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Convergence of the Mann iteration algorithm for a class of pseudocontractive mappings. (English) Zbl 1193.65085
Summary: Let $K$ be a nonempty, closed and convex subset of a real Banach space $E$. Let $T: K\to K$ be a strictly pseudocontractive map in the sense of Browder and Petryshyn. For a fixed $x_0\in K$, define a sequence $\{x_n\}$ by $$ x_{n+1}=(1-\alpha_n)x_n+\alpha_nTx_n, $$ where $\{\alpha_n\}$ is a real sequence defined in $[0,1]$ satisfying the following conditions: (i) $\sum^\infty_{n=1}\alpha_n=\infty$, (ii) $\sum^\infty_{n=1}\alpha^2_n<\infty$. Then $\liminf_{n\to\infty}\Vert x_n-Tx_n\Vert=0$. If, in addition, $T$ is demicompact, then $\{x_n\}$ converges strongly to some fixed point of $T$.

MSC:
65J15Equations with nonlinear operators (numerical methods)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
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References:
[1] Borwein, D.; Borwein, J. M.: Fixed point iterations for real functions. J. math. Anal. appl. 157, 112-126 (1991) · Zbl 0742.26006
[2] Browder, F. E.; Petryshyn, W. E.: Construction of fixed points of nonlinear mappings in Hilbert space. J. math. Anal. appl. 20, 197-228 (1967) · Zbl 0153.45701
[3] Chidume, C. E.: Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings. Proc. amer. Math. soc. 99, No. 2, 283-288 (1987) · Zbl 0646.47037
[4] Chidume, C. E.: Fixed point iterations for certain classes of nonlinear mappings, II. J. nigerian math. Soc. 8, 11-23 (1989)
[5] Chidume, C. E.: Iterative approximation of fixed points of Lipschitz pseudocontractive maps. Proc. amer. Math. soc. 129, No. 8, 2245-2251 (2001) · Zbl 0979.47038
[6] Chidume, C. E.; Mutangadura, S.: An example on the Mann iteration method for Lipschitz pseudocontractions. Proc. amer. Math. soc. 129, No. 8, 2359-2363 (2001) · Zbl 0972.47062
[7] Chidume, C. E.; Zegeye, H.: Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps. Proc. amer. Math. soc. 132, 831-840 (2003) · Zbl 1051.47041
[8] Chidume, C. E.; Moore, C.: Fixed point iteration for pseudocontractive maps. Proc. amer. Math. soc. 127, No. 4, 1163-1170 (1999) · Zbl 0913.47052
[9] Gatica, J. A.; Kirk, W. A.: Fixed point theorems for Lipschitzian pseudocontractive mappings. Proc. amer. Math. soc. 36, 111-115 (1972) · Zbl 0254.47076
[10] Hicks, T. L.; Kubicek, J. R.: On Mann iteration process in Hilbert space. J. math. Anal. appl. 59, 706-721 (1979)
[11] Ishikawa, S.: Fixed points by new iteration method. Proc. amer. Math. soc. 44, No. 1, 147-150 (1974) · Zbl 0286.47036
[12] Kato, T.: Nonlinear semigroups and evolution equations. J. math. Soc. Japan 19, 508-520 (1967) · Zbl 0163.38303
[13] Mann, W. R.: Mean value methods in iteration. Proc. amer. Math. soc. 4, 506-510 (1953) · Zbl 0050.11603
[14] Marino, G.; Xu, H. K.: Weak and strong convergence theorems for strictly pseudocontractions in Hilbert spaces. J. math. Anal. appl. 329, 336-349 (2007) · Zbl 1116.47053
[15] Maruster, S.: The solution by iteration of nonlinear equations. Proc. amer. Math. soc. 66, 69-73 (1977) · Zbl 0355.47037
[16] Osilike, M. O.; Udomene, A.: Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder -- petryshyn type. J. math. Anal. appl. 256, 431-445 (2001) · Zbl 1009.47067
[17] Qihou, L.: Iterative sequences for asymptotically quasi-nonexpansive mappings. J. math. Anal. appl. 259, 1-7 (2001) · Zbl 1033.47047