Fixed point theorems for uniformly Lipschitzian semigroups in Hilbert spaces (Q1098395)

Let E be a Banach space with norm \(\| \cdot \|\) and let C be a nonempty bounded closed convex subset of E. A mapping \(T: C\to C\) is called uniformly k-Lipschitzian on C if \(\| T^ nx-T^ ny\| \leq k\| x-y\|\) for all x,y\(\in C\), \(n=1,2,... \). It is a consequence of a result of \textit{Lifshits} [Veronezh. Gos. Univ. Trudy Mat. Fak. 16, 23- 28 (1975)] that such a mapping always has a fixed point if \(k<\sqrt{2}\). \textit{D. J. Downing} and \textit{W. O. Ray} [Can. Math. Bull. 25, 210-214 (1982; Zbl 0438.47059)] extended this result by replacing \(\{T^ n\}\) with a left reversible semigroup. In this paper the result is extended to a uniformly k-Lipschitzian semigroup such that the space of right uniformly continuous functions on the semigroup has a left invariant mean. The authors also recover a simple proof of the Downing-Ray result.