Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under (μ,ν)-Dichotomies
We obtain the existence of stable invariant manifolds for the nonlinear equation x′=L(t)xt+f(t,xt,λ) provided that the linear delay equation x′=L(t)xt admits a nonuniform (μ,ν)-dichotomy and f is a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent o...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/989526 |
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| _version_ | 1832557013459009536 |
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| author | Lijun Pan |
| author_facet | Lijun Pan |
| author_sort | Lijun Pan |
| collection | DOAJ |
| description | We obtain the existence of stable invariant manifolds for the nonlinear equation x′=L(t)xt+f(t,xt,λ) provided that the linear delay equation x′=L(t)xt admits a nonuniform (μ,ν)-dichotomy and f is a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameter λ. Namely, the stable invariant manifolds are Lipschitz in the parameter λ. In addition, we also show that nonuniform (μ,ν)-contraction persists under sufficiently small nonlinear perturbations. |
| format | Article |
| id | doaj-art-4065a95b56134c049fccff8dee31e6b9 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-4065a95b56134c049fccff8dee31e6b92025-02-03T05:43:43ZengWileyJournal of Mathematics2314-46292314-47852014-01-01201410.1155/2014/989526989526Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under (μ,ν)-DichotomiesLijun Pan0School of Mathematics, Jia Ying University, Meizhou, Guangdong 514015, ChinaWe obtain the existence of stable invariant manifolds for the nonlinear equation x′=L(t)xt+f(t,xt,λ) provided that the linear delay equation x′=L(t)xt admits a nonuniform (μ,ν)-dichotomy and f is a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameter λ. Namely, the stable invariant manifolds are Lipschitz in the parameter λ. In addition, we also show that nonuniform (μ,ν)-contraction persists under sufficiently small nonlinear perturbations.http://dx.doi.org/10.1155/2014/989526 |
| spellingShingle | Lijun Pan Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under (μ,ν)-Dichotomies Journal of Mathematics |
| title | Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under (μ,ν)-Dichotomies |
| title_full | Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under (μ,ν)-Dichotomies |
| title_fullStr | Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under (μ,ν)-Dichotomies |
| title_full_unstemmed | Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under (μ,ν)-Dichotomies |
| title_short | Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under (μ,ν)-Dichotomies |
| title_sort | parameter dependence of stable invariant manifolds for delay differential equations under μ ν dichotomies |
| url | http://dx.doi.org/10.1155/2014/989526 |
| work_keys_str_mv | AT lijunpan parameterdependenceofstableinvariantmanifoldsfordelaydifferentialequationsundermndichotomies |