AbstractThis 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.
Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix
