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Almost stability of iterative procedures for the infinite mappings of uniformly Lipschitzian and asymptotically nonexpansive in the intermediate sense. (English) Zbl 1112.47054

Let \(C\) be a (nonempty) closed convex subset of a real Banach space \(E\) and \(T:=(T_i:C\to C\mid i\in \mathbb N)\) be a family of maps such that (a) \(T_i\) is asymptotically nonexpansive in the intermediate sense \(\forall i\in \mathbb N\), (b) \(F(T):=\cap_{i}F(T_i)\neq \emptyset\), and (c) \(T_1\) is strongly successively pseudocontractive with respect to some \(k\in (0,1)\). The convergence (to some \(q\in F(T)\)) and stability of the iterative scheme \(x_{n+1}=(1-\alpha_{n(1)})x_n+\alpha_{n(1)}T_1^nU_{n(2)}x_n\), \(n\geq 1\), where \(U_{n(i)}=(1-\alpha_{n(i)})I+\alpha_{n(i)}T_i^nU_{n(i+1)}\), \(i=1,\dots,n\), are discussed under certain regularity assumptions involving these data.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H10 Fixed-point theorems
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