical experiments that our iterative algorithms converge faster than
these existing iterative algorithms in terms of iteration numbers.
Share and Cite
ISRP Style
Chunxiang Zong, Yuchao Tang, Iterative methods for solving the split common fixed point problem of demicontractive mappings in Hilbert spaces, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 8, 960970
AMA Style
Zong Chunxiang, Tang Yuchao, Iterative methods for solving the split common fixed point problem of demicontractive mappings in Hilbert spaces. J. Nonlinear Sci. Appl. (2018); 11(8):960970
Chicago/Turabian Style
Zong, Chunxiang, Tang, Yuchao. "Iterative methods for solving the split common fixed point problem of demicontractive mappings in Hilbert spaces." Journal of Nonlinear Sciences and Applications, 11, no. 8 (2018): 960970
Keywords
 Split common fixed point problem
 demicontractive mappings
 cyclic iteration method
 simultaneous iteration method
MSC
References

[1]
H. H. Bauschke, J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367–426.

[2]
H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York (2011)

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

[4]
C. Byrne , A unified treatment of some iterative algorithms in signal procesing and image reconstruction, Inverse Problems, 20 (2004), 103–120

[5]
L.C. Ceng, Q. H. Ansari, J.C. Yao, Mann type iterative methods for finding a common solution of split feasibility and fixed point problems, Positivity, 16 (2012), 471–495.

[6]
Y. Censor, T. Elfving , A multiprojection algorithm using bregman projections in a product space, Numer. Algorithms, 8 (1994), 221–239.

[7]
Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiplesets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071–2084.

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

[9]
S. S. Chang, H. W. Joseph Lee, C. K. Chan, L. Wang, L. J. Qin, Split feasibility problem for quasinonexpansive multivalued mappings and total asymptotically strict pseudocontracive mapping, Appl. Math. Comput., 219 (2013), 10416–10424.

[10]
S. S. Chang, L. Wang, Y. K. Tang, L. Yang , The split common fixed point problems for total asymptotically strictly pseudocontracive mappings, J. Appl. Math., 2012 (2012), 13 pages.

[11]
H.H. Cui, M.L. Su, F.H. Wang, Damped projection method for split common fixed point problems, J. Inequal. Appl., 2013 (2013), 10 pages.

[12]
H.H. Cui, F.H. Wang, Iterative methods for the split common fixed point problem in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 8 pages.

[13]
Q.L. Dong, Y.H. Yao, S.N. He, Weak convergence theorems of the modified relaxed projection algorithms for the split feasibility problem in Hilbert spaces, Optim. Lett., 8 (2014), 1031–1046.

[14]
Y.Y. Huang, C.C. Hong , Approximating common fixed points of averaged selfmappings with application to the split feasibility problem and maximal monotone operators in Hilbert spaces , Fixed Point Theory Appl., 2013 (2013), 20 pages.

[15]
R. Kraikaew, S. Saejung, On split common fixed point problems, J. Math. Anal. Appl., 415 (2014), 513–524.

[16]
A. Moudafi, The split common fixed point problem for demicontractive mappings, Inverse Problems, 26 (2010), 6 pages.

[17]
A. Moudafi , A note on the split common fixedpoint problem for quasinonexpansive operators, Nonlinear Anal., 74 (2011), 4083–4087.

[18]
A. Moudafi, Alternating CQalgorithms for convex feasibility and split fixedpoint problems, J. Nonlinear Convex Anal., 15 (2014), 809–818.

[19]
Y.C. Tang, L.W. Liu, Several iterative algorithms for solving the split common fixed point problem of directed operators with applications, Optimization, 65 (2016), 53–65.

[20]
Y.C. Tang, J.G. Peng, L.W. Liu , A cyclic algorithm for the split common fixed point problem of demicontractive mappings in Hilbert spaces, Math. Model. Anal., 17 (2012), 457–466.

[21]
Y.C. Tang, J.G. Peng, L.W. Liu, A cyclic and simultaneous iterative algorithm for the multiple split common fixed point problem of demicontractive mappings, Bull. Korean. Math. Soc., 51 (2014), 1527–1538.

[22]
D. V. Thong, D. V. Hieu , An inertial method for solving split common fixed point problems, J. Fixed Point Theory Appl., 19 (2017), 3029–3051.

[23]
F.H. Wang, H.H. Cui, Convergence of a cyclic algorithm for the split common fixed point problem without continuity assumption, Math. Model. Anal., 18 (2013), 537–542.

[24]
F.H. Wang, H.K. Xu, Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74 (2011), 4105–4111.

[25]
H.K. Xu, A variable krasnoselskiimann algorithm and the multipleset split feasibility problem, Inverse Problems, 22 (2006), 2021–2034.

[26]
H.K. Xu, Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces, Inverse Problems, 26 (2010), 17 pages.

[27]
L.J. Zhu, Y.C. Liou, S. M. Kang, Y.H. Yao, Algorithmic and analytical approach to the split common fixed points problem, Fixed Point Theory Appl., 2014 (2014), 10 pages.

[28]
L.J. Zhu, Y.C. Liu, J.C. Yao, Y.H. Yao , New algorithms for designed for the split common fixed point problem of quasipseudocontractions, J. Inequal. Appl., 2014 (2014), 13 pages.