# Proving Fixed-Point Theorems Employing Fuzzy \((\sigma,\mathcal{Z})\)-Contractive-Type Mappings
## 2 Mathematical Preliminaries  
In this section, we present some introductory material from the theory of fuzzy metric spaces needed to prove our results. **Definition 1** ([30]). : _Let \(*:[0,1]\times[0,1]\to[0,1]\) be a binary operation. We say that \(*\) is a continuous \(t\)-norm if the following assumptions are fulfilled: \((N1)*\) is associative and commutative; \((N2)*\) is continuous; \((N3)\,r_{1}*r_{2}\leq r_{3}*r_{4}\) whenever \(r_{1}\leq r_{3}\) and \(r_{2}\leq r_{4}\); \((N4)\,r_{1}*1=r_{1}\); for all \(r_{1},r_{2},r_{3},r_{4}\in[0,1]\)._  
Three primary continuous t-norms examples are: \(r_{1}*_{1}r_{2}=r_{1}\cdot r_{2},r_{1}*_{2}r_{2}=\min\{r_{1},r_{2}\}\), and \(r_{1}*_{3}r_{2}=\max\{r_{1}+r_{2}-1,0\}\), which are known as the product, minimum, and Lukasiewicz t-norms, respectively. By modifying the concept of fuzzy metric space introduced in [4], [PERSON] and [PERSON] attempted the following definition:  
**Definition 2** ([6]). : _Let \(\mathcal{K}\) be a non-empty set and \(\mathcal{M}:\mathcal{K}^{2}\times(0,\infty)\to[0,1]\) be a fuzzy set. The ordered triple \((\mathcal{K},\mathcal{M},*)\) is called a fuzzy metric space (FMS), where \(*\) is a continuous t-norm if the following assumptions are fulfilled (for all \(\alpha,\beta,\gamma\in\mathcal{K}\) and \(t,s>0\)): \((\ref{eq:G1})\)\(\mathcal{M}(\alpha,\beta,t)>0\); \((\ref{eq:G2})\)\(\mathcal{M}(\alpha,\beta,t)=1\) if and only if \(\alpha=\beta\); \((\ref{eq:G3})\)\(\mathcal{M}(\alpha,\beta,t)=\mathcal{M}(\beta,\alpha,t)\); \((\ref{eq:G4})\)\(\mathcal{M}(\alpha,\gamma,t)*\mathcal{M}(\gamma,\beta,s)\leq\mathcal{M}(\alpha, \beta,t+s)\); \((\ref{eq:G5})\)\(\mathcal{M}(\alpha,\beta,\cdot):(0,\infty)\to(0,1]\) is continuous._  
Let \((\mathcal{K},\mathcal{M},*)\) be a fuzzy metric space. For \(t>0\), the open ball \(\mathfrak{B}(x,r,t)\) with center \(x\in\mathcal{K}\) and radius \(r\in(0,1)\) is defined by  
\[\mathfrak{B}(x,r,t)=\{y\in\mathcal{K}:\mathcal{M}(x,y,t)>1-r\}.\]  
A subset \(A\subset\mathcal{K}\) is called open if for each \(x\in A\), there exist \(t>0\) and \(r\in(0,1)\) such that \(\mathfrak{B}(x,r,t)\subset A\). The family of all open subsets of \(\mathcal{K}\) is a topology on \(\mathcal{K}\), called the topology induced by the fuzzy metric \(\mathcal{M}\). **Definition 3** ([5; 6]). : _A sequence \(\{\alpha_{n}\}\) in an FMS, \((\mathcal{K},\mathcal{M},*)\) is called: (a) Convergent and converges to \(\alpha\in\mathcal{K}\), if \(\lim_{n\to\infty}\mathcal{M}(\alpha_{n},\alpha,t)=1\), for each \(t>0\); (b) \(\mathcal{M}\)-Cauchy, if for each \(\epsilon\in(0,1)\) and \(t>0\), there is \(n_{0}\in\mathbb{N}\) such that \(\mathcal{M}(\alpha_{m},\alpha_{n},t)>1-\epsilon\), for each \(m,n\geq n_{0}\); (c) \(\mathcal{G}\)-Cauchy, if \(\lim_{n\to\infty}\mathcal{M}(\alpha_{n},\alpha_{n+p},t)=1\), for each \(t>0\) and \(p\in\mathbb{N}\)._  
**Lemma 1** ([5; 6]). : _Let \((\mathcal{K},\mathcal{M},*)\) be a fuzzy metric space: \((1)\)\(\mathcal{M}\) is continuous on \(\mathcal{K}^{2}\times(0,\infty)\); \((2)\)\(\mathcal{M}(\alpha,\beta,\cdot)\) is a non-decreasing function on \((0,\infty)\), for each \(\alpha,\beta\in\mathcal{K}\); \((3)\) The limit of a convergent sequence in \((\mathcal{K},\mathcal{M},*)\) is unique._  
A fuzzy metric space \((\mathcal{K},\mathcal{M},*)\) is called an \(\mathcal{M}\)-complete (\(G\)-complete) FMS, if every \(\mathcal{M}\)-Cauchy (\(\mathcal{G}\)-Cauchy) sequence of \(\mathcal{K}\) converges in \(\mathcal{K}\). Let \(\mathcal{Z}\) be the set of all \(\xi:(0,1]\times(0,1]\to\mathbb{R}\), which satisfy the condition:  
\[\xi(l,s)>s,\ \forall\ l,s\in(0,1).\]  
**Example 1** ([28]). : _Consider the functions \(\xi_{i}:(0,1]\times(0,1]\to\mathbb{R}\), \(i=1,2,3\) which are defined by:_  
1. \(\xi_{1}(l,s)=\frac{1}{\xi}\)_;_
2. \(\xi_{2}(l,s)=\frac{1}{s+1}+l\)_;_
3. \(\xi_{3}(l,s)=\left\{ \begin{array}{cl}l,&\text{ if }l>s,\\ \sqrt{s}&\text{ if }s\geq l,\end{array} \right. ;\)__  
_for all \(l,s\in(0,1]\). Then, \(\xi_{i}\in\mathcal{Z}\), \(i=1,2,3\)._  
**Remark 1** ([28]).