The main purpose of this paper is to investigate the existence and uniqueness of globally attractive almost periodic solution of system (1.1).
First of all, we investigate the attractivity of the solution of (1.1).
Theorem 4.1
Assume that
\((H_{1})\)
and
\((H_{2})\)
hold; suppose further that
-
\((H_{3})\)
:
-
\(\gamma^{-}m_{1}>1\)
and
-
\((H_{4})\)
:
-
\(\alpha^{-}+c^{-}m_{2}>b^{+}+\frac{1}{e^{2}}\beta^{+}\), \(a^{-}>c^{+}M_{1}\),
hold, then system (1.1) is globally attractive. That is, for any positive solutions
\((x(n),u(n))\)
and
\((p(n),q(n))\)
of system (1.1), we have
\(\lim_{n\rightarrow\infty} (x(n)-p(n) )=0\), \(\lim_{n\rightarrow\infty} (u(n)-q(n) )=0\).
Proof
For any solutions \((x(n),u(n))\) and \((p(n),q(n))\) of system (1.1), it follows from Theorem 3.1 that
$$\begin{aligned}& m_{1}\leq\liminf_{n\rightarrow\infty}x(n)\leq\limsup _{n\rightarrow \infty}x(n)\leq M_{1},\quad\quad m_{1}\leq \liminf_{n\rightarrow\infty}p(n)\leq\limsup_{n\rightarrow \infty}p(n)\leq M_{1}, \\& m_{2}\leq\liminf_{n\rightarrow\infty}u(n)\leq\limsup _{n\rightarrow \infty}u(n)\leq M_{2},\quad\quad m_{2}\leq \liminf_{n\rightarrow\infty}q(n)\leq\limsup_{n\rightarrow \infty}q(n)\leq M_{2}. \end{aligned}$$
For any positive constant \(\varepsilon>0\) small enough, there exists an integer \(n_{0}\) such that, for all \(n\geq n_{0}\),
$$ \begin{aligned} &m_{1}-\varepsilon\leq x(n),\quad\quad p(n) \leq M_{1}+\varepsilon, \\ &m_{2}-\varepsilon\leq u(n), \quad\quad q(n)\leq M_{2}+ \varepsilon. \end{aligned} $$
(4.1)
Using the mean value theorem, we get
$$ x(n)e^{-x(n)}-p(n)e^{-p(n)}= \bigl(1-\theta(n) \bigr)e^{-\theta(n)} \bigl(x(n)-p(n) \bigr), $$
(4.2)
where \(\theta(n)\) lies between \(x(n)\) and \(p(n)\).
Let
$$V_{1}(n)= \bigl\vert x(n)-p(n) \bigr\vert + \bigl\vert u(n)-q(n) \bigr\vert . $$
Then, from system (1.1) and (4.2), we get
$$\begin{aligned} \Delta V_{1}(n) =&V_{1}(n+1)-V_{1}(n) \\ \leq&-\alpha(n) \bigl\vert x(n)-p(n) \bigr\vert \\ &{}+\beta(n) \bigl\vert x(n-\tau_{1})e^{-\gamma(n) x(n-\tau_{1})}-p(n-\tau _{1})e^{-\gamma(n) p(n-\tau_{1})} \bigr\vert \\ &{}-c(n)u(n) \bigl\vert x(n)-p(n) \bigr\vert +c(n)p(n) \bigl\vert u(n)-q(n) \bigr\vert \\ &{}-a(n) \bigl\vert u(n)-q(n) \bigr\vert +b(n) \bigl\vert x(n- \tau_{2})-p(n-\tau_{2}) \bigr\vert \\ \leq& -\alpha(n) \bigl\vert x(n)-p(n) \bigr\vert \\ &{}+\beta(n) \bigl\vert \bigl(1-\theta(n-\tau_{1}) \bigr)e^{-\theta(n-\tau_{1})} \bigl(x(n-\tau_{1})-p(n-\tau_{1}) \bigr) \bigr\vert \\ &{}-c(n)u(n) \bigl\vert x(n)-p(n) \bigr\vert +c(n)p(n) \bigl\vert u(n)-q(n) \bigr\vert \\ &{}-a(n) \bigl\vert u(n)-q(n) \bigr\vert +b(n) \bigl\vert x(n- \tau_{2})-p(n-\tau_{2}) \bigr\vert . \end{aligned}$$
(4.3)
According to \((H_{3})\), (4.3), and the fact that \(\max_{x\in[1,+\infty ]}(1-x)e^{-x}= \frac{1}{e^{2}}\), for \(n\geq n_{0}+\tau\), we have
$$\begin{aligned} \Delta V_{1}(n) \leq& -\alpha(n) \bigl\vert x(n)-p(n) \bigr\vert \\ &{}+ \frac{1}{e^{2}}\beta(n) \bigl\vert x(n-\tau_{1})-p(n- \tau_{1}) \bigr\vert \\ &{}-c(n)u(n) \bigl\vert x(n)-p(n) \bigr\vert +c(n)p(n) \bigl\vert u(n)-q(n) \bigr\vert \\ &{}-a(n) \bigl\vert u(n)-q(n) \bigr\vert +b(n) \bigl\vert x(n- \tau_{2})-p(n-\tau_{2}) \bigr\vert . \end{aligned}$$
(4.4)
Let
$$\begin{aligned}& V_{2}(n) = \sum_{u=n-\tau_{1}}^{n-1} \frac{1}{e^{2}}\beta(u+\tau _{1}) \bigl\vert x(u)-p(u) \bigr\vert , \\& V_{3}(n) = \sum_{u=n-\tau_{2}}^{n-1}b(u+ \tau_{2}) \bigl\vert x(u)-p(u) \bigr\vert . \end{aligned}$$
Then
$$ \begin{aligned} &\Delta V_{2}(n)= \frac{1}{e^{2}} \beta(n+\tau_{1}) \bigl\vert x(n)-p(n) \bigr\vert - \frac {1}{e^{2}}\beta(n) \bigl\vert x(n-\tau_{1})-p(n- \tau_{1}) \bigr\vert , \\ &\Delta V_{3}(n)= b(n+\tau_{2}) \bigl\vert x(n)-p(n) \bigr\vert -b(n) \bigl\vert x(n-\tau_{2})-p(n-\tau_{2}) \bigr\vert . \end{aligned} $$
(4.5)
Define
$$V(n)=V_{1}(n)+V_{2}(n)+V_{3}(n). $$
Then it follows from (4.4) and (4.5) that
$$\begin{aligned} \Delta V(n) \leq& \biggl(-\alpha(n)+ \frac{1}{e^{2}}\beta(n+\tau _{1})-c(n)u(n)+b(n+\tau_{2}) \biggr) \bigl\vert x(n)-p(n) \bigr\vert \\ &{}+ \bigl(c(n)p(n)-a(n) \bigr) \bigl\vert u(n)-q(n) \bigr\vert . \end{aligned}$$
(4.6)
From (4.1) and (4.6), for \(n>n_{0}+\tau\), we obtain
$$\begin{aligned} \Delta V(n) \leq& \biggl(-\alpha^{-}+ \frac{1}{e^{2}}\beta ^{+}-c^{-}(m_{2}-\varepsilon)+b^{+} \biggr) \bigl\vert x(n)-p(n) \bigr\vert \\ &{}+ \bigl(c^{+}(M_{1}+\varepsilon)-a^{-} \bigr) \bigl\vert u(n)-q(n) \bigr\vert . \end{aligned}$$
(4.7)
From condition \((H_{4})\) and the above ε, we can choose δ small enough such that
$$ \delta=\min \biggl\{ \alpha^{-}+c^{-}(m_{2}- \varepsilon)-b^{+}-\frac {1}{e^{2}}\beta^{+}, a^{-}-c^{+}(M_{1}+\varepsilon) \biggr\} >0. $$
(4.8)
From (4.7) and (4.8), we obtain
$$ \Delta V(n)\leq-\delta \bigl( \bigl\vert x(n)-p(n) \bigr\vert + \bigl\vert u(n)-q(n) \bigr\vert \bigr). $$
(4.9)
Summing both sides of the above inequalities from \(n_{0}+\tau\) to n, we have
$$\sum_{s=n_{0}+\tau}^{n} \bigl(V(s+1)-V(s) \bigr) \leq-\delta\sum_{s=n_{0}+\tau}^{n} \bigl( \bigl\vert x(s)-p(s) \bigr\vert + \bigl\vert u(s)-q(s) \bigr\vert \bigr), $$
which implies
$$V(n+1)+\delta\sum_{s=n_{0}+\tau}^{n} \bigl( \bigl\vert x(s)-p(s) \bigr\vert + \bigl\vert u(s)-q(s) \bigr\vert \bigr)\leq V(n_{0}+\tau), $$
that is,
$$\sum_{s=n_{0}+\tau}^{n} \bigl( \bigl\vert x(s)-p(s) \bigr\vert + \bigl\vert u(s)-q(s) \bigr\vert \bigr)\leq \frac {V(n_{0}+\tau)}{\delta}. $$
It follows from (4.1) that \(V_{i}(n_{0}+\tau)\) (\(i=1,2,3 \)) are all bounded. Hence
$$\sum_{s=n_{0}+\tau}^{n} \bigl( \bigl\vert x(s)-p(s) \bigr\vert + \bigl\vert u(s)-q(s) \bigr\vert \bigr)\leq \frac {V(n_{0}+\tau)}{\delta}< +\infty, $$
which means that
$$\sum_{s=n_{0}+\tau}^{+\infty} \bigl( \bigl\vert x(s)-p(s) \bigr\vert + \bigl\vert u(s)-q(s) \bigr\vert \bigr)\leq \frac{V(n_{0}+\tau)}{\delta}< +\infty. $$
This implies that \(\lim_{n\rightarrow\infty} (\vert x(n)-p(n)\vert +\vert u(n)-q(n)\vert )=0\), or \(\lim_{n\rightarrow\infty} (x(n)-p(n) )=0\), \(\lim_{n\rightarrow\infty} (u(n)-q(n) )=0\). This completes the proof of Theorem 4.1. □
Next, we investigate the existence and uniqueness of an almost periodic sequence solution of system (1.1) by using almost periodic functional hull theory.
Let \(\{\mu_{k}\}\) be any integer valued sequence such that \(\mu _{k}\rightarrow\infty\) as \(k\rightarrow\infty\). According to Lemma 2.6, taking a subsequence if necessary, we have \(\alpha(n+\mu _{k})\rightarrow\alpha^{*}(n)\), \(\beta(n+\mu_{k})\rightarrow\beta ^{*}(n)\), \(\gamma(n+\mu_{k})\rightarrow\gamma^{*}(n)\), \(a(n+\mu _{k})\rightarrow a^{*}(n)\), \(b(n+\mu_{k})\rightarrow b^{*}(n)\), \(c(n+\mu _{k})\rightarrow c^{*}(n)\), as \(k\rightarrow\infty\) for \(n\in Z\). Then we get a hull equation of system (1.1) as follows:
$$ \begin{aligned} &\Delta x(n)=-\alpha^{*}(n)x(n)+ \beta^{*}(n)x(n-\tau_{1})e^{-\gamma ^{*}(n)x(n-\tau_{1})}-c^{*}(n)x(n)u(n), \\ &\Delta u(n)=-a^{*}(n)u(n)+b^{*}(n)x(n- \tau_{2}), \end{aligned} $$
(4.10)
By the almost periodic theory, we can conclude that if system (1.1) satisfies \((H_{1})\)-\((H_{4})\), then the hull equation (4.10) of system (1.1) also satisfies \((H_{1})\)-\((H_{4})\).
From Theorem 3.4 in [29], the following lemma can easily be obtained.
Lemma 4.2
If each hull equation of system (1.1) has a unique strictly positive solution, then the almost periodic difference system (1.1) has a unique strictly positive almost periodic solution.
Theorem 4.3
If the almost periodic difference system (1.1) satisfies
\((H_{1})\)-\((H_{4})\), then the almost periodic difference system (1.1) admits a uniqueness of globally attractive almost periodic sequence solution.
Proof
By Lemma 4.2, we only need to prove that each hull equation of system (1.1) has a unique strictly positive solution. We prove that the existence of a strictly positive solution of any hull equations of system (1.1).
By the almost periodicity of \(\{\alpha(n)\}\), \(\{\beta(n)\}\), \(\{\gamma (n)\}\), \(\{a(n)\}\), \(\{b(n)\} \), and \(\{c(n)\}\), there exists an integer valued sequence \(\{\delta_{k}\}\) with \(\delta_{k}\rightarrow\infty\) as \(k\rightarrow\infty\) such that \(\alpha(n+\delta_{k})\rightarrow\alpha ^{*}(n)\), \(\beta(n+\delta_{k})\rightarrow\beta^{*}(n)\), \(\gamma (n+\delta_{k})\rightarrow\gamma^{*}(n)\), \(a(n+\delta_{k})\rightarrow a^{*}(n)\), \(b(n+\delta_{k})\rightarrow b^{*}(n)\), \(c(n+\delta _{k})\rightarrow c^{*}(n)\), as \(k\rightarrow\infty\) for \(n\in Z\). Suppose that \(X(n)=(x(n),u(n))\) is any positive solution of hull equation (4.10). Since \((H_{1})\) and \((H_{2})\) hold, combined with the proof of Theorem 3.1, we have
$$ \begin{aligned} &m_{1}\leq\liminf_{n\rightarrow\infty}x(n) \leq\limsup_{n\rightarrow \infty}x(n)\leq M_{1}, \\ &m_{2}\leq\liminf_{n\rightarrow\infty}u(n)\leq\limsup _{n\rightarrow \infty}u(n)\leq M_{2}. \end{aligned} $$
(4.11)
Therefore
$$\begin{aligned}& 0< \inf_{n\in Z^{+}}x(n)\leq\sup_{n\in Z^{+}}x(n)< \infty, \\& 0< \inf_{n\in Z^{+}}u(n)\leq\sup_{n\in Z^{+}}u(n)< \infty. \end{aligned}$$
Let ε be an arbitrary small positive number. It follows from (4.11) that there exists a positive integer \(N_{0}\) such that
$$m_{1}-\varepsilon\leq x(n)\leq M_{1}+\varepsilon,\quad \quad m_{2}-\varepsilon\leq u(n)\leq M_{2}+\varepsilon,\quad n>N_{0}. $$
Define \(x_{k}(n)=x(n+\delta_{k})\) and \(u_{k}(n)=u(n+\delta_{k})\) for all \(n\geq N_{0}+\tau-\delta_{k}\), \(k\in Z^{+}\). For any positive integer q, it is easy to see that there exist sequences \(\{x_{k}(n):k\geq q\}\) and \(\{ u_{k}(n):k\geq q\}\) such that the sequences \(\{x_{k}(n)\}\) and \(\{ u_{k}(n)\}\) have subsequences, denoted by \(\{x_{k}(n)\}\) and \(\{ u_{k}(n)\}\) again, converging on any finite interval of Z as \(k\rightarrow\infty\). Thus we have sequences \(\{y(n)\}\) and \(\{v(n)\}\) satisfying
$$x_{k}(n)\rightarrow y(n), \quad\quad u_{k}(n)\rightarrow v(n),\quad \text{for } n\in Z \text{ as } k\rightarrow\infty. $$
This, combined with
$$\begin{aligned}& \Delta x_{k}(n) = -\alpha^{*}(n+\tau_{k})x_{k}(n)+ \beta^{*}(n+\tau _{k})x_{k}(n- \tau_{1})e^{-\gamma^{*}(n+\tau_{k})x_{k}(n-\tau_{1})} \\& \hphantom{\Delta x_{k}(n) =}{} -c^{*}(n+\tau_{k})x_{k}(n)u_{k}(n), \\& \Delta u_{k}(n) = -a^{*}(n+\tau_{k})u_{k}(n)+b^{*}(n+ \tau _{k})x_{k}(n-\tau_{2}), \end{aligned}$$
give us
$$\begin{aligned}& \Delta y(n) = -\alpha^{*}(n)y(n)+\beta^{*}(n)y(n- \tau_{1})e^{-\gamma ^{*}(n)y(n-\tau_{1})}-c^{*}(n)y(n)v(n), \\& \Delta v(n) = -a^{*}(n)v(n)+b^{*}(n)y(n- \tau_{2}). \end{aligned}$$
We can easily see that \((y(n),v(n))\) is a solution of hull equation (4.10) and
$$m_{1}-\varepsilon\leq y(n)\leq M_{1}+\varepsilon,\quad \quad m_{2}-\varepsilon\leq v(n)\leq M_{2}+\varepsilon,\quad n\in Z. $$
Since ε is an arbitrary small positive number, it follows that
$$m_{1}\leq y(n)\leq M_{1},\quad\quad m_{2}\leq v(n)\leq M_{2}, \quad n\in Z. $$
that is,
$$0< \inf_{n\in Z}y(n)\leq\sup_{n\in Z}y(n)< \infty, \quad\quad 0< \inf_{n\in Z}y(n)\leq\sup_{n\in Z}y(n)< \infty. $$
This implies that each hull equation of the almost periodic difference system (1.1) has at least one strictly positive solution.
Now we prove the uniqueness of the strictly positive solution of each hull equation (4.10). Suppose that the hull equation (4.10) has two arbitrary strictly positive solutions \((x^{*}(n),u^{*}(n))\) and \((p^{*}(n),q^{*}(n))\). Similar to the proof of Theorem 4.1, we define a Lyapunov functional,
$$ V^{*}(n)=V_{1}^{*}(n)+V_{2}^{*}(n)+V_{3}^{*}(n), $$
(4.12)
where
$$\begin{aligned}& V_{1}^{*}(n) = \bigl\vert x^{*}(n)-p^{*}(n) \bigr\vert + \bigl\vert u^{*}(n)-q^{*}(n) \bigr\vert , \\& V_{2}^{*}(n) = \sum_{u=n-\tau_{1}}^{n-1} \frac{1}{e^{2}}\beta(u+\tau _{1}) \bigl\vert x^{*}(u)-p^{*}(u) \bigr\vert , \\& V_{3}^{*}(n) = \sum_{u=n-\tau_{2}}^{n-1}b(u+ \tau_{2}) \bigl\vert x^{*}(u)-p^{*}(u) \bigr\vert . \end{aligned}$$
Calculating the difference of \(V^{*}\) along the solution of the hull equation (4.10), similar to the discussion of (4.9), one has
$$ \Delta V^{*}(n)\leq-\delta \bigl( \bigl\vert x^{*}(n)-p^{*}(n) \bigr\vert + \bigl\vert u^{*}(n)-q^{*}(n) \bigr\vert \bigr), \quad \text{for } n\in Z. $$
(4.13)
We immediately see that \(V^{*}(n)\) is a non-increasing function on Z. Summing both sides of (4.13) from n to 0, we have
$$\delta\sum_{s=n}^{0} \bigl( \bigl\vert x^{*}(s)-p^{*}(s) \bigr\vert + \bigl\vert u^{*}(s)-q^{*}(s) \bigr\vert \bigr)\leq V^{*}(n)-V^{*}(1), \quad \text{for } n< 0. $$
Note that \(V^{*}(n)\) is bounded. Hence we have \(\sum_{s=-\infty }^{0} (\vert x^{*}(s)-p^{*}(s)\vert +\vert u^{*}(s)-q^{*}(s)\vert )<+\infty\), which implies that
$$ \lim_{n\rightarrow-\infty} \bigl(x^{*}(n)-p^{*}(n) \bigr)=0, \quad\quad \lim_{n\rightarrow-\infty} \bigl(u^{*}(n)-q^{*}(n) \bigr)=0. $$
(4.14)
Let ε be an arbitrary small positive number. It follows from (4.14) that there exists a positive integer \(N_{1}>0\) such that
$$\bigl\vert x^{*}(n)-p^{*}(n) \bigr\vert < \frac{\varepsilon}{Q}, \quad\quad \bigl\vert u^{*}(n)-q^{*}(n) \bigr\vert < \frac {\varepsilon}{Q}, \quad n< -N_{1}, $$
where \(Q=2+\frac{\tau_{1}\beta^{+}}{e^{2}}+\tau_{2}b^{+}\). Therefore, for \(n<-N_{1}\),
$$V_{1}^{*}(n)\leq\frac{\varepsilon}{Q}+ \frac{\varepsilon}{Q}, \quad\quad V_{2}^{*}(n)\leq\frac{\tau_{1}\beta^{+}}{e^{2}} \frac{\varepsilon}{Q},\quad\quad V_{3}^{*}(n)\leq \tau_{2}b^{+} \frac{\varepsilon}{Q}. $$
It follows from (4.12) and the above inequalities that
$$V^{*}(n)\leq Q \frac{\varepsilon}{Q}=\varepsilon, \quad n< -N_{1}, $$
so \(\lim_{n\rightarrow-\infty}V^{*}(n)=0\). Notice that \(V^{*}(n)\) is a non-increasing function on Z, and then \(V^{*}(n)\equiv0\). That is \(x^{*}(n)=p^{*}(n)\), \(u^{*}(n)=q^{*}(n)\), for all \(n\in Z\). Therefore, each hull equation of system (1.1) has a unique strictly positive solution.
In view of the above discussion, any hull equation of system (1.1) has a unique strictly positive solution. By Theorem 3.1 and Theorem 4.1, the almost periodic system (1.1) has a uniqueness of globally attractive almost periodic solution. The proof is completed. □