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Hyers-Ulam-Rassias stability of high-dimensional quaternion impulsive fuzzy dynamic equations on time scales
1. | Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China |
2. | Department of Mathematics, Texas A & M University-Kingsville, 700 University Blvd., TX 78363-8202, Kingsville, TX, USA |
3. | Distinguished University Professor of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA |
In this paper, the Hyers-Ulam-Rassias stability of high-dimensional quaternion fuzzy dynamic equations with impulses is first considered on time scales. Some fundamental calculus results of the high-dimensional fuzzy quaternion functions in fuzzy quaternion space are established. Based on it, some sufficient conditions are obtained to guarantee the Hyers-Ulam-Rassias stability of the quaternion impulsive fuzzy dynamic equations in high-dimensional case. Moreover, several examples are provided to show the feasibility of our main results on various types of time scales.
References:
[1] |
R. P. Agarwal, S. Arshad, D. O'Regan and V. Lupulescu,
Fuzzy fractional integral equations under compactness type condition, Fract. Calc. Appl. Anal., 15 (2012), 572-590.
doi: 10.2478/s13540-012-0040-1. |
[2] |
B. Bede and S. G. Gal,
Almost periodic fuzzy-number-valued functions, Fuzzy Sets Syst., 147 (2004), 385-403.
doi: 10.1016/j.fss.2003.08.004. |
[3] |
B. Bede and L. Stefanini,
Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst., 230 (2013), 119-141.
doi: 10.1016/j.fss.2012.10.003. |
[4] |
M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications. Birkhauser, Boston, 2001.
doi: 10.1007/978-1-4612-0201-1. |
[5] |
J. J. Buckley,
Fuzzy complex numbers, Fuzzy Sets and Syst., 33 (1989), 333-345.
doi: 10.1016/0165-0114(89)90122-X. |
[6] |
S. Bernstein, Topics in Clifford Analysis, Special Volume in Honor of Wolfgang Sprößig, Trends in Mathematics, Birkhäuser, 2019. |
[7] |
D. Cheng, K. I. Kou and Y. H. Xia,
A unified analysis of linear quaternion dynamic equations on time scales, J. Appl. Anal. Compt., 8 (2018), 172-201.
doi: 10.11948/2018.172. |
[8] |
Z. Cai and K. I. Kou,
Laplace transform: A new approach in solving linear quaternion differential equations, Math. Meth. Appl. Sci., 41 (2018), 4033-4048.
doi: 10.1002/mma.4415. |
[9] |
P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 1994.
doi: 10.1142/2326. |
[10] |
T. Fang and J. Sun,
Stability of complex-valued impulsive system with delay, Applied Mathematics and Computation, 240 (2014), 102-108.
doi: 10.1016/j.amc.2014.04.062. |
[11] |
T. Fang and J. Sun,
Stability of complex-valued impulsive and switching system and application to the Lü system, Nonlinear Anal.: Hybrid Syst., 14 (2014), 38-46.
doi: 10.1016/j.nahs.2014.04.004. |
[12] |
S. Georgiev and J. Morais,
An introduction to the Hilger quaternion numbers, AIP Conference Proceedings, 1558 (2013), 550-553.
doi: 10.1063/1.4825549. |
[13] |
D. H. Hyers,
On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27 (1941), 222-224.
doi: 10.1073/pnas.27.4.222. |
[14] |
D. H. Hyers, G. Isac and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, 1998.
doi: 10.1007/978-1-4612-1790-9. |
[15] |
S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten (PhD Thesis), Universität Würzburg, 1989. |
[16] |
S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
![]() ![]() |
[17] |
A. Khastan and R. Rodríguez-López,
On periodic solutions to first order linear fuzzy differential equations under differential inclusions' approach, Inf. Sci., 322 (2015), 31-50.
doi: 10.1016/j.ins.2015.06.003. |
[18] |
A. Khastan,
New solutions for first order linear fuzzy difference equations, J. Comput. Appl. Math., 312 (2017), 156-166.
doi: 10.1016/j.cam.2016.03.004. |
[19] |
O. Kaleva,
A note on fuzzy differential equations, Nonlinear Anal., 64 (2006), 895-900.
doi: 10.1016/j.na.2005.01.003. |
[20] |
K. I. Kou and Y. Xia,
Linear quaternion differential equations: Basic theory and fundamental results, Studies. Appl. Math., 141 (2018), 3-45.
doi: 10.1111/sapm.12211. |
[21] |
Z. Li, C. Wang, R. P. Agarwal and D. O'Regan,
Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on time scales, Stud. Appl. Math., 146 (2021), 139-210.
doi: 10.1111/sapm.12344. |
[22] |
Z. Li and C. Wang,
Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales, Open Math., 18 (2020), 353-377.
doi: 10.1515/math-2020-0021. |
[23] |
R. Liu, J. Wang and D. O'Regan,
Ulam type stability of first-order linear impulsive fuzzy differential equations, Fuzzy Sets and Syst., 400 (2020), 34-89.
doi: 10.1016/j.fss.2019.10.007. |
[24] |
V. Lupulescu,
Fractional calculus for interval-valued functions, Fuzzy Sets Syst., 265 (2015), 63-85.
doi: 10.1016/j.fss.2014.04.005. |
[25] |
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
doi: 10.1142/0906. |
[26] |
R. P. A. Moura, F. B. Bergamaschi, R. H. N. Santiago and B. R. C. Bedregal, Fuzzy quaternion numbers, FUZZ-IEEE, (2013), 1–6.
doi: 10.1109/FUZZ-IEEE.2013.6622400. |
[27] |
T. Miura, S. Miyajima and S. E. Takahasi,
A characterization of Hyers-Ulam stability of first order linear differential operators, J. Math. Anal. Appl., 286 (2003), 136-146.
doi: 10.1016/S0022-247X(03)00458-X. |
[28] |
R. E. Moore, Interval Analysis, Prentice Hall, New Jersey, 1966. |
[29] |
J. J. Nieto and R. Rodríguez-López,
Bounded solutions for fuzzy differential and integral equations, Chaos Solitons & Frac., 27 (2006), 1376-1386.
doi: 10.1016/j.chaos.2005.05.012. |
[30] |
D. Ramot, R. Milo, M. Friedman and A. Kandel,
Complex fuzzy sets, IEEE Transactions on Fuzzy Syst., 10 (2002), 171-186.
|
[31] |
T. M. Rassias,
On the stability of linear mappings in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297-300.
doi: 10.1090/S0002-9939-1978-0507327-1. |
[32] |
I. A. Rus,
Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math., 26 (2010), 103-107.
|
[33] |
R. Rodríguez-López,
Monotone method for fuzzy differential equations, Fuzzy Sets and Syst., 159 (2008), 2047-2076.
doi: 10.1016/j.fss.2007.12.020. |
[34] |
R. Rodríguez-López,
Periodic boundary value problems for impulsive fuzzy differential equations, Fuzzy Sets and Syst., 159 (2008), 1384-1409.
doi: 10.1016/j.fss.2007.09.005. |
[35] |
L. Stefanini,
A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets Syst., 161 (2010), 1564-1584.
doi: 10.1016/j.fss.2009.06.009. |
[36] |
Y. Shen,
On the Ulam stability of first order linear fuzzy differential equations under generalized differentiability, Fuzzy Sets Syst., 280 (2015), 27-57.
doi: 10.1016/j.fss.2015.01.002. |
[37] |
D. E. Tamir, L. Jin and A. Kandel,
A new interpretation of complex membership grade, J. Intelligent and Fuzzy Syst., 26 (2011), 285-312.
doi: 10.1002/int.20454. |
[38] |
S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8 Interscience Publishers, New York-London, 1960. |
[39] |
C. Wang, R. P. Agarwal and D. O'Regan,
Calculus of fuzzy vector-valued functions and almost periodic fuzzy vector-valued functions on time scales, Fuzzy Sets and Syst., 375 (2019), 1-52.
doi: 10.1016/j.fss.2018.12.008. |
[40] |
C. Wang, R. P. Agarwal and D. O'Regan,
$\Pi$-semigroup for invariant under translations time scales and abstract weighted pseudo almost periodic functions with applications, Dyna. Syst. Appl., 25 (2016), 1-28.
|
[41] |
C. Wang, R. P. Agarwal and R. Sakthivel,
Almost periodic oscillations for delay impulsive stochastic Nicholson's blowflies timescale model, Comput. Appl. Math., 37 (2018), 3005-3026.
doi: 10.1007/s40314-017-0495-0. |
[42] |
C. Wang, G. Qin, R. P. Agarwal and D. O'Regan, $\Diamond_{\alpha}$-Measurability and combined measure theory on time scales, Applic. Anal., 2020.
doi: 10.1080/00036811.2020.1820997. |
[43] |
C. Wang, R. P. Agarwal and D. O'Regan,
$n_0$-order $\Delta$-almost periodic functions and dynamic equations, Applic. Anal., 97 (2018), 2626-2654.
doi: 10.1080/00036811.2017.1382689. |
[44] |
C. Wang, R. P. Agarwal, D. O'Regan and R. Sakthivel, Theory of Translation Closedness for Time Scales, , Springer, Developments in Mathematics Series, Volume 62, Switzerland, 2020.
doi: 10.1007/978-3-030-38644-3. |
[45] |
C. Wang and R. P. Agarwal,
Almost periodic solution for a new type of neutral impulsive stochastic Lasota-Wazewska timescale model, Appl. Math. Lett., 70 (2017), 58-65.
doi: 10.1016/j.aml.2017.03.009. |
[46] |
C. Wang and R. P. Agarwal,
Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations, Discrt. Contin. Dynam. Syst. B, 25 (2020), 781-798.
doi: 10.3934/dcdsb.2019267. |
[47] |
J. Wang, M. Fěckan and Y. Zhou,
Ulam's type stability of impulsive ordinary differential equations, J. Math. Anal. Appl., 395 (2012), 258-264.
doi: 10.1016/j.jmaa.2012.05.040. |
[48] |
X. X. Hu and K. I. Kou,
Quaternion Fourier and linear canonical inversion theorems, Math. Meth. Appl. Sci., 40 (2017), 2421-2440.
doi: 10.1002/mma.4148. |
[49] |
Y. Xia, C. Jahanchahi and D. P. Mandic,
Quaternion-valued echo state networks, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 663-673.
doi: 10.1109/TNNLS.2014.2320715. |
[50] |
Z. P. Yang, T. Z. Xu and M. Qi,
Ulam-Hyers stability for fractional differential equations in quaternionic analysis, Adv. Appl. Clifford Algebras, 26 (2016), 469-478.
doi: 10.1007/s00006-015-0576-3. |
[51] |
Z. P. Yang, T. Z. Xu and M. Qi,
The Cauchy problem for quaternion fuzzy fractional differential equations, J. Intelligent & Fuzzy Syst., 29 (2015), 451-461.
doi: 10.3233/IFS-151612. |
show all references
References:
[1] |
R. P. Agarwal, S. Arshad, D. O'Regan and V. Lupulescu,
Fuzzy fractional integral equations under compactness type condition, Fract. Calc. Appl. Anal., 15 (2012), 572-590.
doi: 10.2478/s13540-012-0040-1. |
[2] |
B. Bede and S. G. Gal,
Almost periodic fuzzy-number-valued functions, Fuzzy Sets Syst., 147 (2004), 385-403.
doi: 10.1016/j.fss.2003.08.004. |
[3] |
B. Bede and L. Stefanini,
Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst., 230 (2013), 119-141.
doi: 10.1016/j.fss.2012.10.003. |
[4] |
M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications. Birkhauser, Boston, 2001.
doi: 10.1007/978-1-4612-0201-1. |
[5] |
J. J. Buckley,
Fuzzy complex numbers, Fuzzy Sets and Syst., 33 (1989), 333-345.
doi: 10.1016/0165-0114(89)90122-X. |
[6] |
S. Bernstein, Topics in Clifford Analysis, Special Volume in Honor of Wolfgang Sprößig, Trends in Mathematics, Birkhäuser, 2019. |
[7] |
D. Cheng, K. I. Kou and Y. H. Xia,
A unified analysis of linear quaternion dynamic equations on time scales, J. Appl. Anal. Compt., 8 (2018), 172-201.
doi: 10.11948/2018.172. |
[8] |
Z. Cai and K. I. Kou,
Laplace transform: A new approach in solving linear quaternion differential equations, Math. Meth. Appl. Sci., 41 (2018), 4033-4048.
doi: 10.1002/mma.4415. |
[9] |
P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 1994.
doi: 10.1142/2326. |
[10] |
T. Fang and J. Sun,
Stability of complex-valued impulsive system with delay, Applied Mathematics and Computation, 240 (2014), 102-108.
doi: 10.1016/j.amc.2014.04.062. |
[11] |
T. Fang and J. Sun,
Stability of complex-valued impulsive and switching system and application to the Lü system, Nonlinear Anal.: Hybrid Syst., 14 (2014), 38-46.
doi: 10.1016/j.nahs.2014.04.004. |
[12] |
S. Georgiev and J. Morais,
An introduction to the Hilger quaternion numbers, AIP Conference Proceedings, 1558 (2013), 550-553.
doi: 10.1063/1.4825549. |
[13] |
D. H. Hyers,
On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27 (1941), 222-224.
doi: 10.1073/pnas.27.4.222. |
[14] |
D. H. Hyers, G. Isac and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, 1998.
doi: 10.1007/978-1-4612-1790-9. |
[15] |
S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten (PhD Thesis), Universität Würzburg, 1989. |
[16] |
S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
![]() ![]() |
[17] |
A. Khastan and R. Rodríguez-López,
On periodic solutions to first order linear fuzzy differential equations under differential inclusions' approach, Inf. Sci., 322 (2015), 31-50.
doi: 10.1016/j.ins.2015.06.003. |
[18] |
A. Khastan,
New solutions for first order linear fuzzy difference equations, J. Comput. Appl. Math., 312 (2017), 156-166.
doi: 10.1016/j.cam.2016.03.004. |
[19] |
O. Kaleva,
A note on fuzzy differential equations, Nonlinear Anal., 64 (2006), 895-900.
doi: 10.1016/j.na.2005.01.003. |
[20] |
K. I. Kou and Y. Xia,
Linear quaternion differential equations: Basic theory and fundamental results, Studies. Appl. Math., 141 (2018), 3-45.
doi: 10.1111/sapm.12211. |
[21] |
Z. Li, C. Wang, R. P. Agarwal and D. O'Regan,
Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on time scales, Stud. Appl. Math., 146 (2021), 139-210.
doi: 10.1111/sapm.12344. |
[22] |
Z. Li and C. Wang,
Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales, Open Math., 18 (2020), 353-377.
doi: 10.1515/math-2020-0021. |
[23] |
R. Liu, J. Wang and D. O'Regan,
Ulam type stability of first-order linear impulsive fuzzy differential equations, Fuzzy Sets and Syst., 400 (2020), 34-89.
doi: 10.1016/j.fss.2019.10.007. |
[24] |
V. Lupulescu,
Fractional calculus for interval-valued functions, Fuzzy Sets Syst., 265 (2015), 63-85.
doi: 10.1016/j.fss.2014.04.005. |
[25] |
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
doi: 10.1142/0906. |
[26] |
R. P. A. Moura, F. B. Bergamaschi, R. H. N. Santiago and B. R. C. Bedregal, Fuzzy quaternion numbers, FUZZ-IEEE, (2013), 1–6.
doi: 10.1109/FUZZ-IEEE.2013.6622400. |
[27] |
T. Miura, S. Miyajima and S. E. Takahasi,
A characterization of Hyers-Ulam stability of first order linear differential operators, J. Math. Anal. Appl., 286 (2003), 136-146.
doi: 10.1016/S0022-247X(03)00458-X. |
[28] |
R. E. Moore, Interval Analysis, Prentice Hall, New Jersey, 1966. |
[29] |
J. J. Nieto and R. Rodríguez-López,
Bounded solutions for fuzzy differential and integral equations, Chaos Solitons & Frac., 27 (2006), 1376-1386.
doi: 10.1016/j.chaos.2005.05.012. |
[30] |
D. Ramot, R. Milo, M. Friedman and A. Kandel,
Complex fuzzy sets, IEEE Transactions on Fuzzy Syst., 10 (2002), 171-186.
|
[31] |
T. M. Rassias,
On the stability of linear mappings in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297-300.
doi: 10.1090/S0002-9939-1978-0507327-1. |
[32] |
I. A. Rus,
Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math., 26 (2010), 103-107.
|
[33] |
R. Rodríguez-López,
Monotone method for fuzzy differential equations, Fuzzy Sets and Syst., 159 (2008), 2047-2076.
doi: 10.1016/j.fss.2007.12.020. |
[34] |
R. Rodríguez-López,
Periodic boundary value problems for impulsive fuzzy differential equations, Fuzzy Sets and Syst., 159 (2008), 1384-1409.
doi: 10.1016/j.fss.2007.09.005. |
[35] |
L. Stefanini,
A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets Syst., 161 (2010), 1564-1584.
doi: 10.1016/j.fss.2009.06.009. |
[36] |
Y. Shen,
On the Ulam stability of first order linear fuzzy differential equations under generalized differentiability, Fuzzy Sets Syst., 280 (2015), 27-57.
doi: 10.1016/j.fss.2015.01.002. |
[37] |
D. E. Tamir, L. Jin and A. Kandel,
A new interpretation of complex membership grade, J. Intelligent and Fuzzy Syst., 26 (2011), 285-312.
doi: 10.1002/int.20454. |
[38] |
S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8 Interscience Publishers, New York-London, 1960. |
[39] |
C. Wang, R. P. Agarwal and D. O'Regan,
Calculus of fuzzy vector-valued functions and almost periodic fuzzy vector-valued functions on time scales, Fuzzy Sets and Syst., 375 (2019), 1-52.
doi: 10.1016/j.fss.2018.12.008. |
[40] |
C. Wang, R. P. Agarwal and D. O'Regan,
$\Pi$-semigroup for invariant under translations time scales and abstract weighted pseudo almost periodic functions with applications, Dyna. Syst. Appl., 25 (2016), 1-28.
|
[41] |
C. Wang, R. P. Agarwal and R. Sakthivel,
Almost periodic oscillations for delay impulsive stochastic Nicholson's blowflies timescale model, Comput. Appl. Math., 37 (2018), 3005-3026.
doi: 10.1007/s40314-017-0495-0. |
[42] |
C. Wang, G. Qin, R. P. Agarwal and D. O'Regan, $\Diamond_{\alpha}$-Measurability and combined measure theory on time scales, Applic. Anal., 2020.
doi: 10.1080/00036811.2020.1820997. |
[43] |
C. Wang, R. P. Agarwal and D. O'Regan,
$n_0$-order $\Delta$-almost periodic functions and dynamic equations, Applic. Anal., 97 (2018), 2626-2654.
doi: 10.1080/00036811.2017.1382689. |
[44] |
C. Wang, R. P. Agarwal, D. O'Regan and R. Sakthivel, Theory of Translation Closedness for Time Scales, , Springer, Developments in Mathematics Series, Volume 62, Switzerland, 2020.
doi: 10.1007/978-3-030-38644-3. |
[45] |
C. Wang and R. P. Agarwal,
Almost periodic solution for a new type of neutral impulsive stochastic Lasota-Wazewska timescale model, Appl. Math. Lett., 70 (2017), 58-65.
doi: 10.1016/j.aml.2017.03.009. |
[46] |
C. Wang and R. P. Agarwal,
Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations, Discrt. Contin. Dynam. Syst. B, 25 (2020), 781-798.
doi: 10.3934/dcdsb.2019267. |
[47] |
J. Wang, M. Fěckan and Y. Zhou,
Ulam's type stability of impulsive ordinary differential equations, J. Math. Anal. Appl., 395 (2012), 258-264.
doi: 10.1016/j.jmaa.2012.05.040. |
[48] |
X. X. Hu and K. I. Kou,
Quaternion Fourier and linear canonical inversion theorems, Math. Meth. Appl. Sci., 40 (2017), 2421-2440.
doi: 10.1002/mma.4148. |
[49] |
Y. Xia, C. Jahanchahi and D. P. Mandic,
Quaternion-valued echo state networks, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 663-673.
doi: 10.1109/TNNLS.2014.2320715. |
[50] |
Z. P. Yang, T. Z. Xu and M. Qi,
Ulam-Hyers stability for fractional differential equations in quaternionic analysis, Adv. Appl. Clifford Algebras, 26 (2016), 469-478.
doi: 10.1007/s00006-015-0576-3. |
[51] |
Z. P. Yang, T. Z. Xu and M. Qi,
The Cauchy problem for quaternion fuzzy fractional differential equations, J. Intelligent & Fuzzy Syst., 29 (2015), 451-461.
doi: 10.3233/IFS-151612. |
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