On the unique solution of a class of absolute value equations $ Ax-B|Cx| = d $

Hongyu Zhou; Shiliang Wu; Hongyu Zhou; Shiliang Wu

doi:10.3934/math.2021517

AIMS Mathematics

2021, Volume 6, Issue 8: 8912-8919. doi: 10.3934/math.2021517

Previous Article Next Article

Research article Special Issues

On the unique solution of a class of absolute value equations $ Ax-B|Cx| = d $

Hongyu Zhou ¹,
Shiliang Wu ^{2
,
,}

1.
School of Computer and Information Engineering, Anyang Normal University, Anyang, Henan 455000, China
2.
School of Mathematics, Yunnan Normal University, Kunming, Yunnan 650500, China

Received: 05 May 2021 Accepted: 09 June 2021 Published: 15 June 2021
MSC : 90C05, 90C30, 65F10

In this paper, a class of absolute value equations (AVE) $ Ax-B|Cx| = d $ with $ A, B, C\in \mathbb{R}^{n\times n} $ is considered, which is a generalized form of the published works by Wu ^[1], Wu and Shen ^[2] and Mezzadri ^[3]. Some conditions to guarantee the unique solution of the above AVE are gained. The corresponding results of the above published works are generalized.
- sufficient condition,
- unique solution,
- absolute value equation
Citation: Hongyu Zhou, Shiliang Wu. On the unique solution of a class of absolute value equations $ Ax-B|Cx| = d $[J]. AIMS Mathematics, 2021, 6(8): 8912-8919. doi: 10.3934/math.2021517

Related Papers:

Abstract

In this paper, a class of absolute value equations (AVE) $ Ax-B|Cx| = d $ with $ A, B, C\in \mathbb{R}^{n\times n} $ is considered, which is a generalized form of the published works by Wu ^[1], Wu and Shen ^[2] and Mezzadri ^[3]. Some conditions to guarantee the unique solution of the above AVE are gained. The corresponding results of the above published works are generalized.

References

[1]	S. L. Wu, The unique solution of a class of the new generalized absolute value equation, Appl. Math. Lett., 116 (2021), 107029. doi: 10.1016/j.aml.2021.107029
[2]	S. L. Wu, S. Q. Shen, On the unique solution of the generalized absolute value equation, Optim. Lett., (2020).
[3]	F. Mezzadri, On the solution of general absolute value equations, Appl. Math. Lett., 107 (2020) 106462.
[4]	L. Caccetta, B. Qu, G. L. Zhou, A globally and quadratically convergent method for absolute value equations, Comput. Optim. Appl., 48 (2011), 45–58. doi: 10.1007/s10589-009-9242-9
[5]	O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett., 3 (2009), 101–108. doi: 10.1007/s11590-008-0094-5
[6]	J. Rohn, An algorithm for solving the absolute value equations, Electron. J. Linear Algebra, 18 (2009), 589–599.
[7]	D. K. Salkuyeh, The Picard-HSS iteration method for absolute value equations, Optim. Lett., 8 (2014), 2191–2202. doi: 10.1007/s11590-014-0727-9
[8]	S. L. Wu, C. X. Li, A special Shift-splitting iterative method for the absolute value equations, AIMS Math., 5 (2020), 5171–5183. doi: 10.3934/math.2020332
[9]	C. X. Li, S. L. Wu, Modified SOR-like iteration method for absolute value equations, Math. Probl. Eng., 2020 (2020), 9231639.
[10]	P. Guo, S. L. Wu, C. X. Li, On SOR-like iteration method for solving absolute value equations, Appl. Math. Lett., 97 (2019), 107–113. doi: 10.1016/j.aml.2019.03.033
[11]	Y. Y. Lian, C. X. Li, S. L. Wu, Weaker convergent results of the generalized Newton method for the generalized absolute value equations, J. Comput. Appl. Math., 338 (2018), 221–226.
[12]	C. X. Li, A preconditioned AOR iterative method for the absolute value equations, Int. J. Comput. Methods, 14 (2017), 1750016. doi: 10.1142/S0219876217500165
[13]	C. X. Li, A modified generalized Newton method for the absolute value equations, J. Optim. Theory Appl., 170 (2016), 1055–1059. doi: 10.1007/s10957-016-0956-4
[14]	J. Rohn, Systems of linear interval equations, Linear Algebra Appl., 126 (1989), 39–78. doi: 10.1016/0024-3795(89)90004-9
[15]	J. Rohn, A theorem of the alternatives for the equation $Ax+B\|x\| = b$, Linear Multilinear A., 52 (2004), 421–426. doi: 10.1080/0308108042000220686
[16]	S. L. Wu, C. X. Li, A note on unique solvability of the absolute value equation, Optim. Lett., 14 (2020), 1957–1960. doi: 10.1007/s11590-019-01478-x
[17]	O. L. Mangasarian, R. R. Meyer, Absolute value equations, Linear Algebra Appl., 419 (2006), 359–367.
[18]	S. L. Wu, C. X. Li, The unique solution of the absolute value equations, Appl. Math. Lett., 76 (2018), 195–200. doi: 10.1016/j.aml.2017.08.012
[19]	O. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appl., 44 (2009), 363–372. doi: 10.1007/s10589-007-9158-1
[20]	M. Hladík, Bounds for the solutions of absolute value equations, Comput. Optim. Appl., 69 (2018), 243–266. doi: 10.1007/s10589-017-9939-0

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)