Global continuation of monotone wavefronts

In this paper, we establish efficient existence criteria for monotone traveling fronts u = phi(nu center dot x + ct), phi(-infinity) = 0, phi(+infinity) = kappa of the monostable (and, in general, nonquasi-monotone) delayed reaction-diffusion equations u(t)(t, x) - delta u(t, x) = f(u(t, x), u(t - h, x)). The function f is of class C-1,C-gamma and it is assumed to satisfy f(0, 0) = f(kappa, kappa) = 0 together with other monostability restrictions. Our theory covers several important cases including Mackey-Glass-type diffusive equations and Kolmogorov-Petrovskii-Piskunov-Fisher-type equations. The proofs are based on a variant of the Hale-Lin functional-analytic approach to heteroclinic solutions where Lyapunov-Schmidt reduction is realized in a 'mobile' weighted space of C-2-smooth functions. This method requires a detailed analysis of a family of associated linear differential Fredholm operators: at this stage, the discrete Lyapunov functionals by Mallet-Paret and Sell are essential to the method.