Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings

Chidume, C.; Li, Jinlu; Udomene, A.

doi:10.1090/S0002-9939-04-07538-0

Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings
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by C. E. Chidume, Jinlu Li and A. Udomene PDF

Proc. Amer. Math. Soc. 133 (2005), 473-480 Request permission

Abstract:

Let $E$ be a real Banach space with a uniformly Gâteaux differentiable norm possessing uniform normal structure, $K$ be a nonempty closed convex and bounded subset of $E$, $T: K \longrightarrow K$ be an asymptotically nonexpansive mapping with sequence $\{k_n\}_n\subset [1, \infty )$. Let $u\in K$ be fixed, $\{t_n\}_n \subset (0, 1)$ be such that $\lim \limits _{n\to \infty }t_n = 1$, $t_nk_n < 1$, and $\lim \limits _{n\to \infty }\frac {k_n - 1}{k_n-t_n} =0$. Define the sequence $\{z_n\}_n$ iteratively by $z_0\in K$, $z_{n+1}= (1-\frac {t_n}{k_n})u + \frac {t_n}{k_n}T^nz_n, \>n= 0, 1, 2, ..._.$ It is proved that, for each integer $n \geq 0$, there is a unique $x_n \in K$ such that $x_n= (1-\frac {t_n}{k_n})u + \frac {t_n}{k_n}T^nx_n.$ If, in addition, $\lim \limits _{n\to \infty }\|x_n - Tx_n\| = 0$ and $\lim \limits _{n\to \infty }\|z_n - Tz_n\| = 0$, then $\{z_n\}_n$ converges strongly to a fixed point of $T$.

References

Asuman G. Aksoy and Mohamed A. Khamsi, Nonstandard methods in fixed point theory, Universitext, Springer-Verlag, New York, 1990. With an introduction by W. A. Kirk. MR 1066202, DOI 10.1007/978-1-4612-3444-9
W. L. Bynum, Normal structure coefficients for Banach spaces, Pacific J. Math. 86 (1980), no. 2, 427–436. MR 590555, DOI 10.2140/pjm.1980.86.427
Ronald Bruck, Tadeusz Kuczumow, and Simeon Reich, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq. Math. 65 (1993), no. 2, 169–179. MR 1240164, DOI 10.4064/cm-65-2-169-179
S. S. Chang, Some results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 129 (2001), no. 3, 845–853. MR 1802003, DOI 10.1090/S0002-9939-00-05988-8
C. E. Chidume, Convergence theorems for asymptotically pseudocontractive mappings, Nonlinear Anal. 49 (2002), no. 1, Ser. A: Theory Methods, 1–11. MR 1887908, DOI 10.1016/S0362-546X(00)00240-6
Ioana Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Mathematics and its Applications, vol. 62, Kluwer Academic Publishers Group, Dordrecht, 1990. MR 1079061, DOI 10.1007/978-94-009-2121-4
K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171–174. MR 298500, DOI 10.1090/S0002-9939-1972-0298500-3
K. Goebel and W. A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973), 135–140. MR 336468, DOI 10.4064/sm-47-2-134-140
Kazimierz Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. MR 1074005, DOI 10.1017/CBO9780511526152
Benjamin Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957–961. MR 218938, DOI 10.1090/S0002-9904-1967-11864-0
Teck-Cheong Lim and Hong Kun Xu, Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal. 22 (1994), no. 11, 1345–1355. MR 1280202, DOI 10.1016/0362-546X(94)90116-3
Naoki Shioji and Wataru Takahashi, A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Arch. Math. (Basel) 72 (1999), no. 5, 354–359. MR 1680439, DOI 10.1007/s000130050343
Naoki Shioji and Wataru Takahashi, Strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces, J. Approx. Theory 97 (1999), no. 1, 53–64. MR 1676298, DOI 10.1006/jath.1996.3251
Jürgen Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991), no. 2, 407–413. MR 1117571, DOI 10.1016/0022-247X(91)90245-U
Jürgen Schu, Approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 112 (1991), no. 1, 143–151. MR 1039264, DOI 10.1090/S0002-9939-1991-1039264-7
Kok-Keong Tan and Hong Kun Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122 (1994), no. 3, 733–739. MR 1203993, DOI 10.1090/S0002-9939-1994-1203993-5
Hong-Kun Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66 (2002), no. 1, 240–256. MR 1911872, DOI 10.1112/S0024610702003332