Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics
The paper provides conditions sufficient for the existence of strictly increasing solutions of the second-order nonautonomous difference equation x(n+1)=x(n)+(n/(n+1))2(x(n)-x(n-1)+h2f(x(n))), n∈N, where h>0 is a parameter and f is Lipschitz continuous and has three real zero...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2010-01-01
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| Series: | Advances in Difference Equations |
| Online Access: | http://dx.doi.org/10.1155/2010/714891 |
| _version_ | 1818118698872864768 |
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| author | Lukáš Rachůnek Irena Rachůnková |
| author_facet | Lukáš Rachůnek Irena Rachůnková |
| author_sort | Lukáš Rachůnek |
| collection | DOAJ |
| description | The paper provides conditions sufficient for the existence of strictly increasing solutions of the second-order nonautonomous difference equation x(n+1)=x(n)+(n/(n+1))2(x(n)-x(n-1)+h2f(x(n))), n∈N, where h>0 is a parameter and f is Lipschitz continuous and has three real zeros L0<0<L. In particular we prove that for each sufficiently small h>0 there exists a solution {x(n)}n=0∞ such that {x(n)}n=1∞ is increasing, x(0)=x(1)∈(L0,0), and lim⁡n→∞x(n)>L. The problem is motivated by some models arising in hydrodynamics. |
| first_indexed | 2024-12-11T04:58:27Z |
| format | Article |
| id | doaj.art-92a2de96b5bf40a18e50704d4fbf29c5 |
| institution | Directory Open Access Journal |
| issn | 1687-1839 1687-1847 |
| language | English |
| last_indexed | 2024-12-11T04:58:27Z |
| publishDate | 2010-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-92a2de96b5bf40a18e50704d4fbf29c52022-12-22T01:20:14ZengSpringerOpenAdvances in Difference Equations1687-18391687-18472010-01-01201010.1155/2010/714891Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in HydrodynamicsLuk&#225;&#353; Rach&#367;nekIrena Rach&#367;nkov&#225;The paper provides conditions sufficient for the existence of strictly increasing solutions of the second-order nonautonomous difference equation x(n+1)=x(n)+(n/(n+1))2(x(n)-x(n-1)+h2f(x(n))), n∈N, where h>0 is a parameter and f is Lipschitz continuous and has three real zeros L0<0<L. In particular we prove that for each sufficiently small h>0 there exists a solution {x(n)}n=0∞ such that {x(n)}n=1∞ is increasing, x(0)=x(1)∈(L0,0), and lim⁡n→∞x(n)>L. The problem is motivated by some models arising in hydrodynamics.http://dx.doi.org/10.1155/2010/714891 |
| spellingShingle | Luk&#225;&#353; Rach&#367;nek Irena Rach&#367;nkov&#225; Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics Advances in Difference Equations |
| title | Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics |
| title_full | Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics |
| title_fullStr | Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics |
| title_full_unstemmed | Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics |
| title_short | Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics |
| title_sort | strictly increasing solutions of nonautonomous difference equations arising in hydrodynamics |
| url | http://dx.doi.org/10.1155/2010/714891 |
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