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Abstract

We prove the following theorem: Let p > 1 and let E be a real p-uniformly convex Banach space, and C a nonempty bounded closed convex subset of E. If $T = {T_s: C → C: s ∈ G = [0,∞)}$ is a Lipschitzian semigroup such that $g = lim inf_{G ∋ α → ∞} inf_{G ∋ δ ≥ 0} 1/α ʃ^α_0 ∥T_{β+δ}∥^p dβ < 1 + c$, where c > 0 is some constant, then there exists x ∈ C such that $T_sx = x$ for all s ∈ G.