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Iterative approximation of fixed points of Lipschitz pseudocontractive maps. (English) Zbl 0979.47038
Let \(E\) be a \(q\)-uniformly smooth Banach space with a weakly sequentially continuous duality map and \(T\) be a Lipschitzian pseudocontractive selfmapping of a nonempty closed bounded and assume \(\chi\) be in \(K\). The author’s gives an iterative approximation method for a fixed point of \(T\). If \(E\) is a Hilbert space the approximation converges to the fixed point closest to \(\chi\).

MSC:
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J05 Equations involving nonlinear operators (general)
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
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[1] Ronald E. Bruck Jr., A strongly convergent iterative solution of 0\in \?(\?) for a maximal monotone operator \? in Hilbert space, J. Math. Anal. Appl. 48 (1974), 114 – 126. · Zbl 0288.47048 · doi:10.1016/0022-247X(74)90219-4 · doi.org
[2] C. E. Chidume and Chika Moore, Fixed point iteration for pseudocontractive maps, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1163 – 1170. · Zbl 0913.47052
[3] Klaus Deimling, Zeros of accretive operators, Manuscripta Math. 13 (1974), 365 – 374. · Zbl 0288.47047 · doi:10.1007/BF01171148 · doi.org
[4] Joseph Diestel, Geometry of Banach spaces — selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. · Zbl 0307.46009
[5] Benjamin Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957 – 961. · Zbl 0177.19101
[6] Troy L. Hicks and John D. Kubicek, On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl. 59 (1977), no. 3, 498 – 504. · Zbl 0361.65057 · doi:10.1016/0022-247X(77)90076-2 · doi.org
[7] Shiro Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147 – 150. · Zbl 0286.47036
[8] W. R. Mann; Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. · Zbl 0050.11603
[9] G. Müller and J. Reinermann, Fixed point theorems for pseudo-contractive mappings and a counterexample for compact maps, Comment. Math. Univ. Carolinae 18 (1977), no. 2, 281 – 298. · Zbl 0349.47047
[10] Qi Hou Liu, On Naimpally and Singh’s open questions, J. Math. Anal. Appl. 124 (1987), no. 1, 157 – 164. · Zbl 0625.47044 · doi:10.1016/0022-247X(87)90031-X · doi.org
[11] Qi Hou Liu, The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings, J. Math. Anal. Appl. 148 (1990), no. 1, 55 – 62. · Zbl 0729.47052 · doi:10.1016/0022-247X(90)90027-D · doi.org
[12] Rainald Schöneberg, On the structure of fixed point sets of pseudo-contractive mappings. II, Comment. Math. Univ. Carolinae 18 (1977), no. 2, 299 – 310. · Zbl 0358.47034
[13] Jürgen Schu, Approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 112 (1991), no. 1, 143 – 151. · Zbl 0734.47037
[14] Jürgen Schu, Approximating fixed points of Lipschitzian pseudocontractive mappings, Houston J. Math. 19 (1993), no. 1, 107 – 115. · Zbl 0804.47057
[15] Zong Ben Xu and G. F. Roach, Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl. 157 (1991), no. 1, 189 – 210. · Zbl 0757.46034 · doi:10.1016/0022-247X(91)90144-O · doi.org
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