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Chang, S. S.; Wang, Lin; Wang, X. R.; Chan, C. K.

Strong convergence theorems for Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces. (English) Zbl 1364.47013

Appl. Math. Comput. 228, 38-48 (2014).
Summary: The purpose of this article is by using the shrinking projection method introduced by Takahashi, Kubota and Takeuchi to propose an iteration algorithm for Bregman total quasi-\(\phi\)-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of reflexive Banach spaces. As applications, we apply our results to a system of equilibrium problems and zero point problem of maximal monotone mappings in reflexive Banach spaces. The results presented in the paper improve and extend the corresponding results of S. Reich and S. Sabach [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 1, 122–135 (2010; Zbl 1226.47089)], S. Suantai et al. [Comput. Math. Appl. 64, No. 4, 489–499 (2012; Zbl 1252.65100)], W. Nilsrakoo and S. Saejung [Appl. Math. Comput. 217, No. 14, 6577–6586 (2011; Zbl 1215.65104)], X. Qin et al. [Appl. Math. Lett. 22, No. 7, 1051–1055 (2009; Zbl 1179.65061)], Z. Wang et al. [J. Comput. Appl. Math. 235, No. 8, 2364–2371 (2011; Zbl 1213.65082)], Y. Su et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 12, 3890–3906 (2010; Zbl 1215.47091)], C. Martinez-Yanes and H.-K. Xu [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 64, No. 11, 2400–2411 (2006; Zbl 1105.47060)] and others.

Cited in 13 Documents

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

Bregman total quasi-asymptotically nonexpansive mapping; Bregman strongly nonexpansive mapping; Legendre function; totally convex function; Bregman quasi-asymptotically nonexpansive mapping; Bregman quasi-nonexpansive mapping; Bregman projection

Citations:

Zbl 1226.47089; Zbl 1252.65100; Zbl 1215.65104; Zbl 1179.65061; Zbl 1213.65082; Zbl 1215.47091; Zbl 1105.47060
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Full Text: DOI

References:

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