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Fixed points of Lipschitzian semigroups in Banach spaces. (English) Zbl 0894.47044

Let \(E\) be a real \(p\)-uniformly convex Banach space \((p>1)\), \(C\) a bounded closed convex subset of \(E\), and \(T=\{T_s: 0\leq s<\infty\}\) a Lipschitzian semigroup such that \[ \liminf_{\alpha\to\infty} \inf_{\delta\geq 0} {1\over\alpha} \int^\alpha_0\| T_{\beta+\delta}\|^p d\beta< 1+c, \] where \(c\) is some positive constant related to the geometry of \(E\). The author proves then that there is some \(x\in C\) with \(T_sx= x\) for all \(s\geq 0\).

MSC:

47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H20 Semigroups of nonlinear operators
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