Neutral Functional Differential Equations with Applications to Compartmental Systems

Abstract

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

Translation-invariant monotone systems II: Almost periodic/automorphic case

“…Novo, Obaya and Sanz [35] have shown that the omega limit set is a uniformly stable minimal set which admits a fiber distal flow extension if its semiorbit is uniformly stable, which not only extends the classical extension result in Shen and Yi [39], but also drops the distal assumption in the abstract result in [29]. Later, together with their topological result with this Lyapunov integer-valued function method, they have investigated a series of functional differential equations with infinite delay and obtained one covering property of the base space ( [35,33]), some of which are even new in autonomous/periodic cases. The idea in [38] is also along these lines.…”

Human Brain Networks: Spiking Neuron Models, Multistability, Synchronization, Thermodynamics, Maximum Entropy Production, and Anesthetic Cascade Mechanisms

“…In this section, we develop a stochastic field theory as in [84] for capturing neural activity in order to analyze system multistability. In the next section, we extend the results of this section to additionally address time delay functional models [85] in order to account for time delay and memory effects in inhibitory and excitatory networks.…”

Asymptotic behavior of solutions of nonautonomous neutral dynamical systems

“…As explained before, this paper provides a contribution to the dynamical theory of monotone recurrent skew-product semiflows. We consider a monotone structure on Ω × BC determined by an exponential ordering and we enhance the theory started in [15], [18] and [17], where the 1-covering property of omega-limit sets of relatively compact trajectories was proved.…”

“…We omit the proof of the following result, which can be easily adapted to this case from Theorem 2.5 of Muñoz-Villarragut [27]. As stated before, given a continuous function x ∈ C(R, R m ), x t (•) denotes the continuous function…”

“…Later, motivated by the applicability to the study of the long-term dynamics of compartmental systems, the paper by Muñoz-Villarragut et al [17] is the starting point of an important effort to extend the previous results to nonautonomous neutral functional differential equations (NFDEs for short) with infinite delay.…”