一類半線性隨機(jī)微分方程的均方漸近概自守溫和解

姚慧麗　霍貴珍　孫海彤　王晶囡

摘要：均方概自守型函數(shù)理論在隨機(jī)微分方程中的應(yīng)用越來(lái)越引起數(shù)學(xué)研究者的關(guān)注，這類方程的均方漸近概自守解比均方概自守解的應(yīng)用范圍更加廣泛。對(duì)一類半線性隨機(jī)微分方程的均方漸近概自守溫和解進(jìn)行探討。利用Banach壓縮映射原理，結(jié)合均方漸近概自守隨機(jī)過(guò)程的定義和性質(zhì)、Cauchy-Schwarz不等式、Lipschitz條件、It等距積分，討論了該類隨機(jī)微分方程的均方漸近概自守溫和解的存在唯一性。

關(guān)鍵詞：均方漸近概自守溫和解;半線性隨機(jī)微分方程;Banach壓縮映射原理

DOI：10.15938/j.jhust.2022.04.020

中圖分類號(hào)： O175

文獻(xiàn)標(biāo)志碼： A

文章編號(hào)： 1007-2683（2022）04-0154-07

Square-Mean Asymptotically Almost Automorphic Mild Solutions

to a Class of Semi-linear Stochastic Differential Equations

YAO Hui-li，HUO Gui-zhen，SUN Hai-tong，WANG Jing-nan

（School of Science，Harbin University of Science and Technology，Harbin 150080，China）

Abstract：The applications of the theories of square-mean almost automorphic type functions have attracted more and more attention by mathematics researchers， square-mean asymptotically almost automorphic solutions of this class of differential equations have a wider range of applications than square-mean almost automorphic solutions.Square-mean asymptotically almost automorphic mild solutions to a class of semi-linear stochastic differential equations are investigated. The existence and uniqueness of square-mean asymptotically almost automorphic mild solutions for this kind of equation are discussed by using the principle of Banach compressed image， combining with the definition and properties of square-mean asymptotically almost automorphic stochastic processes， Cauchy-Schwarz inequality， Lipschtiz conditions and Ito integrals isometry.

Keywords：square-mean asymptotically almost automorphic mild solutions; semi-linearstochastic differential equations; principle of Banach compressed image

0引言

概自守函數(shù)、漸近概自守函數(shù)以及偽概自守函數(shù)（統(tǒng)稱為概自守型函數(shù)）的定義分別由BOCHNER S、N′GUEREKATA G M、XIAO T J， LIANG J， ZHANG J給出[1-3]。概自守型函數(shù)理論的產(chǎn)生推廣了概周期型函數(shù)的應(yīng)用范圍，并在各類方程中得到了應(yīng)用[4-10]，為了更好的描述自然界中的隨機(jī)現(xiàn)象，2010年，F(xiàn)U M M， LIU Z X提出了均方概自守隨機(jī)過(guò)程的概念[11]，這一概念是對(duì)概自守函數(shù)的推廣。之后，均方偽概守隨機(jī)過(guò)程和均方漸近概自守隨機(jī)過(guò)程的概念也相繼被給出[ 12-13 ] 。自均方概自守型隨機(jī)過(guò)程有關(guān)理論被提出以來(lái)，國(guó)內(nèi)外數(shù)學(xué)工作者將其應(yīng)用到一類將隨機(jī)性納入了數(shù)學(xué)描述中的模型中即隨機(jī)微分方程中，研究了此種方程的均方概自守解[14-16]和均方偽概自守解的存在及唯一性[17-18]。在文[14]中，CHANG Y K， ZHAO Z H， N′GUEREKATA G M.對(duì)下列一類半線性隨機(jī)微分方程

1預(yù)備知識(shí)

2主要結(jié)論

參 考 文 獻(xiàn)：

[1]BOCHNER S. A New Approach to Almost Periodicity[J]. Proceedings of the National Academy of Sciences of the United States of America， 1962， 48（12）：2039.

[2]N′GUEREKATA G M. Some Remarks on Asymptotically Almost Automorphic Function[J]. Rivista Di Matematica Della Università Di Parma， 1988， 13（4）： 301.

[3]XIAO T J， LIANG J， ZHANG J. Pseudo Almost Automorphic Solutions to Semi-linear Differential Equations in Banach Spaces[J]. Semigroup Forum， 2008， 76（3）： 518.

[4]GOLDSTEIN J A， N′GUEREKATA G M. Almost Automorphic Solution of Semi-linear Evolution Equations[J]. Proc.Amer.Math.Soc.133， 2005，2401.

[5]EZZINBI K， N′GUEREKATA G M. Massera Type Theorem for Almost Automorphic Solutions of Functional Differential Equations of Neutral Type[J]. Journal of Mathematical Analysis and Applications，2006， 316：707.

[6]DIAGANA T， N′GUEREKATA G M. Amost Automorphic Solutions to Some Classes of Partial Evolution Equations[J]. Applied.Mathematics Letters，2007，20（4）：462.

[7]M′HAMDI M S. Pseudo Almost Automorphic Solutions for Multidirectional Associative Memory Neural Network with Mixed Delays[J]. Neural processing letters， 2019， 49（3）： 1567.

[8]AOUITI C， DRIDI F. Weighted Pseudo Almost Automorphic Solutions for Neutral Type Fuzzy Cellular Neural Networks with Mixed Delays and D Operator in Clifford Algebra[J]. International Journal of Systems Science， 2020（3）： 1.

[9]ZABSONRE I， MBAINADJI D. Pseudo Almost Automorphic Solutions of Class r in α-norm under the Light of Measure Theory[J]. Nonautonomous Dynamical Systems， 2020， 7（1）： 81.

[10]AOUITI C， M′HAMDI M S， TOUATI A. Pseudo Almost Automorphic Solutions of Recurrent Neural Networks with Time-Varying Coefficients and Mixed Delays[J]. Neural Processing Letters， 2016， 45（1）：1.

[11]FU M M， LIU Z X. Square-mean Almost Automorphic Solutions for Some Stochastic Differential Equations[J]. Proc.Amer.Math.Soc， 2010，138（10）：3689.

[12]CHEN Z， LIN W. Square-mean Pseudo Almost Automorphic Process and Its Application to Stochastic evolution Equations[J]. Journal of Functional Analysis，2011，261（1）：69.

[13]YAN Z， ZHANG H W.Square-mean Asymptotically Almost Automorphic Solutions for Non-local Neutral Stochastic Functional Integro-differential Equations in Hilbert Spaces[J]. Electronic Journal of Mathematical Analysis and Applications，2013，1（1）：15.

[14]CHANG Y K， ZHAO Z H， N′GUEREKATA G M. Square-mean Almost Automorphic Mild Solutions to Non-autonomous Stochastic Differential Equations in Hilbert Spaces[J]. Advances in Difference Equations， 2011， 61（2）： 384.

[15]XI L，HAN Y L， LIU B F. Square-mean Almost Automorphic Solutions to Some Stochastic Evolution Equations I： Autonomous Case[J]. Acta Mathematicae Applicatae Sinica， English Series， 2015， 31（3）： 577.

[16]LI L J. Existence of Square-Mean Almost Automorphic Solutions to Stochastic Functional Integro-differential Equations in Hilbert Spaces[J]. Abstract and Applied Analysis， 2014： 1.

[17]GU Y， REN Y， SAKTHIVEL R. Square-mean Pseudo Almost Automorphic Mild Solutions for Stochastic Evolution Equations Driven by G-Brownian Motion[J]. Stochastic Analysis & Applications， 2016， 34（3）：528.

[18]YAN Z M， ZHANG H W. Existence of Stepanov-Like Square-Mean Pseudo Almost Periodic Solutions to Partial Stochastic Neutral Differential Equations[J]. Annals of Functional Analysis， 2015， 6（1）： 116.

[19]張著洪.關(guān)于閉算子及其共軛的分?jǐn)?shù)次冪的評(píng)注[J].貴州大學(xué)學(xué)報(bào)（自然科學(xué)版），1997（4）：202.ZHANG Zhuhong. Comments on the Fractional Power of Closed Operators and Their Conjugates[J].Journal of Guizhou University （Natural Sciences），1997（4）：202.

[20]姚慧麗， 劉婷， 張士晶. 一類隨機(jī)微分方程的均方漸近概自守溫和解[J]. 哈爾濱理工大學(xué)學(xué)報(bào)， 2016， 21（3）： 114.YAO Huili， LIU Ting， ZHANG Shijing. Square-mean Asymptotically Almost Automorphic Mild Solutions for a Class of Stochastic Differential Equations[J]. Journal of Harbin University of Science and Technology， 2016， 21（3）： 114.

（編輯：溫澤宇）