Let be a Hilbert space and be a nonempty closed convex subset of . Then a mapping is called 2-generalized hybrid if there are , , , such that
This is a new and broad class of nonlinear mappings, covering several known classes such as nonexpansive mappings, nonspreading mappings, hybrid mappings, -generalized hybrid mappings and quasi-nonexpansive mappings. Theorem 3.1 asserts that, provided is a nonempty closed convex subset of a Hilbert space , a 2-generalized hybrid mapping has a fixed point in if and only if is bounded for some . 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 -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).