strict negativity of upper right Dini derivative?

Question

I'm trying to prove the global asymptotic stability of an equilibrium of a system of delay differential equations. After adopting the Lyapunov functional approach, I arrive at the upper right Dini derivative along the solution curves: $$ D^{+}V(t)\leq\sum\limits_{k=1}^{n}\xi_k\left\{-|y_k(t)+x_k^{\ast}|+ \sum\limits_{j=1}^{n}c_j|a_{kj}||y_j(t)| +h_k\left( \sum\limits_{j=1}^{n}|a_{kj}|\int\limits_{0}^{\infty}|y_j(t-u)+x_j^{\ast}|f_{kj}(u)du\right)\right\}, $$ where $h_k:\mathbb{R}\mapsto [-1,1]$ is a hyperbolic tangent function, $x_k^{\ast}$, $k=1,\ldots,n$, are real constants, $f_{kj}:\mathbb{R}\mapsto (0,1]$, $k,j=1,\ldots,n$ is a sequence of delay kernels, $c_j,\xi_j$, $j=1,\ldots, n$, are positive real parameters, $a_{kj}\in\mathbb{R}$, and the unknown $y_k$, $k=1,\ldots,n$, satisfies the system of delay differential equations $$ y_k^{\prime}(t)= -\left(y_k(t)+x_k^{\ast}\right)+h_k\left(\sum\limits_{j=1}^{n}a_{kj}\int\limits_{0}^{\infty}\left(y_j(t-u)+x_j^{\ast}\right)f_{kj}(u)du\right)\;,\;\;k=1,\ldots,n, $$ An important condition arising from the theory of M-matrices is that $$ -\xi_j+\sum\limits_{k=1}^{n}c_k|a_{kj}|\xi_k<0\;,\;\;j=1,\ldots,n. $$ In view of all of the above information, I wish to establish that, under certain conditions, $D^{+}V(t)<0$. I have heard and read that Razumikhin-type arguments are an option, but I'm not exactly sure how such arguments may be applied to the present problem.