## Weak and strong convergence of a scheme with errors for two nonexpansive mappings.(English)Zbl 1086.47050

Let $$C$$ be a (nonempty) bounded closed convex subset of a Banach space $$E$$ and $$(S,T)$$ be a pair of asymptotically nonexpansive selfmaps of $$C$$. The weak and strong convergence of the iterative scheme $$x_{n+1}=a_nSy_n+b_nx_n+c_nu_n,$$ $$y_n=a'_nTx_n+b'_nx_n+c'_nv_n$$ $$(n\geq 1)$$ is discussed; here, $$(a_n)$$, $$(b_n)$$ $$(c_n)$$, $$(a'_n)$$, $$(b'_n)$$, $$(c'_n)$$ are sequences in $$[0,1]$$ with certain regularity properties and $$(u_n)$$, $$(v_n)$$ are bounded sequences in $$C$$.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 49M05 Numerical methods based on necessary conditions 65J15 Numerical solutions to equations with nonlinear operators
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### References:

 [2] Chidume, C. E.; Chika Moore, Fixed points iteration for pseudocontractive maps, Proc. Amer. Math. Soc., 127, 4, 1163-1170 (1999) · Zbl 0913.47052 [3] Das, G.; Debata, J. P., Fixed points of quasi-nonexpansive mappings, Indian J. Pure Appl. Math., 17, 1263-1269 (1986) · Zbl 0605.47054 [4] Maiti, M.; Gosh, M. K., Approximating fixed points by Ishikawa iterates, Bull. Austral. Math. Soc., 40, 113-117 (1989) · Zbl 0667.47030 [5] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 591-597 (1967) · Zbl 0179.19902 [6] Schu, J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43, 153-159 (1991) · Zbl 0709.47051 [7] Senter, H. F.; Dotson, W. G., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44, 2, 375-380 (1974) · Zbl 0299.47032 [8] Takahashi, W.; Tamura, T., Convergence theorems for a pair of nonexpansive mappings, J. Convex Analysis, 5, 1, 45-58 (1998) · Zbl 0916.47042 [9] 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 [10] 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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