Some results on a finite family of Bregman quasi-strict pseudo-contractions
- Department of Foundation, Shandong Yingcai University, Jinan, China.
- School of Control Science and Engineering, Shandong University, Jinan, China.
The aim of this article is to establish a common fixed point theorem for a finite family of Bregman quasi-strict pseudocontractions
in a reflexive Banach space. Applications to equilibrium problems, variational inequality problems, and zero point
problems are provided.
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Zi-Ming Wang, Airong Wei, Some results on a finite family of Bregman quasi-strict pseudo-contractions, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 3, 975--989
Wang Zi-Ming, Wei Airong, Some results on a finite family of Bregman quasi-strict pseudo-contractions. J. Nonlinear Sci. Appl. (2017); 10(3):975--989
Wang, Zi-Ming, Wei, Airong. "Some results on a finite family of Bregman quasi-strict pseudo-contractions." Journal of Nonlinear Sciences and Applications, 10, no. 3 (2017): 975--989
- Bregman mapping
- generalized projection
- variational inequality
- hybrid method.
Y. I. Alber, 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 Appl. Math., Dekker, New York, 178 (1996), 15–50.
H. H. Bauschke, J. M. Borwein, P. L. Combettes, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math., 3 (2001), 615–647.
B. A. Bin Dehaish, A. Latif, H. O. Bakodah, X.-L. Qin, A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces, J. Inequal. Appl., 2015 (2015), 14 pages.
B. A. Bin Dehaish, X.-L. Qin, A. Latif, H. O. Bakodah, Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321–1336.
D. Butnariu, A. N. Iusem, Totally convex functions for fixed points computation and infinite dimensional optimization, Applied Optimization, Kluwer Academic Publishers, Dordrecht (2000)
D. Butnariu, E. Resmerita, Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal., 2006 (2006 ), 39 pages.
G. Cai, C. S. Hu , Strong convergence theorems of modified Ishikawa iterative process with errors for an infinite family of strict pseudo-contractions, Nonlinear Anal., 71 (2009), 6044–6053.
Y. Censor, A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl., 34 (1981), 321–353.
S. Y. Cho, W.-L. Li, S. M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013 (2013 ), 14 pages.
S. Y. Cho, X.-L. Qin, On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems, Appl. Math. Comput., 235 (2014), 430–438.
Y. Hao, Some results on a modified Mann iterative scheme in a reflexive Banach space, Fixed Point Theory Appl., 2013 (2013), 14 pages.
Y. Hao, S. Y. Cho, Fixed point iterations of a pair of hemirelatively nonexpansive mappings, Fixed Point Theory Appl., 2010 (2010), 14 pages.
Y. Haugazeau, Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes, Ph.D. Thesis, Université de Paris, Paris (1968)
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 ), 15 pages.
J. K. Kim, S. Y. Cho, X.-L. Qin, Some results on generalized equilibrium problems involving strictly pseudocontractive mappings, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 2041–2057.
S. Reich, S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal., 10 (2009), 471–485.
S. Reich, S. Sabach, Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim., 31 (2010), 22–44.
S. Reich, S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear Anal., 73 (2010), 122–135.
R. T. Rockafellar , Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497–510.
G. C. Ugwunnadi, B. Ali, I. Idris, M. S. Minjibir, Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in Banach spaces, Fixed Point Theory Appl., 2014 (2014 ), 16 pages.
Z.-M. Wang, Strong convergence theorems for Bregman quasi-strict pseudo-contractions in reflexive Banach spaces with applications, Fixed Point Theory Appl., 2015 (2015 ), 17 pages.
Z.-M. Wang, X.-M. Zhang, Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems, J. Nonlinear Funct. Anal., 2014 (2014 ), 25 pages.
H. Zhang, Iterative processes for fixed points of nonexpansive mappings, Commun. Optim. Theory, 2013 (2013 ), 7 pages.