Dynamic behavior of a nonlinear rational difference equation and generalization
<p>Abstract</p> <p>This paper is concerned about the dynamic behavior for the following high order nonlinear difference equation <it>x</it> <sub> <it>n </it> </sub>= (<it>x</it> <sub> <it>n</it>-<it>k </it&g...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2011-01-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://www.advancesindifferenceequations.com/content/2011/1/36 |
| Summary: | <p>Abstract</p> <p>This paper is concerned about the dynamic behavior for the following high order nonlinear difference equation <it>x</it> <sub> <it>n </it> </sub>= (<it>x</it> <sub> <it>n</it>-<it>k </it> </sub>+ <it>x</it> <sub> <it>n-m </it> </sub>+ <it>x</it> <sub> <it>n</it>-<it>l</it> </sub>)/(<it>x</it> <sub> <it>n</it>-<it>k</it> </sub> <it>x</it> <sub> <it>n</it>-<it>m </it> </sub>+ <it>x</it> <sub> <it>n</it>-<it>m</it> </sub> <it>x</it> <sub> <it>n</it>-<it>l </it> </sub>+1) with the initial data <inline-formula> <m:math name="1687-1847-2011-36-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow> <m:mo class="MathClass-open">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mo class="MathClass-bin">-</m:mo> <m:mi>l</m:mi> </m:mrow> </m:msub> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mo class="MathClass-bin">-</m:mo> <m:mi>l</m:mi> <m:mo class="MathClass-bin">+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo class="MathClass-punc">,</m:mo> <m:mo class="MathClass-op">…</m:mo> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mo class="MathClass-bin">-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">}</m:mo> </m:mrow> <m:mo class="MathClass-rel">∈</m:mo> <m:msubsup> <m:mrow> <m:mi>ℝ</m:mi> </m:mrow> <m:mrow> <m:mo class="MathClass-bin">+</m:mo> </m:mrow> <m:mrow> <m:mi>l</m:mi> </m:mrow> </m:msubsup> </m:math> </inline-formula> and 1 ≤ <it>k </it>≤ <it>m </it>≤ <it>l</it>. The convergence of solution to this equation is investigated by introducing a new sequence, which extends and includes corresponding results obtained in the references (Li in J Math Anal Appl 312:103-111, 2005; Berenhaut et al. Appl. Math. Lett. 20:54-58, 2007; Papaschinopoulos and Schinas J Math Anal Appl 294:614-620, 2004) to a large extent. In addition, some propositions for generalized equations are reported.</p> |
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| ISSN: | 1687-1839 1687-1847 |