Asymptotic behavior of periodic nonexpansive evolution operators in uniformly convex Banach spaces (Q1822343)

Let \(\{C_ t\}_{t\geq 0}\) be a family of closed convex subsets of a Banach space X and let \(\{\) U(t,s)\(\}\) be a family of mappings U(t,s): \(C_ s\to C_ t\) such that \[ U(t,s)U(s,r)=U(t,r),\quad U(r,r)=I, \] \[ \| U(t,s)x-U(t,s)y\| \leq \| x-y\|, \] \[ C_{t+T}=C_ t,\quad U(t+T,s+T)=U(t,s) \] for \(0\leq r\leq s\leq t\) and \(x,y\in C_ s\). We also assume that \(U(t,0)x_ 0\) is bounded in \(t\geq 0\) for some \(x_ 0\in C_ 0\). It is proved that: (i) Suppose that X is uniformly convex and has a Fréchet differentiable norm. Then for each \(x\in C_ 0\), there exists \(z\in C_ 0\) such that \(u_ n(t)=U(nT+t,0)x\) is weakly almost convergent as \(n\to \infty\) to U(t,0)z, \(0\leq t\leq T.\) (ii) Suppose that X is uniformly convex and \(x\in C_ 0\) is such that \(\lim_{n\to \infty}\| u_{n+k}(t)-u_ n(t)\| =\eta (t)\) exists uniformly in k. Then there exists \(z\in C_ 0\) such that \(u_ n(t)\) is strongly convergent as \(n\to \infty\) to U(t,0)z, \(0\leq t\leq T.\) The assertion (i) was obtained independently by \textit{R. E. Bruck} [Proc. Sympos. Pure Math. 45, Part I, 227-235 (1986)].