X. Qin and Y. Su proved a strong convergence theorems of modified Ishikawa iteration by CQ method for relatively nonexpansive mappings in a Banach space [Xiaolong Qin, Yongfu Su, Nonlinear Anal. 67 (2007), no. 6, 1958-1965]. The result of this paper extends and improves the result of X. Qin and Y. Su in the two respects: (1). By using the monotone CQ method to modify the CQ method, so that the new method of proof is used. (2). Relax the restriction on $T$ from uniformly continuous to continuous. The result of this paper also extends and improves the recent ones announced by Nakajo, Takahashi, Kim, Martinez-Yanes, Xu and some others.
References
K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl, 279 (2003), 372–379. MR1974031 10.1016/S0022-247X(02)00458-4 1035.47048K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl, 279 (2003), 372–379. MR1974031 10.1016/S0022-247X(02)00458-4 1035.47048
T.-H. Kim and H.-K. Xu, Strong convergence of modified Mann iterations for asymptotically mappings and semigroups, Nonlinear Anal, 64 (2006), 1140–1152. MR2196814T.-H. Kim and H.-K. Xu, Strong convergence of modified Mann iterations for asymptotically mappings and semigroups, Nonlinear Anal, 64 (2006), 1140–1152. MR2196814
C. Martinez–Yanesa and H.-K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal, 64 (2006), 2400–2411. MR2215815C. Martinez–Yanesa and H.-K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal, 64 (2006), 2400–2411. MR2215815
S.-Y. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, Journal of Approximation Theory, 134 (2005), 257–266. MR2142300 10.1016/j.jat.2005.02.007 1071.47063S.-Y. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, Journal of Approximation Theory, 134 (2005), 257–266. MR2142300 10.1016/j.jat.2005.02.007 1071.47063
Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in: A. G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York, 1996, pp. 15–50. MR1386667 0883.47083Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in: A. G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York, 1996, pp. 15–50. MR1386667 0883.47083
Y. I. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J., 4 (1994), 39–54. MR1274188 0851.47043Y. I. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J., 4 (1994), 39–54. MR1274188 0851.47043
S. Kamimura and W. Takahashi, Strong convergence of a proximal–type algorithm in a Banach space, SIAM J. Optim. 13 (2002), 938–945. MR1972223 10.1137/S105262340139611X 1101.90083S. Kamimura and W. Takahashi, Strong convergence of a proximal–type algorithm in a Banach space, SIAM J. Optim. 13 (2002), 938–945. MR1972223 10.1137/S105262340139611X 1101.90083
I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990. MR1079061 0712.47043I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990. MR1079061 0712.47043
D. Butnariu, S. Reich and A.J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal, 7 (2001), 151–174. MR1875804 1010.47032D. Butnariu, S. Reich and A.J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal, 7 (2001), 151–174. MR1875804 1010.47032
D. Butnariu, S. Reich and A.J. Zaslavski, Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numer. Funct. Anal. Optim, 24 (2003), 489–508. MR1995998 10.1081/NFA-120023869 1071.47052D. Butnariu, S. Reich and A.J. Zaslavski, Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numer. Funct. Anal. Optim, 24 (2003), 489–508. MR1995998 10.1081/NFA-120023869 1071.47052
Y. Censor and S. Reich, Iterationsof paracontractions and firmly nonexpansive operators with applicationsto feasibility and optimization, Optimization, 37 (1996), 323–339. MR1402641 10.1080/02331939608844225 0883.47063Y. Censor and S. Reich, Iterationsof paracontractions and firmly nonexpansive operators with applicationsto feasibility and optimization, Optimization, 37 (1996), 323–339. MR1402641 10.1080/02331939608844225 0883.47063
X. Qin and Y. Su, Strong convergence theorems for relatively nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), no. 6, 1958–1965. MR2326043X. Qin and Y. Su, Strong convergence theorems for relatively nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), no. 6, 1958–1965. MR2326043
