Abstract
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.
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1 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, 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, 15–17, 27]). Obviously, these new results generalize the earlier work of 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 \(\mu\equiv\nu\)).
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 \((\mu,\nu)\)-pap solutions of the following models with delays:
where functions
are continuous and \(\tau_{ij}\), \(\sigma_{ij}\), and \(\nu_{ij}\) are positive constants.
The paper is organized as follows: in Sect. 2 we collect key definitions, examples, and basic results. In Sect. 3 we discuss the existence, the stability, and the uniqueness of double measure pseudo almost periodic solutions of system (1.1). Finally, in Sect. 4 we present an application which illustrates the effectiveness of our results.
2 Preliminaries
Definition 2.1
(see [5])
Let f be a continuous function on \(\mathbb{R}\) with values in \(\mathbb{R}^{n} \). Then f is said to be almost periodic, denoted by \(f \in\mathcal{AP}(\mathbb {R},\mathbb{R}^{n})\), if for all \(\varepsilon>0\), there exists a number \(l(\varepsilon)>0\) such that every interval I of length \(l(\varepsilon)\) contains a point \(\tau\in\mathbb{R} \) with the property that
The space \(\mathcal{AP}(\mathbb{R},\mathbb{R}^{n})\) equipped with the norm
is then a Banach space. Let \(\mathcal{B}\) be the Lebesque σ-field on \(\mathbb{R}\) and define a collection \(\mathcal{M}\) of measures on \(\mathcal{B}\)
Let X be a Banach space, and denote by \(\mathcal{BC}(\mathbb{R},X)\) the Banach space of bounded continuous functions from \(\mathbb{R}\) to X, equipped with the supremum norm \(\| f \|_{\infty} = \sup_{t \in\mathbb{R}} \| f(t) \|\). In order to be able to introduce double measure pseudo almost periodic functions, we need the following ergodic spaces:
and
Definition 2.2
(see [11])
If \(\mu,\nu\in\mathcal{M}\), then \(f \in \mathcal{BC}(\mathbb{R},\mathbb{R}^{n})\) is said to be \((\mu,\nu )\)-pseudo almost periodic, abbreviated as \((\mu,\nu)\)-pap, denoted by \(f\in\mathcal{PAP} (\mathbb{R},\mathbb{R}^{n}, \mu,\nu)\), if there exists a decomposition
We also introduce the following notation \(\mathcal{PAP} (\mathbb{R},\mathbb{R}^{n}, \mu):=\mathcal{PAP} (\mathbb{R},\mathbb{R}^{n}, \mu,\mu)\).
Definition 2.3
(see [11])
If \(\mu, \nu\in\mathcal{M}\) and \(f(t,u):\mathbb{R} \times\mathbb{R} \rightarrow\mathbb{R}^{n}\) is continuous, then \(f(t,u)\) is said to be \((\mu, \nu)\)-pseudo almost periodic int, uniformly with respect tou, abbreviated as \((\mu, \nu)\)-papu, denoted by \(f\in\mathcal{PAPU}(\mathbb{R}\times\mathbb{R},\mathbb {R}^{n},\mu,\nu)\), if
Example 2.1
Let \(\mu\in\mathcal{M}\) and
Then \(G\in\mathcal{PAPU}(\mathbb{R}\times\mathbb{R},\mathbb {R},\mu)\).
We shall need the following two conditions:
- (M.1):
For every measure \(\mu\in\mathcal{M}\) and every \(\tau\in\mathbb{R}\), there exist \(\beta>0 \) and a bounded interval I such that, for every \(A \in\mathcal{B}\),
$$A \cap I = \emptyset \quad\implies\quad\mu_{\tau}(A):= \mu \bigl(\{a+\tau: a \in A \} \bigr) \leq\beta\mu(A). $$- (M.2):
Measures \(\mu, \nu\in\mathcal{M} \) satisfy the following condition:
$$\lim\sup_{r \rightarrow\infty} \frac{\mu([-r,r])}{\nu([-r,r])} < \infty. $$
Lemma 2.2
(see [11])
Let\(\mu, \nu\in\mathcal{M} \)and suppose that conditions(M.1)and(M.2)hold. Then
decomposition (2.1) above is unique;
\(( \mathcal{PAP} ( \mathbb{R}, \mathbb{R}^{n} , \mu, \nu), \| \cdot \|_{\infty}) \)is a Banach space; and
\(\mathcal{PAP}(\mathbb{R},\mathbb{R}^{n},\mu,\nu)\)is translation invariant.
3 Double measure pseudo almost periodic solutions
We introduce the following notations:
and the following conditions:
- (M.3):
For all \(1 \leq i,j,l \leq n\),
$$\{d_{ij}, a_{ij}, b_{ijl}, I_{i}\} \subset\mathcal{PAP} (\mathbb {R},\mathbb{R},\mu,\nu) . $$- (M.4):
For all \(i\in\{1,2,\ldots, n\}\),
$$\bigl[t\mapsto c_{i}(t) \bigr]\in \mathcal{AP}(\mathbb{R},\mathbb{R}) \quad\mbox{and}\quad \inf_{t\in\mathbb{R}} \bigl\{ c_{i}(t) \bigr\} =c_{i}^{\ast}>0. $$- (M.5):
For all \(p>1\) and \(1\leq j \leq n\),
$$f_{j},g_{j}, h_{j} \in\mathcal{PAP} (\mathbb{R} \times\mathbb{R},\mathbb {R},\mu,\nu) $$and there exist positive continuous functions
$$L_{j}^{f},L_{j}^{g},L_{j}^{h} \in L^{p}(\mathbb{R},d\mu)\cap L^{p}(\mathbb{R},dx) $$such that, for all \(t, u, v \in\mathbb{R}\),
$$\begin{gathered} \bigl\vert f_{j}(t,u)-f_{j}(t,v) \bigr\vert < L_{j}^{f}(t) \vert u-v \vert , \\ \bigl\vert g_{j}(t,u)-g_{j}(t,v) \bigr\vert < L_{j}^{g}(t) \vert u-v \vert , \\ \bigl\vert h_{j}(t,u)-h_{j}(t,v) \bigr\vert < L_{j}^{h}(t) \vert u-v \vert .\end{gathered} $$
In addition, we also assume that for \(1\leq j\leq n\):
- (M.6):
- $$q_{0} := \max_{ i\in\{1,2,\ldots,n\} } \biggl\{ \frac{\sum_{j=1}^{n} [ \bar{d}_{ij} \Vert L_{j}^{f} \Vert _{p} + \bar{a}_{ij} \Vert L_{j}^{g} \Vert _{p} + \sum_{l=1}^{n} \bar{b}_{ijl} ( \Vert L_{j}^{h} \Vert _{p} \Vert h_{l} \Vert _{\infty} + \Vert L_{l}^{h} \Vert _{p} \Vert h_{j} \Vert _{\infty}) ]}{(qc^{*}_{i})^{\frac{1}{q}} } \biggr\} < 1. $$
Next, define
Remark 3.1
If \(q_{0}<1\), then \(p_{0}<1\).
Lemma 3.2
Suppose that measures\(\mu,\nu\in\mathcal{M}\)satisfy the following requirements:
\(p>1\)and condition(M.2)holds;
\(\varLambda\in\mathcal{C}(\mathbb{R}\times\mathbb{R},\mathbb {R})\)is a Lipschitz function such that\(L^{\varLambda}\in L^{p}(\mathbb{R},d\mu)\); and
\(y \in\mathcal{PAP} (\mathbb{R},\mathbb{R},\mu,\nu)\).
Then\([s\mapsto\varLambda(s,y(s-\theta))]\in\mathcal{PAP} (\mathbb {R},\mathbb{R},\mu,\nu)\), where\(\theta\in\mathbb{R}\).
Proof
Since \(y \in\mathcal{PAP} (\mathbb{R},\mathbb{R},\mu,\nu)\), it follows that
Let
where
Applying [14], we can conclude that \(\varPsi_{1}\in\mathcal{AP}(\mathbb{R},\mathbb{R})\).
Next, we prove that \(\varPsi_{2} \in \mathcal{E}(\mathbb{R},\mathbb{R},\mu,\nu)\). Let \(z>0\), then we have
Since condition (M.2) holds and \(y_{2}\in \mathcal{E}(\mathbb{R},\mathbb{R},\mu,\nu)\), we get
Therefore
This completes the proof of Lemma 3.2. □
If measures μ and ν are equal, then hypothesis (M.2) is satisfied and we can deduce the following corollary.
Corollary 3.3
Suppose that measure\(\mu\in \mathcal{M}\)satisfies the following conditions:
\(p>1\);
\(\varLambda\in\mathcal{C}(\mathbb{R}\times\mathbb{R},\mathbb {R})\)is a Lipschitz function such that\(L^{{\varLambda}}\in L^{p}(\mathbb{R},d\mu)\); and
\(y \in\mathcal{PAP} (\mathbb{R},\mathbb{R},\mu)\).
Then\([s\mapsto\varLambda(s,y(s-\theta))]\in\mathcal{PAP} (\mathbb {R},\mathbb{R},\mu)\), where\(\theta\in\mathbb{R}\).
Lemma 3.4
Let \(\mu,\nu\in\mathcal{M}\) and suppose that
Then
Proof
Since \(y,z\in\mathcal{PAP}(\mathbb{R}, \mathbb{R},\mu,\nu)\), it follows that
Then
We shall show that \(y_{1} z_{1} \in\mathcal{AP}( \mathbb{R} , \mathbb{R} )\). Letting \(\varphi_{0}\in\mathcal{AP}( \mathbb{R} , \mathbb{R} )\), we see that
Then \(\varphi_{0}^{2}\in\mathcal{AP}( \mathbb{R} , \mathbb{R} ) \), so it follows that
since
Note that
so we can conclude that indeed \(y_{1} z_{1} \in\mathcal{AP}( \mathbb{R} , \mathbb{R} ) \).
Next, we shall prove that
Indeed, for \(z>0\), we have
Since \(y_{2},z_{2}\in\mathcal{E} ( \mathbb{R} , \mathbb{R} , \mu,\nu ) \), this completes the proof of Lemma 3.4. □
Next, we define the nonlinear operator Γ as follows: for any \(\varphi= ( \varphi_{1},\ldots, \varphi_{n} ) \in \mathcal{PAP}(\mathbb{R}, \mathbb{R}^{n} , \mu,\nu)\),
and
Lemma 3.5
Suppose that conditions(M.1)–(M.6)hold. ThenΓmaps\(\mathcal{PAP}(\mathbb{R}, \mathbb{R}^{n} , \mu,\nu)\)into itself.
Proof
Let \(\varphi= ( \varphi_{1},\ldots, \varphi_{n} ) \in\mathcal{PAP}(\mathbb{R}, \mathbb{R}^{n} , \mu,\nu)\). Then the function
is double measure pseudo almost periodic for all \(1 \leq i \leq n\), by Lemmas 2.2, 3.2, and 3.4.
Hence, for all \(1 \leq i \leq n\), we have
Therefore
We have to prove that \(\varGamma_{i}\circ F_{i}^{1} \in\mathcal{AP} ( \mathbb{R} , \mathbb{R} ) \), \(i\in\{1,2,3,\ldots,n\}\). To this end, note that
Therefore \(\varGamma_{i} \circ F_{i}^{1} \in\mathcal{AP} ( \mathbb {R} , \mathbb{R}^{n} )\), \(i\in\{1,2,3,\ldots,n\}\).
On the other hand, we can prove that \(\varGamma_{i} \circ F_{i}^{2} \in \mathcal{E} ( \mathbb{R}, \mathbb{R} , \mu,\nu)\) for \(i\in\{1,2,3,\ldots,n\}\). To this end, note that
Using Fubini’s theorem, we get
for all \(z>0\). Since \(F_{i}^{2} \in\mathcal{E} ( \mathbb{R}, \mathbb{R} , \mu,\nu) \), it follows by Lemma 2.2 and the dominated convergence theorem that
We can thus conclude that
hence
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\((\mu,\nu)\)-pap solution in\(\mathbb{E}\), where
and
Proof
We have
and
Let
Then, for every \(\varphi\in\mathbb{E} \), we obtain the following:
where
Therefore \(\varGamma\circ\varphi\in\mathbb{E}\).
Next, for all \(\phi, \psi\in\mathbb{E} \), we get the following:
where \(i= 1,\ldots, n \). Therefore \(\|( \varGamma\circ{\phi}) - (\varGamma\circ{\psi} )\|_{\infty}\leq q_{0}\| \phi- \psi\|_{\infty}\).
Note that since \(q_{0} < 1 \), Γ is a contraction and possesses a unique fixed point z, which is a \((\mu,\nu)\)-pap solution of system (1.1) in the region \(\mathbb{E}\). 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.
Corollary 3.7
Suppose that conditions(M.1)and(M.3)–(M.6)hold. Then system (1.1) admits a uniqueμ-pap solution in
In the sequel, we shall assume that the functions \(L^{f}_{j}\), \(L^{g}_{j}\), and \(L^{h}_{j}\) are constant. By analogy, we can prove the same results as above. In addition, by the following modifications of conditions (M.5) and (M.6), the exponential stability of the solution can be obtained:
- (M.7):
For all \(1\leq j\leq n\), there exist constants
$$L_{j}^{f}, L_{j}^{g}, L_{j}^{h}, M_{j}^{f}, M_{j}^{g},M_{j}^{h}\in \mathbb{R}^{*}_{+} $$such that, for all \(t, x_{1}, x_{2} \in\mathbb{R}\),
$$\begin{gathered} \bigl\vert f_{j} (t,x_{1}) - f_{j} (t,x_{2}) \bigr\vert \leq L_{j}^{f} \vert x_{1}-x_{2} \vert , \qquad \bigl\vert f_{j} (t,x_{1}) \bigr\vert \leq M_{j}^{f}, \\ \bigl\vert g_{j} (t,x_{1}) - g_{j} (t,x_{2}) \bigr\vert \leq L_{j}^{g} \vert x_{1}-x_{2} \vert , \qquad \bigl\vert g_{j} (t,x_{1}) \bigr\vert \leq M_{j}^{g}, \\ \bigl\vert h_{j} (t,x_{1}) - h_{j} (t,x_{2}) \bigr\vert \leq L_{j}^{h} \vert x_{1}-x_{2} \vert , \qquad \bigl\vert h_{j} (t,x_{1}) \bigr\vert \leq M_{j}^{h}, \end{gathered}$$and
$$f_{j}(t,0)=g_{j}(t,0)=h_{j}(t,0)=0. $$- (M.8):
There exists a nonnegative constant \(q_{1}\) such that
$$q_{1}: = \max_{ i\in\{1,2,\ldots,n\} } \biggl\{ \frac{ \sum_{j=1}^{n}[ \bar {d}_{ij} L_{j}^{f} + \bar{a}_{ij} L_{j}^{g} + \sum_{l=1}^{n} \bar{b}_{ijl} (L_{j}^{h} M_{l}^{h} + M_{j}^{h} L_{l}^{h} )]}{c_{i}^{\ast}} \biggr\} < 1. $$We let
$$p_{1} := \max_{ i\in\{1,2,\ldots,n\} } \biggl\{ \frac{\sum_{j=1}^{n} [\bar {d}_{ij} L_{j}^{f} + \bar{a}_{ij} L_{j}^{g } + \sum_{l=1}^{n} \bar {b}_{ijl} L_{j}^{h}M_{l}^{h}]}{c_{i}^{\ast}} \biggr\} $$and
$$\varphi_{0}(t) := \biggl( \int_{- \infty}^{t} e^{-\int_{s}^{t}c_{1}(u)\, du} I_{1} (s) \,ds,\ldots, \int_{- \infty}^{t} e^{-\int _{s}^{t}c_{n}(u)\,du}I_{n}(s) \,d s \biggr)^{T}. $$
Theorem 3.8
Suppose that conditions(M.1)–(M.4)and(M.7)–(M.8)hold. Then system (1.1) admits a unique\((\mu,\nu)\)-pap solution in\(\mathbb{F}\), where
Proof
The following inequality holds:
Therefore \(\varGamma\circ{\varphi} \in\mathbb{F}\). Next, for all \(\phi, \psi\in\mathbb{F} \),
Since \(q_{1}< 1 \), it follows that Γ possesses a unique fixed point z which is a \((\mu,\nu )\)-pap solution of system (1.1) in the region \(\mathbb{F}\). This completes the proof of Theorem 3.8. □
If \(\mu=\nu\), we can deduce the following result.
Corollary 3.9
Suppose that conditions(M.1), (M.3)–(M.4), and(M.7)–(M.8)hold. Then system (1.1) admits a uniqueμ-pap solution in
Theorem 3.10
Suppose that conditions(M.1)–(M.4)and(M.7)–(M.8)hold. Then system (1.1) has a unique globally exponentially stable\((\mu,\nu)\)-pap solution.
Proof
System (1.1) has a unique \((\mu,\nu)\)-pap solution
and \(u(t) = ( u_{1}(t),\ldots, u_{n} (t))^{T}\) is the initial value.
Let \(x(t)= (x_{1}(t),\ldots, x_{n}(t))^{T} \) be an arbitrary solution of system (1.1) with initial value \(\varphi^{*} (t) = ( \varphi_{1}^{*}(t) ,\ldots, \varphi_{n}^{*} (t))^{T}\). Let \(y_{i}(t) = x_{i}(t) -z_{i}(t) \), \(\varphi_{i}(t)= \varphi _{i}^{*}(t) - u_{i}(t)\) for \(i=1,\ldots,n\). Then
where \(i\in\{1,2,3,\ldots,n\}\). Let \(F_{i}\) be defined by
By condition (M.8), we have
Thus there exists \(\varepsilon_{i}^{*} > 0 \) such that \(F_{i}(\varepsilon_{i}^{*} )=0 \) and \(F_{i}( \varepsilon_{i} > 0 )\) if \(\varepsilon_{i} \in( 0, \varepsilon_{i}^{*} )\).
Let \(\eta= \min\{ \varepsilon_{1}^{*} ,\ldots, \varepsilon_{n}^{*} \} \). Then \(F_{i}(\eta) \geq0\) if \(i =1 ,\ldots, n\). Next, there exists a nonnegative λ such that
so for all \(i\in\{ 1,\ldots,n\}\),
Multiplying (3.3)–(3.5) by \(e^{\int_{0}^{s} c_{i}(u) \,du} \) and integrating on \([0,t]\), we get
Let
Clearly, \(M > 1\), and
where \(0 < \lambda< \min\{ \eta, c^{*}_{1} , c^{*}_{2} ,\ldots, c^{*}_{n} \} \) is as in (3.6). Also,
To prove inequality (3.7), we first show that, for any \(u> 1\), the following inequality holds:
Indeed, if (3.8) were false, there would exist some \(t_{1} > 0 \) and \(i \in\{ 1,\ldots, n\} \) such that
and
So we could obtain
Hence we could conclude that \(\| y( t_{1} ) \|_{\infty} < u M \| \varphi\|_{\infty} e^{- \lambda t_{1}} \), which contradicts inequality (3.8). Note that \(u \rightarrow1 \), so (3.7) holds. Therefore system (1.1) has a unique globally exponentially stable \((\mu,\nu)\)-pap solution. This completes the proof of Theorem 3.10. □
If \(\mu=\nu\), then hypothesis (M.2) is satisfied, and we can deduce the following corollary:
Corollary 3.11
Suppose that conditions(M.1), (M.3)–(M.4), and(M.7)–(M.8)hold. Then system (1.1) has a unique globally exponentially stableμ-pap solution.
4 An application to neural networks
Neural networks have attracted a lot of attention in recent years, and especially the special case of the so-called 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 Hopfield neural networks. Many excellent results about their dynamic characteristics have been obtained, e.g., [2–4, 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 this paper, we prove the existence, the exponential stability, and the uniqueness of \((\mu,\nu)\)-pap solutions of the following models of high-order Hopfield neural networks (HOHNNs) with delays:
where \(i\in\{1,\ldots,n\}\).
n—number of neurons in neural network;
\(x_{i} (t)\)—ith neuron at time t;
\(f_{j}\), \(g_{j}\), \(h_{j}\)—activation function of jth 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 ith neuron;
\(\tau_{ij} \geq0\), \(\sigma_{ij} \geq0\), \(\nu_{ij} \geq0\)—transmission delays.
The initial conditions associated with system (4.1) are of the form
In our paper we have generalized the previous results by using the notion of double measure and working with two-variable functions.
Example 4.1
Consider the following model:
where \(c_{1}=c_{2}=2\), \(g_{1}(t)= g_{2}(t)= \sin t \). Then
Measures μ and ν are defined by the following double weights, respectively:
and
Then we have
We now prove that \(\mu\in\mathcal{M}\) satisfies condition (M.1). Indeed,
which implies that
so by [5], \(\nu\in\mathcal{M}\) satisfies condition (M.1). Since
it follows that condition (M.2) is also satisfied. We set
Therefore
Using Theorems 3.8 and 3.10, we can now see that model (4.2) has a unique \((\mu,\nu)\)-pap solution which is globally exponentially stable on
Abbreviations
- ap:
-
almost periodic
- apu:
-
almost periodic uniformly
- pap:
-
pseudo almost periodic
- papu:
-
pseudo almost periodic uniformly
- HOHNN:
-
high-field Hopfield neural network
References
Ait Dads, E.H., Ezzinbi, K., Miraoui, M.: \((\mu,\nu)\)-Pseudo almost automorphic solutions for some nonautonomous differential equations. Int. J. Math. 26(11), Article ID 1550090 (2015)
Alimi, A.M., Aouiti, C., Chérif, F., M’hamdi, M.S.: Dynamics and oscillations of generalized high-order Hopfield neural networks with mixed delays. Neurocomputing 321, 274–295 (2018)
Aouiti, C., M’hamdi, M.S., Chérif, F.: The existence and the stability of weighted pseudo almost periodic solution of high-order Hopfield neural network. In: International Conference on Artificial Neural Networks, pp. 478–485 (2016)
Arbi, A., Chérif, F., Aouiti, C., Touati, A.: Dynamics of new class of Hopfield neural networks with time-varying and distributed delays. Acta Math. Sci. 36(3), 891–912 (2016)
Blot, J., Cieutat, P., Ezzinbi, K.: New approach for weighted pseudo almost periodic functions under the light of measure theory, basic result and applications. Appl. Anal. 92(3), 493–526 (2013)
Bochner, S.: Continuous mappings of almost automorphic and almost periodic functions. Proc. Natl. Acad. Sci. USA 52, 907–910 (1964)
Brahmi, H., Ammar, B., Chérif, F., Alimi, A.M.: Stability and exponential synchronization of high-order Hopfield neural networks with mixed delays. Cybern. Syst. 48(1), 49–69 (2016)
Chérif, F., Miraoui, M.: New results for a Lasota–Wazewska model. Int. J. Biomath. 12(2), Article ID 1950019 (2019)
Cieutat, P., Fatajou, S., N’Guérékata, G.M.: Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations. Appl. Anal. 89(1), 11–27 (2010)
Diagana, T.: Weighted pseudo almost periodic solution to some differentiable equations. Nonlinear Anal. 68, 2250–2260 (2008)
Diagana, T., Ezzinbi, K., Miraoui, M.: Pseudo almost periodic and pseudo-almost automorphic solutions to some evolution equations involving theoretical measure theory. CUBO 16(2), 1–31 (2014)
Ezzinbi, K., Miraoui, M.: μ-Pseudo almost periodic and automorphic solutions in the α-norm for some partial functional differential equations. Numer. Funct. Anal. Optim. 36(8), 991–1012 (2015)
Ezzinbi, K., Miraoui, M., Rebey, A.: Measure pseudo-almost periodic solutions in the α-norm to some neutral partial differential equations with delay. Mediterr. J. Math. 13(5), 3417–3431 (2016)
M’hamdi, M.S., Aouiti, C., Touati, A., Alimi, A.M., Snasel, V.: Weighted pseudo almost-periodic solutions of shunting inhibitory cellular neural networks with mixed delays. Acta Math. Sci. 36(6), 1662–1682 (2016)
Miraoui, M.: Existence of μ-pseudo almost periodic solutions to some evolution equations. Math. Methods Appl. Sci. 40(13), 4716–4726 (2017)
Miraoui, M.: Pseudo almost automorphic solutions for some differential equations with reflection of the argument. Numer. Funct. Anal. Optim. 38(3), 376–394 (2017)
Miraoui, M., Yaakoubi, N.: Measure pseudo almost periodic solutions of shunting inhibitory cellular neural networks with mixed delays. Numer. Funct. Anal. Optim. 40(5), 571–585 (2019)
N’Guérékata, G.M.: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer Academic, New York (2001)
Papageorgiou, N.S., Radulescu, V.D., Repovš, D.D.: Periodic solutions for a class of evolution inclusions. Comput. Math. Appl. 75, 3047–3065 (2018)
Papageorgiou, N.S., Radulescu, V.D., Repovš, D.D.: Periodic solutions for implicit evolution inclusions. Evol. Equ. Control Theory 8(3), 621–631 (2019)
Papageorgiou, N.S., Radulescu, V.D., Repovš, D.D.: Nonlinear Analysis—Theory and Methods. Springer Monographs in Mathematics. Springer, Cham (2019)
Qiu, F., Cui, B., Wu, W.: Global exponential stability of high order recurrent neural network with time-varying delays. Appl. Math. Model. 33, 198–210 (2009)
Radulescu, V.D., Repovš, D.D.: Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis. Chapman & Hall/CRC, Boca Raton (2015)
Xiao, B., Meng, H.: Existence and exponential stability of positive almost-periodic solutions for high-order Hopfield neural networks. Appl. Math. Model. 33, 532–542 (2009)
Yu, Y., Cai, M.: Existence and exponential stability of almost periodic solutions for high-order Hopfield neural networks. Math. Comput. Model. 47, 943–951 (2008)
Zhang, C.Y.: Pseudo almost periodic solutions of some differential equations. J. Math. Anal. Appl. 151, 62–76 (1994)
Zhao, H.Y.: Pseudo almost periodic solutions for a class of differential equation with delays depending on state. Adv. Nonlinear Anal. 9, 1251–1258 (2020)
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Miraoui, M., Repovš, D.D. Dynamics and oscillations of models for differential equations with delays. Bound Value Probl 2020, 54 (2020). https://doi.org/10.1186/s13661-020-01348-x
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Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-020-01348-x