Existence of periodic solutions for first-order totally nonlinear neutral differential equations with variable delay

We use a modification of Krasnoselskii's fixed point theorem due to Burton (see [Liapunov functionals, fixed points and stability by Krasnoselskii's theorem, Nonlinear Stud. 9 (2002), 181-190], Theorem 3) to show that the totally nonlinear neutral differential equation with variable delay x'(t) = -a(t)h(x(t)) + c(t)x' (t - g(t))Q' (x (t - g(t))) G (t, x(t), x(t - g(t))), has a periodic solution. We invert this equation to construct a fixed point mapping expressed as a sum of two mappings such that one is compact and the other is a large contraction. We show that the mapping fits very nicely for applying the modification of Krasnoselskii's theorem so that periodic solutions exist.