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, 1-11

Abstract:

Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H , and ℑ = { T t : t ∈ G } a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L ( x ) = { z ∈ H : inf s ∈ G sup t ∈ G ‖ T t s   x − z ‖ = inf t ∈ G ‖ T t   x − z ‖ } for each x ∈ C and L ( ℑ ) = ∩ x ∈ C   L ( x ) . In this paper, we prove that ∩ s ∈ G conv ¯ { T t s   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 P T s = P for all s ∈ G and P ( x ) ∈ conv ¯ { T s   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:hin:jnlaaa:953291

DOI: 10.1155/S1085337599000056

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