Uncertain fractional differential equation driven by Liu process plays an important role in describing uncertain dynamic systems. This paper investigates the continuous dependence of solution on the parameters and initial values, respectively, for uncertain fractional differential equations involving the Caputo fractional derivative in measure sense. Several continuous dependence theorems are obtained based on uncertainty theory by employing the generalized Gronwall inequality, in which the coefficients of uncertain fractional differential equation are required to satisfy the Lipschitz conditions. Several illustrative examples are provided to verify the validity of the obtained results.
LiuB., Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems2(1) (2008), 3–16.
4.
LiuB., Theory and practice of uncertain programming, 2nd edition, Springer-VerlagBerlin, (2009).
5.
ChenX. and LiuB., Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making9(1) (2010), 69–81.
6.
TaoN. and ZhuY., Attractivity and stability analysis of uncertain differential systems, International Journal of Bifurcation and Chaos25(2) (2015), 1550022 1-1550022 10.
7.
WangX., NingY., MoughalT.A. and ChenX., Adams-Simpson method for solving uncertain differential equation, Applied Mathematics and Computation271 (2015), 209–219.
8.
YaoK. and ChenX., A numerical method for solving uncertain differential equations, Journal of Intelligent and Fuzzy Systems25 (2013), 825–832.
9.
YaoK., GaoJ. and GaoY., Some stability theorems of uncertain differential equation, Fuzzy Optimization and Decision Making12(1) (2013), 3–13.
10.
GaoY. and YaoK., Continuous dependence theorems on solutions of uncertain differential equations, Applied Mathematical Modelling38(11-12) (2014), 3031–3037.
11.
ZhangJ., LiD. and ChenW., Continuous dependence on solutions of uncertain differential equations via uncertain measure, Journal of Intelligent and Fuzzy Systems36(6) (2019), 6455–6465.
12.
ChenX., LiuY. and RalescuD., Uncertain stock model with periodic dividends, Fuzzy Optimization and Decision Making12(1) (2013), 111–123.
13.
LiB., ZhuY., SunY., GraceA. and TeoK., Multi-period portfolio selection problem under uncertain environment with bankruptcy constraint, Applied Mathematical Modeling56 (2018), 539–550.
14.
ZhuY., Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems41(7) (2010), 535–547.
KilbasA., SrivastavaH., TrujilloJ., Theory and application of fractional differential equation, Elsevier Science B.V, Amsterdam, (2006).
17.
PodlubnyI., Fractional differential equation, Academic Press, San Diego, (1999).
18.
ValérioD., TenreiroJ. and KiryakovaM., Some pioneers of the applications of fractional calculus, Fractional Calculus and Applied Analysis17(2) (2014), 552–578.
19.
DiethelmK., GarrappaR., GiustiA. and StynesM., Why fractional derivatives with nonsingular kernels should not be used, Fractional Calculus and Applied Analysis23(3) (2020), 610–634.
20.
YeH., GaoJ. and DingY., A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications328 (2007), 1075–1081.
21.
AgarwalR., LakshmikanthamV. and NietoJ., On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis - Theory Methods and Applications72 (2010), 2859–2862.
22.
AllahviranlooT., GouyandehZ. and ArmandA., Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal of Intelligent and Fuzzy Systems26 (2014), 1481–1490.
23.
AllahviranlooT., SalahshourS. and AbbasbandyS., Explicit solutions of fractional differential equations with uncertainty, Soft Computing16 (2012), 297–302.
24.
SalahshouraS., AllahviranlooT. and AbbasbandyS., Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Communications in Nonlinear Science and Numerical Simulation17(3) (2012), 1372–1381.
25.
AhmadianA., SalahshourS. and ChanC., Fractional differential systems: a fuzzy solution based on operational matrix of shifted Chebyshev polynomials and its applications, IEEE Transactions on Fuzzy Systems25(1) (2017), 218–236.
26.
LupulescuV., Fractional calculus for interval-valued functions, Fuzzy Sets and Systems265 (2015), 63–85.
27.
NgoV., Fuzzy fractional functional integral and differential equations, Fuzzy Sets and Systems280 (2015), 58–90.
28.
NajariyanM. and ZhaoY., Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives, IEEE Transactions on Fuzzy Systems26(4) (2018), 2273–2288.
29.
SakthivelR., RevathiP. and RenY., Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Analysis-Theory Methods and Applications81 (2013), 70–86.
30.
ChadhaA., PandeyD.N. and BahugunaD., Faedo-Galerkin approximate solutions of a neutral stochastic fractional differential equation with finite delay, Journal of Computational and Applied Mathematics347 (2019), 238–256.
31.
XuW., XuW. and LuK., An averaging principle for stochastic differential equations of fractional order 0 < α <1, Fractional Calculus and Applied Analysis23(3) (2020), 908–919.
32.
ZhuY., Uncertain fractional differential equations and an interest rate model, Mathematical Methods in the Applied Sciences38(15) (2015), 3359–3368.
33.
ZhuY., Exitence and uniquence of the solution to uncertain fractional differential equation, Journal of Uncertainty Analysis and Applications3(5) (2015), 1–11.
34.
LuZ., YanH. and ZhuY., European option pricing model based on uncertain fractional differential equation, Fuzzy Optimization and Decisision Making18(2) (2019), 199–217.
35.
LuZ. and ZhuY., Numerical approach for solution to an uncertain fractional differential equation, Applied Mathematics and Computation343 (2019), 137–148.
36.
LuQ. and ZhuY., Finite-time stability of uncertain fractional difference equations, Fuzzy Optimization and Decisision Making19(2) (2020), 239–249.
37.
LuQ. and ZhuY., Comparison theorems and distributions of solutions to uncertain fractional difference equations,Article 84, Journal of Computational and Applied Mathematics376 (1128), 1–13.
38.
WangJ., ZhuY., Solution of linear uncertain fractional order delay differential equations, Soft Computing doi: 10.1007/s00500-020-05037-w, 2020
39.
LuZ., ZhuY. and LiB., Critical value-based Asian option pricing model for uncertain financial markets, Physica A: Statistical Mechanics and its Applications525 (2019), 694–703.
40.
JinT. and ZhuY., First hitting time about solution for an uncertain fractional differential equation and application to an uncertain risk index model, Chaos, Solitons and Fractals137 (2020), Article 109836, 1–7.
