ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Existence and Uniqueness of Solutions for Fuzzy Mixed Type of Delay Differential Equations

Journal of Applied Nonlinear Dynamics 10(1) (2021) 187--196 | DOI:10.5890/JAND.2021.03.012

D. Prasantha Bharathi , T. Jayakumar, T. Muthukumar, S. Vinoth

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore-641 020, Tamil Nadu, India

Abstract

In this paper we are defining the fuzzy type of mixed delay differential equations. The necessity of studying the mixed delay differential equation in terms of fuzzy is that the single real valued solution can be ordered as the set of fuzzy valued solution. So it is very important in establishing the existence of solution for the fuzzy mixed type of delay differential equations using necessary theorems and lemmas. In addition we are also proving that the existing solution is unique.

Acknowledgments

All the authors would like to thank the anonymous reviewers for their valuable comments and suggestions.

References

1.  [1] Lupulescu, V. and Abbas, U. (2012), Fuzzy delay differential equations, Fuzzy Optimization and Decision Making, 11, 99-111.
2.  [2] Abbasbandy, S. and Allahviranloo, T. (2002), Numerical solution of fuzzy differential equation by Taylor method, Journal of Computational Methods in Applied mathematics, 2, 113-124.
3.  [3] Abbasbandy, S. and Allahviranloo, T. (2002), Numerical solution of fuzzy differential equation, Mathematical and Computational Applications, 7, 41-52.
4.  [4] Abbasbandy, S. and Allahviranloo, T. (2004), Numerical Solution of fuzzy differential equation by Runge-Kutta Method, Nonlinear Studies, 11 117-129.
5.  [5] Pederson, S. and Sambandham, M. (2008), The Runge-Kutta method for hybrid fuzzy differential equations, Nonlinear Analysis Hybrid Systems, 2, 626-634.
6.  [6] Park, J.Y. and Han, H.K. (1999), Existence and uniqueness theorem for a solution of fuzzy differential equations, Internat. J. Math. $\&$ Math. Sci., 22(2), 271-279.
7.  [7] Park, J.Y., and Jeong, J.U. (1993), Common fixed points of fuzzy mappings, Fuzzy Sets and Systems, 59(2), 231-235.
8.  [8] Park, J.Y., Kwun, Y. C., and Jeong, J. U., (1995), Existence of solutions of fuzzy integral equations in Banach spaces, Fuzzy Sets and Systems, 72(3), 373-378.
9.  [9] Ge, X.T. and Zhu Y.G. (2012), Existence and uniqueness theorem for uncertain delay differential equations, Journal of Computational Information Systems, 8(20), 8341-8347.
10.  [10] Gao, Y. (2012), Existence and Uniqueness Theorem on Uncertain Differential Equations with Local Lipschitz Condition, Journal of Uncertain Systems, 6(3), 223-232.
11.  [11] Chen, X., Liu, B. (2010), Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, 9(1), 69-81.
12.  [12] Najeeb AL Rawi, S., Kadhim Salih, R., and Mohammed, A.A. (2006), Numerical Solution of $N^{th}$ order linear delay differential equation using Runge-Kutta method, Um Salama Science journal, 3(1), 140-146.
13.  [13] Bellen, A. and Zennaro, M. (2003), Numerical methods for delay differential equations, Numerical mathematics and scientific computation, Oxford science publications, Clarendon Press.
14.  [14] Jayakumar, T., Parivallal, A. and Prasantha Bharathi, D. (2016), Numerical Solutions of Fuzzy delay differential equations by fourth order Runge Kutta Method, Advanced Fuzzy Sets and Systems, 21 , 135-161.
15.  [15] Dhandapani, P.B., Baleanu, D., Thippan, J., and Sivakumar, V. (2019), Fuzzy Type RK4 Solutions to Fuzzy Hybrid Retarded Delay Differential Equations, Front. Phys., 7(168), 1-6.
16.  [16] Bharathi, D.P., Jayakumar, T., and Vinoth, S. (2018), Numerical solution of fuzzy pure multiple retarded delay differential equations, International Journal of Research in Advent Technology, 6 (12), 3693-3698.
17.  [17] Bharathi, D.P., Jayakumar, T., and Vinoth, S. (2019), Numerical solution of fuzzy pure multiple neutral delay differential equations, International Journal of Advanced Scientific Research and Management, 4(1), 172-178.
18.  [18] Bharathi, D.P., Jayakumar, T., and Vinoth, S. (2019), Numerical Solution of Fuzzy Mixed Delay Differential Equations Via Runge-Kutta Method of Order Four, International Journal of Applied Engineering Research, 14(3), (Special Issue) 70-74.
19.  [19] Bharathi, D.P., Jayakumar, T., and Vinoth, S. (2019), Numerical Solution of Fuzzy Neutral Delay Differential Equations An Application of Runge Kutta Method of Order Four Via Runge-Kutta Method of Order Four, Journal of Emerging Technologies and Innovative Research, 6(2), 765-768.
20.  [20] Bharathi, D.P., Jayakumar, T., and Vinoth, S. (2019), Numerical Solution of Fuzzy Multiple Hybrid Single Retarded Delay Differential Equations, International Journal of Recent Technology and Engineering, 8(3), 1946-1949.
21.  [21] Bharathi, D.P., Jayakumar, T., and Vinoth, S. (2019), Numerical Solutions of Fuzzy Multiple Hybrid Single Neutral Delay Differential Equations, International Journal of Scientific $\&$ Technology Research, 8(09), 520-523.
22.  [22] Seikkala, S. (1987), On the fuzzy initial value problem, Fuzzy Sets and Systems, 24, 319-330.
23.  [23] Bukley, J.J. and Feuring. T, (2000), Fuzzy differential equations, Fuzzy Sets and Systems, 110, 43-54.
24.  [24] Zadeh, L.A. (1965), Fuzzy sets, Information and Control, 8(3), 338-353.