In this paper, we initiate to investigate the existence and uniqueness of solutions to initial value problems for fuzzy fractional Schrödinger equations involving the Caputo’s H-derivative. Continuous dependence on initial values of the solution is also considered. Our results are based on a successive approximation method and the Banach contraction mapping principle. Two examples are presented for our new results.
SchrödingerE., An undulatory theory of the mechanics of atomsand molecules, Physical Review28(1926), 1049–1070.
2.
EdwardsM. and K.Burnett, Numerical solution of the nonlinear Schrödinger equation for small samples of trapped neutral atoms, Physical Review A51(1995), 1382–1386.
RidaS., El-SherbinyH. and A.Arafa, On the solution of thefractional nonlinear Schrödinger equation, Physics Letters A372(2008), 553–558.
5.
LiX., HanZ. and X.Li, Boundary value problems of fractionalqdifference Schrödinger equations, Applied Mathematics Letters46(2015) 100–105.
6.
GolmankhanehA. and A.Jafarian, About fuzzy Schrödingerequation, in: International Conference on Fractional Differ entiation and Its Applications, IEEE, 2014, pp. 1–5. DOI: 526 10.1109/ICFDA.2014.6967457.
7.
AllahviranlooT., KianiN. and M.Barkhordari, Toward the existenceand uniqueness of solutions of second-order fuzzy differential equations, Information Sciences179(2009), 1207–1215.
8.
HoaN., The initial value problem for interval-valued second-orderdifferential equations under generalized Hdifferentiability, Information Sciences311(2015), 119–148.
9.
VuH. and L.Dong, Initial value problem for second-order randomfuzzy differential equations, Advances in Difference Equations2015(2015), 373.
10.
TapaswiniS., ChakravertyS. and T.Allahviranloo, A new approach tonth order fuzzy differential equations, Computational Mathematics and Modeling28(2017), 278–300.
11.
SalahshourS., AllahviranlooT. and S.Abbasbandy, Existence anduniqueness results for fractional differential equations withuncertainty, Advances in Difference Equations2012(2012), 112.
12.
SalahshourS., AhmadianA., SenuN., BaleanuD. and P.Agarwal, Onanalytical solutions of the fractional differential equation with uncertainty: Application to the Basset problem, Entropy17(2015), 885–902.
13.
ChehlabiM. and T.Allahviranloo, Concreted solutions to fuzzylinear fractional differential equations, Applied Soft Computing44(2016), 108–116.
14.
ArmandA., AllahviranlooT., AbbasbandyS. and Z.Gouyandeh, Fractional relaxation-oscillation differential equations via fuzzyvariational iteration method, Journal of Intelligent and Fuzzy Systems32(2017), 363–371.
15.
NajafiaN. and T.Allahviranloo, Semi-analytical methods for solvingfuzzy impulsive fractional differential equations, Journal of Intelligent and Fuzzy Systems33(2017), 3539–3560.
16.
AhmadianA., SalahshourS., Ali-AkbariM., IsmailF. and D.Baleanu, A novel approach to approximate fractional derivative with uncertain conditions, Chaos, Solitons and Fractals104(2017), 68–76.
17.
AlinezhadM. and T.Allahviranloo, On the solution of fuzzy fractional optimal control problems with the Caputo derivative, Information Sciences421(2017), 218–236.
18.
LakshmikanthamV., MohapatraR.Theory of Fuzzy Differential Equations and Inclusions, Taylor and Francis, London, 2003.
19.
LakshmikanthamV., GnanaB., VasundharaD.Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, 2006.
20.
GomesL., BarrosL., BedeB.Fuzzy Differential Equations in Various Approaches, Springer International Publishing, Berlin, 2015.
21.
AnastassiouG. and S.Gal, On a fuzzy trigonometric approximationtheorem of Weierstrass-type, The Journal of Fuzzy Mathematics3(2001), 701–708.
22.
PuriM. and D.Ralescu, Differentials of fuzzy functions, Journal of Mathematical Analysis and Applications91(1983), 552–558.
23.
LupulescuV., On a class of fuzzy functional differential equations, Fuzzy Sets and Systems160(2009), 1547–1562.
24.
PuriM. and D.Ralescu, Fuzzy random variables, Journal of Mathematical Analysis and Applications114(1986), 409–422.
25.
KalevaO., Fuzzy differential equations, Fuzzy Sets and Systems24(1987), 301–317.
26.
KhastanA., NietoJ. and R.Rodríguez-López, Schauder fixedpoint theorem in semilinear spaces and its application to fractional differential equations with uncertainty, Fixed Point Theory and Applications21(2014), 1–14.
27.
BedeB. and S.Gal, Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems147(2004), 385–403.
28.
BedeB. and S.Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems151(2005), 581–599.
29.
AllahviranlooT., ArmandA. and Z.Gouyandeh, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal of Intelligent and Fuzzy Systems26(2014), 1481–1490.
30.
YeH., GaoJ. and Y.Ding, A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications328(2007), 1075–1081.
31.
KadakU. and F.Başar, Power series of fuzzy numbers with realor fuzzy coefficients, Filomat26(2012), 519–528.
