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Eventually Periodic Solutions of a Max-Type System of Difference Equations of Higher Order.

1. Introduction

Our purpose in this paper is to study eventual periodicity of the following max-type system of difference equations of higher order:

[mathematical expression not reproducible], (1)

with

A [greater than or equal to] B > 0, (2)

where k [member of] N [equivalent to] {1, 2, ...} and the initial conditions [x.sub.-k], [y.sub.-k], [x.sub.-k+1], [y.sub.-k+1], ..., [x.sub.-1], [y.sub.-1] [member of] [R.sub.+] [equivalent to] (0,+[infinity]).

Recently, there has been a great interest in studying max-type systems of difference equations. For example, Fotiades and Papaschinopoulos [1] studied the following max-type system of difference equations:

[mathematical expression not reproducible], (3)

and they showed that every positive solution of (3) is eventually periodic.

In 2012, Stevic [2] obtained in an elegant way the general solution to the following max-type system of difference equations:

[mathematical expression not reproducible], (4)

for the case [x.sub.0], [y.sub.0] [greater than or equal to] A > 0 and [y.sub.0]/[x.sub.0] [greater than or equal to] max{A, 1/A}.

In [3], we have recently studied the following max-type system of difference equations:

[mathematical expression not reproducible], (5)

where A, B [member of] [R.sub.+], m,r,t [member of] N, and the initial conditions [x.sub.-d], [y.sub.-d], [x.sub.-d+1],[y.sub.-d+1], ..., [x.sub.-1],[y.sub.-1] [member of] [R.sub.+] with d = max{m,r,t},

and we showed that every positive solution of (5) is eventually periodic with period 2m.

When m = r = t =1 and A = B, (5) reduces to the max-type system of difference equations

[mathematical expression not reproducible]. (6)

Yazlik et al. [4] in 2015 obtained in an elegant way the general solution of (6).

In 2014, Stevic et al. [5] investigated the behavior of positive solutions of the following max-type system of difference equations:

[mathematical expression not reproducible], (7)

where B, p [member of] [R.sub.+]. It is proved that system (7) is permanent when p [member of] (0,4).

For some other results on solutions of many max-type difference equations and systems, such as eventual periodicity, the boundedness character, and attractivity, see, for example, [6-20] and the related references therein.

2. Main Results and Proofs

In this section, we study the eventual periodicity of solutions of system (1). Let [{([x.sub.n], [y.sub.n])}.sub.n[greater than or equal to]-k] be a solution of (1) with the initial conditions [x.sub.-k], [y.sub.-k], [x.sub.-k+1], [y.sub.-k+1], ..., [x.sub.-1], [y.sub.-1] [member of] [R.sub.+]. From (1), it immediately follows that, for any n [member of] [N.sub.0],

[x.sub.n] [greater than or equal to] A, (8)

[y.sub.n] [greater than or equal to] B. (9)

Lemma 1. If [x.sub.n] = A eventually, then [y.sub.n] is a periodic sequence with period 2 eventually. If [y.sub.n] = B eventually, then [x.sub.n] is a periodic sequence with period 2 eventually.

Proof. Assume that [x.sub.n] = A eventually. By (1), we see

[y.sub.n] = max {B, A/[[y.sub.n] - 1]} eventually, (10)

which implies

[y.sub.n][y.sub.n-1] [greater than or equal to] A eventually. (11)

We have from (9), (10), and (11) that

B [less than or equal to] [y.sub.n] = max {B, A[y.sub.n-2]/[y.sub.n-1][y.sub.-2]} [less than or equal to] max{B, [y.sub.n-2]} = [y.sub.n-2] eventually. (12)

Then, for any i [member of] {0.1}, [y.sub.2n+i] is eventually nonincreasing.

We claim that, for every i [member of] {0.1}, [y.sub.2n+i] is a constant sequence eventually. Assume on the contrary that, for some i [member of] {0.1}, [y.sub.2n+i] is not a constant sequence eventually. Then, by (10) and (12), we see that there exists a sequence of positive integers [k.sub.1] < [k.sub.2] < ... such that, for an[y.sub.n] [member of] N,

[mathematical expression not reproducible], (13)

which implies [mathematical expression not reproducible] for any n [member of] N. This is a contradiction. Thus, [y.sub.n] is a periodic sequence with period 2 eventually. The other case is treated similarly/dually, so we omit the detail. The proof is complete.

Lemma 2. If AB [greater than or equal to] 1, then [x.sub.2(n+1)k+i] [less than or equal to] [x.sub.2nk+i] for any n [greater than or equal to] k+1and i [member of] [N.sub.0].

Proof. Assume that AB [greater than or equal to] 1 Then, for 2(n + 1)k + i [greater than or equal to] k + 1 by (1) and (8), we have

[mathematical expression not reproducible] (14)

and by (2), (8), and (9), we have

[mathematical expression not reproducible] (15)

Thus,

[mathematical expression not reproducible]. (16)

The proof is complete.

Theorem 3. Let AB > 1. Then, [x.sub.n] = A eventually and [y.sub.n] is a periodic sequence with period 2 eventually.

Proof. Assume that AB > 1. For any i [member of] {0.1, ..., 2k - 1} and n [member of] [N.sub.0], let [lim.sub.n[right arrow][x.sub.2nk+i] = [A.sub.i] since [x.sub.2nk+i] is eventually nonincreasing. Thus, by using (15) and by letting n [greater than or equal to] [infinity] in (14),it is obtained that [x.sub.n] = A eventually. By Lemma 1, we see that [y.sub.n] is a periodic sequence with period 2 eventually. The proof is complete.

Now, we assume AB = 1, and by Lemma 2, we can assume that, for any i [member of] [N.sub.0],

[mathematical expression not reproducible]. (17)

Then, [A.sub.i] [greater than or equal to] A and [A.sub.2k+i] = [A.sub.i] for any i [member of] [N.sub.0].

Lemma 4. If AB = 1 and [A.sub.i] > A for some i [member of] [N.sub.0], then, for any s [member of] N, [x.sub.2nk+s+i] and [y.sub.2nk-k+s+i] are constant sequences eventually.

Proof. Assume that AB = 1 and [A.sub.i] > A for some i [member of] [N.sub.0]. Since [A.sub.i] > A, it follows from (14), (15), and Lemma 2 that

[mathematical expression not reproducible]. (18)

By (9), (17), and (18), we have

[mathematical expression not reproducible]. (19)

This implies

[mathematical expression not reproducible]. (20)

Note that [mathematical expression not reproducible] eventually; it follows that

[mathematical expression not reproducible]. (21)

Since

[mathematical expression not reproducible], (22)

we see that [x.sub.2nk+1+i] = A eventually. This with (21) implies

[mathematical expression not reproducible]. (23)

From (23), it follows that

[mathematical expression not reproducible]. (24)

By [mathematical expression not reproducible] eventually and (24), it follows that

[mathematical expression not reproducible]. (25)

If [x.sub.2nk+2+i] > A eventually, then, in a similar fashion, we can obtain the following:

(1) [x.sub.2nk+3+i] = A eventually and [y.sub.2nk-k+3+i] = B eventually.

(2) [x.sub.2nk+4+i] and [y.sub.2nk-k+4+i] are constant sequences eventually.

If [x.sub.2nk+2+i] = A eventually, then [y.sub.2nk-k+2+i] = A/B eventually and

[mathematical expression not reproducible], (26)

[mathematical expression not reproducible]. (27)

From (27), we see that if A = B = 1, then by [x.sub.2(n-1)k+3+i] [greater than or equal to] A we have

[x.sub.2nk+3+i] = [x.sub.2(n-1)k+3+i] eventually, (28)

and if A > B, then

[x.sub.2nk+3+i] = A eventually (29)

since

[mathematical expression not reproducible]. (30)

Using arguments similar to ones developed in the above given proof, we can show that, for any s [member of] N, [x.sub.2nk+s+i] and [y.sub.2nk-k+s+i] are constant sequences eventually. The proof is complete.

Theorem 5. Assume AB = 1 > B. Then, one of the following statements holds:

(i) [x.sub.n] and [y.sub.n] are periodic sequences with period 2k eventually.

(ii) [x.sub.n] = A eventually and [y.sub.n] is a periodic sequence with period 2 eventually.

Proof. Assume that A=1/B > B. If [A.sub.i] > A for some i [member of] [N.sub.0], then by Lemma 4 we see that, for any s [member of] N, [x.sub.2nk+s+i] and [y.sub.2nk-k+s+i] are constant sequences eventually, which implies that [x.sub.n] and [y.sub.n] are periodic sequences with period 2k eventually. In the following, we assume that [A.sub.i] = A for any i [member of] [N.sub.0].

We claim that [x.sub.n] = A eventually. Indeed, if, for some i [member of] [N.sub.0], [x.sub.2nk+i] > A, then by (1) we have

[mathematical expression not reproducible], (31)

[mathematical expression not reproducible], (32)

[mathematical expression not reproducible]. (33)

Then, we see that

[x.sub.2nk+1+i] = A eventually. (34)

From (31), we have [y.sub.2nk-k+i] [greater than or equal to] [A.sup.2] eventually, and with (34) and B = 1/A, it follows that

[mathematical expression not reproducible]. (35)

Thus, [x.sub.2nk+2+i] and [y.sub.2nk-k+2+i] are constant sequences eventually. Using arguments similar to ones developed in the proof of Lemma 4, we can show that, for any s [member of] N, [x.sub.2nk+s+i] and [y.sub.2nk-k+s+i] are constant sequences eventually. Thus, [x.sub.2nk+i] = [A.sub.i] = A eventually; this is a contradiction.

By Lemma 1, we see that [x.sub.n] = A eventually and [y.sub.n] is a periodic sequence with period 2 eventually. Theorem 5 is proven.

Theorem 6. If A [less than or equal to] 1 and B [less than or equal to]1, then (1) has a solution which is not periodic eventually.

Proof. We claim that the equation

[z.sub.n] = [z.sub.n-k]/[z.sub.n-1] (36)

has a nonincreasing solution. Indeed, write

[mathematical expression not reproducible]. (37)

It is easy to see that M [subset] E since, for any ([x.sub.1], ..., [x.sub.k]) [member of] M, we have [x.sub.1] [greater than or equal to] ... [greater than or equal to][x.sub.k] [greater than or equal to][square root of [x.sub.1]] and [x.sub.k] [greater than or equal to] [x.sup.2.sub.k]/[x.sub.k-1] [greater than or equal to] [x.sub.1]/[x.sub.k-1]. Define F : M [right arrow] E by

F([x.sub.1], ..., [x.sub.k]) = ([u.sub.1], ..., [u.sub.k]) [equivalent to] ([x.sub.2], ..., [x.sub.k], [x.sub.1]/[x.sub.k]) (38)

for all ([x.sub.1], ..., [x.sub.k]) [member of] M.

We claim that F is well defined. Indeed, from (38) and the definition of M, we have

[mathematical expression not reproducible]. (39)

Thus, ([u.sub.1], ..., [u.sub.k]) [member of] E.

We also claim that F is a bijection from M to E. Indeed, let z = ([z.sub.1], ..., [z.sub.k]), y = ([y.sub.1], ..., [y.sub.k]) [member of] M with z [not equal to] y; we have F(y) [not equal to] F(z). On the other hand, for any u = ([u.sub.1], ..., [u.sub.k]) [member of] E, we have

[u.sub.1] [greater than or equal to] ... [greater than or equal to] [u.sub.k] [greater than or equal to] [u.sub.1]/[u.sub.k-1]. (40)

Choose

x = ([x.sub.1], ..., [x.sub.k]) [equivalent to] ([u.sub.k][u.sub.k-1], [u.sub.1], ... [u.sub.k-1]). (41)

It follows from (40) and (41) that

[mathematical expression not reproducible], (42)

which implies x [member of] M, and by (38) we obtain F(x) = u.

Furthermore, since [F.sup.-1] ([u.sub.1], ..., [u.sub.k]) = ([u.sub.k][u.sub.k-1], [u.sub.1], ..., [u.sub.k-1]) is continuous, F is a homeomorphism from M to E.

Since M [subset] E and F is a homeomorphism from M onto E, it follows that [F.sup.-1](M) [subset] [F.sup.-1](E) = M. By induction, we have

p = (1, 1, ..., 1) [member of] [F.sup.-n] (M) [subset] [F.sup.-n+1] (M) (43)

for every n [member of] N. Because M is an unbounded connected closed set, we know that [F.sup.-n](M) is an unbounded connected closed set for every n [member of] N. Let

[mathematical expression not reproducible]. (44)

Then, S [not equal to] [empty set] and S is also an unbounded connected set.

Now, let [{[z.sub.n]}.sup.[infinity].sub.n=-k] be a solution of (36) with the initial conditions ([z.sub.-k], ..., [z.sub.-1]) [member of] S - {p}; we can show that, for all n [member of] N,

[F.sup.n] ([z.sub.-k], ..., [z.sub.-1]) = ([z.sub.n-k], ..., [z.sub.n-1]) [member of] M-{p}. (45)

Thus, [z.sub.n] > [z.sub.n+1] > 1 for any n [greater than or equal to] -k and [lim.sub.n [right arrow] [infinity]] [z.sub.n] = 1. It is easy to see that [{([z.sub.n], [z.sub.n])}.sup.[infinity].sub.n=-k] is also a solution of (1) which is not periodic eventually. Theorem 6 is proven.

Theorem 7. If AB < 1 < A, then (1) has an unbounded solution.

Proof. Let [{([x.sub.n], [y.sub.n])}.sup.[infinity].sub.n=-k] be a solution of (1) with the initial values [x.sub.i] = A and [y.sub.i] = B for i [member of] {-k, -k+ 1, ..., -1}.

If k = 2m for some m [member of] N, then it is easyto see that, for i [member of] {0,1, ..., m-1}, we have ([x.sub.2i], [y.sub.2i]) = (A, A/B), ([x.sub.2i+1], [y.sub.2i+1]) = (A, B). Thus,

[mathematical expression not reproducible]. (46)

By induction, we can obtain that, for any n [member of] N and i [member of] {0, 1, ..., m - 1},

[mathematical expression not reproducible]. (47)

If k = 2m + 1 for some m [member of] N, then we also can obtain that, for any n [member of] N,

[mathematical expression not reproducible]. (48)

It is easy to see that [{([x.sub.n], [y.sub.n])}.sup.[infinity].sub.n=-k] is an unbounded solution of (1). Theorem 7 is proven.

https://doi.org/10.1155/2018/8467682

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This project was supported by the NNSF of China (Grant no. 11761011), NSF of Guangxi (Grants nos. 2016GXNSFBA380235 and 2016GXNSFAA380286), and YMTBAPP of Guangxi Colleges (Grant no. 2017KY0598).

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Guangwang Su (iD), Taixiang Sun, and Bin Qin

College of Information and Statistics, Guangxi University of Finance and Economics, Nanning 530003, China

Correspondence should be addressed to Guangwang Su; s1g6w3@163.com

Received 24 January 2018; Accepted 21 March 2018; Published 24 April 2018

Academic Editor: Douglas R. Anderson
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Title Annotation:Research Article
Author:Su, Guangwang; Sun, Taixiang; Qin, Bin
Publication:Discrete Dynamics in Nature and Society
Date:Jan 1, 2018
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