# Proving Fixed-Point Theorems Employing Fuzzy \((\sigma,\mathcal{Z})\)-Contractive-Type Mappings
## 2 Mathematical Preliminaries
: _From the definition of \(\xi\), it is obvious that \(\xi(t,t)>t\), for all \(t\in(0,1)\)._  
Employing the function \(\xi\) that satisfies the above condition, [PERSON] et al. unified and extended the contractive-type mappings introduced in [7; 8; 9; 29] by introducing the following interesting class of mappings:  
**Definition 4** ([28]). : _Let \(T\) be a self-mapping of an FMS \((\mathcal{K},\mathcal{M},*)\). The mapping \(T\) is said to be fuzzy \(\mathcal{Z}\)-contractive if there is \(\xi\in\mathcal{Z}\) such that_  
\[\mathcal{M}(T\alpha,T\beta,t)\geq\xi(\mathcal{M}(T\alpha,T\beta,t),\mathcal{M} (\alpha,\beta,t)),\]  
_for each \(\alpha,\beta\in\mathcal{K}\) with \(T\alpha\
eq T\beta\) and \(t>0\)._  
Let \(\mathcal{K}\) be a nonempty set, \(x_{0}\in\mathcal{K}\) and \(T:\mathcal{K}\to\mathcal{K}\). A sequence \(\{x_{n}\}\subseteq\mathcal{K}\) is called a Picard sequence of \(T\) based at \(x_{0}\) if  
\[x_{n}=Tx_{n-1}=T^{n}x_{0},\ \ \forall\ n\in\mathbb{N}.\]  
**Definition 5** ([28]). : _Let \(T\) be a self-mapping of an FMS \((\mathcal{K},\mathcal{M},*)\) and \(\xi\in\mathcal{Z}\). Assume that \(\{\alpha_{n}\}\) is any Picard sequence for all \(n\in\mathbb{N}\). The quadruple \((\mathcal{K},\mathcal{M},T,\xi)\) is said to have the property \((S)\) if for each \(n\in\mathbb{N}\) and \(t>0\) with_  
\[\inf_{m\sim n}\mathcal{M}(\alpha_{n},\alpha_{m},t)\leq\inf_{m>n} \mathcal{M}(\alpha_{n+1},\alpha_{m+1},t)\] \[\text{implies} \lim_{n\to\infty}\inf_{m>n}\xi(\mathcal{M}(\alpha_{n+1},\alpha_{m +1},t),\mathcal{M}(\alpha_{n},\alpha_{m},t))=1.\]  
**Definition 6** ([28]). : _Let \(T\) be a self-mapping of an FMS \((\mathcal{K},\mathcal{M},*)\) and \(\xi\in\mathcal{Z}\). Assume that \(\{\alpha_{n}\}\) is any Picard sequence for all \(n\in\mathbb{N}\). The quadruple \((\mathcal{K},\mathcal{M},T,\xi)\) is said to have the property \((S^{\prime})\) if for each \(n\in\mathbb{N}\) and \(t>0\) with \(0<\lim_{n\to\infty}\inf_{m>n}\mathcal{M}(\alpha_{n},\alpha_{m},t)<1\) and_  
\[\inf_{m\sim n}\mathcal{M}(\alpha_{n},\alpha_{m},t)\leq\inf_{m>n} \mathcal{M}(\alpha_{n+1},\alpha_{m+1},t)\] \[\text{implies} \lim_{n\to\infty}\inf_{m>n}\xi(\mathcal{M}(\alpha_{n+1},\alpha_{m +1},t),\mathcal{M}(\alpha_{n},\alpha_{m},t))=1.\]  
Notice that the condition \((S^{\prime})\) is weaker than the condition \((S)\) (see [28], Example 3.18). [PERSON] et al. [28] proved the following theorem as a consequence of their study. **Theorem 1**. : _Let \((\mathcal{K},\mathcal{M},*)\) be an \(\mathcal{M}\)-complete FMS and \(T:\mathcal{K}\to\mathcal{K}\) be a fuzzy \(\mathcal{Z}\)-contractive mapping with respect to \(\xi\in\mathcal{Z}\). If the quadruple \((\mathcal{K},\mathcal{M},T,\xi)\) has the property \((S)\), then \(T\) admits a unique fixed point._  
Let \(\Psi\) denote the set of all \(\psi:(0,1]\to(0,1]\), which have the following properties:\((\mathbb{V}1)\)\(\psi\) is a left continuous and non-decreasing function;  
\((\mathbb{V}2)\)\(\psi(r)>r\), for all \(r\in(0,1)\). **Lemma 2** ([31]). : _If \(\psi\in\mathbb{V}\), then \(\psi(1)=1\)._  
Inspired by the work of [PERSON] et al. [26], [PERSON] and [PERSON] [27] employed a function \(\sigma:\mathcal{K}\times\mathcal{K}\times(0,\infty)\to(0,\infty)\) instead of \(\sigma:\mathcal{K}\times\mathcal{K}\to[0,\infty)\) and introduced the following definition:  
**Definition 7** ([27]). : _Let \(T\) be a self-mapping of an FMS \((\mathcal{K},\mathcal{M},*)\). We say that \(T\) is a \(\sigma\)-\(\psi\)-fuzzy contractive mapping if there exist two functions \(\sigma:\mathcal{K}\times\mathcal{K}\times(0,\infty)\to(0,\infty)\) and \(\psi\in\mathbb{V}\) such that_  
\[\mathcal{M}(\alpha,\beta,t)>0\ \implies\ \sigma(\alpha,\beta,t)\mathcal{M}(T \alpha,T\beta,t)\geq\psi(\mathcal{M}(\alpha,\beta,t)),\]  
_for all \(\alpha,\beta\in\mathcal{K},t>0\) with \(\alpha\
eq\beta\)._  
**Definition 8** ([27]). : _Let \(T\) be a self-mapping of an FMS and \(\sigma:\mathcal{K}\times\mathcal{K}\times(0,\infty)\to(0,\infty)\).