## Strongly convergent iterative schemes for a sequence of nonlinear mappings.(English)Zbl 1162.47049

This article deals with the following iterative scheme
$\begin{cases} x_1 = x \in C, \\ C_1 = C, \\ y_n = T_nx_n, \\ C_{n+1} = \{z \in C_n: \;\langle x_n - z,J(y_n - x_n) \rangle \leq a_n\|x_n - y_n\|^2\}, \\ x_{n+1} = P_{C_{n+1}}x \end{cases}$
in a smooth, strictly convex, and reflexive Banach space with the Kadec–Klee property ($$P_Q$$ is a metric projection in $$E$$). It is assumed that $$\{T_n\}$$ is a countable family of mappings of a nonempty closed convex subset $$C$$ of E into itself such that $$F = \bigcap_{n=1}^\infty F(T_n) \neq \emptyset$$ and that $$\{T_n\}$$ satisfies the condition
$\langle x - z,J(T_nx - x) \rangle \leq a_n\|x - T_nx\|^2, \qquad x \in C, \;z \in F(T_n), \;n \in {\mathbb N},$
for some $$\{a_n\} \subset (-\infty,0)$$, $$\sup_{n \in {\mathbb N}} a_n < 0$$ (here, $$F(T) = \{x: \;Tx = x\}$$).
It is proved the strong convergence of the sequence $$x_n$$ to $$P_Fx$$ under some additional assumptions about $$C$$ and $$F$$ (in particular, that the relations $$\{z_n\} \subset C$$, $$z \in C$$, $$x_n \to z$$, and $$T_nz_n \to z$$ imply that $$z \in F$$). The case when $$E$$ is real Hilbert space is considered as a particular case. Furthermore, the case when $$\{T_n\}$$ is a family of maximal monotone operators and an application to the feasibility problem are considered.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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