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:

$$\begin{aligned}& x_{i}'(t)= -c_{i}(t) x_{i}(t) + \sum_{j=1}^{n} d_{ij} (t) f_{j} \bigl(t,x_{j}(t) \bigr) + \sum _{j=1}^{n} a_{ij}(t) g_{j} \bigl(t,x_{j} (t- \tau_{ij}) \bigr) \\& \phantom{x_{i}'(t)=}{}+ \sum_{j=1}^{n} \sum _{l=1}^{n} b_{ijl}(t) h_{j} \bigl(t,x_{j}(t- \sigma_{ij}) \bigr) h_{l} \bigl(t,x_{l}(t- \nu_{ij}) \bigr) + I_{i}(t), \\& x_{i}(s)= \varphi_{i} (s) ,\quad s \in(- \theta, 0 ] , \quad i \in \{1,\ldots,n\}, \end{aligned}$$
(1.1)

where functions

$$c_{i}, I_{i}, d_{ij}, a_{ij},b_{ijl}: \mathbb{R}\to\mathbb{R} \quad\mbox{and}\quad f_{j},g_{j},h_{j}: \mathbb{R}\times\mathbb{R}\to\mathbb{R},\quad i,j,l\in \{1,\ldots,n\} $$

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

$$\bigl\Vert f(t + \tau) - f(t) \bigr\Vert < \varepsilon\quad\mbox{for all } t \in\mathbb{R}. $$

The space \(\mathcal{AP}(\mathbb{R},\mathbb{R}^{n})\) equipped with the norm

$$ \Vert f \Vert _{\infty}:=\max_{1\leq i \leq n}\sup _{t\in\mathbb{R}} \bigl\vert f_{i}(t) \bigr\vert $$

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 \(\mathcal{B}\)

$$\begin{aligned} \mathcal{M}={}& \bigl\{ \mu\mbox{ is a positive measure on } \mathcal{B};\\ & \mu( \mathbb{R})=+\infty, \mbox{and } \mu \bigl([s,t] \bigr)< \infty, \mbox{for all } s,t \in\mathbb{R}, s\leq t \bigr\} .\end{aligned} $$

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:

$$\mathcal{E} \bigl( \mathbb{R}, \mathbb{R}^{n}, \mu,\nu \bigr) := \biggl\{ f \in \mathcal{BC} \bigl(\mathbb{R}, \mathbb{R}^{n} \bigr): \lim_{z \rightarrow \infty} \frac{1}{\nu([-z,z])} \int_{-z}^{z} \bigl\Vert f (t) \bigr\Vert \,d \mu(t) = 0 \biggr\} $$

and

$$\mathcal{E} \bigl( \mathbb{R}, \mathbb{R}^{n}, \mu \bigr) := \mathcal{E} \bigl( \mathbb{R}, \mathbb{R}^{n}, \mu,\mu \bigr)= \biggl\{ f \in\mathcal{BC} \bigl(\mathbb{R}, \mathbb{R}^{n} \bigr): \lim _{z \rightarrow\infty} \frac {1}{\mu([-z,z])} \int_{-z}^{z} \bigl\Vert f (t) \bigr\Vert \,d \mu(t) = 0 \biggr\} . $$

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

$$ f = g + \varphi, \quad\mbox{where } \varphi\in \mathcal{E} \bigl( \mathbb{R},\mathbb{R}^{n}, \mu,\nu \bigr) \mbox{ and } g \in \mathcal{AP} \bigl(\mathbb{R},\mathbb{R}^{n} \bigr). $$
(2.1)

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

$$f=g+h, \quad\mbox{where } g \in\mathcal{APU} \bigl(\mathbb{R}\times\mathbb {R}, \mathbb{R}^{n} \bigr) \mbox{ and } h \in\mathcal{E}U \bigl(\mathbb{R} \times \mathbb{R},\mathbb {R}^{n},\mu \bigr). $$

Example 2.1

Let \(\mu\in\mathcal{M}\) and

$$G(t)= \bigl[\sin(t)+\sin(\sqrt{2}t) \bigr]\cos(x)+\frac{\sin(x)}{1+t^{2}},\quad t \in \mathbb{R}. $$

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:

$$\begin{gathered} \sup_{t \in\mathbb{R} } \bigl\{ \bigl\vert d_{ij} (t) \bigr\vert \bigr\} := \bar{d}_{ij},\qquad \sup_{t\in\mathbb{R}} \bigl\{ \bigl\vert I_{i} (t) \bigr\vert \bigr\} : = \bar{I}_{i}, \\ \sup_{t \in\mathbb{R} } \bigl\{ \bigl\vert a_{ij} (t) \bigr\vert \bigr\} := \bar{a}_{ij},\qquad \sup_{t\in\mathbb{R}} \bigl\{ \bigl\vert b_{ijl} (t) \bigr\vert \bigr\} := \bar{b}_{ijl},\end{gathered} $$

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 \(1\leq j\leq n\):

$$f_{j}(t,0)=g_{j}(t,0)=h_{j}(t,0)=0 \quad\mbox{for all } t\in\mathbb{R}. $$
(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

$$\begin{gathered} L:= \max_{ i\in\{1,2,\ldots,n\} } \biggl\{ \frac{\bar{I}_{i}}{c_{i}^{*}} \biggr\} ,\\ p_{0}: = \max_{ i\in\{1,2,\ldots,n\} } \biggl\{ \frac{ \sum_{j=1}^{n} [\bar{d}_{ij} \Vert L_{j}^{f} \Vert _{p}+ \bar{a}_{ij} \Vert L_{j}^{g } \Vert _{p} + \sum_{l=1}^{n} \bar{b}_{ijl} \Vert L_{j}^{h} \Vert _{p} \Vert h_{l} \Vert _{\infty } ]}{ (qc^{*}_{i})^{\frac{1}{q}}} \biggr\} .\end{gathered} $$

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

$$y=y_{1}+y_{2},\quad \mbox{where } y_{1}\in \mathcal{AP}(\mathbb{R},\mathbb{R}) \mbox{ and } y_{2}\in\mathcal{E}( \mathbb{R},\mathbb{R},\mu,\nu). $$

Let

$$\begin{aligned} \varPsi(t) =&\varLambda \bigl(t,y_{1}(t-\theta) \bigr)+ \bigl[\varLambda \bigl(t,y_{1}(t-\theta)+y_{2}(t-\theta) \bigr)- \varLambda \bigl(t,y_{1}(t-\theta) \bigr) \bigr] = \varPsi_{1}(t)+\varPsi_{2}(t), \end{aligned}$$

where

$$\varPsi_{1}(t)=\varLambda \bigl(t,y_{1}(t-\theta) \bigr) \quad\mbox{and}\quad \varPsi _{2}(t)=\varLambda \bigl(t,y_{1}(t- \theta)+y_{2}(t-\theta) \bigr)- \varLambda \bigl(t,y_{1}(t- \theta) \bigr). $$

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

$$\begin{aligned} &\frac{1}{\nu([-z,z])} \int_{-z}^{z} \bigl\vert \varPsi_{2}(t) \bigr\vert \,d\mu(t) \\ &\quad= \frac{1}{\nu([-z,z])} \int_{-z}^{z} \bigl\vert \varLambda \bigl(t,y_{1}(t- \theta)+y_{2}(t-\theta) \bigr) -\varLambda \bigl(t, y_{1}(t- \theta) \bigr) \bigr\vert \,d\mu(t) \\ &\quad\leq\frac{1}{\nu([-z,z])} \int_{-z}^{z}L^{\varLambda }(t) \bigl\vert y_{2}(t- \theta) \bigr\vert \,d\mu(t). \end{aligned}$$

Since condition (M.2) holds and \(y_{2}\in \mathcal{E}(\mathbb{R},\mathbb{R},\mu,\nu)\), we get

$$\begin{aligned} &\frac{1}{\nu([-z,z])} \int_{-z}^{z} \bigl\vert \varPsi_{2}(t) \bigr\vert \,d\mu(t) \\ &\quad\leq \frac{1}{\nu([-z,z])} \int_{-z}^{z}L^{\varLambda}(t) \bigl\vert y_{2}(t- \theta) \bigr\vert \, d\mu(t) \\ &\quad\leq\frac{ \Vert y_{2} \Vert _{\infty}}{\nu([-z,z])} \int _{-z}^{z}L^{\varLambda}(t)\,d\mu(t) \\ &\quad\leq\frac{ \Vert y_{2} \Vert _{\infty}}{\nu([-z,z])} \biggl[ \int _{-z}^{z} \bigl(L^{\varLambda}(t) \bigr)^{p}\,d\mu(t) \biggr]^{\frac{1}{p}} \biggl[ \int_{-z}^{z}\,d\mu(t) \biggr]^{\frac{1}{q}},\quad \mbox{where } \frac{1}{p}+\frac{1}{q}=1 \\ &\quad\leq\frac{ \Vert y_{2} \Vert _{\infty}}{\nu([-z,z])^{\frac{1}{p}}} \bigl\Vert L^{\varLambda} \bigr\Vert _{p} \biggl[ \frac{\mu([-z,z])}{\nu([-z,z])} \biggr]^{\frac {1}{q}}\rightarrow0,\quad \text{as }z\rightarrow+\infty. \end{aligned}$$

Therefore

$$\bigl[t\mapsto\varPsi_{2}(t) \bigr]\in \mathcal{E}(\mathbb{R}, \mathbb{R},\mu,\nu) \quad\mbox{and}\quad \bigl[s\mapsto \varLambda \bigl(s,y(s-\theta) \bigr) \bigr]\in\mathcal{PAP} (\mathbb{R},\mathbb {R},\mu,\nu). $$

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

$$y,z\in\mathcal{PAP}(\mathbb{R}, \mathbb{R},\mu,\nu). $$

Then

$$y \times z\in\mathcal{PAP}( \mathbb{R}, \mathbb{R}, \mu,\nu). $$

Proof

Since \(y,z\in\mathcal{PAP}(\mathbb{R}, \mathbb{R},\mu,\nu)\), it follows that

$$y = y_{1} + y_{2} \quad\mbox{and}\quad z= z_{1} + z_{2}, \quad\mbox{where } y_{1},z_{1}\in\mathcal{AP}( \mathbb{R} , \mathbb{R} ) \mbox{ and } y_{2},z_{2}\in \mathcal{E} ( \mathbb{R} , \mathbb{R} , \mu,\nu). $$

Then

$$y \times z= y_{1} z_{1} + y_{2} z_{1} + y_{1} z_{2} + y_{2} z_{2}. $$

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

$$\begin{aligned} \bigl\Vert \varphi_{0}^{2} ( t ) - \varphi_{0}^{2} (t+ \tau) \bigr\Vert =& \bigl\Vert \varphi_{0}(t) +\varphi_{0} (t + \tau) \bigr\Vert \cdot \bigl\Vert \varphi_{0} (t)- \varphi_{0} (t + \tau) \bigr\Vert \leq2 \Vert \varphi_{0} \Vert _{\infty }\cdot \varepsilon. \end{aligned}$$

Then \(\varphi_{0}^{2}\in\mathcal{AP}( \mathbb{R} , \mathbb{R} ) \), so it follows that

$$( y_{1} + z_{1} )^{2} \in\mathcal{AP}( \mathbb{R} , \mathbb{R} ) \quad\mbox{and}\quad ( y_{1} - z_{1} )^{2} \in \mathcal{AP}( \mathbb{R} , \mathbb{R} ), $$

since

$$( y_{1} + z_{1} ) \in\mathcal{AP}( \mathbb{R} , \mathbb{R} ) \quad\mbox{and}\quad ( y_{1} - z_{1} ) \in\mathcal{AP}( \mathbb{R} , \mathbb{R} ). $$

Note that

$$y_{1} \times z_{1} = \frac{1}{4} \bigl( ( y_{1} + z_{1} )^{2} - ( y_{1} - z_{1})^{2} \bigr), $$

so we can conclude that indeed \(y_{1} z_{1} \in\mathcal{AP}( \mathbb{R} , \mathbb{R} ) \).

Next, we shall prove that

$$y_{2} z_{1} + y_{1} z_{2} + y_{2} z_{2} \in\mathcal{E} ( \mathbb{R} , \mathbb{R} , \mu, \nu). $$

Indeed, for \(z>0\), we have

$$\begin{aligned}& \frac{1}{\nu( [-z,z]) } \int_{-z}^{z} \bigl\vert ( y_{1} z_{2} + y_{2} z_{1} + y_{2} z_{2}) (t) \bigr\vert \,d \mu(t) \\& \quad\leq \frac{ \Vert y_{1} \Vert _{\infty}}{ \nu([ -z,z]) } \int_{-z}^{z} \bigl\vert z_{2} (t) \bigr\vert \,d \mu(t)+ \frac{ \Vert z_{1} \Vert _{\infty}}{ \nu([ -z,z]) } \int_{-z}^{z} \bigl\vert y_{2} (t) \bigr\vert \,d \mu(t) \\& \qquad{}+ \frac{ \Vert y_{2} \Vert _{\infty}}{ \nu([ -z,z])} \int_{-z}^{z} \bigl\vert z_{2} (t) \bigr\vert \,d \mu(t). \end{aligned}$$

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 \(\varphi= ( \varphi_{1},\ldots, \varphi_{n} ) \in \mathcal{PAP}(\mathbb{R}, \mathbb{R}^{n} , \mu,\nu)\),

$$( \varGamma\circ\varphi) (t) :=x_{\varphi} (t) = \biggl( \int_{- \infty }^{t} F_{1}(s) e^{-\int_{s}^{t}c_{1}(u)\,du} \,ds,\ldots, \int_{-\infty }^{t}F_{n}(s)e^{-\int_{s}^{t}c_{n}(u)\,du} \,ds \biggr)^{T} $$

and

$$\begin{aligned} F_{i}(s) ={}& \sum_{j=1}^{n} d_{ij}(s) f_{j} \bigl( s,\varphi_{j}(s) \bigr) + \sum_{j=1}^{n} a_{ij} (s) g_{j} \bigl(s,\varphi_{j}(s- \tau_{ij}) \bigr) \\ &+ \sum_{j=1}^{n} \sum _{l=1}^{n} b_{i j l} (s) h_{j} \bigl(s, \varphi _{j}(s- \sigma_{ij}) \bigr) h_{l} (s,s - \nu_{ij}) + I_{i}(s).\end{aligned} $$

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

$$\begin{aligned} F_{i} : s \mapsto& \sum_{i=1}^{n} d_{ij} (s) f_{j} \bigl(s,\varphi_{j} (s) \bigr) + \sum_{j=1}^{n} a_{ij}(s) g_{j} \bigl(s,\varphi_{j} (s- \tau_{ij}) \bigr) \\ &{} + \sum_{j=1}^{n}\sum _{l=1}^{n} b_{ijl}(s) h_{j}(s,s- \sigma_{ij} ) h_{l} \bigl( \varphi_{l} (s,s - \nu_{ij} ) \bigr) + I_{i}(s) \end{aligned}$$
(3.1)

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

$$F_{i} = F_{i}^{1} + F_{i}^{2},\quad \mbox{where } F_{i}^{1} \in \mathcal{AP} ( \mathbb{R} , \mathbb{R} ) \mbox{ and } F_{i}^{2} \in \mathcal{E} ( \mathbb{R}, \mathbb{R} , \mu,\nu). $$

Therefore

$$\begin{aligned} (\varGamma_{i} \circ\varphi) ( t) = & \int_{- \infty}^{t} e^{-\int _{s}^{t}c_{i}(u)\,du} F_{i}^{1}(s) \,ds + \int_{- \infty}^{t} e^{-\int_{s}^{t}c_{i}(u)\,du} F_{i}^{2}(s) \,ds \\ = & \bigl(\varGamma_{i}\circ F_{i}^{1} \bigr) (t) + \bigl( \varGamma_{i}\circ F_{i}^{2} \bigr) (t). \end{aligned}$$
(3.2)

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

$$\begin{aligned} \bigl\vert \bigl(\varGamma_{i} \circ F_{i}^{1} \bigr) (t+ \tau)- \bigl( \varGamma_{i}\circ F_{i}^{1} \bigr) (t) \bigr\vert =& \biggl\vert \int_{- \infty}^{t + \tau} e^{-\int_{s}^{t+ \tau }c_{i}(u)\,du} F_{i}^{1}(s) \,d s- \int_{- \infty}^{t} e^{-\int _{s}^{t}c_{i}(u)\,du} F_{i}^{1}(s) \,d s \biggr\vert \\ \leq& \biggl\vert \int_{0}^{+ \infty} e^{-y c_{i}^{\ast}} F_{i}^{1}(t + \tau - y )\, d y - \int_{0}^{+ \infty} e^{-y c_{i}^{\ast}} F_{i}^{1} ( t- y ) \,d y \biggr\vert \\ \leq& \int_{0}^{+ \infty} e^{- y c_{i}^{\ast}} \bigl\vert F_{i}^{1} ( t + \tau- y ) - F_{i}^{1} ( t- y ) \bigr\vert \,d y \\ \leq& \varepsilon \int_{0}^{+ \infty} e^{- y c_{i}^{\ast}} \,dy = \frac{\varepsilon}{c_{i}^{\ast}}. \end{aligned}$$

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

$$\int_{-z}^{z} \bigl\vert \bigl( \varGamma_{i}\circ F_{i}^{2} \bigr) (t) \bigr\vert \,d\mu(t) = \int _{-z}^{z} \biggl\vert \int_{-\infty}^{t} e^{-\int_{s}^{t}c_{i}(u)\,du} F_{i}^{2} (s) \,d s \biggr\vert \,d\mu(t). $$

Using Fubini’s theorem, we get

$$\begin{aligned} \frac{1}{\nu([-z,z])} \int_{-z}^{z} \bigl\vert \bigl( \varGamma_{i}\circ F_{i}^{2} \bigr) (t) \bigr\vert \,d \mu(t) =& \frac{1}{\nu([-z,z])} \int_{-z}^{z} \biggl\vert \int_{-\infty}^{t} e^{-\int_{s}^{t}c_{i}(u)\,du} F_{i}^{2} (s) \,d s \biggr\vert \,d \mu(t) \\ \leq&\frac{1}{\nu([-z,z])} \int_{-z}^{z} \int_{0}^{\infty} e^{-y c_{i}^{\ast}} \bigl\vert F_{i}^{2} (t - y ) \bigr\vert \,d s \,d \mu(t) \\ \leq& \frac{1}{\nu([-z,z])} \int_{0}^{\infty} \int_{-z}^{z} e^{-y c_{i}^{\ast}} \bigl\vert F_{i}^{2} (t - y ) \bigr\vert \,d s \,d \mu(t) \end{aligned}$$

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

$$\varGamma_{i}\circ F_{i}^{2} \in\mathcal{E} ( \mathbb{R}, \mathbb{R} , \mu,\nu) \quad\mbox{for all } i\in\{1,2,3,\ldots,n\}. $$

We can thus conclude that

$$\varGamma_{i} \circ\varphi\in \mathcal{PAP} ( \mathbb{R} , \mathbb{R} , \mu,\nu) \quad\mbox{for all } i\in\{1,2,3,\ldots,n\}, $$

hence

$$\varGamma\circ\varphi\in\mathcal{PAP} \bigl( \mathbb{R} , \mathbb{R}^{n}, \mu,\nu \bigr). $$

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

$$\mathbb{E} = \biggl\{ \psi\in\mathcal{PAP} \bigl(\mathbb{R}, \mathbb{R}^{n} , \mu,\nu \bigr): \Vert \psi- \varphi_{0} \Vert _{\infty} \leq\frac{p_{0}L}{1-p_{0}} \biggr\} $$

and

$$\varphi_{0}(t) = \biggl( \int_{- \infty}^{t} e^{-\int_{s}^{t}c_{1}(u)\, du} I_{1} (s) \,ds,\ldots, \int_{- \infty}^{t} e^{-\int _{s}^{t}c_{n}(u)\,du}I_{n}(s) \,d s \biggr)^{T}. $$

Proof

We have

$$\begin{aligned} \Vert \varphi_{0} \Vert _{\infty} =& \max _{ i\in\{1,2,\ldots,n\} }\sup_{t \in\mathbb{R}} \biggl( \biggl\vert \int_{-\infty}^{t} e^{-\int_{s}^{t}c_{i}(u)\,du }I_{i} (s) \,d s \biggr\vert \biggr) \leq\max_{ i\in\{1,2,\ldots,n\} } \biggl( \frac{\bar{I}_{i}}{c_{i}^{\ast}} \biggr) := L \end{aligned}$$

and

$$\begin{aligned} \Vert \varphi \Vert _{\infty} \leq& \Vert \varphi- \varphi_{0} \Vert _{\infty} + \Vert \varphi \Vert _{\infty} \leq \Vert \varphi- \varphi_{0} \Vert _{\infty} +L \leq\frac{L}{1-p_{0}}. \end{aligned}$$

Let

$$\mathbb{E} = \mathbb{E} (\varphi_{0} , p_{0})= \biggl\{ \varphi\in\mathcal {PAP} \bigl(\mathbb{R}, \mathbb{R}^{n} , \mu,\nu \bigr): \Vert \varphi- \varphi _{0} \Vert _{\infty} \leq \frac{p_{0}L}{1-p_{0}} \biggr\} . $$

Then, for every \(\varphi\in\mathbb{E} \), we obtain the following:

$$\begin{aligned} \bigl\Vert (\varGamma\circ\varphi) - \varphi_{0} \bigr\Vert _{\infty} =& \max_{ i\in\{1,2,\ldots,n\} } \sup_{t \in\mathbb{R }} \Biggl\{ \Biggl\vert \int_{- \infty}^{t} e^{-\int_{s}^{t}c_{i}(u)\,du}\\ &{}\times \sum _{j=1}^{n} \Biggl[ d_{ij}(s) f_{j} \bigl(s, \varphi_{j}(s) \bigr) + a_{ij} (s) g_{j} \bigl( s, \varphi_{j}(s - \tau_{ij} ) \bigr) \\ &{}+\sum_{l=1}^{n} b_{ijl}(s) h_{j} \bigl(s, \varphi_{j} (s - \sigma_{ij} ) \bigr) h_{l} \bigl( s,\varphi_{l}(s - \nu_{ij} ) \bigr) \Biggr] \,ds \Biggr\vert \Biggr\} \\ \leq& \max_{ i\in\{1,2,\ldots,n\} } \sup_{t \in\mathbb{R}} \Biggl\{ \int _{- \infty}^{t} e^{-\int_{s}^{t}c_{i}(u)\,du} \sum _{j=1}^{n} \Biggl[ \bar {d}_{ij} L_{j}^{f}(s) \Vert \varphi \Vert _{ \infty} + \bar{a}_{ij} L_{j}^{g}(s) \Vert \varphi \Vert _{\infty} \\ &{}+ \sum_{l=1}^{n} \bar{b}_{ijl} L_{j}^{h}(s) \Vert h_{l} \Vert _{\infty} \Vert \varphi \Vert _{\infty} \Biggr] \,ds \Biggr\} \\ \leq& \max_{ i\in\{1,2,\ldots,n\} } \biggl\{ \frac{ \sum_{j=1}^{n} [\bar {d}_{ij} \Vert L_{j}^{f} \Vert _{p}+ \bar{a}_{ij} \Vert L_{j}^{g } \Vert _{p} + \sum_{l=1}^{n} \bar{b}_{ijl} \Vert L_{j}^{h} \Vert _{p} \Vert h_{l} \Vert _{\infty}]}{ (qc^{*}_{i})^{\frac{1}{q}}} \biggr\} \Vert \varphi \Vert _{ \infty} \\ \leq& p_{0} \Vert \varphi \Vert _{\infty} \leq p_{0}\bigl( \Vert \varphi-\varphi_{0} \Vert _{\infty} + \Vert \varphi_{0} \Vert _{\infty}\bigr) \leq\frac{p_{0} L}{1- p_{0}}, \end{aligned}$$

where

$$p_{0} = \max_{ i\in\{1,2,\ldots,n\} } \biggl\{ \frac{ \sum_{j=1}^{n} [ \bar{d}_{ij} \Vert L_{j}^{f} \Vert _{\infty}+ \bar{a}_{ij} \Vert L_{j}^{g } \Vert _{\infty} + \sum_{l=1}^{n} \bar{b}_{ijl} \Vert L_{j}^{h} \Vert _{\infty} \Vert h_{l} \Vert _{\infty} ]}{ (qc^{*}_{i})^{\frac{1}{q}}} \biggr\} < 1. $$

Therefore \(\varGamma\circ\varphi\in\mathbb{E}\).

Next, for all \(\phi, \psi\in\mathbb{E} \), we get the following:

$$\begin{aligned} & \bigl\vert (\varGamma_{i}\circ{\phi}) (t) - ( \varGamma_{i}\circ{ \psi}) (t) \bigr\vert \\ &\quad\leq{ \int_{-\infty}^{t}} e^{-\int_{s}^{t}c_{i}(u)\,du} \sum _{j=1}^{n} \Biggl\vert d_{ij} (s) \bigl(f_{j} \bigl(s,\phi_{j}(s) \bigr) - f_{j} \bigl(s,\psi _{j}(s) \bigr) \bigr) \\ &\quad\quad{}+ a_{ij}(s) \bigl( g_{j} \bigl(s,\phi_{j}(s- \tau_{ij}) \bigr) - g_{j} \bigl(s,\psi _{j}(s- \tau_{ij} ) \bigr) \bigr) \\ &\quad\quad{}+ \sum_{l=1}^{n} b_{ijl}(s) \bigl(h_{j} \bigl(s,\phi_{j}(s- \sigma _{ij}) \bigr)h_{l} \bigl(s,\phi_{l}(s- \nu_{ij}) \bigr)\\ &\qquad{}- h_{j} \bigl(s,\psi_{j}(s- \sigma _{ij}) \bigr) h_{l} \bigl(s,\psi_{l}(s-\nu_{ij} ) \bigr) \bigr) \Biggr\vert \,ds \\ &\quad\leq \int_{-\infty}^{t} e^{{-\int_{s}^{t}c_{i}(u)\,du}}\sum _{j=1}^{n} \Biggl[\bar{d}_{ij} L_{j}^{f}(s) \sup_{t \in\mathbb{R}} \bigl\vert \phi_{j} (t) - \psi_{j}(t) \bigr\vert + \bar{a}_{ij} L_{j}^{g}(s) \sup _{t \in \mathbb{R}} \bigl\vert \phi_{j} (t) - \psi_{j}(t) \bigr\vert \\ &\qquad+ \sum_{l=1}^{n} b_{ijl}(s) \bigl\vert h_{j} \bigl(s,\phi_{j} (s- \sigma_{ij} ) \bigr)h_{l} \bigl(s,\phi_{l}(s- \nu_{ij}) \bigr) - h_{j} \bigl(s,\psi_{j}(s- \sigma_{ij} ) \bigr) h_{l} \bigl(s,\phi_{l}(s - \nu_{ij} ) \bigr) \\ &\qquad+ h_{j} \bigl(s,\psi_{j} \bigl(s - \sigma_{ij} (s) \bigr) \bigr) h_{l} \bigl(s, \phi_{l} (s- \nu_{ij} ) \bigr) - h_{j} \bigl(s, \psi_{j} (s-\sigma_{ij} ) \bigr) h_{l} \bigl(s, \psi _{l}(s- \nu_{ij} ) \bigr) \bigr\vert \Biggr] \,ds \\ &\quad \leq \int^{+ \infty}_{0} e^{-c^{*}_{i}y}\,dy \sum _{j=1}^{n} \Biggl[ \bar{d}_{ij} L_{j}^{f}(s) \sup_{t \in\mathbb{R} } \bigl\vert \phi_{j}(t) - \psi_{j} (t) \bigr\vert + \bar{a}_{ij} L_{j}^{g}(s) \sup _{t\in\mathbb{R}} \bigl\vert \phi_{j}(t) - \psi_{j} (t) \bigr\vert \\ &\qquad+ \sum_{l=1}^{n} \bar{b}_{ijl}(s) \bigl( L_{j}^{h}(s) \Vert h_{l} \Vert _{\infty } + L_{l}^{h}(s) \Vert h_{j} \Vert _{\infty} \bigr) \sup_{t \in\mathbb{R} } \bigl\vert \phi _{j}(t) - \psi_{j} (t) \bigr\vert \Biggr]\,ds \\ &\quad\leq \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}} } \Vert \phi- \psi \Vert _{\infty} \\ &\quad\leq q_{0} \Vert \phi- \psi \Vert _{\infty}, \end{aligned}$$

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

$$\mathbb{E} = \biggl\{ \psi\in\mathcal{PAP} \bigl(\mathbb{R}, \mathbb{R}^{n} , \mu \bigr): \Vert \psi- \varphi_{0} \Vert _{\infty} \leq\frac{p_{0}L}{1-p_{0}} \biggr\} . $$

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

$$\mathbb{F} = \biggl\{ \psi\in\mathcal{PAP} \bigl(\mathbb{R}, \mathbb{R}^{n} , \mu,\nu \bigr): \Vert \psi- \varphi_{0} \Vert _{\infty} \leq\frac{p_{1}L}{1-p_{1}} \biggr\} . $$

Proof

The following inequality holds:

$$\begin{aligned} \bigl\Vert ( \varGamma\circ{\varphi}) - \varphi_{0} \bigr\Vert _{\infty} \leq \frac {p_{1} L}{1- p_{1}}. \end{aligned}$$

Therefore \(\varGamma\circ{\varphi} \in\mathbb{F}\). Next, for all \(\phi, \psi\in\mathbb{F} \),

$$\bigl\Vert ( \varGamma\circ{\phi}) -( \varGamma\circ{\psi}) \bigr\Vert _{\infty}\leq q_{1} \Vert \phi- \psi \Vert _{\infty}. $$

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

$$\mathbb{F} = \biggl\{ \psi\in\mathcal{PAP} \bigl(\mathbb{R}, \mathbb{R}^{n} , \mu \bigr): \Vert \psi- \varphi_{0} \Vert _{\infty} \leq\frac{p_{1}L}{1-p_{1}} \biggr\} . $$

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

$$z(t)= \bigl( z_{1}(t),\ldots, z_{n} (t) \bigr)^{T} \in\mathbb{E} $$

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

$$\begin{aligned} y_{i}'(t) =& -c_{i}(t) y_{i}(t) \end{aligned}$$
(3.3)
$$\begin{aligned} &{}+ \sum_{j=1}^{n} (d_{ij} (t) \bigl( f_{j} \bigl(t,x_{j}(t) \bigr)-f_{j} \bigl(t,z_{j}(t) \bigr) \bigr) \\ &{}+ a_{ij}(t) \bigl[ g_{j} \bigl(t,x_{j} (t- \tau_{ij} ) \bigr)-g_{j} \bigl(t,z_{j} (t- \tau_{ij} ) \bigr) \bigr] \end{aligned}$$
(3.4)
$$\begin{aligned} &{}+ \sum_{l=1}^{n} b_{ijl}(t) \bigl[ h_{j} \bigl(t,x_{j}(t- \sigma_{ij}) h_{l} \bigl(t,x_{l}(t- \nu_{ij}) \bigr) \\ &{} - h_{j} \bigl(t,z_{j}(t-\sigma _{ij}) \bigr)h_{l} \bigl(t,z_{l}(t-\nu_{ij}) \bigr) \bigr] \bigr), \end{aligned}$$
(3.5)

where \(i\in\{1,2,3,\ldots,n\}\). Let \(F_{i}\) be defined by

$$F_{i}(w)= c^{*}_{i}-w- \sum_{j=1}^{n} \Biggl[\bar{d}_{ij} L_{j}^{f} + \bar{a}_{ij}L_{j}^{g} e^{w \tau_{ij}} + \sum _{l=1}^{n} \bar{b}_{ijl} \bigl( L_{j}^{h} e^{w \sigma_{ij}} M_{l}^{h}+ M_{j}^{h}L_{l}^{h} e^{w \nu_{ij}} \bigr) \Biggr]. $$

By condition (M.8), we have

$$F_{i}(0) = c^{*}_{i} - \sum_{j=1}^{n} \Biggl[ \bar{d}_{ij} L_{j}^{f} + \bar{a}_{ij} L_{j}^{g} + \sum _{l=1}^{n} \bar{b}_{ijl} \bigl(L_{j}^{h} M_{l}^{h} + M_{j}^{h} L_{l}^{h} \bigr) \Biggr] > 0. $$

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

$$0 < \lambda< \min \bigl\{ \eta, c_{1}^{*},\ldots, c_{n}^{*} \bigr\} \quad\mbox{and}\quad F_{i}( \lambda ) > 0 , $$

so for all \(i\in\{ 1,\ldots,n\}\),

$$ \frac{1}{c^{*}_{i} - \lambda} \Biggl[\sum_{j=1}^{n} \bigl( \bar{c}_{ij} L_{j}^{f} + \bar{d}_{ij} L_{j}^{g} e^{ \lambda\tau_{ij}} \bigr) + \sum_{j=1}^{n} \sum _{l=1}^{n} \bar{b}_{ijl} \bigl( L_{j}^{h} e^{\lambda \sigma_{ij}} M_{l}^{h}+ M_{j}^{h}L_{l}^{h} e^{\lambda\nu_{ij}} \bigr) \Biggr]< 1. $$
(3.6)

Multiplying (3.3)–(3.5) by \(e^{\int_{0}^{s} c_{i}(u) \,du} \) and integrating on \([0,t]\), we get

$$\begin{aligned} y_{i}(t) = & \varphi_{i} (0) e^{- \int_{0}^{t} c_{i}(u) \,du }+ \int_{0}^{t} e^{-\int_{s}^{t} c_{i}(u) \,du} \sum _{j=1}^{n} \Biggl( d_{ij} (s) \bigl[ f_{j} \bigl(s,y_{j}(s) + z_{j} (s) \bigr) - f_{j} \bigl(s,z_{j}(s) \bigr) \bigr] \\ &{}+ a_{ij} (s) \bigl[ g_{j} \bigl(s, y_{j} (s- \tau_{ij} ) + z_{j} (s-\tau _{ij} ) \bigr) - g_{j} \bigl(s, z_{j}(s- \tau_{ij} ) \bigr) \bigr] \\ &{}+ \sum_{l=1}^{n} b_{ijl}(s) \bigl[ h_{j} \bigl(s,y_{j}(s- \sigma_{ij})+z_{j}(s- \sigma_{ij}) \bigr) h_{l} \bigl(s,y_{j}(t- \nu_{ij})+z_{j}(t- \nu_{ij}) \bigr) \\ &{}- h_{j} \bigl(z_{j}(t-\sigma _{ij}) \bigr)h_{l} \bigl(z_{l}(t-\nu_{ij}) \bigr) \bigr] \Biggr) \,ds. \end{aligned}$$

Let

$$M= \max_{1\leq i \leq n} \frac{c_{i}^{*}}{\sum_{j=1}^{n} [( \bar{d}_{ij} L_{j}^{f} + \bar{ a}_{ij} L_{j}^{g} )+ \sum_{l=1}^{n} \bar{b}_{ijl} ( L_{j}^{h} M_{l}^{h} + M_{j}^{h} L_{l}^{h} ) ]}. $$

Clearly, \(M > 1\), and

$$ \frac{1}{M}-\frac{1}{c^{*}_{i} - \lambda} \Biggl[\sum_{j=1}^{n} \bigl( \bar {c}_{ij} L_{j}^{f} + \bar{d}_{ij} L_{j}^{g} e^{ \lambda\tau_{ij}} \bigr) + \sum_{j=1}^{n} \sum _{l=1}^{n} \bar{b}_{ijl} \bigl( L_{j}^{h} e^{\lambda \sigma_{ij}} M_{l}^{h}+ M_{j}^{h}L_{l}^{h} e^{\lambda\nu_{ij}} \bigr) \Biggr]\leq0, $$

where \(0 < \lambda< \min\{ \eta, c^{*}_{1} , c^{*}_{2} ,\ldots, c^{*}_{n} \} \) is as in (3.6). Also,

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert _{\infty} \leq M \Vert \varphi \Vert _{\infty} e^{-\lambda t},\quad t > 0. \end{aligned}$$
(3.7)

To prove inequality (3.7), we first show that, for any \(u> 1\), the following inequality holds:

$$ \bigl\Vert y(t) \bigr\Vert _{\infty} < u M \Vert \varphi \Vert _{\infty} e^{- \lambda t} ,\quad t > 0. $$
(3.8)

Indeed, if (3.8) were false, there would exist some \(t_{1} > 0 \) and \(i \in\{ 1,\ldots, n\} \) such that

$$ \bigl\Vert y(t_{1}) \bigr\Vert _{\infty} = \bigl\Vert y_{i} (t_{1}) \bigr\Vert _{\infty} = u M \Vert \varphi \Vert _{\infty} e^{- \lambda t_{1}} $$

and

$$ \bigl\Vert y(t) \bigr\Vert _{\infty} \leq u M \Vert \varphi \Vert _{\infty} e^{-\lambda t} \quad\mbox{for every } t \in( - \infty, t_{1}]. $$

So we could obtain

$$\begin{aligned} \bigl\vert y(t_{1}) \bigr\vert \leq& \Vert \varphi \Vert _{\infty} e^{-t_{1} c^{*}_{i}} + \int _{0}^{t_{1}} e^{(- t_{1} - s ) c^{*}_{i}} \sum _{j=1}^{n} \Biggl[ \bar{d}_{ij} L_{j}^{f} \bigl\Vert y_{j}(s) \bigr\Vert _{\infty}+ \bar{a}_{ij} L_{j}^{g} \bigl\Vert y_{j} (s - \tau_{ij} ) \bigr\Vert _{\infty} \\ &{}+ \sum_{l=1}^{n} \bar{b}_{ijl} \bigl( L_{j}^{h}M_{l}^{h} \bigl\Vert y_{j} (s - \sigma_{ij} ) \bigr\Vert _{\infty}+ M_{j}^{h}L_{l}^{h} \bigl\Vert y_{j} (s - \nu_{ij} ) \bigr\Vert _{\infty} \bigr) \Biggr] \,ds \\ \leq& \Vert \varphi \Vert _{\infty}e^{- t_{1} c_{i}^{*}} + \int_{0}^{t_{1}} e^{-( t_{1} - s ) c^{*}_{i}}uM \sum _{j=1}^{n} \Biggl[\bar{d}_{ij} L_{j}^{f} \Vert \varphi \Vert _{\infty}e^{- \lambda s} + \bar{a}_{ij} L_{j}^{g} \Vert \varphi \Vert _{\infty} e^{-\lambda(s -\tau_{ij})} \\ &{}+ \sum_{l=1}^{n} \bar{b}_{ijl} \bigl( L_{j}^{h}M_{l}^{h} \Vert \varphi \Vert _{\infty} e^{-\lambda(s -\sigma_{ij})}+ M_{j}^{h}L_{l}^{h} \Vert \varphi \Vert _{\infty} e^{-\lambda(s -\nu_{ij})} \bigr) \Biggr] \,ds \\ \leq& \Vert \varphi \Vert _{\infty}e^{- t_{1} c_{i}^{*}} + \int_{0}^{t_{1}} e^{-( t_{1} - s ) c^{*}_{i}}u M \Vert \varphi \Vert _{\infty}e^{-\lambda s} \sum_{j=1}^{n} \Biggl[\bar{d}_{ij}L_{j}^{f} + \bar{a}_{ij} L_{j}^{g} e^{\lambda\tau_{ij}} \\ &{}+ \sum_{l=1}^{n} \bar{b}_{ijl} \bigl( L_{j}^{h}M_{l}^{h} e^{\lambda \sigma_{ij}}+ M_{j}^{h}L_{l}^{h} e^{\lambda\nu_{ij}} \bigr) \Biggr] \,ds \\ \leq& u M \Vert \varphi \Vert _{\infty}e^{ - \lambda t_{1} } \Biggl[ e^{( \lambda- a_{i*}) t_{1}} \Biggl( \frac{1}{M}-\frac{1}{c^{*}_{i} - \lambda } \Biggl[\sum _{j=1}^{n} \Biggl\{ \bar{c}_{ij} L_{j}^{f} + \bar{d}_{ij} L_{j}^{g} e^{ \lambda\tau_{ij}} \\ &{}+ \sum_{l=1}^{n} \bar{b}_{ijl} \bigl( L_{j}^{h} e^{\lambda \sigma_{ij}} M_{l}^{h}+ M_{j}^{h}L_{l}^{h} e^{\lambda\nu_{ij}} \bigr) \Biggr\} \Biggr] \Biggr) \\ &{}+ \frac{1}{c^{*}_{i} - \lambda} \Biggl[\sum_{j=1}^{n} \bigl( \bar{d}_{ij} L_{j}^{f} + \bar{a}_{ij} L_{j}^{g} e^{ \lambda\tau_{ij}} \bigr) + \sum_{j=1}^{n} \sum _{l=1}^{n} \bar{b}_{ijl} \bigl( L_{j}^{h} e^{\lambda \sigma_{ij}} M_{l}^{h}+ M_{j}^{h}L_{l}^{h} e^{\lambda\nu_{ij}} \bigr) \Biggr] \Biggr] \\ \leq& u M \Vert \varphi \Vert _{\infty}e^{- \lambda t_{1}} \frac{1}{c^{*}_{i} - \lambda} \Biggl[\sum_{j=1}^{n} \Biggl( \bigl( \bar{c}_{ij} L_{j}^{f} + \bar{d}_{ij} L_{j}^{g} e^{ \lambda\tau_{ij}} \bigr)\\ &{} + \sum_{l=1}^{n} \bar {b}_{ijl} \bigl( L_{j}^{h} e^{\lambda \sigma_{ij}} M_{l}^{h}+ M_{j}^{h}L_{l}^{h} e^{\lambda\nu_{ij}} \bigr) \Biggr) \Biggr] \\ =& u M \Vert \varphi \Vert _{\infty}e^{- \lambda t_{1}}. \end{aligned}$$

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:

$$\begin{aligned} x_{i}'(t) =& -c_{i}(t) x_{i}(t) + \sum_{j=1}^{n} d_{ij} (t) f_{j} \bigl(t,x_{j}(t) \bigr) + \sum _{j=1}^{n} a_{ij}(t) g_{j} \bigl(t,x_{j} (t- \tau_{ij}) \bigr) \\ &{}+ \sum_{j=1}^{n} \sum _{l=1}^{n} b_{ijl}(t) h_{j} \bigl(t,x_{j}(t- \sigma_{ij}) \bigr) h_{l} \bigl(t,x_{l}(t- \nu_{ij}) \bigr) + I_{i}(t), \end{aligned}$$
(4.1)

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

$$x_{i}(s)= \varphi_{i} (s) ,\quad s \in{}({-} \theta, 0 ], \quad i= 1,2, \ldots,n . $$

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:

$$\begin{aligned} x_{i}'(t) =& -c_{i} (t) x_{i}(t) + \sum_{j=1}^{2} d_{ij} (t) f_{j} \bigl(t,x_{j}(t) \bigr) + \sum _{j=1}^{2} a_{ij}(t) g_{j} \bigl(t,x_{j} (t- 1 ) \bigr) \\ &{}+ \sum_{j=1}^{2} \sum _{l=1}^{2} b_{ijl}(t) h_{j} \bigl(t,x_{j}(t-1) \bigr) h_{l} \bigl(t,x_{l}(t- 1) \bigr) + I_{i}(t),\quad 1 \leq i \leq2, \end{aligned}$$
(4.2)

where \(c_{1}=c_{2}=2\), \(g_{1}(t)= g_{2}(t)= \sin t \). Then

$$L^{g_{1}}=L^{g_{2}}=M^{g_{1}}=M^{g_{2}}=1,\qquad \tau_{i j } = \sigma_{i j } = \nu_{i j } = 1 . $$

Measures μ and ν are defined by the following double weights, respectively:

$$ \rho_{1}(t)=e^{\sin(t)},\quad t\in\mathbb{R}, $$

and

$$ \rho_{2}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} e^{t} & \mbox{if } t\leq0, \\ 1 & \mbox{if } t>0.\end{array}\displaystyle \right . $$

Then we have

$$ \frac{2r}{e}\leq\mu \bigl([-r,r] \bigr)= \int_{-r}^{r} e^{\sin(t)}\,dt\leq2er. $$

We now prove that \(\mu\in\mathcal{M}\) satisfies condition (M.1). Indeed,

$$\sin(\tau +a)\leq2+\sin(a) \quad\mbox{for all } \tau\in\mathbb{R}, a\in A, $$

which implies that

$$\mu(\tau+ A)\leq e^{2}\mu(A) \quad\mbox{for all } \tau\in\mathbb{R}, $$

so by [5], \(\nu\in\mathcal{M}\) satisfies condition (M.1). Since

$$ \limsup_{r\rightarrow+\infty}\frac{\mu ([-r,r])}{\nu([-r,r])}= \limsup_{r \rightarrow+\infty}\frac{\int_{-r}^{r}\rho_{1}(t)\,dt}{ \int_{-r}^{r}\rho _{2}(t)\,dt}< \infty, $$

it follows that condition (M.2) is also satisfied. We set

$$\begin{aligned}& \bigl(d_{ij}(t) \bigr)_{1 \leq i,j \leq2} = \begin{pmatrix} \frac{2 \sin t + e^{-t}}{10} & \frac{\cos t}{10}\\ \frac{\sin\sqrt{2} t+ e^{-t}}{10} &\frac{2 \cos\sqrt{2}t + e^{-t}}{10} \end{pmatrix} , \\& \bigl(a_{ij}(t) \bigr)_{1 \leq i,j \leq2} = \begin{pmatrix} \frac{ \cos t + e^{-t}}{10} & \frac{ \sin t}{10}\\ \frac{4 \cos t + e^{-t}}{10} & \frac{\sin t +e^{-t}}{10} \end{pmatrix} , \\& \bigl(I_{i}(t) \bigr)_{1 \leq i,j \leq2} = \begin{pmatrix}\frac{8 \cos\sqrt{5}t}{10} \\ \frac{5 \sin t + e^{-t}}{10} \end{pmatrix} , \\& (b_{1jl}) (t))_{1 \leq j,l \leq2} = \begin{pmatrix} 0 & \frac{3 \sin\sqrt{3} t + e^{-t}}{10} \\ 0 & 0 \end{pmatrix} , \\& \bigl(b_{2jl}(t) \bigr)_{1 \leq j,l \leq2} = \begin{pmatrix} 0 & \frac{2 \cos\sqrt{5} t+ e^{-t}}{10} \\ 0 & 0 \end{pmatrix} . \end{aligned}$$

Therefore

$$L= \frac{4}{10},\qquad p_{1} =\frac{75}{100} < 1,\quad \mbox{and}\quad q_{1}=\frac{9}{10}< 1. $$

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

$$\mathbb{G}= \biggl\{ \varphi\in\mathcal{PAP} \bigl(\mathbb{R}, \mathbb{R}^{n} , \mu,\nu \bigr): \Vert \varphi- \varphi_{0} \Vert _{\infty} \leq\frac {12}{10} \biggr\} . $$