姚慧麗 張悅嬌

摘 要：均方概周期型函數(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)工作者的關(guān)注，其中隨機(jī)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(dòng)點(diǎn)定理、線性算子解析半群理論及均方漸近概周期隨機(jī)過(guò)程的概念和基本性質(zhì)，研究了實(shí)可分的Hilbert空間上的一類隨機(jī)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(jī)微分方程;Banach不動(dòng)點(diǎn)定理;線性算子解析半群

DOI：10.15938/j.jhust.2019.04.024

中圖分類號(hào)： O175

文獻(xiàn)標(biāo)志碼： A

文章編號(hào)： 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(duì)該理論進(jìn)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(jī)過(guò)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(jī)過(guò)程的概念[14]，之后，一些文獻(xiàn)中對(duì)隨機(jī)微分方程的均方漸近概周期溫和解的存在性和唯一性進(jìn)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(jī)泛函微分方程的均方概周期解的存在唯一性，方程如下：

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（編輯：溫澤宇）