Unified relational-theoretic approach in metric-like spaces with an application

Reena Jain; Hemant Kumar Nashine; Jung Rye Lee; Choonkil Park; Reena Jain; Hemant Kumar Nashine; Jung Rye Lee; Choonkil Park

doi:10.3934/math.2021520

AIMS Mathematics

2021, Volume 6, Issue 8: 8959-8977. doi: 10.3934/math.2021520

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Research article

Unified relational-theoretic approach in metric-like spaces with an application

1.
Mathematics Division, SASL, VIT Bhopal University, Madhya Pradesh, 466114, India
2.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632014, TN, India
3.
Department of Mathematics and Applied Mathematics, University of Johannesburg, Kingsway Campus, Auckland Park 2006, South Africa
4.
Department of Data Science, Daejin University Kyunggi 11159, Korea
5.
Research Institute for Natural Sciences, Hanyang University Seoul 04763, Korea

Received: 22 April 2021 Accepted: 09 June 2021 Published: 15 June 2021
MSC : 45J05, 47H10, 54H25

In this paper, we introduce a modified implicit relation and obtain some new fixed point results for $ \sigma $-implicit type contractive conditions in relational metric-like spaces. We present some nontrivial examples to illustrative facts and compare our results with the related work. We also discuss sufficient conditions for the existence of a unique positive definite solution of the non-linear matrix equation $ \mathcal{U} = \mathcal{D} + \sum_{i = 1}^{m}\mathcal{A}_{i}^{*}\mathcal{G} \mathcal{(U)}\mathcal{A}_{i} $, where $ \mathcal{D} $ is an $ n\times n $ Hermitian positive definite matrix, $ \mathcal{A}_{1} $, $ \mathcal{A}_{2} $, $\dots$, $ \mathcal{A}_{m} $ are $ n \times n $ matrices, and $ \mathcal{G} $ is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Finally, we discuss a couple of examples, convergence and error analysis, average CPU time analysis and visualization of solution in surface plot.
- positive definite matrix,
- nonlinear matrix equation,
- fixed point, relational metric space,
- metric-like space
Citation: Reena Jain, Hemant Kumar Nashine, Jung Rye Lee, Choonkil Park. Unified relational-theoretic approach in metric-like spaces with an application[J]. AIMS Mathematics, 2021, 6(8): 8959-8977. doi: 10.3934/math.2021520

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Abstract

In this paper, we introduce a modified implicit relation and obtain some new fixed point results for $ \sigma $-implicit type contractive conditions in relational metric-like spaces. We present some nontrivial examples to illustrative facts and compare our results with the related work. We also discuss sufficient conditions for the existence of a unique positive definite solution of the non-linear matrix equation $ \mathcal{U} = \mathcal{D} + \sum_{i = 1}^{m}\mathcal{A}_{i}^{*}\mathcal{G} \mathcal{(U)}\mathcal{A}_{i} $, where $ \mathcal{D} $ is an $ n\times n $ Hermitian positive definite matrix, $ \mathcal{A}_{1} $, $ \mathcal{A}_{2} $, $\dots$, $ \mathcal{A}_{m} $ are $ n \times n $ matrices, and $ \mathcal{G} $ is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Finally, we discuss a couple of examples, convergence and error analysis, average CPU time analysis and visualization of solution in surface plot.

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