FUZZY FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS IN PARTIALLY ORDERED METRIC SPACES

Document Type: Research Paper

Authors

1 Division of Computational Mathematics and Engineering, Insti- tute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam; Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

2 Department of Mathematics, Hanoi University of Education, Vietnam

3 Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam; Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Abstract

In this paper, we consider fuzzy fractional partial differential equations under Caputo generalized Hukuhara differentiability. Some new results on the existence and uniqueness of two types of fuzzy solutions are studied via  weakly contractive mapping in the partially ordered metric space. Some application examples are presented to illustrate our main results.

Keywords


[1] S. Abbas, M. Benchohra and G. M. N'Guerekata, Topics in fractional DEs, Springer, Berlin,
Heidelberg, New York, Hong Kong, London, Milan, Paris, Tokyo, 2012.
[2] R. Alikhani and F. Bahrami, Global solutions of fuzzy integro-di erential equations under
generalized di erentiability by the method of upper and lower solutions, Inf. Sci., 295 (2015),
600-608.
[3] T. Allahviranloo, Z. Gouyandeh and A. Armand, Fuzzy fractional di erential equations under
generalized fuzzy Caputo derivative, J. Intell. Fuzzy Syst., 26 (2014), 1481-1490.
[4] T. Allahviranloo, Z. Gouyandeh, A. Armand and A. Hasanoglu, On fuzzy solutions for heat
equation based on generalized Hukuhara di erentiability, Fuzzy Sets Syst., 265 (2015), 1-23.
[5] B. Bede and L. Stefanini, Generalized di erentiability of fuzzy-valued functions, Fuzzy Sets
Syst., 230 (2013), 119-141.

[6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geo-
physical J. Int., 13 (1967), 529-539.
[7] J. Harjani and K. Sadarangani, Generalized contractions in partially ordered metric spaces
and applications to ordinary di erential equations, Nonlinear Anal. (TMA), 72 (2010), 1188-
1197.
[8] N. V. Hoa, Fuzzy fractional functional di erential equations under Caputo gH-
di erentiability, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1134-1157.
[9] N. V. Hoa, Fuzzy fractional functional integral and di erential equations, Fuzzy Sets Syst.,
280 (2015), 58-90.
[10] A. Khastan, J. J. Nieto and R. Rodrguez-Lopez, Schauder xed-point theorem in semilinear
spaces and its application to fractional di erential equations with uncertainty, Fixed Point
Theory Appl., 2014 (2014): 21.
[11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional
di erential equations, Elsevier Science B.V, Amsterdam, 2006.
[12] H. V. Long, N. T. K. Son, N. T. M. Ha and L. H. Son, The existence and uniqueness of fuzzy
solutions for hyperbolic partial di erential equations, Fuzzy Optim. Decis. Mak., 13 (2014),
435-462.
[13] H. V. Long, N. T. K. Son and H. T. T. Tam, Global existence of solutions to fuzzy partial
hyperbolic functional di erential equations with generalized Hukuhara derivatives, J. Intell.
Fuzzy Syst., 29 (2015), 939-954.
[14] H. V. Long, N. T. K. Son and H. T. T. Tam, The solvability of fuzzy fractional partial
di erential equations under Caputo gH-di erentiability, Fuzzy Sets Syst., 309 (2017), 35-63.
[15] V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets Syst., 265 (2015),
63-85.
[16] M. T. Malinowski, Random fuzzy fractional integral equations - Theoretical foundations,
Fuzzy Sets Syst., 265 (2015) 39-62.
[17] J. J. Nieto and R. Rodrguez-Lopez, Applications of contractive-like mapping principles to
fuzzy equations, Revista Matematica Complutense, 19 (2006), 361-383.
[18] E. J. Villamizar-Roa, V. Angulo-Castillo and Y. Chalco-Cano, Existence of solutions to fuzzy
di erential equations with generalized Hukuhara derivative via contractive-like mapping prin-
ciples, Fuzzy Sets Syst., 265 (2015), 24-38.
[19] H. Vu and N. V. Hoa, On impulsive fuzzy functional di erential equations, Iranian Journal
of Fuzzy Systems, 13(4) (2016), 79-94.