Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 21, pp. 1-21.

Title: Existence of periodic solutions and stability for a nonlinear system of neutral differential equations

Authors: Yang Li (Southwest Jiaotong Univ., Chengdu, China)
         Guiling Chen (Southwest Jiaotong Univ., Chengdu, China)

Abstract: 
In this article, we study the existence and uniqueness of periodic solutions, and stability of the zero solution to the nonlinear neutral system
$$
 \frac{d}{dt}x(t)=A(t)h\big(x(t-\tau_1(t))\big)+\frac{d}{dt}Q\big(t,x(t-\tau_2(t))\big)
 +G\big(t,x(t),x(t-\tau_2(t))\big).
$$
We use integrating factors to transform the neutral differential equation into
an equivalent integral equation. Then we construct appropriate mappings and employ
Krasnoselskii's fixed point theorem to show the existence of a periodic solution.
We also use the contraction mapping principle to show the existence of a unique
periodic solution and the asymptotic stability of the zero solution. Our results generalize the corresponding results in the existing literature. An example is given to illustrate our results.


Submitted December 12, 2023. Published March 04, 2024
Math Subject Classifications: 34K13, 34K20, 34K40.
Key Words: Neutral equation; periodic solution; existence; uniqueness;
           stability; Krasnoselskii's fixed point theorem; contraction mapping principle.