Iterative methods for fixed point problems and generalized split feasibility problems in Banach spaces
Volume 11, Issue 2, pp 198--217
http://dx.doi.org/10.22436/jnsa.011.02.03
Publication Date: January 23, 2018
Submission Date: August 25, 2017
Revision Date: November 23, 2017
Accteptance Date: December 06, 2017
-
3046
Downloads
-
6832
Views
Authors
Yanlai Song
- College of Science, Zhongyuan University of Technology, 450007 Zhengzhou, China.
Abstract
In this paper, we study
the Halpern type iterative algorithm to approximate a common solution of fixed point problems of an infinite family of
demimetric mappings and generalized split feasibility problems with firmly nonexpansive-like mappings in Banach spaces.
We also prove strong convergence theorems for a common solution of the above-said problems by the proposed iterative algorithm and discuss some applications of our
results.
The methods in this paper are novel and different from those given
in many other paper. And the results are the extension and improvement
of the recent results in the literature.
Share and Cite
ISRP Style
Yanlai Song, Iterative methods for fixed point problems and generalized split feasibility problems in Banach spaces, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 2, 198--217
AMA Style
Song Yanlai, Iterative methods for fixed point problems and generalized split feasibility problems in Banach spaces. J. Nonlinear Sci. Appl. (2018); 11(2):198--217
Chicago/Turabian Style
Song, Yanlai. "Iterative methods for fixed point problems and generalized split feasibility problems in Banach spaces." Journal of Nonlinear Sciences and Applications, 11, no. 2 (2018): 198--217
Keywords
- Banach space
- generalized split feasibility problem
- fixed point
- metric resolvent
- demimetric mapping
MSC
References
-
[1]
S. Akashi, W. Takahashi , Weak convergence theorem for an infinite family of demimetric mappings in a Hilbert space, J. Nonlinear Convex Anal., 10 (2016), 2159–2169.
-
[2]
K. Aoyama, F. Kohsaka , Existence of fixed points of firmly nonexpansive-like mappings in Banach spaces, Fixed Point Theory Appl., 2010 (2010), 15 pages.
-
[3]
K. Aoyama, F. Kohsaka, W. Takahashi , Strong convergence theorems for a family of mappings of type (P) and applications, Nonlinear Anal. Optim., 2009 (2009), 17 pages.
-
[4]
K. Aoyama, F. Kohsaka, W. Takahashi , Three generalizations of firmly nonexpansive mappings: their relations and continuous properties, J. Nonlinear Convex Anal., 10 (2009), 131–147.
-
[5]
F. E. Browder, Nonlinear maximal monotone operators in Banach spaces, Math. Ann., 175 (1968), 89–113.
-
[6]
C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441–453.
-
[7]
Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space , Numer. Algorithms, 8 (1994), 221–239.
-
[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, L. Wang, Y. Zhao , On a class of split equality fixed point problems in Hilbert spaces, J. Nonlinear Var. Anal., 1 (2017), 201–212.
-
[10]
C. Chidume, Ş. Măruşter , Iterative methods for the computation of fixed points of demicontractive mappings, J. Comput. Appl. Math., 234 (2010), 861–882.
-
[11]
S. Y. Cho , Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space, J. Appl. Anal. Comput., 8 (2018), 19–31.
-
[12]
M. Hojo, W. Takahashi, I. Termwuttipong , Strong convergence theorems for 2-generalized hybrid mappings in Hilbert spaces , Nonlinear Anal., 75 (2012), 2166–2176.
-
[13]
P. Kocourek, W. Takahashi, J.-C. Yao, Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese J. Math., 14 (2010), 2497–2511.
-
[14]
P. E. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469–479.
-
[15]
G. Marino, H.-K. Xu , Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336–346.
-
[16]
S. Matsushita, W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2004 (2004), 10 pages.
-
[17]
E. Naraghirad, L. J. Lin , Strong convergence theorems for generalized nonexpansive mappings on star-shaped set with applications , Fixed Point Theory Appl., 2014 (2014), 24 pages.
-
[18]
X. Qin, J.-C. Yao, Projection splitting algorithm for nonself operator, J. Nonlinear Convex Anal., 18 (2017), 925–935.
-
[19]
R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497–510.
-
[20]
R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), 75–88.
-
[21]
R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), 97–116.
-
[22]
Y. Shehu, O. S. Iyiola, C. D. Enyi , An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces , Numer. Algorithms, 72 (2016), 835–864.
-
[23]
W. Takahashi, Convex Analysis and Approximation of Fixed Points, Yokohama Publishers, Yokohama (2000)
-
[24]
W. Takahashi , Nonlinear Functional Analysis, Yokohama Publ., Yokohama (2000)
-
[25]
S. Takahashi, W. Takahashi , The split common null point problem and the shrinking projection method in Banach spaces, Optimization, 65 (2016), 281–287.
-
[26]
S. Takahashi, W. Takahashi, M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 147 (2010), 27–41.
-
[27]
W. Takahashi, J.-C. Yao , Strong convergence theorems by hybrid methods for the split common null point problem in Banach spaces, Fixed Point Theory Appl., 2015 (2015), 13 pages.
-
[28]
K.-K. Tan, H.-K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301–308.
-
[29]
H.-K. Xu , Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
-
[30]
Y.-H.Yao, H.-Y. Zhou, Y.-C. Liou , Weak and strong convergence theorems for an asymptotically k-strict pseudo-contraction and a mixed equilibrium problem, J. Korean Math. Soc., 46 (2009), 561–576.
-
[31]
Z.-T. Yu, L.-J. Lin, C.-S. Chuang , Mathematical programming with multiple sets split monotone variational inclusion constraints, Fixed Point Theory Appl., 2014 (2014), 27 pages.