Is there an exponential stability version of the Lyapunov-Razumikhin Theorem for retarded functional differential equations? i.e. analogous to ODEs, given a FDE $\dot{x}(t)=f(x_t)$ does the following imply global exponential stability?
For a positive definite $V:\mathbb{R}^n\to\mathbb{R}^+$ with $c_1||x||^p\leq V(x)\leq c_2||x||^p$ it holds, $$\dot{V}(x_t) \leq -\alpha V(x(t)), \quad\text{if } V(x(t+\theta))\leq \beta V(x(t)) \quad\forall \theta \in [-\tau,0], \quad \text{where }\alpha>0, \beta>1.$$ Where $x(t)\in\mathbb{R}^n$ and $x_t(\cdot)= x(t+\cdot)\in \mathcal{C}\left([-\tau,0]\right)$.
In case someone is interested in it: The answer to my above question is yes and a proof can be found here. Furthermore, the rate of decay is at least $$\gamma = \min\left(\alpha, \log( \beta)/\tau\right)/p.$$
However, the proof is quite a lot more complicated as for ODEs or Krasovskii functionals.