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Strongly convergent iterative schemes for a sequence of nonlinear mappings. (English) Zbl 1162.47049
This article deals with the following iterative scheme $$\cases x_1 = x \in C, \\ C_1 = C, \\ y_n = T_nx_n, \\ C_{n+1} = \{z \in C_n: \ \langle x_n - z,J(y_n - x_n) \rangle \le a_n\|x_n - y_n\|^2\}, \\ x_{n+1} = P_{C_{n+1}}x \endcases$$ 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 $F = \bigcap_{n=1}^\infty F(T_n) \ne \emptyset$ and that $\{T_n\}$ satisfies the condition $$\langle x - z,J(T_nx - x) \rangle \le a_n\|x - T_nx\|^2, \qquad x \in C, \ z \in F(T_n), \ n \in {\Bbb N},$$ for some $\{a_n\} \subset (-\infty,0)$, $\sup_{n \in {\Bbb N}} a_n < 0$ (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.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
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