Eventually Periodic Solutions of a Max-Type Difference Equation
We study the following max-type difference equation xn=max{An/xn-r,xn-k}, n=1,2,…, where {An}n=1+∞ is a periodic sequence with period p and k,r∈{1,2,…} with gcd(k,r)=1 and k≠r, and the initial conditions x1-d,x2-d,…,x0 are real numbers with d=max{r,k}. We show that if p=1 (or p≥2 and k is odd), th...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2014/219437 |
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| author | Taixiang Sun Jing Liu Qiuli He Xin-He Liu Chunyan Tao |
| author_facet | Taixiang Sun Jing Liu Qiuli He Xin-He Liu Chunyan Tao |
| author_sort | Taixiang Sun |
| collection | DOAJ |
| description | We study the following max-type difference equation xn=max{An/xn-r,xn-k}, n=1,2,…, where {An}n=1+∞ is a periodic sequence with period p and k,r∈{1,2,…} with gcd(k,r)=1 and k≠r, and the initial conditions x1-d,x2-d,…,x0 are real numbers with d=max{r,k}. We show that if p=1 (or p≥2 and k is odd), then every well-defined solution of this equation is eventually periodic with period k, which generalizes the results of (Elsayed and Stevic´ (2009), Iričanin and Elsayed (2010), Qin et al. (2012), and Xiao and Shi (2013)) to the general case. Besides, we construct an example with p≥2 and k being even which has a well-defined solution that is not eventually periodic. |
| format | Article |
| id | doaj-art-1772f123926d4b3a9fb1d6acacb7c68d |
| institution | Kabale University |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-1772f123926d4b3a9fb1d6acacb7c68d2025-02-03T01:07:12ZengWileyThe Scientific World Journal2356-61401537-744X2014-01-01201410.1155/2014/219437219437Eventually Periodic Solutions of a Max-Type Difference EquationTaixiang Sun0Jing Liu1Qiuli He2Xin-He Liu3Chunyan Tao4College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, ChinaCollege of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, ChinaCollege of Electrical Engineering, Guangxi University, Nanning, Guangxi 530004, ChinaCollege of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, ChinaCollege of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, ChinaWe study the following max-type difference equation xn=max{An/xn-r,xn-k}, n=1,2,…, where {An}n=1+∞ is a periodic sequence with period p and k,r∈{1,2,…} with gcd(k,r)=1 and k≠r, and the initial conditions x1-d,x2-d,…,x0 are real numbers with d=max{r,k}. We show that if p=1 (or p≥2 and k is odd), then every well-defined solution of this equation is eventually periodic with period k, which generalizes the results of (Elsayed and Stevic´ (2009), Iričanin and Elsayed (2010), Qin et al. (2012), and Xiao and Shi (2013)) to the general case. Besides, we construct an example with p≥2 and k being even which has a well-defined solution that is not eventually periodic.http://dx.doi.org/10.1155/2014/219437 |
| spellingShingle | Taixiang Sun Jing Liu Qiuli He Xin-He Liu Chunyan Tao Eventually Periodic Solutions of a Max-Type Difference Equation The Scientific World Journal |
| title | Eventually Periodic Solutions of a Max-Type Difference Equation |
| title_full | Eventually Periodic Solutions of a Max-Type Difference Equation |
| title_fullStr | Eventually Periodic Solutions of a Max-Type Difference Equation |
| title_full_unstemmed | Eventually Periodic Solutions of a Max-Type Difference Equation |
| title_short | Eventually Periodic Solutions of a Max-Type Difference Equation |
| title_sort | eventually periodic solutions of a max type difference equation |
| url | http://dx.doi.org/10.1155/2014/219437 |
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