Hyers-Ulam stability of linear quaternion-valued differential equations
DOI:
https://doi.org/10.58997/ejde.2023.21Abstract
In this article, we study the Hyers-Ulam stability of the first-order linear quaternion-valued differential equations. We transfer a linear quaternion-valued differential equation into a real differential system. The Hyers-Ulam stability results for the linear quaternion-valued differential equations are obtained according to the equivalent relationship between the vector 2-norm and the quaternion module.
For more information see https://ejde.math.txstate.edu/Volumes/2023/21/abstr.html
References
D. R. Anderson, M. Onitsuka; Hyers-Ulam stability for differential systems with 2×2 constant coefficient matrix, Results in Mathematics, 77 (2022), Art. 136. https://doi.org/10.1007/s00025-022-01671-y
D. C. Brody, E. M. Graefe; On complexified mechanics and coquaternions, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 072001. https://doi.org/10.1088/1751-8113/44/7/072001
L. Borsten, D. Dahanayake, M. J. Duff, et al.; Black holes, qubits and octonions, Physics Reports, 471(2009), 113-219. https://doi.org/10.1016/j.physrep.2008.11.002
D. Chen, M. Fěckan, J. Wang; On the stability of linear quaternion-valued differential equations, Qualitative Theory of Dynamical Systems, 21 (2022), Art. 9. https://doi.org/10.1007/s12346-022-00599-6
D. Chen, M. Fěckan, J. Wang; Hyers-Ulam stability for linear quaternion-valued differential equations with constant coefficient, Rocky Mountain Journal of Mathematics, 52 (2022), 1237-1250. https://doi.org/10.1216/rmj.2022.52.1237
D. Chen, M. Fěckan, J. Wang; Investigation of controllability and observability for linear quaternion-valued systems from its complex-valued systems, Qualitative Theory of Dynamical Systems, 21 (2022), Art. 66. https://doi.org/10.1007/s12346-022-00599-6
C. M. Carlevaro, R. M. Irastorza, F. Vericat; Quaternionic representation of the genetic code, Biosystems, 141 (2016), 10-19. https://doi.org/10.1016/j.biosystems.2015.12.009
T. Fu, K. Kou, J. Wang; Representation of solutions to linear quaternion differential equations with delay, Qualitative Theory of Dynamical Systems, 21 (2022), Art. 118. https://doi.org/10.1007/s12346-022-00648-0
S. Huang, F. He; On the second Lyapunov method for quaternionic differential equations, Qualitative Theory of Dynamical Systems, 20 (2021), Art. 41. https://doi.org/10.1007/s12346-021-00476-8
R. A. Horn, C. R. Johnson; Matrix Analysis, Cambridge University Press, New York, 2012, ISBN 978-0-521-83940-2.
S. M. Jung; Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, Journal of Mathematical Analysis and Applications, 320 (2006), 549- 561. https://doi.org/10.1016/j.jmaa.2005.07.032
S. M. Jung; Hyers-Ulam stability of the first-order matrix differential equations, Journal of Function Spaces, 2015 (2015), 614745. https://doi.org/10.1186/s13662-015-0507-6
S. M. Jung; Hyers-Ulam stability of linear differential equations of first order, Applied Mathematics Letters, 17 (2004), 1135-1140. https://doi.org/10.1016/j.aml.2003.11.004
T. Jiang, Z. Jiang, S. Ling; An algebraic method for quaternion and complex least squares coneigen-problem in quantum mechanics, Applied Mathematics and Computation, 249 (2014), 222-228. https://doi.org/10.1016/j.amc.2014.10.075
K. Kou, Y. Xia; Linear quaternion differential equations: Basic theory and fundamental results, Studies in Applied Mathematics, 141 (2018), 3-45. https://doi.org/10.1111/sapm.12211
K. Kou, W. Liu, Y. Xia; Solve the linear quaternion-valued differential equations having multiple eigenvalues, Journal of Mathematical Physics, 60 (2019), 023510.
M. Kobayashi; Split quaternion-valued twin-multistate Hopfield neural networks, Advances in Applied Clifford Algebras, 30 (2020), Art. 30. https://doi.org/10.1007/s00006-020-01056-w
J. Lv, K. Kou, J. Wang; Hyers-Ulam stability of linear quaternion-valued differential equations with constant coefficients via Fourier transform, Qualitative Theory of Dynamical Systems, 21 (2022), Art. 116. https://doi.org/10.1007/s12346-022-00649-z
T. Politi; A formula for the exponential of a real skew-symmetric matrix of order 4, BIT Numerical Mathematics, 41 (2001), 842-845. https://doi.org/10.1023/A:1021960405660
A. Pratap, R. Raja, J. Alzabut, et al.; Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field, Mathematical Methods in the Applied Sciences, 43 (2020), 6223-6253. https://doi.org/10.1002/mma.6367
J. Prǒskova; Description of protein secondary structure using dual quaternions, Journal of Molecular Structure, 1076 (2014), 89-93. https://doi.org/10.1016/j.molstruc.2014.07.031
L. Suo, M. Fěckan, J. Wang; Quaternion-valued linear impulsive differential equations, Qual- itative Theory of Dynamical Systems, 20 (2021), Art. 33. https://doi.org/10.1007/s12346-021-00467-9
L. Suo, M. Fěckan, J. Wang; Existence of periodic solutions to quaternion-valued impulsive differential equations, Qualitative Theory of Dynamical Systems, 22 (2023), Art. 1. https://doi.org/10.1007/s12346-022-00693-9
Y. Xia, K. Kou, Y. Liu; Theory and applications of quaternion-valued differential equations, Science Press, Beijing, 2021, ISBN 978-7-03-069056-2.
Y. Xia, H. Huang, K. Kou; An algorithm for solving linear nonhomogeneous quaternion- valued differential equations and some open problems, Discrete and Continuous Dynamical Systems-Series S, 15 (2022), 1685-1697. https://doi.org/10.3934/dcdss.2021162
M. Yoshida, Y. Kuroe, T. Mori; Models of hopfield-type quaternion neural networks and their energy functions, International Journal of Neural Systems, 15 (2005), 129-135. https://doi.org/10.1142/S012906570500013X
Y. Zou, M. Fěckan, J. Wang; Hyers-Ulam stability of linear recurrence with constant coefficients over the quaternion skew yield, Qualitative Theory of Dynamical Systems, 22 (2023), Art. 3. https://doi.org/10.1007/s12346-022-00695-7
X. Zhang; Global structure of quaternion polynomial differential equations, Communications in Mathematical Physics, 303 (2011), 301-316. https://doi.org/10.1007/s00220-011-1196-y
M. Zahid, A. Younus, M. E. Ghoneim, et al.; Quaternion-valued exponential matrices and its fundamental properties, International Journal of Modern Physics B, 37 (2022), 2350027. https://doi.org/10.1142/S0217979223500273
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Copyright (c) 2023 Jiaojiao Lv, Jinrong Wang, Rui Liu
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