Degenerate strongly continuous operator groups
The degenerate Cauchy problem for the linear differential-operator equation \[ Lu'(t) =M u(t), \] \(\ker L\not=\{0\}\) on the whole \(\mathbb{R}\) is considered. Here \(L\in {\mathcal L}(U,F)\) is bounded linear operator and \(M\in {\mathcal Cl}(U,F)\) is closed linear one in Banach spaces \(U, F\). It is proved that if \(M\) is \((L,p)\)-biradial: \[ \begin{aligned}\max\{\|R_{(\lambda,p)}^n\|, \|L_{(\lambda,p)}^n\|\}\leq \frac K{{\prod}_{k=0}^{p}(|\lambda_k|-a)^n}, \quad \lambda=(\lambda_1,...\lambda_p), \lambda_k\in \mathbb{R},|\lambda_k|>a, \lambda_k\in\rho^L(M),\end{aligned} \] \[ \begin{aligned} R_{(\lambda,p)}:={\prod}_{k=0}^{p}(\lambda_k L-M)^{-1}L, \quad L_{(\lambda,p)}:={\prod}_{k=0}^{p}L(\lambda_k L-M)^{-1}),\end{aligned} \] then for the Cauchy problem there exists an exponentially bounded strongly continuous solving group on a subspace \(\tilde{U}\) of \(U\). Under some additional estimates the decomposition of \(\tilde{U}=U\) is obtained. The results extends well known results for the equation \( u'(t) =A u(t)\) and continue previous ones of author.
scientific article; zbMATH DE number 1541589