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http://dx.doi.org/10.4134/CKMS.2010.25.4.583

CONVERGENCE THEOREMS FOR TWO FAMILIES OF WEAK RELATIVELY NONEXPANSIVE MAPPINGS AND A FAMILY OF EQUILIBRIUM PROBLEMS  

Zhang, Xin (DEPARTMENT OF MATHEMATICS TIANJIN POLYTECHNIC UNIVERSITY)
Su, Yongfu (DEPARTMENT OF MATHEMATICS TIANJIN POLYTECHNIC UNIVERSITY)
Publication Information
Communications of the Korean Mathematical Society / v.25, no.4, 2010 , pp. 583-607 More about this Journal
Abstract
The purpose of this paper is to prove strong convergence theorems for common fixed points of two families of weak relatively nonexpansive mappings and a family of equilibrium problems by a new monotone hybrid method in Banach spaces. Because the hybrid method presented in this paper is monotone, so that the method of the proof is different from the original one. We shall give an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping in Banach space $l^2$. Our results improve and extend the corresponding results announced in [W. Takahashi and K. Zembayashi, Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory Appl. (2008), Article ID 528476, 11 pages; doi:10.1155/2008/528476] and [Y. Su, Z. Wang, and H. Xu, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Anal. 71 (2009), no. 11, 5616?5628] and some other papers.
Keywords
relatively nonexpansive; weak relatively nonexpansive; common fixed point; generalized projection; equilibrium problem; monotone hybrid algorithm, maximal monotone operator;
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1 H. Zegeye and N. Shahzad, Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings, Nonlinear Anal. 70 (2009), no. 7, 2707–2716.
2 N. Shioji and W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3641–3645.
3 Y. Su, J. Gao, and H. Zhou, Monotone CQ algorithm of fixed points for weak relatively nonexpansive mappings and applications, J. Math. Res. Exposition 28 (2008), no. 4, 957–967.
4 Y. Su, D. Wang, and M. Shang, Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings, Fixed Point Theory Appl. 2008 (2008), Art. ID 284613, 8 pp.   DOI
5 Y. Su, Z. Wang, and H. Xu, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Anal. 71 (2009), no. 11, 5616–5628.   DOI   ScienceOn
6 W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, 2000.
7 W. Takahashi, Y. Takeuchi, and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 341 (2008), no. 1, 276–286.   DOI   ScienceOn
8 W. Takahashi and K. Zembayashi, Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory Appl. 2008 (2008), Art. ID 528476, 11 pp.
9 W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009), no. 1, 45–57.   DOI   ScienceOn
10 L. Wei, Y. J. Cho, and H. Zhou, A strong convergence theorem for common fixed points of two relatively nonexpansive mappings and its applications, J. Appl. Math. Comput. 29 (2009), no. 1-2, 95–103.   DOI
11 R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel) 58 (1992), no. 5, 486–491.   DOI   ScienceOn
12 H. Zegeye and N. Shahzad, Strong convergence for monotone mappings and relatively weak nonexpansive mappings, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.03.058.   DOI   ScienceOn
13 Y. Kimura, On Mosco convergence for a sequence of closed convex subsets of Banach spaces, Banach and function spaces, 291–300, Yokohama Publ., Yokohama, 2004.
14 Y. Kimura and W. Takahashi, On a hybrid method for a family of relatively nonexpansive mappings in a Banach space, J. Math. Anal. Appl. 357 (2009), no. 2, 356–363.   DOI   ScienceOn
15 P. Kumam, Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turkish J. Math. 33 (2009), no. 1, 85–98.
16 S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005), no. 2, 257–266.   DOI   ScienceOn
17 S. Matsushita and W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2004 (2004), no. 1, 37–47.   DOI
18 U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (1969), 510–585.
19 X. Qin, Y. J. Cho, and S. M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225 (2009), no. 1, 20–30.   DOI   ScienceOn
20 K. Nakajo and W. Takahashi, Approximation of a zero of maximal monotone operators in Hilbert spaces, Nonlinear analysis and convex analysis, 303–314, Yokohama Publ., Yokohama, 2003.
21 X. Qin, Y. J. Cho, S. M. Kang, and H. Zhou, Convergence of a modified Halpern-type iteration algorithm for quasi-Á-nonexpansive mappings, Appl. Math. Lett. 22 (2009), no. 7, 1051–1055.   DOI   ScienceOn
22 S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), no. 1, 287–292.
23 R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75–88.
24 G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers Group, Dordrecht, 1993.
25 E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), no. 1-4, 123–145.
26 D. Butnariu, Y. Censor, and S. Reich, Iterative averaging of entropic projections for solving stochastic convex feasibility problems, Comput. Optim. Appl. 8 (1997), no. 1, 21–39.   DOI
27 D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publishers, Dordrecht, 2000.
28 D. Butnariu, S. Reich, and A. J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal. 7 (2001), no. 2, 151–174.
29 Y. J. Cho, H. Zhou, and G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47 (2004), no. 4-5, 707–717.   DOI   ScienceOn
30 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.   DOI   ScienceOn
31 K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, 83. Marcel Dekker, Inc., New York, 1984.
32 B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957–961.
33 T. Ibaraki, Y. Kimura, and W. Takahashi, Convergence theorems for generalized projections and maximal monotone operators in Banach spaces, Abstr. Appl. Anal. 2003 (2003), no. 10, 621–629.   DOI   ScienceOn
34 M. M. Israel, Jr. and S. Reich, Extension and selection problems for nonlinear semigroups in Banach spaces, Math. Japon. 28 (1983), no. 1, 1–8.
35 S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2002), no. 3, 938–945.   DOI   ScienceOn
36 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, 15–50, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, 1996.