This paper discusses differential mean value theorems based on uncertainty differential equation. Firstly, we propose some uncertain differential mean value theorems driven by canonical process, in which finite interval and infinite interval cases are considered respectively. Then the stability for uncertain differential mean value theorem is discussed under the Lipschitz conditions. Finally, some examples are given.
WienerN., Differential space, Journal of Mathematical Physics2 (1923), 131–174.
2.
ItoK., Stochastic integral, In: Proceedings of the Japan Academy (1944), 519–524.
3.
KalmanR.E. and BucyR.S., New results in linear filtering and prediction theory, Journal of Basic Engineering83 (1961), 95–108.
4.
BlackF. and ScholesM., The pricing of options and corporate liabilities, Journal Polititical Economy81 (1973), 637–654.
5.
KalevaO., Fuzzy differential equations, Fuzzy Sets and Systems24 (1987), 301–317.
6.
DiamondP., Stability and periodicity in fuzzy differential equations, IEEE Transactions on Fuzzy Systems8 (2000), 583–590.
7.
BedeB. and GalS.G., Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems151 (2005), 581–599.
8.
YangL. and LiuB., On continuity theorem for characteristic function of fuzzy variable, Journal of Intelligent and Fuzzy Systems17 (2006), 325–332.
9.
BedeB., RudasI.J. and BencsikA.L., First order linear fuzzy differential equations under generalized differentiability, Information Sciences17 (2007), 1648–1662.
10.
Chalco-CanoY. and Roman-FloresH., On new solutions of fuzzy differential equations, Chaos Solitons and Fractals38 (2008), 112–119.
11.
ShenY. and WangF., A fixed point approach to the Ulam stability of fuzzy differential equations under generalized differentiability, Journal of Intelligent and Fuzzy Systems30 (2016), 3253–3260.
12.
Abu ArqubO.,
AL-SmadiM.,
MomaniS. and HayatT., Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method, Softe Computing20 (2016), 3283–3302.
13.
ShenY., ChenW. and WangJ., Fuzzy Laplace transform method for the Ulam stability of linear fuzzy differential equations of first order with constant coefficients, Journal of Intelligent and Fuzzy Systems32 (2017), 671–680.
YaoK., GaoJ. and GaoY., Some stability theorems of uncer- tain differential equation, Fuzzy Optimization and Decision Making12 (2013), 3–13.
22.
YaoK., KeH. and ShengY., Stability in mean for uncertain differential equation, Fuzzy Optimization and Decision Making14 (2015), 365–379.
23.
ShengY. and GaoJ., Exponential stability of uncertain differential equation, Soft Computing20 (2016), 3673–3678.
24.
LiuH., KeH. and FeiW., Almost sure stability for uncer- tain differential equation, Fuzzy Optimization and Decision Making13 (2014), 463–473.
25.
YangX., NiY. and ZhangY., Stability in inverse distribution for uncertain differential equations, Journal of Intelligent and Fuzzy Systems32 (2017), 2051–2059.
26.
MaW., LiuL. and ZhangX., Stability in-th moment for uncertain differential equation with jumps, Journal of Intelligent and Fuzzy Systems33 (2017), 1375–1384.
27.
TuckerT.W., Rethinking rigor in calculus: The role of the Mean Value Theorem, American Mathematical Monthly104 (1997), 231–240.
