this is my first question here, hope I am in the right place :)
Recently I have been looking at the proof of theorem 4.2 on Razumikhin stability for RFDEs in the book by Jack Hale and Lunel Verduyn: Introduction to functional-differential equations, Applied Mathematical Sciences, 99. Springer-Verlag (1993).
However, I found the arguments in the proof quite elusive. Anyone out there who can provide me with a more "intuitive" insight on why condition (4.4) is needed to complete the proof?
EDIT - maybe I can make the question more specific.
Theorem 4.2 in the book by Hale basically establishes some conditions under which the origin of the state space of a system in the form
$ \dot{x} = f(t, x_t(t)), \quad x_t(0)=\phi(\theta) \qquad \theta \in [-r, 0]$
is stable, where $x_t$ is the retarded state vector, which is actually a function of the delay
$x_t(\theta) := x(t+\theta)$
Generally speaking, Razumikhin-type theorems are sufficient conditions which link some given properties of a Lyapunov-like function to the stability of the trajectories of the considered system. These methods can be used as a simpler alternative to the Lyapunov-Krasovskii results, which are a direct extension of the Lyapunov stability theorems based on functionals.
In particular, the Razumikhin condition for (uniform) stability of $x=0$ is quite simple to prove (it is a straightforward extension of the Lyapunov-Krasovskii stability theorem), while the conditions for asymptotic stability are quite elusive (at least for me). In particular, the so-called "Razumikhin condition" requires that
$\dot{V}(t,\phi(0)) \le -w(|\phi(0)|) if V(t+\theta, \phi(\theta)) \le V(t,\phi(0))$
where $w, p$ are nonnegative, nondecreasing functions such that $w(s) \neq 0$ for $s \neq 0$, $w(0) = 0$ and $p(s)>s$ for $s>0$. What is puzzling me is the need for the last condition on $p(s)$.