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An iterative algorithm on approximating fixed points of pseudocontractive mappings. (English) Zbl 1295.47104
Summary: Let $$E$$ be a real reflexive Banach space with a uniformly GĂ˘teaux differentiable norm. Let $$K$$ be a nonempty bounded closed convex subset of $$E$$, and assume that every nonempty closed convex bounded subset of $$K$$ has the fixed point property for non-expansive self-mappings. Let $$f : K \rightarrow K$$ be a contractive mapping and $$T : K \rightarrow K$$ be a uniformly continuous pseudocontractive mapping with $$F(T) \neq \emptyset$$. Let $$\{\lambda_n\} \subset (0, 1/2)$$ be a sequence satisfying the following conditions: (i) $$\lim_{n \rightarrow \infty} \lambda_n = 0$$; (ii) $$\sum^\infty_{n=0} \lambda_n = \infty$$. Define the sequence $$\{x_n\}$$ in $$K$$ by $$x_0 \in K$$, $$x_{n+1} = \lambda_n f(x_n) + (1 - 2\lambda_n) x_n + \lambda_n Tx_n$$ for all $$n \geq 0$$. Under some appropriate assumptions, we prove that the sequence $$\{x_n\}$$ converges strongly to a fixed point $$p \in F(T)$$ which is the unique solution of the variational inequality $$\langle f(p) - p, j(z - p)\rangle \leq 0$$ for all $$z \in F(T)$$.
Reviewer: Reviewer (Berlin)

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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##### References:
 [1] S. S. Chang, “On the convergence problems of Ishikawa and Mann iterative processes with error for \Phi -pseudo contractive type mappings,” Applied Mathematics and Mechanics, vol. 21, no. 1, pp. 1-12, 2000. · Zbl 0958.47041 · doi:10.1007/BF02458533 [2] C. E. Chidume and C. Moore, “The solution by iteration of nonlinear equations in uniformly smooth Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 215, no. 1, pp. 132-146, 1997. · Zbl 0906.47050 · doi:10.1006/jmaa.1997.5628 [3] Q. H. Liu, “The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings,” Journal of Mathematical Analysis and Applications, vol. 148, no. 1, pp. 55-62, 1990. · Zbl 0729.47052 · doi:10.1016/0022-247X(90)90027-D [4] W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506-510, 1953. · Zbl 0050.11603 · doi:10.2307/2032162 [5] M. O. Osilike, “Iterative solution of nonlinear equations of the \Phi -strongly accretive type,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 259-271, 1996. · Zbl 0860.65039 · doi:10.1006/jmaa.1996.0203 [6] S. Reich, “Iterative methods for accretive sets,” in Nonlinear Equations in Abstract Spaces, pp. 317-326, Academic Press, New York, NY, USA, 1978. · Zbl 0495.47034 [7] Y. Yao, G. Marino, and Y. C. Liou, “A hybrid method for monotone variational inequalities involving pseudocontractions,” Fixed Point Theory and Applications, vol. 2011, Article ID 180534, 8 pages, 2011. · Zbl 1215.49021 · doi:10.1155/2011/180534 · eudml:227684 [8] Y. Yao, Y. C. Liou, and S. M. Kang, “Iterative methods for k-strict pseudo-contractive mappings in Hilbert spaces,” Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 19, no. 1, pp. 313-330, 2011. · Zbl 1216.47104 · eudml:228954 [9] Y. Yao, Y. C. Liou, and J.-C. Yao, “New relaxed hybrid-extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problems,” Optimization, vol. 60, no. 3, pp. 395-412, 2011. · Zbl 1296.47104 · doi:10.1080/02331930903196941 [10] Y. Yao, Y. J. Cho, and Y. C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242-250, 2011. · Zbl 1266.90186 · doi:10.1016/j.ejor.2011.01.042 [11] S. Ishikawa, “Fixed points and iteration of a nonexpansive mapping in a Banach space,” Proceedings of the American Mathematical Society, vol. 59, no. 1, pp. 65-71, 1976. · Zbl 0352.47024 · doi:10.2307/2042038 [12] C. E. Chidume and S. A. Mutangadura, “An example of the Mann iteration method for Lipschitz pseudocontractions,” Proceedings of the American Mathematical Society, vol. 129, no. 8, pp. 2359-2363, 2001. · Zbl 0972.47062 · doi:10.1090/S0002-9939-01-06009-9 [13] C. E. Chidume and H. Zegeye, “Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 132, no. 3, pp. 831-840, 2004. · Zbl 1051.47041 · doi:10.1090/S0002-9939-03-07101-6 [14] N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641-3645, 1997. · Zbl 0888.47034 · doi:10.1090/S0002-9939-97-04033-1 [15] C. H. Morales and J. S. Jung, “Convergence of paths for pseudocontractive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 128, no. 11, pp. 3411-3419, 2000. · Zbl 0970.47039 · doi:10.1090/S0002-9939-00-05573-8 [16] S. S. Chang, Y. J. Cho, and H. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Science, Huntington, NY, USA, 2002. · Zbl 1070.47054 [17] T. H. Kim and H. K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis, Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51-60, 2005. · Zbl 1091.47055 · doi:10.1016/j.na.2004.11.011 [18] S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung, and S. M. Kang, “Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 224, no. 1, pp. 149-165, 1998. · Zbl 0933.47040 · doi:10.1006/jmaa.1998.5993
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