Geodesic
On the Dichotomy of Linear Autonomous Systems of Functional-Differential Equations
Matematičeskie zametki, Tome 95 (2014) no. 1, pp. 129-135

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We study the exponential dichotomy of solutions to the Cauchy problem for a subclass of linear autonomous systems of functional-differential equations of neutral type by reducing this problem to the same problem for a difference equation of the form $x_{n}=\Gamma x_{n-1}$ with a compact operator $\Gamma$, in Banach space. The spectrum of $\Gamma$ is described. The dichotomy of the solutions of the Cauchy problem is a consequence of the dichotomy of the spectrum. In a particular case, the result obtained provides a criterion for the dichotomy for a linear autonomous system of functional-differential equations of retarded type of general form.
Keywords: dichotomy of a linear autonomous system, Cauchy problem, system of functional-differential equations, functional-differential equation of retarded (neutral) type, Banach space, spectrum. 
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     author = {R. K. Romanovskii and E. M. Nazaruk},
     title = {On the {Dichotomy} of {Linear} {Autonomous} {Systems} of {Functional-Differential} {Equations}},
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R. K. Romanovskii; E. M. Nazaruk. On the Dichotomy of Linear Autonomous Systems of Functional-Differential Equations. Matematičeskie zametki, Tome 95 (2014) no. 1, pp. 129-135. http://geodesic.mathdoc.fr/item/MZM_2014_95_1_a10/