# On random fuzzy fractional partial integro-differential equations under Caputo generalized Hukuhara differentiability

- 72 Downloads
- 3 Citations

## Abstract

This paper is devoted to the study of the solvability of random fuzzy fractional partial integro-differential equations under Caputo generalized Hukuhara differentiability. The notions of random fuzzy variables and fuzzy stochastic processes are developed for multivariable functions. The existence and uniqueness of two types of integral solutions generated from Darboux problem for nonlinear wave equations are proved using the successive approximations method and Gronwall’s inequality for stochastic processes. The continuous dependence on the data, the boundedness, and the stability with probability one of integral solutions are established to confirm the well posedness of our model. Some computational examples are presented to illustrate the theoretical results.

## Keywords

Random fuzzy variable Random fuzzy fractional partial integro-differential equations Caputo gH derivatives Gronwall’s inequality Integral solution## Mathematics Subject Classification

35R13 35R60 45R05 26A33 26E50## Notes

### Acknowledgements

The author is greatly indebted to Editor-in-Chiefs (Prof. Jose E. Souza de Cursi), Associate Editor (Prof. José Tenreiro Machado), and anonymous reviewers for their comments and valuable suggestions that greatly improve the quality and clarity of the paper.

## References

- Allahviranloo T, Gouyandeh Z, Armand A, Hasanoglu A (2015) On fuzzy solutions for heat equation based on generalized Hukuhara differentiability. Fuzzy Sets Syst 265:1–23MathSciNetCrossRefMATHGoogle Scholar
- Aubin J, Frankowska H (1990) Set-valued analysis. Birkhauser, BostonMATHGoogle Scholar
- Bede B (2013) Mathematics of fuzzy sets and fuzzy logic. Springer, BerlinCrossRefMATHGoogle Scholar
- Bede B, Stefanini L (2013) Generalized differentiability of fuzzy valued functions. Fuzzy Sets Syst 230:119–141MathSciNetCrossRefMATHGoogle Scholar
- Bitsadze AV (1988) Some classes of partial differential equations, Advanced studies in contemporary mathematics, vol 4. Gordon and Breach Science Publishers, New York (translated from the Russian by Zahavi H)Google Scholar
- Buckley J, Feuring T (1999) Introduce to fuzzy partial differential equations. Fuzzy Sets Syst 105:241–248CrossRefMATHGoogle Scholar
- Castaing C, Valadier M (1997) Lecture notes in mathematics: convex analysis and measurable multifunctions. Springer, BerlinMATHGoogle Scholar
- Chen L, Wu R, Pan D (2011) Mean square exponential stability of impulsive stochastic fuzzy cellular neural networks with distributed delays. Expert Syst Appl 38:6294–6299CrossRefGoogle Scholar
- Durikovic V (1968) On the uniqueness of solutions and the convergence of successive approximations in the Darboux problem for certain differential equations of the type \(u_{xy}=f(x, y, u, u_x, u_y)\). Arch Math 4:223–235MathSciNetMATHGoogle Scholar
- Fei W (2013) Existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non-Lipschitzian condition. Nonlinear Anal (TMA) 76:202–214MathSciNetCrossRefMATHGoogle Scholar
- Fei W, Xia D (2013) On solutions to stochastic set differential equations of Itô type under the non-Lipschitzian condition. Dyn Syst Appl 22:137–156MATHGoogle Scholar
- Feng Y (1999) Mean-square integral and differential of fuzzy stochastic processes. Fuzzy Sets Syst 102:271–280MathSciNetCrossRefMATHGoogle Scholar
- Feng Y (2000) Fuzzy stochastic differential systems. Fuzzy Sets Syst 115:351–363MathSciNetCrossRefMATHGoogle Scholar
- Gomes LT, Barros LC, Bede B (2015) Fuzzy differential equations in various approaches. SpringerGoogle Scholar
- Guo R, Guo D (2009) Random fuzzy variable foundation for Grey differential equation modeling. Soft Comput 13:185–201CrossRefMATHGoogle Scholar
- Hai S, Gong Z, Lic H (2016) Generalized differentiability for n-dimensional fuzzy-number-valued functions and fuzzy optimization. Inf Sci 374:151–163CrossRefGoogle Scholar
- Hung NT (1978) A note on the extension principle for fuzzy set. J Math Anal Appl 64:369–380MathSciNetCrossRefMATHGoogle Scholar
- Khastan A, Neito JJ, Rodríguez-López R (2014) Fuzzy delay differential equations under generalized differentiability. Inf Sci 275:145–167MathSciNetCrossRefMATHGoogle Scholar
- Kwakernaak H (1978) Fuzzy random variables: definition and theorems. Inf Sci 15:1–29MathSciNetCrossRefMATHGoogle Scholar
- Lakshmikantham V, Mohapatra RN (2003) Theory of fuzzy differential equations and inclusions. Taylor and Francis Publishers, LondonCrossRefMATHGoogle Scholar
- Li J, Wang J (2012) Fuzzy set-valued stochastic Lebesgue integral. Fuzzy Sets Syst 200:48–64MathSciNetCrossRefMATHGoogle Scholar
- Long HV, Son NTK, Ha NM, Son LH (2014a) The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations. Fuzzy Optim Decis Mak 13(4):435–462MathSciNetCrossRefGoogle Scholar
- Long HV, Son NTK, Tam HT, Cuong BC (2014b) On the existence of fuzzy solutions for partial hyperbolic functional differential equations. Int J Comput Intell Syst 7(6):1159–1173CrossRefGoogle Scholar
- Long HV, Son NTK, Tam HT (2015) Global existence of solutions to fuzzy partial hyperbolic functional differential equations with generalized Hukuhara derivatives. J Intell Fuzzy Syst 29(2):939–954MathSciNetCrossRefMATHGoogle Scholar
- Long HV, Son NK, Tam HT (2017a) The solvability of fuzzy fractional partial differential equations under Caputo gH-differentiability. Fuzzy Sets Syst 309:35–63MathSciNetCrossRefMATHGoogle Scholar
- Long HV, Son NTK, Hoa NV (2017b) Fuzzy fractional partial differential equations in partially ordered metric spaces. Iran J Fuzzy Syst 14:107–126MathSciNetMATHGoogle Scholar
- Long HV, Nieto JJ, Son NTK (2017c) New approach to study nonlocal problems for differential systems and partial differential equations in generalized fuzzy metric spaces. Fuzzy Sets Syst. doi: 10.1016/j.fss.2016.11.008
- Long HV, Son NTK, Tam HTT, Yao J-C (2017d) Ulam stability for fractional partial integro-differential equation with uncertainty. Acta Math Vietnam. doi: 10.1007/s40306-017-0207-2
- Lupulescu V, Dong LS, Hoa NV (2015) Existence and uniqueness of solutions for random fuzzy fractional integral and differential equations. J Intell Fuzzy Syst 29(1):27–42MathSciNetCrossRefMATHGoogle Scholar
- Majumder P, Mondal SP, Bera UK, Maiti M (2016) Application of generalized Hukuhara derivative approach in an economic production quantity model with partial trade credit policy under fuzzy environment. Oper Res Perspect 3:77–91MathSciNetCrossRefGoogle Scholar
- Malinowski MT (2009) On random fuzzy differential equations. Fuzzy Sets Syst 160:3152–3165MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2010) Existence theorems for solutions to random fuzzy differential equations. Nonlinear Anal (TMA) 73:1515–1532MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2012a) Strong solutions to stochastic fuzzy differential equations of Itô type. Math Comput Model 55:918–928CrossRefMATHGoogle Scholar
- Malinowski MT (2012b) Itô type stochastic fuzzy differential equations with delay. Syst Control Lett 61:692–701CrossRefMATHGoogle Scholar
- Malinowski MT (2012c) Random fuzzy differential equations under generalized Lipschitz condition. Nonlinear Anal (RWA) 13:860–881MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2012d) Interval Cauchy problem with a second type Hukuhara derivative. Inf Sci 213:94–105MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2013a) On a new set-valued stochastic integral with respect to semimartingales and its applications. J Math Anal Appl 408:669–680MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2013b) Some properties of strong solutions to stochastic fuzzy differential equations. Inf Sci 252:62–80MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2013c) On equations with a fuzzy stochastic integral with respect to semimartingales. Adv Intell Syst Comput 190:93–101MATHGoogle Scholar
- Malinowski MT (2013d) Approximation schemes for fuzzy stochastic integral equations. Appl Math Comput 219(24):11278–11290MathSciNetMATHGoogle Scholar
- Malinowski MT (2015a) Random fuzzy fractional integral equations—theoretical foundations. Fuzzy Sets Syst 265:39–62MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2015b) Fuzzy and set-valued stochastic differential equations with local Lipschitz condition. IEEE Trans Fuzzy Syst 23(5):1891–1898CrossRefGoogle Scholar
- Malinowski MT (2015c) Set-valued and fuzzy stochastic differential equations in M-type 2 Banach spaces. Tohoku Math J 67:349–381MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2015d) Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition. Open Math 13:106–134MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2015e) Fuzzy stochastic differential equations of decreasing fuzziness: approximate solutions. J Intell Fuzzy Syst 29:1087–1107MathSciNetCrossRefMATHGoogle Scholar
- Malinowski MT (2016a) Bipartite fuzzy stochastic differential equations with global Lipschitz condition. Math Probl Eng 2016:3830529-1–3830529-13Google Scholar
- Malinowski MT (2016b) Stochastic fuzzy differential equations of a nonincreasing type. Commun Nonlinear Sci Numer Simul 33:99–117MathSciNetCrossRefGoogle Scholar
- Malinowski MT (2016c) Fuzzy stochastic differential equations of decreasing fuzziness: non Lipschitz coefficients. J Intell Fuzzy Syst 31:13–25CrossRefMATHGoogle Scholar
- Mazandarani M, Najariyan M (2014a) Differentiability of type-2 fuzzy number-valued functions. Commun Nonlinear Sci Numer Simul 19:710–725MathSciNetCrossRefGoogle Scholar
- Mazandarani M, Najariyan M (2014b) Type-2 fuzzy fractional derivatives. Commun Nonlinear Sci Numer Simul 19:2354–2372MathSciNetCrossRefGoogle Scholar
- Mazandarani M, Pariz N, Kamyad AV (2017) Granular differentiability of fuzzy-number-valued functions. IEEE Tran Fuzzy Syst 99:1–14. doi: 10.1109/TFUZZ.2017.2659731
- Najariyan M, Farahi MH (2015) A new approach for solving a class of fuzzy optimal control systems under generalized Hukuhara differentiability. J Frankl Inst 352:1836–1849MathSciNetCrossRefGoogle Scholar
- Puri ML, Ralescu DA (1983) Differentials for fuzzy functions. J Math Anal Appl 91:552–558MathSciNetCrossRefMATHGoogle Scholar
- Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114:409–422MathSciNetCrossRefMATHGoogle Scholar
- Wang T, Tong S, Li Y (2012) Robust adaptive fuzzy control for a class of stochastic nonlinear systems with dynamical uncertainties. J Frankl Inst 349:3121–3141MathSciNetCrossRefMATHGoogle Scholar
- Wu LG, Wang ZD (2009) Fuzzy filtering of nonlinear fuzzy stochastic systems with time-varying delay. Signal Process 89:1739–1753CrossRefMATHGoogle Scholar
- Xia Z, Li J, Li J (2014) Passivity-based resilient adaptive control for fuzzy stochastic delay systems with Markovian switching. J Frankl Inst 351:3818–3836MathSciNetCrossRefMATHGoogle Scholar
- Zheng C-D, Wang Y, Wang Z (2014) Stability of stochastic fuzzy Markovian jumping neural networks with leakage delay under impulsive perturbations. J Frankl Inst 351:1728–1755MathSciNetCrossRefGoogle Scholar