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摘 要：均方概周期型函数理论在随机微分方程中的应用越来越引起数学工作者的关注，其中随机微分方程的均方渐近概周期解比均方概周期解的应用范围更加广泛。利用Banach不动点定理、线性算子解析半群理论及均方渐近概周期随机过程的概念和基本性质，研究了实可分的Hilbert空间上的一类随机微分方程的均方渐近概周期温和解的存在性和唯一性。
关键词：均方渐近概周期温和解;随机微分方程;Banach不动点定理;线性算子解析半群
DOI：10.15938/j.jhust.2019.04.024
中图分类号： O175
文献标志码： A
文章编号： 1007-2683（2019）04-0143-06
Abstract：The application of the theory of square-mean almost periodic type functions to stochastic differential equations has attracted more and more attention by researchers.The square-mean asymptotically almost periodic solutions of stochastic differential equations have a wider range of applications than square-mean almost periodic solutions.In this paper，the existence and uniqueness of the square-mean asymptotically almost periodic mild solutions of a class of stochastic differential equations in real separable Hilbert spaces are discussed， using the Banach fixed point theorem，analytic semigroup theory of linear operators and the concept and basic properties of the square-mean asymptotically almost periodic stochastic processes.
Keywords：square-mean asymptotically almost periodic mild solutions;stochastic differential equations; Banach fixed point theorem; analytic semigroup theory of linear operators
0 引 言
在1925-1926年間，丹麦数学家BOHR H提出并建立了概周期函数理论[1-2]。随后，BOCHNER、NEVMANN、ZHANG CHUANYI等对该理论进行推广[3-5]，并将其应用于物理、生物和力学等诸多领域[6-10]。2007年BEZANDRY和DIAGANA提出了均方概周期随机过程[11]，并应用于微分方程解的求解中[12-13]。2011年曹俊飞给出了均方渐近概周期随机过程的概念[14]，之后，一些文献中对随机微分方程的均方渐近概周期温和解的存在性和唯一性进行了研究，并取得了一定的成果[15-17]。文[18]研究了一类中立型随机泛函微分方程的均方概周期解的存在唯一性，方程如下：
参 考 文 献：
[1] BOHR H.Zur Theorie Der Fastperiodischen [J].Acta.Math.， 1925， 45：19.
[2] BOHR H.Almost Periodic Functions[M].Chelsea：New York， 1951.
[3] BOCHNER H.Abstrakte Fastperiodische Funktionen[J].Acta.Math.， 1933， 61（1）：149.
[4] BOCHNER H，NEUMANN J V.Almost Periodic Functions in Groups[J].Trans. Amer. Math.Soc.， 1935， 37（1）：21.
[5] ZHANG Chuanyi.Almost Periodic Type Functions and Ergodicity[M].Beijing：Science Press， 2003.
[6] BEZANDRY P H，DIAGANA T.Existence of Quadratic-Mean Almost Periodic Solutions to Some Stochastic Hyperbolic Differential Equations[J].Electronic Journal of Differential Equations， 2009， 111.
[7] ZHAO Zhihan，CHANG Yongkui，LI Wensheng.Asymptotically Almost Periodic，Almost Periodic and Pseudo Almost Periodic Mild Solutions for Neutral Differential Equations[J].Nonlinear Analysis.Real World Applications， 2010， 4（11）：3037. [8] JOS Paulo C.DOS Santos，SANDRO M.GUZZO，MARCOS N.RABELO.Asymptotically Almost Periodic Solutions for Abstract Partial Neutral Integro-Differential Equations[J].Advances in Difference Equations， 2010， 1（210）：26.
[9]TOUFIK Guendouzi，KHADEM Mehdi.Almost Periodic Mild Solutions for Stochastic Delay Functional. Differential Equations Driven by a Fractional Brownian Motion[J].Romanian Journal of Mathematics and Computer Science， 2014， 1（4）：12.
[10]ZHANG Aiping.Pseudo Almost Periodic Solutions for SICNNs with Oscillating Leakage Coefficients and Complex Deviating Arguments[J].Neural Processing Letters， 2017， 45（1）：183.
[11]BEZANDRY P H，DIAGANA T.Existence of Almost Periodic Solutions to Some StochasticDifferential Equations[J].Applicable Analysis， 2007， 86（7）：819.
[12]XIA Zhihan.Pseudo Almost Periodicity of Fractional Integro-Differential Equations with Impulsive Effects in Banach Spaces[J].Czechoslovak Mathematical Journal， 2017， 1（67）.
[13]KERBOUA，MOURAD. Quadratic Mean Almost Periodic Mild Solutions to a Fractional Stochastic Differential Equation in Hilbert Spaces[J].Nonlinear Evol.Equ.Appl.，2016（4）：123.
[14]CAO Junfei，YANG Qigui，HUANG Zaitang，et al.Asymptotically Almost Periodic Solutions of Stochastic Functional Differential Equations[J].Applied Mathematics and Computation， 2011， 5（218）：1499.
[15]姚慧麗，王建伟.一类随机微分方程的均方渐近概周期解[J].哈尔滨理工大学学报，2014， 19（6）：118.
[16]SAKTHIVEL R，REVATHI J，MAHMUDOV N I.Asymptotic Stability of Fractional Stochastic Neutral Differential Equation with Infinite Delays[J].Abstract and Applied Analysis， 2013.
[17] LIU Aimin，LIU Yongjian，LIU Qun.Asymptotically Almost Periodic Solutions for a Class of a Stochastic Functional Differential Equations[J].ABSTRACT AND APPLIED ANALYSIS，2014.
[18]CHANG Yongkui，MA Ruyun，ZHAO Zhihan.Almost Periodic Solutions to a Stochastic Differential Equation in Hilbert Space[J].Results in Mathematics， 2013， 63（1/2）：435.
[19]PAZY A.Semigroups of Linear Operators and Applications to Partial Differential Equations[J].Springer-Verlag，New York， 1983.
[20]BEZANDRY P，DIAGANA T. Almost Periodic Stochastic Processes[M].New York：Springer，April.，2011：1.
（编辑：温泽宇）
关键词：均方渐近概周期温和解;随机微分方程;Banach不动点定理;线性算子解析半群
DOI：10.15938/j.jhust.2019.04.024
中图分类号： O175
文献标志码： A
文章编号： 1007-2683（2019）04-0143-06
Abstract：The application of the theory of square-mean almost periodic type functions to stochastic differential equations has attracted more and more attention by researchers.The square-mean asymptotically almost periodic solutions of stochastic differential equations have a wider range of applications than square-mean almost periodic solutions.In this paper，the existence and uniqueness of the square-mean asymptotically almost periodic mild solutions of a class of stochastic differential equations in real separable Hilbert spaces are discussed， using the Banach fixed point theorem，analytic semigroup theory of linear operators and the concept and basic properties of the square-mean asymptotically almost periodic stochastic processes.
Keywords：square-mean asymptotically almost periodic mild solutions;stochastic differential equations; Banach fixed point theorem; analytic semigroup theory of linear operators
0 引 言
在1925-1926年間，丹麦数学家BOHR H提出并建立了概周期函数理论[1-2]。随后，BOCHNER、NEVMANN、ZHANG CHUANYI等对该理论进行推广[3-5]，并将其应用于物理、生物和力学等诸多领域[6-10]。2007年BEZANDRY和DIAGANA提出了均方概周期随机过程[11]，并应用于微分方程解的求解中[12-13]。2011年曹俊飞给出了均方渐近概周期随机过程的概念[14]，之后，一些文献中对随机微分方程的均方渐近概周期温和解的存在性和唯一性进行了研究，并取得了一定的成果[15-17]。文[18]研究了一类中立型随机泛函微分方程的均方概周期解的存在唯一性，方程如下：
参 考 文 献：
[1] BOHR H.Zur Theorie Der Fastperiodischen [J].Acta.Math.， 1925， 45：19.
[2] BOHR H.Almost Periodic Functions[M].Chelsea：New York， 1951.
[3] BOCHNER H.Abstrakte Fastperiodische Funktionen[J].Acta.Math.， 1933， 61（1）：149.
[4] BOCHNER H，NEUMANN J V.Almost Periodic Functions in Groups[J].Trans. Amer. Math.Soc.， 1935， 37（1）：21.
[5] ZHANG Chuanyi.Almost Periodic Type Functions and Ergodicity[M].Beijing：Science Press， 2003.
[6] BEZANDRY P H，DIAGANA T.Existence of Quadratic-Mean Almost Periodic Solutions to Some Stochastic Hyperbolic Differential Equations[J].Electronic Journal of Differential Equations， 2009， 111.
[7] ZHAO Zhihan，CHANG Yongkui，LI Wensheng.Asymptotically Almost Periodic，Almost Periodic and Pseudo Almost Periodic Mild Solutions for Neutral Differential Equations[J].Nonlinear Analysis.Real World Applications， 2010， 4（11）：3037. [8] JOS Paulo C.DOS Santos，SANDRO M.GUZZO，MARCOS N.RABELO.Asymptotically Almost Periodic Solutions for Abstract Partial Neutral Integro-Differential Equations[J].Advances in Difference Equations， 2010， 1（210）：26.
[9]TOUFIK Guendouzi，KHADEM Mehdi.Almost Periodic Mild Solutions for Stochastic Delay Functional. Differential Equations Driven by a Fractional Brownian Motion[J].Romanian Journal of Mathematics and Computer Science， 2014， 1（4）：12.
[10]ZHANG Aiping.Pseudo Almost Periodic Solutions for SICNNs with Oscillating Leakage Coefficients and Complex Deviating Arguments[J].Neural Processing Letters， 2017， 45（1）：183.
[11]BEZANDRY P H，DIAGANA T.Existence of Almost Periodic Solutions to Some StochasticDifferential Equations[J].Applicable Analysis， 2007， 86（7）：819.
[12]XIA Zhihan.Pseudo Almost Periodicity of Fractional Integro-Differential Equations with Impulsive Effects in Banach Spaces[J].Czechoslovak Mathematical Journal， 2017， 1（67）.
[13]KERBOUA，MOURAD. Quadratic Mean Almost Periodic Mild Solutions to a Fractional Stochastic Differential Equation in Hilbert Spaces[J].Nonlinear Evol.Equ.Appl.，2016（4）：123.
[14]CAO Junfei，YANG Qigui，HUANG Zaitang，et al.Asymptotically Almost Periodic Solutions of Stochastic Functional Differential Equations[J].Applied Mathematics and Computation， 2011， 5（218）：1499.
[15]姚慧麗，王建伟.一类随机微分方程的均方渐近概周期解[J].哈尔滨理工大学学报，2014， 19（6）：118.
[16]SAKTHIVEL R，REVATHI J，MAHMUDOV N I.Asymptotic Stability of Fractional Stochastic Neutral Differential Equation with Infinite Delays[J].Abstract and Applied Analysis， 2013.
[17] LIU Aimin，LIU Yongjian，LIU Qun.Asymptotically Almost Periodic Solutions for a Class of a Stochastic Functional Differential Equations[J].ABSTRACT AND APPLIED ANALYSIS，2014.
[18]CHANG Yongkui，MA Ruyun，ZHAO Zhihan.Almost Periodic Solutions to a Stochastic Differential Equation in Hilbert Space[J].Results in Mathematics， 2013， 63（1/2）：435.
[19]PAZY A.Semigroups of Linear Operators and Applications to Partial Differential Equations[J].Springer-Verlag，New York， 1983.
[20]BEZANDRY P，DIAGANA T. Almost Periodic Stochastic Processes[M].New York：Springer，April.，2011：1.
（编辑：温泽宇）