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\((\mu ,\nu )\)-Pseudo-almost automorphic solutions for high-order Hopfield bidirectional associative memory neural networks

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Abstract

This article is concerned with a high-order Hopfield bidirectional associative memory neural networks with time-varying coefficients and mixed delays. Sufficient conditions are derived for the existence, the uniqueness and the exponential stability of \((\mu ,\nu )\)-pseudo-almost automorphic solutions of the considered model. Banach fixed-point theorem is applied for the existence and the uniqueness results. Global exponential stability is derived via differential inequalities. Finally, two examples are provided to support the feasibility of the theoretical results.

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References

  1. 1.

    Abbas S, Yonghui XIA (2013) Existence and attractivity of \(k\)-almost automorphic sequence solution of a model of cellular neural networks with delay. Acta Math Sci 33(1):290–302

  2. 2.

    Abbas S, Mahto L, Hafayed M, Alimi AM (2014) Asymptotic almost automorphic solutions of impulsive neural network with almost automorphic coefficients. Neurocomputing 142:326–334

  3. 3.

    Abbas S, Xia Y (2015) Almost automorphic solutions of impulsive cellular neural networks with piecewise constant argument. Neural Process Lett 42(3):691–702

  4. 4.

    Ait Dads EH, Ezzinbi K, Miraoui M (2015) \((\mu,\nu )\)-Pseudo almost automorphic solutions for some non-autonomous differential equations. Int J Math 26(11):1550090

  5. 5.

    Alimi AM, Aouiti C, Chérif F, Dridi F, M’hamdi MS (2018) Dynamics and oscillations of generalized high-order Hopfield Neural Networks with mixed delays. Neurocomputing. https://doi.org/10.1016/j.neucom.2018.01.061

  6. 6.

    Ammar B, Brahmi H, Chérif F (2017) On the weighted pseudo-almost periodic solution for BAM networks with delays. Neural Process Lett. https://doi.org/10.1007/s11063-017-9725-0

  7. 7.

    Aouiti C (2018) Oscillation of impulsive neutral delay generalized high-order Hopfield neural networks. Neural Comput Appl 29(9):477–495

  8. 8.

    Aouiti C (2016) Neutral impulsive shunting inhibitory cellular neural networks with time-varying coefficients and leakage delays. Cogn Neurodynamics 10(6):573–591

  9. 9.

    Aouiti C, M’hamdi MS, Chérif F (2017) New results for impulsive recurrent neural networks with time-varying coefficients and mixed delays. Neural Process Lett 46(2):487–506

  10. 10.

    Aouiti C, Coirault P, Miaadi F, Moulay E (2017) Finite time boundedness of neutral high-order Hopfield neural networks with time delay in the leakage term and mixed time delays. Neurocomputing 260:378–392

  11. 11.

    Aouiti C, M’hamdi MS, Touati A (2017) Pseudo Almost Automorphic Solutions of Recurrent Neural Networks with Time-Varying Coefficients and Mixed Delays. Neural Process Lett 45(1):121–140

  12. 12.

    Aouiti C, M’hamdi MS, Cao J, Alsaedi A (2017) Piecewise Pseudo Almost Periodic Solution for Impulsive Generalised High-Order Hopfield Neural Networks with Leakage Delays. Neural Process Lett 45(2):615–648

  13. 13.

    Aouiti C, Dridi F (2018) Piecewise asymptotically almost automorphic solutions for impulsive non-autonomous high-order Hopfield neural networks with mixed delays. Neural Comput Appl. https://doi.org/10.1007/s00521-018-3378-4

  14. 14.

    Aouiti C, Gharbia IB, Cao J, M’hamdi MS, Alsaedi A (2018) Existence and global exponential stability of pseudo almost periodic solution for neutral delay BAM neural networks with time-varying delay in leakage terms. Chaos Solitons Fractals 107:111–127

  15. 15.

    Blot J, Cieutat P, Ezzinbi K (2012) Measure theory and pseudo almost automorphic functions: new developments and applications. Nonlinear Anal Theory Methods Appl 75(4):2426–2447

  16. 16.

    Cao J, Wang L (2002) Exponential stability and periodic oscillatory solution in BAM networks with delays. IEEE Trans Neural Netw 13(2):457–463

  17. 17.

    Cao J, Liang J, Lam J (2004) Exponential stability of high-order bidirectional associative memory neural networks with time delays. Physica D Nonlinear Phenom 199(3):425–436

  18. 18.

    Chang CY, Chung PC (2000) Two-layer competitive based Hopfield neural network for medical image edge detection. Opt Eng 39(3):695–704

  19. 19.

    Chang CA, Angkasith V (2001) Using Hopfield neural networks for operational sequencing for prismatic parts on NC machines. Eng Appl Artif Intell 14(3):357–368

  20. 20.

    Chang YK, Luo XX (2014) Existence of \(\mu\)-pseudo almost automorphic solutions to a neutral differential equation by interpolation theory. Filomat 28(3):603–614

  21. 21.

    Chen A, Huang L, Cao J (2003) Existence and stability of almost periodic solution for BAM neural networks with delays. Appl Math Comput 137(1):177–193

  22. 22.

    Choi E, Schuetz A, Stewart WF, Sun J (2016) Using recurrent neural network models for early detection of heart failure onset. J Am Med Inform Assoc 24(2):361–370

  23. 23.

    Diagana T (2013) Almost automorphic type and almost periodic type functions in abstract spaces. Springer, New York

  24. 24.

    Ait Dads EH, Drisi N, Ezzinbi K, Ziat M (2016) Exponential dichotomy and \((\mu , \nu )\)-Pseudo almost automorphic solutions for some ordinary differential equations. Commun Optim Theory 2016:6

  25. 25.

    Ezzinbi K, Fatajou S, N’Guérékata GM (2009) Pseudo-almost-automorphic solutions to some neutral partial functional differential equations in Banach spaces. Nonlinear Anal Theory Methods Appl 70(4):1641–1647

  26. 26.

    Gui R, Yang Z (2006) Application of Hopfield neural network for extracting Doppler spectrum from ocean echo. Radio Sci 41(4):RS4S90-1–RS4S90-6

  27. 27.

    Huo HF, Li WT, Tang S (2009) Dynamics of high-order BAM neural networks with and without impulses. Appl Math Comput 215(6):2120–2133

  28. 28.

    Jagannatha AN, Yu H (2016, June). Bidirectional RNN for medical event detection in electronic health records. In: Proceedings of the conference. Association for Computational Linguistics. North American Chapter. Meeting (Vol 2016, p 473). NIH Public Access

  29. 29.

    Kavitha V, Wang PZ, Murugesu R (2013) Existence of weighted pseudo almost automorphic mild solutions to fractional integro-differential equations. J Fract Calc Appl 4(1):37–55

  30. 30.

    Liang J, Zhang J, Xiao TJ (2008) Composition of pseudo almost automorphic and asymptotically almost automorphic functions. J Math Anal Appl 340(2):1493–1499

  31. 31.

    Li Y, Zhao L, Yang L (2015) \(C^1\)-Almost periodic solutions of BAM neural networks with time-varying delays on time scales. Sci World J. https://doi.org/10.1155/2015/727329

  32. 32.

    Ma F, Chitta R, Zhou J, You Q, Sun T, Gao J (2017, August). Dipole: Diagnosis prediction in healthcare via attention-based bidirectional recurrent neural networks. In: Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining (pp 1903-1911). ACM

  33. 33.

    Marcus CM, Westervelt RM (1989) Stability of analog neural networks with delay. Phys Rev A 39(1):347

  34. 34.

    Miraoui M (2017) \(\mu\)-Pseudo-Almost Automorphic Solutions for some Differential Equations with Reflection of the Argument. Numer Funct Anal Optim 38(3):376–394

  35. 35.

    Pu Z, Rao R (2018) Exponential stability criterion of high-order BAM neural networks with delays and impulse via fixed point approach. Neurocomputing 292:63–71

  36. 36.

    Ren F, Cao J (2007) Periodic oscillation of higher-order bidirectional associative memory neural networks with periodic coefficients and delays. Nonlinearity 20(3):605

  37. 37.

    Sammouda R, Mathkour HB (2015) Lung region segmentation using artificial neural network hopfield model for cancer diagnosis in thorax CT images. Autom Control Intell Syst 3(2):19–25

  38. 38.

    Sammouda RS, Wang X, Basilion JP (2015) Hopfield Neural Network for the segmentation of Near Infrared Fluorescent images for diagnosing prostate cancer. In: 2015 6th International conference on information and communication systems (ICICS), pp 111-118. IEEE

  39. 39.

    Singh YP, Yadav SV, Gupta A, Khare A (2009) Bi directional associative memory neural network method in the character recognition. J Theor Appl Inf Technol 5(4):382–386

  40. 40.

    Wang J, Jiang H, Hu C (2014) Existence and stability of periodic solutions of discrete-time Cohen-Grossberg neural networks with delays and impulses. Neurocomputing 142:542–550

  41. 41.

    Xu C, Chen L, Guo T (2018) Anti-periodic oscillations of bidirectional associative memory (BAM) neural networks with leakage delays. J Inequal Appl. https://doi.org/10.1186/s13660-018-1658-2

  42. 42.

    Yang W, Yu W, Cao J, Alsaadi FE, Hayat T (2017) Almost automorphic solution for neutral type high-order Hopfield BAM neural networks with time-varying leakage delays on time scales. Neurocomputing 267:241–260

  43. 43.

    Zhang L, Si L (2007) Existence and exponential stability of almost periodic solution for BAM neural networks with variable coefficients and delays. Appl Math Comput 194(1):215–22

  44. 44.

    Zhou H, Zhou Z, Jiang W (2015) Almost periodic solutions for neutral type BAM neural networks with distributed leakage delays on time scales. Neurocomputing 157:223–230

  45. 45.

    Zhou H, Alzabut J (2017) Existence and stability of neutraltype BAM neural networks with time delay in the neutral and leakage terms on time scales. Glob J Pure Appl Math 13(2):589–616

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Correspondence to Chaouki Aouiti.

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Appendices

Appendix A Proof of Lemma 1

Proof

By definition, we can express \(\phi\) as \(\phi =f+g\) such that \(f\in AA({\mathbb {R}},{\mathbb {R}})\) and \(g \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu ).\)

Clearly, \(\phi (.-h)=f(.-h)+g(.-h).\)

It is easily seen that \(f(.-h)\in AA({\mathbb {R}},{\mathbb {R}}).\) For each \(h\in {\mathbb {R}},\) letting \(\mu _h=\mu (\{t+h,t\in A\})\) for \(A \in B,\) it follows from (H4) that \(\mu\) and \(\mu _h\) are equivalent. For \(h \ge 0,\) we have

$$\begin{aligned}&\frac{1}{\nu ([-r,r])} \int _{-r}^{r} | g(t-h)| d\mu (t )\\&\quad =\frac{1}{\nu ([-r,r])} \int _{-r-h}^{r-h} | g(t)| d\mu _{h} (t) \\&\quad =\frac{1}{\nu ([-r,r])} \left( \int _{-r-h}^{r+h} | g(t)| d\mu _{h} (t) - \int _{r-h}^{r+h} | g(t)| d\mu _{h}(t) \right) \\&\quad \le \frac{1}{\nu ([-r,r])} \int _{-r-h}^{r+h}| g(t)| d\mu _{h} (t). \end{aligned}$$

For \(h \le 0,\) we have

$$\begin{aligned}&\frac{1}{\nu ([-r,r])} \int _{-r}^{r} | g(t-h)| d\mu (t )\\&\quad =\frac{1}{\nu ([-r,r])} \int _{-r-h}^{r-h} | g(t)| d\mu _{h}(t) \\&\quad =\frac{1}{\nu ([-r,r])} \left( \int _{-r+h}^{r-h} | g(t)| d\mu _{h} (t))-\int _{-r+h}^{-r-h} | g(t)| d\mu _{h} (t) \right) \\&\quad \le \frac{1}{\nu ([-r,r])} \int _{-r+h}^{r-h}| g(t)| d\mu _{h}(t) . \end{aligned}$$

So we obtain

$$\begin{aligned}&\frac{1}{\nu ([-r,r])} \int _{-r}^{r} | g(t-h)| d\mu (t)\\&\quad \le \frac{1}{\nu ([-r,r])} \int _{-r-\mid h \mid }^{r+ \mid h \mid } |g(t)| d\mu _{h}(t) \\&\quad \le \frac{\nu ([-r-|h|,r+|h|])}{\nu ([-r,r])}\frac{1}{\nu ([-r-|h|,r+|h|])}\\&\qquad \times \int _{-r-\mid h \mid }^{r+ \mid h \mid } |g(t)| d\mu _h(t)\\&\quad \le \frac{\nu ([-r-|h|,r+|h|])}{\nu ([-r,r])}\frac{\beta }{\nu ([-r-|h|,r+|h|])}\\&\qquad \times \int _{-r-\mid h \mid }^{r+ \mid h \mid } |g(t)| d\mu (t)\\ \end{aligned}$$

Since \(\nu\) satisfies (H4) and \(g \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu ),\) then

$$\begin{aligned} \lim _{r\mapsto \infty }\frac{1}{\nu ([-r,r])} \int _{-r}^{r} | g(t-h)| d\mu (t)=0. \end{aligned}$$

The space \(PAA({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\) is translation invariant. \(\square\)

Appendix B Proof of Lemma 2

Proof

By definition, we can write \(\phi = \phi _{1}+ \phi _{2} , \psi =\psi _{1}+\psi _{2}\) where \(\phi _{1} , \psi _{1} \in AA({\mathbb {R}},{\mathbb {R}})\) and \(\phi _{2} , \psi _{2} \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\).

Clearly, \(\phi \psi =\phi _{1}\psi _{1}+\phi _{1}\psi _{2}+\phi _{2}\psi _{1} +\phi _{2}\psi _{2},\)\(\phi _{1}\psi _{1} \in AA({\mathbb {R}},{\mathbb {R}})\) is hold.

For \([\phi _{1}\psi _{2}+\phi _{2}\psi _{1} +\phi _{2}\psi _{2}],\) we have

$$\begin{aligned}&\lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{-r}^{r} |\phi _{1} \psi _{2} + \varphi _{2} \psi _{1} + \varphi _{2} \psi _{2} | d\mu (t)\\&\quad \le \lim _{r\longrightarrow \infty }\frac{1}{\nu ([-r,r])} \int _{-r}^{r} ( \parallel \phi _{1} \parallel _{\infty } | \psi _{2} | + |\phi _{2}| \Vert \psi _{1} \Vert _{\infty } \\&\qquad + \parallel \phi _{2} \parallel _{\infty } | \psi _{2}| ) d\mu (t)= 0, \end{aligned}$$

from which \([\phi _{1}\psi _{2}+\phi _{2}\psi _{1} +\varphi _{2}\psi _{2}] \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\).

Then, \(\phi \times \psi \in PAA({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\). \(\square\)

Appendix C Proof of Lemma 3

Proof

We have \(y = y_{1}+ y_{2},\) where \(y_{1} \in AA({\mathbb {R}},{\mathbb {R}})\) and \(y_{2} \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu ).\) Let

$$\begin{aligned} {\hat{E}}(t)&= {} g(y(t-\tau ))\\&= g(y_{1}(t-\tau ))+ [g(y_{1}(t-\tau )+y_{2}(t-\tau ))\\ & \quad - g(y_{1}(t-\tau ))]\\&= {\hat{E}}_{1}(t) + {\hat{E}}_{2}(t). \end{aligned}$$

Since \(y_{1}\) is almost automorphic, so for every sequence of real numbers \(\left( s_{n}^{\prime }\right)\), we can extract a subsequence \(\left( s_{n}\right)\) such that \(\lim \nolimits _{n\rightarrow +\infty }y_{1}\left( t+s_{n}\right) =y_{1}^{1}\left( t\right) ,\)\(\lim \nolimits _{n\rightarrow +\infty }y_{1}^{1}\left( t-s_{n}\right) =y_{1}\left( t\right) ,\) for \(t \in {\mathbb {R}}\). Clearly,

$$\begin{aligned}&\mid {\hat{E}}_{1}(t +s_{n})- g(y_{1}^{1}(t - \tau ))\mid \\&\quad \le \mid g(y_{1}(t - \tau +s_n))- g(y_{1}^{1}(t - \tau )) \mid \\&\quad \le m\mid y_{1}(t - \tau +s_n)- y_{1}^{1}(t - \tau ) \mid \rightarrow 0, n\rightarrow \infty \end{aligned}$$

Reasoning in a similar way, we can show easily that for \(n > N,\)\(\{ y_{1}^{1}(t - \tau -s_{n} )\}\) converges to \(y_{1}(t - \tau )\) uniformly on \({\mathbb {R}}.\)

So \(y_{1}(t - \tau )\) is almost automorphic . \({\hat{E}}_{1}(t)\in AA({\mathbb {R}},{\mathbb {R}})\).

Now, we show that \({\hat{E}}_{2}(t) \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\)

$$\begin{aligned}&\lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} | {\hat{E}}_{2}(t) | d\mu (t) \\&\quad =\lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} | g(y_{1}(t-\tau )\\&\qquad + y_{2}(t-\tau ))- g(y_{1}(t-\tau ))| d\mu (t) \\&\quad \le \lim _{r\longrightarrow \infty } \frac{ m_j}{\nu ([-r,r])} \int _{[-r,r]} | y_{2}(t-\tau ))| d\mu (t)=0. \end{aligned}$$

Thus, \({\hat{E}}_{2}(t) \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\). So, \(E(t) \in PAA({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\). \(\square\)

Appendix D Proof of Lemma 4

Proof

We know that

$$\begin{aligned} \phi _{ij}(t)= & {} \int \limits _{-\infty }^{t} N_{ij}(t-s) h_{i}(x_{i}(s)) {\mathrm{d}}s=\int \limits _{0}^{\infty } N_{ij}(s) h_{i}(x_{i}(t-s)) {\mathrm{d}}s.\\ \end{aligned}$$

First, we have

$$\begin{aligned} |\phi _{ij}(t) |\le & {} \int \limits _{-\infty }^{t} N_{ij}(t-s) |h_{i}^{2}(x_{i}(s))| {\mathrm{d}}s\\\le & {} \int \limits _{-\infty }^{t} N_{ij}(t-s) |h_{i}^{2}| {\mathrm{d}}s = |h_{i}^{2}| \end{aligned}$$

which proves that the integral \(\int \nolimits _{-\infty }^{t} N_{ij}(t-s) h_{i}^{2}(x_{i}(s)) {\mathrm{d}}s\) is absolutely convergent and the function \(\phi _{ij}\) is bounded.

Here, we have to prove the continuity of the function \(\phi _{ij}\).

Let \((w_{n})_{n}\) be a sequence of real numbers such that \(\lim \nolimits _{n\longrightarrow \infty } w_{n} = 0\).

The continuity of the function \(x_{i}(.)\) implies that for all \(\epsilon > 0\), \(\exists N \in \mathbb {N}\) such that for all \(n \ge N,\)\(\mid x_{i}(s+w_{n})- x_{i}(s)\mid \le \frac{\epsilon }{ D_{i}}.\) We have

$$\begin{aligned}&\mid \phi _{ij}(t+w_{n})- \phi _{ij}(t)\mid \\&\quad = \mid \int \limits _{-\infty }^{t+w_{n}} N_{ij}(t+w_{n}-s) h_{i}^{2}(x_{i}(s)) {\mathrm{d}}s- \int \limits _{-\infty }^{t} N_{ij}(t-s)h_{i}^{2}(x_{i}(s)) {\mathrm{d}}s \mid \\&\quad \le \int \limits _{-\infty }^{t} N_{ij}(t-s)\mid h_{i}^{2}(x_{i}(s+w_{n}))-h^{2}_{i} (x_{i}(s)) \mid {\mathrm{d}}s \\&\quad \le \int \limits _{-\infty }^{t} N_{ij}(t-s) D_{i} \mid x_{i}(s+w_{n})- x_{i}(s) \mid {\mathrm{d}}s \\&\quad \le \int \limits _{-\infty }^{t} N_{ij}(t-s) {\mathrm{d}}s D_{i}\frac{\epsilon }{ D_{i}} \le \epsilon ; \end{aligned}$$

it remains to be proven whether the function \(\phi _{ij} \in PAA({\mathbb {R}}, {\mathbb {R}},\mu ,\nu ).\) Since \(h_{i}^{2}(x_{i}(.))\in PAA({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\), we can write \(h_{i}^{2}(x_{i}(s))= u_{i}(s)+v_{i}(s)\) such that \(u_{i} (.)\in AA({\mathbb {R}},{\mathbb {R}})\) and \(v_{i}(.) \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\). Consequently,

$$\begin{aligned} \phi _{ij}(t)= & {} \int \limits _{-\infty }^{t} N_{ij}(t-s) [u_{i}(s)+v_{i}(s)] {\mathrm{d}}s \\= & {} \int \limits _{-\infty }^{t} N_{ij}(t-s) u_{i}(s) {\mathrm{d}}s + \int \limits _{-\infty }^{t} N_{ij}(t-s) v_{i}(s) {\mathrm{d}}s \\= & {} \phi _{ij}^{1}(t)+ \phi _{ij}^{2}(t). \end{aligned}$$

Let \(\left( s_{n}^{\prime }\right)\) be a sequence of real numbers; we can extract a subsequence \(\left( s_{n}\right)\) of \(\left( s_{n}^{\prime }\right)\) such that for all \(t\in {\mathbb {R}}:\)\(\lim \nolimits _{n\rightarrow +\infty }u_{i}\left( t+s_{n}\right) =u_{i}^{1}\left( t\right) ,\) and, \(\lim \nolimits _{n\rightarrow +\infty }u_{i}^{1}\left( t-s_{n}\right) =u_{i}\left( t\right) ,\)

Let \(\phi _{ij}^{1*}(t) = \int _{-\infty }^{t} N_{ij}(t-s) u_{i}^{1}(s) {\mathrm{d}}s.\) Hence,

$$\begin{aligned}&\mid \phi _{ij}^{1}(t+s_{n})- \phi _{ij}^{1*}(t)\mid \\&\quad =\, \mid \int \limits _{-\infty }^{t+s_{n}} N_{ij}(t+s_{n}-s) u_{i}(s) {\mathrm{d}}s - \int \limits _{-\infty }^{t} N_{ij}(t-s) u_{i}^{1}(s) {\mathrm{d}}s \mid \\&\quad =\, \mid \int \limits _{-\infty }^{t} N_{ij}(t-s) u_{i}(s+s_{n}) {\mathrm{d}}s - \int \limits _{-\infty }^{t} N_{ij}(t-s) u_{i}^{1}(s) {\mathrm{d}}s \mid \\&\quad \le \int \limits _{-\infty }^{t} N_{ij}(t-s) \mid u_{i}(s+s_{n}) - u_{i}^{1}(s) \mid {\mathrm{d}}s \\&\quad \le\, \epsilon \mid u_{i}(s+s_{n}) - u_{i}^{1}(s) \mid . \end{aligned}$$

We obtain immediately that \(\lim \nolimits _{n\rightarrow +\infty } \phi _{ij}^{1}(t+s_{n}) = \phi _{ij}^{1*}(t),\; \forall \; t \in {\mathbb {R}}.\)

Reasoning in a similar way, we can show easily that \(\lim \nolimits _{n\rightarrow +\infty } \phi _{ij}^{1*}(t- s_{n}) = \phi _{ij}^{1}(t), \;\forall \; t \in {\mathbb {R}},\) which implies that for all , \(1\le i\le n,\) we have \(\phi _{ij}^{1} \in AA({\mathbb {R}},{\mathbb {R}}).\)

Second, let us show that for all \(1\le i\le n,\)\(1\le j\le p,\) the function \((\phi _{ij}^{2}(t)) \in \xi ({\mathbb {R}},{\mathbb {R}}^n,\mu ,\nu )\).

We obtain the following estimate

$$\begin{aligned}&\lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]}| \phi _{ij}^{2}(t)| d\mu (t)\\&\quad \le \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int \limits _{0}^{\infty } N_{ij}(s) | v_{i}(t-s) | {\mathrm{d}}s\right) d\mu (t) \\&\quad \le \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int \limits _{0}^{\infty } N_{ij}(s) \left( \int \limits _{[-r,r]} | v_{i}(t-s) | d\mu (t)\right) {\mathrm{d}}s \\&\quad = \int \limits _{0}^{\infty } N_{ij}(s)\left( \lim _{r\longrightarrow \infty }\frac{1}{\nu ([-r,r])} \int \limits _{[-r,r]} | v_{i}(t-s) | d\mu (t)\right) {\mathrm{d}}s \\&\quad =0 \end{aligned}$$

which implies that \(\phi _{ij}^{2}(t) \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\).

Consequently, \(\phi _{ij}\) is \((\mu ,\nu )\)-pseudo-almost automorphic function. \(\square\)

Appendix E Proof of Lemma 5

Proof

By using Lemma 1-4, for all \(1 \le i \le n,\)\(1 \le j \le p,\) the functions \(F_{i}(.)\) and \(G_{j}(.)\) are \((\mu ,\nu )\)-pseudo-almost automorphic.

Consequently, for all, \(1 \le i \le n,\)\(1 \le j \le p\), we can write \(F_{i}=F_{i}^{1}+F_{i}^{2}\) and \(G_{j}=G_{j}^{1}+G_{j}^{2}\) such that \(F_{i}^{1},G_{j}^{1} \in AA({\mathbb {R}},{\mathbb {R}}),\) and \(F_{i}^{2},G_{j}^{2} \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu ).\)

First, we prove that the function \((\varGamma _i F_i)(t)= \int \nolimits _{-\infty }^{t} e^{-\int \limits _{s}^{t} a_{i}^{1}(u){\mathrm{d}}u} F_i(s){\mathrm{d}}s\) is \((\mu ,\nu )\)-pseudo-almost automorphic function.

$$\begin{aligned} (\varGamma _i F_i)(t)= & {} \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(u) {\mathrm{d}}u} F_{i}^{1}(s){\mathrm{d}}s + \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(u) {\mathrm{d}}u } F_{i}^{2}(s){\mathrm{d}}s \\=\, & {} (\varGamma _{i}F_{i}^{1})(t) +(\varGamma _{i}F_{i}^{2})(t), \end{aligned}$$

Let \(\left( s_{n}^{\prime }\right) _{n \in \mathbb {N}}\) be a sequence of real numbers. By hypothesis, we can extract a subsequence \((s_{n})_{n \in \mathbb {N}}\) of \(\left( s_{n}^{\prime }\right) _{n \in \mathbb {N}}\) such that \(\lim \nolimits _{n\rightarrow +\infty }a_{i}^{1} \left( t+s_{n}\right) =a_{i}^{1*}\left( t\right) , \;\; \lim \nolimits _{n\rightarrow +\infty }a_{i}^{1*}\left( t-s_{n}\right) =a_{i}^{1}\left( t\right) ,\;\forall \; t\in {\mathbb {R}}.\)

\(\lim \nolimits _{n\rightarrow +\infty }F_{i}^{1} \left( t+s_{n}\right) = F_{i}^{1*}\left( t\right) , \;\; \lim \nolimits _{n\rightarrow +\infty }F_{i}^{1*}\left( t-s_{n}\right) = F_{i}^{1}\left( t\right) ,\;\forall \; t\in {\mathbb {R}}.\)

Pose \((\varGamma _{i}^{1}F_{i}^{1*})(t)=\int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1*}(u) {\mathrm{d}}u } F_{i}^{1*}(s){\mathrm{d}}s\). We write

$$\begin{aligned}&(\varGamma _{i}F_{i}^{1})(t+s_{n})-(\varGamma _{i}^{1}F_{i}^{1*})(t)\\&\quad =\int \limits _{-\infty }^{t+s_{n}} e^{-\int \limits _{s}^{t+s_{n}}a_{i}^{1}(u) {\mathrm{d}}u } F_{i}^{1}(s){\mathrm{d}}s - \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a^{1*}_{i}(u) {\mathrm{d}}u } F_{i}^{1*}(s){\mathrm{d}}s \\&\quad = \int \limits _{-\infty }^{t+s_{n}} e^{-\int \limits _{s-s_{n}}^{t}a_{i}^{1}(\rho +s_{n}) d\rho } F_{i}^{1}(s){\mathrm{d}}s -\int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a^{1*}_{i}(u) {\mathrm{d}}u } F_{i}^{1*}(s){\mathrm{d}}s \\&\quad = \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(\rho +s_{n}) d\rho } F_{i}^{1}(s+s_{n}){\mathrm{d}}s - \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(\rho +s_{n}) d\rho } F_{i}^{1*}(s){\mathrm{d}}s \\&\qquad + \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(\rho +s_{n}) d\rho } F_{i}^{1*}(s){\mathrm{d}}s -\int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a^{1*}_{i}(u) {\mathrm{d}}u } F_{i}^{1*}(s){\mathrm{d}}s \\&\quad = \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u } (F_{i}^{1}(s+s_{n}) - F_{i}^{1*}(s)){\mathrm{d}}s \\&\qquad + \int \limits _{-\infty }^{t} \left( e^{-\int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u } - e^{-\int \limits _{s}^{t}a^{1*}_{i}(u) {\mathrm{d}}u } \right) F_{i}^{1*}(s){\mathrm{d}}s. \\ \end{aligned}$$

Then, there exists \(\theta \in ]0,1[\) such that

$$\begin{aligned}&|(\varGamma _{i} F_{i}^{1} )(t+s_{n})-(\varGamma _{i} ^{1}F_{i}^{1*} )(t) |\\&\quad \le |F_{i}^{1*}|_\infty \int \limits _{-\infty }^{t}\left( e^{-\int \limits _{s}^{t}a_{i}^{1}\left( u +s_{n}\right) {\mathrm{d}}u }-e^{-\int \limits _{s}^{t}a_{i}^{1*}\left( u \right) {\mathrm{d}}u }\right) {\mathrm{d}}s\\&\qquad +\int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a_{i}^{1}\left( u +s_{n}\right) {\mathrm{d}}u }\left| F_{i}^{1} \left( s+s_{n}\right) -F_{i}^{1*}\left( s\right) \right| {\mathrm{d}}s \\&\quad \le \int \limits _{-\infty }^{t}\left( e^{- \left[ \int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u + \theta \left( \int \limits _{s}^{t}a_{i}^{1*}( u) {\mathrm{d}}u-\int \limits _{s}^{t}a_{i}^{1}(u+s_{n}) {\mathrm{d}}u\right) \right] } \right. \\&\left. \qquad \times \int _s^t |a_{i}^{1*}(u)-a_{i}^{1}(u+s_{n})|{\mathrm{d}}u \right) {\mathrm{d}}s |F_{i}^{1*}|_\infty \\&\qquad +\int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u }\left| F_{i}^{1} ( s+s_{n}) -F_{i}^{1*}(s) \right| {\mathrm{d}}s\\&\le \int \limits _{-\infty }^{t}\left( e^{-\left[ \int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u + \theta \left( \int \limits _{s}^{t}a_{i}^1(u) {\mathrm{d}}u-\int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u\right) \right] }\right. \\&\left. \quad \times \int _s^t |a_{i}^{1*}(u)-a_{i}^{1}(u+s_{n})|{\mathrm{d}}u \right) {\mathrm{d}}s |F_{i}^{1*}|_\infty \\&\qquad +\int \limits _{-\infty }^{t}e^{-a_{i*}^{1}(t-s)}\left| F_{i}^{1} \left( s+s_{n}\right) -F_{i}^{1*}\left( s\right) \right| {\mathrm{d}}s\\&\quad \le \int \limits _{-\infty }^{t}\left\{ e^{-a_{i*}^{1}(t-s)} e^{- \theta \left( \int \limits _{s}^{t}\left| a_{i}^1*( u )-a_{i}^{1}(u +s_{n})\right| {\mathrm{d}}u \right) }\right. \\&\left. \qquad \times \int _s^t \left| a_{i}^{1*}(u)-a_{i}^{1}(u+s_{n})\right| {\mathrm{d}}u \right\} {\mathrm{d}}s |F_{i}^{1*}|_\infty \\&\qquad + \int \limits _{-\infty }^{t}e^{-a_{i*}^{1}(t-s)} \left| F_{i}^{1} (s+s_{n}) -F_{i}^{1*}(s) \right| {\mathrm{d}}s\\&\quad \le |F_{i}^{1*}|_\infty \int \limits _{-\infty }^{t} \left\{ e^{-a_{i*}^{1}(t-s)} \int _s^t |a_{i}^{1*}(u)-a_{i}^{1}(u +s_{n})|{\mathrm{d}}u \right\} {\mathrm{d}}s \\&\qquad + \int \limits _{-\infty }^{t}e^{-a_{i*}^{1}(t-s)} \left| F_{i}^{1} (s+s_{n}) -F_{i}^{1*}(s) \right| {\mathrm{d}}s\\&\quad = \int \limits _{-\infty }^{t}\varPhi _{i}(t,s){\mathrm{d}}s+\int \limits _{-\infty }^{t}\Psi _{i}(t,s){\mathrm{d}}s, \end{aligned}$$

such that \(\varPhi _{i}(t,s)=e^{-a_{i*}^{1}(t-s)}|F_{i}^{1*}|_\infty \int \nolimits _s^t |a_{i}^{1*}(u)-a_{i}^{1}(u+s_{n})|{\mathrm{d}}u,\) and \(\Psi _{i}(t,s)=e^{-a_{i*}^{1}(t-s)} \left| F_{i}^{1} \left( s+s_{n}\right) -F_{i}^{1*}\left( s\right) \right| .\)

By the Lebesgue dominated convergence theorem, we have for all \(t\in {\mathbb {R}},\)\(\lim \nolimits _{n\rightarrow +\infty }(\varGamma _{i} F_{i}^{1} )(t+s_{n})=(\varGamma _{i} ^{1}F_{i}^{1*} )(t).\)

The same approach proves that, \(\forall \; t\in {\mathbb {R}}\)\(\lim \nolimits _{n\rightarrow +\infty }(\varGamma _{i}^{1}F_{i}^{1*})(t-s_{n})=(\varGamma _{i} F_{i}^{1})(t),\) so \((\varGamma _{i}F_{i}^{1}) \in AA({\mathbb {R}},{\mathbb {R}}).\)

Reasoning in a similar way, we can show that \((\varGamma _{j}G_{j}^{1}) \in AA({\mathbb {R}},{\mathbb {R}}).\) Now, we focus on \((\varGamma _{i}F_{i}^{2}) \in \xi ({\mathbb {R}},{\mathbb {R}}^n,\mu ,\nu ).\)

$$\begin{aligned}&\lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} | (\varGamma _{i}F_{i}^{2})(t) | d\mu (t)\\&\quad = \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]}\left| \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1} (u) {\mathrm{d}}u} F_{i}^{2}(s) {\mathrm{d}}s \right| d\mu (t) \\&\quad \le \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \int \limits _{-\infty }^{t} e^{-(t-s)a_{i*}^{1}} |F_{i}^{2}(s)| {\mathrm{d}}s d\mu (t) \\&\quad \le E_{1} + E_{2}, \text{ such } \text{ that }\\&\quad E_{1} = \lim \limits _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int _{-r}^{t} | e^{-(t-s)a_{i*}^{1}} F_{i}^{2}(s) | {\mathrm{d}}s \right) d\mu (t), \\&\quad E_{2} = \lim \limits _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int \limits _{- \infty }^{-r} e^{-(t-s)a_{i*}^{1}} |F_{i}^{2}(s)| {\mathrm{d}}s\right) d \mu (t).\\ \end{aligned}$$

Let \(m= t-s,\) then by Fubini's theorem, we obtain

$$\begin{aligned} E_{1}= & {} \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int _{-r}^{t} e^{-(t-s)a_{i*}^{1}}| F_{i}^{2}(s) | {\mathrm{d}}s \right) d\mu (t) \\\le & {} \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int \limits _{0}^{+ \infty } e^{-m a_{i*}^{1}}| F_{i}^{2}(t-m) |{\mathrm{d}}m \right) d\mu (t) \\\le & {} \int \limits _{0}^{+ \infty } e^{-m a_{i*}^{1}} \left( \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} | F_{i}^{2}(t-m) |d \mu (t) \right) {\mathrm{d}}m \\= & {} \int \limits _{0}^{+ \infty } e^{-m a_{i*}^{1}} \left( \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int \limits _{-r-m}^{r-m} | F_{i}^{2}(t) | d\mu _m(t) \right) {\mathrm{d}}m \\\le & {} \int _{0}^{+ \infty } e^{-m a_{i*}^{1}} \left( \lim _{r\longrightarrow \infty } \frac{\nu ([-r-m,r+m] ) }{\nu ([-r,r]}\frac{\beta }{\nu ([-r-m,r+m])} \right. \\&\times \int \limits _{[-r-m,r+m] } | F_{i}^{2}(t) | d\mu (t) ) {\mathrm{d}}m. \end{aligned}$$

Since \(\nu\) satisfies (H4) and \(F_{i}^{2} \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\), then \(E_1=0.\)

On the other hand, let, \(| F_{i}^{2} |_{\infty }= \sup \nolimits _{t \in {\mathbb {R}}} | F_{i}^{2} (t)|< \infty\) then

$$\begin{aligned} E_{2}= & {} \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int \limits _{- \infty }^{-r} | e^{-(t-s)a_{i*}^{1}} F_{i}^{2}(s)| {\mathrm{d}}s\right) d\mu (t) \\= & {} \lim _{r\longrightarrow \infty }\frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int \limits _{- \infty }^{-r} e^{-(t-s)a_{i*}^{1}} |F_{i}^{2}(s)| {\mathrm{d}}s \right) d\mu (t) \\\le & {} \lim _{r\longrightarrow \infty }\frac{1}{\nu ([-r,r])} \int \limits _{- \infty }^{- r} e^{s a_{i*}^{1}} | F_{i}^{2}(s)| {\mathrm{d}}s\int \limits _{-r}^{r} e^{- t a_{i*}^{1}} d\mu (t) \\= & {} \lim _{r\longrightarrow \infty } \frac{|F_{i}^{2}|_{\infty }}{ a_{i*}^{1}} e^{- 2r a_{i*}^{1}}.\\&\text{ So }\\&\lim \limits _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{-r}^{r} \left| \int \limits _{-\infty }^{t} e^{-(t-s)a_{i*}^{1}} F_{i}^{2}(s) {\mathrm{d}}s \right| d\mu (t) =0. \end{aligned}$$

Consequently, the function \((\varGamma _{i}F_{i}^{2})\in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )\).

The same approach proves that, \(\forall \; t\in {\mathbb {R}}\), \((\varGamma _{j}G_{j}^{2}) \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu ).\)

Therefore, \(\varGamma _{Z}\) maps \(PAA({\mathbb {R}},{\mathbb {R}}^{n+p},\mu ,\nu )\) into itself. \(\square\)

Appendix F Proof of Theorem 3

Proof

We have

$$\begin{aligned} \Vert Z_{0}\Vert= & {} \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{ 1 \le i\le n}\left\{ \left| \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a^{1}_{i}(u){\mathrm{d}}u}I_{i}(s){\mathrm{d}}s \right| \right\} ;\right. \\&\left. \sup _{t \in {\mathbb {R}}} \max _{ 1 \le j\le p} \left\{ \left| \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a^{2}_{j}(u){\mathrm{d}}u}J_{j}(s){\mathrm{d}}s\right| \right\} \right\} \\\le & {} \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{ 1 \le i\le n}\left\{ \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a^{1}_{i}(u){\mathrm{d}}u} \left| I_{i}(s)\right| {\mathrm{d}}s\right\} ,\right. \\&\left. \sup _{t \in {\mathbb {R}}} \max _{1 \le j\le p}\left\{ \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a^{2}_{j}(u){\mathrm{d}}u} \left| J_{j}(s)\right| {\mathrm{d}}s \right\} \right\} \\\le & {} \max \left\{ \max \limits _{ 1 \le i\le n}\left\{ \frac{I_{i}^{*}}{a_{i*}^{1}}\right\} ;\max \limits _{ 1 \le j\le p}\left\{ \frac{J_{j}^{*}}{a_{j*}^{2}}\right\} \right\} =\varpi , \end{aligned}$$

after,

$$\begin{aligned} \Vert Z\Vert \le \Vert Z-Z_{0}\Vert _{\infty }+\Vert Z_{0}\Vert _{\infty }\le & {} \frac{\lambda }{1-\lambda }\varpi +\varpi \\= & {} \frac{\varpi }{1-\lambda }.\\ \end{aligned}$$

Set \(S^{*}=\left\{ Z \in PAA({\mathbb {R}},{\mathbb {R}}^{n+p},\mu ,\nu ) ; \Vert Z-Z_{0}\Vert _{\infty }\le \frac{\lambda }{1-\lambda }\varpi \right\} .\) Clearly, \(S^{*}\) is a closed convex subset of \(PAA({\mathbb {R}},{\mathbb {R}}^{n+p},\mu ,\nu ).\) For any \(Z \in S^{*}\) by using the estimate just obtained, we obtain

$$\begin{aligned}&\Vert \varGamma _{Z}-Z_{0}\Vert \\&\quad = \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{1 \le i\le n} \left\{ \left| \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t} a_{i}^{1}(u) {\mathrm{d}}u} \left( \sum \limits _{j=1}^{p}b^{1}_{ji}(s) f_{j}^{1}(\psi _{j}(s))\right. \right. \right. \right. \\&\qquad +\sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}c^{1}_{jil}(s) g_{j}^{1}(\psi _{j}(s-\tau ))g^{1}_{l}(\psi _{l}(s-\tau ))\\&\qquad +\sum \limits _{j=1}^{p}d^{1}_{ji}(s) \int \limits _{-\infty }^{s} K_{ji}(s-m) h_{j}^{1}(\psi _{j}(m)){\mathrm{d}}m\\&\qquad + \sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}r^{1}_{jil}(s)\int \limits _{-\infty }^{s} P_{jil}(s-m) k^{1}_{j}(\psi _{j}(m)){\mathrm{d}}m\\&\left. \left. \qquad \times \int \limits _{-\infty }^{t} Q_{jil}(s-m) k^{1}_{l}(\psi _{l}(m)){\mathrm{d}}m ){\mathrm{d}}s)\right| \right\} ,\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{ 1 \le j \le p}\left\{ \left| \int _{-\infty }^{t}e^{-\int _{s}^{t} a_{j}^{2}(u){\mathrm{d}}u}\left( \sum \limits _{i=1}^{n}b^{2}_{ij}(s) f_{i}^{2}(\varphi _{i}(s))\right. \right. \right. \\&\qquad + \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2}_{ijl}(s)g_{i}^{2}(\varphi _{i}(s-\gamma )g^{2}_{l}(\varphi _{l}(s-\gamma ))\\&\qquad +\sum \limits _{i=1}^{n} d^{2}_{ij}(s)\int \limits _{-\infty }^{s}N_{ij}(s-m) h_{i}^{2}(\varphi _{i}(m)){\mathrm{d}}m\\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n}r^{2}_{ijl}(s) \int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) k^{2}_{i}(\varphi _{i}(m)){\mathrm{d}}m \\&\left. \left. \left. \left. \qquad \times \int \limits _{-\infty }^{s}\tilde{Q}_{ijl}(s-m) k^{2}_{l}(\varphi _{l}(m)){\mathrm{d}}m){\mathrm{d}}s\right) \right| \right\} \right\} \\&\quad \le \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{1 \le i\le n} \left\{ \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t} a_{i}^{1}(u) {\mathrm{d}}u} \left( \sum \limits _{j=1}^{p}\left| b^{1}_{ji}(s)\right| \left| f_{j}^{1}(\psi _{j}(s))\right| \right. \right. \right. \\&\qquad +\sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}\left| c^{1}_{jil}(s)\right| \left| g_{j}^{1}(\psi _{j}(s-\tau ) g^{1}_{l}(\psi _{l}(s-\tau ))\right| \\&\qquad +\sum \limits _{j=1}^{p}\left| d^{1}_{ji}(s)\right| \int \limits _{-\infty }^{s} K_{ji}(s-m) \left| h_{j}^{1}(\psi _{j}(m))\right| {\mathrm{d}}m\\&\qquad + \sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}\left| r^{1}_{jil}(s)\right| \int \limits _{-\infty }^{s} P_{jil}(s-m)\left| k^{1}_{j}(\psi _{j}(m))\right| {\mathrm{d}}m\\&\left. \left. \qquad \times \int \limits _{-\infty }^{t} Q_{jil}(s-m)\left| k^{1}_{l}(\psi _{l}(m))\right| {\mathrm{d}}m){\mathrm{d}}s\right) \right\} ;\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{ 1 \le j \le p}\left\{ \int _{-\infty }^{t}e^{-\int _{s}^{t} a_{j}^{2}(u){\mathrm{d}}u} \left( \sum \limits _{i=1}^{n}\left| b^{2}_{ij}(s)\right| \left| f_{i}^{2}(\varphi _{i}(s))\right| \right. \right. \\&\qquad + \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n} \left| c^{2}_{ijl}(s)\right| \left| g_{i}^{2}(\varphi _{i}(s-\gamma ) g^{2}_{l}(\varphi _{l}(s-\gamma ))\right| \\&\qquad +\sum \limits _{i=1}^{n}\left| d^{2}_{ij}(s)\right| \int \limits _{-\infty }^{s}N_{ij}(s-m) \left| h_{i}^{2}(\varphi _{i}(m)\right| {\mathrm{d}}m\\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n}\left| r^{2}_{ijl}(s)\right| \int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) \left| k^{2}_{i} (\varphi _{i}(m))\right| {\mathrm{d}}m\\&\left. \left. \qquad \times \int \limits _{-\infty }^{s}\tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\varphi _{l}(m))\right| {\mathrm{d}}m){\mathrm{d}}s)\right\} \right\} \\&\quad \le \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{1 \le i\le n} \left\{ \int _{-\infty }^{t}e^{-a^{1}_{i*}(t-s)} \left( \sum \limits _{j=1}^{p}b^{1*}_{ji}\left| f_{j}^{1}(\psi _{j}(s))\right| \right. \right. \right. \\&\qquad +\sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1*}_{jil}\left| g_{j}^{1}(\psi _{j}(s-\tau )g^{1}_{l}(\psi _{l}(s-\tau ))\right| \\&\qquad +\sum \limits _{j=1}^{p}d^{1*}_{ji} \int \limits _{-\infty }^{s} K_{ji}(s-m) \left| h_{j}^{1}(\psi _{j}(m))\right| {\mathrm{d}}m\\&\qquad + \sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}r^{1*}_{jil}\int \limits _{-\infty }^{s} P_{jil}(s-m) \left| k^{1}_{j}(\psi _{j}(m))\right| {\mathrm{d}}m\\&\left. \left. \qquad \times \int \limits _{-\infty }^{t} Q_{jil}(s-m) \left| k^{1}_{l}(\psi _{l}(m))\right| ){\mathrm{d}}s\right) \right\} ;\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{1\le j\le p}\left\{ \int \limits _{-\infty }^{t}e^{-a^{2}_{j*}(t-s)} \left( \sum \limits _{i=1}^{n}b^{2*}_{ij}\left| f_{i}^{2}(\varphi _{i}(s))\right| \right. \right. \\&\qquad + \sum \limits _{i=1}^{n}\sum \limits _{i=1}^{n}c^{2*}_{ijl} \left| g_{i}^{2}(\varphi _{i}(s-\gamma )g^{2}_{l}(\varphi _{l}(s-\gamma )) \right| \\&\qquad +\sum \limits _{i=1}^{n} d^{2*}_{ij}\int \limits _{-\infty }^{s}N_{ij} (s-m) \left| h_{i}^{2}(\varphi _{i}(m)\right| {\mathrm{d}}m\\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n}r^{2*}_{ijl} \int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m)\left| k^{2}_{i}(\varphi _{i}(m))\right| {\mathrm{d}}m\\&\left. \left. \qquad \times \int \limits _{-\infty }^{s}\tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\varphi _{l}(m))\right| {\mathrm{d}}m){\mathrm{d}}s)\right\} \right\} \\ \end{aligned}$$
$$\begin{aligned}&\quad \le \max \left\{ \sup _{t \in {\mathbb {R}}}\max _{1 \le i\le n} \left\{ \int _{-\infty }^{t}e^{-a^{1}_{i*}(t-s)} \sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}\left( b^{1*}_{ji}l_{j}+c^{1*}_{jil}m_{j}e_{l}+d^{1*}_{ij}d_{j} \right. \right. \right. \\&\left. \left. \qquad +\,r^{1*}_{jil}\alpha _{j}w_{l}\right) {\mathrm{d}}s\right\} ;\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{1 \le j\le p} \left\{ \int _{-\infty }^{t}e^{-a^{2}_{j*}(t-s)} \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}\left( b^{2*}_{ij}L_{i}+c^{2*}_{ijl}M_{i}E_{l}\right. \right. \\&\left. \left. \left. \qquad +\,d^{2*}_{ij}D_{i}+r^{2*}_{ijl}\beta _{i}W_{l}\right) {\mathrm{d}}s\right\} \right\} \Vert Z\Vert \\&\quad = \max \left\{ \max _{1\le i\le n}\left\{ \frac{1}{a^{1}_{i*}} \sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}(b^{1*}_{ji}l_{j} +c^{1*}_{jil}m_{j}e_{l}+d^{1*}_{ji}d_{j}+r^{1*}_{jil}\alpha _{j}w_{l}) \right\} ;\right. \\ &\left.\qquad \max_{ 1 \leq j\leq p} \left\{\frac{1}{a^{2}_{j*}} \sum\limits_{i=1}^{n}\sum\limits_{l=1}^{n}( b^{2*}_{ij}L_{i} +c^{2*}_{ijl}M_{i}E_{l}+d^{2*}_{ij}D_{i}+r^{2*}_{ijl}\beta_{i}W_{l})\right\}\right\}\| Z\|\\ &\quad= \lambda \| Z\|\leq \frac{\lambda}{1-\lambda} \varpi, \end{aligned}$$

then, \(\varGamma _{Z}\in S^{*}.\) The mapping \(\varGamma\) is a self-mapping from \(S^*\) to \(S^*.\)

In view of (H1), for any \(Z_{1},Z_{2} \in S^{*},\;\) such that \(Z_{1}=(\varphi _1,...,\varphi _n,\psi _1,...,\psi _p)^{T}\) and \(Z_{2}=(\phi _1,...,\phi _n,\chi _1,...,\chi _p)^{T},\) we have

$$\begin{aligned}&\Vert \varGamma _{Z_{1}}-\varGamma _{Z_{2}}\Vert \\&\quad \le \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{1\le i\le n}\left\{ \left| \int \limits _{-\infty }^{t}e^{-a^{1}_{i*} (t-s)}\left[ \left( \sum \limits _{j=1}^{p} b^{1}_{ji}(s) \left( f_{j}^{1}(\psi _{j}(s))-f_{j}^{1}(\chi _{j}(s))\right) \right. \right. \right. \right. \right. \\&\qquad +\sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1}_{jil}(s) \left( g_{j}^{1}(\psi _{j}(s-\tau )g_{l}^{1}(\psi _{l}(s-\tau ) -g_{j}^{1}(\chi _{j}(s-\tau )g_{l}^{1}(\chi _{l}(s-\tau ))\right. \\&\qquad +\sum \limits _{j=1}^{p}d^{1}_{ji}(s) \int \limits _{-\infty }^{s}K_{ji}(s-m) \left( (h_{j}^{1}(\psi _{j}(m))-h_{j}^{1}(\chi _{j}(m))\right) {\mathrm{d}}m\\&\qquad + \sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}r^{1}_{jil}(s) \left( \int \limits _{-\infty }^{s} P_{jil}(s-m) k^{1}_{j}(\psi _{j}(m)){\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(s-m) k^{1}_{l}(\psi _{l}(m)){\mathrm{d}}m\right. \\&\left. \left. \left. \left. \qquad -\int \limits _{-\infty }^{s} P_{jil}(s-m) k^{1}_{j}(\chi _{j}(m)){\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(s-m) k^{1}_{l}(\chi _{l}(m)){\mathrm{d}}m \right) \right] {\mathrm{d}}s\right| \right\} ;\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{ 1 \le j\le p} \left\{ \left| \int \limits _{-\infty }^{t}e^{-a^{2}_{j*}(t-s)} \left[ \sum \limits _{i=1}^{n} b^{2}_{ij}(s) \left( f_{i}^{2}(\varphi _{i}(s))-f_{i}^{2}(\phi _{i}(s))\right) \right. \right. \right. \\&\qquad +\sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2}_{ijl}(s) \left( g_{i}^{2}(\varphi _{i}(s-\gamma ) g_{l}^{2}(\varphi _{l}(s-\gamma ))-g_{i}^{2} (\phi _{i}(s-\gamma )g_{l}^{2}(\phi _{l}(s-\gamma ) \right) \\&\qquad +\sum \limits _{i=1}^{n} d^{2}_{ij}(s)\int \limits _{-\infty }^{s} N_{ij}(s-m)\left( h_{i}^{2}(\varphi _{i}(m))-h_{i}^{2}(\phi _{i}(m)) \right) {\mathrm{d}}m\\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n}r^{2}_{ijl}(s) \left( \int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m)k^{2}_{i}(\varphi _{i}(m)){\mathrm{d}}m\int \limits _{-\infty }^{s}\tilde{Q}_{ijl}(s-m) k^{2}_{l}(\varphi _{i}(m)){\mathrm{d}}m \right. \\&\left. \left. \left. \left. \left. \qquad -\int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) k^{2}_{i}(\phi _{i}(m)){\mathrm{d}}m\int \limits _{-\infty }^{s}\tilde{Q}_{ijl} (s-m) k^{2}_{l}(\phi _{i}(m)){\mathrm{d}}m\right) \right] {\mathrm{d}}s\right| \right\} \right\} . \end{aligned}$$

Let

$$\begin{aligned} \Lambda (s,\psi ,\chi )= & {} \sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1}_{jil}(s) \left( g_{j}^{1}(\psi _{j}(s-\tau )g_{l}^{1}(\psi _{l}(s-\tau )\right. \\&\left. -\,g_{j}^{1}(\chi _{j}(s-\tau )g_{l}^{1}(\chi _{l}(s-\tau )\right) \\ \left| \Lambda (s,\psi ,\chi )\right|= & {} \left| \sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1}_{jil}(s) \left( g_{j}^{1}(\psi _{j}(s-\tau )g_{l}^{1}(\psi _{l}(s-\tau )\right. \right. \\&\left. -\,g_{j}^{1}(\chi _{j}(s-\tau )g_{l}^{1}(\chi _{l}(s-\tau ))\right| \\\le & {} \sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1*}_{jil} \left| g_{j}^{1}(\psi _{j}(s-\tau )\right| \left| g_{l}^{1}(\psi _{l}(s-\tau )-g_{l}^{1}(\chi _{l}(s-\tau ) \right| \\&+ \left| g_{j}^{1}(\psi _{j}(s-\tau )-g_{j}^{1}(\chi _{j}(s-\tau ) \right| \left| g_{l}^{1}(\chi _{l}(s-\tau )\right| , \end{aligned}$$

and

$$\begin{aligned} \varOmega (s,\varphi ,\phi )= & {} \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2}_{ijl}(s) \left( g_{i}^{2}(\varphi _{i}(s-\gamma ) g_{l}^{2}(\varphi _{l}(s-\gamma )\right. \\&\left. -\,g_{i}^{2}(\phi _{i}(s-\gamma )g_{l}^{2}(\phi _{l}(s-\gamma )\right) \\ \left| \varOmega (s,\varphi ,\phi )\right|= & {} \left| \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2}_{ijl}(s) \left( g_{i}^{2}(\varphi _{i}(s-\gamma )g_{l}^{2}(\varphi _{l}(s-\gamma )\right. \right. \\&\left. \left. -\,g_{i}^{2}(\phi _{i}(s-\gamma )g_{l}^{2}(\phi _{l}(s-\gamma )\right) \right| \\\le & {} \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2*}_{ijl} \left| g_{i}^{2}(\varphi _{i}(s-\gamma )\right| \\&\times \left| g_{l}^{2}(\varphi _{l}(s-\gamma )-g_{l}^{2}(\phi _{l} (s-\gamma ) \right| \\&+ \left| g_{i}^{2}(\varphi _{i}(s-\gamma )-g_{i}^{2} (\phi _{i}(s-\gamma )\right| \left| g_{l}^{2}(\phi _{l}(s-\gamma )\right| .\\ \end{aligned}$$

Therefore,

$$\begin{aligned}&\Vert \varGamma _{Z_{1}}-\varGamma _{Z_{2}}\Vert \\&\quad \le \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{ 1 \le i\le n}\left\{ \int \limits _{-\infty }^{t}e^{-a^{1}_{i*} (t-s)}\left[ \sum \limits _{j=1}^{p}\left| b^{1}_{ji}(s)\right| \left| f_{j}^{1}(\psi _{j}(s))-f_{j}^{1}(\chi _{j}(s))\right| \right. \right. \right. \\&\qquad +\sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1}_{jil} \left| g_{j}^{1}(\psi _{j}(s-\tau )\right| \left| g_{l}^{1}(\psi _{l}(s-\tau ) -g_{l}^{1}(\chi _{l}(s-\tau ) \right| \\&\qquad + \left| g_{j}^{1}(\psi _{j}(s-\tau )-g_{j}^{1}(\chi _{j}(s-\tau ) \right| \left| g_{l}^{1}(\chi _{l}(s-\tau )\right| \\&\qquad +\sum \limits _{j=1}^{p}\left| d^{1}_{ji}(s) \right| \int \limits _{-\infty }^{s}K_{ji}(s-m) \left| h_{j}^{1}(\psi _{j}(m))-h_{j}^{1}(\chi _{j}(m))\right| {\mathrm{d}}m \\&\qquad + \sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p} \left| r^{1}_{jil}(s)\right| \left( \int \limits _{-\infty }^{s} P_{jil}(s-m)\left| k^{1}_{j}(\psi _{j}(m)) \right| {\mathrm{d}}m \right. \\&\qquad \times \int \limits _{-\infty }^{t} Q_{jil}(s-m) \left| k^{1}_{l}(\psi _{l}(m))\right| {\mathrm{d}}m\\&\qquad -\int \limits _{-\infty }^{s} P_{jil}(s-m) \left| k^{1}_{j}(\chi _{j}(m))\right| {\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(s-m)\left| k^{1}_{l}(\psi _{l}(m))\right| {\mathrm{d}}m \\&\qquad +\int \limits _{-\infty }^{s} P_{jil}(s-m) \left| k^{1}_{j}(\chi _{j}(m))\right| {\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(s-m)\left| k^{1}_{l}(\psi _{l}(m))\right| {\mathrm{d}}m \\&\left. \left. \left. \qquad -\int \limits _{-\infty }^{s} P_{jil}(s-m) \left| k^{1}_{j}(\chi _{j}(m))\right| {\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(s-m) \left| k^{1}_{l}(\chi _{l}(m))\right| {\mathrm{d}}m\right) \right] {\mathrm{d}}s\right\} ;\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{ 1 \le j\le p} \left\{ \int \limits _{-\infty }^{t}e^{-a^{2}_{j*}(t-s)} \left[ \sum \limits _{i=1}^{n}\left| b^{2}_{ij}(s)\right| \left| f_{i}^{2}(\varphi _{i}(s))-f_{i}(\phi _{i}(s))\right| \right. \right. \\&\qquad +\sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2}_{ijl} \left| g_{i}^{2}(\varphi _{i}(s-\gamma )\right| \left| g_{l}^{2}(\varphi _{l}(s-\gamma )-g_{l}^{2} (\phi _{l}(s-\gamma ) \right| \\&\qquad + \left| g_{i}^{2}(\varphi _{i}(s-\gamma )-g_{i}^{2} (\phi _{i}(s-\gamma )\right| \left| g_{l}^{2}(\phi _{l}(s-\gamma )\right| \\&\qquad +\sum \limits _{i=1}^{n}\left| d^{2}_{ij}(s)\right| \int \limits _{-\infty }^{s} N_{ij}(s-m)\left| h_{i}^{2}(\varphi _{i}(m))-h_{i}^{2}(\phi _{i}(m))\right| {\mathrm{d}}m \\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n} \left| r^{2}_{ijl}(s)\right| \left( \int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) \left| k^{2}_{i}(\varphi _{i}(m))\right| {\mathrm{d}}m\right. \\&\qquad \times \int \limits _{-\infty }^{s}\tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\varphi _{l}(m))\right| {\mathrm{d}}m\\&\qquad -\int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) \left| k^{2}_{i}(\phi _{i}(m))\right| {\mathrm{d}}m\int \limits _{-\infty }^{s} \tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\varphi _{l}(m))\right| {\mathrm{d}}m\\&\qquad +\int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) \left| k^{2}_{i}(\phi _{i}(m))\right| {\mathrm{d}}m\int \limits _{-\infty }^{s} \tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\varphi _{l}(m))\right| {\mathrm{d}}m\\&\left. \left. \left. \left. \qquad -\int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) \left| k^{2}_{i}(\phi _{i}(m))\right| {\mathrm{d}}m\int \limits _{-\infty }^{s} \tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\phi _{l}(m))\right| {\mathrm{d}}m\right) \right] {\mathrm{d}}s \right\} \right\} \\&\quad \le \max \left\{ \max _{1 \le i \le n}\frac{1}{a_{i*}^{1}}\sum _{j=1}^{p}\sum _{l=1}^{p}( b_{ji}^{1*}l_{j} +c_{jil}^{1*}(e_{j}m_{l}+m_{j}e_{l})+ d_{ji}^{1*}d_{j}\right. \\&\qquad + r_{jil}^{1*}(\alpha _jw_l +w_j \alpha _l ); \max _{1 \le j \le p}\frac{1}{a_{j*}^{2}}\sum _{i=1}^{n}\sum _{l=1}^{n}(b_{ij}^{2*}L_{i} +c_{ijl}^{2*}(E_{i}M_{l}+M_{i}E_{l})\\&\left. \qquad + d_{ij}^{2*}D_{i} +r_{ijl}^{2*}( \beta _i W_l+W_i\beta _l)\right\} \Vert \ Z_{1}-Z_{2}\Vert \le \bar{\lambda }\Vert \ Z_{1}-Z_{2}\Vert , \end{aligned}$$

which prove that \(\varGamma\) is a contraction mapping. Then, by the Banach fixed-point theorem, \(\varGamma\) has a unique fixed point which corresponds to the solution of (2) in \(S^{*} \subset PAA({\mathbb {R}},{\mathbb {R}}^{n+p},\mu ,\nu )\). \(\square\)

Appendix G Proof of Theorem 4

Proof

Suppose that \(Z(t)=(x_{1}(t),\ldots ,x_{n}(t),y_{1}(t),\ldots ,y_{p}(t))^{T},\) is an arbitrary solution of system (2) with initial value \(\phi (t)=( \varphi _{1}(t) ,\ldots ,\varphi _{n}(t),\psi _{1}(t) ,\ldots ,\psi _{p}(t))^{T}.\)

It follows from Theorem 3 that system (2) has one and only one (\(\mu ,\nu )\)-pseudo-almost automorphic solution \(Z^{*}(t)=(x_{1}^{*}(t),\ldots ,x_{n}^{*}(t),y_{1}^{*}(t),\ldots ,y_{p}^{*}(t))^{T} \in S^{*},\) with initial value \(\phi ^{*}(t)=( \varphi _{1}^{*}(t) ,\ldots ,\varphi _{n}^{*}(t),\psi _{1}^{*}(t) ,\ldots ,\psi _{p}^{*}(t))^{T}.\)

Let \(u_{i}(t)=x_{i}(t)-x_{i}^{*}(t),\) and \(v_{j}(t)=y_{j}(t)-y_{j}^{*}(t),\;i=1 \ldots n, j=1 \ldots p.\) We can write that

$$\begin{aligned}&u'_{i}(t)+a_{i}^{1}(t) u_{i}(t) \nonumber \\&\quad = \sum _{j=1}^{p} b_{ji}^{1}(t) \left[ f_{j}^{1} (v_{j}(t)+y_{j}^{*}(t))-f_{j}^{1} (y_{j}^{*}(t))\right] \nonumber \\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} c_{jil}^{1}(t) \left[ g_{j}^{1}(v_{j}(t-\tau )+ y_{j}^{*}(t-\tau )) \right. \nonumber \\&\left. \qquad \times g_{l}^{1}(v_{l}(t-\tau )+ y_{l}^{*}(t-\tau )) -g_{j}^{1}(y_{j}^{*}(t-\tau ) g_{l}^{1}(y_{l}^{*}(t-\tau )) \right] \nonumber \\&\qquad +\sum _{j=1}^{n} d_{ji}^{1}(t)\int _{-\infty }^{t} K_{ji}(t-m) \nonumber \\&\qquad \times \left[ h_{j}^{1}(v_{j}(m)+y_{j}^{*}(m))-h_{j}^{1} (y_{j}^{*}(m))\right] {\mathrm{d}}m \nonumber \\&\qquad +\sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}r^{1}_{jil}(t) \left[ \int \limits _{-\infty }^{t} P_{jil}(t-m) k^{1}_{j}(v_{j}(m)+y_{j}^*(m)){\mathrm{d}}m \right. \nonumber \\&\qquad \times \int \limits _{-\infty }^{t} Q_{jil}(t-m) k^{1}_{l}(v_{l} (m)+y_{l}^*(m)){\mathrm{d}}m \nonumber \\&\left. \qquad -\int \limits _{-\infty }^{t} P_{jil}(t-m) k^{1}_{j}(y_{j}^*(m)){\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(t-m) k^{1}_{l}(y_{l}^*(m)){\mathrm{d}}m \right] , \end{aligned}$$
(4)
$$\begin{aligned}&v'_{j}(t)+a_{j}^{2}(t) v_{j}(t)\nonumber \\&\quad = \sum _{i=1}^{p} b_{ij}^{2}(t) \left[ f_{i}^{2} (u_{i}(t)+x_{i}^{*}(t))-f_{i}^{2} (x_{i}^{*}(t))\right] \nonumber \\&\qquad + \sum _{i=1}^{n} \sum _{l=1}^{n} c_{ijl}^{2}(t) \left[ g_{i}^{2}(u_{i}(t-\gamma )+x_{i}^{*}(t-\gamma )) \right. \nonumber \\&\left. \qquad \times g_{l}^{2}(u_{l}(t-\gamma )+ x_{l}^{*}(t-\gamma ))- g_{i}^{2}(x_{i}^{*}(t-\gamma ) g_{l}^{2}(x_{l}^{*}(t-\gamma ))\right] \nonumber \\&\qquad +\sum _{i=1}^{n} d_{ij}^{2}(t)\int _{-\infty }^{t} N_{ij}(t-m)\nonumber \\&\qquad \times \left[ h_{i}^{2}(u_{i}(m)+x_{i}^{*}(m))-h_{i}^{2} (x_{i}^{*}(m))\right] {\mathrm{d}}m\nonumber \\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n}r^{2}_{ijl}(t) \left[ \int \limits _{-\infty }^{t} \tilde{P}_{ijl}(t-m) k^{2}_{i}(u_{i}(m)+x_{i}^*(m)){\mathrm{d}}m \right. \nonumber \\&\qquad \times \int \limits _{-\infty }^{t} \tilde{Q}_{ijl}(t-m) k^{2}_{l}(u_{l}(m)-x_{l}^*(m)){\mathrm{d}}m \nonumber \\&\left. \qquad -\int \limits _{-\infty }^{t} \tilde{P}_{ijl}(t-m) k^{2}_{i} (x_{i}^*(m)){\mathrm{d}}m\int \limits _{-\infty }^{t} \tilde{Q}_{ijl} (t-m) k^{2}_{l}(x_{l}^*(m)){\mathrm{d}}m\right] \end{aligned}$$
(5)

Let \(F_{i},G_{j}\) be defined by

$$\begin{aligned} F_{i}(w)= & {} a_{i*}^{1}-w- \sum \limits _{j=1}^{p}\left( b_{ji}^{1*} l_{j}-\sum \limits _{l=1}^{p}c_{jil}^{1*}(e_{l} m_{j} + e_{j} m_{l}) e^{w \tau }\right. \\&-d_{ji}^{1*} d_{j} \int _{0}^{+\infty } K_{ji}(m) e^{w m} {\mathrm{d}}m- \sum \limits _{l=1}^{p}r^{1*}_{jil}\left( \alpha _j w_l\int \limits _{-\infty }^{t} P_{jil}(m)e^{wm}{\mathrm{d}}m\right. \\&\left. + w_j\alpha _l \int \limits _{-\infty }^{t} Q_{jil}(m) e^{wm}{\mathrm{d}}m \right) ,\\ \end{aligned}$$

and

$$\begin{aligned} G_{j}(w)= & {} a_{j*}^{2}-w- \sum \limits _{i=1}^{n} \left( b_{ij}^{2*} L_{i}-\sum \limits _{l=1}^{n}c_{ijl}^{2*} (E_{i} M_{l}+ E_{l} M_{i}) e^{w\gamma }\right. \\&-d_{ij}^{2*} D_{i} \int _{0}^{+\infty } N_{ij}(m) e^{w m} {\mathrm{d}}m- \sum \limits _{l=1}^{n}r^{2*}_{ijl}\left( \beta _i W_l \int \limits _{-\infty }^{t} \tilde{P}_{ijl}(m)e^{wm}{\mathrm{d}}m \right. \\&\left. +\,W_j\beta _l\int \limits _{-\infty }^{t} \tilde{Q}_{ijl}(m) e^{wm}{\mathrm{d}}m \right) , \end{aligned}$$

for \(i=1,\ldots ,n,\)\(j=1,\ldots ,p,\)\(w\in [0,+\infty [,\) by (H2) and \(\bar{\lambda } < 1\), we obtain that

$$\begin{aligned} \left\{ \begin{array}{lll} F_{i}(0)&{}=&{}a_{i*}^{1}- \sum \limits _{j=1}^{p}\left( b_{ji}^{1*} l_{j}+ \sum \limits _{l=1}^{p}c_{jil}^{1*}(e_{l} m_{j}+ e_{j} m_{l})\right. \\ &{}&{}+d_{ji}^{1*} d_{j}+ \sum \limits _{l=1}^{p}r^{1*}_{jil}(\alpha _j w_l+w_j\alpha _l))> 0, \\ G_{j}(0)&{}=&{}a_{j*}^{2}- \sum \limits _{i=1}^{n}\left( b_{ij}^{2*} L_{i}+\sum \limits _{l=1}^{n} c_{ijl}^{2*}(E_{i} M_{l}+ E_{l} M_{i})\right. \\ &{}&{}+d_{ij}^{2*} D_{i}+\sum \limits _{l=1}^{n}r^{2*}_{ijl}(\beta _i W_l+W_j\beta _l) )> 0. \end{array}\right. \end{aligned}$$

Both, \(F_{i}(.)\) and \(G_{j}(.)\) are continuous on \([0,\infty [\) such that \(F_{i}(w) \longrightarrow - \infty\) when \(w\longmapsto +\infty\), there exist \(\varepsilon _{i}^{*} > 0\) such that \(F_{i}(\varepsilon _{i}^{*}) = 0\) and \(F_{i}(\varepsilon _{i}) >0\) for \(\varepsilon _{i} \in (0,\varepsilon _{i}^{*})\), also \(G_{j}(w) \longrightarrow - \infty\) when \(w \longmapsto +\infty\), \(\exists \zeta _{j}^{*} > 0\) such that \(G_{j}(\zeta _{j}^{*}) = 0\) and \(G_{j}(\zeta _{j}) > 0\) for \(\zeta _{j} \in (0,\zeta _{j}^{*})\).

By choosing \(\eta = \min \{ \varepsilon _{1}^{*},\ldots ,\varepsilon _{n}^{*},\zeta _{1}^{*},\ldots , \zeta _{p}^{*}\}\), we obtain

$$\begin{aligned} \left\{ \begin{array}{cc} F_{i}(\eta ) \ge 0 ,\; i=1,\ldots ,n,\\ G_{j}(\eta ) \ge 0 ,\, j=1,\ldots ,p. \end{array} \right. \end{aligned}$$

So, we can choose a positive constant \(\lambda\) such that \(0<\lambda < \min \{ \eta , a_{1*}^{1},\ldots ,a_{n*}^{1},a_{1*}^{2},\ldots ,a_{p*}^{2},\lambda _{0} \}\) such that \(F_{i}(\lambda )\) and \(G_{j}(\lambda )\) are nonnegative ,which implies that, for \(i=1,\ldots ,n,\)\(j=1,\ldots ,p,\)

$$\begin{aligned} \left\{ \begin{array}{lll} &{}\frac{1}{a_{i*}^{1}-\lambda } \sum \limits _{j=1}^{p}\left( b_{ji}^{1*} l_{j} +\sum \limits _{l=1}^{p}c_{jil}^{1*}(e_{l} m_{j}+ e_{j} m_{l}) e^{\lambda \tau }\right. \\ &{}+d_{ji}^{1*} d_{j}\int _{0}^{+\infty } K_{ji}(m) e^{\lambda m} {\mathrm{d}}m \\ &{}+ \sum \limits _{l=1}^{p}r^{1*}_{jil}\left( \alpha _j w_l \int \nolimits _{-\infty }^{t} P_{jil}(m)e^{wm}{\mathrm{d}}m \right. \\ &{}\left. \left. + w_j\alpha _l\int \nolimits _{-\infty }^{t} Q_{jil}(m) e^{wm}{\mathrm{d}}m \right) \right)< 1, \\ &{}\frac{1}{a_{j*}^{2}-\lambda } \sum \limits _{i=1}^{n}\left( b_{ij}^{2*} L_{i} +\sum \limits _{l=1}^{n}c_{ijl}^{2*}(E_{i} M_{l}+ E_{l} M_{i})e^{\lambda \gamma } \right. \\ &{}+ d_{ij}^{2*} D_{i} \int _{0}^{+\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m \\ &{}+ \sum \limits _{l=1}^{n}r^{2*}_{ijl}\left( \beta _i W_l \int \nolimits _{-\infty }^{t} \tilde{P}_{ijl}(m)e^{wm}{\mathrm{d}}m \right. \\ &{}+W_j\beta _l \int \nolimits _{-\infty }^{t} \tilde{Q}_{ijl}(m) e^{wm}{\mathrm{d}}m) < 1. \end{array}\right. \end{aligned}$$

Let

$$\begin{aligned} \left\{ \begin{array}{lll} N= \max \nolimits _{1 \le i\le n} \left( \frac{a_{i*}^{1}}{\sum \nolimits _{j=1}^{p}(b_{ji}^{1*} l_{j}+\sum \nolimits _{l=1}^{p}c_{jil}^{1*} (e_{l} m_{j}+ e_{j} m_{l}) +d_{ji}^{1*} d_{j}+\sum \nolimits _{l=1}^{p}r_{jil}^{1*} (\alpha _{j}w_{l} + w_{j} \alpha _{l}))}\right) ,\\ \bar{N}=\max \nolimits _{1 \le j\le p} \left( \frac{a_{j*}^{2}}{\sum \nolimits _{i=1}^{n}(b_{ij}^{2*} L_{i}+\sum \nolimits _{l=1}^{n}c_{ijl}^{2*} (E_{i} M_{l}+ E_{l} M_{i})+d_{ij}^{2*} D_{i}+\sum \nolimits _{l=1}^{n}r_{ijl}^{2*} ( \beta _{l}W_{i}+ W_{l} \beta _{i}))}\right) . \end{array}\right. \end{aligned}$$

Clearly \(N> 1\), \(\bar{N}> 1,\) and we take \(M=\max \{N ,\bar{N}\}> 1.\)

$$\begin{aligned} \left\{ \begin{array}{l} \left(\frac{1}{M} - \frac{1}{a_{i*}^{1}-\lambda } \sum \limits _{j=1}^{p}\left( b_{ji}^{1*} l_{j}+\sum \limits _{l=1}^{p}c_{jil}^{1*} (e_{l} m_{j}+ e_{j} m_{l}) e^{\lambda \tau }\right. \right. \\ + d_{ji}^{1*} d_{j} \int _{0}^{+\infty } K_{ji}(m) e^{\lambda m} {\mathrm{d}}m +\sum \limits _{l=1}^{p}r^{1*}_{jil}(\alpha _j w_l \int \limits _{-\infty }^{t} P_{jil}(m)e^{\lambda m}{\mathrm{d}}m\\ \left. \left. \left. + w_j\alpha _l\int \limits _{-\infty }^{t} Q_{jil}(m) e^{\lambda m}{\mathrm{d}}m \right) \right) \right) \le 0\\ \left( \frac{1}{M} - \frac{1}{a_{j*}^{2}-\lambda } \sum \limits _{i=1}^{n} \left( b_{ij}^{2*} L_{i} +\sum \limits _{l=1}^{n}c_{ijl}^{*2} (E_{i} M_{l}+ E_{l} M_{i}) e^{v \gamma }\right. \right. \\ +d_{ij}^{2*} D_{i} \int _{0}^{+\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m+ \sum \limits _{l=1}^{n}r^{2*}_{ijl}\left( \beta _i W_l \int \limits _{-\infty }^{t} \tilde{P}_{ijl}(m)e^{\lambda m}{\mathrm{d}}m \right. \\ \left. \left. \left. +W_j\beta _l\int \limits _{-\infty }^{t} \tilde{Q}_{ijl}(m) e^{\lambda m}{\mathrm{d}}m\right) \right) \right) \le 0 \end{array}\right. \end{aligned}$$

where \(0<\lambda < \min \{\eta , a_{1*}^{1},\ldots ,a_{n*}^{1},a_{1*}^{2},\ldots ,a_{p*}^{2}, \lambda _{0} \}\).

Besides, \(\;\forall \; t \in (-\infty ,0]\)

$$\begin{aligned} \left\{ \begin{array}{lll} \parallel u\parallel &{}\le &{}Me^{-\lambda t}\parallel \phi -\phi ^* \parallel ,\\ \parallel v\parallel &{}\le &{}Me^{-\lambda t}\parallel \phi -\phi ^* \parallel , \end{array}\right. \end{aligned}$$
(6)

We claim that, \(\text{ for }\; t>0,\)

$$\begin{aligned} \left\{ \begin{array}{lll} \parallel u\parallel &{}\le &{}Me^{-\lambda t}\parallel \phi -\phi ^* \parallel ,\\ \parallel v\parallel &{}\le &{}Me^{-\lambda t}\parallel \phi -\phi ^* \parallel ,\; \end{array}\right. \end{aligned}$$
(7)

If (7) is false, then there must be some \(t_{1} > 0\) some \(i \in \{ 1,\ldots ,n\},\)\(j \in \{ 1,\ldots ,p\},\) for any \(p > 1\) and some k such that

$$\begin{aligned} \left\{ \begin{array}{lll} \parallel u(t_1) \parallel &{}=&{} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1}},\\ \parallel v(t_1) \parallel &{}=&{} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1}}, \end{array}\right. \end{aligned}$$
(8)

and

$$\begin{aligned} \left\{ \begin{array}{lll} \parallel u(t) \parallel &{}\le &{} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t}, \; \forall \; t \in (-\infty , t_{1}]\\ \parallel v(t) \parallel &{}\le &{} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t}, \; \forall \; t \in (-\infty , t_{1}]. \end{array}\right. \end{aligned}$$
(9)

Now,we have the following:

$$\begin{aligned}&\mid u_{i}(t_{1})\mid \\&\quad \le \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{i*}^{1}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{i*}^{1}}\left( \sum _{j=1}^{p} b_{ji}^{1*} l_{j} \parallel v_{j}(s)\parallel \right. \\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j}) \parallel v_{j}(s-\tau )\parallel \\&\qquad + \sum _{j=1}^{p} d_{ji}^{1*} \int _{-\infty }^{s} K_{ji}(s-m) d_{j} \parallel v_{j}(m)\parallel {\mathrm{d}}m \\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l} \int \limits _{-\infty }^{s} P_{jil}(s-m) \parallel v_{j}(m)\parallel {\mathrm{d}}m \right. \\&\left. \qquad + w_{j}\alpha _{l} \int \limits _{-\infty }^{s} Q_{jil}(s-m) \parallel v_{j}(m)\parallel {\mathrm{d}}m\right) {\mathrm{d}}s \\&\quad \le \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{i*}^{1}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{i*}^{1}}\\&\qquad \times \left( \sum _{j=1}^{p} b_{ji}^{1*} l_{j} p M \parallel \phi -\phi ^*\parallel e^{-\lambda s} \right. \\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j}) p M \parallel \phi -\phi ^*\parallel e^{-\lambda (s-\tau )} \\&\qquad +\sum _{j=1}^{p} d_{ji}^{1*}d_{j} \int _{0}^{\infty } K_{ji}(m) p M \parallel \phi -\phi ^*\parallel e^{- \lambda (s-m)} {\mathrm{d}}m \\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l} \int \limits _{0}^{\infty } P_{jil}(m) p M \parallel \phi -\phi ^*\parallel e^{-\lambda (s-m)} {\mathrm{d}}m\right. \\&\left. \qquad +\, w_{j}\alpha _{j} \int \limits _{0}^{\infty } Q_{jil}(m) p M \parallel \phi -\phi ^*\parallel e^{-\lambda (s-m)}) {\mathrm{d}}m\right) {\mathrm{d}}s \\&\quad \le \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{i*}^{1}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{i*}^{1}}p M \parallel \phi -\phi ^*\parallel e^{-\lambda s}\\&\qquad \times \left( \sum _{j=1}^{p} b_{ji}^{1*} l_{j} + \sum _{j=1}^{p}\sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j}) e^{\lambda \tau }\right. \\&\qquad + \sum _{j=1}^{p} d_{ji}^{1*}d_{j} \int _{0}^{\infty } K_{ji}(m) e^{ \lambda m} {\mathrm{d}}m\\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l}\int \limits _{0}^{\infty } P_{jil}(m) e^{\lambda m} {\mathrm{d}}m\right. \\&\left. \qquad + w_{j}\alpha _{j} \int \limits _{0}^{\infty } Q_{jil}(m) e^{\lambda m} {\mathrm{d}}m \right) {\mathrm{d}}s \\&\quad \le p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} }[e^{(\lambda - a_{i*}^{1})t_{1}}\left( \frac{1}{M}-\frac{1}{a_{i*}^{1}-\lambda }\right. \\&\qquad \times \sum _{j=1}^{p}\left( b_{ji}^{1*}l_{j} + \sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j})e^{\lambda \tau }\right. \\&\qquad + d_{ji}^{1*} d_{j} \int _{0}^{\infty } K_{ji}(m) e^{\lambda m} {\mathrm{d}}m \\&\qquad +\sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l}\int \limits _{0}^{\infty } P_{jil}(m) e^{\lambda m} {\mathrm{d}}m \right. \\&\left. \qquad + w_{j}\alpha _{j} \int \limits _{0}^{\infty } Q_{jil}(m) e^{\lambda m} {\mathrm{d}}m)\right) \\&\qquad +\left( \frac{1}{a_{i*}^{1}-\lambda }\left( \sum _{j=1}^{p}( b_{ji}^{1*} l_{j} + \sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j})e^{\lambda \tau }\right. \right. \\&\qquad + d_{ji}^{1*} d_{j} \int _{0}^{\infty } K_{ji}(m) e^{\lambda m} {\mathrm{d}}m \\&\left. \qquad +\sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l}\int \limits _{0}^{\infty } P_{jil}(m) e^{\lambda m} {\mathrm{d}}m+ w_{j}\alpha _{j} \int \limits _{0}^{\infty } Q_{jil}(m) e^{\lambda m} {\mathrm{d}}m\right) \right) \\&\quad \le p M\parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} } \left( \frac{1 }{a_{i*}^{1} - \lambda } \sum _{j=1}^{p}\left( b_{ji}^{1*} l_{j} \sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j}) e^{\lambda \tau }\right. \right. \\&\qquad + d_{ji}^{1*} d_{j} \int _{0}^{\infty } K_{ji}(m) e^{\lambda m} {\mathrm{d}}m \\&\left. \left. \qquad +\sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l}\int \limits _{0}^{\infty } P_{jil}(m) e^{\lambda m} {\mathrm{d}}m + w_{j}\alpha _{j} \int \limits _{0}^{\infty } Q_{jil}(m) e^{\lambda m} {\mathrm{d}}m\right) \right) \right. \\&\quad < p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1}}. \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \mid v_{j}(t_{1})\mid\le & {} \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{j*}^{2}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{j*}^{2}} \left( \sum _{i=1}^{n} b_{ij}^{2*} L_{i} \parallel u_{i}(s)\parallel \right. \\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i}) \parallel u_{i}(s-\gamma )\parallel \\&+ \sum _{i=1}^{n} d_{ij}^{2*} \int _{-\infty }^{s} N_{ij}(s-m) D_{i} \parallel u_{i}(m)\parallel {\mathrm{d}}m \\&+\sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) \parallel u_{i}(s)\parallel {\mathrm{d}}m \right. \\&\left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) \parallel u_{i}(s)\parallel {\mathrm{d}}m\right) {\mathrm{d}}s\\\le & {} \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{j*}^{2}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{j*}^{2}}\left( \sum _{i=1}^{n} b_{ij}^{2*} L_{i} p M \right. \\&\times \parallel \phi -\phi ^* \parallel e^{-\lambda s} + \sum _{i=1}^{n}\sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i}) q M \\&\times \parallel \phi -\phi ^*\parallel e^{-\lambda (s-\gamma )}\\&+ \sum _{i=1}^{n} d_{ij}^{2*} \int _{0}^{\infty } N_{ij}(m) D_{i} p M \parallel \phi -\phi ^* \parallel e^{- \lambda (s-m)} {\mathrm{d}}m\\&+\sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) p M \parallel \phi -\phi ^* \parallel e^{- \lambda (s-m)} {\mathrm{d}}m \right) {\mathrm{d}}s\\&\left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) p M \parallel \phi -\phi ^* \parallel e^{- \lambda (s-m)} {\mathrm{d}}m \right) {\mathrm{d}}s\\\le & {} \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{j*}^{2}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{j*}^{2}}p M \parallel \phi -\phi ^* \parallel e^{\lambda s}\\&\times \left( \sum _{i=1}^{n} b_{ij}^{2*} L_{i} +\sum _{i=1}^{n}\sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i}) e^{\lambda \gamma } \right. \\&+\sum _{i=1}^{n} d_{ij}^{2*} \int _{0}^{\infty } N_{ij}(m) D_{i} e^{\lambda m} {\mathrm{d}}m\\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m)e^{-\lambda m} {\mathrm{d}}m \right. \\&\left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right) {\mathrm{d}}s\\\le & {} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} }\left[ \frac{e^{(\lambda - a_{j*}^{2}) t_{1} }}{p M} + \frac{1}{a_{j*}^{2} - \lambda }\right. \\&\times \left( \sum _{i=1}^{n} b_{ij}^{2*} L_{i}+ \sum _{i=1}^{n}\sum _{l=1}^{n} c_{ijl}^{2*}(M_{i}E_{l}+M_{l}E_{i}) e^{\lambda \gamma }\right. \\&+ \sum _{i=1}^{n} d_{ij}^{2*} D_{i} \int _{0}^{\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m\\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right. \\&\left. \left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m\right) \left( 1- e^{(\lambda - a_{j*}^{2}) t_{1} }\right) \right] \\\le & {} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} }[e^{(\lambda - a_{j*}^{2})t_{1}}\left( \frac{1}{M}-\frac{1}{a_{j*}^{2}-\lambda }\right. \\&\times \sum _{i=1}^{n}\left( b_{ij}^{2*}L_{i} + \sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i})e^{\lambda \gamma }+ d_{ij}^{2*} D_{i} \int _{0}^{\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m \right. \\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right. \\&\left. \left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m\right) \right) \\&+\left( \frac{1}{a_{j*}^{2}-\lambda }\left( \sum _{i=1}^{n}\left( b_{ij}^{1*} L_{i} + \sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i})e^{\lambda \gamma }\right. \right. \right. \\&+ d_{ij}^{2*} D_{i} \int _{0}^{\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m\\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right. \\&\left. \left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right) \right) \\\le & {} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} } \left( \frac{1 }{a_{j*}^{2} - \lambda } \sum _{i=1}^{n}( b_{ij}^{2*} L_{i}\right. \\&+ \sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i}) e^{\lambda \gamma }\\&+ d_{ij}^{2*} D_{i} \int _{0}^{\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m \\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right. \\&\left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) e^{-\lambda m} {\mathrm{d}}m \right) \\< & {} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} }. \end{aligned}$$

which contradicts 8, then 7 holds.

Letting \(p \longrightarrow 1,\) then 7 holds.

Hence, the \((\mu ,\nu )\)-pseudo-almost automorphic solution \(Z(t)=(x(t),y(t))^T\) of system (2) is globally exponentially stable. \(\square\)

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Aouiti, C., Dridi, F. \((\mu ,\nu )\)-Pseudo-almost automorphic solutions for high-order Hopfield bidirectional associative memory neural networks. Neural Comput & Applic 32, 1435–1456 (2020). https://doi.org/10.1007/s00521-018-3651-6

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Keywords

  • \((\mu , \nu )\)-Pseudo-almost automorphic function
  • High-order BAM neural networks
  • Global exponential stability

Mathematics Subject Classification

  • 34C27
  • 37B25
  • 92C20