# Convex \((\alpha,\beta)\)-Generalized Contraction and Its Applications in Matrix Equations
## 1 Introduction  
Matrix equations arise as essential mathematical tools in various fields, such as ladder networks, control theory, stochastic filtering, dynamic programming, and statistics. Understanding the solutions of linear and nonlinear matrix equations is crucial for advancing research and practical applications in these areas [1]. This paper aims to investigate the existence and convergence of solutions for such matrix equations. We commence our study by focusing on a linear matrix equation of the form:  
\[U-\Gamma_{1}^{*}U\Gamma_{1}-\cdots-\Gamma_{m}U\Gamma_{m}=Q, \tag{1}\]  
where \(\Gamma_{1},\ldots,\Gamma_{m}\) are arbitrary matrices of order \(n\times n\), and \(Q\) is a matrix of order \(n\times n\). Our objective is to ensure the existence of solutions for Equation (1), which also possess the property of being positive definite. To achieve this, we employ the Krasnosel'skii iterative method, a proven technique that converges to solutions of linear matrix equations [2]. Moving on, we delve into the realm of nonlinear matrix equations and consider the following equation:  
\[U=Q\pm\sum_{j=1}^{m}\Gamma_{j}^{*}F(U)\Gamma_{j}, \tag{2}\]  
where \(F\) is a continuous mapping from the set of all positive definite matrices to itself. We impose certain assumptions on \(F\), such as order-preserving or order-reversing properties. Our objective is to determine solutions to Equation (2) under these assumptions. To tackle the challenge of solving both linear and nonlinear matrix equations, we draw inspiration from the concept of Picard-Banach contractions. Picard operators, which are a specific type of mapping, have been widely recognized as valuable tools in nonlinear analysis, pioneered by [PERSON] [3]. The fundamental idea behind the Banach contraction principle (BCP) is that within a complete metric space \((X,\Omega)\), any mapping \(\Phi:X\to X\) satisfying the contraction condition \(\Omega(\Phi(\xi),\Phi(\varrho))\leq\beta\Omega(\xi,\varrho)\) for all \(\xi,\varrho\in X\), possesses a unique fixed point. Building upon the foundations laid by previous researchers, such as [PERSON] and [PERSON] [2], who utilized the Banach contraction principle in partially ordered sets to obtain positive definite solutions for matrix equations, and [PERSON] and [PERSON] [4], who applied similar fixed point theorems to find solutions of differential equations, we explore the potential of convex \((\alpha,\beta)\)-generalized contraction mappings in geodesic spaces. This perspective enables us to ensure the existence of solutions for both linear and nonlinear matrix equations under appropriate assumptions. Over the past nine decades, the Picard-Banach fixed point theorem and its various extensions have become invaluable tools in solving a wide range of nonlinear problems, including differential equations, integral equations, integro-differential equations, optimization problems, and variational inequalities. This is evident from the extensive body of literature surrounding these topics, including monographs by authors such as [PERSON], [PERSON], and [PERSON], as well as numerous references [5; 6; 7]. The following results are related to integral type contractive conditions [8; 9; 10]. See also [11; 12; 13]. Recent research by [PERSON] and [PERSON] [14] and [PERSON] [15] has focused on convex orbital \(\beta\)-Lipschitz mappings and convex orbital \((\alpha,\beta)\)-Lipschitz mappings, respectively, in Hilbert spaces (see also [16]). They demonstrated the existence of fixed points within these classes of mappings and established connections with the admissible perturbations approach. Motivated by their work, we consider convex \((\alpha,\beta)\)-generalized contraction mappings in the setting of geodesic spaces. Our aim is to ensure the solution of Equation (2) under appropriate assumptions.