We study the monotone skew-product semiflow generated by a family of neutral functional differential equations with infinite delay and stable D-operator. The stability properties of D allow us to introduce a new order and to take the neutral family to a family of functional differential equations with infinite delay. Next, we establish the 1-covering property of omega-limit sets under the componentwise separating property and uniform stability. Finally, the obtained results are applied to the study of the long-term behavior of the amount of material within the compartments of a neutral compartmental system with infinite delay. 1. Introduction. After the pioneering work of Hale and Meyer [11], the theory of neutral functional differential equations (NFDE) aroused considerable interest and a fast development ensued. At present a wide collection of theoretical and practical results make up the main body of the theory of NFDEs (see Hale [10], Hale and Verduyn Lunel [12], Kolmanovskii and Myshkis [18], and Salamon [22], among many others). In particular, a substantial number of results for delayed functional differential equations (FDEs) have been generalized for NFDEs solving new and challenging problems in these extensions.In this paper we provide a dynamical theory for nonautonomous monotone NFDEs with infinite delay and autonomous stable D-operator along the lines of the results by Jiang and Zhao [17] and Novo, Obaya, and Sanz [20]. We assume some recurrence properties on the temporal variation of the NFDE. Thus, its solutions induce a skewproduct semiflow with minimal flow on the base. In particular, the uniform almost periodic and almost automorphic cases are included in this formulation. The skewproduct formalism permits the analysis of the dynamical properties of the trajectories using methods of ergodic theory and topological dynamics.Novo et al.[20] study the existence of recurrent solutions of nonautonomous FDEs with infinite delay using the phase space BU ⊂ C((−∞, 0], R m ) of bounded and uniformly continuous functions with the supremum norm. Assuming some technical conditions on the vector field, it is shown that every bounded solution is relatively compact for the compact-open topology, and its omega-limit set admits a flow extension. An alternative method for the study of recurrent solutions of almost periodic FDEs with infinite delay makes use of a fading memory Banach phase space (see Hino, Murakami, and Naiko [13] for an axiomatic definition and main properties). Since this kind of space contains BU and, under natural assumptions, the restriction of the norm

We study monotone skew-product semiflows generated by families of nonautonomous neutral functional differential equations with infinite delay and stable D-operator, when the exponential ordering is considered. Under adequate hypotheses of stability for the order on bounded sets, we show that the omega-limit sets are copies of the base to explain the long-term behavior of the trajectories. The application to the study of the amount of material within the compartments of a neutral compartmental system with infinite delay, shows the improvement with respect to the standard ordering.

We study neutral functional differential equations with stable linear non-autonomous D-operator. The operator of convolution D transforms BU into BU . We show that, if D is stable, then D is invertible and, besides, D and D −1 are uniformly continuous for the compact-open topology on bounded sets. We introduce a new transformed exponential order and, under convenient assumptions, we deduce the 1-covering property of minimal sets. These conclusions are applied to describe the amount of material in a class of compartmental systems extensively studied in the literature.

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