摘 要：研究了一類在Hilbert空間中具有分?jǐn)?shù)Brown運(yùn)動(dòng)的分?jǐn)?shù)階中立型隨機(jī)微分方程，利用Picard逐步逼近法得到了其非Lipschitz條件和弱化的線性增長條件下粘性解的新的存在唯一性的充分條件.所提的研究方法使得先前一些研究結(jié)果得到了拓展.最后通過具有分?jǐn)?shù)Brown運(yùn)動(dòng)的隨機(jī)非線性波動(dòng)方程驗(yàn)證了理論的有效性.

關(guān)鍵詞：分?jǐn)?shù)階隨機(jī)微分方程;分?jǐn)?shù)Brown運(yùn)動(dòng);粘性解;存在唯一性

中圖分類號(hào)：O175.14 文獻(xiàn)標(biāo)志碼：A文章編號(hào)：1000-2367（2025）03-0104-08

1 預(yù)備知識(shí)

2 解的存在唯一性

3 例 子

4 結(jié) 論

本文討論了一類具有分?jǐn)?shù)Brown運(yùn)動(dòng)的分?jǐn)?shù)階中立型隨機(jī)微分方程.運(yùn)用逐步逼近法給出了非Lipschitz條件和弱化的線性增長條件下粘性解的存在性和唯一性的充分條件，其中Lipschitz條件視為其特殊情形.并通過具有分?jǐn)?shù)Brown運(yùn)動(dòng)的隨機(jī)非線性波動(dòng)方程的應(yīng)用驗(yàn)證了理論的有效性.本文的方法避免了使用較強(qiáng)的Lipschitz條件，對于研究隨機(jī)非線性系統(tǒng)的解的存在唯一性較為有效.該方法亦可用于討論時(shí)滯中立型隨機(jī)微分方程解的存在唯一性和穩(wěn)定性，為研究其他中立型隨機(jī)微分方程提供了理論依據(jù).

參 考 文 獻(xiàn)

［1］ GOVINDAN T E.Almost sure exponential stability for stochastic neutral partial functional differential equations［J］.Stochastics，2005，77：139-154.

［2］BAO J H，HOU Z T.Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients［J］.Comput Math Appl，2010，59：207-214.

［3］JIANG F，SHEN Y.A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients［J］.Comput Math Appl，2011，61：1590-1594.

［4］ BAO H B，CAO J D.Existence of solutions for fractional stochastic impulsive neutral functional differential equations with infinite delay［J］.Advances in Difference Equations，2017（1）：66.

［5］AHMADOVA A，MAHMUDOV N I.Existence and uniqueness results for a class of fractional stochastic neutral differential equations［J］.Chaos，Solitons and Fractals，2020，139：110253.

［6］LI Y J，WANG Y J.The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay［J］.Journal of Differential Equations，2019，266：3514-3558.

［7］REN Y，XIA N M.Existence，uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay［J］.Applied Mathematics and Computation，2009，210：72-79.

［8］CARABALLO T，GARRIDO-ATIENZA M J，TANIGUCHI T.The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion［J］.Nonlinear Anal TMA，2011，74：3671-3684.

［9］BOUFOUSSI B，HAJJI S.Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space［J］.Statiscics and Probability Letters，2012，82：1549-1558.

［10］DENG S F，SHU X B，MAO J Z.Existence and exponential stability for impulsive neutral stochastic functional dierential equations driven by fBm with noncompact semigroup via Mnch fixed point［J］.Journal of Mathematical Analysis and Applications，2018，467：398-420.

［11］PAZY A.Semigroups of Linear Operators and Applications to Partial Differential Equations［M］.Berlin：Springer-Verlag，1983.

［12］PODLUBNY I.Fractional Differential Equations［C］//Mathematics in Science and Engineering.［S.l.：s.n.］，1999.

［13］MAINARDI F.Fractional Calculus and Waves in Linear Viscoelasticity： An Introduction to Mathematical Models［M］.London：Imperial College Press，2010.

［14］MAO X R.Stochastic Differential Equations and Applications（Second Edition）［M］.New York：Academic Press，2006.

［15］TANIGUCHI T.The existence and asymptotic behaviour of solutions to non-Lipschitz stochastic functional evolution equations driven by Poisson jumps［J］.Stochastics，2010，82：339-363.

Existence and uniqueness of solutions to fractional neutral stochastic

differential equations with fractional brownian motionLi Guoping¹， Han Ting^1，2

（1. Xinhua College， Ningxia University， Yinchuan 750021， China;

2. College of Mathematics and Statistics， Ningxia University， Yinchuan 750021， China）

Abstract： In this paper， we consider a class of fractional-order neutral stochastic" differential" equations with fractional Brownian motion by using picard step by step in a Hilbert space. A novel sufficient condition for the existence and uniqueness of mild solutions is obtained in conditions of the non-Lipschitz condition and the weakened linear growth. The result generalizes a few previous known results. An application to the stochastic nonlinear wave equation with fractional Brownian motion is given to illustrate the validity of the obtained theory.

Keywords： fractional" stochastic" differential" equations; fractional Brownian motion; mild solution; existence and uniqueness

［責(zé)任編校 陳留院 楊浦］