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In this chapter, we want to study the second-order autonomous abstract Cauchy problem of the form \( {\left\{ \begin{array}{ll} \ddot{u}(t)=Au(t),& \quad t\ge 0,\\ u(0)=x\in X,& \\ \dot{u}(0)=y\in X,& \end{array}\right. } \) for some linear operator \((A,D(A))\) on a Banach space X. In the classical Banach space setting, the so-called cosine families serve as solutions to this problem.

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Bi-continuous Cosine Families

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  • 发表日期：

    2026年

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  • Christian Budde

摘要

In this chapter, we want to study the second-order autonomous abstract Cauchy problem of the form \( {\left\{ \begin{array}{ll} \ddot{u}(t)=Au(t),& \quad t\ge 0,\\ u(0)=x\in X,& \\ \dot{u}(0)=y\in X,& \end{array}\right. } \) for some linear operator \((A,D(A))\) on a Banach space X. In the classical Banach space setting, the so-called cosine families serve as solutions to this problem.