# Convex \((\alpha,\beta)\)-Generalized Contraction and Its Applications in Matrix Equations
## 2 Preliminaries  
Consider a pair of points, \(\xi\) and \(\varrho\), within the metric space \((X,\Omega)\). We define a path, denoted as \(\zeta:[0,1]\to X\), to connect \(\xi\) and \(\varrho\) if the conditions are met:  
\[\zeta(0)=\xi\text{ and }\zeta(1)=\varrho.\]  
A path \(\zeta\) is categorized as a geodesic if the following criteria are satisfied for all \(s\) and \(t\) in the interval \([0,1]\):  
\[\Omega(\zeta(s),\zeta(t))=\Omega(\zeta(0),\zeta(1))|s-t|.\]  
When each pair of points \(\xi\) and \(\varrho\) in \(X\) can be linked by a geodesic, the metric space \((X,\Omega)\) obtains the classification of a geodesic space. Further, if the geodesics in a geodesic space are unique, the space falls under the classification of a Busemann space, according to [17]. Notable instances of these spaces include normed spaces, CAT(0)-spaces, Hadamard manifolds, and the Hilbert open unit ball equipped with the hyperbolic metric, as shown in References [18; 19]. [PERSON] [19] introduced a precise formulation of hyperbolic spaces, which is presented below. Definition 1: If a function \(W:X\times X\times[0,1]\to X\) exists such that \((X,\Omega)\) is a metric space and \((X,\Omega,W)\) satisfies the following conditions, then it is referred to as a hyperbolic metric space:_(i)_ \(\Omega(z,W(\xi,\varrho,\theta))\leq(1-\theta)\Omega(z,\xi)+\theta\Omega(z,\varrho)\)_;_  
_(ii)_ \(\Omega(W(\xi,\varrho,\theta),W(\xi,\varrho,\overline{\theta}))=[\theta- \overline{\theta}]\Omega(\xi,\varrho)\)_;_  
_(iii)_ \(W(\xi,\varrho,\theta)=W(\varrho,\xi,1-\theta)\)_;_  
_(iv)_ \(\Omega(W(\xi,z,\theta),W(\varrho,\zeta,\theta))\leq(1-\theta)\Omega(\xi, \varrho)+\theta\Omega(z,\xi)\)_,_  
_for all \(\xi,\varrho,z,\xi\in X\) and \(\theta,\overline{\theta}\in[0,1]\)._  
**Remark 1**. : _If \(W(\xi,\varrho,\theta)=(1-\theta)\xi+\theta\varrho\) for all \(\xi,\varrho\in X,\theta\in[0,1]\), then these spaces include all normed linear spaces._  
For \(\xi,\varrho\in X\),  
\[[\xi,\varrho]=\{(1-\theta)\xi\oplus\theta\varrho:\theta\in[0,1]\}\]  
denotes geodesic segments. A map \(x:[a,b]\to X\) is an affinely reparametrized geodesic if there exist an interval \([c,d]\) and a geodesic \(x^{\prime}:[c,d]\to X\) such that \(x=x^{\prime}\) o \(\psi\), where \(\psi:[a,b]\to[c,d]\) is the unique affine homeomorphism between the intervals \([a,b]\) and \([c,d]\), or \(x\) is a constant path. A geodesic space \((X,\Omega)\) is a Busemann space if for any two affinely reparametrized geodesics \(x:[a,b]\to X\) and \(x^{\prime}:[c,d]\to X\), and the map \(D_{x,x^{\prime}}:[a,b]\times[c,d]\to\mathbb{R}\) is defined as  
\[D_{x,x^{\prime}}(s,t)=\Omega(x(s),x^{\prime}(t))\]  
is convex, see [17]. A unique mapping for convexity, denoted as \(W\), exists such that the triple \((X,\Omega,W)\) forms a uniquely geodesic \(W\)-hyperbolic space when \((X,\Omega)\) is a Busemann space. This implies that for any distinct pair of points \(\xi\) and \(\varrho\) in \(X\), and for any value of \(\theta\) within the interval \([0,1]\), there exists a sole element \(\zeta\in X\) (specifically denoted as \(\zeta=W(\xi,\varrho,\theta)\)) that satisfies the following conditions:  
\[\Omega(\xi,\zeta)=\theta\Omega(\xi,\varrho)\text{ and }\Omega(\varrho,\zeta)=(1- \theta)\Omega(\xi,\varrho).\]  
**3. Convex (\(\alpha,\beta\))-Generalized Contraction Mapping**  
In this section, we delve into the following concept:  
**Definition 2**. : _Let \((X,\Omega,W)\) be a \(W\)-hyperbolic space, and \(\Phi:X\to X\) a mapping. The mapping \(\Phi\) is called convex orbital \((\alpha,\beta)\)-contraction mapping if there exists \(\alpha,\beta\in(0,1)\) such that_  
\[\Omega(\Phi(W(\xi,\Phi(\xi),\alpha),\Phi(\varrho)))\leq(1-\alpha)\beta d(\xi, \varrho)+\alpha\beta\Omega(\Phi(\xi),\varrho) \tag{3}\]  
_for all \(\xi,\varrho\in X\)._  
If we consider \(\alpha=0\), then convex \((\alpha,\beta)\)-generalized contraction is contraction mapping. Thus, we take \(\alpha\in(0,1)\). **Theorem 1**.