## Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications.(English)Zbl 1113.47055

Let $$C$$ be a nonempty closed convex subset of a uniformly convex Banach space $$E$$. A mapping $$T:C\rightarrow C$$ is called uniformly $$L$$-Lipschitzian on $$C$$ if, for some $$L>0$$, $\left\| T^n x-T^n y\right\| \leq L \left\| x-y\right\|$ for all $$x,y \in C$$ and for all $$n=1,2,3\dots$$. Denote by $$\text{Fix}(T)$$ the set of all fixed points of $$T$$. If $$\text{Fix}(T)\neq \varnothing$$, then $$T$$ is called asymptotically quasi-nonexpansive if, for a sequence $$\{k_n\}\subset [0,\infty)$$ with $$\lim\limits_{n\rightarrow \infty} k_n=0$$, we have $\left\| T^n x-p\right\| \leq (1+k_n) \left\| x-p\right\|$ for all $$x \in C$$, $$p \in \text{Fix}(T)$$ and for all $$n=1,2,3\dots$$. It is proved (Theorem 2) that if $$S,T: C\rightarrow C$$ are uniformly $$L$$-Lipschitzian and asymptotically quasi-nonexpansive mappings which satisfy a certain condition (A’), then a two-step iteration process of Ishikawa type associated with $$S$$ and $$T$$ converges to a common fixed point of $$S$$ and $$T$$.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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### References:

 [1] Das, G.; Debata, J.P., Fixed points of quasi-nonexpansive mappings, Indian J. pure appl. math., 17, 1263-1269, (1986) · Zbl 0605.47054 [2] Fukhar-ud-din, H.; Khan, S.H., Convergence of two-step iterative scheme with errors for two asymptotically nonexpansive mappings, Int. J. math. math. sci., 37, 1965-1971, (2004) · Zbl 1086.47048 [3] Igbokwe, D.I., Approximation of fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces, J. inequal. pure appl. math., 3, 1, (2002), Article 3; online: · Zbl 1009.47061 [4] Khan, S.H.; Fukhar-ud-din, H., Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear anal., 61, 8, 1295-1301, (2005) · Zbl 1086.47050 [5] Khan, S.H.; Takahashi, W., Approximating common fixed points of two asymptotically nonexpansive mappings, Sci. math. jpn., 53, 1, 143-148, (2001) · Zbl 0985.47042 [6] Liu, L.S., Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces, J. math. anal. appl., 194, 1, 114-125, (1995) · Zbl 0872.47031 [7] Maiti, M.; Ghosh, M.K., Approximating fixed points by Ishikawa iterates, Bull. austral. math. soc., 40, 113-117, (1989) · Zbl 0667.47030 [8] Qihou, L., Iterative sequences for asymptotically quasi-nonexpansive mappings with errors member, J. math. anal. appl., 259, 18-24, (2001) · Zbl 1001.47034 [9] Schu, J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. austral. math. soc., 43, 153-159, (1991) · Zbl 0709.47051 [10] Senter, H.F.; Dotson, W.G., Approximatig fixed points of nonexpansive mappings, Proc. amer. math. soc., 44, 2, 375-380, (1974) · Zbl 0299.47032 [11] Takahashi, W., Iterative methods for approximation of fixed points and their applications, J. oper. res. soc. Japan, 43, 1, 87-108, (2000) · Zbl 1004.65069 [12] Takahashi, W.; Tamura, T., Convergence theorems for a pair of nonexpansive mappings, J. convex anal., 5, 1, 45-58, (1998) · Zbl 0916.47042 [13] Tan, K.K.; Xu, H.K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. math. anal. appl., 178, 301-308, (1993) · Zbl 0895.47048 [14] Xu, Y., Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. math. anal. appl., 224, 91-101, (1998) · Zbl 0936.47041
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