\(C\)-cosine functions and the abstract Cauchy problem. I
This article deals with \(C\)-cosine functions generated by \(A\) and the second-order abstract Cauchy problem \[ u''(t)= Au(t)+ f(t)\qquad (t\in\mathbb{R}), \] \[ u(0)= x,\qquad u'(0)=y \] in \(C(\mathbb{R},X)\), where \(X\) is a Banach space, \(A\) a closed linear operator with domain \(D(A)\) and range \(R(A)\), \(f\in C(\mathbb{R},X)\). The main result is a characterization of the generator \(A\) in terms of the solvability properties of the associated abstract Cauchy problem; this result covers the cases when the \(C\)-cosine function \(C(\cdot)\) is not obligatory exponentially bounded and \(A\) densely defined. The results of the authors are similar to classical theorems of \textit{N. Tanaka} and \textit{I. Miyadera} [J. Math. Anal. Appl. 170, No. 1, 196-206 (1992; Zbl 0812.47044)] about \(C\)-semigroups and associated abstract Cauchy problems of first order.
scientific article; zbMATH DE number 1046950