Strict Stability of Fuzzy Differential Equations by Lyapunov Functions
Abstract:In this study, we have investigated the strict stability
of fuzzy differential systems and we compare the classical notion of
strict stability criteria of ordinary differential equations and the notion
of strict stability of fuzzy differential systems. In addition that, we
present definitions of stability and strict stability of fuzzy differential
equations and also we have some theorems and comparison results.
Strict Stability is a different stability definition and this stability
type can give us an information about the rate of decay of the
solutions. Lyapunov’s second method is a standard technique used
in the study of the qualitative behavior of fuzzy differential systems
along with a comparison result that allows the prediction of behavior
of a fuzzy differential system when the behavior of the null solution
of a fuzzy comparison system is known. This method is a usefull
for investigating strict stability of fuzzy systems. First of all, we
present definitions and necessary background material. Secondly, we
discuss and compare the differences between the classical notion
of stability and the recent notion of strict stability. And then, we
have a comparison result in which the stability properties of the null
solution of the comparison system imply the corresponding stability
properties of the fuzzy differential system. Consequently, we give
the strict stability results and a comparison theorem. We have used
Lyapunov second method and we have proved a comparison result
with scalar differential equations.
 Aumann, R.J. Integrals of set-valued functions, J. Math. Anal. Appl. 12
 Bernfeld, S. and Lakshmikantham, V. An Introduction to Nonlinear
Boundary V alue Problems. Academic Press, New York, 1974.
 Bede, B. and Gal, S. G. Generalizations of the differentiability of
fuzzy-number valued functions with applications to fuzzy differential
equations, Fuzzy Sets and Systems, Vol. 151, No. 3, (2005) 581-589.
 Bede, B., Rudas, I. J. and Bencsik, A. L. First order linear fuzzy
differential equations under generalized differentiability,Information
Sciences, vol. 177, no. 7, (2007) pp. 1648-1662.
 Bede, B., Stefanini, L. Generalized differentiability of fuzzy-valued
functions, Fuzzy Sets and Systems, Vol. 230, November, (2013) pp.
 Buckley, J.J. and Feuring, T.H. Fuzzy differential equations,Fuzzy Sets
and Systems,Vol. 110 , (2000) pp. 43–54.
 Gomes, L. T., Barros, L. C., Bede, B. Fuzzy Differential Equations in
Various Approaches, Springer International Publishing (2015).
 Buckley, J. J. and Feuring, T. H. Fuzzy differential equations, Fuzzy Sets
and Systems, 110 (2000), 43 -54.
 Chen, Z., Fu, X., The variational Lyapunov function and strict stability
theory for differential systems, Nonlinear Analysis 64, 1931 – 1938,
 Ding, Z., Ming, M. and Kandel, A. Existence of solutions of Fuzzy
differential equations, Inform. Sci., 99 (1997), 205 - 217.
 Dubois, D. and Prade, H. Towards fuzzy differential calculus, Part I,
Part II, Part III, Fuzzy Sets and Systems, 8 (1982), 1 - 17, 105 - 116, 225
 Kaleva, O. Fuzzy differential equations. Fuzzy Sets and Systems 24
 Kaleva, O. On the calculus of fuzzy valued mappings, Appl. Math. Lett.,
3 (1990), 55 - 59.
 Kaleva, O. The Cauchy problem for fuzzy differential equations. Fuzzy
Sets and Systems 35 (1990) 389–396.
 Lakshmikantham, V. and Leela, S. Differential and Integral
Inequalities, Vol. I. Academic Press, New York, 1969.
 Lakshmikantham, V. and Leela, S. Fuzzy differential systems and the
new concept of stability. Nonlinear Dynamics and Systems Theory, 1 (2)
 Lakshmikantham, V. and Leela, S. A new concept unifying Lyapunov
and orbital stabilities. Communications in Applied Analysis, (2002), 6 (2).
 Lakshmikantham, V. and Leela, S. Stability theory of fuzzy differential
equations via differential inequalities. Math. Inequalities and Appl. 2
 Lakshmikantham, V., Leela, S. and Martynyuk, A.A. Practical
Stability of Nonlinear System. World Scientific Publishing, NJ,
 Lakshmikantham, V., Leela, S. and Martynyuk, A. A. Stability
Analysis of Nonlinear System. Marcel Dekker, New York, 1989.
 Lakshmikantham, V. and Mohapatra, R. Basic properties of solutions of
fuzzy differential equations. Nonlinear Studies 8 (2001) 113–124.
 Lakshmikantham, V. and Mohapatra, R. N. : Strict Stability of
Differential Equations, Nonlinear Analysis, Volume 46, Issue 7, Pages
 Lakshmikantham, V. and Mohapatra, R. N. Theory of Fuzzy
Differential Equations. Taylor and Francis Inc. New York, 2003.
 Lakshmikantham, V. and Vatsala, A.S., Differential inequalities with
time difference and application, Journal of Inequalities and Applications
3, (1999) 233-244.
 Li, A., Feng, E. and Li, S., Stability and boundedness criteria for
nonlinear differential systems relative to initial time difference and
applications. Nonlinear Analysis: Real World Applications 10(2009)
 Liu, K., Yang, G., Strict Stability Criteria for Impulsive Functional
Differential Systems, Journal of Inequalities and Applications ,
2008:243863 doi:10.1155/2008/243863, (2008).
 Lyapunov, A. Sur les fonctions-vecteurs completement additives. Bull.
Acad. Sci. URSS, Ser. Math 4 (1940) 465-478.
 Massera, J. L. The meaning of stability. Bol. Fac. Ing. Montevideo 8
 Nieto, J. J. The Cauchy problem for fuzzy differential equations. Fuzzy
Sets and Systems, (102 (1999), 259 - 262.
 Park, J. Y. and Hyo, K. H. Existence and uniqueness theorem for a
solution of Fuzzy differential equations, Inter. J. Math.and Math. Sci, 22
 Puri, M. L. D. and Ralescu, A. Differential of Fuzzy functions, J. Math.
Anal. Appl, 91 (1983), 552 - 558.
 Rojas K., Gomez D., Monteroa J., Rodrigueza J. Tinguaro, Strictly stable
families of aggregation operators, Fuzzy Sets and Systems, 228, 44–63
 Rojas K., Gomez D., Monteroa J., Rodrigueza J. Tinguaro, Valdivia A.,
Paiva F., Development of child’s home environment indexes based on
consistent families of aggregation operators with prioritized hierarchical
information, Fuzzy Sets and Systems 241, 41–60, (2014).
 Shaw, M. D. and Yakar, C., Generalized variation of parameters
with initial time difference and a comparison result in term
Lyapunov-like functions, International Journal of Non-linear Differential
Equations-Theory-Methods and Applications 5, (1999). 86-108.
 Shaw, M. D. and Yakar, C., Stability criteria and slowly growing
motions with initial time difference, Problems of Nonlinear Analysis in
Engineering Systems 1, (2000) 50-66.
 Song, S. J., Guo, L. and Feng, C. H. Global existence of solutions of
Fuzzy differential equations, Fuzzy Sets and Systems, 115 (2000), 371 -
 Song, S.J. and Wu, C. Existence and Uniqueness of solutions to Cauchy
problem of Fuzzy differential equations, Fuzzy Sets and Systems, 110
(2000), 55 - 67.
 Yakar, C. Boundedness criteria with initial time difference in terms of
two measures, Dynamics of Continuous, Discrete & Impulsive Systems.
Series A, vol. 14, supplement 2, (2007) 270–274, .
 Yakar C. and C¸ ic¸ek M. and G¨ucen, M.B. ”Boundedness and Lagrange
stability of fractional order perturbed system related to unperturbed
systems with initial time difference in Caputo’s sense.” Advances in
difference Equations (2011):54. doi:10.1186/1687-1847-2011-54. ISSN:
 Yakar C., C¸ ic¸ek M. and G¨ucen M. B. “Practical Stability
in Terms of Two Measures for Fractional Order Dynamic
Systems in Caputo’s Sense with Initial Time Difference” Journal
of the Franklin Institute.PII: S0016-0032(13)00377-3 DOI:
http://dx.doi.org/10.1016/j.jfranklin.2013.10.009. Ref.: FI1903. (2013).
 Yakar C., C¸ ic¸ek M. and G¨ucen M. B. “Practical Stability, Boundedness
Criteria and Lagrange Stability of Fuzzy Differential Systems” Journal
of Computers and Mathematics with Applications. 64 (2012) 2118-2127.
Doi: 10.1016/j.camwa.2012.04.008. (2012).
 Yakar, C. Strict stability criteria of perturbed systems with respect
to unperturbed systems in terms of initial time difference. Complex
Analysis and Potential Theory, World Scientific, Hackensack, NJ,
USA (2007) 239–248.
 Yakar, C. and Shaw, M. D. A comparison result and Lyapunov stability
criteria with initial time difference. Dynamics of Continuous, Discrete &
Impulsive Systems. Series A, vol. 12, no. 6, (2005) 731–737.
 Yakar, C. and Shaw, M. D. Initial time difference stability in terms of two
measures and a variational comparison result. Dynamics of Continuous,
Discrete & Impulsive Systems. Series A, vol. 15, no. 3, (2008) 417–425,
 Yakar, C. and Shaw, M. D. Practical stability in terms of two measures
with initial time difference. Nonlinear Analysis: Theory, Methods &
Applications, vol. 71, no. 12, (2009) e781–e785.
 Yoshizawa, T. Stability Theory by Lyapunov’s second Method, The
Mathematical Society of Japan, Tokyo, 1966.
 Zadeh, L. A. Fuzzy Sets, Inform. Control., 8 (1965), 338 - 353.
 Zhang, Y. Criteria for boundedness of Fuzzy differential equations,
Math. Ineq. Appl., 3 (2000), 399 -410.
 Zhang, Y., Sun J., Strict stability of impulsive functional differential
equations, J. Math. Anal. Appl., 301, 237–248 (2005).