1. bookVolume 29 (2021): Issue 2 (June 2021)
Journal Details
License
Format
Journal
eISSN
1844-0835
First Published
17 May 2013
Publication timeframe
1 time per year
Languages
English
access type Open Access

Fundamental solution matrix and Cauchy properties of quaternion combined impulsive matrix dynamic equation on time scales

Published Online: 08 Jul 2021
Volume & Issue: Volume 29 (2021) - Issue 2 (June 2021)
Page range: 107 - 130
Received: 11 Jan 2021
Accepted: 19 Jan 2021
Journal Details
License
Format
Journal
eISSN
1844-0835
First Published
17 May 2013
Publication timeframe
1 time per year
Languages
English
Abstract

In this paper, we establish some basic results for quaternion combined impulsive matrix dynamic equation on time scales for the first time. Quaternion matrix combined-exponential function is introduced and some basic properties are obtained. Based on this, the fundamental solution matrix and corresponding Cauchy matrix for a class of quaternion matrix dynamic equation with combined derivatives and bi-directional impulses are derived.

Keywords

MSC 2010

[1] S. Adler, Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York, 1994. Search in Google Scholar

[1] R.P. Agarwal, C. Wang, D. O’Regan, Recent development of time scales and related topics on dynamic equations. Mem. Differential Equations Math. Phys. 67 (2016) 131-135. Search in Google Scholar

[3] F.M. Atici, G.Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 18, (2002) 75-99. Search in Google Scholar

[4] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications. Birkhauser, Boston, 2001.10.1007/978-1-4612-0201-1 Search in Google Scholar

[5] Z. Cai, K.I. Kou, Laplace transform: a new approach in solving linear quaternion differential equations, Math. Meth. Appl. Sci. 41, (2018) 4033-4048. Search in Google Scholar

[6] D. Cheng, K.I. Kou, Y.H. Xia, A unified analysis of linear quaternion dynamic equations on time scales, J. Appl. Anal. Comput. 8, (2018) 172-201. Search in Google Scholar

[7] D.J. Gibbon, D.D. Holm, R.M. Kerr, I. Roulstone, Quaternions and particle dynamics in the Euler fluid equations, Nonlinearity, 19, (2006) 1969-1983.10.1088/0951-7715/19/8/011 Search in Google Scholar

[8] W.R. Hamilton, Elements of Quaternions. London, UK: Longmans, Green, & Co, 1866. Search in Google Scholar

[9] S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannig-faltigkeiten, Ph.D. Thesis, Universit¨at Würzburg, 1988. Search in Google Scholar

[10] K.I. Kou, Y.H. Xia, Linear quaternion differential equations: basic theory and fundamental results, Stud. Appl. Math. 141, (2018) 3-45. Search in Google Scholar

[11] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.10.1142/0906 Search in Google Scholar

[12] Z. Li, C. Wang, R.P. Agarwal, The non-eigenvalue form of Liouville’s formula and α-matrix exponential solutions for combined matrix dynamic equations on time scales, Mathematics, 7(10), 962; https://doi.org/10.3390/math7100962 (2019).10.3390/math7100962 Search in Google Scholar

[13] Z. Li, C. Wang, R.P. Agarwal, D. O’Regan, Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on time scales, Stud. Appl. Math., 146, (2021) 139-210.10.1111/sapm.12344 Search in Google Scholar

[14] Z. Li, C. Wang, R.P. Agarwal, R. Sakthivel, Hyers-Ulam-Rassias stability of quaternion multidimensional fuzzy nonlinear difference equations with impulses, Iranian J. Fuzzy Syst., 10.22111/IJFS.2021.5950 (2021). Search in Google Scholar

[15] Z. Li, C. Wang, Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales, Open Math., 18, (2020) 353-377.10.1515/math-2020-0021 Search in Google Scholar

[16] G. Qin, C. Wang, Lebesgue-Stieltjes combined ♢α-measure and integral on time scales, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RAC-SAM. https://doi.org/10.1007/s13398-021-01000-y, 115:50 (2021).10.1007/s13398-021-01000-y Search in Google Scholar

[17] J.W. Rogers, Q. Sheng, Notes on the diamond-α dynamic derivative on time scales, J. Math. Anal. Appl. 326, (2007) 228-241. Search in Google Scholar

[18] Q. Sheng, M. Fadag, J. Henderson, J.M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal.: RWA. 7, (2006) 395-413. Search in Google Scholar

[19] C. Wang, R.P. Agarwal, Almost periodic dynamics for impulsive delay neural networks of a general type on almost periodic time scales, Commun. Nonlinear Sci. Numer. Simulat. 36, (2016) 238-251. Search in Google Scholar

[20] C. Wang, R.P. Agarwal, D. O’Regan, R. Sakthivel, A computation method of Hausdorff distance for translation time scales, 99, (2020) 1218-1247.10.1080/00036811.2018.1529303 Search in Google Scholar

[21] C. Wang, R.P. Agarwal, Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations, Adv. Differ. Equa. 2015:312, (2015) 1-9. Search in Google Scholar

[22] C. Wang, R.P. Agarwal, D. O’Regan, Periodicity, almost periodicity for time scales and related functions. Nonauton. Dyn. Syst. 3, (2016) 24-41. Search in Google Scholar

[23] C. Wang, R.P. Agarwal, D. O’Regan, δ-almost periodic functions and applications to dynamic equations, Mathematics 7, 525, doi.org/10.3390/math7060525 (2019).10.3390/math7060525 Search in Google Scholar

[24] C. Wang, R. Sakthivel, G.M. N’Guérékata, S-almost automorphic solutions for impulsive evolution equations on time scales in shift operators, Mathematics, 8(6), 1028; https://doi.org/10.3390/math8061028 (2020).10.3390/math8061028 Search in Google Scholar

[25] 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. Search in Google Scholar

[26] C. Wang, Almost periodic solutions of impulsive BAM neural networks with variable delays on time scales. Commun. Nonlinear Sci. Numer. Simul. 19, (2014) 2828-2842. Search in Google Scholar

[27] C. Wang, R.P. Agarwal, A Further study of almost periodic time scales with some notes and applications Abstr. Appl. Anal., Article ID 267384, (2014) 1-12.10.1155/2014/267384 Search in Google Scholar

[28] C. Wang, R.P. Agarwal, Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive ∇-dynamic equations on time scales, Adv. Differ. Equa. 2014:153, (2014) 1-29. Search in Google Scholar

[29] C. Wang, R.P. Agarwal, D. O’Regan, Π-semigroup for invariant under translations time scales and abstract weighted pseudo almost periodic functions with applications. Dyn. Syst. Appl. 25, (2016) 1-28. Search in Google Scholar

[30] C. Wang, R.P. Agarwal, A classification of time scales and analysis of the general delays on time scales with applications. Math. Meth. Appl. Sci. 39, (2016) 1568-1590. Search in Google Scholar

[31] C. Wang, R.P. Agarwal, Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations, Discret. Continu. Dynam. Syst. B. 25, (2020) 781-798. Search in Google Scholar

[32] C. Wang, R.P. Agarwal, D. O’Regan, R. Sakthivel, Local pseudo almost automorphic functions with applications to semilinear dynamic equations on changing-periodic time scales, Bound Value Probl. 133, doi:10.1186/s13661-019-1247-4 (2019).10.1186/s13661-019-1247-4 Search in Google Scholar

[33] C. Wang, R.P. Agarwal, D. O’Regan, Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales, Math. Slovaca. 68, (2018) 1397-1420. Search in Google Scholar

[34] C. Wang, R.P. Agarwal, Changing-periodic time scales and decomposition theorems of time scales with applications to functions with local almost periodicity and automorphy, Adv. Differ. Equa. 2015:296 (2015) 1-21. Search in Google Scholar

[35] C. Wang, R.P. Agarwal, D. O’Regan, A matched space for time scales and applications to the study on functions, Adv. Differ. Equa. 2017:305, (2017) 1-28. Search in Google Scholar

[36] C. Wang, R.P. Agarwal, D. O’Regan, n0-order ∆-almost periodic functions and dynamic equations, Applic. Anal. 97, (2018) 2626-2654. Search in Google Scholar

[37] C. Wang, R.P. Agarwal, D. O’Regan, Weighted pseudo δ-almost automorphic functions and abstract dynamic equations, Georgian Math. J. doi: https://doi.org/10.1515/gmj-2019-2066 (2019) (In press).10.1515/gmj-2019-2066 Search in Google Scholar

[38] C. Wang, R.P. Agarwal, R. Sakthivel, Almost periodic oscillations for delay impulsive stochastic Nicholson’s blowflies timescale model, Comput. Appl. Math. 37, (2018) 3005-3026. Search in Google Scholar

[39] C. Wang, 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. Search in Google Scholar

[40] C. Wang, R. Sakthivel, Double almost periodicity for high-order Hopfield neural networks with slight vibration in time variables, Neurocomputing, 282, (2018) 1-15.10.1016/j.neucom.2017.12.008 Search in Google Scholar

[41] C. Wang, R.P. Agarwal, D. O’Regan, R. Sakthivel, Theory of Translation Closedness for Time Scales, Developments in Mathematics, Vol. 62, Springer, Switzerland, 2020.10.1007/978-3-030-38644-3 Search in Google Scholar

[42] C. Wang, G. Qin, R.P. Agarwal, D. O’Regan, ♢α-Measurability and combined measure theory on time scales, Applic. Anal., https://doi.org/10.1080/00036811.2020.1820997, 2020 (In press).10.1080/00036811.2020.1820997 Search in Google Scholar

[43] C. Wang, R.P. Agarwal, D. O’Regan, Matrix measure on time scales and almost periodic analysis of the impulsive Lasota-Wazewska model with patch structure and forced perturbations, Math. Meth. Appl. Sci. 39, (2016) 5651-5669. Search in Google Scholar

[44] C. Wang, Piecewise pseudo almost periodic solution for impulsive non-autonomous highorder Hopfield neural networks with variable delays, Neurocomputing, 171, (2016) 1291-1301.10.1016/j.neucom.2015.07.054 Search in Google Scholar

[45] C. Wang, R.P. Agarwal, Uniformly rd-piecewise almost periodic functions with applications to the analysis of impulsive ∆-dynamic system on time scales, Appl. Math. Comput. 259, (2015) 271-292. Search in Google Scholar

[46] C. Wang, R.P. Agarwal, D. O’Regan, Weighted piecewise pseudo double-almost periodic solution for impulsive evolution equations. J. Nonlinear Sci. Appl. 10, (2017) 3863-3886.10.22436/jnsa.010.07.41 Search in Google Scholar

[47] C. Wang, R.P. Agarwal, D. O’Regan, Compactness criteria and new impulsive functional dynamic equations on time scales, Adv. Differ. Equa. 2016:197, (2016) 1-41. Search in Google Scholar

[48] C. Wang, R.P. Agarwal, Exponential dichotomies of impulsive dynamic systems with applications on time scales, Math. Meth. Appl. Sci. 38, (2015) 3879-3900. Search in Google Scholar

[49] C. Wang, R.P. Agarwal, D. O’Regan, R. Sakthivel, Discontinuous generalized double-almost-periodic functions on almost-complete-closed time scales, Bound Value Probl. 165, doi.org/10.1186/s13661-019-1283-0 (2019).10.1186/s13661-019-1283-0 Search in Google Scholar

[50] C. Wang, Z. Li, R. P. Agarwal, D. O’Regan, Coupled-jumping timescale theory and applications to time-hybrid dynamic equations, convolution and Laplace transforms. Dynam. Syst. Appl., 30 (2021) 461-508. Search in Google Scholar

[51] P. Wilczynski, Quaternionic-valued ordinary differential equations. The Riccati equation, J. Differ. Equ. 247, (2009) 2163-2187. Search in Google Scholar

[52] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra and its Applications, 251, (1997) 21-57.10.1016/0024-3795(95)00543-9 Search in Google Scholar

[53] J. Zhu, J. Sun, Existence and uniqueness results for quaternion-valued nonlinear impulsive differential systems, J. Syst. Sci. Compl. 31, (2018) 596-607. Search in Google Scholar

[54] J. Zhu, J. Sun, Global exponential stability of Clifford-valued recurrent neural networks, Neurocomputing, 173, (2016) 685-689.10.1016/j.neucom.2015.08.016 Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo