Conditions for finite-dimensionality of a monodromy operator for periodic sytems with aftereffect (Q2386951)

A linear periodic system of differential equations with aftereffect is described by \[ \frac{dx(t)}{dt}=\int_{-\tau}^{0}d_s \eta(t,s)x(t+s),\quad t\in \mathbb{R}^+=[0,+\infty). \tag{1} \] The matrix function \(\eta:(-\infty, +\infty)\times [-r,0]\rightarrow \mathbb{R}^{n\times n}\) is \(\omega\)-periodic with respect to the first argument and Lebesgue-measurable on the set \([0,\omega]\times [-r,0], \eta(\cdot,0)=0, \omega>r>0\). For almost all \(t\in [0,\omega]\), a finite variation \(V(t)=Var_{[-r,0]}\eta(t,\cdot)\) exists and the function \(V\) is integrable on \([0,\omega]\). Under the above conditions for an initial moment \(t_0=0\) and an arbitrary initial function \(\phi\in C([-r,0],\mathbb{R}^n)\), equation (1) has a unique solution, denoted by \(x(t,\phi), t\geq -r\). Denote by \((x_t(\phi)(s)=x(t+s, \phi), s\in [-r,0])\), \(t\geq 0\), the solution of (1) which satisfies for \(t=t_0=0\) the initial value \(x_0=\phi\). By the fundamental theory of FDE [\textit{J. Hale}, Theory of functional differential equations. 2nd ed. Applied Mathematical Sciences. Vol. 3. New York-Heidelberg-Berlin: Springer-Verlag (1977; Zbl 0352.34001)], the monodromy operator, defined by the formula \(U\phi=x_{\omega}(\phi)\), acts in the space \(C([-r,0], \mathbb{R}^n)\) and is completely continuous. In this paper, using the formula for a general solution of a system of differential equations with aftereffect, the authors derive the representation of the monodromy operator as follows \[ (U\phi)(\vartheta)=V(\omega+\vartheta, 0)\phi(0)+\int_{-r}^{-0}d_{\beta} \Bigl(\int_{0}^{\omega+\vartheta}V(\omega+\vartheta, \alpha) \eta(\alpha, \beta-\alpha)d\alpha\Bigr)\phi(\beta),\quad \vartheta\in [-r,0]. \tag{2} \] Here, the matrix function \(V(\cdot,\cdot)\) is locally absolutely continuous with respect to the first argument \(t\) on the half interval \([\alpha, +\infty)\) for every fixed value of the second argument \(\alpha\in [0, +\infty)\), has a finite variation with respect to the second argument \(\alpha\) on \([0, t]\) for every fixed value of \(t\in (0, +\infty)\), satisfies \[ \frac{\partial V(t, \alpha)}{\partial t}=\int_{-r}^{0}d_{\tau} \eta(t, \tau)V(t+\tau, \alpha), V(t,\alpha)=I_n-\int_{\alpha}^{t}V(t, \tau)\eta(\tau, \alpha-\tau)d\tau, 0\leq \alpha\leq t<\infty, \] and \(V(t, \alpha)=0\) for \(\alpha-r\leq t<\alpha, V(\alpha, \alpha)=I_n, \alpha \in \mathbb{R}^+\). Here, \(I_n\) is the unit matrix of dimension \(n\times n\). Necessary and sufficient conditions are derived for the monodromy operator acting in the space \(C([-r, 0], \mathbb{R}^n)\) to be of a finite dimension.