On the generalized Bregman projection operator in reflexive Banach spaces. (English) Zbl 1491.47061
Summary: In this paper, we study the generalized Bregman \(f\)-projection operator in reflexive Banach spaces. After providing some properties of the generalized Bregman \(f\)-projection operator, we propose an iterative algorithm to finding a common fixed point of a finite family of Bregman relatively nonexpansive mappings in reflexive Banach spaces using the generalized Bregman \(f\)-projection operator. An application of our algorithm to finding a common zero of a finite family of maximal monotone operators will also be exhibited.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H05 | Monotone operators and generalizations |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
Bregman distance; generalized Bregman \(f\)-projection; Bregman relatively nonexpansive mappings; iterative algorithm; common fixed point; maximal monotone operatorsReferences:
| [1] | Alber, Ya, Generalized projection operators in Banach spaces: properties and applications, Funct. Differ. Equ., 1, 1-21 (1994) · Zbl 0882.47046 |
| [2] | Alber, Ya.: Metric and generalized projection operators in Banach spaces: properties and applications, theory and applications of nonlinear operators of accretive and monotone type. Lecture Notes in Pure and Applied Mathematics, vol. 178. Dekker, New York, pp. 15-50 (1996) · Zbl 0883.47083 |
| [3] | Ambrosetti, A.; Prodi, G., A Primer of Nonlinear Analysis (1993), Cambridge: Cambridge University Press, Cambridge · Zbl 0781.47046 |
| [4] | Barbu, V., Precupanu, Th: Convexity and optimization in Banach spaces, Revised Edition, Translated from the Romanian, Editura Academiei. Bucharest; Sijthoff and Noordhoff International Publishers, Alphen aan den Rijn (1978) · Zbl 0379.49010 |
| [5] | Bauschke, HH; Borwein, JM; Combettes, PL, Bregman monotone optimization algorithms, SIAM J. Control Optim., 42, 596-636 (2003) · Zbl 1049.90053 · doi:10.1137/S0363012902407120 |
| [6] | Bauschke, HH; Borwein, JM; Combettes, PL, Essential smoothness, essential strict convexity, and legendre functions in Banach spaces, Commun. Contemp. Math., 3, 615-647 (2001) · Zbl 1032.49025 · doi:10.1142/S0219199701000524 |
| [7] | Bonnans, JF; Shapiro, A., Perturbation Analysis of Optimization Problems (2000), New York: Springer, New York · Zbl 0966.49001 |
| [8] | Bregman, LM, A relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys., 7, 200-217 (1967) · doi:10.1016/0041-5553(67)90040-7 |
| [9] | Bruck, RE; Reich, S., Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houst. J. Math., 3, 459-470 (1977) · Zbl 0383.47035 |
| [10] | Butnariu, D.; Iusem, AN, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization (2000), Dordrecht: Kluwer, Dordrecht · Zbl 0960.90092 |
| [11] | Butnariu, D.; Iusem, AN; Resmerita, E., Total convexity for powers of the norm in uniformly convex Banach spaces, J. Convex Anal., 7, 319-334 (2000) · Zbl 0971.46005 |
| [12] | Butnariu, D.; Iusem, AN; Zălinescu, C., On uniform convexity, total convexity and convergence of the proximal point and outer Bregman projection algorithms in Banach spaces, J. Convex Anal., 10, 35-61 (2003) · Zbl 1091.90078 |
| [13] | Butnariu, D.; Kassay, G., A proximal-projection method for finding zeroes of setvalued operators, SIAM J. Control Optim., 47, 2096-2136 (2008) · Zbl 1208.90133 · doi:10.1137/070682071 |
| [14] | Butnariu, D.; Reich, S.; Zaslavski, AJ, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal., 7, 151-174 (2001) · Zbl 1010.47032 · doi:10.1515/JAA.2001.151 |
| [15] | Butnariu, D., Resmerita, E.: Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal., Art. 1-39 (2006) · Zbl 1130.47046 · doi:10.1155/AAA/2006/84919 |
| [16] | Censor, Y.; Reich, S., Iteratoins of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization, 37, 323-339 (1996) · Zbl 0883.47063 · doi:10.1080/02331939608844225 |
| [17] | Deimling, K., Nonlinear Functional Analysis (1985), Berlin: Springer, Berlin · Zbl 0559.47040 |
| [18] | Eskandani, GZ; Raeisi, M.; Rassias, TM, A hybrid extragradient method for solving pseudomonotone equilibrium problems using Bregman distance, J. Fixed Point Theory Appl., 20, 132 (2018) · Zbl 06969123 · doi:10.1007/s11784-018-0611-9 |
| [19] | Fan, J.; Liu, X.; Li, J., Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces, Nonlinear Anal., 70, 3997-4007 (2009) · Zbl 1219.47110 · doi:10.1016/j.na.2008.08.008 |
| [20] | Khan, SA; Suantai, S.; Cholamjiak, W., shrinking projection methods involving inertial forward-backward spliting methods for inclusion problems., Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 113, 2, 645-656 (2019) · Zbl 07086838 · doi:10.1007/s13398-018-0504-1 |
| [21] | Kim, JK, Convergence theorems of iterative sequences for generalized equilibrium problems involving strictly pscudocontractive mappings in Hilbert spaces, J. Comut. Anal. Appl., 18, 454-471 (2015) · Zbl 1325.47124 |
| [22] | Li, J., The metric projection and its application to solving inequalities in Banach spaces, Fixed Point Theory, 5, 2, 285-298 (2004) · Zbl 1076.46057 |
| [23] | Li, J., The generalized projection operator on reflexive Banach spaces and its application, J. Math. Anal. Appl., 306, 377-388 (2005) · Zbl 1129.47043 · doi:10.1016/j.jmaa.2004.11.007 |
| [24] | Li, X.; Huang, NJ; O’Regan, D., Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications, Comput. Math. Appl., 60, 1322-1331 (2010) · Zbl 1201.65091 · doi:10.1016/j.camwa.2010.06.013 |
| [25] | Liu, Y.: A modified hybrid method for solving variational inequality problems in Banach spaces. J. Nonlinear Funct. Anal. 2017, Article ID 31 (2017) |
| [26] | Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279, 372-379 (2003) · Zbl 1035.47048 · doi:10.1016/S0022-247X(02)00458-4 |
| [27] | Naraghirad, E., Bregman best proximity points for Bregman asymptotic cyclic contraction mappings in Banach spaces, J. Nonlinear Var. Anal., 3, 27-44 (2019) · Zbl 1479.47051 |
| [28] | Naraghirad, E., Yao, J.C.: Bregman weak relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. (2013). 10.1186/1687-1812-2013-141 · Zbl 1423.47046 |
| [29] | Pang, C.T., Naraghirad, E., Wen, C.F.: Bregman \(f\)-projection operator with application to variational inequalites in Banach spaces. Abstr. Appl. Anal., Hindawi Publishing Corporation, Article ID 594285 (2014) · Zbl 1473.47025 |
| [30] | Phelps, R.R.: Convex Functions, Monotone Operators, and Differentiability, 2nd edn. Lecture Notes in Mathematics, vol. 1364, Springer, Berlin (1993) · Zbl 0921.46039 |
| [31] | Plubtieng, S., Ungchittrakool, K.: Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 19, Art. ID 583082 (2008) · Zbl 1173.47051 |
| [32] | Qin, X.; Cho, SY, Convergence analysis of a monotone projection algorithm in reflexive Banach spaces, Acta Math. Sci., 37, 488-502 (2017) · Zbl 1389.47163 · doi:10.1016/S0252-9602(17)30016-4 |
| [33] | Raeisi, M.; Eskandani, GZ, A hybrid extragradient method for a general split equality problem involving resolvents and pseudomonotone bifunctions in Banach spaces, Calcolo, 56, 43 (2019) · Zbl 07142732 · doi:10.1007/s10092-019-0341-4 |
| [34] | Raeisi, M.; Eskandani, GZ; Eslamian, M., A general algorithm for multiplesets split feasibility problem involving resolvents and Bregman mappings, Optimization, 68, 309-327 (2018) · Zbl 06865936 · doi:10.1080/02331934.2017.1396603 |
| [35] | Reem, D.; Reich, S., Solutions to inexact resolvent inclusion problems with applications to nonlinear analysis and optimization, Rend. Circ. Mat. Palermo, 2, 67, 337-371 (2018) · Zbl 1401.90233 |
| [36] | Reich, S.: A weak convergence theorem for the alternating method with Bregman distances. In: Theory and Applications of Nonlinear Operators, pp. 313-318. Marcel Dekker, New York (1996) · Zbl 0943.47040 |
| [37] | Reich, S.; Sabach, S., A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal., 10, 471-485 (2009) · Zbl 1180.47046 |
| [38] | Reich, S.; Sabach, S., Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim., 31, 22-44 (2010) · Zbl 1200.47085 · doi:10.1080/01630560903499852 |
| [39] | Reich, S.; Zaslavski, AJ, There are many totally convex functions, J. Convex Anal., 13, 623-632 (2006) · Zbl 1119.46055 |
| [40] | Reich, S., Sabach, S.: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces, fixed-point algorithms for inverse problems in science and engineering, Springer Optim. Appl., vol. 49. Springer, New York, pp. 301-316 (2011) · Zbl 1245.47015 |
| [41] | Sabach, S., Products of finitely many resolvents of maximal monotone mappings in reflexive Banach spaces, SIAM J. Optim., 21, 1289-1308 (2011) · Zbl 1237.47073 · doi:10.1137/100799873 |
| [42] | Takahashi, S.; Takahashi, W., The split common null point problem and the shrinking projection method in two Banach spaces, Linear Nonlinear Anal., 1, 297-304 (2015) · Zbl 1338.47109 |
| [43] | Takahashi, W.; Takeuchi, Y.; Kubota, R., Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341, 1, 276286 (2008) · Zbl 1134.47052 · doi:10.1016/j.jmaa.2007.09.062 |
| [44] | Wu, KQ; Huang, NJ, The generalized \(f\)-projection operator with an application, Bull. Aust. Math. Soc., 73, 307-317 (2006) · Zbl 1104.47053 · doi:10.1017/S0004972700038892 |
| [45] | Wu, KQ; Huang, NJ, Properties of the generalized \(f\)-projection operator and its applications in Banach spaces, Comput. Math. Appl., 54, 399-406 (2007) · Zbl 1151.47057 · doi:10.1016/j.camwa.2007.01.029 |
| [46] | Zhao, X.; Ng, KF; Li, C.; Yao, JC, Linear regularity and linear convergence of projection-based methods for solving convex feasibility problems, Appl. Math. Optim., 78, 613-641 (2018) · Zbl 1492.47089 · doi:10.1007/s00245-017-9417-1 |
| [47] | Zălinescu, C., Convex Analysis in General Vector Spaces (2002), Singapore: World Scientific Publishing, Singapore · Zbl 1023.46003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
