Consider the linear neutral functional differential equation of the formwhere the function u(·) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator we prove that the solution semigroup for this equation is hyperbolic provided that the differential operator B generates a hyperbolic semigroup on X.
Consider the linear partial neutral functional differential equations with nonautonomous past of the formwhere the function u(· , ·) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Φ we prove that the solution semigroup for this system of equations is hyperbolic (or admits an exponential dichotomy) provided that the backward evolution family U = (U (t, s)) t≤s≤0 generated by A(s) is uniformly exponentially stable and the operator B generates a hyperbolic semigroup (e tB ) t≥0 on X. Furthermore, under the positivity conditions on (e tB ) t≥0 , U, F and Φ we prove that the above-mentioned solution semigroup is positive and then show a sufficient condition for the exponential stability of this solution semigroup.
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