A Theorem of Rolewicz's type in solid function spaces

Buşe, Constantin ; Dragomir, Sever S.

Glasgow mathematical journal, Tome 44 (2002) no. 1, pp. 125-135

Résumé

Let \bf{R}_+ be the set of all non-negative real numbers, \bf(I}\in \{ \bf(R}, \bf(R}_+\} and {\cal U}_\bf(I}=\{ U(t, s): t\ge s\in I\} be a strongly measurable and exponentially bounded evolution family of bounded linear operators acting on a Banach space X. Let \phi :\bf(R}_+\to \bf(R}_+ be a strictly increasing function and E be a normed function space over \bf(I} satisfying some properties; see Section 2. We prove that if\phi \circ (\chi _{[s, \infty )}(\cdot )||U(\cdot , s)x||)defines an element of the space E for every s\in \bf(I} and all x\in X and if there exists M>0 such that\sup \limits _{s\in \bf(I}}|\phi \circ (\chi _{[s, \infty )} (\cdot )||U(\cdot , s)x||)|_{E}= M<\infty \quad \forall x\in X\ {\text with}\ ||x||\le 1,then {\cal U}_\bf(I} is uniformly exponentially stable. In particular if \psi :\bf(R}_+\to \bf(R}_+ is a nondecreasing function such that \psi (t)>0, for all t>0, and if there exists K>0 such that\sup \limits _{s\in \bf(I}}\int \limits _s^{\infty }\psi (||U(t, s)x||)dt=K <\infty ,\quad \forall x\in X\ \text(with)\ ||x||\le 1, then{\cal U}_\bf(I} is uniformly exponentially stable. For \bf(I}=\bf(R}_+, \psi continuous and {\cal U}_{\bf(R}_+} strongly continuous this last result is due to S. Rolewicz. Some related results for periodic evolution families are also proved.

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DOI : 10.1017/S001708950201008X