Piecewise asymptotically almost automorphic solutions for impulsive non-autonomous high-order Hopfield neural networks with mixed delays

Abstract

This paper is concerned with an impulsive non-autonomous high-order Hopfield neural network with mixed delays. Under proper conditions, we studied the existence, the uniqueness and the global exponential stability of asymptotic almost automorphic solutions for the suggested system. Our method was mainly based on the Banach’s fixed-point theorem and the generalized Gronwall–Bellman inequality. Moreover, four examples are presented to demonstrate the effectiveness of the proposed findings.

Appendices

Appendix 1: Proof of the Lemma 1

Proof

Let $\varphi (.)\in AAA(\mathbb {R},\mathbb {R})$, it can be written as $\varphi (.)=\varphi _1 (.)+\varphi _2 (.)$ where $\varphi _1 (.) \in AA(\mathbb {R},\mathbb {R})$ and $\varphi _2 (.)\in PC_0(\mathbb {R},\mathbb {R}).$

First, we know that the space $AA(\mathbb {R},\mathbb {R})$ is translation invariant, then for $h\in \mathbb {R},$ we have $\varphi _1(.-h) \in AA(\mathbb {R},\mathbb {R}).$

Second, we prove that $\varphi _2(.-h) \in PC_0(\mathbb {R},\mathbb {R}).$

For $\varphi _2 (.)\in PC_0(\mathbb {R},\mathbb {R}),$ we have: $\varphi _2 (.) \in PC(\mathbb {R},\mathbb {R}),$ such that $\varphi _2(t)$ is continuous at t for any $t \notin \{ t_i , i \in \mathbb {Z}\},$$\varphi _2(t_i^+),\varphi _2(t_i^-)$ exists and $\varphi _2(t_i^-)=\varphi _2(t_i).$

Therefore, for $h\in \mathbb {R},$$\varphi _2(t-h)$ is continuous at $(t-h)$ for any $(t-h)\notin \{ t_i , i \in \mathbb {Z}\},$$\varphi _2((t_i-h)^+),\varphi _2((t_i-h)^-)$ exist and $\varphi _2((t_i-h)^-)=\varphi _2(t_i-h).$ Then, $\varphi _2(t-h) \in PC(\mathbb {R},\mathbb {R}).$

On the other hand, we have $\lim \nolimits _{t\longrightarrow \infty } \Vert \varphi _2(t)\Vert =0, \;$ then for h in $\mathbb {R},\lim \nolimits _{t\longrightarrow \infty } \Vert \varphi _2(t-h)\Vert =0.$ This completes the proof. $\square$

Appendix 2: Proof of Lemma 2

Proof

By definition, we can write $\varphi =\varphi _1+\varphi _2,$$\psi =\psi _1+\psi _2$ where $\varphi _1,\psi _1 \in AA(\mathbb {R},\mathbb {R}),$$\varphi _2,\psi _2 \in PC_0(\mathbb {R},\mathbb {R}).$

Obviously, $\varphi \times \psi =\varphi _1 \times \psi _1 +\varphi _1 \times \psi _2 +\varphi _2 \times \psi _1 +\varphi _2 \times \psi _2,$ we have $\varphi _1 \times \psi _1 \in AA(\mathbb {R},\mathbb {R}).$

On the other hand, $\varphi _1 \times \psi _2 +\varphi _2 \times \psi _1 +\varphi _2 \times \psi _2 \in PC(\mathbb {R},\mathbb {R}),$ and

$$\begin{aligned}&\Vert \varphi _1 \times \psi _2 +\varphi _2 \times \psi _1 +\varphi _2 \times \psi _2 \Vert \\ &\quad \le \Vert \varphi _1\Vert _\infty \times \Vert \psi _2\Vert +\Vert \varphi _2\Vert \times \Vert \psi _1\Vert _{\infty } +\Vert \varphi _2\Vert _\infty \times \Vert \psi _2\Vert , \end{aligned}$$

which implies that $\varphi _1 \times \psi _2 +\varphi _2 \times \psi _1 +\varphi _2 \times \psi _2 \in PC_0(\mathbb {R},\mathbb {R}).$

Then, $\varphi \times \psi \in AAA(\mathbb {R},\mathbb {R}).$ This completes the proof. $\square$

Appendix 3: Proof of Lemma 3

Proof

By definition, we have $\phi (.)=\phi _1 (.)+\phi _2 (.)$ where $\phi _1 (.)\in AA(\mathbb {R},\mathbb {R}^n), \phi _2 (.)\in PC_0(\mathbb {R},\mathbb {R}^n).$ Let

$$\begin{aligned} G(t) &= f(\phi (t-\varsigma )) \\ & = f(\phi _1(t-\varsigma )) \\ &+\bigg [ f(\phi _1(t-\varsigma ))+\phi _2(t-\varsigma )) -f(\phi _1(t-\varsigma ))\bigg ] \\ & = G_1(t)+G_2(t) \end{aligned}$$

(16)

First, 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 $\lim \nolimits _{n\rightarrow +\infty } \phi _1\left( t-\varsigma +s_{n}\right) =\phi _1^1(t-\varsigma ),\; \forall t \in \mathbb {R}$ and $\lim \nolimits _{n\rightarrow +\infty } \phi _1^1\left( t-\varsigma -s_{n}\right) =\phi _1(t-\varsigma ),\; \forall t \in \mathbb {R}.$ Obviously,

$$\begin{aligned}&|G_1(t+s_n)-f(\phi _1^1(t-\varsigma ))|\\ &\quad =|f(\phi _1(t-\varsigma +s_{n}))-f(\phi _1^1(t-\varsigma ))| \\ &\quad \le l^j_f|\phi _1(t-\varsigma +s_{n})-\phi _1^1(t-\varsigma )| \rightarrow 0, \; n\rightarrow +\infty . \end{aligned}$$

Therefore, $\lim \nolimits _{t\longrightarrow \infty }f(\phi _1(t-\varsigma +s_{n}))=f(\phi _1^1(t-\varsigma )).$

By the same way, we have: $\lim \nolimits _{t\longrightarrow \infty }f(\phi _1^1(t-\varsigma -s_{n}))=f(\phi _1(t-\varsigma )).$

Then $G_1(.) \in AA(\mathbb {R},\mathbb {R}^n).$

Second, we prove that $G_2(.)\in PC_0(\mathbb {R},\mathbb {R}^n)$.

It is clear that $G_2(.)\in PC(\mathbb {R},\mathbb {R}^n)$, also we have:

$$\begin{aligned} G_2(t)& = f(\phi _1(t-\varsigma )+\phi _2(t-\varsigma ))-f(\phi _1(t-\varsigma ))\\ |G_2(t)|& = |f(\phi _1(t-\varsigma )+\phi _2(t-\varsigma ))-f(\phi _1(t-\varsigma ))|\\ & \le l^j_f|\phi _2(t-\varsigma )|, \end{aligned}$$

since $\phi _2(.) \in PC_0(\mathbb {R},\mathbb {R}^n),$ we have $\lim \nolimits _{t\longrightarrow \infty } |\phi _2(t-\varsigma )|=0,$ then $G_2(.)\in PC_0(\mathbb {R},\mathbb {R}^n).$ The proof is completed. $\square$

Appendix 4: Proof of Lemma 4

Proof

Let $\phi _{j}(.) \in AAA(\mathbb {R},\mathbb {R}^n),$ from Lemma 3 we obtain $h_{j}(\phi _{j}(.) )\in AAA(\mathbb {R},\mathbb {R}^n).$

Let $h_{j}(\phi _{j}(.))=u_j(.)+v_j(.),$ where $u_j (.)\in AA(\mathbb {R},\mathbb {R}^n)$ and $v_j (.)\in PC_0(\mathbb {R},\mathbb {R}^n),$ then

$$\begin{aligned} \varPhi _{ij}(t)& = \int \limits _{-\infty }^{t} K_{ij}(t-s) h_{j}(\phi _{j}(s)) \,{\text{d}}s \\ & = \int \limits _{-\infty }^{t} K_{ij}(t-s) u_{j}(s) \,{\text{d}}s+ \int \limits _{-\infty }^{t} K_{ij}(t-s)v_{j}(s) \,{\text{d}}s \\ & = \varPhi _{ij}^1(t)+\varPhi _{ij}^2(t) \end{aligned}$$

(17)

First, let us show that $\varPhi _{ij}^1(t) \in AA(\mathbb {R},\mathbb {R}^n).$

For each sequence $(s'_n)$ there exists a subsequence $(s_n)$ such that $\theta (t)= \lim \nolimits _{n\longrightarrow \infty } u_j(t+s_n)$ is well defined for every $t \in \mathbb {R}$ and $\theta (t-s_n)= \lim \nolimits _{n\longrightarrow \infty } u_j(t)$ is well defined for every $t \in \mathbb {R}.$

In addition, we have

$$\begin{aligned} \varPhi _{ij}^1(t+s_n)& = \int \limits _{-\infty }^{t+s_n} K_{ij}(t+s_n-s) u_{j}(s) \,{\text{d}}s\\ & = \int \limits _{-\infty }^{t} K_{ij}(t-s) u_{j}(s+s_n) \,{\text{d}}s \end{aligned}$$

One has $\Vert K_{ij}(t-s) u_{j}(s+s_n)\Vert \le K^+ e^{-\nu ^K(t-s)}\Vert u_{j}(t)\Vert$ it follows that $\int \nolimits _{-\infty }^{t} K_{ij}(t-s) u_{j}(s+s_n) \,{\text{d}}s\le \frac{K^+}{\nu ^K}\Vert u_{j}(t)\Vert .$

Then using the Lebesgue-dominated convergence theorem, we obtain $\lim \nolimits _{n\longrightarrow \infty } \varPhi _{ij}^1(t+s_n)=\int \nolimits _{-\infty }^{t} K_{ij}(t-s) \theta _j(s)\,{\text{d}}s.$

Analogously, we get $\varPhi _{ij}^1(t)=\lim \nolimits _{n\longrightarrow \infty } \int \nolimits _{-\infty }^{t-s_n} K_{ij}(t-s_n-s) \theta _j(s)\,{\text{d}}s$.

Second, let us show that $\varPhi _{ij}^2(t) \in PC_0(\mathbb {R},\mathbb {R}^n).$

It is not difficult to see that $\varPhi _{ij}^2(t) \in PC(\mathbb {R},\mathbb {R}^n).$ We have

$$\begin{aligned} \varPhi _{ij}^2(t) & = \int \limits _{-\infty }^{t} K_{ij}(t-s)v_{j}(s) \,{\text{d}}s\\ & = \int \limits _{-\infty }^{0} K_{ij}(t-s)v_{j}(s) \,{\text{d}}s+\int \limits _{0}^{t} K_{ij}(t-s)v_{j}(s) \,{\text{d}}s\\ \end{aligned}$$

since $v_j \in PC_0(\mathbb {R},\mathbb {R}^n),$ for every $\varepsilon >0$ there exist a constant $N>0$ such that $\Vert v_j (s)\Vert \le \varepsilon$ for all $s\ge N$ and for all $t\ge 2N$, we obtain

$$\begin{aligned}&\Vert \varPhi _{ij}^2(t) \Vert \\ &\quad =\Vert \int \limits _{-\infty }^{0} K_{ij}(t-s)v_{j}(s) \,{\text{d}}s +\int \limits _{0}^{\frac{t}{2}} K_{ij}(t-s)v_{j}(s) \,{\text{d}}s \\ &\qquad +\int \limits _{\frac{t}{2}}^{t} K_{ij}(t-s)v_{j}(s) \,{\text{d}}s\Vert \\ &\quad \le \int \limits _{-\infty }^{0}K^+e^{-\nu ^K(t-s)}\Vert v_j(s)\Vert \,{\text{d}}s+\int \limits _{0}^{\frac{t}{2}}K^+e^{-\nu ^K(t-s)}\Vert v_j(s)\Vert \,{\text{d}}s \\ &\qquad +\int \limits _{\frac{t}{2}}^{t}K^+e^{-\nu ^K(t-s)}\Vert v_j(s)\Vert \,{\text{d}}s \\ &\quad \le \frac{K^+}{\nu ^K}e^{-\nu ^K t}\Vert v_j\Vert _\infty + \frac{K^+}{\nu ^K}e^{-\frac{\nu ^K}{2} t}\Vert v_j\Vert _\infty +\frac{K^+}{\nu ^K}\varepsilon . \end{aligned}$$

(18)

where $\Vert v_j\Vert _\infty = \sup \nolimits _{s \in \mathbb {R}}\Vert v_j(s)\Vert .$

Consequently $\varPhi _{ij}^2(.) \in PC_0(\mathbb {R},\mathbb {R}^n).$ The proof is completed. $\square$

Appendix 5: Proof of Lemma 6

Proof

Step 1 Noting $(\varPsi _{U_{\phi }})_{i}(s):=\int \nolimits _{-\infty }^{t} W(t,s)(U_\phi )_i(s)\,{\text{d}}s.$

First, by Lemmas 1–4, the function $(U_{\phi })_{i}$ belongs to $AAA(\mathbb {R},\mathbb {R}).$ This ensures the existence of two functions $\varLambda _{i}$ in $AA(\mathbb {R},\mathbb {R})$ and $\varOmega _{i}$ in $PC_0(\mathbb {R},\mathbb {R})$ such that for all $1\le i,j\le n,$ it can be expressed as $(U_{\phi })_{i}(.)=\varLambda _{i}(.)+\varOmega _{i}(.).$

One can write $\varPsi$ as follows:

$(\varPsi _{U_{\phi }})_i(t):=\int \limits _{-\infty }^{t} W(t,s)\varLambda _{i}(s)\,{\text{d}}s+\int \limits _{-\infty }^{t} W(t,s) \varOmega _{i}(s)\,{\text{d}}s.$

Let us study the almost automorphicity of

$(\varPsi \varLambda _{i}): t\mapsto \int \limits _{-\infty }^{t} W(t,s)\varLambda _{i}(s) \,{\text{d}}s.$

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 }\varLambda _{i}\left( t+s_{n}\right) =\varLambda _{i}^{1}\left( t\right) ,$$\forall \; t\in \mathbb {R},$

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

Let $(\varPsi ^1\varLambda _{i})(t)= \int \nolimits _{-\infty }^{t} W(t,s)\varLambda _{i}^1(s) \,{\text{d}}s,$ it follows that

$$\begin{aligned}&|(\varPsi \varLambda _{i})(t+s_n)-(\varPsi ^1\varLambda _{i})(t)| \\ &\quad =|\int \limits _{-\infty }^{t+s_n} W(t+s_n,s)\varLambda _{i}(s) \,{\text{d}}s-\int \limits _{-\infty }^{t} W(t,s)\varLambda _{i}^1(s) \,{\text{d}}s | \\ &\quad \le |\int \limits _{-\infty }^{t} W(t,s)\varLambda _{i}(t+s_n) \,{\text{d}}s -\int \limits _{-\infty }^{t} W(t,s)\varLambda _{i}^1(s) \,{\text{d}}s | \\ &\quad \le \int \limits _{-\infty }^{t}| W(t,s)||\varLambda _{i}(t+s_n)-\varLambda _{i}^1(s)|\,{\text{d}}s \\ &\quad \le \int \limits _{-\infty }^{t} K e^{-\delta (t-s)}|\varLambda _{i}(t+s_n)-\varLambda _{i}^1(s)|\,{\text{d}}s. \end{aligned}$$

(19)

Based on the Lebesgue-dominated convergence theorem, we have for all $t \in \mathbb {R}$

$$\begin{aligned} \lim \nolimits _{n\rightarrow +\infty }(\varPsi \varLambda _{i})(t+s_n)=(\varPsi ^1\varLambda _{i})(t). \end{aligned}$$

By a similar way, we prove that

$$\begin{aligned} \lim \nolimits _{n\rightarrow +\infty }(\varPsi ^1\varLambda _{i})(t-s_n)=(\varPsi \varLambda _{i})(t), \end{aligned}$$

which implies that $(\varPsi \varLambda _{i}) \in AA(\mathbb {R},\mathbb {R}^{n}).$

Second, we turn our attention to $(\varPsi \varOmega _{i}): t\mapsto \int \limits _{-\infty }^{t} W(t,s)\varOmega _{i}(s) \,{\text{d}}s.$ It is easy to prove that $(\varPsi \varOmega _{i})\in PC(\mathbb {R},\mathbb {R}).$

We have $\lim \nolimits _{t\rightarrow +\infty } \int \nolimits _{-\infty }^{t} W(t,s) \varOmega _{i}(s)\,{\text{d}}s=0.$ Since $\varOmega _{i} \in PC_0(\mathbb {R},\mathbb {R}),$ then $\lim \nolimits _{t\rightarrow +\infty } |\int \nolimits _{-\infty }^{t} W(t,s) \varOmega _{i}(s)|\,{\text{d}}s=0.$

By the Lebesgue-dominated convergence theorem, we have

$\lim \limits _{t\rightarrow +\infty } \int \limits _{-\infty }^{t} W(t,s) \varOmega _{i}(s)\,{\text{d}}s=0.$

Hence, the function $\varPsi \varOmega _{i}$ belongs to $PC_0(\mathbb {R},\mathbb {R}).$

Step 2 Proving that $\sum \limits _{t_k<t} W(t,t_k)(I_k(\phi _i(t_k))+\omega _k)$ belongs to $AAA(\mathbb {R},\mathbb {R}).$

From the assumption (H7), $I_k(\phi _i(t_k))\in AAA(\mathbb {R},\mathbb {R}).$ By definition, it can be expressed as

$$\begin{aligned} I_k(\phi _i(t_k))=I_{k1}(\phi _i(t_k))+I_{k2}(\phi _i(t_k)), \end{aligned}$$

such that $I_{k1}(\phi _i(t_k)) \in AA(\mathbb {R},\mathbb {R}),$$I_{k2}(\phi _i(t_k))=0.$ Then:

$$\begin{aligned}&\sum \limits _{t_k<t} W(t,t_k)(I_k(\phi _i(t_k))+\omega _k)\\ &\quad =\sum \limits _{t_k<t} W(t,t_k)(I_{k1}(\phi _i(t_k)) +\omega _k)+\sum \limits _{t_k<t} W(t,t_k)(I_{k2}(\phi _i(t_k))). \end{aligned}$$

For every real sequence $(t_{n})_{n\in \mathbb {N}}$, there exists a subsequence $(t_{n_{k}})_{n_{k} \in \mathbb {N}}$ such that $\lim \limits _{n_k\rightarrow +\infty }I_{k1}(\phi _i(t_k+t_{n_{k}}))=I_{k1}^1(\phi _i(t_k))$ and

$\lim \limits _{n_k\rightarrow +\infty }I_{k1}^1(\phi _i(t_k-t_{n_{k}}))=I_{k1}(\phi _i(t_k)).$

Now, we have

$$\begin{aligned}&\sum \limits _{t_k<t+t_{n_{k}}}W (t+t_{n_{k}},t_k)(I_{k1}(\phi _i(t_k)+\omega _k)\\ &\quad =\sum \limits _{t_k<t}W (t+t_{n_{k}},t_k+t_{n_{k}})(I_{k1}(\phi _i(t_k+t_{n_{k}}))+\omega _k), \end{aligned}$$

then

$$\begin{aligned}&\lim \limits _{n_k\rightarrow +\infty } \sum \limits _{t_k<t}W (t+t_{n_{k}},t_k+t_{n_{k}})(I_{k1}(\phi _i(t_k+t_{n_{k}}))+\omega _k) \\ &\quad =\sum \limits _{t_k < t} W(t,t_k)(I_{k1}^1( \phi _i(t_k ))+\omega _k) \end{aligned}$$

(20)

Similarly

$$\begin{aligned}&\sum \limits _{t_k<t-t_{n_{k}}}W (t-t_{n_{k}},t_k)(I_{k1}^1(\phi _i(t_k)+\omega _k)\\ &\quad = \sum \limits _{t_k<t}W (t-t_{n_{k}},t_k-t_{n_{k}})(I_{k1}^1(\phi _i(t_k-t_{n_{k}}))+\omega _k), \end{aligned}$$

then

$$\begin{aligned}&\lim \limits _{n_k\rightarrow +\infty }\sum \limits _{t_k<t}W (t-t_{n_{k}},t_k-t_{n_{k}})(I_{k1}^1(\phi _i(t_k-t_{n_{k}}))+\omega _k) \\ &\quad =\sum \limits _{t_k < t} W(t,t_k)(I_{k1}( \phi _i(t_k ))+\omega _k). \end{aligned}$$

(21)

Then, $\sum \limits _{t_k<t} W(t,t_k)(I_{k1}(\phi _i(t_k))+\omega _k) \in AA(\mathbb {R},\mathbb {R}).$

On the other hand,

$$\begin{aligned} \lim \limits _{t\rightarrow +\infty }\sum \limits _{t_k< t} |W(t,t_k)||I_{k2}(\phi _i(t_k))| & = \lim \limits _{t\rightarrow +\infty }|I_{k2}|\sum \limits _{t_k < t} |W(t,t_k)|=0, \end{aligned}$$

as $\sum \nolimits _{t_k< t} |W(t,t_k)|<\infty .$

By Steps 1 and 2 we have:

$$\begin{aligned} \varTheta _\phi (t):= \int \limits _{-\infty }^{t} W(t,s)(U_\phi )_i(s)\,{\text{d}}s+ \sum \limits _{t_k<t} W(t,t_k)(I_k(\phi _i(t_k))+\omega _k) \end{aligned}$$

maps $AAA(\mathbb {R},\mathbb {R})$ into itself. $\square$

Appendix 6: Proof of the Theorem 1

Proof

Let us calculate the norm of $\phi _0$. One has

$$\begin{aligned}&\Vert \phi _{0}\Vert \\ &\quad =\sup _{t \in \mathbb {R}}\bigg \{ \max _{ 1 \le i\le n}\big \{ \big |\int \limits _{-\infty }^{t}|W(t,s)||\gamma _{i}(s)|\,{\text{d}}s +\sum \limits _{t_k< t} |W(t,t_k)||\omega _k|\bigg \} \\ &\quad \le \sup _{t \in \mathbb {R}} \bigg \{\max _{ 1 \le i\le n}\big \{\int \limits _{-\infty }^{t}Ke^{-\delta (t-s)}\big |\gamma _{i}(s)\big |\,{\text{d}}s +\sum \limits _{t_k < t} Ke^{-\delta (t-t_k)}|\omega _k|\big \} \bigg \} \\ &\quad \le K \bar{\gamma }(\frac{1}{\delta }+\frac{1}{1-e^{-\delta }}) =\bar{R}, \end{aligned}$$

(22)

such that $\bar{\gamma }\ge \max \bigg \{\max \limits _{1 \le i \le n}|\gamma _i(t)|, \max \limits _{1 \le k \le n} |\omega _k|\bigg \}.$

After, $\Vert \phi \Vert _{\infty }\le \Vert \phi -\phi _{0}\Vert +\Vert \phi _{0}\Vert \le \frac{r}{1-r}\bar{R}+\bar{R }=\frac{\bar{R}}{1-r}.$

Set $S^{*}=\bigg \{ \phi \in AAA(\mathbb {R},\mathbb {R}^{n}) ; \Vert \phi -\phi _{0}\Vert \le \frac{r}{1-r}\bar{R} \bigg \}.$

Clearly, $S^{*}$ is a closed convex subset of $AAA(\mathbb {R},\mathbb {R}^{n}).$ Therefore, for any $\phi \in S^{*}$ by using the estimate just obtained, we see that

$$\begin{aligned}&\Vert \varTheta _{\phi }-\phi _{0}\Vert \\ &\quad \le \sup _{t \in \mathbb {R}} \bigg \{\max _{1 \le i\le n}\int \limits _{-\infty }^{t}\bigg (|W(t,s)|\bigg [ \sum \limits _{j=1}^{n}|a_{ij}(s)||f_j(\phi _j(s-\varsigma _{j}))| \\ &\qquad + \sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}|b_{ijl}(s)||g_j(\phi _j(s-\sigma _{j}))||g_l(\phi _l(s-\upsilon _{l}))| \\ &\qquad +\sum \limits _{j=1}^{n} |d_{ij}(s)|\int \limits _{-\infty }^{s} |K_{ij}(s-m)|| h_{j}(\phi _{j}(m))|\,{\text{d}}m \\ &\qquad +\sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}|r_{ijl}(s)|\int \limits _{-\infty }^{s}|P_{ijl}(s-m)||k_{j}(\phi _{j}(m))|\,{\text{d}}m \\ &\qquad \times \int \limits _{-\infty }^{s}|Q_{ijl}(s-m)||k_{l}(\phi _{l}(m))|\,{\text{d}}m \bigg ]\,{\text{d}}s\bigg ) \\ &\qquad +\sum \limits _{t_k< t} |W(t,t_k)||\omega _k(\phi (t_k))|\bigg \} \\ &\quad \le \sup _{t \in \mathbb {R}} \bigg \{\max _{1 \le i\le n}\bigg (\int \limits _{-\infty }^{t}Ke^{-\delta (t-s)}\bigg [ \sum \limits _{j=1}^{n}a_{ij}^*|f_j(\phi _j(s-\varsigma _{j}))| \\ &\qquad + \sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}b_{ijl}^*|g_j(\phi _j(s-\sigma _{j}))||g_l(\phi _l(s-\upsilon _{l}))| \\ &\qquad +\sum \limits _{j=1}^{n} d_{ij}^* \int \limits _{-\infty }^{s} |K_{ij}(s-m)|| h_{j}(\phi _{j}(m))|\,{\text{d}}m \\ &\qquad +\sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}r_{ijl}^*\int \limits _{-\infty }^{s}|P_{ijl}(s-m)||k_{j}(\phi _{j}(m))|\,{\text{d}}m \\ &\qquad \times \int \limits _{-\infty }^{s}|Q_{ijl}(s-m)||k_{l}(\phi _{l}(m))|\,{\text{d}}m\bigg ]\,{\text{d}}s\bigg ) \\ &\qquad + \sum \limits _{t_k< t} Ke^{-\delta (t-s)} |\omega _k(\phi (t_k))|\bigg \} \\ &\quad \le \sup _{t \in \mathbb {R}} \bigg \{\max _{1 \le i\le n}\bigg (\int \limits _{-\infty }^{t}Ke^{-\delta (t-s)}\bigg [ \sum \limits _{j=1}^{n}a_{ij}^* l_f^j|\phi _j(s- \varsigma _{j})| \\ &\qquad + \sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}b_{ijl}^* l_g^j e^l|\phi _j(s-\sigma _{j})|+\sum \limits _{j=1}^{n} d_{ij}^* l_h^j \frac{K^+}{\nu ^K}|\phi _{j}(s)| \\ &\qquad +\sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}r_{ijl}^* \frac{P^+}{\nu ^P} \frac{Q^+}{\nu ^Q} l_k^j M^l|\phi _j(s)| \bigg ]\,{\text{d}}s\bigg ) \\ &\qquad +\sum \limits _{t_k< t} Ke^{-\delta (t-s)} L |(\phi (t_k))|\bigg \} \\ &\quad \le \sup _{t \in \mathbb {R}}\bigg \{\int \limits _{-\infty }^{t}Ke^{-\delta (t-s)}\max _{1 \le i\le n}\bigg [ \sum \limits _{j=1}^{n}a_{ij}^* l_f^j + \sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}b_{ijl}^* l_g^j e^l \\ &\qquad +\sum \limits _{j=1}^{n} d_{ij}^* l_h^j \frac{K^+}{\nu ^+}+\sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}r_{ijl}^* \frac{P^+}{\nu ^P} \frac{Q^+}{\nu ^Q} l_k^j M^l \bigg ]\,{\text{d}}s \\ &\qquad +\sum \limits _{t_k < t} Ke^{-\delta (t-s)} L \bigg \}\Vert \ \phi \Vert \\ &\quad \le \bigg \{ \frac{K}{\delta } \max _{1 \le i\le n}\bigg [\sum \limits _{j=1}^{n}a_{ij}^* l_f^j + \sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}b_{ijl}^* l_g^j e^l \\ &\qquad +\sum \limits _{j=1}^{n} d_{ij}^* l_h^j \frac{K^+}{\nu ^K} +\sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}r_{ijl}^* \frac{P^+}{\nu ^P} \frac{Q^+}{\nu ^Q} l_k^j M^l\bigg ] \\ &\qquad + \frac{K L}{1-e^{-\delta }} \bigg \}\Vert \ \phi \Vert =r\Vert \ \phi \Vert \end{aligned}$$

(23)

then, $\varTheta _{\phi }\in S^{*}.$

Now our aim is to prove that $\varTheta$ is a contraction. For any $\phi _{1},\phi _{2} \in S^{*},$ we have

$$\begin{aligned}&\Vert \varTheta _{\phi _{1}}-\varTheta _{\phi _{2}}\Vert \\ &\quad \le \sup _{t \in \mathbb {R}} \bigg \{\max _{1 \le i\le n}(\int \limits _{-\infty }^{t}|W(t,s)| |U_{\phi _{1}}(s)-U_{\phi _{2}}(s)|\,{\text{d}}s) \\ &\qquad +\sum \limits _{t_k< t}|W(t,t_k)||I_k(\phi _1(t_k))-I_k(\phi _2(t_k))|\bigg \} \\ &\quad \le \sup _{t \in \mathbb {R}} \bigg \{\max _{1 \le i\le n}\bigg (\int \limits _{-\infty }^{t}Ke^{-\delta (t-s)} \\ &\qquad \times \bigg [\sum \limits _{j=1}^{n}a_{ij}^* l_f^j\bigg |\phi _1(s-\varsigma _{j})-\phi _2(s-\varsigma _{j})\bigg | \\ &\qquad + \sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}b_{ijl}^*\bigg |g_j(\phi _1(s-\sigma _{j})) g_l(\phi _1(s-\sigma _{l})) \\ &\qquad -g_j(\phi _2(s-\sigma _{j})) g_l(\phi _1(s-\sigma _{l}) \\ &\qquad +g_j(\phi _2(s-\sigma _{j})) g_l(\phi _1(s-\sigma _{l}) \\ &\qquad -g_j(\phi _2(s-\sigma _{j})) g_l(\phi _2(s-\sigma _{l})\bigg | \\ &\qquad +\sum \limits _{j=1}^{n} d_{ij}^* l_h^j \frac{K^+}{\nu ^K}\bigg | \phi _{1}(s)-\phi _{2}(s)\bigg | \\ &\qquad + \sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}r_{ijl}^*\bigg |\int \limits _{-\infty }^{s}P_{ijl}(s-m)h_j(\phi _{1}(m))\,{\text{d}}m \\ &\qquad \times \int \limits _{-\infty }^{s}Q_{ijl}(s-m)k_l(\phi _{1}(m))\,{\text{d}}m \\ &\qquad -\int \limits _{-\infty }^{s}P_{ijl}(s-m)h_j(\phi _{2}(m))\,{\text{d}}m \\ &\qquad \times \int \limits _{-\infty }^{s}Q_{ijl}(s-m)k_l(\phi _{1}(m))\,{\text{d}}m \\ &\qquad +\int \limits _{-\infty }^{s}P_{ijl}(s-m)h_j(\phi _{2}(m))\,{\text{d}}m \\ &\qquad \times \int \limits _{-\infty }^{s}Q_{ijl}(s-m)k_l(\phi _{1}(m))\,{\text{d}}m \\ &\qquad -\int \limits _{-\infty }^{s}P_{ijl}(s-m)h_j(\phi _{2}(m))\,{\text{d}}m \\ &\qquad \times \int \limits _{-\infty }^{s}Q_{ijl}(s-m)k_l(\phi _{2}(m))\,{\text{d}}m\bigg |\bigg ]\,{\text{d}}s\bigg ) \\ &\qquad +\sum \limits _{t_k< t}Ke^{-\delta (t-s)}L\bigg |\phi _1(t_k)-\phi _2(t_k)\bigg |\bigg \} \\ &\quad \le \max _{1 \le i\le n}\bigg \{\int \limits _{-\infty }^{t}Ke^{-\delta (t-s)} \bigg [\sum \limits _{j=1}^{n}a_{ij}^* l_f^j \\ &\qquad + \sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}b_{ijl}^*(l_g^je^l+l_g^l e^j) +\sum \limits _{j=1}^{n} d_{ij}^* l_h^j \frac{K^+}{\nu ^K} \\ &\qquad + \sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}r_{ijl}^* \frac{P^+}{\nu ^P} \frac{Q^+}{\nu ^Q} (l_k^j M^l+l_k^l M^j)\bigg ]\,{\text{d}}s \\ &\qquad +\sum \limits _{t_k < t}Ke^{-\delta (t-s)}L\bigg \} ||\phi _1-\phi _2|| \\ &\quad \le \bigg \{ \frac{K}{\delta }\max _{1 \le i\le n}\bigg [\sum \limits _{j=1}^{n}a_{ij}^* l_f^j + \sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}b_{ijl}^*(l_g^je^l+l_g^l e^j) \\ &\qquad +\sum \limits _{j=1}^{n} d_{ij}^* l_h^j \frac{K^+}{\nu ^K} + \sum \limits _{j=1}^{n} \sum \limits _{l=1}^{n}r_{ijl}^* \frac{P^+}{\nu ^P} \frac{Q^+}{\nu ^Q} (l_k^j M^l+l_k^l M^j) \bigg ] \\ &\qquad +\frac{KL}{1-e^{-\delta }}\bigg \} ||\phi _1-\phi _2||=\bar{r}||\phi _1-\phi _2|| \end{aligned}$$

(24)

which prove that $\varTheta$ is a contraction mapping.

By virtue of the Banach’s fixed-point theorem, $\varTheta$ has a unique fixed point which corresponds to the solution of (3) in $S^{*}.$$\square$

Appendix 7: Proof of Theorem 2

Proof

First, using Lemma 6, $\varTheta$ has a fixed point $\phi .$ Let $I^*_k(\phi (t_k))=I_k(\phi (t_k))+\omega _k.$

Hence, for all $t\in \mathbb {R},$ the fixed point $\phi$ satisfies the following integral system:

$\phi (t):= \int \limits _{-\infty }^{t} W(t,s)U_\phi (s)\,{\text{d}}s+ \sum \limits _{t_k<t} W(t,t_k)(I^*_k(\phi (t_k))).$

Fixed $t_0,$$t_0\ne t_i,$$i \in \mathbb {Z},$ we have

$$\begin{aligned} \phi (a)= \int \limits _{-\infty }^{a} W(a,s)U_\phi (s)\,{\text{d}}s+ \sum \limits _{t_k<a} W(a,t_k)(I^*_k(\phi (t_k))). \end{aligned}$$

Therefore

$$\begin{aligned} \phi (t) & = \int \limits _{-\infty }^{a} W(t,s)U_\phi (s)\,{\text{d}}s+ \sum \limits _{t_k<a} W(T,t_k)(I^*_k(\phi (t_k))) \\ &\quad + \int \limits _{a}^{t} W(t,s)U_\phi (s)\,{\text{d}}s+ \sum \limits _{a<t_k<t} W(t,t_k)(I^*_k(\phi (t_k))) \\ & = W(t,a)\phi (a)+\int \limits _{a}^{t} W(t,s)U_\phi (s)\,{\text{d}}s \\ &\quad+\sum \limits _{a<t_k<t} W(t,t_k)(I^*_k(\phi (t_k))). \end{aligned}$$

(25)

Second, by Theorem 1, we know that system (3) has an asymptotically almost automorphic solution u(t), by using integral form of system (3), if $t>\sigma ,\;\; \sigma \ne t_k,\; \; k \in \mathbb {Z}$

$$\begin{aligned} u (t) & = W(t,\sigma )u(\sigma )+\int \limits _{\sigma }^{t} W(t,s)U_u(s)\,{\text{d}}s \\ &\quad+\sum \limits _{\sigma<t_k<t} W(t,t_k)(I^*_k(u(t_k))), \end{aligned}$$

(26)

Let $u(t)=u(t,\sigma ,\phi _1)$ and $v(t)=v(t,\sigma ,\phi _2)$ be two solutions of (3), then

$$\begin{aligned} u (t) & = W(t,\sigma )u(\sigma )+\int \limits _{\sigma }^{t} W(t,s)U_u(s)\,{\text{d}}s\\ &\quad+\sum \limits _{\sigma<t_k<t} W(t,t_k)(I^*_k(u(t_k)))\\ v(t) & = W(t,\sigma )v(\sigma )+\int \limits _{\sigma }^{t} W(t,s)U_v(s)\,{\text{d}}s\\ &\quad+\sum \limits _{\sigma<t_k<t} W(t,t_k)(I^*_k(v(t_k))). \end{aligned}$$

Therefore,

$$\begin{aligned}&\Vert u(t)-v(t)\Vert \\ &\quad \le \Vert W(t,\sigma )[\phi _1-\phi _2]\Vert +\Vert \int \limits _{\sigma }^{t}W(t,s)[U_u(s)- U_v (s)]\,{\text{d}}s \Vert \\ &\qquad +\Vert \sum \limits _{\sigma<t_k<t} W(t,t_k) [I^*_k(u(t_k))-I^*_k(v(t_k))] \Vert \\ &\quad \le \Vert W(t,\sigma )[\phi _1-\phi _2]\Vert + \int \limits _{\sigma }^{t}\Vert W(t,s)\Vert \Vert U_u(s)- U_v (s)\Vert \,{\text{d}}s \\ &\qquad +\sum \limits _{\sigma<t_k<t}\Vert W(t,t_k)\Vert \Vert I^*_k(u(t_k))-I^*_k(v(t_k)) \Vert \\ &\quad \le Ke^{-\delta (t-\sigma )} \Vert \phi _1-\phi _2\Vert \\ &\qquad + \int \limits _{\sigma }^{t}Ke^{-\delta (t-\sigma )} \sum \limits _{i=1}^{n}\bigg [\sum \limits _{j=1}^{n}a_{ij}^*l_j^f+\sum \limits _{j=1}^{n}\sum \limits _{l=1}^{n} b_{ijl}^*( l_g^je^l+ l_g^le^j ) \\ &\qquad +\sum \limits _{j=1}^{n}d_{ij}^*\frac{K^+}{\nu ^K}l_h^j +\sum \limits _{j=1}^{n}\sum \limits _{l=1}^{n}r_{ijl}^* \frac{P^+}{\nu ^P} \frac{Q^+}{\nu ^Q} (l_k^j M^l+l_k^l M^j) \bigg ] \\ &\qquad \times \Vert u(s)- v(s)\Vert \,{\text{d}}s+\sum \limits _{\sigma<t_k<t} Ke^{-\delta (t-t_k)} L \Vert u(t_k)-v(t_k) \Vert \end{aligned}$$

(27)

Then

$$\begin{aligned}&e^{\delta t}\Vert u(t)-v(t)\Vert \\ &\quad \le Ke^{\delta \sigma } \Vert \phi _1-\phi _2\Vert + \int \limits _{\sigma }^{t}K \sum \limits _{i=1}^{n}\left[ \sum \limits _{j=1}^{n}a_{ij}^*l_j^f\right. \\ &\qquad +\sum \limits _{j=1}^{n}\sum \limits _{l=1}^{n} b_{ijl}^*\left( l_g^je^l+ l_g^le^j \right) +\sum \limits _{j=1}^{n}d_{ij}^*\frac{K^+}{\nu ^K}l_h^j \\ &\left. \qquad +\sum \limits _{j=1}^{n}\sum \limits _{l=1}^{n}r_{ijl}^* \frac{P^+}{\nu ^P} \frac{Q^+}{\nu ^Q} \left( l_k^j M^l+l_k^l M^j\right) \right] e^{\delta s} \Vert u(s)- v(s)\Vert \,{\text{d}}s \\ &\qquad +\sum \limits _{\sigma<t_k<t} Ke^{\delta t_k} L \Vert u(t_k)-v(t_k) \Vert \end{aligned}$$

(28)

Let $y(t)=e^{\delta t}\Vert u(t)-v(t)\Vert ,$ Eq. (28) can be rewritten in the following form:

$$\begin{aligned} y(t) & \le K y(\sigma )+ \int \limits _{\sigma }^{t}K \sum \limits _{i=1}^{n}\left[ \sum \limits _{j=1}^{n}a_{ij}^*l_j^f\right. \\ &+\sum \limits _{j=1}^{n}\sum \limits _{l=1}^{n} b_{ijl}^*\left( l_g^je^l+ l_g^le^j\right) +\sum \limits _{j=1}^{n}d_{ij}^*\frac{K^+}{\nu ^K}l_h^j \\ &\left. +\sum \limits _{j=1}^{n}\sum \limits _{l=1}^{n}r_{ijl}^* \frac{P^+}{\nu ^P} \frac{Q^+}{\nu ^Q} \left( l_k^j M^l+l_k^l M^j\right) \right] y(s)\,{\text{d}}s \\ &+\sum \limits _{\sigma<t_k<t} K L \; y(t_k). \end{aligned}$$

(29)

By the generalized Gronwall–Bellman inequality, we have

$$\begin{aligned} y(t) & \le K y(\sigma ) \prod _{\sigma<t_k<t}(1+KN_1) e^{ \int \limits _{\sigma }^{t} KN_2\,{\text{d}}s } \\ & = K y(\sigma ) \prod _{\sigma<t_k<t}(1+KN_1) e^{KN_2(t-\sigma )} \end{aligned}$$

(30)

$$\begin{aligned} N_1 & = KL,\;N_2=\sum \limits _{i=1}^{n}\left[ \sum \limits _{j=1}^{n}a_{ij}^*l_j^f\right. \\ &+\sum \limits _{j=1}^{n}\sum \limits _{l=1}^{n} b_{ijl}^*\left( l_g^je^l+ l_g^le^j \right) +\sum \limits _{j=1}^{n}d_{ij}^*\frac{K^+}{\nu ^K}l_h^j \\ &\left. + \sum \limits _{j=1}^{n}\sum \limits _{l=1}^{n}r_{ijl}^* \frac{P^+}{\nu ^P} \frac{Q^+}{\nu ^Q} \left( l_k^j M^l+l_k^l M^j\right) \right] . \end{aligned}$$

(31)

Since $\vartheta =\inf \nolimits _{k\in \mathbb {Z}}(t_{k+1}-t_k)>0$, we have

$$y(t)\le Ky(\sigma )(1+KN_1)^{\frac{t-\sigma }{\vartheta }}e^{KN_2(t-\sigma )}= Ky(\sigma )e^{\zeta (t-\sigma )},$$

(32)

$$\begin{aligned} \zeta & = \frac{\ln (1+KL)}{\vartheta }+K\sum \limits _{i=1}^{n}\left[ \sum \limits _{j=1}^{n}a_{ij}^*l_j^f\right. \\ &+\sum \limits _{j=1}^{n}\sum \limits _{l=1}^{n} b_{ijl}^*\left( l_g^je^l+ l_g^le^j\right) \sum \limits _{j=1}^{n}d_{ij}^*\frac{K^+}{\nu ^K}l_h^j \\ &\left. + \sum \limits _{j=1}^{n}\sum \limits _{l=1}^{n}r_{ijl}^* \frac{P^+}{\nu ^P} \frac{Q^+}{\nu ^Q} (l_k^j M^l+l_k^l M^j)\right] . \end{aligned}$$

(33)

That is $\Vert u(t)-v(t)\Vert \le K\Vert \phi _1-\phi _2\Vert e^{(\zeta -\delta )(t-\sigma )}.$

Since $(\zeta -\delta )<0,$ then system (3) has an exponential stable asymptotically almost automorphic solution. This completes the proof. $\square$