*(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 ${\alpha}_{1}$, ${\alpha}_{2}$, ${\beta}_{1}$, ${\beta}_{2}\in \mathbb{R}$ such that

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 $T:C\to C$ 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 |