Nonlinear ergodic theorems for a semitopological semigroup of non‐Lipschitzian mappings without convexity

G. Li and J. K. Kim

Abstract and Applied Analysis, 1999, vol. 4, issue 1, 49-59

Abstract: Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H, and ℑ = {Tt : t ∈ G} a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L(x) = {z ∈ H : infs∈Gsupt∈G‖Tts x − z‖ = inft∈G‖Tt x − z‖} for each x ∈ C and L(ℑ) = ∩x∈C L(x). In this paper, we prove that ∩s∈Gconv¯{Tts x:t∈G}∩L(ℑ) is nonempty for each x ∈ C if and only if there exists a unique nonexpansive retraction P of C into L(ℑ) such that PTs = P for all s ∈ G and P(x)∈conv¯{Ts x:s∈G} for every x ∈ C. Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non‐Lipschitzian mappings without convexity.

Date: 1999
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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:4:y:1999:i:1:p:49-59

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