中立型非线性动力系统周期解在移位$δ_{\pm}$中的存在性
Existence of periodic solutions in shifts $δ_{\pm}$ for neutral nonlinear dynamic systems
摘要：
在这项研究中，我们专注于周期解的存在性中立型非线性动力系统的时滞% \[ x^Δ(t)=A(t)x(t)+Q^Δ\left(t,x\left(δ_{-}(s,t)\right) \right) +G\left(t,x(t),x\left(δ_{-}(s,t)\right) \right) . \]我们利用新的周期性概念的移位算子，这使我们能够扩展的概念的周期性的时间尺度的可加性要求$t\pm T\in\mathbb{T}$的所有$t\in\mathbb{T}$和一个固定的$T>0,$可能不成立。更重要的是，新的概念将很容易处理的时间尺度，不是周期性的传统方式，如; $\overline{q^{\mathbb{Z}}}$和$\cup_{k=1}^{\infty}\left[ 3^{\pm k},2.3^{\pm k}\right] \cup\left\{0\right\} .$因此，我们开发了一个工具，使调查周期解的$q$差分系统。由于我们处理的是系统，为了将方程转化为积分系统，我们借助于齐次Floquet系统的转移矩阵$y^Δ(t)=A(t)y(t)$，然后利用Krasnoselskii的不动点定理得到一个不动点.
英文摘要：
In this study, we focus on the existence of a periodic solution for the neutral nonlinear dynamic systems with delay% \[ x^Δ(t)=A(t)x(t)+Q^Δ\left(t,x\left(δ_{-}(s,t)\right) \right) +G\left(t,x(t),x\left(δ_{-}(s,t)\right) \right) . \] We utilize the new periodicity concept in terms of shifts operators, which allows us to extend the concept of periodicity to time scales where the additivity requirement $t\pm T\in\mathbb{T}$ for all $t\in\mathbb{T}$ and for a fixed $T>0,$ may not hold. More, importantly, the new concept will easily handle time scales that are not periodic in the conventional way such as; $\overline{q^{\mathbb{Z}}}$ and $\cup_{k=1}^{\infty}\left[ 3^{\pm k},2.3^{\pm k}\right] \cup\left\{0\right\} .$ Hence, we develop a tool that enables the investigation of periodic solutions of $q$-difference systems. Since we are dealing with systems, in order to convert our equation to an integral systems, we resort to the transition matrix of the homogeneous Floquet system $y^Δ(t)=A(t)y(t)$ and then make use of Krasnoselskii's fixed point theorem to obtain a fixed point.
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