Browse > Article
http://dx.doi.org/10.5666/KMJ.2012.52.4.433

Strong Convergence of Modified Iteration Processes for Relatively Weak Nonexpansive Mappings  

Boonchari, Daruni (Department of Mathematics, Mahasarakham University, Centre of Excellence in Mathematics, CHE)
Saejung, Satit (Department of Mathematics, Khon Kaen University, Centre of Excellence in Mathematics, CHE)
Publication Information
Kyungpook Mathematical Journal / v.52, no.4, 2012 , pp. 433-441 More about this Journal
Abstract
We adapt the concept of shrinking projection method of Takahashi et al. [J. Math. Anal. Appl. 341(2008), 276-286] to the iteration scheme studied by Kim and Lee [Kyungpook Math. J. 48(2008), 685-703] for two relatively weak nonexpansive mappings. By letting one of the two mappings be the identity mapping, we also obtain strong convergence theorems for a single mapping with two types of computational errors. Finally, we improve Kim and Lee's convergence theorem in the sense that the same conclusion still holds without the uniform continuity of mappings as was the case in their result.
Keywords
relatively nonexpansive mapping; relatively weak nonexpansive mapping; shrinking projection method;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 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.
2 K. Aoyama, F. Kohsaka and W. Takahashi, Strong convergence theorems by shrinking and hybrid projection methods for relatively nonexpansive mappings in Banach spaces, Proceeding of the 5th international conference on nonlinear analysis and convex analysis, Taiwan, (2007), 7-26.
3 S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13(2002), 938-945.   DOI   ScienceOn
4 J. Kang, Y. Su and X. Zhang, Hybrid algorithm for xed points of weak relatively nonexpansive mappings and applications, Nonlinear Anal., 4(4)(2010), 755-765.
5 T. H. Kim and H. J. Lee, Strong convergence of modified iteration processes for relatively nonexpansive mappings, Kyungpook Math. J., 48(4)(2008), 685-703.   과학기술학회마을   DOI   ScienceOn
6 S. Matsushita and W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., (1)(2004), 37-47.
7 S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonex- pansive mappings in a Banach space, J. Approx. Theory, 134(2)(2005), 257-266.   DOI   ScienceOn
8 K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279(2)(2003), 372-379.   DOI   ScienceOn
9 S. Reich, A weak convergence theorem for the alternating method with Bregman distances, Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., Dekker, New York, 178(1996), 313-318.
10 Y. Su, D. Wang and M. Shang, Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings, Fixed Point Theory Appl., Art. ID 284613, (2008), 8 pp
11 W. Takahashi, Convex Analysis and Approximation Fixed points, Yokohama Publishers, Yokohama, (2000), (Japanese).
12 W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, (2000).
13 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(1)(2008), 276-286.   DOI   ScienceOn
14 H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16(12)(1991), 1127-1138.   DOI   ScienceOn
15 X. Zhang and Y. Su, Convergence theorems for two families of weak relatively non-expansive mappings and a family of equilibrium problems, Commun. Korean Math. Soc., 25(2010), 583-607.   과학기술학회마을   DOI   ScienceOn
16 Y. Xu and Y. Su, On the weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., Art. ID 189751, (2010), 7 pp.
17 H. Zegeye and N. Shahzad, Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings, Nonlinear Anal., 70(7)(2009), 2707-2716.   DOI   ScienceOn