  We prove existence of transition fronts for a large class of
reaction-diffusion equations in one dimension, with inhomogeneous monostable
reactions. We construct these as perturbations of corresponding front-like
solutions to the linearization of the PDE at $u=0$. While a close relationship
of the solutions to the two PDEs has been well known and exploited for KPP
reactions (and our method is an extension of such ideas from [15]), to the best
of our knowledge this is the first time such an approach has been used in the
construction and study of fronts for non-KPP monostable reactions.
