August  2013, 6(4): 1043-1063. doi: 10.3934/dcdss.2013.6.1043

Iterative methods for approximating fixed points of Bregman nonexpansive operators

1. 

Departamento de Análisis Matemático

2. 

Universidad de Sevilla

3. 

Apdo. 1160, 41080 Sevilla

4. 

Department of Mathematics

5. 

The Technion-Israel Institute of Technology

6. 

32000 Haifa

7. 

The Technion --- Israel Institute of Technology

Received  July 2011 Revised  March 2012 Published  December 2012

Diverse notions of nonexpansive type operators have been extended to the more general framework of Bregman distances in reflexive Banach spaces. We study these classes of operators, mainly with respect to the existence and approximation of their (asymptotic) fixed points. In particular, the asymptotic behavior of Picard and Mann type iterations is discussed for quasi-Bregman nonexpansive operators. We also present parallel algorithms for approximating common fixed points of a finite family of Bregman strongly nonexpansive operators by means of a block operator which preserves the Bregman strong nonexpansivity. All the results hold, in particular, for the smaller class of Bregman firmly nonexpansive operators, a class which contains the generalized resolvents of monotone mappings with respect to the Bregman distance.
Citation: Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043
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show all references

References:
[1]

Linear Algebra Appl., 120 (1989), 165-175. doi: 10.1016/0024-3795(89)90375-3.  Google Scholar

[2]

Comm. Contemp. Math., 3 (2001), 615-647. doi: 10.1142/S0219199701000524.  Google Scholar

[3]

SIAM J. Control Optim., 42 (2003), 596-636. doi: 10.1137/S0363012902407120.  Google Scholar

[4]

Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

[5]

Nonlinear Anal., 56 (2004), 715-738. doi: 10.1016/j.na.2003.10.010.  Google Scholar

[6]

$2^{nd}$, Lecture Notes in Mathematics 1912, Springer, Berlin, 2007.  Google Scholar

[7]

J. Nonlinear Convex Anal., 12 (2011), 161-184  Google Scholar

[8]

Springer, New York, 2000.  Google Scholar

[9]

Soviet Math. Dokl., 6 (1965), 688-692. Google Scholar

[10]

USSR Comput. Math. Math. Phys., 7 (1967), 200-217.  Google Scholar

[11]

Houston J. Math., 3 (1977), 459-470.  Google Scholar

[12]

J. Comput. Appl. Math., 53 (1994), 33-42. doi: 10.1016/0377-0427(92)00123-Q.  Google Scholar

[13]

Comput. Optim. Appl., 8 (1997), 21-39. doi: 10.1023/A:1008654413997.  Google Scholar

[14]

Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-011-4066-9.  Google Scholar

[15]

Abstr. Appl. Anal., 2006 (2006), Art. ID 84919, 1-39. doi: 10.1155/AAA/2006/84919.  Google Scholar

[16]

SIAM Rev., 23 (1981), 444-466. doi: 10.1137/1023097.  Google Scholar

[17]

J. Optim. Theory Appl., 34 (1981), 321-353. doi: 10.1007/BF00934676.  Google Scholar

[18]

Optimization, 37 (1996), 323-339. doi: 10.1080/02331939608844225.  Google Scholar

[19]

Lecture Notes in Mathematics 1965, Springer, London, 2009.  Google Scholar

[20]

Ric. Sci. (Roma), 9 (1938), 326-333. Google Scholar

[21]

Springer, New York, 2003.  Google Scholar

[22]

Israel J. Math., 22 (1975), 81-86.  Google Scholar

[23]

Marcel Dekker, New York, 1984.  Google Scholar

[24]

Bull. Internat. Acad. Polon. Sci. Lett. Sér. A Sci. Math., 35 (1937), 355-357. Google Scholar

[25]

SIAM J. Optim., 21 (2011), 1319-1344. doi: 10.1137/110820002.  Google Scholar

[26]

Int. J. Comput. Numer. Anal. Appl., 5 (2004), 59-66.  Google Scholar

[27]

Uspehi Math. Nauk., 10 (1955), 123-127. Google Scholar

[28]

Springer, New York, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

[29]

Proc. Amer. Math. Soc., 4 (1953), 506-510.  Google Scholar

[30]

J. Math. Anal. Appl., 67 (1979), 274-276. doi: 10.1016/0022-247X(79)90024-6.  Google Scholar

[31]

in "Theory and Applications of Nonlinear Operators of Accretive and Monotone Type" Marcel Dekker, New York, (1996), 313-318.  Google Scholar

[32]

J. Nonlinear Convex Anal., 10 (2009), 471-485.  Google Scholar

[33]

Nonlinear Analysis, 73 (2010), 122-135. doi: 10.1016/j.na.2010.03.005.  Google Scholar

[34]

in "Fixed-Point Algorithms for Inverse Problems in Science and Engineering" Springer, New York, (2011), 299-314. doi: 10.1007/978-1-4419-9569-8_15.  Google Scholar

[35]

Springer, New York, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

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