Extended Branciari quasi-b-distance spaces, implicit relations and application to nonlinear matrix equations

This study introduces extended Branciari quasi-b-distance spaces, a novel implicit contractive condition in the underlying space, and basic fixed-point results, a weak well-posed property, a weak limit shadowing property and generalized Ulam–Hyers stability. The given notions and results are exemplified by suitable models. We apply these results to obtain a sufficient condition ensuring the existence of a unique positive-definite solution of a nonlinear matrix equation (NME) $\mathcal{X}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G(X)}\mathcal{A}_{i}$ X = Q + ∑ i = 1 k A i ∗ G ( X ) A i , where $\mathcal{Q}$ Q is an $n\times n$ n × n Hermitian positive-definite matrix, $\mathcal{A}_{1}$ A 1 , $\mathcal{A}_{2}$ A 2 , …, $\mathcal{A}_{m}$ A m are $n \times n$ n × n matrices, and $\mathcal{G}$ G is a nonlinear self-mapping of the set of all Hermitian matrices that are continuous in the trace norm. We demonstrate this sufficient condition for the NME $\mathcal{X}= \mathcal{Q} +\mathcal{A}_{1}^{*}\mathcal{X}^{1/3} \mathcal{A}_{1}+\mathcal{A}_{2}^{*}\mathcal{X}^{1/3} \mathcal{A}_{2}+ \mathcal{A}_{3}^{*}\mathcal{X}^{1/3}\mathcal{A}_{3}$ X = Q + A 1 ∗ X 1 / 3 A 1 + A 2 ∗ X 1 / 3 A 2 + A 3 ∗ X 1 / 3 A 3 , and visualize this through convergence analysis and a solution graph.

A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation

The goal of this article is to investigate a new solver in the form of an iterative method to solve X+A∗X−1A=I as an important nonlinear matrix equation (NME), where A,X,I are appropriate matrices. The minimal and maximal solutions of this NME are discussed as Hermitian positive definite (HPD) matrices. The convergence of the scheme is given. Several numerical tests are also provided to support the theoretical discussions.

Inversion free Iterative Method for Finding Symmetric Solution of the Nonlinear Matrix Equation 𝑿 − 𝑨∗𝑿𝒒𝑨 = 𝑰 (𝒒 ≥ 𝟐)

In this paper, we propose the inversion free iterative method to find symmetric solution of thenonlinear matrix equation 𝑿 − 𝑨∗𝑿𝒒𝑨 = 𝑰 (𝒒 ≥ 𝟐), where 𝑋 is an unknown symmetricsolution, 𝐴 is a given Hermitian matrix and 𝑞 is a positive integer. The convergence of theproposed method is derived. Numerical examples demonstrate that the proposed iterative methodis quite efficient and converges well when the initial guess is sufficiently close to the approximatesolution. Keywords: Symmetric solution, nonlinear matrix equation, inversion free, iterative method

An implicit relation, relational theoretic approach under w-distance and application to nonlinear matrix equations

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 $\mathcal{U}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G}\mathcal{(U)}\mathcal{A}_{i}$ U = Q + ∑ i = 1 k A i ∗ G ( U ) A i , where $\mathcal{Q}$ Q is an $n\times n$ n × n Hermitian positive definite matrix, $\mathcal{A}_{1}$ A 1 , $\mathcal{A}_{2}$ A 2 , …, $\mathcal{A}_{m}$ A m are $n \times n$ n × n matrices and $\mathcal{G}$ 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.

Elegant Iterative Methods for Solving a Nonlinear Matrix Equation X-A* eX A=I

The nonlinear matrix equation was solved by Gao (2016) via standard fixed point method. In this paper, three more elegant iterative methods are proposed to find the approximate solution of the nonlinear matrix equation namely: Newton’s method; modified fixed point method and a combination of Newton’s method and fixed point method. The convergence of Newton’s method and modified fixed point method are derived. Comparative numerical experimental results indicate that the new developed algorithms have both less computational time and good convergence properties when compared to their respective standard algorithms. Keywords: Hermitian positive definite solution; nonlinear matrix equation; modified fixed point method; iterative method