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Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces spaces. (English) Zbl 1217.47098

Let $H$ be a Hilbert space and $C$ be a nonempty closed convex subset of $H$. Then a mapping $T:C\to C$ is called 2-generalized hybrid if there are ${\alpha }_{1}$, ${\alpha }_{2}$, ${\beta }_{1}$, ${\beta }_{2}\in ℝ$ such that

$\begin{array}{c}{\alpha }_{1}{∥{T}^{2}x-Ty∥}^{2}+{\alpha }_{2}{∥Tx-Ty∥}^{2}+\left(1-{\alpha }_{1}-{\alpha }_{2}\right){∥x-Ty∥}^{2}\hfill \\ \hfill \le {\beta }_{1}{∥{T}^{2}x-y∥}^{2}+{\beta }_{2}{∥Tx-y∥}^{2}+\left(1-{\beta }_{1}-{\beta }_{2}\right){∥x-y∥}^{2}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}x,y\in C·\end{array}$

This is a new and broad class of nonlinear mappings, covering several known classes such as nonexpansive mappings, nonspreading mappings, hybrid mappings, $\left(\alpha ,\beta \right)$-generalized hybrid mappings and quasi-nonexpansive mappings. Theorem 3.1 asserts that, provided $C$ is a nonempty closed convex subset of a Hilbert space $H$, a 2-generalized hybrid mapping $T:C\to C$ has a fixed point in $C$ if and only if $\left\{{T}^{n}z\right\}$ is bounded for some $z\in C$. Several known fixed point results for the subclasses of 2-generalized hybrid mappings mentioned above are proved as consequences of Theorem 3.1. An even broader class of nonlinear mappings, that of $n$-generalized hybrid mappings, is mentioned. An analogue of Theorem 3.1 can be proved for this class as well.

Two other important results for the class of 2-generalized hybrid mappings are proved: a nonlinear ergodic theorem of Baillon’s type (Theorem 4.1) and a weak convergence theorem of Mann’s type (Theorem 5.3).

##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H05 Monotone operators (with respect to duality) and generalizations 47H25 Nonlinear ergodic theorems