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Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. (English) Zbl 1218.47131
Let $C$ be a nonempty closed convex subset of a real Banach space $X$ whose norm is uniformly Gâteaux differentiable and let $T : C\to C$ be a continuous pseudocontraction with nonempty fixed point set $F(T)$. A sequence $(x_n)$ is defined iteratively which converges strongly to a fixed point of $T$. For all the continuous pseudocontractive mappings for which it is possible to construct the sequence $(x_n)$, the obtained result improves and extends a recent result of {\it Y.-H. Yao}, {\it Y.-C. Liou} and {\it R.-D. Chen} [Nonlinear Anal., Theory Methods Appl. 67, No. 12, A, 3311--3317 (2007; Zbl 1129.47059)].

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H05 Monotone operators (with respect to duality) and generalizations
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##### References:
 [1] Browder, F. E.: Existence of periodic solutions for nonlinear equations of evolution. Proc. natl. Acad. sci. USA 53, 1100-1103 (1965) · Zbl 0135.17601 [2] Browder, F. E.: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. sympos. Math. amer. Math. soc. Providence RI 18 (1976) · Zbl 0327.47022 [3] Browder, F. E.; Petryshyn, W. V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. math. Anal. appl. 20, 197-228 (1967) · Zbl 0153.45701 [4] Jr., R. E. Bruck: A strongly convergent iterative solution of $0\inUx$for a maximal monotone operator U in Hilbert spaces. J. math. Anal. appl. 48, 114-126 (1974) [5] Chen, R. D.; Song, Y. S.; Zhou, H. Y.: Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings. J. math. Anal. appl. 314, 701-709 (2006) · Zbl 1086.47046 [6] Chidume, C. E.; Zegeye, H.: Approximate point sequences and convergence theorems for Lipschitz pseudo-contractive maps. Proc. amer. Math. soc. 132, 831-840 (2004) · Zbl 1051.47041 [7] Deimling, K.: Zeros of accretive operators. Manuscripta math. 13, 365-374 (1974) · Zbl 0288.47047 [8] Gossez, J. P.; Dozo, E. Lami: Some geometric properties related to the fixed point theory for nonexpansive mappings. Pacific J. Math. 40, 565-573 (1972) · Zbl 0223.47025 [9] Ishikawa, S.: Fixed point by a new iteration. Proc. amer. Math. soc. 44, 147-150 (1974) · Zbl 0286.47036 [10] Kazor, W.: Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups. J. math. Anal. appl. 272, 565-574 (2002) · Zbl 1058.47049 [11] Liu, L. S.: Ishikawa and Mann iterative processes with errors for nonlinear operator equations in Banach spaces. J. math. Anal. appl. 194, 114-125 (1995) · Zbl 0872.47031 [12] Morales, C. H.; Jung, J. S.: Convergence of paths for pseudocontractive mappings in Banach spaces. Proc. amer. Math. soc. 128, 3411-3419 (2000) · Zbl 0970.47039 [13] Opial, Z.: Weak convergence of successive approximations for nonexpansive mappings. Bull. amer. Math. soc. 73, 591-597 (1967) · Zbl 0179.19902 [14] Osilike, M. O.: Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps. J. math. Anal. appl. 294, 73-81 (2004) · Zbl 1045.47056 [15] Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. math. Anal. appl. 75, 287-292 (1980) · Zbl 0437.47047 [16] Song, Y. S.; Chen, R. D.; Zhou, H. Y.: Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear anal. 66, 1016-1024 (2007) · Zbl 1121.47057 [17] Takanashi, W.: Nonlinear functional analysis--fixed point theory and its applications. (2000) [18] Xu, H. K.; Ori, R. G.: An implicit iteration process for nonexpansive mappings. Numer. funct. Anal. optim. 22, 767-773 (2001) · Zbl 0999.47043 [19] Yao, Y. H.; Liou, Y. C.; Chen, R. D.: Strong convergence of an iterative algorithm for pseudocontractive mappings in Banach spaces. Nonlinear anal. 67, 3311-3317 (2007) · Zbl 1129.47059 [20] Zhou, H. Y.: Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces. Nonlinear anal. (2007)