First order linear fuzzy dynamic equations on time scales

Document Type: Original Manuscript

Authors

1 Department of Mathematics Institute for Advanced Studies in Basic Sciences

2 Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran.

Abstract

In this paper, we study the concept of generalized differentiability for fuzzy-valued functions on time scales. Using
the derivative of the product of two functions, we provide solutions to first order linear fuzzy dynamic equations. We
present some examples to illustrate our results.


Keywords


[1] O. Abu Arqub, M. AL-Smadi, S. Momani, T. Hayat, Numerical solutions of fuzzy differential equations using
reproducing kernel Hilbert space method
, Soft Computing, 20 (2016), 3283–3302.
[2] O. Abu Arqub, M. AL-Smadi, S. Momani, T. Hayat,
Application of reproducing kernel algorithm for solving secondorder two-point fuzzy boundary value problems, Soft Computing, 21 (2017), 7191–7206.
[3] O. Abu Arqub,
Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential
equations
, Neural Computing & Applications, 28 (2017), 1591–1610.
[4] R.P. Agarwal,
Difference Equations and Inequalities: Theory, Methods, and Applications, Marcel Dekker, New York,
1992.
[5] S.E. Amrahov, A. Khastan, N. Gasilov, A.G. Fatullayev,
Relationship between Bede{Gal differentiable set-valued
functions and their associated support functions,
Fuzzy Sets and Systems, 295 (2016), 57–71
[6] B. Bede, S.G. Gal,
Generalizations of the differentiability of fuzzy number value functions with applications to fuzzy
differential equations
, Fuzzy Sets and Systems, 151 (2005), 581–599.
[7] B. Bede, I. J. Rudas, A. L. Bencsik,
First order linear fuzzy differential equations under generalized differentiability,
Information Sciences,
177 (2007), 1648–1662.
[8] B. Bede,
Mathematics of Fuzzy Sets and Fuzzy Logic, Springer, London, 2013.
[9] M. Bohner, A. Peterson,
Dynamic equations on time scales: An introduction with applications, Springer Science &
Business Media, 2012.
[10] K. A. Chrysafis, B. K. Papadopoulos, G. Papaschinopoulos,
On the fuzzy difference equations of finance, Fuzzy
Sets and Systems,
159 (2008), 3259–3270.
[11] E. Deeba, A. Korvin, E. L. Koh,
A fuzzy difference equation with and Application, Journal of Difference Equations
and applications,
2 (1996), 365–374.
[12] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 1994.
[13] O. S. Fard, T. A. Bidgoli,
Calculus of fuzzy functions on time scales (I), Soft Computing, 19 (2014), 293–305.
[14] O. S. Fard, D. F. M. Torres, M. R. Zadeh,
A Hukuhara approach to study of hybrid fuzzy systems on time scales,
Applicable Analysis and Discrete Mathematics,
10 (2016), 152–167.
[15] G. S. Guseinov,
Integration on time scales, Journal of Mathematical Analysis and Applications, 285 (2003), 107–
127.
[16] S. Hilger,
Analysis on measure chains - a unified approach to continuous and discrete calculus, Results in Mathematics, 18 (1990), 18–56.
[17] S. Hong,
Differentiability of multivalued functions on time scales and applications to multivalued dynamic equations,
Nonlinear Analysis: Theory, Methods & Applications,
71 (2009), 3622–3637.
[18] S. Hong, Y. Peng,
Almost periodicity of set-valued functions and set dynamic equations on time scales, Information
Sciences,
330 (2016), 157–174.
[19] S. Hong, J. Gao, Y. Peng,
Solvability and stability of impulsive set dynamic equations on time scales, Abstract
Appl. Anal., vol. 2014, Article ID 610365, 19 pages, 2014. doi:10.1155/2014/610365.
[20] A. Khastan, J.J. Nieto, R. Rodr´ıguez-L´opez,
Variation of constant formula for first order fuzzy differential equations, Fuzzy Sets and Systems, 177 (2011), 20–33.
[21] A. Khastan, R. Rodr´ıguez-L´opez,
On the solutions to first order linear fuzzy differential equations, Fuzzy Sets and
Systems,
295 (2016), 114–135.
[22] A. Khastan,
New solutions for first order linear fuzzy difference equations, Journal of Computational and Applied
Mathematics,
312 (2017), 156–166.
[23] V. Lakshmikantham, A. S. Vatsala,
Basic Theory of Fuzzy Difference Equations, Journal of Fuzzy Difference
Equations,
8 (2002), 957–968.
[24] W. T. Li, H. R. Sun,
Dynamics of rational difference equation, Applied Mathematics and Computation, 163 (2005),
577–591.
[25] V. Lupulescu,
Hukuhara differentiability of interval-valued functions and interval differential equations on time
scales
, Information Sciences, 248 (2013), 50–67.
[26] M. Najariyan, M. Mazandarani, V.E. Balas,
Solving first order linear fuzzy differential equations system, In:
Balas V., Jain L., Balas M. (eds) Soft Computing Applications. SOFA 2016. Advances in Intelligent Systems and
Computing, vol 634. Springer, Cham, 2018.
[27] J. J. Nieto, R. Rodr´ıguez-L´opez, D. Franco,
Linear first order fuzzy differential equations, International Journal
of Uncertainty Fuzziness Knowledge-Based Syst,
14 (2006), 687–709.
[28] C. Vasavi, G. S. Kumar, M. S. N. Murty,
Generalized differentiability and integrability for fuzzy set-valued functions
on time scales
, Soft Computing, 20 (2016), 1093–1104.
[29] C. Vasavi, G. S. Kumar, M. S. N. Murty,
Fuzzy dynamic equations on time scales under second type Hukuhara
delta derivative
, International Journal of Chemical Science, 14 (2016), 49–66.
[30] C. Vasai, G.S. Kumar, M. S. N. Murty,
Fuzzy Hukuhara delta differential and applications to fuzzy dynamic
equations on time scales
, Journal of Uncertain Systems, 10 (2016), 163–180.
[31] Q. Zhang, L. Yang, D. Liao,
On the first fuzzy Riccati difference equation, Information Sciences, 270 (2014),
226–236.