### AIMS Mathematics

2021, Issue 8: 8959-8977. doi: 10.3934/math.2021520
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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沈阳化工大学材料科学与工程学院 沈阳 110142

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