summary: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.
展开▼
机译：摘要：由于伯顿，我们对Krasnoselskii不动点定理进行了修改（请参阅[Liapunov函数，不动点和稳定性，由Krasnoselskii定理，Nonlinear Stud。9（2002），181--190]，定理3表示）可变延迟的非线性中立微分方程\ begin {equation *} x'（t）= -a（t）h（x（t））+ c（t）x'（tg（t））Q'（x（tg （t）））+ G（t，x（t），x（tg（t））），\ end {equation *}具有周期解。我们将这个方程式反过来构造一个定点映射，表示为两个映射之和，这样一个映射很紧凑，而另一个则是大收缩。我们表明，映射非常适合于应用Krasnoselskii定理的修改，从而存在周期解。
展开▼
