Long-term behavior of a cyclic max-type system of difference equations
We study the long-term behavior of positive solutions of the cyclic system of difference equations $$ x^{(i)}_{n+1}=\max\Big\{\alpha,\frac{(x^{(i+1)}_n)^p}{(x^{(i+2)}_{n-1})^q}\Big\}, \quad i=1,\ldots,k,\; n\in\mathbb{N}_0, $$ where $k\in\mathbb{N}$, $\min\{\alpha, p, q\}>0$ and where we r...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-09-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/234/abstr.html |
| _version_ | 1818248508633776128 |
|---|---|
| author | Tatjana Stevic Bratislav Iricanin |
| author_facet | Tatjana Stevic Bratislav Iricanin |
| author_sort | Tatjana Stevic |
| collection | DOAJ |
| description | We study the long-term behavior of positive solutions of the cyclic system of
difference equations
$$
x^{(i)}_{n+1}=\max\Big\{\alpha,\frac{(x^{(i+1)}_n)^p}{(x^{(i+2)}_{n-1})^q}\Big\},
\quad i=1,\ldots,k,\; n\in\mathbb{N}_0,
$$
where $k\in\mathbb{N}$, $\min\{\alpha, p, q\}>0$ and where we
regard that $x^{(i_1)}_n=x^{(i_2)}_n$ when $i_1\equiv i_2$
(mod $k$). We determine the set of parameters $\alpha$, p
and q in $(0,\infty)^3$ for which all such solutions are
bounded. In the other cases we show that the system has unbounded
solutions. For the case p=q we give some sufficient conditions
which guaranty the convergence of all positive solutions. The main
results in this paper generalize and complement some recent ones. |
| first_indexed | 2024-12-12T15:21:43Z |
| format | Article |
| id | doaj.art-0db5795be214469eb19207888c677f05 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-12T15:21:43Z |
| publishDate | 2015-09-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-0db5795be214469eb19207888c677f052022-12-22T00:20:22ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912015-09-012015234,112Long-term behavior of a cyclic max-type system of difference equationsTatjana Stevic0Bratislav Iricanin1 Belgrade Univ., Beograd, Serbia Belgrade Univ., Beograd, Serbia We study the long-term behavior of positive solutions of the cyclic system of difference equations $$ x^{(i)}_{n+1}=\max\Big\{\alpha,\frac{(x^{(i+1)}_n)^p}{(x^{(i+2)}_{n-1})^q}\Big\}, \quad i=1,\ldots,k,\; n\in\mathbb{N}_0, $$ where $k\in\mathbb{N}$, $\min\{\alpha, p, q\}>0$ and where we regard that $x^{(i_1)}_n=x^{(i_2)}_n$ when $i_1\equiv i_2$ (mod $k$). We determine the set of parameters $\alpha$, p and q in $(0,\infty)^3$ for which all such solutions are bounded. In the other cases we show that the system has unbounded solutions. For the case p=q we give some sufficient conditions which guaranty the convergence of all positive solutions. The main results in this paper generalize and complement some recent ones.http://ejde.math.txstate.edu/Volumes/2015/234/abstr.htmlMax-type system of difference equationscyclic systempositive solutionsboundedness characterglobal attractivity |
| spellingShingle | Tatjana Stevic Bratislav Iricanin Long-term behavior of a cyclic max-type system of difference equations Electronic Journal of Differential Equations Max-type system of difference equations cyclic system positive solutions boundedness character global attractivity |
| title | Long-term behavior of a cyclic max-type system of difference equations |
| title_full | Long-term behavior of a cyclic max-type system of difference equations |
| title_fullStr | Long-term behavior of a cyclic max-type system of difference equations |
| title_full_unstemmed | Long-term behavior of a cyclic max-type system of difference equations |
| title_short | Long-term behavior of a cyclic max-type system of difference equations |
| title_sort | long term behavior of a cyclic max type system of difference equations |
| topic | Max-type system of difference equations cyclic system positive solutions boundedness character global attractivity |
| url | http://ejde.math.txstate.edu/Volumes/2015/234/abstr.html |
| work_keys_str_mv | AT tatjanastevic longtermbehaviorofacyclicmaxtypesystemofdifferenceequations AT bratislaviricanin longtermbehaviorofacyclicmaxtypesystemofdifferenceequations |