Bounded solutions and Hyers–Ulam stability of quasilinear dynamic equations on time scales
We consider the quasilinear dynamic equation in a Banach space on unbounded above and below time scales T with rd-continuous, regressive right-hand side.We define the corresponding Green-type map. Using the integral functional technique, we find a new simpler, but at the same time, more general suf...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Vilnius University Press
2023-02-01
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| Series: | Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://www.zurnalai.vu.lt/nonlinear-analysis/article/view/31603 |
| _version_ | 1797897047847731200 |
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| author | Andrejs Reinfelds Dzintra Šteinberga |
| author_facet | Andrejs Reinfelds Dzintra Šteinberga |
| author_sort | Andrejs Reinfelds |
| collection | DOAJ |
| description |
We consider the quasilinear dynamic equation in a Banach space on unbounded above and below time scales T with rd-continuous, regressive right-hand side.We define the corresponding Green-type map. Using the integral functional technique, we find a new simpler, but at the same time, more general sufficient condition for the existence of a bounded solution on the time scales expressed in terms of integrals of the Green-type map. We construct previously unknown linear scalar differential equation, which does not possess exponentially dichotomy, but for which the integral of the corresponding Green-type map is uniformly bounded. The existence of such example allows, on the one hand, to obtain the new sufficient condition for the existence of bounded solution and, on the other hand, to prove Hyers–Ulam stability for a much broader class of linear dynamic equations even in the classical case.
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| first_indexed | 2024-04-10T07:52:28Z |
| format | Article |
| id | doaj.art-ca90f25407ba4f1287a22061358ec2d8 |
| institution | Directory Open Access Journal |
| issn | 1392-5113 2335-8963 |
| language | English |
| last_indexed | 2024-04-10T07:52:28Z |
| publishDate | 2023-02-01 |
| publisher | Vilnius University Press |
| record_format | Article |
| series | Nonlinear Analysis |
| spelling | doaj.art-ca90f25407ba4f1287a22061358ec2d82023-02-23T09:45:21ZengVilnius University PressNonlinear Analysis1392-51132335-89632023-02-0128210.15388/namc.2023.28.31603Bounded solutions and Hyers–Ulam stability of quasilinear dynamic equations on time scalesAndrejs Reinfelds0Dzintra Šteinberga1University of LatviaUniversity of Latvia We consider the quasilinear dynamic equation in a Banach space on unbounded above and below time scales T with rd-continuous, regressive right-hand side.We define the corresponding Green-type map. Using the integral functional technique, we find a new simpler, but at the same time, more general sufficient condition for the existence of a bounded solution on the time scales expressed in terms of integrals of the Green-type map. We construct previously unknown linear scalar differential equation, which does not possess exponentially dichotomy, but for which the integral of the corresponding Green-type map is uniformly bounded. The existence of such example allows, on the one hand, to obtain the new sufficient condition for the existence of bounded solution and, on the other hand, to prove Hyers–Ulam stability for a much broader class of linear dynamic equations even in the classical case. https://www.zurnalai.vu.lt/nonlinear-analysis/article/view/31603dynamic equations on time scalesGreen-type mapbounded solutionperiodic solution, Hyers–Ulam stability |
| spellingShingle | Andrejs Reinfelds Dzintra Šteinberga Bounded solutions and Hyers–Ulam stability of quasilinear dynamic equations on time scales Nonlinear Analysis dynamic equations on time scales Green-type map bounded solution periodic solution, Hyers–Ulam stability |
| title | Bounded solutions and Hyers–Ulam stability of quasilinear dynamic equations on time scales |
| title_full | Bounded solutions and Hyers–Ulam stability of quasilinear dynamic equations on time scales |
| title_fullStr | Bounded solutions and Hyers–Ulam stability of quasilinear dynamic equations on time scales |
| title_full_unstemmed | Bounded solutions and Hyers–Ulam stability of quasilinear dynamic equations on time scales |
| title_short | Bounded solutions and Hyers–Ulam stability of quasilinear dynamic equations on time scales |
| title_sort | bounded solutions and hyers ulam stability of quasilinear dynamic equations on time scales |
| topic | dynamic equations on time scales Green-type map bounded solution periodic solution, Hyers–Ulam stability |
| url | https://www.zurnalai.vu.lt/nonlinear-analysis/article/view/31603 |
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