# Proving Fixed-Point Theorems Employing Fuzzy \((\sigma,\mathcal{Z})\)-Contractive-Type Mappings
## 3 Main Results
Assume that \(T:\mathcal{K}\to\mathcal{K}\) is a fuzzy \((\sigma,\mathcal{Z})\)-contractive mapping and the following properties hold:_  
\((a)\): \(T\) _is_ \(\sigma\)_-admissible;_ \((b^{\prime})\): _The quadruple_ \((\mathcal{K},\mathcal{M},T,\xi)\) _owns the property_ \((S^{\prime})\)_;_ \((c)\): _There exists_ \(\alpha_{0}\in\mathcal{K}\) _with_ \(\sigma(\alpha_{0},Tx_{0},t)\leq 1\)_, for each_ \(t>0\)_;_ \((d)\): _For each sequence_ \(\{\alpha_{n}\}\) _of_ \(\mathcal{K}\) _with the property that_ \(\sigma(\alpha_{n},\alpha_{n+1},t)\leq 1\)_, for each_ \(t>0\)_, there exists_ \(k_{0}\in\mathbb{N}\) _such that_ \(\sigma(\alpha_{n},\alpha_{m},t)\leq 1\)_, for each_ \(m,n\in\mathbb{N}\) _with_ \(m>n\geq k_{0}\)_,_ \(t>0\)_;_ \((e^{\prime})\): _If_ \(\{\alpha_{n}\}\) _is a sequence in_ \(\mathcal{K}\) _such that_ \(\lim_{n\to\infty}\alpha_{n}=\alpha\in\mathcal{K}\) _and_ \(\sigma(\alpha_{n},\alpha_{n+1},t)\leq 1\)_, for each_ \(n\in\mathbb{N}\) _and_ \(t>0\)_, then_ \(\sigma(\alpha_{n},\alpha_{t},t)\leq 1\)_._ _In addition, assume that_ \(\lim_{n\to\infty}\inf_{m>n}\mathcal{M}(T^{n}\alpha,T^{m}\alpha,t)>0\)_, for all_ \(x\in\mathcal{K}\) _and_ \(t>0\)_. Then,_ \(T\) _admits a fixed point._  
Proof. : Following the same lines of the proof of Theorem 4 and taking into account that the quadruple \((\mathcal{K},\mathcal{M},T,\xi)\) owns the property \((S^{\prime})\) instead of the property \((S)\) with the fact that \(\lim_{n\to\infty}\inf_{m>n}\mathcal{M}(T^{n}\alpha,T^{m}\alpha,t)>0\), for all \(x\in\mathcal{K}\) and \(t>0\), we obtain the required result. Next, we discuss the uniqueness of the fixed point in Theorems 3-5. In order to ensure the uniqueness of the fixed point, we add one more sufficient condition to the hypothesis of the theorems. Precisely, we take into account the following condition:  
\((h)\) For each \(\alpha,\beta\in Fix(T)\), we have \(\sigma(\alpha,\beta,t)\leq 1\), for all \(t>0\). **Theorem 6**. : _In addition to the hypothesis of Theorems 3-5, assume that the condition \((h)\) holds. Then, the fixed point of \(T\) is unique._  
Proof. : Theorems 3-5 ensure the existence of a fixed point of \(T\). Assume that \(\gamma_{1}\) and \(\gamma_{2}\) are two distinct fixed points, that is, \(T\gamma_{1}=\gamma_{1}\
eq\gamma_{2}=T\gamma_{2}\). Then, there exists \(s>0\) such that \(\mathcal{M}(\gamma_{1},\gamma_{2},s)<1\). As \(T\) is a fuzzy \((\sigma,\mathcal{Z})\)-contractive mapping, in view of the definition of \(\xi\) and condition \((h)\), we have  
\[\mathcal{M}(\gamma_{1},\gamma_{2},s)\geq\sigma(\gamma_{1},\gamma_{2},s) \mathcal{M}(\gamma_{1},\gamma_{2},s)\geq\xi(\mathcal{M}(\gamma_{1},\gamma_{2 },s),\mathcal{M}(\gamma_{1},\gamma_{2},s))>\mathcal{M}(\gamma_{1},\gamma_{2}, s),\]  
a contradiction. Therefore, \(\mathcal{M}(\gamma_{1},\gamma_{2},t)=1\), for all \(t>0\), that is \(\gamma_{1}=\gamma_{2}\). **Remark 3**. : _Observe that the mappings defined in Examples 5 and 6 satisfy the condition \((h)\), and hence, according to Theorem 6, \(T\) admits a unique fixed point._  
**Corollary 1** ([28]). : _Let \((\mathcal{K},\mathcal{M},*)\) be an \(\mathcal{M}\)-complete FMS and \(T:\mathcal{K}\to\mathcal{K}\). Assume that \(T\) is a fuzzy \(\mathcal{Z}\)-contractive mapping and the quadruple \((\mathcal{K},\mathcal{M},T,\xi)\) owns the property \((S)\). Then, \(T\) possesses a unique fixed point._  
Proof. : The existence of the fixed point follows from Remark 2, Part \((a)\), and Theorem 4, and the uniqueness of the fixed point follows from Theorem 6. **Corollary 2** ([28]). : _Let \((\mathcal{K},\mathcal{M},*)\) be an \(\mathcal{M}\)-complete FMS and \(T:\mathcal{K}\to\mathcal{K}\). Assume that \(T\) is a fuzzy \(\mathcal{Z}\)-contractive mapping and the quadruple \((\mathcal{K},\mathcal{M},T,\xi)\) owns the property \((S^{\prime})\). In addition, assume that \(\lim_{n\to\infty}\inf_{m>n}\mathcal{M}(\alpha_{n},\alpha_{m},t)>0\), for all \(x\in\mathcal{K}\) and \(t>0\). Then, \(T\) possesses a unique fixed point._  
Proof.