Browse > Article
http://dx.doi.org/10.4134/JKMS.j190268

A PARALLEL ITERATIVE METHOD FOR A FINITE FAMILY OF BREGMAN STRONGLY NONEXPANSIVE MAPPINGS IN REFLEXIVE BANACH SPACES  

Kim, Jong Kyu (Department of Mathematics Education Kyungnam University)
Tuyen, Truong Minh (Department of Mathematics and Informatics Thainguyen University of Sciences)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.3, 2020 , pp. 617-640 More about this Journal
Abstract
In this paper, we introduce a parallel iterative method for finding a common fixed point of a finite family of Bregman strongly nonexpansive mappings in a real reflexive Banach space. Moreover, we give some applications of the main theorem for solving some related problems. Finally, some numerical examples are developed to illustrate the behavior of the new algorithms with respect to existing algorithms.
Keywords
Bregman distance; Bregman firmly nonexpansive mapping; Bregman strongly nonexpansive mapping; Bregman projection;
Citations & Related Records
연도 인용수 순위
  • Reference
1 L. L. Duan, A. F. Shi, L. Wei, and R. P. Agarwal, Construction techniques of projection sets in hybrid methods for infinite weakly relatively nonexpansive mappings with applications, J. Nonlinear Funct. Anal. 2019 (2019), Article ID 14.
2 G. Z. Eskandani, M. Raeisi, and J. K. Kim, A strong convergence theorem for Bregman quasi-noexpansive mappings with applications, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113 (2019), no. 2, 353-366. https://doi.org/10.1007/s13398-017-0481-9   DOI
3 B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957-961. https://doi.org/10.1090/S0002-9904-1967-11864-0   DOI
4 G. Kassay, S. Reich, and S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces, SIAM J. Optim. 21 (2011), no. 4, 1319-1344. https://doi.org/10.1137/110820002   DOI
5 J. K. Kim, Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-$\phi$-nonexpansive mappings, Fixed Point Theory Appl. 2011 (2011), 10, 15 pp. https://doi.org/10.1186/1687-1812-2011-10
6 J. K. Kim, Convergence theorems of iterative sequences for generalized equilibrium problems involving strictly pseudocontractive mappings in Hilbert spaces, J. Comput. Anal. Appl. 18 (2015), no. 3, 454-471.
7 J. K. Kim and N. Buong, An iteration method for common solution of a system of equilibrium problems in Hilbert spaces, Fixed Point Theory Appl. 2011 (2011), Art. ID 780764, 15 pp. https://doi.org/10.1155/2011/780764
8 J. K. Kim, N. Buong, and J. Y. Sim, A new iterative method for the set of solutions of equilibrium problems and of operator equations with inverse-strongly monotone mappings, Abstr. Appl. Anal. 2014 (2014), Art. ID 595673, 8 pp. https://doi.org/10.1155/2014/595673
9 F. Kohsaka and W. Takahashi, Proximal point algorithms with Bregman functions in Banach spaces, J. Nonlinear Convex Anal. 6 (2005), no. 3, 505-523.
10 J. K. Kim and W. H. Lim, A new iterative algorithm of pseudomonotone mappings for equilibrium problems in Hilbert spaces, J. Inequal. Appl. 2013 (2013), 128, 16 pp. https://doi.org/10.1186/1029-242X-2013-128
11 P.-E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), no. 7-8, 899-912. https://doi.org/10.1007/s11228-008-0102-z   DOI
12 V. Martin-Marquez, S. Reich, and S. Sabach, Bregman strongly nonexpansive operators in reflexive Banach spaces, J. Math. Anal. Appl. 400 (2013), no. 2, 597-614. https://doi.org/10.1016/j.jmaa.2012.11.059   DOI
13 J.-J. Moreau, Sur la fonction polaire d'une fonction semi-continue superieurement, C. R. Acad. Sci. Paris 258 (1964), 1128-1130.
14 E. Naraghirad and J.-C. Yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2013 (2013), 141, 43 pp. https://doi.org/10.1186/1687-1812-2013-141
15 S. Suantai, Y. J. Cho, and P. Cholamjiak, Halpern's iteration for Bregman strongly nonexpansive mappings in re exive Banach spaces, Comput. Math. Appl. 64 (2012), no. 4, 489-499. https://doi.org/10.1016/j.camwa.2011.12.026   DOI
16 S. Reich, Review of Geometry of Banach spaces, duality mappings and nonlinear problems by Ioana Cioranescu, Bull. Amer. Math. Soc. 26 (1992), 367-370. https://doi.org/10.1090/S0273-0979-1992-00287-2   DOI
17 S. Reich, A weak convergence theorem for the alternating method with Bregman distances, in Theory and applications of nonlinear operators of accretive and monotone type, 313-318, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, 1996.
18 S. Reich and S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal. 10 (2009), no. 3, 471-485.
19 E. Resmerita, On total convexity, Bregman projections and stability in Banach spaces, J. Convex Anal. 11 (2004), no. 1, 1-16.
20 R. T. Rockafellar, Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc. 123 (1966), 46-63. https://doi.org/10.2307/1994612   DOI
21 W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), no. 2, 417-428. https://doi.org/10.1023/A:1025407607560   DOI
22 T. M. Tuyen, Parallel iterative methods for Bregman strongly nonexpansive operators in reflexive Banach spaces, J. Fixed Point Theory Appl. 19 (2017), no. 3, 1695-1710. https://doi.org/10.1007/s11784-016-0325-9   DOI
23 T. M. Tuyen, Parallel iterative methods for solving systems of generalized mixed equilibrium problems in re exive Banach spaces, Optimization 66 (2017), no. 4, 623-639. https://doi.org/10.1080/02331934.2016.1277999   DOI
24 A. Y. Wang and Z. M. Wang, A simple hybrid Bregman projection algorithms for a family of countable Bregman quasi-strict pseudo-contractions, Nonlinear Funct. Anal. Appl. 22 (2017), no. 5, 1001-1011.
25 S. Reich, A projection method for solving nonlinear problems in reflexive Banach spaces, J. Fixed Point Theory Appl. 9 (2011), no. 1, 101-116. https://doi.org/10.1007/s11784-010-0037-5   DOI
26 S. Reich and S. Sabach, Two strong convergence theorems for a proximal method in re exive Banach spaces, Numer. Funct. Anal. Optim. 31 (2010), no. 1-3, 22-44. https://doi.org/10.1080/01630560903499852
27 S. Reich and S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in re exive Banach spaces, Nonlinear Anal. 73 (2010), no. 1, 122-135. https://doi.org/10.1016/j.na.2010.03.005   DOI
28 S. Reich and S. Sabach, Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in re exive Banach spaces, in Fixed-point algorithms for inverse problems in science and engineering, 301-316, Springer Optim. Appl., 49, Springer, New York, 2011. https://doi.org/10.1007/978-1-4419-9569-8_15
29 S. Reich, Three strong convergence theorems regarding iterative methods for solving equilibrium problems in re exive Banach spaces, in Optimization theory and related topics, 225-240, Contemp. Math., 568, Israel Math. Conf. Proc, Amer. Math. Soc., Providence, RI, 2012. https://doi.org/10.1090/conm/568/11285   DOI
30 H.-K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006), no. 2, 631-643. https://doi.org/10.1016/j.jmaa.2005.04.082   DOI
31 C. Zalinescu, Convex analysis in general vector spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. https://doi.org/10.1142/9789812777096
32 H. Zegeye, Convergence theorems for Bregman strongly nonexpansive mappings in reflexive Banach spaces, Filomat 28 (2014), no. 7, 1525-1536. https://doi.org/10.2298/FIL1407525Z   DOI
33 H. H. Bauschke, J. M. Borwein, and P. L. Combettes, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001), no. 4, 615-647. https://doi.org/10.1142/S0219199701000524   DOI
34 F. E. Browder, Existence and approximation of solutions of nonlinear variational inequalities, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1080-1086. https://doi.org/10.1073/pnas.56.4.1080   DOI
35 J. Zhao and S. Wang, Strong convergence for Bregman relatively nonexpansive mapping in reflexive Banach spaces and applications, Nonlinear Funct. Anal. Appl. 20 (2015), no. 3, 365-379.
36 Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in Theory and applications of nonlinear operators of accretive and monotone type, 15-50, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, 1996.
37 A. Ambrosetti and G. Prodi, A primer of nonlinear analysis, Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1993.
38 H. H. Bauschke, J. M. Borwein, and P. L. Combettes, Bregman monotone optimization algorithms, SIAM J. Control Optim. 42 (2003), no. 2, 596-636. https://doi.org/10.1137/S0363012902407120   DOI
39 E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), no. 1-4, 123-145.
40 J. F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. https://doi.org/10.1007/978-1-4612-1394-9
41 D. Butnariu and A. N. Iusem, Totally convex functions for fixed points computation and infinite dimensional optimization, Applied Optimization, 40, Kluwer Academic Publishers, Dordrecht, 2000. https://doi.org/10.1007/978-94-011-4066-9
42 Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34 (1981), no. 3, 321-353. https://doi.org/10.1007/BF00934676   DOI
43 D. Butnariu and G. Kassay, A proximal-projection method for finding zeros of set-valued operators, SIAM J. Control Optim. 47 (2008), no. 4, 2096-2136. https://doi.org/10.1137/070682071   DOI
44 D. Butnariu and E. Resmerita, Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006 (2006), Art. ID 84919, 39 pp. https://doi.org/10.1155/AAA/2006/84919
45 L.-C. Ceng and J.-C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008), no. 1, 186-201. https://doi.org/10.1016/j.cam.2007.02.022   DOI
46 Y. Censor and S. Reich, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization 37 (1996), no. 4, 323-339. https://doi.org/10.1080/02331939608844225   DOI
47 V. Darvish, Strong convergence theorem for a system of generalized mixed equilibrium problems and finite family of Bregman nonexpansive mappings in Banach spaces, Opsearch 53 (2016), no. 3, 584-603. https://doi.org/10.1007/s12597-015-0245-2   DOI
48 C. E. Chidume, A. Adamu, and L. C. Okereke, A Krasnoselskii type algorithm for approximating solutions of variational inequality problems and convex feasibility problems, J. Nonlinear Var. Anal. 2 (2018), 203-218.   DOI
49 I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Mathematics and its Applications, 62, Kluwer Academic Publishers Group, Dordrecht, 1990. https://doi.org/10.1007/978-94-009-2121-4