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A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense. (English) Zbl 0981.47037

Let \(X\) be a uniformly convex real Banach space, \(C\) a nonempty bounded closed convex subset of \(X\). If the inequality \[ \limsup_{t\to\infty} \sup_{x,y\in C}(\|T(t)x- T(t)y\|-\|x-y\|)\leq 0, \] holds, then the \(C_0\)-semigroup \(\{T(t)\}_{t\geq 0}\) of (nonlinear) selfmappings of \(C\) is called “asymptotically nonexpansive in the intermediate sense”. One says that a function \(u: [0,\infty)\to C\) is an “almost orbit” of \(\{T(t)\}_{t\geq 0}\) if \[ \lim_{s\to \infty} \Biggl(\sup_{t\in [0,\infty)}\|u(t+ s)- T(t) u(s)\|\Biggr)= 0. \] A long part of the paper is devoted to proving that if \(X^*\) has the Kadec-Klee property and \(\{T(t)\}_{t\geq 0}\) is asymptotically nonexpansive in the intermediate sense, then every continuous almost orbit \(u\) of \(\{T(t)\}_{t\geq 0}\) is weakly almost convergent to some \(y\in \text{Fix}(T)\) (the set of common fixed points of \(T(t)\)), i.e. \(w\)-\(\lim_{t\to\infty} {1\over t} \int^t_0 u(h+\tau) d\tau= y\) uniformly in \(h\geq 0\).

MSC:

47H20 Semigroups of nonlinear operators
47D06 One-parameter semigroups and linear evolution equations
46B20 Geometry and structure of normed linear spaces
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