Almost exponential stability and exponential stability of resolvent operator families (Q512193)

A new concept of almost exponential stability of the resolvent operator family \(\{ R(t), \, t\geq 0 \},\) \[ R(t) x = x+ \int_0^t a(t-\tau) A R(\tau) x\,d\tau, \, x\in {\mathcal D}(A),\quad A: {\mathcal D}(A ) \subseteq X\rightarrow X \] is constructed. The suggested concept is stronger than uniform stability and weaker than exponential stability. Using the rescaling technique and the contour integral method, the authors demonstrate that the analytic resolvent is almost exponentially stable or exponentially stable under special conditions applied on the resolvent kernel.