Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients
We study the Hyers-Ulam stability in a Banach space X of the system of first order linear difference equations of the form xn+1=Axn+dn for n∈N0 (nonnegative integers), where A is a given r×r matrix with real or complex coefficients, respectively, and (dn)n∈N0 is a fixed sequence in Xr. That is, we...
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Wiley
2015-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2015/269356 |
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| author | Bing Xu Janusz Brzdęk |
| author_facet | Bing Xu Janusz Brzdęk |
| author_sort | Bing Xu |
| collection | DOAJ |
| description | We study the Hyers-Ulam stability in a Banach space X of the system of first order linear difference equations of the form xn+1=Axn+dn for n∈N0 (nonnegative integers), where A is a given r×r matrix with real or complex coefficients, respectively, and (dn)n∈N0 is a fixed sequence in Xr. That is, we investigate the sequences (yn)n∈N0 in Xr such that δ∶=supn∈N0yn+1-Ayn-dn<∞ (with the maximum norm in Xr) and show that, in the case where all the eigenvalues of A are not of modulus 1, there is a positive real constant c (dependent only on A) such that, for each such a sequence (yn)n∈N0, there is a solution (xn)n∈N0 of the system with supn∈N0yn-xn≤cδ. |
| format | Article |
| id | doaj-art-da198d49ffb94c0db091aeb75d95a68d |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-da198d49ffb94c0db091aeb75d95a68d2025-08-20T03:24:22ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2015-01-01201510.1155/2015/269356269356Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant CoefficientsBing Xu0Janusz Brzdęk1Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, ChinaDepartment of Mathematics, Pedagogical University, Podchorążych 2, 30-084 Kraków, PolandWe study the Hyers-Ulam stability in a Banach space X of the system of first order linear difference equations of the form xn+1=Axn+dn for n∈N0 (nonnegative integers), where A is a given r×r matrix with real or complex coefficients, respectively, and (dn)n∈N0 is a fixed sequence in Xr. That is, we investigate the sequences (yn)n∈N0 in Xr such that δ∶=supn∈N0yn+1-Ayn-dn<∞ (with the maximum norm in Xr) and show that, in the case where all the eigenvalues of A are not of modulus 1, there is a positive real constant c (dependent only on A) such that, for each such a sequence (yn)n∈N0, there is a solution (xn)n∈N0 of the system with supn∈N0yn-xn≤cδ.http://dx.doi.org/10.1155/2015/269356 |
| spellingShingle | Bing Xu Janusz Brzdęk Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients Discrete Dynamics in Nature and Society |
| title | Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients |
| title_full | Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients |
| title_fullStr | Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients |
| title_full_unstemmed | Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients |
| title_short | Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients |
| title_sort | hyers ulam stability of a system of first order linear recurrences with constant coefficients |
| url | http://dx.doi.org/10.1155/2015/269356 |
| work_keys_str_mv | AT bingxu hyersulamstabilityofasystemoffirstorderlinearrecurrenceswithconstantcoefficients AT januszbrzdek hyersulamstabilityofasystemoffirstorderlinearrecurrenceswithconstantcoefficients |