Nonlinear ergodic theorems for a semitopological semigroup ofnon‐Lipschitzian mappings without convexity
G. Li, J. K. Kim · Abstract and Applied Analysis · 1999
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 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 for every x ∈ C. Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non‐Lipschitzian mappings without convexity.