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

Abstract

<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. </abstract>