Boundedness character of a fourth-order system of difference equations
- Stevo Stević^{1, 2}Email author,
- Bratislav Iričanin^{3} and
- Zdeněk Šmarda^{4, 5}
https://doi.org/10.1186/s13662-015-0644-y
© Stević et al. 2015
Received: 27 July 2015
Accepted: 23 September 2015
Published: 13 October 2015
Abstract
The boundedness character of positive solutions of the following system of difference equations: \(x_{n+1}=A+\frac{y^{p}_{n}}{x_{n-3}^{r}}\), \(y_{n+1}=A+\frac {x^{p}_{n}}{y_{n-3}^{r}}\), \(n\in{\mathbb{N}}_{0}\), when \(\min\{A,r\}>0\) and \(p\ge0\), is studied.
Keywords
MSC
1 Introduction
Motivated by these two lines of investigations Stević has proposed recently studying symmetric and close to symmetric systems of difference equations which, among others, stem from special cases of (2).
2 Main results
In this section we prove the main results in this paper, all of which are related to the boundedness character, that is, the boundedness of all positive solutions of system (3) or the existence of an unbounded solution of the system depending on the values of parameters A, p, and r.
Theorem 1
Assume that \(\min\{A, p, r\}>0\) and \(27p^{4}<256r\). Then all positive solutions of system (3) are bounded.
Proof
Remark 1
Note that if \(a_{k}=p\) for some \(k\in{\mathbb{N}}\), then \(a_{k+1}\), \(b_{k+1}\), and \(c_{k+1}\) are not defined. However, if this happens then above mentioned index l is chosen to be this k. For such chosen l is obtained an upper bound for positive solutions of system (3) in the way described in the proof of Theorem 1.
Theorem 2
Assume that \(\min\{A, p, r\}>0\), \(27p^{4}\ge256r\), and \(p\geq4/3\) (where at least one of these two inequalities is strict), or \(r< p-1<1/3\). Then system (3) has positive unbounded solutions.
Proof
Now assume that \(r< p-1<1/3\). Then \(P(1)=1-p+r<0\) and since (20) holds, we again see that there is \(\lambda_{1}>1\) such that \(P(\lambda_{1})=0\).
Theorem 3
Assume that \(\min\{A, p, r\}>0\) and \(p=r+1\). Then system (3) has positive unbounded solutions.
Proof
Theorem 4
Assume that \(\min\{A, r\}>0\) and \(p\in(0,1)\). Then every positive solution of system (3) is bounded.
Proof
If \(z_{4}\in(0,x^{*}]\) condition (31) implies that \((z_{n})_{n\geq4}\) is nondecreasing and bounded above by \(x^{*}\), and if \(z_{4}\geq x^{*}\) that it is nonincreasing and bounded below by \(x^{*}\). Hence \((z_{n})_{n\geq4}\) is bounded, which along with (30) implies the boundedness of \((x_{n})_{n\geq4}\) and \((y_{n})_{n\geq4}\), from which the result easily follows. □
In the next theorem we use the fact that the comparison equation is a linear first order difference equation, which is solvable in closed form. For recent application of this and related equations see, for example, [4, 22, 23, 25–29, 33].
Theorem 5
Assume that \(p=1\), \(r>0\), and \(A>\sqrt[r]{2}\). Then every positive solution of system (3) is bounded.
Proof
From the proof of Theorem 4 we see that any positive solution \((x_{n},y_{n})_{n\ge-3}\) of system (3) satisfies (28) with \(p=1\).
Remark 2
- (a)
\(r\leq27p^{4}/256\), \(1< p< r+1\), \(r<1/3\);
- (b)
\(r\leq27p^{4}/256\), \(p=1\), \(A\in(0,\sqrt[r]{2}]\),
Declarations
Acknowledgements
The work of Stevo Stević is supported by the Serbian Ministry of Education and Science projects III 41025 and III 44006. The work of Bratislav Iričanin is supported by the Serbian Ministry of Education and Science projects III 41025 and OI 171007. The work of Zdeněk Šmarda was realized in CEITEC - Central European Institute of Technology with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund. He was also supported by the project FEKT-S-14-2200 of Brno University of Technology.
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Authors’ Affiliations
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