Positive definite solution of non-linear matrix equations through fixed point technique

Sourav Shil; Hemant Kumar Nashine; Sourav Shil; Hemant Kumar Nashine

doi:10.3934/math.2022348

AIMS Mathematics

2022, Volume 7, Issue 4: 6259-6281. doi: 10.3934/math.2022348

Previous Article Next Article

Research article

Positive definite solution of non-linear matrix equations through fixed point technique

Sourav Shil ,
Hemant Kumar Nashine ^,

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, TN, India

Received: 26 September 2021 Revised: 11 January 2022 Accepted: 11 January 2022 Published: 18 January 2022
MSC : 45J05, 47H10, 54H25

The goal of this study is to solve a non-linear matrix equation of the form $ \mathcal{X} = \mathcal{Q} + \sum\limits_{i = 1}^{m} \mathcal{B}_{i}^{*}\mathcal{G} (\mathcal{X})\mathcal{B}_{i} $, where $ \mathcal{Q} $ is a Hermitian positive definite matrix, $ \mathcal{B}_{i}^{*} $ stands for the conjugate transpose of an $ n\times n $ matrix $ \mathcal{B}_{i} $ and $ \mathcal{G} $ an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive definite matrices such that $ \mathcal{G}(O) = O $. We explore the necessary and sufficient criteria for the existence of a unique positive definite solution to a particular matrix problem. For the said reason, we develop some fixed point results for $ \mathcal{FG} $-contractive mappings on complete metric spaces equipped with any binary relation (not necessarily a partial order). We give adequate examples to confirm the fixed-point results and compare them to early studies, as well as four instances that show the convergence analysis of non-linear matrix equations using graphical representations.
- positive definite matrix,
- nonlinear matrix equation,
- fixed point,
- relational metric space
Citation: Sourav Shil, Hemant Kumar Nashine. Positive definite solution of non-linear matrix equations through fixed point technique[J]. AIMS Mathematics, 2022, 7(4): 6259-6281. doi: 10.3934/math.2022348

Related Papers:

Abstract

The goal of this study is to solve a non-linear matrix equation of the form $ \mathcal{X} = \mathcal{Q} + \sum\limits_{i = 1}^{m} \mathcal{B}_{i}^{*}\mathcal{G} (\mathcal{X})\mathcal{B}_{i} $, where $ \mathcal{Q} $ is a Hermitian positive definite matrix, $ \mathcal{B}_{i}^{*} $ stands for the conjugate transpose of an $ n\times n $ matrix $ \mathcal{B}_{i} $ and $ \mathcal{G} $ an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive definite matrices such that $ \mathcal{G}(O) = O $. We explore the necessary and sufficient criteria for the existence of a unique positive definite solution to a particular matrix problem. For the said reason, we develop some fixed point results for $ \mathcal{FG} $-contractive mappings on complete metric spaces equipped with any binary relation (not necessarily a partial order). We give adequate examples to confirm the fixed-point results and compare them to early studies, as well as four instances that show the convergence analysis of non-linear matrix equations using graphical representations.

References

[1]	M. Ahmadullah, J. Ali, M. Imdad, Unified relation-theoretic metrical fixed point theorems under an implicit contractive condition with an application, Fixed Point Theory Appl., 2016 (2016), 42. https://doi.org/10.1186/s13663-016-0531-6 doi: 10.1186/s13663-016-0531-6
[2]	M. Ahmadullah, M. Imdad, M. Arif, Relation-theoretic metrical coincidence and common fixed point theorems under nonlinear contractions, Appl. Gen. Topol., 19 (2018), 65–84. https://doi.org/10.4995/agt.2018.7677 doi: 10.4995/agt.2018.7677
[3]	M. Ahmadullah, M. Imdad, Unified relation-theoretic fixed point results via $F_\mathcal{R}$-suzuki-contractions with an application, Fixed Point Theory, 21 (2020), 19–34. https://doi.org/10.24193/fpt-ro.2020.1.02 doi: 10.24193/fpt-ro.2020.1.02
[4]	A. Alam, M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory Appl., 17 (2015), 693–702. https://doi.org/10.1007/s11784-015-0247-y doi: 10.1007/s11784-015-0247-y
[5]	R. Bhatia, Matrix analysis, New York: Springer, 1997. https://doi.org/10.1007/978-1-4612-0653-8
[6]	H. Ben-El-Mechaiekh, The Ran–Reurings fixed point theorem without partial order: A simple proof, J. Fixed Point Theory Appl., 16 (2014), 373–383. https://doi.org/10.1007/s11784-015-0218-3 doi: 10.1007/s11784-015-0218-3
[7]	L. B. Ćirić, A generalization of Banach’s contraction principle. Proc. Am. Math. Soc., 45 (1974), 267–273.
[8]	L. B. Ćirić, N. Cakić, M. Rajović, J. S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl., 2008 (2008), 131294. https://doi.org/10.1155/2008/131294 doi: 10.1155/2008/131294
[9]	D. Dukić, Z. Kadelburg, S. Radenović, Fixed points of Geraghty-type mappings in various generalized metric spaces, Abstr. Appl. Anal., 2011 (2011), 561245. https://doi.org/10.1155/2011/561245 doi: 10.1155/2011/561245
[10]	M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604–608. https://doi.org/10.1090/S0002-9939-1973-0334176-5 doi: 10.1090/S0002-9939-1973-0334176-5
[11]	C. R. Johnson, Positive definite matrices, Am. Math. Mon., 77 (1970), 259–264. https://doi.org/10.1080/00029890.1970.11992465 doi: 10.1080/00029890.1970.11992465
[12]	B. Kolman, R. C. Busby, S. Ross, Discrete mathematical structures, New Delhi: PHI Pvt. Ltd., 2000.
[13]	S. Lipschutz, Schaum's outlines of theory and problems of set theory and related topics, New York: McGraw-Hill, 1964.
[14]	R. D. Maddux, Relation algebras: Studies in logic and foundations of mathematics Volume 150, Amsterdam: Elsevier, 2006.
[15]	J. Matkowski, Integrable solutions of functional equations, Dissertationes Math., 127 (1975), 1–68.
[16]	J. Matkowski, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Am. Math. Soc., 62 (1977), 344–348. https://doi.org/10.1090/S0002-9939-1977-0436113-5 doi: 10.1090/S0002-9939-1977-0436113-5
[17]	J. J. Nieto, R. R. López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223–239. https://doi.org/10.1007/s11083-005-9018-5 doi: 10.1007/s11083-005-9018-5
[18]	J. J. Nieto, R. R. López, R. Rodríguez-López, Fixed point theorems in ordered abstract spaces, Proc. Am. Math. Soc., 135 (2007), 2505–2517.
[19]	V. Parvaneh, N. Hussain, Z. Kadelburg, Generalized Wardowski type fixed point theorems via $\alpha$-admissible $FG$-contractions in $b$-metric spaces, Acta Math. Sci., 36 (2016), 1445–1456. https://doi.org/10.1016/S0252-9602(16)30080-7 doi: 10.1016/S0252-9602(16)30080-7
[20]	A. C. M. Ran, M. C. B. Reurings, On the nonlinear matrix equation $X + A^F(X)A = Q$: Solutions and perturbation theory, Linear Algebra Appl., 346* (2002), 15–26. https://doi.org/10.1016/S0024-3795(01)00508-0 doi: 10.1016/S0024-3795(01)00508-0
[21]	A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435–1443.
[22]	B. Samet, M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications, Commun. Math. Anal., 13 (2012), 82–97.
[23]	M. Turinici, Abstract comparison principles and multivariable Gronwall-Bellman inequalities, J. Math. Anal. Appl., 117 (1986), 100–127. https://doi.org/10.1016/0022-247X(86)90251-9 doi: 10.1016/0022-247X(86)90251-9
[24]	M. Turinici, Fixed points for monotone iteratively local contractions, Demonstr. Math., 19 (1986), 171–180.
[25]	D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 94. https://doi.org/10.1186/1687-1812-2012-94 doi: 10.1186/1687-1812-2012-94

Reader Comments

Your name:*

Email:*
© 2022 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)