AN INERTIAL SHRINKING PROJECTION ALGORITHM FOR SPLIT COMMON FIXED POINT PROBLEMS

Zheng Zhou; Bing Tan; Songxiao Li

doi:10.11948/20190330

2020 Volume 10 Issue 5

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Zheng Zhou, Bing Tan, Songxiao Li. AN INERTIAL SHRINKING PROJECTION ALGORITHM FOR SPLIT COMMON FIXED POINT PROBLEMS[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2104-2120. doi: 10.11948/20190330

Citation:

Zheng Zhou, Bing Tan, Songxiao Li. AN INERTIAL SHRINKING PROJECTION ALGORITHM FOR SPLIT COMMON FIXED POINT PROBLEMS[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2104-2120. doi: 10.11948/20190330

AN INERTIAL SHRINKING PROJECTION ALGORITHM FOR SPLIT COMMON FIXED POINT PROBLEMS

1.
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, China

Corresponding author: Email: zhouzheng2272@163.com (Z. Zhou)

Abstract

Abstract

In this paper, the purpose is to introduce and study a new modified shrinking projection algorithm with inertial effects, which solves split common fixed point problems in Banach spaces. The corresponding strong convergence theorems are obtained without the assumption of semi-compactness on mappings. Finally, some numerical examples are presented to illustrate the results in this paper.
- Split common fixed point problem /
- shrinking projection method /
- inertial method /
- strong convergence

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References

[1]	F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 2001, 9(1-2), 3-11. Google Scholar
[2]	C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 2002, 18(2), 441-453. doi: 10.1088/0266-5611/18/2/310 CrossRef Google Scholar
[3]	C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal., 2012, 13(4), 759-775. Google Scholar
[4]	L. Ceng, A. Petruşel, C. Wen and J. C. Yao, Inertial-like subgradient extragradient methods for variational inequalities and fixed points of asymptotically nonexpansive and strictly pseudocontractive mappings, Mathematics, 2019, 7(9), 860. doi: 10.3390/math7090860 CrossRef Google Scholar
[5]	Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 2006, 51(10), 2353-2365. doi: 10.1088/0031-9155/51/10/001 CrossRef Google Scholar
[6]	Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 1994, 8(2), 221-239. doi: 10.1007/BF02142692 CrossRef Google Scholar
[7]	Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 2005, 21(6), 2071. doi: 10.1088/0266-5611/21/6/017 CrossRef Google Scholar
[8]	Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 2012, 59(2), 301-323. doi: 10.1007/s11075-011-9490-5 CrossRef Google Scholar
[9]	Y. Censor, A. Motova and A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl., 2007, 327(2), 1244-1256. doi: 10.1016/j.jmaa.2006.05.010 CrossRef Google Scholar
[10]	Y. Censor and A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 2009, 16(2), 587-600. Google Scholar
[11]	S. Y. Cho, Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space, J. Appl. Anal. Comput., 2018, 8(1), 19-31. Google Scholar
[12]	S. Y. Cho, B. A. Bin Dehaish and X. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 2017, 7(2), 427-438. Google Scholar
[13]	I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Springer Science & Business Media, 2012. Google Scholar
[14]	P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 2005, 6(1), 117-136. Google Scholar
[15]	V. H. Dang, New inertial algorithm for a class of equilibrium problems, Numer. Algorithms, 2019, 80(4), 1413-1436. doi: 10.1007/s11075-018-0532-0 CrossRef Google Scholar
[16]	L. Liu, A hybrid steepest descent method for solving split feasibility problems involving nonexpansive mappings, J. Nonlinear Convex Anal., 2019, 20(3), 471-488. Google Scholar
[17]	L. Liu, X. Qin and R. P. Agarwal, Iterative methods for fixed points and zero points of nonlinear mappings with applications, Optimization, 2019, 1-21. Google Scholar
[18]	Z. Ma, L. Wang and S. Chang, On the split feasibility problem and fixed point problem of quasi-$\phi$-nonexpansive mapping in Banach spaces, Numer. Algorithms, 2019, 80(4), 1203-1218. doi: 10.1007/s11075-018-0523-1 CrossRef Google Scholar
[19]	P. E. Maingé and A. Moudafi, Convergence of new inertial proximal methods for DC programming, SIAM J. Optim., 2008, 19(1), 397-413. doi: 10.1137/060655183 CrossRef Google Scholar
[20]	B. Martinet, Brève communication. Régularisation d'inéquations variationnelles par approximations successives, Revue française d'informatique et de recherche opérationnelle. Série rouge, 1970, 4(R3), 154-158. Google Scholar
[21]	X. Qin and N. T. An, Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets, Comput. Optim. Appl., 2019, 74(3), 821-850. doi: 10.1007/s10589-019-00124-7 CrossRef Google Scholar
[22]	X. Qin, A. Petrusel and J. C. Yao, CQ iterative algorithms for fixed points of nonexpansive mappings and split feasibility problems in Hilbert spaces, J. Nonlinear Convex Anal., 2018, 19(1), 157-165. Google Scholar
[23]	X. Qin and J. C. Yao, A viscosity iterative method for a split feasibility problem, J. Nonlinear Convex Anal., 2019, 20(8), 1497-1506. Google Scholar
[24]	Y. Shehu, O. Iyiola and E. Akaligwo, Modified inertial methods for finding common solutions to variational inequality problems, Fixed Point Theory, 2019, 20(2), 683-702. doi: 10.24193/fpt-ro.2019.2.45 CrossRef Google Scholar
[25]	Y. Shehu, F. Ogbuisi and O. Iyiola, Convergence analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces, Optimization, 2016, 65(2), 299-323. doi: 10.1080/02331934.2015.1039533 CrossRef Google Scholar
[26]	W. Takahashi, Mann and Halpern iterations for the split common fixed point problem in Banach spaces, Linear Nonlinear Anal., 2017, 3, 1-18. Google Scholar
[27]	W. Takahashi, Weak and strong convergence theorems for new demimetric mappings and the split common fixed point problem in Banach spaces, Numer. Funct. Anal. Optim., 2018, 39(10), 1011-1033. doi: 10.1080/01630563.2018.1466803 CrossRef Google Scholar
[28]	W. Takahashi and 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(1), 87. doi: 10.1186/s13663-015-0324-3 CrossRef Google Scholar
[29]	J. Tang, S. Chang, L. Wang and X. Wang, On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces, J. Inequal. Appl., 2015, 2015(1), 305. Google Scholar
[30]	H. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 1991, 16(12), 1127-1138. doi: 10.1016/0362-546X(91)90200-K CrossRef Google Scholar
[31]	X. Zhang, L. Wang, Z. Ma and L. Qin, The strong convergence theorems for split common fixed point problem of asymptotically nonexpansive mappings in Hilbert spaces, J. Inequal. Appl., 2015, 2015(1), 1. Google Scholar