一类具有泊松跳的脉冲中立型随机泛函微分方程的存在性及稳定性研究

Abstract

Abstract:

In this paper, we consider the existence, uniqueness and exponential stability of mild solutions for a class of impulsive neutral stochastic functional differential equations with Poisson jumps.By using successive approximation and Picard iteration method, the existence of mild solutions in Hilbert space is proved. Secondly, the sufficient conditions for the mean square exponential stability and almost certain exponential stability are given by Banach's fixed point principle.

Key words: Existence and uniqueness, Netral stochastic functional differential equations, Mild solution, Poisson jumps, Exponential stability

CLC Number:

O211.4

He Xuyang,Mao Mingzhi,Zhang Tengfei. Existence and Stability of a Class of Impulsive Neutral Stochastic Functional Differential Equations with Poisson Jump[J].Acta mathematica scientia,Series A, 2023, 43(4): 1221-1243.

Trendmd

References 31

[1]	Hale J K, Lunel S M V. Introduction to Functional-Differential Equations. Applied Mathematical Sciences, New York: Springer-Verlag, 1993
[2]	Liu K. The fundamental solution and its role in the optimal control of infinite dimensional neutral systems. Appl Math Optim, 2009, 60(1): 1-38 doi: 10.1007/s00245-009-9065-1
[3]	Wu S J, Meng X Z. Boundedness of nonlinear differential systems with impulsive effect on random moments. Acta Math Appl Sin Engl Ser, 2004, 20(1): 147-154 doi: 10.1007/s10255-004-0157-z
[4]	Wu F K, Mao X R, Szpruch L. Almost sure exponential stability of numerical solutions for stochastic delay differential equations. Numer Math, 2010, 115(4): 681-697 doi: 10.1007/s00211-010-0294-7
[5]	沈轶, 张玉民, 廖晓昕. 中立型随机泛函微分方程的稳定性. 数学物理学报, 2005, 25(3): 323-330
	Shen Y, Zhang Y M, Liao X X. Stability for neutral stochastic functional differential equations. Acta Math Sci, 2005, 25(3): 323-330
[6]	崔静, 梁秋菊, 毕娜娜. 分数布朗运动驱动的带脉冲的中立性随机泛函微分方程的渐近稳定. 数学物理学报, 2019, 39(3): 570-581
	Cui J, Liang Q J, Bi N N. Asymptotic stability of impulsive neutral stochastic functional differential equation driven by fractional Brownian motion. Acta Math Sci, 2019, 39(3): 570-581
[7]	Song Y, Zeng Z. Razumikhin-type theorems on th moment boundedness of neutral stochastic functional differential equations with Makovian switching. J Franklin Inst B, 2018, 355(5): 8296-8312 doi: 10.1016/j.jfranklin.2018.09.019
[8]	Chang M. On Razumikhin-type stability conditions for stochastic functional differential equations. Math Modelling, 1984, 5(5): 299-307 doi: 10.1016/0270-0255(84)90007-1
[9]	沈轶, 廖晓昕. 随机中立型泛函微分方程指数稳定的Razumikhin型定理. 科学通报, 1998, 21: 2272-2275
	Shen Y, Liao X X. The Razumikhin type theorem for exponential stability of stochastic neutral type functional differential equations. Chinese Science Bulletin, 1998, 21: 2272-2275
[10]	Ren Y, Xia N. Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay. Appl Math Comput, 2008, 210(1): 72-79
[11]	Yang Z, Xu D. Stability analysis and design of impulsive control systems with time delay. IEEE Trans Automat Control, 2007, 52(8): 48-54
[12]	Guo Y, Zhu Q, Wang F. Stability analysis of impulsive stochastic functional differential equations. Commun Nonlinear Sci Numer Simul, 2020, 82(C): 5-13
[13]	Zhao X. Mean square Hyers-Ulam stability of stochastic differential equations driven by Brownian motion. Adv Difference Equ, 2016, 2016(1): 271 doi: 10.1186/s13662-016-1002-4
[14]	Li S, Shu L X, Shu X B, Xu F. Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays. Stochastic, 2019, 91(6): 857-872 doi: 10.1080/17442508.2018.1551400
[15]	Ferhat M, Blouhi T. Existence and uniqueness results for systems of impulsive functional stochastic differential equations driven by fractional Brownian motion with multiple delay. Topol Methods Nonlinear Anal, 2018, 52: 449-476
[16]	Wu F, Yin G, Mei H. Stochastic functional differential equations with infinite delay: existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity. J Diff Equ, 2017, 62(3): 1226-1252
[17]	Banupriya K, Anguraj A. Successive approximation of neutral functional impulsive stochastic differential equations with Poisson jumps. Dyn Contin Discrete Impuls Syst Ser B Appl and Algorithms, 2021, 28: 215-229
[18]	Boufoussi B, Hajji S. Successive approximation of neutral functional stochastic differential equations in Hilbert spaces. Ann Math Blaise Pascal, 2010, 17(1): 183-197 doi: 10.5802/ambp.282
[19]	Chen G, van Gaans O, Lunel S V. Existence and exponential stability of a class of impulsive neutral stochastic partial differential equations with delays and Poisson jumps. Statist Probab Lett, 2018, 141: 7-18 doi: 10.1016/j.spl.2018.05.017
[20]	Deng S F, Shu X B, Mao J Z. Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Mönch fixed point. J Math Anal Appl, 2018, 467(1): 398-420 doi: 10.1016/j.jmaa.2018.07.002
[21]	Guo Y C, Chen M Q, Shu X B, Xu F. The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm. Stoch Anal Appl, 2021, 39(4): 643-666 doi: 10.1080/07362994.2020.1824677
[22]	Faizullah F, Zhu Q, Ullah R. The existence-uniqueness and exponential estimate of solutions for stochastic functional differential equations driven by G-Brownian motion. Math Methods Appl Sci, 2021, 44: 1639-1650 doi: 10.1002/mma.v44.2
[23]	Xiao G, Wang J, O'Regan D. Existence and stability of solutions to neutral conformable stochastic functional differential equations. Qual Theory Dyn Syst, 2022, 21(1): 7-22 doi: 10.1007/s12346-021-00538-x
[24]	Tan J G, Tan Y H, Guo Y F, Feng J F. Almost sure exponential stability of numerical solutions for stochastic delay Hopfield neural networks with jumps. Phys A: Statist Mech Appl, 2020, 545: 37-82
[25]	Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl Math Sci vol. New York: Springer Verlag, 1983
[26]	Da Prato G, Zabczyk J. Stochastic Equations in Infinite Dimensions. Cambridge: Cambridge University Press, 1992, 45
[27]	Luo Q, Mao X, Shen Y. New criteria on exponential stability of neutral stochastic differential delay equations. Syst Control Lett, 2006, 55(10): 826-834 doi: 10.1016/j.sysconle.2006.04.005
[28]	Li R, Pang W, Leung P. Exponential stability of numerical solutions to stochastic delay Hopfield neural networks. Appl Math Comput, 2009, 73(4): 920-926
[29]	Rathinasamy A. The split-step $\theta $-methods for stochastic delay Hopfield neural networks. Appl Math Model, 2021, 36: 3477-3485 doi: 10.1016/j.apm.2011.10.020
[30]	Jiang F, Shen Y. Stability in the numerical simulation of stochastic delayed Hopfield neural networks. Neural Comput Appl, 2013, 22(7/8): 1493-1498 doi: 10.1007/s00521-012-0935-0
[31]	Liu L, Zhu Q. Almost sure exponential of numerical solutions to stochastic delay Hopfield neural networks. Appl Math Comput, 2015, 266: 698-712

Existence and Stability of a Class of Impulsive Neutral Stochastic Functional Differential Equations with Poisson Jump

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 31

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Li Yongxiang,Wei Qilin. Existence Results of Periodic Solutions for Semilinear Evolution Equation in Banach Spaces and Applications [J]. Acta mathematica scientia,Series A, 2023, 43(3): 702-712.
[2]	Song Yuying, Fan Hongxia. Existence and Regularity of Solutions for a Class of Neutral Stochastic Evolution Equations [J]. Acta mathematica scientia,Series A, 2023, 43(1): 238-248.
[3]	Wenjie Liu,Shengli Xie. Existence Results of Mild Solutions for Impulsive Neutral Measure Differential Equations with Infinite Delay [J]. Acta mathematica scientia,Series A, 2022, 42(6): 1671-1681.
[4]	Qiang Li,Lishan Liu. Existence of Periodic Mild Solutions for Fractional Evolution Equations with Periodic Impulses [J]. Acta mathematica scientia,Series A, 2022, 42(5): 1433-1450.
[5]	Shaowen Yao,Wenjie Li,Zhibo Cheng. Nondegeneracy and Uniqueness of Periodic Solution for Third-Order Nonlinear Differential Equations [J]. Acta mathematica scientia,Series A, 2022, 42(2): 454-462.
[6]	Dongxia Fan,Dongxia Zhao,Na Shi,Tingting Wang. The PDP Feedback Control and Stability Analysis of a Diffusive Wave Equation [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1088-1096.
[7]	Xingshou Huang,Ricai Luo,Wusheng Wang. Stability Analysis for a Class Neural Network with Proportional Delay Based on the Gronwall Integral Inequality [J]. Acta mathematica scientia,Series A, 2020, 40(3): 824-832.
[8]	Haide Gou. Existence of L-Quasi Mild Solutions for Damped Elastic Systems in Banach Spaces [J]. Acta mathematica scientia,Series A, 2020, 40(2): 408-421.
[9]	Chao Yang,Runjie Li. Existence and Stability of Periodic Solution for a Lasota-Wazewska Model with Discontinuous Harvesting [J]. Acta mathematica scientia,Series A, 2019, 39(4): 785-796.
[10]	Yongzhen Yun,Youhui Su,Weimin Hu. Existence and Uniqueness of Solutions to a Class of Anti-Periodic Boundary Value Problem of Fractional Differential Equations with p-Laplacian Operator [J]. Acta mathematica scientia,Series A, 2018, 38(6): 1162-1172.
[11]	Xiong Jun, Li Junmin, He Chao. Fuzzy Boundary Control Design for a Class of First-Order Hyperbolic PDEs [J]. Acta mathematica scientia,Series A, 2017, 37(3): 469-477.
[12]	Zhang Liping, Liu Dongyi, Zhang Guoshan. Exponential Stabilization of a Timoshenko Beam System with Internal Disturbances [J]. Acta mathematica scientia,Series A, 2017, 37(1): 185-198.
[13]	Deng Jiqin, Deng Ziming. Existence and Uniqueness of Solutions for Nonlocal Cauchy Problem for Fractional Evolution Equations [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1157-1164.
[14]	Huang Zuda. Positive Pseudo Almost Periodic Solutions for a Delayed Nicholson's Blowflies Model with a Feedback Control [J]. Acta mathematica scientia,Series A, 2016, 36(3): 558-568.
[15]	Jiang Ani. Pseudo Almost Periodic Solutions for a Model of Hematopoiesis with an Oscillating Circulation Loss Rate [J]. Acta mathematica scientia,Series A, 2016, 36(1): 80-89.