Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations
This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equations dxt=Atxt+Ft,xt,xtdt+H(t,xt,xt)∘dW(t). A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mi...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/934534 |
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| _version_ | 1832548868601937920 |
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| author | Aimin Liu Yongjian Liu Qun Liu |
| author_facet | Aimin Liu Yongjian Liu Qun Liu |
| author_sort | Aimin Liu |
| collection | DOAJ |
| description | This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equations dxt=Atxt+Ft,xt,xtdt+H(t,xt,xt)∘dW(t). A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup {Tt}t≥0 is essentially removed, which is generated by the linear densely defined operator A∶D(A)⊂L2(ℙ,ℍ)→L2(ℙ,ℍ), only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results. |
| format | Article |
| id | doaj-art-261375ca501a42ba954e7c46e19d4d1b |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-261375ca501a42ba954e7c46e19d4d1b2025-02-03T06:12:46ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/934534934534Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential EquationsAimin Liu0Yongjian Liu1Qun Liu2Educational Technology Center, Yulin Normal University, Yulin 537000, ChinaSchool of Mathematics and Information Science, Yulin Normal University, Yulin 537000, ChinaSchool of Mathematics and Information Science, Yulin Normal University, Yulin 537000, ChinaThis work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equations dxt=Atxt+Ft,xt,xtdt+H(t,xt,xt)∘dW(t). A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup {Tt}t≥0 is essentially removed, which is generated by the linear densely defined operator A∶D(A)⊂L2(ℙ,ℍ)→L2(ℙ,ℍ), only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results.http://dx.doi.org/10.1155/2014/934534 |
| spellingShingle | Aimin Liu Yongjian Liu Qun Liu Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations Abstract and Applied Analysis |
| title | Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations |
| title_full | Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations |
| title_fullStr | Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations |
| title_full_unstemmed | Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations |
| title_short | Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations |
| title_sort | asymptotically almost periodic solutions for a class of stochastic functional differential equations |
| url | http://dx.doi.org/10.1155/2014/934534 |
| work_keys_str_mv | AT aiminliu asymptoticallyalmostperiodicsolutionsforaclassofstochasticfunctionaldifferentialequations AT yongjianliu asymptoticallyalmostperiodicsolutionsforaclassofstochasticfunctionaldifferentialequations AT qunliu asymptoticallyalmostperiodicsolutionsforaclassofstochasticfunctionaldifferentialequations |