We study the existence of traveling waves for nonlocal KPP–Fisher equations with diffusive delay \(\begin{aligned} \frac{\partial u(x,t)}{\partial t}=\frac{\partial ^2u (x,t-\tau _1)}{\partial x^2}+u(x,t)\left( 1-\int \nolimits _{-\infty }^{\infty }\textrm{d}\mu (y)u(x-y,t-\tau _2)\right) , \end{aligned}\) where \(\mu \) is a nondecreasing function with bounded variation. The existence of traveling waves is proved using the method of monotone iteration developed recently for general reaction–diffusion equations with delays in both reaction and diffusion terms. We extend this iteration method to start with very rough upper and lower solutions that we call supper and subsolutions. We show that for small delays \(\tau _1\) and \(\tau _2\) traveling waves connecting 0 and 1 exist. The obtained results appear to be new.
