# Convex \((\alpha,\beta)\)-Generalized Contraction and Its Applications in Matrix Equations
## 6 Solutions to Nonlinear Matrix Equations  
In this section, our focus is on investigating the following nonlinear matrix equations, given as follows:  
\[U=Q\pm\sum\limits_{j=1}^{m}\Gamma_{j}^{*}F(U)\Gamma_{j} \tag{22}\]  
Here, \(F:\mathcal{P}(n)\rightarrow\mathcal{P}(n)\) represents a continuous mapping. For a comprehensive understanding of these equation classes, refer to [21]. We explore various scenarios based on different conditions on the mapping \(F\), which are discussed as follows:  
**Case 1. ** Assuming that \(F\) is order-preserving, we examine the following equation:  
\[U=Q+\sum\limits_{j=1}^{m}\Gamma_{j}^{*}F(U)\Gamma_{j}. \tag{23}\]  
To simplify our analysis, we define the mapping \(G(U)\) as:  
\[G(U)=Q+\sum\limits_{j=1}^{m}\Gamma_{j}^{*}F(U)\Gamma_{j}.\]  
The mapping \(G\) is well-defined on \(\mathcal{P}(n)\) and preserves the order of matrices. For any matrix \(U\in\mathcal{P}(n)\), it holds that \(Q\leq G(U)\). Particularly, \(Q\leq G(Q)\) since \(Q\) is positive definite. Since \(G\) is order-preserving, we have the following inequality chain:  
\[Q\leq G(Q)\leq G^{2}(Q)\leq G^{3}(Q)\leq\ldots\]  
Consequently, the sequence \(\{G^{j}(Q)\}\) is increasing. **Proposition 1**. : _Assume the existence of a matrix \(U_{0}\) satisfying \(G(U_{0})\leq U_{0}\). In this case, the mapping \(G\) preserves the set \(\{U:Q\leq U\leq U_{0}\}\) within itself. The sequence \(\{G^{j}(Q)\}\) gradually approaches a certain point denoted as \(U_{-}\), which represents the smallest solution of Equation (23). Moreover, the sequence \(\{G^{j}(U_{0})\}\) exhibits a decreasing pattern, constituting the largest solution within the interval \([Q,U_{0}]\)._  
The subsequent theorem guarantees the uniqueness of the solution to Equation (23). **Theorem 7**. : _Suppose there is a positive number \(M\) for which \(\sum\limits_{j=1}^{m}\Gamma_{j}\Gamma_{j}^{*}<MI_{n}\). Assume that for all \(U,V\in\mathcal{H}(n)\) with \(U\leq V\), we have_  
\[\left|\operatorname{tr}\left\{F\left(\frac{1}{2}U+\frac{1}{2}Q+ \frac{1}{2}\sum\limits_{i=1}^{m}\Gamma_{i}^{*}F(U)\Gamma_{i}\right)-F(V) \right\}\right| \leq \frac{1}{2M}|\operatorname{tr}(U-V)|\] \[+ \frac{1}{2M}\left|\operatorname{tr}\left\{Q+\sum\limits_{i=1}^{ m}\Gamma_{i}^{*}F(U)\Gamma_{i}-F(V)\right\}\right|\]  
_Then, (23) has a unique solution which is positive definite. Moreover, for a given \(U_{1}\in\mathcal{H}(n)\) with \(U_{1}\leq Q+\sum\limits_{i=1}^{m}\Gamma_{i}^{*}F(U_{1})\Gamma_{i}\) the sequence \(\{U_{k}\}\) defined as_  
\[U_{k+1}=\frac{1}{2}U_{k}+\frac{1}{2}G(U_{k})\text{ for all }k\in\mathbb{N}\]  
_converges (in sense of norm \(\|.\|_{1}\)) to the solution of (23)._  
Proof. : Let \(U,V\in\mathcal{H}(n)\) with \(U\leq V\)\[\|G\left(\frac{1}{2}U+\frac{1}{2}G(U)\right)-G(V)\|_{1} = \text{tr}\Bigg{\{}\sum_{j=1}^{m}\Gamma_{j}^{*}\bigg{(}F\left(\frac{1 }{2}U+\frac{1}{2}G(U)\right)-F(V)\bigg{)}\Gamma_{j}\Bigg{\}}\] \[= \sum_{j=1}^{m}\text{tr}\bigg{\{}\Gamma_{j}\Gamma_{j}^{*}\bigg{(}F \left(\frac{1}{2}U+\frac{1}{2}G(U)\right)-F(V)\bigg{)}\bigg{\}}\] \[= \text{tr}\Bigg{\{}\sum_{j=1}^{m}\Gamma_{j}\Gamma_{j}^{*}\bigg{(}F \left(\frac{1}{2}U+\frac{1}{2}G(U)\right)-F(V)\bigg{)}\Bigg{\}}\] \[U=Q+\sum\limits_{j=1}^{m}\Gamma_{j}^{*}F(U)\Gamma_{j}. \tag{23}\]0 \[U=Q+\sum\limits_{j=1}^{m}\Gamma_{j}^{*}F(U)\Gamma_{j}. \tag{23}\]1 \[U=Q+\sum\limits_{j=1}^{m}\Gamma_{j}^{*}F(U)\Gamma_{j}. \tag{23}\]2  
From the assumptions in the theorem,  
\[U=Q+\sum\limits_{j=1}^{m}\Gamma_{j}^{*}F(U)\Gamma_{j}. \tag{23}\]3  
Thus, all the assumptions of Corollary 2 are satisfied. Hence, (23) has a unique solution which is positive definite. **Case 2. ** Consider the following equation  
\[U=Q+\sum\limits_{j=1}^{m}\Gamma_{j}^{*}F(U)\Gamma_{j}. \tag{23}\]4  
We can define  
\[U=Q+\sum\limits_{j=1}^{m}\Gamma_{j}^{*}F(U)\Gamma_{j}. \tag{23}\]5  
If we assume that \(F\) is order-reversing in (24), then the mapping \(G\) becomes order-preserving. Furthermore, we consider the existence of a matrix \(U_{0}\) that satisfies \(U_{0}\leq Q\) and \(U_{0}\leq G(U_{0})\).