Geodesic
Existence of periodic solutions for first-order totally nonlinear neutral differential equations with variable delay
Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 2, pp. 215-225.

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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 \begin{equation*} 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))), \end{equation*} 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.
Classification : 34K13, 34K20, 34K40, 45D05, 45J05
Keywords: periodic solution; nonlinear neutral differential equation; large contraction; integral equation 
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Ardjouni, Abdelouaheb; Djoudi, Ahcène. Existence of periodic solutions for first-order totally nonlinear neutral differential equations with variable delay. Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 2, pp. 215-225. http://geodesic.mathdoc.fr/item/CMUC_2014__55_2_a7/