Document Type : Research Paper
Authors
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 prove the existence and uniqueness of solution to the impulsive fuzzy functional differential equations under generalized Hukuhara differentiability via the principle of contraction mappings. Some examples are provided to illustrate the result.
Keywords
[1] T. Allahviranloo, S. Abbasbandy, S. Salahshour and A. Hakimzadeh, A new method for
solving fuzzy linear di erential equations, Computing, 92 (2010), 181{197.
[2] T. Allahviranloo, S. Salahshour and S. Abbasbandy, Explicit solutions of fractional di eren-
tial equations with uncertainty, Soft Computing, 16 (2011), 297{302.
[3] T. Allahviranloo, S. Abbasbandy, O. Sedaghgatfar and P. Darabi, A new method for solving
fuzzy integro-di erential equation under generalized di erentiability, Neural Computing and
Applications, 21 (2011), 191{196.
[4] L. C. Barros, R. C. Bassanezi and P. A. Tonelli, Fuzzy modelling in population dynamics,
Ecological Modelling, 128 (2000), 27{33.
[5] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive di erential equations and inclusions,
Hindawi Publishing Corporation, USA, 2006.
[6] M. Benchohra, J. J. Nieto and A. Ouahab, Fuzzy solutions for impulsive di erential equations,
Communications in Applied Analysis, 11 (2007), 379{394.
[7] J. J. Buckley and T. Feuring, Fuzzy di erential equations, Fuzzy Sets and Systems, 110
(2000), 43 { 54.
[8] V. J. Devi and A. S. Vatsala, Method of vector lyapunov functions for impulsive fuzzy systems,
Dynamic Systems and Applications, 13 (2004), 521{531.
[9] L. S. Dong, H. Vu and N. V. Hoa, The formulas of the solution for linear-order random fuzzy
di erential equations, Journal of Intelligent & Fuzzy Systems, 28 (2015), 795{807.
[10] M. Guo, X. Xue and R. Li, Impulsive functional di erential inclusions and fuzzy population
models, Fuzzy Sets and Systems, 138 (2003), 601{615.
[11] N. V. Hoa, Fuzzy fractional functional di erential equations under Caputo gH-
di erentiability, Communications in Nonlinear Science and Numerical Simulation, 22 (2015),
1134-1157.
[12] N. V. Hoa, Fuzzy fractional functional integral and di erential equations, Fuzzy Sets and
Systems, 280 (2015), 58-90.
[13] N. V. Hoa and N. D. Phu, Fuzzy functional integro-di erential equations under generalized
H-di erentiability, Journal of Intelligent & Fuzzy Systems, 26 (2014), 2073{2085.
[14] N. V. Hoa, N. D. Phu, T. T. Tung and L. T. Quang, Interval-valued functional integro-
di erential equations, Advance in Di erence Equations, (2014), 2014:177.
[15] N. V. Hoa, P. V. Tri, T. T. Dao and I. Zelinka, Some global existence results and stability
theorem for fuzzy functional di erential equations, Journal of Intelligent & Fuzzy Systems,
28 (2015), 393{409.
[16] O. Kaleva, Fuzzy di erential equations, Fuzzy Sets and Systems, 24 (1987), 301{317.
[17] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of impulsive di erential
equations, World Scienti c, 1989.
solving fuzzy linear di erential equations, Computing, 92 (2010), 181{197.
[2] T. Allahviranloo, S. Salahshour and S. Abbasbandy, Explicit solutions of fractional di eren-
tial equations with uncertainty, Soft Computing, 16 (2011), 297{302.
[3] T. Allahviranloo, S. Abbasbandy, O. Sedaghgatfar and P. Darabi, A new method for solving
fuzzy integro-di erential equation under generalized di erentiability, Neural Computing and
Applications, 21 (2011), 191{196.
[4] L. C. Barros, R. C. Bassanezi and P. A. Tonelli, Fuzzy modelling in population dynamics,
Ecological Modelling, 128 (2000), 27{33.
[5] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive di erential equations and inclusions,
Hindawi Publishing Corporation, USA, 2006.
[6] M. Benchohra, J. J. Nieto and A. Ouahab, Fuzzy solutions for impulsive di erential equations,
Communications in Applied Analysis, 11 (2007), 379{394.
[7] J. J. Buckley and T. Feuring, Fuzzy di erential equations, Fuzzy Sets and Systems, 110
(2000), 43 { 54.
[8] V. J. Devi and A. S. Vatsala, Method of vector lyapunov functions for impulsive fuzzy systems,
Dynamic Systems and Applications, 13 (2004), 521{531.
[9] L. S. Dong, H. Vu and N. V. Hoa, The formulas of the solution for linear-order random fuzzy
di erential equations, Journal of Intelligent & Fuzzy Systems, 28 (2015), 795{807.
[10] M. Guo, X. Xue and R. Li, Impulsive functional di erential inclusions and fuzzy population
models, Fuzzy Sets and Systems, 138 (2003), 601{615.
[11] N. V. Hoa, Fuzzy fractional functional di erential equations under Caputo gH-
di erentiability, Communications in Nonlinear Science and Numerical Simulation, 22 (2015),
1134-1157.
[12] N. V. Hoa, Fuzzy fractional functional integral and di erential equations, Fuzzy Sets and
Systems, 280 (2015), 58-90.
[13] N. V. Hoa and N. D. Phu, Fuzzy functional integro-di erential equations under generalized
H-di erentiability, Journal of Intelligent & Fuzzy Systems, 26 (2014), 2073{2085.
[14] N. V. Hoa, N. D. Phu, T. T. Tung and L. T. Quang, Interval-valued functional integro-
di erential equations, Advance in Di erence Equations, (2014), 2014:177.
[15] N. V. Hoa, P. V. Tri, T. T. Dao and I. Zelinka, Some global existence results and stability
theorem for fuzzy functional di erential equations, Journal of Intelligent & Fuzzy Systems,
28 (2015), 393{409.
[16] O. Kaleva, Fuzzy di erential equations, Fuzzy Sets and Systems, 24 (1987), 301{317.
[17] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of impulsive di erential
equations, World Scienti c, 1989.
[18] V. Lakshmikantham, T. Gnana Bhaskar and Devi J. Vasundhara, Theory of set di erential
equations in a metric space, Cambridge Scienti c Publishing, UK, 2006.
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Mathematical Inequalities & Applications, 4 (2001), 239{246.
[20] V. Lupulescu, On a class of fuzzy functional di erential equations, Fuzzy Sets and Systems,
160 (2009), 1547{1562.
[21] R. N. Mohapatra and V. Lakshmikantham, Theory of fuzzy di erential equations and inclu-
sions, CRC Press, Singapore, 2003.
[22] J. J. Nieto, A. Khastan and K. Ivaz, Numerical solution of fuzzy di erential equations under
generalized di erentiability, Nonlinear Analysis: Hybrid Systems, 3 (2009), 700{707.
[23] J. J. Nieto and R. Rodrguez-Lopez, Periodic boundary value problem for non-Lipschitzian
impulsive functional di erential equations, Journal of Mathematical Analysis and Applica-
tions, 318 (2006), 593-610.
[24] R. Rodrguez-Lopez, Periodic boundary value problems for impulsive fuzzy di erential equa-
tions, Fuzzy Sets and Systems, 159 (2008), 1384{1409.
[25] A. M. Samoilenko and N. A. Perestyuk, Impulsive di erential equations, World Scienti c,
Singapore, 1995.
[26] P. V. Tri, N. V. Hoa and N. D. Phu, Sheaf fuzzy problems for functional di erential equations,
Advance in Di erence Equation, 2014 2014:156
[27] A. S. Vatsala, Impulsive hybrid fuzzy di erential equations, Facta Univ. Ser Mech, Automatic
Control Robot, 3 (2003), 851{859.
[28] H. Vu, L. S. Dong and N. V. Hoa, Random fuzzy functional integro-di erential equations
under generalized Hukuhara di erentiability , Journal of Intelligent & Fuzzy Systems, 27
(2014), 1491-1506.
[29] H. Vu and L. S. Dong, Random set-valued functional di erential equations with the second
type hukuhara derivative, Di erential Equations & Applications, 5 (2013), 501{518.
[30] H. Vu and L. S. Dong, Initial value problem for second-order random fuzzy di erential equa-
tions, Advances in Di erence Equations, 2015 2015:373
equations in a metric space, Cambridge Scienti c Publishing, UK, 2006.
[19] V. Lakshmikantham and F. A. McRae, Basic results for fuzzy impulsive di erential equations,
Mathematical Inequalities & Applications, 4 (2001), 239{246.
[20] V. Lupulescu, On a class of fuzzy functional di erential equations, Fuzzy Sets and Systems,
160 (2009), 1547{1562.
[21] R. N. Mohapatra and V. Lakshmikantham, Theory of fuzzy di erential equations and inclu-
sions, CRC Press, Singapore, 2003.
[22] J. J. Nieto, A. Khastan and K. Ivaz, Numerical solution of fuzzy di erential equations under
generalized di erentiability, Nonlinear Analysis: Hybrid Systems, 3 (2009), 700{707.
[23] J. J. Nieto and R. Rodrguez-Lopez, Periodic boundary value problem for non-Lipschitzian
impulsive functional di erential equations, Journal of Mathematical Analysis and Applica-
tions, 318 (2006), 593-610.
[24] R. Rodrguez-Lopez, Periodic boundary value problems for impulsive fuzzy di erential equa-
tions, Fuzzy Sets and Systems, 159 (2008), 1384{1409.
[25] A. M. Samoilenko and N. A. Perestyuk, Impulsive di erential equations, World Scienti c,
Singapore, 1995.
[26] P. V. Tri, N. V. Hoa and N. D. Phu, Sheaf fuzzy problems for functional di erential equations,
Advance in Di erence Equation, 2014 2014:156
[27] A. S. Vatsala, Impulsive hybrid fuzzy di erential equations, Facta Univ. Ser Mech, Automatic
Control Robot, 3 (2003), 851{859.
[28] H. Vu, L. S. Dong and N. V. Hoa, Random fuzzy functional integro-di erential equations
under generalized Hukuhara di erentiability , Journal of Intelligent & Fuzzy Systems, 27
(2014), 1491-1506.
[29] H. Vu and L. S. Dong, Random set-valued functional di erential equations with the second
type hukuhara derivative, Di erential Equations & Applications, 5 (2013), 501{518.
[30] H. Vu and L. S. Dong, Initial value problem for second-order random fuzzy di erential equa-
tions, Advances in Di erence Equations, 2015 2015:373
)