姚慧麗 張悅嬌

摘 要：均方概周期型函數(shù)理論在隨機微分方程中的應(yīng)用越來越引起數(shù)學(xué)工作者的關(guān)注，其中隨機微分方程的均方漸近概周期解比均方概周期解的應(yīng)用范圍更加廣泛。利用Banach不動點定理、線性算子解析半群理論及均方漸近概周期隨機過程的概念和基本性質(zhì)，研究了實可分的Hilbert空間上的一類隨機微分方程的均方漸近概周期溫和解的存在性和唯一性。

關(guān)鍵詞：均方漸近概周期溫和解;隨機微分方程;Banach不動點定理;線性算子解析半群

DOI：10.15938/j.jhust.2019.04.024

中圖分類號： O175

文獻標(biāo)志碼： A

文章編號： 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，丹麥數(shù)學(xué)家BOHR H提出并建立了概周期函數(shù)理論[1-2]。隨后，BOCHNER、NEVMANN、ZHANG CHUANYI等對該理論進行推廣[3-5]，并將其應(yīng)用于物理、生物和力學(xué)等諸多領(lǐng)域[6-10]。2007年BEZANDRY和DIAGANA提出了均方概周期隨機過程[11]，并應(yīng)用于微分方程解的求解中[12-13]。2011年曹俊飛給出了均方漸近概周期隨機過程的概念[14]，之后，一些文獻中對隨機微分方程的均方漸近概周期溫和解的存在性和唯一性進行了研究，并取得了一定的成果[15-17]。文[18]研究了一類中立型隨機泛函微分方程的均方概周期解的存在唯一性，方程如下：

參 考 文 獻：

[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，王建偉.一類隨機微分方程的均方漸近概周期解[J].哈爾濱理工大學(xué)學(xué)報，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.

（編輯：溫澤宇）