Uniform stability of \((a,k)\)-regularized families (Q2856523)

Let \(a(t)\in L_{loc}^1([0,+\infty))\), \(k(t)\in C([0,+\infty))\cap L^1_{loc}\) be of subexponential growth and both functions be 1-regular. By \(\hat{a}\) and \(\hat{k}\) we denote a Laplace transform. We say that \(k\) satisfies the (H)-condition if there exists \(M>0\) such that \(|\rho\hat{k}(i\rho)|^{-1}\leq M\) for all \(|\rho|\geq1\) (here, \(\hat{k}(i\rho)=\lim_{\lambda\to i\rho}\hat{k}(\lambda)\)). Let \(\{S(t)\}_{t\geq0}\subset B(H)\) be an \((a,k)\)-regularized family in a Hilbert space \(H\) generated by closed operator \(A\). Suppose that \(k(t)\) satisfies the (H)-condition, \(\hat{a}^{-1}(\lambda)\in\rho(A)\) (\(\mathrm{Re }\lambda\geq0\), \(\lambda\neq 0\)), NEWLINE\[NEWLINE\sup\left\|\frac{\hat{k}(\lambda)}{\hat{a}(\lambda)}\left(\hat{a}^{-1}(\lambda)-A\right)^{-1}\right\|\leq\infty,NEWLINE\]NEWLINE and, for all \(x\in H\), the limit NEWLINE\[NEWLINE\lim_{\lambda\to0}\frac{\hat{k}(\lambda)}{\hat{a}(\lambda)}\left(\hat{a}^{-1}(\lambda)-A\right)^{-1}xNEWLINE\]NEWLINE exists. Then \(\{S(t)\}_{t\geq0}\) is uniformly stable, i.e., \(\lim_{t\to+\infty}\|S(t)\|=0\).