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doi: 10.3934/dcdss.2021041

Hyers-Ulam-Rassias stability of high-dimensional quaternion impulsive fuzzy dynamic equations on time scales

Chao Wang 1, , Zhien Li 1, and  Ravi P. Agarwal 2,3,,

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

* Corresponding author: Ravi P. Agarwal

Received  January 2021 Published  April 2021

Fund Project: This work is supported by Youth Fund of NSFC (No. 11961077)

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.

Keywords: Impulsive fuzzy dynamic equations, Hyers-Ulam-Rassias stability, Quaternion space, Time Scales.
Mathematics Subject Classification: Primary: 34A07, 34N05; Secondary: 11R52.
Citation: Chao Wang, Zhien Li, Ravi P. Agarwal. Hyers-Ulam-Rassias stability of high-dimensional quaternion impulsive fuzzy dynamic equations on time scales. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021041
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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S. Georgiev and J. Morais, An introduction to the Hilger quaternion numbers, AIP Conference Proceedings, 1558 (2013), 550-553.  doi: 10.1063/1.4825549.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets Syst., 265 (2015), 63-85.  doi: 10.1016/j.fss.2014.04.005.  Google Scholar

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V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. doi: 10.1142/0906.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. E. Moore, Interval Analysis, Prentice Hall, New Jersey, 1966.  Google Scholar

[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.  Google Scholar

[30]

D. Ramot, R. Milo, M. Friedman and A. Kandel, Complex fuzzy sets, IEEE Transactions on Fuzzy Syst., 10 (2002), 171-186.   Google Scholar

[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.  Google Scholar

[32]

I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math., 26 (2010), 103-107.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8 Interscience Publishers, New York-London, 1960.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. J. Buckley, Fuzzy complex numbers, Fuzzy Sets and Syst., 33 (1989), 333-345.  doi: 10.1016/0165-0114(89)90122-X.  Google Scholar

[6]

S. Bernstein, Topics in Clifford Analysis, Special Volume in Honor of Wolfgang Sprößig, Trends in Mathematics, Birkhäuser, 2019. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 1994. doi: 10.1142/2326.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten (PhD Thesis), Universität Würzburg, 1989. Google Scholar

[16] S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[19]

O. Kaleva, A note on fuzzy differential equations, Nonlinear Anal., 64 (2006), 895-900.  doi: 10.1016/j.na.2005.01.003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets Syst., 265 (2015), 63-85.  doi: 10.1016/j.fss.2014.04.005.  Google Scholar

[25]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. doi: 10.1142/0906.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. E. Moore, Interval Analysis, Prentice Hall, New Jersey, 1966.  Google Scholar

[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.  Google Scholar

[30]

D. Ramot, R. Milo, M. Friedman and A. Kandel, Complex fuzzy sets, IEEE Transactions on Fuzzy Syst., 10 (2002), 171-186.   Google Scholar

[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.  Google Scholar

[32]

I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math., 26 (2010), 103-107.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8 Interscience Publishers, New York-London, 1960.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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