## Abdelouaheb Ardjouni, Ahcène Djoudi

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

Comment.Math.Univ.Carolin. 55,2 (2014) 215-225.**Abstract:**We use a modification of Krasnoselskii's fixed point theorem due to Burton (see [{\it Liapunov functionals, fixed points and stability by Krasnoselskii's theorem\/}, Nonlinear Stud. {\bf 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.

**Keywords:** periodic solution; nonlinear neutral differential equation; large contraction; integral equation

**AMS Subject Classification:** 34K20 45J05 45D05

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