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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:CC is called 2-generalized hybrid if there are α 1 , α 2 , β 1 , β 2 such that

α 1 T 2 x-Ty 2 +α 2 Tx-Ty 2 +(1-α 1 -α 2 )x-Ty 2 β 1 T 2 x-y 2 +β 2 Tx-y 2 +(1-β 1 -β 2 )x-y 2 forallx,yC·

This is a new and broad class of nonlinear mappings, covering several known classes such as nonexpansive mappings, nonspreading mappings, hybrid mappings, (α,β)-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:CC has a fixed point in C if and only if {T n z} is bounded for some zC. 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:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H05Monotone operators (with respect to duality) and generalizations
47H25Nonlinear ergodic theorems