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Let S be a commutative semigroup with identity, and let H be a real Hilbert space with inner product 〈 · , · 〉 and norm ‖ · ‖. We also denote by Z, Z, R and R the sets of all integers, nonnegative integers, real numbers and nonnegative real numbers, respectively. Let C be a subset of H. Then a mapping T : C → C is called nonexpansive if ‖Tx−Ty‖ ≤ ‖x−y‖ for all x, y ∈ C. The first nonlinear ergodic theorem for nonexpansive mappings (in a Hilbert space) was established by Baillon [1]: Let C be a nonempty closed convex subset of H and let T be a nonexpansive mapping of C into itself. If T has a fixed point, then the Cesàro means (1/n) ∑n−1 k=0 T x converge weakly as n →∞ to a fixed point y of T. In this case, put y = Px for each x ∈ C. Then P is a nonexpansive retraction of C onto the set Fix(T ) of fixed points of T such that PT = TP = P for all n ∈ Z, and Px ∈ clco{Tx : n ∈ Z} for each x ∈ C, where clco A is the closure of the convex hull of A. In [33, 34], Takahashi proved the existence of such an ergodic retraction for an amenable semigroup of nonexpansive mappings in a Hilbert space. And also Rodé [30] found a sequence of means on the semigroup, generalizing the Cesàro means on the positive integers, such that the corresponding sequence of mappings converges to an ergodic

Nonlinear ergodic theorems for almost nonexpansive curves over commutative semigroups