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Stability in distribution for uncertain delay differential equation. (English) Zbl 1428.34080
Summary: As a type of differential equation, uncertain delay differential equation is driven by Liu process. Stability in measure, stability in mean and stability in moment for uncertain delay differential equation have been proposed. This paper mainly gives a concept of stability in distribution, and proves a sufficient condition for uncertain delay differential equation being stable in distribution as a supplement. Moreover, this paper further discusses their relationships among stability in distribution, stability in measure, stability in mean and stability in moment.

##### MSC:
 34F05 Ordinary differential equations and systems with randomness 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34A07 Fuzzy ordinary differential equations 34D20 Stability of solutions to ordinary differential equations 28E10 Fuzzy measure theory
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##### References:
 [1] Barbacioru, C., Uncertainty functional differential equations for finance, Surv. Math. Appl., 5, 275-284 (2010) · Zbl 1399.34242 [2] Chen, X.; Liu, B., Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim. Decis. Making, 9, 1, 69-81 (2010) · Zbl 1196.34005 [3] Gao, R., Uncertain wave equation with infinite half-boundary, Appl. Math. Comput., 304, 28-40 (2017) · Zbl 1411.35190 [4] Ge, X.; Zhu, Y., Existence and uniqueness theorem for uncertain delay differential equations, J. Comput. Inf. Syst., 8, 20, 8341-8347 (2012) [6] Jia, L.; Yang, X., Existence and uniqueness theorem for uncertain spring vibration equation, J. Intell. Fuzzy Syst., 35, 2, 2607-2617 (2018) [7] Jia, L.; Lio, W.; Yang, X., Numerical method for solving uncertain spring vibration equation, Appl. Math. Comp., 377, 428-441 (2018) · Zbl 1427.34002 [8] Liu, B., Uncertainty Theory (2007), Springer-Verlag: Springer-Verlag Berlin [9] Liu, B., Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2, 1, 3-16 (2008) [10] Liu, B., Some research problems in uncertainty theory, J. Uncertain Syst., 3, 1, 3-10 (2009) [11] Liu, B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty (2010), Springer-Verlag: Springer-Verlag Berlin [12] Liu, B., Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013) [13] Liu, H.; Ke, H.; Fei, W., Almost sure stability for uncertain differential equation, Fuzzy Optim. Decis. Making, 13, 4, 463-473 (2014) · Zbl 1449.60118 [14] Sheng, Y.; Wang, C., Stability in the pth moment for uncertain differential equation, J. Intell. Fuzzy Syst., 26, 3, 1263-1271 (2014) · Zbl 1305.34086 [15] Sheng, Y.; Gao, J., Exponential stability of uncertain differential equation, Soft Comput., 20, 9, 3673-3678 (2016) · Zbl 06773270 [16] Wang, X.; Ning, Y., Stability of uncertain delay differential equations, J. Intell. Fuzzy Syst., 32, 2655-2664 (2017) · Zbl 1381.34008 [17] Yang, X.; Gao, J., Uncertain differential games with application to capitalism, J. Uncertain. Anal. Appl., 1 (2013) [18] Yang, X.; Gao, J., Linear-quadratic uncertain differential games with application to resource extraction problem, IEEE Trans. Fuzzy Syst., 24, 4, 819-826 (2016) [19] Yang, X.; Yao, K., Uncertain partial differential equation with application to heat conduction, Fuzzy Optim. Decis. Making, 16, 3, 379-403 (2017) · Zbl 1428.60092 [20] Yang, X.; Ni, Y., Existence and uniqueness theorem for uncertain heat equation, J. Ambient Intell. Humaniz. Comput., 8, 5, 717-725 (2017) [21] Yang, X.; Ni, Y.; Zhang, Y., Stability in inverse distribution for uncertain differential equations, J. Intell. Fuzzy Syst., 32, 3, 2051-2059 (2017) · Zbl 1385.34006 [22] Yang, X., Solving uncertain heat equation via numerical method, Appl. Math. Computat., 329, 92-104 (2018) · Zbl 1427.80031 [23] Yang, X.; Ni, Y., Extreme values problem of uncertain heat equation, J. Ind. Manag. Optim. (2018) [24] Yao, K., Uncertainty Differential Equation (2016), Springer-Verlag: Springer-Verlag Berlin [25] Yao, K.; Chen, X., A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Syst., 25, 3, 825-832 (2013) · Zbl 1291.65025 [26] Yao, K.; Gao, J.; Gao, Y., Some stability theorems of uncertain differential equation, Fuzzy Optim. Decis. Making, 12, 1, 3-13 (2013) · Zbl 1412.60104 [27] Yao, K.; Ke, H.; Sheng, Y., Stability in mean for uncertain differential equation, Fuzzy Optim. Decis. Making, 14, 3, 365-379 (2015) · Zbl 1463.93250 [28] Zhu, Y., Uncertain optimal control with application to a portfolio selection model, Cybern. Syst., 41, 7, 535-547 (2010) · Zbl 1225.93121
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