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July  2019, 15(3): 1375-1385. doi: 10.3934/jimo.2018099

## Stability in mean for fuzzy differential equation

 College of Mathematics and Information Science, Hebei University, Baoding 071002, China

* Corresponding author: Cuilian You

Received  September 2017 Revised  March 2018 Published  July 2018

Fund Project: The first author is supported by NSFC grant (No.61773150) and Key Lab. of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, 071002, China.

Fuzzy differential equation driven by Liu process is an important tool to deal with dynamic system in fuzzy environment. Stability for a fuzzy differential equation plays a key role in differential equation, which means influence of the state of a system to small changes in the initial state. In order to discuss the influence of different initial value on the solution, this paper proposes a concept of stability in mean for fuzzy differential equation driven by Liu process. Some stability theorems for fuzzy differential equation being stable in mean are given. In addition, the concept of stability in mean for fuzzy differential equation driven by Liu process is extended to the case of multi-dimensional. A sufficient condition for multi-dimensional fuzzy differential equation being stable in mean is also provided in this paper.

Citation: Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099
##### References:
 [1] X. Chen, A new existence and uniqueness theorem for fuzzy differential equation, International Journal of Fuzzy Systems, 13 (2011), 148-151.   Google Scholar [2] W. Dai, Lipschitz continuity of Liu process, 2008. Available from: http://orsc.edu.cn/process/080831.pdf. Google Scholar [3] W. Dai, Reflection principle of Liu process, 2007. Available from: http://orsc.edu.cn/process/071110.pdf. Google Scholar [4] J. Gao and X. Gao, A new stock model for credibilistic option pricing, Journal of Uncertain Systems, 4 (2008), 243-247.   Google Scholar [5] V. H. Le, A note on the asymptotic stability of fuzzy differential equations, Ukrainian Mathematical Journal, 57 (2005), 1066-1076.  doi: 10.1007/s11253-005-0248-x.  Google Scholar [6] B. Liu, Uncertainty Theory, Springer-Verlag, Berlin, 2004. Google Scholar [7] B. Liu, Uncertainty Theory 2$^{nd}$ edition, Springer-Verlag, Berlin, 2007. Google Scholar [8] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16.   Google Scholar [9] B. Liu and Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, 10 (2002), 445-450.   Google Scholar [10] M. Mizukoshi, Stability of fuzzy dynamic systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 17 (2009), 69-83.  doi: 10.1142/S0218488509005747.  Google Scholar [11] Z. Qin and M. Wen, On analytic functions of complex Liu process, Journal of Intelligent and Fuzzy Systems, 28 (2015), 1627-1633.   Google Scholar [12] Z. Qin, M. Bai and R. Dan, A fuzzy control system with application to production planning problems, Information Sciences, 181 (2011), 1018-1027.  doi: 10.1016/j.ins.2010.10.029.  Google Scholar [13] Z. Qin and X. Li, Option pricing formula for fuzzy financial market, Journal of Uncertain Systems, 1 (2008), 17-21.   Google Scholar [14] H. Tian and J. Guo, Stability of fuzzy differential equations, Journal of Taiyuan Normal University(Natural Science Edition), 11 (2012), 7-9.   Google Scholar [15] C. You, H. Huo and W. Wang, Multi-dimensional Liu process, differential and integral, East Asian Mathematical Journal, 29 (2013), 13-22.  doi: 10.7858/eamj.2013.002.  Google Scholar [16] C. You and G. Wang, Properties of a new kind of fuzzy integral, Journal of Hebei University(Natural Science Edition), 31 (2011), 337-340.   Google Scholar [17] C. You, H. Ma and H. Huo, A new kind of generalized fuzzy integrals, Journal of Nonlinear Science and Applications, 3 (2016), 1396-1401.  doi: 10.22436/jnsa.009.03.63.  Google Scholar [18] C. You, W. Wang and H. Huo, Existence and uniqueness theorems for fuzzy differential equations, Journal of Uncertain Systems, 7 (2013), 303-315.   Google Scholar [19] C. You and W. Wang, Some properties of complex fuzzy integral, Mathematical Problems in Engineering, 2015 (2015), Art. ID 290539, 7 pp. doi: 10.1155/2015/290539.  Google Scholar [20] Y. Zhu, A fuzzy optimal control model, Journal of Uncertain Systems, 3 (2009), 270-279.   Google Scholar [21] Y. Zhu, Stability analysis of fuzzy linear differential equations, Fuzzy Optimization and Decision Making, 9 (2010), 169-186.  doi: 10.1007/s10700-010-9080-3.  Google Scholar

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##### References:
 [1] X. Chen, A new existence and uniqueness theorem for fuzzy differential equation, International Journal of Fuzzy Systems, 13 (2011), 148-151.   Google Scholar [2] W. Dai, Lipschitz continuity of Liu process, 2008. Available from: http://orsc.edu.cn/process/080831.pdf. Google Scholar [3] W. Dai, Reflection principle of Liu process, 2007. Available from: http://orsc.edu.cn/process/071110.pdf. Google Scholar [4] J. Gao and X. Gao, A new stock model for credibilistic option pricing, Journal of Uncertain Systems, 4 (2008), 243-247.   Google Scholar [5] V. H. Le, A note on the asymptotic stability of fuzzy differential equations, Ukrainian Mathematical Journal, 57 (2005), 1066-1076.  doi: 10.1007/s11253-005-0248-x.  Google Scholar [6] B. Liu, Uncertainty Theory, Springer-Verlag, Berlin, 2004. Google Scholar [7] B. Liu, Uncertainty Theory 2$^{nd}$ edition, Springer-Verlag, Berlin, 2007. Google Scholar [8] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16.   Google Scholar [9] B. Liu and Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, 10 (2002), 445-450.   Google Scholar [10] M. Mizukoshi, Stability of fuzzy dynamic systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 17 (2009), 69-83.  doi: 10.1142/S0218488509005747.  Google Scholar [11] Z. Qin and M. Wen, On analytic functions of complex Liu process, Journal of Intelligent and Fuzzy Systems, 28 (2015), 1627-1633.   Google Scholar [12] Z. Qin, M. Bai and R. Dan, A fuzzy control system with application to production planning problems, Information Sciences, 181 (2011), 1018-1027.  doi: 10.1016/j.ins.2010.10.029.  Google Scholar [13] Z. Qin and X. Li, Option pricing formula for fuzzy financial market, Journal of Uncertain Systems, 1 (2008), 17-21.   Google Scholar [14] H. Tian and J. Guo, Stability of fuzzy differential equations, Journal of Taiyuan Normal University(Natural Science Edition), 11 (2012), 7-9.   Google Scholar [15] C. You, H. Huo and W. Wang, Multi-dimensional Liu process, differential and integral, East Asian Mathematical Journal, 29 (2013), 13-22.  doi: 10.7858/eamj.2013.002.  Google Scholar [16] C. You and G. Wang, Properties of a new kind of fuzzy integral, Journal of Hebei University(Natural Science Edition), 31 (2011), 337-340.   Google Scholar [17] C. You, H. Ma and H. Huo, A new kind of generalized fuzzy integrals, Journal of Nonlinear Science and Applications, 3 (2016), 1396-1401.  doi: 10.22436/jnsa.009.03.63.  Google Scholar [18] C. You, W. Wang and H. Huo, Existence and uniqueness theorems for fuzzy differential equations, Journal of Uncertain Systems, 7 (2013), 303-315.   Google Scholar [19] C. You and W. Wang, Some properties of complex fuzzy integral, Mathematical Problems in Engineering, 2015 (2015), Art. ID 290539, 7 pp. doi: 10.1155/2015/290539.  Google Scholar [20] Y. Zhu, A fuzzy optimal control model, Journal of Uncertain Systems, 3 (2009), 270-279.   Google Scholar [21] Y. Zhu, Stability analysis of fuzzy linear differential equations, Fuzzy Optimization and Decision Making, 9 (2010), 169-186.  doi: 10.1007/s10700-010-9080-3.  Google Scholar
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