一类半线性随机微分方程的均方渐近概自守温和解
2022-05-30姚慧丽霍贵珍孙海彤王晶囡
姚慧丽 霍贵珍 孙海彤 王晶囡
摘要:均方概自守型函數理论在随机微分方程中的应用越来越引起数学研究者的关注,这类方程的均方渐近概自守解比均方概自守解的应用范围更加广泛。对一类半线性随机微分方程的均方渐近概自守温和解进行探讨。利用Banach压缩映射原理,结合均方渐近概自守随机过程的定义和性质、Cauchy-Schwarz不等式、Lipschitz条件、It等距积分,讨论了该类随机微分方程的均方渐近概自守温和解的存在唯一性。
关键词:均方渐近概自守温和解;半线性随机微分方程;Banach压缩映射原理
DOI:10.15938/j.jhust.2022.04.020
中图分类号: O175
文献标志码: A
文章编号: 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
概自守函数、渐近概自守函数以及伪概自守函数(统称为概自守型函数)的定义分别由BOCHNER S、N′GUEREKATA G M、XIAO T J, LIANG J, ZHANG J给出[1-3]。概自守型函数理论的产生推广了概周期型函数的应用范围,并在各类方程中得到了应用[4-10],为了更好的描述自然界中的随机现象,2010年,FU M M, LIU Z X提出了均方概自守随机过程的概念[11],这一概念是对概自守函数的推广。之后,均方伪概守随机过程和均方渐近概自守随机过程的概念也相继被给出[ 12-13 ] 。自均方概自守型随机过程有关理论被提出以来,国内外数学工作者将其应用到一类将随机性纳入了数学描述中的模型中即随机微分方程中,研究了此种方程的均方概自守解[14-16]和均方伪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.对下列一类半线性随机微分方程
1预备知识
2主要结论
参 考 文 献:
[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]张著洪.关于闭算子及其共轭的分数次幂的评注[J].贵州大学学报(自然科学版),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, 刘婷, 张士晶. 一类随机微分方程的均方渐近概自守温和解[J]. 哈尔滨理工大学学报, 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.
(编辑:温泽宇)