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Strong convergence theorems for relatively nonexpansive mappings in a Banach space. (English) Zbl 1124.47046

In [J. Math. Anal. Appl. 279, No. 2, 372–379 (2003; Zbl 1035.47048)], K. Nakajo and W. Takahashi defined a modified Mann iteration for a single nonexpansive map \(T\) to obtain strong convergence results in Hilbert space. In [Nonlinear Anal., Theory Methods Appl. 64, No. 11 (A), 2400–2411 (2006; Zbl 1105.47060)], C. Martinez–Yanes and H. K. Xu defined modified Ishikawa and Halpern iterations to prove interesting convergence results. In the paper under review, the authors, guided by the first mentioned article, prove strong convergence theorems for relatively nonexpansive mappings in uniformly convex and uniformly smooth Banach space.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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[1] Alber, Ya.I.; Reich, S., An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. math. J., 4, 2, 39-54, (1994) · Zbl 0851.47043
[2] Alber, Ya.I., Metric and generalized projection operators in Banach spaces: properties and applications, (), 15-50 · Zbl 0883.47083
[3] Alber, Ya.I.; Guerre-Delabriere, S., On the projection methods for fixed point problems, Analysis, 21, 17-39, (2001) · Zbl 0985.47044
[4] Butnariu, D.; Reich, S.; Zaslavski, A.J., Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. appl. anal., 7, 151-174, (2001) · Zbl 1010.47032
[5] Butnariu, D.; Reich, S.; Zaslavski, A.J., Weak convergence of orbits of nonlinear operators in reflexsive Banach spaces, Numer. funct. anal. optim., 24, 489-508, (2003) · Zbl 1071.47052
[6] Censor, Y.; Reich, S., Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization, 37, 323-339, (1996) · Zbl 0883.47063
[7] Chidume, C.E.; Mutangadura, S.A., An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. amer. math. soc., 129, 2359-2363, (2001) · Zbl 0972.47062
[8] Cioranescu, I., Geometry of Banach spaces, duality mappings and nonlinear problems, (1990), Kluwer Dordrecht · Zbl 0712.47043
[9] Genel, A.; Lindenstrass, J., An example concerning fixed points, Israel J. math., 22, 81-86, (1975) · Zbl 0314.47031
[10] Halpern, B., Fixed points of nonexpanding maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101
[11] Ishikawa, S., Fixed points by a new iteration method, Proc. amer. math. soc., 44, 147-150, (1974) · Zbl 0286.47036
[12] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. optim., 13, 938-945, (2003) · Zbl 1101.90083
[13] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603
[14] Martinez-Yanes, C.; Xu, H.K., Strong convergence of the CQ method for fixed point iteration processes, Nonlinear anal., 64, 2400-2411, (2006) · Zbl 1105.47060
[15] Matsushita, S.; Takahashi, W., A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. approx. theory, 134, 257-266, (2005) · Zbl 1071.47063
[16] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-379, (2003) · Zbl 1035.47048
[17] Reich, S., A weak convergence theorem for the alternating method with Bregman distance, (), 313-318 · Zbl 0943.47040
[18] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. math. anal. appl., 67, 274-276, (1979) · Zbl 0423.47026
[19] Takahashi, W., Nonlinear functional analysis, (2000), Yokohama-Publishers
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