Nonlinear Ergodic Theorems for a Semitopological Semigroup of Non-lipschitzian Mappings without Convexity
نویسندگان
چکیده
Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H , and = {Tt : t ∈ G} a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L(x)= {z ∈H : infs∈G supt∈G ‖Ttsx−z‖ = inf t∈G ‖Ttx−z‖} for each x ∈ C and L( )= ⋂x∈C L(x). In this paper, we prove that ⋂s∈G conv{Ttsx : t ∈ G}⋂L( ) is nonempty for each x ∈ C if and only if there exists a unique nonexpansive retraction P of C into L( ) such that PTs = P for all s ∈ G and P(x) ∈ conv{Tsx : s ∈ G} for every x ∈ C. Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non-Lipschitzian mappings without convexity.
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