On local integrated C-cosine function and weak solution of second order abstract Cauchy problem
Let \(X\) be a Banach space over \({\mathbb F}\) with norm \(\|\cdot\|\) and \(X^*\) its dual space. Denote by \(B(X)\) the set of all bounded linear operators from \(X\) into itself. For each \(0<T_0\leq \infty\), it is considered the following Cauchy problem \[ ACP_2(A,f,x,y):\quad \begin{cases} u^{\prime\prime}(t)=Au(t)+f(t), \;t\in (0,T_0), \\ u(0)=x, \;u^{\prime}(0)=y, \end{cases} \] where \(x,y\in X\) are given, \(A:D(A)\subset X\rightarrow X\) is a closed linear operator and \(f\) is an \(X\)-valued function defined on a subset of \({\mathbb R}\) containing \((0,T_0)\). The author shows that \(A\) generates a nondegenerate local \(1\)-times (respectively, \(0\)-times) integrated \(C\)-cosine function on \(X\) if and only if \(ACP_2(A,C_g,(\cdot),0,C_x)\) has a unique weak solution in \(C([0,T_0),X)\) (respectively, in \(C^1([0,T_0),X)\)) for all \(x\in X\) and \(g\in L^1_{\text{loc}}([0,T_0),X)\) if and only if \(ACP_2(A,0,0,C_x)\) has a unique weak solution in \(C([0,T_0),X)\) (respectively, in \(C^1([0,T_0),X)\) for all \(x\in X\).
scientific article; zbMATH DE number 5910201