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.


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