Matrix linearization of functional-differential equations of point type and existence and uniqueness of periodic solutions (Q1734709)

The paper is a continuation and generalization of the authors' earlier joint work which is given as a reference [Differ. Equ. 51, No. 12, 1541--1555 (2015; Zbl 1336.34098); translation from Differ. Uravn. 51, No. 12, 1565--1579 (2015)]. Consider the functional differential equation of the form \[\dot{x}(t) = g(t,x(t+\tau_1),\dots,x(t+\tau_s)), \quad t\in \mathbb{R}, \tag{1}\] where the function $ g(.) $ belongs to the space $C^{(r)}(\mathbb{R}\times \mathbb{R}^{n\times s};\mathbb{R}^n )$, $r\in\mathbb{N}_0 \equiv\mathbb{N}\cup\{0\} $. \par The main differences and contribution to the previous work is that the authors consider a more general case of separation of the linearized part and rearrange the functional differential equation in the matrix representation \[\dot{x}(t)= \sum_{j=1}^s A_jx(t+\tau_j+f(t,x(t+\tau_1),\dots,x(t+\tau_s)), \quad t\in\mathbb{R}\tag{2}\] where $ A_j $ is the real $ n\times n $ matrix, $ \tau_j \in [0,2\pi )$, $j\in \{1,2,\dots,s\} $. The authors study the existence of $\omega$-periodic solutions using a complicated matrix linearization.