DICHOTOMY AND POSITIVITY OF NEUTRAL EQUATIONS WITH NONAUTONOMOUS PAST

Consider the linear partial neutral functional differential equations with nonautonomous past of the form partial derivative/partial derivative tF(u(t, .)) = BFu(t, .) + Phi u(t, .), t >= 0, partial derivative/partial derivative tu(t, s) = partial derivative/partial derivative su(t, s) + A(s)u(t, s), t >= 0 >= s, where the function u(. , .) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Phi. 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 Phi. we prove that the above-mentioned solution semigroup is positive and then show a sufficient condition for the exponential stability of this solution semigroup.