Traveling waves for non-local delayed diffusion equations via auxiliary equations

In this paper, we study the existence of traveling wave solutions for a class of delayed non-local reaction–diffusion equations without quasi-monotonicity. The approach is based on the construction of two associated auxiliary reaction–diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space by using the traveling wavefronts of the auxiliary equations. Under monostable assumption, by using the Schauderʹs fixed point theorem, we then show that there exists a constant c*>0 such that for each c>c*, the equation under consideration admits a traveling wavefront solution with speed c, which is not necessary to be monotonic.