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

$\left\{\begin{array}{c}{x}_{1}=x\in C,\hfill \\ {C}_{1}=C,\hfill \\ {y}_{n}={T}_{n}{x}_{n},\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:\phantom{\rule{4pt}{0ex}}〈{x}_{n}-z,J\left({y}_{n}-{x}_{n}\right)〉\le {a}_{n}\parallel {x}_{n}-{y}_{n}{\parallel }^{2}\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}x\hfill \end{array}\right\$
in a smooth, strictly convex, and reflexive Banach space with the Kadec–Klee property (${P}_{Q}$ is a metric projection in $E$). It is assumed that $\left\{{T}_{n}\right\}$ is a countable family of mappings of a nonempty closed convex subset $C$ of E into itself such that $F={\bigcap }_{n=1}^{\infty }F\left({T}_{n}\right)\ne \varnothing$ and that $\left\{{T}_{n}\right\}$ satisfies the condition
$〈x-z,J\left({T}_{n}x-x\right)〉\le {a}_{n}{\parallel x-{T}_{n}x\parallel }^{2},\phantom{\rule{2.em}{0ex}}x\in C,\phantom{\rule{4pt}{0ex}}z\in F\left({T}_{n}\right),\phantom{\rule{4pt}{0ex}}n\in ℕ,$
for some $\left\{{a}_{n}\right\}\subset \left(-\infty ,0\right)$, ${sup}_{n\in ℕ}{a}_{n}<0$ (here, $F\left(T\right)=\left\{x:\phantom{\rule{4pt}{0ex}}Tx=x\right\}$).
It is proved the strong convergence of the sequence ${x}_{n}$ to ${P}_{F}x$ under some additional assumptions about $C$ and $F$ (in particular, that the relations $\left\{{z}_{n}\right\}\subset C$, $z\in C$, ${x}_{n}\to z$, and ${T}_{n}{z}_{n}\to z$ imply that $z\in F$). The case when $E$ is real Hilbert space is considered as a particular case. Furthermore, the case when $\left\{{T}_{n}\right\}$ is a family of maximal monotone operators and an application to the feasibility problem are considered.