Existence of Periodic Solutions for Totally Nonlinear Neutral Differential Equations With Variable Delay - Ardjouni, Abdelouaheb; Djoudi, Ahcéne (Sarajevo Journal of Mathematics, 2024)
We use a modification of Krasnoselskii's fixed point theorem introduced by T. A. Burton (see $\left[ 1\right] $ Theorem $3$) to show that the totally nonlinear neutral differential equation with variable delay \begin{multline*} x^{\prime }(t)=-a(t)x^{3}(t)+c(t)x^{\prime }(t-g(t))Q^{\prime }\left( x\left( t-g(t)\right) \right)\\ +G\left( t,x^{3}(t),x^{3}(t-g(t))\right) , \end{multline*} has a periodic solution. We invert this equation to construct a sum of a compact map and a large contraction which is suitable for applying the modification of Krasnoselskii's theorem. The results of $\left[ 5\right]$ are generalized. Finally, an example is given to illustrate our result. 2000 Mathematics Subject Classification. 34K20, 45J05, 45D05