Solvability of fuzzy fractional stochastic Pantograph differential system

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematics, The Gandhigram Rural Institute - Deemed to be University, Gandhigram - 624 302, Dindigul, Tamil Nadu, India


In this paper, a new type of  equation namely  fuzzy fractional stochastic Pantograph delay differential system  (FSPDDS) is proposed. In our previous work,  a first  extension of fuzzy stochastic differential system into   fuzzy fractional stochastic differential system by using Granular differentiability has been established. Here we study the existence and uniqueness  results for the fuzzy FSPDDS which  are obtained by using  generalized Granular  differentiability  and contraction principle with weaker conditions. This kind of equation is used in many real world problems. Finally, we provide two numerical examples for the effectiveness of the theoretical results.


[1] S. Abbasbandy, T. Allahviranloo, M. R. Balooch Shahryari, S. Salhshour, Fuzzy local fractional differential equations, International Journal of Industrial Mathematics, 4 (2012), 231-246.
[2] R. P. Agarwal, S. Arshad, D. O’Regan, V. Lupulescu, Fuzzy fractional integral equations under compactness type condition, Fractional Calculus and Applied Analysis, 51 (2012), 572-590.
[3] M. Z. Ahmad, M. K. Hasan, B. De Baets, Analytical and numerical solutions of fuzzy differential equations, Information Sciences, 236 (2013), 156-167.
[4] T. Allahviranloo, Fuzzy fractional differential operators and equations, Springer, Cham, 2020.
[5] T. Allahviranloo, A. Armand, Z. Gouyandeh, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal of Intelligent Fuzzy Systems, 26 (2014), 1481-1490.
[6] T. Allahviranloo, S. Salahshour, S. Abbasbandy, Explicit solutions of fractional differential equations with uncertainty, Soft Computing, 6 (2012), 297-302.
[7] S. Arshad, V. Lupulescu, On the fractional differential equations with uncertainty, Nonlinear Analysis, 74 (2011), 3685-3693.
[8] P. Balasubramaniam, S. Muralisankar, Existence and uniqueness of fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions, Computers and Mathematics with Applications, 47(6-7) (2004), 1115-1122.
[9] L. C. Barros, L. T. Gomes, P. A, Tonelli, Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems, 230 (2013), 39-52.
[10] N. V. Hoa, P. V. Tri, T. T. Dao, I. Zelinka, Some global existence results and stability theorem for fuzzy functional differential equations, Journal of Intelligent and Fuzzy Systems, 28 (2015), 393-409.
[11] A. Khastan, J. J. Nieto, R. Rodríguez-López, Fuzzy delay differential equations under generalized differentiability, Information Sciences, 275 (2014), 145-167.
[12] H. V. Long, M. Ali, L. H. Son, M. Khan, D. T. Tu, A novel approach for fuzzy clustering based on neutrosophic association matrix, Computers and Industrial Engineering, 127 (2018), 1-11.
[13] M. T. Malinowski, On random fuzzy differential equations, Fuzzy Sets and Systems, 160 (2009), 3152-3165.
[14] M. T. Malinowski, Fuzzy and set-valued stochastic differential equations with local Lipschitz condition, IEEE Transactions on Fuzzy Systems, 23 (2015), 1891-1898.
[15] M. T. Malinowski, Random fuzzy fractional integral equations-theoretical foundations, Fuzzy Sets and Systems, 265 (2015), 39-62.
[16] M. T. Malinowski, Stochastic fuzzy differential equations of a non-increasing type, Communications in Nonlinear Science and Numerical Simulation, 33 (2015), 99-117.
[17] M. T. Malinowski, Fuzzy stochastic differential equations of decreasing fuzziness: Non-Lipschitz coefficients, Journal of Intelligent and Fuzzy Systems, 31 (2016), 13-25.
[18] M. Mazandarani, N. Pariz, A. V. Kamyad, Granular differentiability of fuzzy-number-valued functions, IEEE Transactions on Fuzzy Systems, 26 (2018), 310-323.
[19] K. S. Miller, B. Ross, An introduction to the fractional calculus and differential equations, John Wiley, New York, 1993.
[20] N. Najafi, T. Allahviranloo, Semi-analytical methods for solving fuzzy impulsive fractional differential equations, Journal of Intelligent and Fuzzy Systems, 33 (2017), 3539-3560.
[21] N. Najafi, T. Allahviranloo, Combining fractional differential transform method and reproducing kernel Hilbert space method to solve fuzzy impulsive fractional differential equations, Computational and Applied Mathematics, 39 (2020), 1-25.
[22] M. Najariyan, Y. Zhao, Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives, IEEE Transactions on Fuzzy Systems, 26 (2018), 2273-2288.
[23] J. J. Niet, R. Rodríguez-López, Some results on boundary value problems for fuzzy differential equations with functional dependence, Fuzzy Sets and Systems, 230 (2013), 92-118.
[24] J. R. Ockendon, A. B. Taylor, The dynamics of a current collection system for an electric locomotive, Proceedings of the Royal Society A, 322 (1971), 447-468.
[25] Y. Ogura, On stochastic differential equations with fuzzy set coefficients, soft methods for handling variability and imprecision, Springer, Berlin, 2008.
[26] J. Y. Park, J. U. Jeong, On random fuzzy functional differential equations, Fuzzy Sets and Systems, 223 (2013), 89-99.
[27] J. Priyadharsini, P. Balasubramaniam, Existence of fuzzy fractional stochastic differential system with impulses, Computational and Applied Mathematics, 39 (2020), 1-21.
[28] M. L. Puri, D. A. Ralescu, Differential of fuzzy functions, Journal of Mathematical Analysis and Applications, 91 (1983), 552-558.
[29] A. Rivaz, O. S. Fard, T. A. Bidgoli, On the existence and uniqueness of solutions for fuzzy fractional differential equations, Tbilisi Mathematical Journal, 10 (2017), 197-205.
[30] S. Salahshour, T. Allahviranloo, S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Communication in Nonliner Science and Numerical Simulation, 17 (2012), 1372-1381.
[31] M. Senol, S. Atpinar, Z. Zararsiz, S. Salhshour, Approximate solution of time-fractional fuzzy partial differential equations, Computation and Applied Mathematics, 38 (2019), 2238-3603.
[32] N. T. K. Son, N. P. Dong, H. V. Long, L. H. Son, A. Khastan, Linear quadratic regulator problem governed by granular neutrosophic fractional differential equations, ISA Transactions, 97 (2019), 296-316.
[33] N. T. K. Son, N. P. Dong, L. H. Son, M. A. Basset, G. Monogaran, H. V. Long, On the stabilizability for a class of linear time-invariant systems under uncertainty, Circuits, Systems, and Signal Processing, 39 (2020), 919-960.
[34] N. T. K. Son, N. P. Dong, L. H. Son, H. V. Long, Towards granular calculus of single-valued neutrosophic functions under granular computing, Multimedia Tools and Applications, 79 (2020), 16845-16881.
[35] N. T. K. Son, H. V. Long, N. P. Dong, Fuzzy delay differential equations under granular differentiability with applications, Computational and Applied Mathematics, 38 (2019), 1-29.
[36] S. Tapaswini, D. Behera, Analysis of imprecisely defined fuzzy space-fractional telegraph equations, Pramana Journal of Physics, 32 (2020), 1-10.
[37] H. Vu, N. Van Hao, On impulsive fuzzy functional differential equations, Iranian Journal of Fuzzy Systems, 4 (2016), 79-94.
[38] L. A. Zadeh, Fuzzy sets, Information Control, 8 (1965), 338-353.