In this work, we consider a new form of fuzzy fractional Volterra integral equations (FFVIEs) involving the generalized kernel functions. By using the monotone iterative technique (MIT) combined with the method of lower and upper solutions, the existence of extremal solutions of FFVIEs is established. Some examples are given to illustrate our main results.
AgarwalR.P., LakshmikanthamV. and NietoJ.J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis, Theory, Methods & Applications72 (2010), 2859–2862.
2.
AgarwalR.P., BaleanuD., NietoJ.J., TorresD.F. and ZhouY., A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, Journal of Computational and Applied Mathematics339 (2018), 3–29.
3.
AhmadianA., SalahshourS. and ChanC.S., Fractional differential systems: A fuzzy solution based on operational matrix of shifted Chebyshev polynomials and its applications, IEEE Transactions on Fuzzy Systems25 (2017), 218–236.
4.
AhmadianA., IsmailF., SalahshourS., BaleanuD. and GhaemiF., Uncertain viscoelastic models with fractional order: A new spectral tau method to study the numerical simulations of the solution, Communications in Nonlinear Science and Numerical Simulation53 (2017), 44–64.
5.
AlikhaniR. and BahramiF., Global solutions for nonlinear fuzzy fractional integral and integro-differential equations, Commun. Nonlinear Sci. Numer. Simulat.18 (2013), 2007–2017.
6.
AllahviranlooT., SalahshourS. and AbbasbandyS., Explicit solutions of fractional differential equations with uncertainty, Soft Computing16 (2012), 297–302.
7.
AllahviranlooT., GouyandehZ. and ArmandA., Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal of Intelligent & Fuzzy Systems26 (2014), 1481–1490.
8.
AnT.V., VuH. and HoaN.V., Applications of contractive-like mapping principles to interval-valued fractional integro-differential equations, Journal of Fixed Point Theory and Applications19 (2017), 2577–99.
9.
AnT.V., VuH. and HoaN.V., A new technique to solve the initial value problems for fractional fuzzy delay differential equations, Advances in Difference Equations2017 (2017), 181.
10.
ArshadS. and LupulescuV., On the fractional differential equations with uncertainty, Nonlinear Analysis, Theory, Methods & Applications7 (2011), 85–93.
11.
BhaskarT.G., LakshmikanthamV. and DeviJ.V., Monotone iterative technique for functional differential equations with retardation and anticipation, Nonlinear Analysis (TMA)66 (2007), 2237–2242.
12.
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.
13.
BedeB. and StefaniniL., Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems230 (2013), 119–141.
14.
FardO.S. and SalehiM., A survey on fuzzy fractional variational problems, Journal of Computational and Applied Mathematics271 (2014), 71–82.
15.
HoaN.V., Fuzzy fractional functional integral and differential equations, Fuzzy Sets and Systems280 (2015), 58–90.
16.
HoaN.V., Fuzzy fractional functional differential equations under Caputo gH-differentiability, Communications in Nonlinear Science and Numerical Simulation22 (2015), 1134–1157.
17.
HoaN.V., LupulescuV. and O’ReganD., Solving interval-valued fractional initial value problem under Caputo gH-fractional differentiability, Fuzzy Sets and Systems309 (2017), 1–34.
18.
HoaN.V., Existence results for extremal solutions of interval fractional functional integro-differential equations, Fuzzy Sets and Systems347 (2018), 29–53.
19.
HoaN.V., VuH. and DucT.M., Fuzzy fractional differential equations under Caputo-Katugampola fractional derivative approach, Fuzzy Sets and Systems375 (2019), 70–99.
20.
KhastanA., NietoJ.J. and Rodríguez-LópezR., Schauder fixed-point theorem in semilinear spaces and its application to fractional differential equations with uncertainty, Fixed Point Theory and Applications2014 (2014), 21.
21.
LaddeG.S., LakshmikanthamV. and VatsalaA.S., Monotone Iterative Technique for Nonlinear Differential Equations, Pittman Publishing Inc., London, 1985.
22.
LongH.V., SonN.T.K. and HoaN.V., Fuzzy fractional partial differential equations in partially ordered metric spaces, Iranian Journal of Fuzzy Systems14 (2017), 107–126.
23.
LongH.V., NietoJ.J. and SonN.T.K., New approach for studying nonlocal problems related to differential systems and partial differential equations in generalized fuzzy metric spaces, Fuzzy Sets and Systems331 (2018), 26–46.
24.
LongH.V. and ThaoH.T.P., Hyers-Ulam stability for nonlocal fractional partial integro-differential equation with uncertainty, Journal of Intelligent & Fuzzy Systems34 (2018), 233–244.
25.
LongH.V. and DongN.P., An extension of Krasnoselskii’s fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty, Journal of Fixed Point Theory and Applications20 (2018), 37.
26.
LongH.V., NietoJ.J. and SonN.T.K., New approach for studying nonlocal problems related to differential systems and partial differential equations in generalized fuzzy metric spaces, Fuzzy Sets and Systems331 (2018), 26–46.
27.
LongH.V., SonN.T.K. and ThaoH.T.P., System of fuzzy fractional differential equations in generalized metric space in the sense of Perov, Iranian Journal of Fuzzy Systems16(2) (2019).
28.
LupulescuV., Fractional calculus for interval-valued functions, Fuzzy Sets and Systems265 (2015), 63–85.
29.
MalinowskiM.T., Random fuzzy fractional integral equations-theoretical foundations, Fuzzy Sets and Systems265 (2015), 39–62.
30.
MazandaraniM. and KamyadA.V., Modified fractional Euler method for solving fuzzy fractional initial value problem, Communications in Nonlinear Science and Numerical Simulation18 (2013), 12–21.
31.
MazandaraniM. and NajariyanM., Type-2 fuzzy fractional derivatives, Communications in Nonlinear Science and Numerical Simulation19 (2014), 2354–2372.
QuangL.T.Q., HoaN.V., PhuN.D. and TungT.T., Existence of extremal solutions for interval-valued functional integro-differential equations, Journal of Intelligent & Fuzzy Systems30 (2016), 3495–3512.
SalahshourS., AllahviranlooT. and AbbasbandyS., Solving fuzzy fractional differentialequations by fuzzy Laplace transforms, Commun. Nonlinear Sci. Numer. Simul.17 (2012), 1372–1381.
36.
SalahshourS., AhmadianA., SenuN., BaleanuD. and AgarwalP., On analytical solutions of the fractional differential equation with uncertainty: Application to the Basset problem, Entropy17 (2015), 885–902.
37.
StefaniniL., A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and Systems161 (2010), 1564–1584.
38.
Villamizar-RoaE.J., Angulo-CastilloV. and Chalco-CanoY., Existence of solutions to fuzzy differential equations with generalized Hukuhara derivative via contractive-like mapping principles, Fuzzy sets and systems265 (2015), 24–38.
39.
VuH., LupulescuV. and HoaN.V., Existence of extremal solutions to interval-valued delay fractional differential equations via monotone iterative technique, Journal of Intelligent & Fuzzy Systems34 (2018), 2177–95.
40.
VuH., AnT.V. and HoaN.V., Ulam-Hyers stability of uncertain functional differential equation in fuzzy setting with Caputo-Hadamard fractional derivative concept, Journal of Intelligent & Fuzzy Systems, (Preprint), pp. 1–15. DOI: 10.3233/JIFS-191025
