Convergence analysis of a novel iteration algorithm for solving split feasibility problems
-
1844
Downloads
-
3344
Views
Authors
Qinwei Fan
- School of Science, Xi’an Polytechnic University, Xi’an, Shaanxi 710048, China.
Abstract
In this paper, our aim is to construct a convergence theorem in Banach spaces via the following Ishikawa recursive algorithm
\[
\begin{cases}
x_{n+1}=(1-\alpha_n)x_n+\alpha_nT_ny_n,\\
y_n=(1-\beta_n)x_n+\beta_nT_nx_n,
\end{cases}
\]
where \(\{\alpha_n\}\), \(\{\beta_n\}\) are sequences in \([0, 1]\) and \(\{T_n\}\) is a sequence of nonexpansive mappings. Moreover, we also apply these results
to solve a split feasibility problem.
Share and Cite
ISRP Style
Qinwei Fan, Convergence analysis of a novel iteration algorithm for solving split feasibility problems, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 2, 647--655
AMA Style
Fan Qinwei, Convergence analysis of a novel iteration algorithm for solving split feasibility problems. J. Nonlinear Sci. Appl. (2017); 10(2):647--655
Chicago/Turabian Style
Fan, Qinwei. "Convergence analysis of a novel iteration algorithm for solving split feasibility problems." Journal of Nonlinear Sciences and Applications, 10, no. 2 (2017): 647--655
Keywords
- Split feasibility problem
- fixed point
- nonexpansive mapping
- weak convergence.
MSC
References
-
[1]
H. Attouchi, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA (1984)
-
[2]
R. E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math., 32 (1979), 107–116.
-
[3]
C. Byren, 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 processing and image reconstruction, Inverse Problems, 20 (2014), 103–120.
-
[5]
Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221–239.
-
[6]
S.-S. Chang, J. K. Kim, Y. J. Cho, J. Y. Sim, Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014 ), 12 pages.
-
[7]
S. Y. Cho, B. A. Bin Dehaish, X.-L. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 7 (2017), 427–438.
-
[8]
S. Y. Cho, W.-L. Li, S. M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013 (2013 ), 14 pages.
-
[9]
S. Y. Cho, X.-L. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014 ), 15 pages.
-
[10]
N.-N. Fang, Y.-P. Gong, Viscosity iterative methods for split variational inclusion problems and fixed point problems of a nonexpansive mapping, Commun. Optim. Theory, 2016 (2016), 15 pages.
-
[11]
D. V. Hieu, L. D. Muu, P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algorithms, 73 (2016), 197–217.
-
[12]
S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150.
-
[13]
J. K. Kim, S. Y. Cho, X.-L. Qin, Some results on generalized equilibrium problems involving strictly pseudocontractive mappings, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 2041–2057.
-
[14]
J. K. Kim, Salahuddin, A system of nonconvex variational inequalities in Banach spaces, Commun. Optim. Theory, 2016 (2016), 19 pages.
-
[15]
M. A. Krasnosel’skii, Two remarks on the method of successive approximations, (Russian) Uspehi Mat. Nauk (N.S.), 10 (1955), 123–127.
-
[16]
S.-T. Lv, Some results on a two-step iterative algorithm in Hilbert spaces, J. Nonlinear Funct. Anal., 2015 (2015 ), 10 pages.
-
[17]
W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510.
-
[18]
A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117–121.
-
[19]
X.-L. Qin, S.-S. Chang, Y. J. Cho, Iterative methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Anal. Real World Appl., 11 (2010), 2963–2972.
-
[20]
X.-L. Qin, S. Y. Cho, L. Wang, A regularization method for treating zero points of the sum of two monotone operators, Fixed Point Theory Appl., 2014 (2014 ), 10 pages.
-
[21]
X.-L. Qin, J.-C. Yao , Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators, J. Inequal. Appl., 2016 (2016 ), 9 pages.
-
[22]
B. Qu, B.-H. Liu, N. Zheng, On the computation of the step-size for the CQ-like algorithms for the split feasibility problem, Appl. Math. Comput., 262 (2015), 218–223.
-
[23]
S. Rathee, Ritika, \(\delta\)-convergence theorems for Mann and Ishikawa iteration procedures with errors in CAT(0) spaces, Commun. Optim. Theory, 2013 (2013 ), 11 pages.
-
[24]
S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274–276.
-
[25]
H.-K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127–1138.
-
[26]
H.-K. Xu, A variable Krasnoselskiı-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021–2034.
-
[27]
Q.-Z. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261–1266.
-
[28]
Y.-H. Yao, R.-D. Chen, H.-Y. Zhou , Iterative process for certain nonlinear mappings in uniformly smooth Banach spaces, Nonlinear Funct. Anal. Appl., 10 (2005), 651–664.
-
[29]
C.-J. Zhang, J.-L. Li, B.-Q. Liu, Strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Comput. Math. Appl., 61 (2011), 262–276.
-
[30]
J.-L. Zhao, Q.-Z. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791–1799.
-
[31]
F. Zhao, L. Yang, Hybrid projection methods for equilibrium problems and fixed point problems of infinite family of multivalued asymptotically nonexpansive mappings, J. Nonlinear Funct. Anal., 2016 (2016 ), 13 pages.
