Research article

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

  • 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.

    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

    Related Papers:

  • 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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