  We study the existence of traveling waves of reaction-diffusion systems with
delays in both diffusion and reaction terms of the form $\partial
u(x,t)/\partial t = \Delta u(x,t-\tau_1)+f(u(x,t),u(x,t-\tau_2))$, where
$\tau_1,\tau_2$ are positive constants. We extend the monotone iteration method
to systems that satisfy typical monotone conditions by thoroughly studying the
sign of the Green function associated with a linear functional differential
equation. Namely, we show that for small positive $r$ the functional equation
$x''(t)-ax'(t+r)-bx(t+r)=f(t)$, where $a\not=0, b>0$ has a unique bounded
solution for each given bounded and continuous $f(t)$. Moreover, if $r>0$ is
sufficiently small, $f(t)\ge 0$ for $t\in {\mathbb R}$, then the unique bounded
solution $x_f(t)\le 0$ for all $t\in {\mathbb R}$. In the framework of the
monotone iteration method that is developed based on this result, upper and
lower solutions are found for Fisher-KPP and Belousov-Zhabotinski equations to
show that traveling waves exist for these equations when delays are small in
both diffusion and reaction terms. The obtained results appear to be new.
