Abstract
We propose a new class of implicit relations and an implicit type contractive condition based on it in the relational metric spaces under w-distance functional. Further we derive fixed points results based on them. Useful examples illustrate the applicability and effectiveness of the presented results. We apply these results to discuss sufficient conditions ensuring the existence of a unique positive definite solution of the nonlinear matrix equation (NME) of the form mathcal{U}=mathcal{Q} + sum_{i=1}^{k}mathcal{A}_{i}^{*} mathcal{G}mathcal{(U)}mathcal{A}_{i}, where mathcal{Q} is an ntimes n Hermitian positive definite matrix, mathcal{A}_{1}, mathcal{A}_{2}, …, mathcal{A}_{m} are n times n matrices and mathcal{G} is a nonlinear self-mapping of the set of all Hermitian matrices which are continuous in the trace norm. In order to demonstrate the obtained conditions, we consider an example together with convergence and error analysis and visualisation of solutions in a surface plot.
Highlights
One of the most visually attractive applications of contraction mapping is found in nonlinear matrix equations
In relational metric spaces with w-distance functional, we investigate a novel class of implicit relations and implicit type contractions
Contraction and rational type contraction mappings are included in the proposed implicit contraction. We use these findings to examine the necessary conditions for the existence of a unique positive definite solution to the nonlinear matrix equation (NME)
Summary
One of the most visually attractive applications of contraction mapping is found in nonlinear matrix equations. Contraction and rational type contraction mappings are included in the proposed implicit contraction We use these findings to examine the necessary conditions for the existence of a unique positive definite solution to the nonlinear matrix equation (NME). We fix the following notation for a relational space (W, R), a self-mapping K on W and an R-directed subset D of W:. 2.2 Relational metric spaces with w-distance The corresponding definitions and lemmas, in the setting of metric spaces endowed with an arbitrary binary relation R, are as follows. Lemma 2.3 ([4, 20]) Let ω be a w-distance on a metric space (W, d) and {θn} be a sequence in W such that for each > 0 there exists N ∈ N such that m > n > N implies ω(θn, θm) < , i.e. limm,n→∞ ω(θn, θm) = 0.
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