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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 $T:C\rightarrow C$ is called $2$-generalized hybrid if there are $\alpha_1$, $\alpha_2$, $\beta_1$, $\beta_2\in \mathbb{R}$ such that $$\multline \alpha_1\left\|T^2x-Ty\right\|^2+\alpha_2\left\|Tx-Ty\right\|^2+(1-\alpha_1-\alpha_2) \left\|x-Ty\right\|^2\\ \leq \beta_1 \left\|T^2x-y\right\|^2+\beta_2\left\|Tx-y\right\|^2+(1-\beta_1-\beta_2)\left\|x-y\right\|^2\ \text{ for all } x,y\in C. \endmultline$$ This is a new and broad class of nonlinear mappings, covering several known classes such as nonexpansive mappings, nonspreading mappings, hybrid mappings, $(\alpha,\beta)$-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 $T:C\rightarrow C$ has a fixed point in $C$ if and only if $\{T^nz\}$ 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).

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H05Monotone operators (with respect to duality) and generalizations
47H25Nonlinear ergodic theorems
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