 Research
 Open Access
 Published:
On the solutions of a maxtype system of difference equations of higher order
Advances in Difference Equations volume 2020, Article number: 213 (2020)
Abstract
In this paper, we study the following maxtype system of difference equations of higher order:
where \(A,B\in (0, +\infty )\), \(t,s\in \{1,2,\ldots \}\) with \(\gcd (s,t)=1\), the initial values \(x_{d},y_{d},x_{d+1},y_{d+1}, \ldots , x_{1}, y_{1}\in (0,+ \infty )\) and \(d=\max \{t,s\}\).
Introduction
Concrete nonlinear difference equations and systems have attracted some recent attention (see, e.g., [1–39]). One of the classes of such equations/systems are maxtype difference equations/systems. For some results of solutions of many maxtype difference equations and systems, such as eventual periodicity, the boundedness character and attractivity, see, e.g. [1–5, 7–9, 11–16, 18–25, 28–30, 32–36, 38, 39] and the references therein. Our purpose in this paper is to study the eventual periodicity of the following maxtype system of difference equation of higher order:
where \(A,B\in {\mathbf{{R}}}_{+}\equiv (0,+\infty )\), \(t,s\in {\mathbf{{N}\equiv }\{\mathbf{1},\mathbf{2},\ldots \}}\) with \(\gcd (s,t)=1\), the initial values \(x_{d},y_{d}, x_{d+1},y_{d+1}, \ldots , x_{1}, y_{1}\in \mathbf{{R}}_{+}\) and \(d=\max \{t,s\}\).
When \(t=1\) and \(s=2\), (1.1) reduces to the maxtype system of difference equations
Fotiades and Papaschinopoulos in [5] showed that every positive solution of (1.2) is eventually periodic.
In 2012, Stević [23] obtained in an elegant way the general solution to the following maxtype system of difference equations:
for the case \(x_{0},y_{0}\geq A>0\) and \(y_{0}/x_{0}\geq \max \{A,1/A\}\).
In [35], Sun and Xi studied the following maxtype system of difference equations:
where \(A,B\in { \mathbf{{R}}}_{+}\), \(m,r,t\in {\mathbf{{N}}}\) and the initial values \(x_{d},y_{d},x_{d+1},y_{d+1}, \ldots , x_{1}, y_{1}\in {\mathbf{{R}}}_{+}\) with \(d=\max \{m,r,t\}\) and showed that every positive solution of (1.4) is eventually periodic with period 2m.
When \(m=r=t=1\) and \(A=B\), (1.4) reduces to the maxtype system of difference equations
Yazlik et al. [39] in 2015 obtained in an elegant way the general solution of (1.5).
In 2012, Stević [24] studied the following maxtype system of difference equations:
where \(s,l,m_{j},k^{(j)}_{i,t}\in {\mathbf{{N}}}\) (\(j,t\in \{1,2,\ldots ,l \}\)) and \(f_{ji}:\mathbf{{R}}_{+}^{l}\times {\mathbf{{ N}}}_{0}\longrightarrow \mathbf{{R}}_{+}\) (\(j\in \{1,\ldots ,l\}\) and \(i\in \{1,\ldots ,m_{j}\}\)), and showed that every positive solution of (1.6) is eventually periodic with (not necessarily prime) period s if \(f_{ji}\) satisfy some conditions.
Moreover, Stević et al. [29] in 2014 investigated the following maxtype system of difference equations:
where \(s,l,m_{j},t_{j},k^{(j)}_{i_{j},h}\in { \mathbf{{N}}}\) (\(j,h\in \{1,2,\ldots ,l\}\)), \((\sigma (1),\ldots ,\sigma (l))\) is a permutation of \((1,\ldots ,l)\) and \(f_{ji_{j}}:\mathbf{{R}}_{+}^{l}\times {\mathbf{{ N}}}_{0}\longrightarrow \mathbf{{R}}_{+}\) (\(j\in \{1,\ldots ,l\}\) and \(i_{j}\in \{1,\ldots ,m_{j}\}\)). They showed that every positive solution of (1.7) is eventually periodic with period sT for some \(T\in \mathbf{{N}}\) if \(f_{ji_{j}}\) satisfy some conditions.
Main results and proofs
In this section, we study the eventual periodicity of positive solutions of system (1.1). Let \(\{(x_{n},y_{n})\}_{n\geq d}\) be a solution of (1.1) with the initial values \(x_{d},y_{d},x_{d+1},y_{d+1},\ldots , x_{1}, y_{1}\in { \mathbf{{R}}}_{+}\).
Lemma 2.1
If\(x_{n}=A\)eventually, then\(y_{n}\)is a periodic sequence with period 2seventually. If\(y_{n}=B\)eventually, then\(x_{n}\)is a periodic sequence with period 2seventually.
Proof
Assume that \(x_{n}=A\) eventually. By (1.1) we see
which implies \(y_{n}y_{ns}\geq A\) eventually and
Then, for any \(0\leq i\leq 2s1\), \(y_{2ns+i}\) is eventually nonincreasing.
We claim that, for every \(0\leq i\leq 2s1\), \(y_{2ns+i}\) is a constant sequence eventually. Assume on the contrary that, for some \(0\leq i\leq 2s1\), \(y_{2ns+i}\) is not a constant sequence eventually. Then there exists a sequence of positive integers \(k_{1}< k_{2}<\cdots \) such that, for any \(n\in { \mathbf{{N}}}\), we have
which implies \(y_{2sk_{n+1}+is}>y_{2sk_{n}+is}\) for any \(n\in { \mathbf{{N}}}\). This is a contradiction. Thus \(y_{n}\) is a periodic sequence with period 2s eventually. The second case follows from the previously proved one by interchanging letters. The proof is complete. □
Lemma 2.2
If\(A\geq B\geq 1/A\), then\(x_{2(n+1)t+i}\leq x_{2nt+i}\)for any\(n\geq t+s\)and\(i\in { {\mathbf{{N}}}}_{0}\). If\(B\geq A\geq 1/B\), then\(y_{2(n+1)t+i}\leq y_{2nt+i}\)for any\(n\geq t+s\)and\(i\in { {\mathbf{{N}}}}_{0}\).
Proof
Assume that \(A\geq B\geq 1/A\). By (1.1) we see that \(x_{n}\geq A\) and \(y_{n}\geq B\) for any \(n\in { \mathbf{{N}}}_{0}\), and
Since \(B/x_{2(n+1)t+is}\leq B/A\leq 1\leq A\) and \(x_{2(n+1)t+is}y_{2(n+1)t+its}\geq AB\geq 1\) for \(2(n+1)t+i\geq t+s\), we obtain
The second case follows from the previously proved one by interchanging letters. The proof is complete. □
Theorem 2.1
Let\(AB>1\). If\(A\geq B\), then\(x_{n}=A\)eventually and\(y_{n}\)is a periodic sequence with period 2seventually. If\(B> A\), then\(y_{n}=B\)eventually and\(x_{n}\)is a periodic sequence with period 2seventually.
Proof
Assume that \(A\geq B\). For any \(0\leq i\leq 2t1\) and \(n\in { \mathbf{{N}}}_{0}\), we have
By Lemma 2.2 we may let \(\lim_{n\longrightarrow \infty }x_{2nt+i}=A_{i}\). Note that
and
Thus we have \(x_{n}=A\) eventually. By Lemma 2.1, we see that \(y_{n}\) is a periodic sequence with period 2s eventually. The second case follows from the previously proved one by interchanging letters. The proof is complete. □
In the following, we assume \(AB=1\). For any \(i\in { \mathbf{{N}}}_{0}\), let
and
Then \(A_{i}\geq A\) and \(B_{i}\geq B\).
Lemma 2.3
If\(A\geq B=1/A\)and\(A_{i}>A\)for some\(i\in { {\mathbf{{N}}}}_{0}\), then, for any\(k\in { {\mathbf{{N}}}}\), \(x_{2nt+ks+i}\)and\(y_{2ntt+ks+i}\)are constant sequences eventually. If\(B\geq A=1/B\)and\(B_{i}>B\)for some\(i\in { {\mathbf{{N}}}}_{0}\), then, for any\(k\in { {\mathbf{{N}}}}\), \(y_{2nt+ks+i}\)and\(x_{2ntt+ks+i}\)are constant sequences eventually.
Proof
Assume that \(A\geq B\) and \(A_{i}>A\) for some \(i\in { \mathbf{{N}}}_{0}\). Since \(A_{i}>A\), it follows from Lemma 2.2 and (1.1) that
From this we have
This implies
and
and
Since
and
we see that \(x_{2nt+s+i} =A\) eventually. Note that
from which it follows that
and
and
If \(x_{2nt+2s+i}>A\) eventually, then, in a similar fashion, we obtain:
 (1)
\(x_{2nt+3s+i} =A\) eventually and \(y_{2ntt+3s+i} =B \) eventually.
 (2)
\(x_{2nt+4s+i}\) and \(y_{2ntt+4s+i}\) are constant sequences eventually.
If \(x_{2nt+2s+i}=A\) eventually, then \(y_{2ntt+2s+i} =A/B\) eventually, and
and
From this we see that if \(A=B\), then
and if \(A>B\), then
since
Using induction and arguments similar to the ones developed in the above given proof, we can show that, for any \(k\in { \mathbf{{N}}}\), \(x_{2nt+ks+i}\) and \(y_{2ntt+ks+i}\) are constant sequences eventually. The second case follows from the previously proved one by interchanging letters. The proof is complete. □
Lemma 2.4
If\(A=1/B>B\)and for some\(i\in { {\mathbf{{N}}}}_{0}\), \(x_{2nt+i}>A\)eventually and\(A_{i}=A\), then, for any\(k\in { {\mathbf{{N}}}}\), \(x_{2nt+ks+i}\)and\(y_{2ntt+ks+i}\)are constant sequences eventually. If\(B=1/A>A\)and for some\(i\in { {\mathbf{{N}}}}_{0}\), \(y_{2nt+i}>B\)eventually and\(B_{i}=B\), then, for any\(k\in { {\mathbf{{N}}}}\), \(y_{2nt+ks+i}\)and\(x_{2ntt+ks+i}\)are constant sequences eventually.
Proof
Assume that \(A=1/B>B\) and for some \(i\in { \mathbf{{N}}}_{0}\), \(x_{2nt+i}>A\) eventually and \(A_{i}=A\). By (1.1) we have
and
and
Then we see that \(x_{2nt+s+i} =A\) eventually. From this and \(y_{2ntt+i}\geq A^{2}\) eventually it follows that
and
and
Thus \(x_{2nt+2s+i}\) and \(y_{2ntt+2s+i}\) are constant sequences eventually. Using arguments similar to the ones developed in the proof of Lemma 2.3, we can show that, for any \(k\in { \mathbf{{N}}}\), \(x_{2nt+ks+i}\) and \(y_{2ntt+ks+i}\) are constant sequences eventually. The second case follows from the previously proved one by interchanging letters. The proof is complete. □
Theorem 2.2

(1)
Assume\(A=1/B>B\). Then one of the following statements holds.
 (i)
\(x_{n}=A\)eventually and\(y_{n}\)is a periodic sequence with period 2seventually.
 (ii)
Ifsis odd, then\(x_{n}\), \(y_{n}\)are periodic sequences with period 2teventually.
 (iii)
Ifsis even, then\(x_{n}\)is a periodic sequence with period 2teventually and\(y_{n}\)is a periodic sequence with period\(2st\)eventually.
 (i)

(2)
Assume\(B=1/A>A\). Then one of the following statements holds.
 (i)
\(y_{n}=B\)eventually and\(x_{n}\)is a periodic sequence with period 2seventually.
 (ii)
Ifsis odd, then\(x_{n}\), \(y_{n}\)are periodic sequences with period 2teventually.
 (iii)
Ifsis even, then\(y_{n}\)is a periodic sequence with period 2teventually and\(x_{n}\)is a periodic sequence with period\(2st\)eventually.
 (i)
Proof
Assume that \(A=1/B>B\). If \(x_{n}=A\) eventually, then by Lemma 2.1 we see that \(y_{n}\) is a periodic sequence with period 2s eventually. Now we assume that \(x_{n}\neq A\) eventually. Then we have \(A_{i}>A\) (or \(x_{2nt+i}>A\) eventually and \(A_{i}=A\)) for some \(0\leq i\leq 2t1\).
If s is odd, then \(\gcd (2t,s)=1\). Thus, for every \(j\in \{0,1,2,\ldots ,2t1\} \), there exist some \(1\leq i_{j}\leq 2t\) and integer \(\lambda _{j}\) such that \(i_{j}s=\lambda _{j} 2t+j\) since \(\{rs:0\leq r\leq 2t1\}=\{0,1,2,\ldots ,2t1\}\) (mod 2t). By Lemma 2.3 and Lemma 2.4 we see that, for any \(k\in { \mathbf{{N}}}\), \(x_{2nt+ks+i}\) and \(y_{2ntt+ks+i}\) are constant sequences eventually. Thus, for any \(0\leq r\leq 2t1\), \(x_{2nt+r}\) and \(y_{2nt+r}\) are constant sequences eventually, which implies that \(x_{n}\), \(y_{n}\) are periodic sequences with period 2t eventually.
In the following, we assume that s is even with \(s=2s'\). Then \(\gcd (t,s')=1\) and t is odd. Thus, for every \(j\in \{0,1,2,\ldots ,t1\} \), there exist some \(1\leq i_{j}\leq t\) and integer \(\lambda _{j}\) such that \(i_{j}s'=\lambda _{j} t+j\) and \(i_{j}s=\lambda _{j} 2t+2j\).
If \(x_{2nt+i}\neq A\) eventually for some \(i\in \{0,2,\ldots \}\) and \(x_{2nt+l}\neq A\) eventually for some \(l\in \{1,3,\ldots \}\), then by Lemma 2.3 and Lemma 2.4 we see that, for any \(k\in { \mathbf{{N}}}\), \(x_{2nt+ks+i}\), \(y_{2ntt+ks+i}\), \(x_{2nt+ks+l}\) and \(y_{2ntt+ks+l}\) are constant sequences eventually. Thus, for any \(0\leq r\leq 2t1\), \(x_{2nt+r}\) and \(y_{2nt+r}\) are constant sequences eventually, which implies that \(x_{n}\), \(y_{n}\) are periodic sequences with period 2t eventually.
If \(x_{2nt+i}\neq A\) eventually for some \(i\in \{0,2,\ldots \}\) and \(x_{2nt+l}=A\) eventually for any \(l\in \{1,3,\ldots \}\), then by Lemma 2.3 and Lemma 2.4 we see that, for any \(k\in { \mathbf{{N}}}\), \(x_{2nt+ks+i}\) and \(y_{2ntt+ks+i}\) are constant sequences eventually. This implies that, for every \(r\in \{0,1,2,\ldots ,2t1\}\), \(x_{2nt+r}\) is constant sequence eventually and for every \(l\in \{1,3,\ldots \}\), \(y_{2nst+l}\) is constant sequence eventually. By (1.1) we see that there exists \(N\in { \mathbf{{N}}}\) such that, for any \(n\geq N\) and \(r\in \{0,2,\ldots \}\),
Then we have \(y_{2nt+r}y_{2nt+rs}\geq A\). Thus, for any \(n\geq N\) and \(l\in \{1,3,\ldots \}\) and \(k\in { \mathbf{{N}}}\),
Then, for every \(n\geq N\) and \(l\in \{1,3,\ldots \}\), we have
We claim that, for every \(n\geq N\) and \(l\in \{1,3,\ldots \}\), \(\{y_{2nt+t+2ks+l}\}_{k\in { \mathbf{{N}}}}\) is a constant sequence eventually. Assume on the contrary that, for some \(n\geq N\) and some \(l\in \{1,3,\ldots \}\), \(\{y_{2nt+t+2ks+l}\}_{k\in { \mathbf{{N}}}}\) is not a constant sequence eventually. Then there exists a sequence of positive integers \(k_{1}< k_{2}<\cdots \) such that, for any \(r\in { \mathbf{{N}}}\), we have
which implies \(y_{2nt+t+2k_{r}s+ls}>y_{2nt+t+2k_{r1}s+ls}\) for any \(r\in { \mathbf{{N}}}\). This is a contradiction. Take \(ps>N\). Then \(y_{2nst+t+l}=y_{2pst+t+2(ntpt)s+l}\) is a constant sequence eventually for any \(l\in \{1,3,\ldots \}\). From the above we see that \(y_{n}\) is a periodic sequence with period \(2st\) eventually.
In a similar fashion, we can show that if \(x_{2nt+i}=A\) eventually for any \(i\in \{0,2,\ldots \}\) and \(x_{2nt+l}\neq A\) eventually for some \(l\in \{1,3,\ldots \}\), then also statement (1(iii)) holds.
The second case follows from the previously proved one by interchanging letters. The proof is complete. □
Now we assume that \(A=B=1\). Then, for any \(0\leq i\leq 2t1\) and \(n\in { \mathbf{{N}}}_{0}\), we have \(1\leq x_{2(n+1)t+i}\leq x_{2nt+i}\) eventually and \(1\leq y_{2(n+1)t+i}\leq y_{2nt+i}\) eventually.
Lemma 2.5
Let\(A=B=1\)and\(s\geq t\). Then the following statements hold.
 (1)
If\(A_{i}=1\), then\(B_{t+i}=1\). If\(B_{i}=1\), then\(A_{t+i}=1\).
 (2)
If\(x_{N}=1\)for some\(N\in { {\mathbf{{N}}}}\)and\(A_{2nt+N+ks}=1\)for any\(k,n\in { {\mathbf{{N}}}}\), then\(x_{2nt+N+ks}=y_{2nt+t+ks+N}=1\)for any\(k,n\in { {\mathbf{{N}}}}\). If\(y_{N}=1\)for some\(N\in { {\mathbf{{N}}}}\)and\(B_{2nt+N+ks}=1\)for any\(k,n\in { {\mathbf{{N}}}}\), then\(y_{2nt+N+ks}=x_{2nt+t+ks+N}=1\)for any\(k,n\in { {\mathbf{{N}}}}\).
 (3)
Ifsis even and\(\gcd (s,t)=1\), then\(1\in \{x_{n}:n\in \{0,2,\ldots \}\}\cup \{y_{t+n}:n\in \{0,2,\ldots \}\}\)and\(1\in \{x_{n}:n\in \{1,3,\ldots \}\}\cup \{y_{t+n}:n\in \{1,3,\ldots \}\}\).
 (4)
\(1\in \{x_{n}:n\in { {\mathbf{{N}}}}\}\cup \{y_{n}:n\in { {\mathbf{{N}}}}\}\).
Proof
(1) Assume that \(A_{i}=1\). Assume on the contrary that \(B_{t+i}>1\). It follows from (1.1) that
This implies
This is a contradiction. The second case follows from the previously proved one by interchanging letters.
(2) If \(x_{N}=1\) for some \(N\in { \mathbf{{N}}}\), then \(x_{2nt+N}=1\) for any \(n\in { \mathbf{{N}}}\). It follows from (1.1) that
and
and
Thus \(x_{2nt+N+s}=y_{2nt+t+s+N}=1\) for any \(n\in { \mathbf{{N}}}\). In a similar fashion, we can show that \(x_{2nt+N+ks}=y_{2nt+t+ks+N}=1\) for any \(k,n\in { \mathbf{{N}}}\). The second case follows from the previously proved one by interchanging letters.
(3) If s is even and \(\gcd (s,t)=1\), then t is odd. Assume on the contrary that \(1\notin \{x_{n}:n\in \{0,2,\ldots \}\}\cup \{y_{t+n}:n\in \{0,2, \ldots \}\}\). Then it follows from (1.1) that, for any \(n\in { \mathbf{{N}}}\),
and
Thus
This is a contradiction.
(4) Case (4) is treated similarly to case (3). The proof is complete. □
Theorem 2.3
Let\(A=B=1\)and\(s\geq t\). Then one of the following statements holds.
 (1)
\(x_{n}=1\)eventually and\(y_{n}\)is a periodic sequence with period 2seventually.
 (2)
\(y_{n}=1\)eventually and\(x_{n}\)is a periodic sequence with period 2seventually.
 (3)
\(x_{n}\), \(y_{n}\)are periodic sequences with period 2teventually.
Proof
If \(x_{n}=1\) (or \(y_{n}=1\)) eventually, then by Lemma 2.1 we see that \(y_{n}\) (or \(x_{n}\)) is a periodic sequence with period 2s eventually. Now we assume that \(x_{n}\neq1\) eventually. Then we have \(A_{i}>1\) for some \(0\leq i\leq 2t1\) or \(\lim_{n\longrightarrow \infty }x_{n}=1\).
If s is odd, then \(\gcd (2t,s)=1\). Thus, for every \(j\in \{0,1,2,\ldots ,2t1\} \), there exist some \(1\leq i_{j}\leq 2t\) and integer \(\lambda _{j}\) such that \(i_{j}s=\lambda _{j} 2t+j\). By Lemma 2.3 and Lemma 2.5 we see that, for any \(k\in { \mathbf{{N}}}\), \(x_{2nt+ks+i}\) and \(y_{2ntt+ks+i}\) are constant sequences eventually, or for some \(N\in { \mathbf{{N}}}\), \(x_{2nt+N+ks}=y_{2nt+t+ks+N}=1\) for any \(k,n\in { \mathbf{{N}}}\), or for some \(N\in { \mathbf{{N}}}\), \(y_{2nt+N+ks}=x_{2nt+t+ks+N}=1\) for any \(k,n\in { \mathbf{{N}}}\). Thus, for any \(0\leq r\leq 2t1\), \(x_{2nt+r}\) and \(y_{2nt+r}\) are constant sequences eventually, which implies that \(x_{n}\), \(y_{n}\) are periodic sequences with period 2t eventually.
In the following, we assume that s is even with \(s=2s'\), then \(\gcd (t,s')=1\) and t is odd. Thus, for every \(j\in \{0,1,2,\ldots ,t1\} \), there exist some \(1\leq i_{j}\leq t\) and integer \(\lambda _{j}\) such that \(i_{j}s=\lambda _{j} 2t+2j\).
If \(A_{i}>1\) for some \(i\in \{0,2,\ldots \}\), then by Lemma 2.3 we see that, for any \(k\in { \mathbf{{N}}}\), \(x_{2nt+ks+i}\) and \(y_{2ntt+ks+i}\) are constant sequences eventually. If \(A_{i}=1\) for any \(i\in \{0,2,\ldots \}\), then by Lemma 2.5 we have \(B_{t+i}=1\) for any \(i\in \{0,2,\ldots \}\) and \(x_{2nt+i+ks}=y_{2ntt+ks+i}=1\) for any \(k\in { \mathbf{{N}}}\) eventually. In a similar fashion, also we can show that, for any \(i\in \{1,3,\ldots \}\), \(x_{2nt+ks+i}\) and \(y_{2ntt+ks+i}\) are constant sequences eventually for any \(k\in { \mathbf{{N}}}\), or \(x_{2nt+i+ks}=y_{2ntt+ks+i}=1\) for any \(k\in { \mathbf{{N}}}\) eventually for any \(i\in \{1,3,\ldots \}\) and \(k\in { \mathbf{{N}}}\). Thus, for any \(0\leq r\leq 2t1\), \(x_{2nt+r}\) and \(y_{2nt+r}\) are constant sequences eventually. This implies that \(x_{n}\), \(y_{n}\) are periodic sequences with period 2t eventually.
Using the previously proved one by interchanging letters, also we can show that if \(y_{n}\neq1\) eventually, then \(x_{n}\), \(y_{n}\) are periodic sequences with period 2t eventually. The proof is complete. □
In Example 3.1 of [37], we showed that the equation
has a positive solution \(z_{n}\) (\(n\geq t\)) with \(1< z_{n+1}< z_{n}\) for any \(n\geq t\) and \(\lim_{n\longrightarrow \infty }z_{n}=1\).
From Example 3.1 of [37], we obtain the following theorem.
Theorem 2.4
Let\(A\leq 1\)and\(B\leq 1\)and\(s< t\). Assume\(z_{n}\) (\(n\geq t\)) is a positive solution of (2.45) with\(1< z_{n+1}< z_{n}\)for any\(n\geq t\)and\(\lim_{n\longrightarrow \infty }z_{n}=1\). Then equation (1.1) have a solution\((x_{n},y_{n})\)with\(1< x_{n+1}=y_{n+1}=z_{n+1}< x_{n}=y_{n}=z_{n}\)for any\(n\geq t\)and\(\lim_{n\longrightarrow \infty }x_{n}=\lim_{n\longrightarrow \infty }y_{n}=1\).
References
 1.
Berenhaut, K.S., Foley, J.D., Stević, S.: Boundedness character of positive solutions of a max difference equation. J. Differ. Equ. Appl. 12, 1193–1199 (2006)
 2.
Cranston, D.M., Kent, C.M.: On the boundedness of positive solutions of the reciprocal maxtype difference equation \(x_{n} =\max \{A^{1}_{n1}/x_{n1}, A^{2}_{n1}/x_{n2}, \ldots , A^{t}_{n1}/x_{nt} \}\) with periodic parameters. Appl. Math. Comput. 221, 144–151 (2013)
 3.
Elsayed, E.M., Iričanin, B.D.: On a maxtype and a mintype difference equation. Appl. Math. Comput. 215, 608–614 (2009)
 4.
Elsayed, E.M., Iričanin, B.D., Stević, S.: On the maxtype equation \(x_{n+1} =\max \{A_{n}/x_{n}, x_{n1}\}\). Ars Comb. 95, 187–192 (2010)
 5.
Fotiades, E., Papaschinopoulos, G.: On a system of difference equations with maximum. Appl. Math. Comput. 221, 684–690 (2013)
 6.
Iričanin, B.D., Stević, S.: Eventually constant solutions of a rational difference equation. Appl. Math. Comput. 215, 854–856 (2009)
 7.
Iričanin, B.D., Stević, S.: Global attractivity of the maxtype difference equation \(x_{n}=\max \{c, x^{p}_{n1}/\prod_{j=2}^{k}x_{nj}^{p_{j}}\}\). Util. Math. 91, 301–304 (2013)
 8.
Liu, W., Stević, S.: Global attractivity of a family of nonautonomous maxtype difference equations. Appl. Math. Comput. 218, 6297–6303 (2012)
 9.
Liu, W., Yang, X., Stević, S.: On a class of nonautonomous maxtype difference equations. Abstr. Appl. Anal. 2011, Article ID 327432 (2011)
 10.
Papaschinopoulos, G., Schinas, J.: On a system of two nonlinear difference equations. J. Math. Anal. Appl. 219, 415–426 (1998)
 11.
Papaschinopoulos, G., Schinas, J., Hatzifilippidis, V.: Global behavior of the solutions of a maxequation and of a system of two maxequations. J. Comput. Anal. Appl. 5, 237–254 (2003)
 12.
Qin, B., Sun, T., Xi, H.: Dynamics of the maxtype difference equation \(x_{n+1} = \max \{A/x_{n}, x_{nk}\}\). J. Comput. Anal. Appl. 14, 856–861 (2012)
 13.
Sauer, T.: Global convergence of maxtype equations. J. Differ. Equ. Appl. 17, 1–8 (2011)
 14.
Shi, Q., Su, X., Yuan, G.: Characters of the solutions to a generalized nonlinear maxtype difference equation. Chin. Q. J. Math. 28, 284–289 (2013)
 15.
Stefanidou, G., Papaschinopoulos, G., Schinas, C.: On a system of max difference equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 14, 885–903 (2007)
 16.
Stević, S.: On the recursive sequence \(x_{n+1}=\max \{c, x^{p}_{n}/x_{n1}^{p}\}\). Appl. Math. Lett. 21, 791–796 (2008)
 17.
Stević, S.: Boundedness character of a class of difference equations. Nonlinear Anal. TMA 70, 839–848 (2009)
 18.
Stević, S.: Global stability of a difference equation with maximum. Appl. Math. Comput. 210, 525–529 (2009)
 19.
Stević, S.: Global stability of a maxtype difference equation. Appl. Math. Comput. 216, 354–356 (2010)
 20.
Stević, S.: On a generalized maxtype difference equation from automatic control theory. Nonlinear Anal. TMA 72, 1841–1849 (2010)
 21.
Stević, S.: Periodicity of max difference equations. Util. Math. 83, 69–71 (2010)
 22.
Stević, S.: Periodicity of a class of nonautonomous maxtype difference equations. Appl. Math. Comput. 217, 9562–9566 (2011)
 23.
Stević, S.: Solutions of a maxtype system of difference equations. Appl. Math. Comput. 218, 9825–9830 (2012)
 24.
Stević, S.: On some periodic systems of maxtype difference equations. Appl. Math. Comput. 218, 11483–11487 (2012)
 25.
Stević, S.: On a symmetric system of maxtype difference equations. Appl. Math. Comput. 219, 8407–8412 (2013)
 26.
Stević, S.: Producttype system of difference equations of secondorder solvable in closed form. Electron. J. Qual. Theory Differ. Equ. 2014, 56 (2014)
 27.
Stević, S.: Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences. Electron. J. Qual. Theory Differ. Equ. 2014, 67 (2014)
 28.
Stević, S., Alghamdi, M.A., Alotaibi, A.: Longterm behavior of positive solutions of a system of maxtype difference equations. Appl. Math. Comput. 235, 567–574 (2014)
 29.
Stević, S., Alghamdi, M.A., Alotaibi, A., Shahzad, N.: Eventual periodicity of some systems of maxtype difference equations. Appl. Math. Comput. 236, 635–641 (2014)
 30.
Stević, S., Alghamdi, M.A., Alotaibi, A., Shahzad, N.: Boundedness character of a maxtype system of difference equations of second order. Electron. J. Qual. Theory Differ. Equ. 2014, 45 (2014)
 31.
Stević, S., Iričanin, B.D.: On a maxtype difference inequality and its applications. Discrete Dyn. Nat. Soc. 2010, Article ID 975740 (2010)
 32.
Sun, T., He, Q., Wu, X., Xi, H.: Global behavior of the maxtype difference equation \(x_{n} =\max \{1/ x_{nm},A_{n}/ x_{nr}\}\). Appl. Math. Comput. 248, 687–692 (2014)
 33.
Sun, T., Liu, J., He, Q., Liu, X.: Eventually periodic solutions of a maxtype difference equation. Sci. World J. 2014, Article ID 219437 (2014)
 34.
Sun, T., Qin, B., Xi, H., Han, C.: Global behavior of the maxtype difference equation \(x_{n+1}=\max \{1/x_{n}, A_{n}/x_{n1}\}\). Abstr. Appl. Anal. 2009, Article ID 152964 (2009)
 35.
Sun, T., Xi, H.: On the solutions of a system of difference equations with maximum. Appl. Math. Comput. 290, 292–297 (2016)
 36.
Sun, T., Xi, H., Han, C., Qin, B.: Dynamics of the maxtype difference equation \(x_{n} =\max \{ 1/ x_{nm} , A_{n}/ x_{nr}\}\). J. Appl. Math. Comput. 38, 173–180 (2012)
 37.
Sun, T., Xi, H., Quan, W.: Existence of monotone solutions of a difference equation. Discrete Dyn. Nat. Soc. 2008, Article ID 917560 (2008)
 38.
Xiao, Q., Shi, Q.: Eventually periodic solutions of a maxtype equation. Math. Comput. Model. 57, 992–996 (2013)
 39.
Yazlik, Y., Tollu, D.T., Taskara, N.: On the solutions of a maxtype difference equation system. Math. Methods Appl. Sci. 38, 4388–4410 (2015)
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions.
Availability of data and materials
None.
Funding
The research was supported by NNSF of China (11761011, 71862003) and NSF of Guangxi (2018GXNSFAA294010) and SF of Guangxi University of Finance and Economics (2019QNB10).
Author information
Affiliations
Contributions
All authors participated in every phase of research conducted for this paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Su, G., Han, C., Sun, T. et al. On the solutions of a maxtype system of difference equations of higher order. Adv Differ Equ 2020, 213 (2020). https://doi.org/10.1186/s13662020026732
Received:
Accepted:
Published:
MSC
 39A10
 39A11
Keywords
 Maxtype system of difference equations
 Solution
 Eventual periodicity