Positive solutions of nonlinear matrix equations via fixed point results in relational metric spaces with w-distance

Abstract

We consider the non-linear matrix equation (NME) of the form
$\mathcal{U}=\mathcal{Q} +
\sum_{i=1}^{k}\mathcal{A}_{i}^{*}\hbar\mathcal{(U)}\mathcal{A}_{i}$,
where $\mathcal{Q}$ 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 $\hbar$ is a
non-linear self-mapping of the set of all Hermitian matrices which
are continuous in the trace norm. We discuss sufficient conditions
ensuring the existence of a unique positive definite solution of
the given NME. In order to do this, we introduce
$\Theta_w$-contractive conditions involving modified simulation
functions in relational metric spaces and derive fixed points
results based on them, followed by two suitable examples. In order
to demonstrate the obtained conditions, we consider three
different sets of matrices. Three different types of examples
(including randomly generated matrix and a complex matrix) are
given, together with convergence and error analysis, as well as
average CPU time analysis with different dimensions bar graphs,
and visualization of solutions in surface plot.