Dynamics and oscillations of models for differential equations with delays

By developing new efficient techniques and using an appropriate fixed point theorem, we derive several new sufficient conditions for the pseudo almost periodic solutions with double measure for some system of differential equations with delays. As an application, we consider certain models for neural networks with delays.


Introduction
Existence of periodic, almost periodic and pseudo almost periodic solutions of differential equations has great significance and is therefore an important problem.Such dynamics can be found in electronic circuits and many other physical and biological systems (see [3,6,9,18,21,19,20,23,26]).Ezzinbi et al. [5] introduced a new and powerful measure-theoretic method to resolve this open problem.Since then, this method has been used for various classes of evolution equations as well as stochastic differential equations and has become very popular.
The notion of measure pseudo almost periodicity was first introduced by Blot et al. [5] (see also [1,8,12,13,15,16,17,27]).Obviously, these new results generalize the earlier work od Diagana [10].Recently, Diagana et al. [11] have introduced the notion of double measure pseudo almost periodicity as a generalization of the measure pseudo almost periodicity.We note that this generalized concept coincides with the latter one (take µ ≡ ν).
In this paper, by applying an appropriate fixed point theorem, we derive some conditions which ensure the existence, the exponential stability, and the uniqueness of (µ, ν)−pap solutions of the following models with delays: a ij (t)g j (t, x j (t − τ ij )) x i (s) = ϕ i (s), s ∈ (−θ, 0], i ∈ {1, ..., n}, where functions c i , I i , d ij , a ij , b ijl : R → R and f j , g j , h j : R × R → R, i, j, l ∈ {1, ..., n} are continuous and τ ij , σ ij , and ν ij are positive constants.The paper is organized as follows: in Section 2 we collect key definitions, examples, and basic results.In Section 3 we discuss the existence, the stability and the uniqueness of double measure pseudo almost periodic solutions of system (1.1).Finally, in Section 4 we present an application which illustrates the effectiveness of our results.

Preliminaries
Definition 2.1.(see [5]) Let f be a continuous function on R with values in R n .Then f is said to be almost periodic, denoted by f ∈ AP(R, R n ), if for all ε > 0, there exists a number l(ε) > 0 such that every interval I of length l(ε) contains a point τ ∈ R with the property that The space AP(R, R n ) equipped with the norm is then a Banach space.Let B be the Lebesque σ-field on R and define a collection M of measures on B M = {µ is a positive measure on B; µ(R) = +∞, and µ([s, t]) < ∞, for all s, t ∈ R, s ≤ t}.
Let X be a Banach space and denote by BC(R, X) the Banach space of bounded continuous functions from R to X, equipped with the supremum norm f ∞ = sup t∈R f (t) .In order to be able to introduce double measure pseudo almost periodic functions, we need the following ergodic spaces (2.1) We also introduce the following notation PAP(R, R n , µ) := PAP(R, R n , µ, µ).

✷
If measures µ and ν are equal, then hypothesis (M.2) is satisfied and we can deduce the following corollary.
Then the function is double measure pseudo almost periodic for all 1 ≤ i ≤ n, by Lemmas 2.2, 3.2 and 3.4.Hence for all 1 ≤ i ≤ n, we have We have to prove that Γ i • F 1 i ∈ AP(R, R), i ∈ {1, 2, 3, ..., n}.To this end, note that Using Fubini's Theorem, we get for all z > 0. Since F 2 i ∈ E(R, R, µ, ν), it follows by Lemma 2.2 and the dominated convergence theorem, that Γ i • F 2 i ∈ E(R, R, µ, ν), for all i ∈ {1, 2, 3, ..., n}.We can thus conclude that This completes the proof of Lemma 3.5.✷ Theorem 3.6.Suppose that conditions (M.1)-(M.6)hold.Then system (1.1) admits a unique (µ, ν)−pap solution in E, where Proof.We have and Then for every ϕ ∈ E, we obtain the following where Therefore Γ • ϕ ∈ E. Next, for all φ, ψ ∈ E, we get the following Note that since q 0 < 1, Γ is a contraction and possesses a unique fixed point z which is a (µ, ν)−pap solution of system (1.1) in the region E.This completes the proof of Theorem 3.6.✷ If the two measures µ and ν are equal, then according to the proof of Theorem 3.6, the following corollary can be deduced.

An application to neural networks
Neural networks have attracted a lot of attention in recent years and especially the special case of the socalled high-order Hopfield neural networks (HOHNNs) which have been intensively investigated by many scholars in recent years, because of their stronger approximation characteristics, larger storage capacity, faster convergence speed, and higher fault tolerance than low-order Hopfeld neural networks.Many excellent results about their dynamic characteristics have been obtained in e.g.[2,3,4,7,14,22,24,25].
Clearly, the study of the oscillations and dynamics of such models is an exciting new topic.
Using the results from of this paper, we prove the existence, the exponential stability, and the uniqueness of (µ, ν)−pap solutions of the following models of high-order Hopfield neural networks (HOHNNs) with delays: where i ∈ {1, ..., n}.n number of neurons in neural network x i (t) i th neuron at time t f j , g j , h j activation function of j th neuron d ij (t), a ij (t), b ijl (t) functions connection weights I i (t) external inputs at time t c i (t) > 0 rate of i th neuron τ ij ≥ 0, σ ij ≥ 0, ν ij ≥ 0 transmission delays The initial conditions associated with system (4.1) are of the form x i (s) = ϕ i (s), s ∈ (−θ, 0], i = 1, 2, ..., n, In our paper we have generalized the previous results by using the notion of double measure and working with two-variable functions.