On unification of the strong convergence theorems for a finite family of total asymptotically nonexpansive mappings in Banach spaces. (English) Zbl 1318.47091
Summary: We unify all known iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically \(I\)-nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced by C. E. Chidume and E. U. Ofoedu [Int. J. Math. Math. Sci. 2009, Article ID 615107, 17 p. (2009; Zbl 1186.47059)]. We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically \(I\)-nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed-convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify known results.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
iterative methods; common fixed points; total asymptotically nonexpansive mappings; strong convergenceCitations:
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