**1. Introduction**

In the sequel, the letter R+ will denote the set of all nonnegative real numbers. Let *S* be a nonempty set and *V* : *S* → *S* be given mappings. A point *j* ∈ *S* is said to be:


Kosjasteh et al. [1] defined a new control function as follows.

**Definition 1** ([1])**.** *Let ζ* : [0, ∞)<sup>2</sup> → R *be a mapping. The mapping ζ is named a simulation function satisfying the following conditions:*

*ζ*1. *ζ* (0, 0) = 0*, ζ*2. *ζ* (*a*, *b*) < *a* − *b, for all a*, *b* > 0*, ζ*3. *if* {*ak*} *and* {*bk*} *are sequences in* R+ *such that* lim*k*→∞ *ak* = lim*k*→∞ *bk* = *l*, *l* ∈ R+*. Thus,* lim sup *ζ* (*ak*, *bk*) < 0.

Argoubi et al. [2] modified the above and so introduced it as follows.

**Definition 2** ([2])**.** *The mapping ζ is a simulation function providing the following:*


*k*→∞

For examples and related results on simulation functions, one may refer to [1–8]. Radenovic and Chandok generalized the simulation function combining the *C*-class function as follows.

**Definition 3** ([4])**.** *A mapping G* : [0, ∞)<sup>2</sup> → R *is named a C-class function if it is continuous and satisfies the following conditions:*


**Definition 4** ([4])**.** *A CG-simulation function is a mapping ζ* : [0, ∞)<sup>2</sup> → R *satisfying the following conditions:*


**Definition 5** ([4])**.** *A mapping G* : [0, ∞)<sup>2</sup> → R *has the property CG, if there exists a CG* ≥ 0 *such that:*


Moreover, using *C*-class function many researchers investigated some new results combining other control functions in different spaces [9].

Suzuki [10] proved the following fixed point theorem using a new contraction, which is known as the Suzuki contraction in literature. Furthermore, many mathematicians generalized this contraction in other spaces.

**Theorem 1** ([10])**.** *Let* (*<sup>S</sup>*, *d*) *be a compact metric space and V* : *S* → *S be a mapping. Suppose that, for all j*, ∈ *S with j* = *,*

$$\frac{1}{2}d\left(\mathfrak{j}, V\mathfrak{j}\right) < d\left(\mathfrak{j}, \ell\right) \quad \Rightarrow \quad d\left(V\mathfrak{j}, V\ell\right) < d\left(\mathfrak{j}, \ell\right).$$

*Then, V has a unique fixed point in S.*

Bindu et al. [11] proved the commonfixed point theorem for Suzuki type mapping in a complete subspace of the partial metric space.

**Theorem 2.** *Let* (*<sup>S</sup>*, *δ*) *be a partial metric space and f* , *g*, *V*, *Z* : *S* → *S be mappings satisfying:*

$$\frac{1}{2}\min\left\{\delta\left(f\mathfrak{j},V\mathfrak{j}\right),\delta\left(\mathfrak{g}\ell,Z\ell\right)\right\} \le \ell\left(f\mathfrak{j},\mathfrak{g}\ell\right) \quad \Rightarrow \quad \Phi\left(V\mathfrak{j},Z\ell\right) \le \mathfrak{a}\left(M\left(\mathfrak{j},\ell\right)\right) - \beta\left(M\left(\mathfrak{j},\ell\right)\right), \quad \mathfrak{j} \le \mathfrak{a}\left(M\left(\mathfrak{j},\mathfrak{j}\right)\right) - \beta\left(M\left(\mathfrak{j},\mathfrak{j}\right)\right)$$

*for all j*, ∈ *S*, *where φ*, *α*, *β* : [0, ∞) → [0, ∞) *are such that φ is an altering distance function, α is continuous, and β is lower-semi continuous α* (0) = *β* (0) = 0 *and φ* (*t*) − *α* (*t*) + *β* (*t*) > 0, *for all t* > 0 *and:*

$$M(\mathfrak{j},\ell) = \max\left\{\delta\left(f\mathfrak{j},\mathfrak{g}\ell\right), \delta\left(f\mathfrak{j},V\ell\right), \delta\left(\mathfrak{g}\ell,Z\ell\right), \frac{\delta\left(f\mathfrak{j},Z\ell\right) + \delta\left(\mathfrak{g}\ell,V\mathfrak{j}\right)}{2}\right\},$$


*Then, f* , *g*, *V*, *Z have a common fixed point.*

Jleli and Samet [12] introduced a Σ-contraction and established fixed point results in generalized metric spaces. Jleli and Samet [12] also introduced a class of Θ such that Σ : (0, ∞) → (1, ∞) of all functions, providing the following conditions:


Σ3. there exist *r* ∈ (0, 1) and *l* ∈ (0, ∞) such that lim*k*→0<sup>+</sup> <sup>Σ</sup>(*k*)−<sup>1</sup> *kr* = *l*.

**Theorem 3** ([12])**.** *Let* (*<sup>S</sup>*, *d*) *be a complete generalized metric space and V* : *S* → *S be a mapping. Suppose that there exist* Σ ∈ Θ *and γ* ∈ (0, 1) *such that:*

$$d\left(V\mathfrak{j}, V\ell\right) \neq 0 \quad \Rightarrow \quad \Sigma\left(d\left(V\mathfrak{j}, V\ell\right)\right) \leq [\Sigma\left(d\left(\mathfrak{j}, \ell\right)\right)]^\gamma.$$

*for all j*, ∈ *S*. *Then, V has a unique fixed point.*

> After that, many authors generalized such a contraction in different spaces [13–17].

Liu et al. [15] modified the class of function Θ exchanging conditions. The class of functions Θ was defined by the set of Σ : (0, ∞) → (1, ∞) satisfying the following conditions:

˜

Σ ˜ 1. Σ is non-decreasing and continuous, Σ ˜ 2. inf *k*∈(0,∞) Σ (*k*) = 1.

**Lemma 1** ([15])**.** *Let* Σ : (0, ∞) → (1, ∞) *be a non-decreasing and continuous function with* inf *k*∈(0,∞) Σ (*k*) = 1 *and* {*ak*} *be a sequence in* (0, <sup>∞</sup>)*. Then, the following condition holds:*

$$\lim\_{k \to \infty} \Sigma\left(a\_k\right) = 1 \quad \Leftrightarrow \quad \lim\_{k \to \infty} a\_k = 0.$$

Zheng et al. [18] denoted new set functions Φ satisfying the following conditions:

Φ1. *ϕ* : [1, ∞) → [1, ∞) is nondecreasing, Φ2. for each *k* > 0, lim *n*→∞ *ϕn* (*k*) = 1, Φ3. *ϕ* is continuous on [1, <sup>∞</sup>).

**Lemma 2** ([18])**.** *If ϕ* ∈ Φ, *then ϕ*(1) = 1 *and ϕ*(*t*) < *t for each t* > 1.

**Definition 6** ([18])**.** *Let* (*<sup>S</sup>*, *d*) *be a metric space and V* : *S* → *S be a mapping. V is said to be a* Σ − *ϕ-contraction if there exist* Σ ∈ Θ *and ϕ* ∈ Φ *such that for any j*, ∈ *S*,

$$
\Sigma\left(d\left(V\_{\mathcal{I}}, V\ell\right)\right) \le \wp\left(\Sigma\left(N\left(\mathfrak{j}, \ell\right)\right)\right),
$$

*where:*

$$N\left(\mathfrak{y},\ell\right) = \max\left\{ d\left(\mathfrak{y},\ell\right), d\left(\mathfrak{y},V\ell\right), d\left(\mathfrak{y},V\ell\right) \right\}.$$

.

**Theorem 4** ([18])**.** *Let* (*<sup>S</sup>*, *d*) *be a complete metric space and V* : *S* → *S be a* Σ − *ϕ-contraction. Then, V has a unique fixed point.*

Motivated by the above, we will establish a generalized Suzuki-simulation-type contractive mapping and obtain fixed point results.

#### **2. Quasi Modular Metric Space**

Girgin and Öztürk [19] introduced a new space, which is named a quasi modular metric space. Furthermore, they gave some topological properties. Moreover, defining non-Archimedean quasi modular metric space, they proved some fixed point theorems and obtained some applications.

**Definition 7** ([19])**.** *A function Q* : (0, ∞) × *S* × *S* → [0, ∞] *is called a quasi modular metric on S if the following hold:*

$$
\eta\_1. \quad \xi = \eta \text{ if and only if } \mathbb{Q}\_m^n(\xi, \eta) = 0 \text{ for all } m > 0;
$$

*q*2. *Qm*<sup>+</sup>*n* (*ξ*, *η*) ≤ *Qm* (*ξ*, *ν*) + *Qn* (*<sup>ν</sup>*, *η*) *for all m*, *n* > 0 *and ξ*, *η*, *ν* ∈ *S.*

*Then, SQ is a quasi modular metric space. If in the above definition, we utilize the condition:*

*q*1 . *Qm* (*ξ*, *ξ*) = 0 *for all m* > 0 *and ξ* ∈ *S*,

*instead of* (*q*1)*, then Q is said to be a quasi pseudo modular metric on S. A quasi modular metric Q on S is called a regular if the following weaker version of* (*q*1) *is satisfied:*

*q*3. *ξ* = *η if and only if Qm* (*ξ*, *η*) = 0 *for some m* > 0.

*Again, Q is called a convex if for m*, *n* > 0 *and ξ*, *η*, *ν* ∈ *S, the inequality holds:*

*q*4. *Qm*<sup>+</sup>*n* (*ξ*, *η*) ≤ *mm*<sup>+</sup>*nQm* (*ξ*, *ν*) + *n m*+*nQn* (*<sup>ν</sup>*, *η*).

**Definition 8** ([19])**.** *In Definition 7, if we replace* (*q*2) *by:*

*q*5. *Q*max{*<sup>m</sup>*,*<sup>n</sup>*} (*ξ*, *η*) ≤ *Qm* (*ξ*, *ν*) + *Qn* (*<sup>ν</sup>*, *η*)

*for all m*, *n* > 0 *and ξ*, *η*, *ν* ∈ *S, then SQ is called a non-Archimedean quasi modular metric space.*

Note that the function *Q*max{*<sup>m</sup>*,*<sup>n</sup>*} is more general than the function *Qm*<sup>+</sup>*n* (*ξ*, *η*), so every non-Archimedean quasi modular metric space is a quasi modular metric space.

**Example 1** ([19])**.** *Let S* = [0, ∞) *and Q be defined by:*

$$Q\_{\mathfrak{m}}\left(\xi,\eta\right) = \begin{cases} \frac{\xi-\eta}{m} & \text{if } \xi \ge \eta\\ 1 & \text{if } \xi < \eta. \end{cases}$$

*Then, SQ is a quasi modular metric space with m* = 13 *and n* = 23 *, but is not modular metric space since Qm* (0, 1) = 1 *and Qm* (1, 0) = 13 *.*

**Remark 1** ([19])**.** *From the above definitions we deduce that:*


Now, we discuss some topological properties of quasi modular metric spaces.

**Definition 9** ([19])**.** *A sequence ξ p in SQ converges to ξ and is called:*


**Definition 10** ([19])**.** *A sequence ξ p in a quasi modular metric space SQ is called:*

*d*. *left (right) Q-K-Cauchy if for every ε* > 0*, there exists pε* ∈ *N such that Qm ξ<sup>r</sup>*, *ξ p* < *ε for all p*,*<sup>r</sup>* ∈ *N with pε* ≤ *r* ≤ *p* (*pε* ≤ *p* ≤ *r*) *and for all m* > 0*.*

*e*. *QE-Cauchy if for every ε* > 0*, there exists pε* ∈ *N such that Qm ξ<sup>p</sup>*, *ξr* < *ε for all p*,*<sup>r</sup>* ∈ *N with p*,*<sup>r</sup>* ≥ *p<sup>ε</sup>.*

**Remark 2** ([19])**.** *From the above definitions, we deduce that:*

*i*. *a sequence is left Q-K-Cauchy with respect to Q if and only if it is right Q-K-Cauchy with respect to Q*−1*;* *ii*. *a sequence is QE-Cauchy if and only if it is left and right Q-K-Cauchy.*

**Definition 11** ([19])**.** *A quasi modular metric space SQ is called:*


#### **3. Common Fixed Point Results**

In the sequel, *Q* is regular and convex and *TZ* denotes the family of all *CG*-simulation functions *ζ* : [0, ∞)<sup>2</sup> → R.

**Definition 12.** *Let SQ be a non-Archimedean quasi modular metric space and V* : *SQ* → *SQ be a mapping. We say that V is a generalized Suzuki-simulation-type contractive mapping if there exist* Σ ∈ Θ ˜ *, ϕ* ∈ Φ *and ζ* ∈ *TZ such that:*

$$\begin{aligned} \frac{1}{2}Q\_m\left(\overline{\xi}, V\overline{\xi}\right) &\leq Q\_m\left(\overline{\xi}, \eta\right) \quad \text{implies} \\\\ \overline{\xi}\left(\Sigma\left(Q\_m\left(V^{\overline{\xi}}, V\eta\right)\right), \rho\left(\Sigma\left(P\left(\overline{\xi}, \eta\right)\right)\right)\right) &\geq C\_G \\\\ \overline{\xi}\left(\overline{\xi}, \eta\right) &\geq C\_G \left(\overline{\xi}, \eta\right) \quad \text{O}\_{\overline{\xi}}\left(\overline{\xi}, V\overline{\xi}\right) \quad \text{O}\_{\overline{\xi}}\left(\overline{\eta}, V\eta\right) \end{aligned} \tag{1}$$

*where:*

$$P\left(\xi,\eta\right) = \max\left\{Q\_{\mathfrak{m}}\left(\xi,\eta\right), Q\_{\mathfrak{m}}\left(\xi,V\xi\right), Q\_{\mathfrak{m}}\left(\eta,V\eta\right)\right\}$$

*for all ξ*, *η* ∈ *SQ.*

**Theorem 5.** *Let SQ be a Q-Smyth-complete non-Archimedean quasi modular metric space and V be the generalized Suzuki-simulation-type contractive mapping. Then, V has a unique fixed point.*

**Proof.** Define a sequence {*ξk*} in *SQ* by:

$$
\xi\_{k+1} = V \xi\_{k\prime} \tag{2}
$$

for all *k* ∈ N. If there exists an *k*0 such that *ξ<sup>k</sup>*0 = *ξ<sup>k</sup>*0+1, then *z* = *ξ<sup>k</sup>*0 becomes a fixed point of *V*. Consequently, we shall assume that *ξk* = *ξ<sup>k</sup>*+<sup>1</sup> for all *k* ∈ N. Therefore, we have:

$$Q\_m\left(\xi\_k, \xi\_{k+1}\right) > 0, \quad \text{for all } n = 0, 1, 2\dots \tag{3}$$

Hence, we have:

$$\frac{1}{2}Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{k},V^{\mathfrak{x}}\_{\xi k}\right) < Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{k},V^{\mathfrak{x}}\_{\xi k}\right) = Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{k},\mathbb{Z}\_{k+1}\right) \quad \text{implies},$$

$$\mathbb{C}\_{G} \le \mathbb{Z}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(V^{\mathfrak{x}}\_{\xi k},V^{\mathfrak{x}}\_{\xi k+1}\right)\right),q\left(\Sigma\left(P\left(\mathfrak{x}\_{k},\mathbb{Z}\_{k+1}\right)\right)\right)\right)$$

$$= \mathbb{Z}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{k+1},\mathbb{Z}\_{k+2}\right)\right),q\left(\Sigma\left(P\left(\mathfrak{x}\_{k},\mathbb{Z}\_{k+1}\right)\right)\right)\right) \tag{4}$$

$$< G\left(q\left(\Sigma\left(P\left(\mathfrak{x}\_{k},\mathbb{Z}\_{k+1}\right)\right)\right),\Sigma\left(Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{k+1},\mathbb{Z}\_{k+2}\right)\right)\right),$$

by Definition 5, we ge<sup>t</sup> that:

$$
\Sigma\left(Q\_m\left(\mathfrak{z}\_{k+1}, \mathfrak{z}\_{k+2}\right)\right) \preccurlyeq \mathfrak{q}\left(\Sigma\left(P\left(\mathfrak{z}\_{k\prime}\mathfrak{z}\_{k+1}\right)\right)\right),
\tag{5}
$$

where:

$$P\left(\xi\_{k},\mathbb{Z}\_{k+1}\right) = \max\left\{Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{k},\mathbb{Z}\_{k+1}\right), Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{k},V^{\mathfrak{x}}\_{\mathbb{X}k}\right), Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{k+1},V^{\mathfrak{x}}\_{\mathbb{X}k+1}\right)\right\}$$

$$= \max\left\{Q\_{\mathfrak{m}}\left(\xi\_{k},\mathbb{Z}\_{k+1}\right), Q\_{\mathfrak{m}}\left(\xi\_{k},\mathbb{Z}\_{k+1}\right), Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{k+1},\mathbb{Z}\_{k+2}\right)\right\}\tag{6}$$

$$= \max\left\{Q\_{\mathfrak{m}}\left(\xi\_{k},\mathbb{Z}\_{k+1}\right), Q\_{\mathfrak{m}}\left(\xi\_{k+1},\mathbb{Z}\_{k+2}\right)\right\}.$$

If:

$$\max\left\{Q\_{m}\left(\mathfrak{z}\_{k},\mathfrak{z}\_{k+1}\right),Q\_{m}\left(\mathfrak{z}\_{k+1},\mathfrak{z}\_{k+2}\right)\right\} = Q\_{m}\left(\mathfrak{z}\_{k+1},\mathfrak{z}\_{k+2}\right)$$

for some *k* ∈ N, then it follows from (5) and Lemma 2 that we get:

$$\Sigma\left(Q\_{\mathfrak{m}}\left(\mathfrak{z}\_{k+1},\mathfrak{z}\_{k+2}\right)\right) \prec \left(\Sigma\left(Q\_{\mathfrak{m}}\left(\mathfrak{z}\_{k+1},\mathfrak{z}\_{k+2}\right)\right)\right) \prec \Sigma\left(Q\_{\mathfrak{m}}\left(\mathfrak{z}\_{k+1},\mathfrak{z}\_{k+2}\right)\right)$$

which is a contradiction. Therefore, we have:

$$P\left(\xi\_{k\prime}\xi\_{k+1}\right) = \mathcal{Q}\_{m}\left(\xi\_{k\prime}\xi\_{k+1}\right)$$

for each *k* ∈ N. Also, by (5), we have

$$
\Sigma\left(Q\_{m}\left(\mathfrak{z}\_{k+1},\mathfrak{z}\_{k+2}\right)\right) \preccurlyeq \mathfrak{q}\left(\Sigma\left(Q\_{m}\left(\mathfrak{z}\_{k},\mathfrak{z}\_{k+1}\right)\right)\right) \dots
$$

Repeating this step, we conclude that:

$$
\begin{split} \Sigma\left(Q\_{\mathfrak{m}}\left(\xi\_{k+1}^{\mathfrak{z}},\xi\_{k+2}^{\mathfrak{z}}\right)\right) &< \varrho\left(\Sigma\left(Q\_{\mathfrak{m}}\left(\xi\_{k}^{\mathfrak{z}},\xi\_{k+1}^{\mathfrak{z}}\right)\right)\right) \\ &< \varrho^{2}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(\xi\_{k-1}^{\mathfrak{z}},\xi\_{k}^{\mathfrak{z}}\right)\right)\right) \\\\ &\vdots \\\\ &< \varrho^{k}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(\xi\_{1}^{\mathfrak{z}},\xi\_{2}^{\mathfrak{z}}\right)\right)\right). \end{split}
$$

for all *k* ∈ N. Taking the limit *k* → ∞ above, by the definition of *ϕ* and property Θ2, we have:

$$\lim\_{k \to \infty} \varphi^k \left( Q\_m \left( \mathfrak{f}\_1, \mathfrak{f}\_2 \right) \right) = 1. \tag{7}$$

Thus, from Lemma 1, it follows that:

$$\lim\_{k \to \infty} Q\_{\mathfrak{m}}\left(\mathfrak{z}\_{k+1}, \mathfrak{z}\_{k+2}\right) = 0,\tag{8}$$

for all *k* ∈ N. Now, we show that {*ξk*} is a left *Q*-*<sup>K</sup>*-Cauchy sequence. Assume the contrary. There exists *ε* > 0 such that we can find two subsequences {*tk*} and {*sk*} of positive integers satisfying the following inequalities:

$$Q\_{\mathcal{W}}\left(\xi\_{t\_k}^{\mathbf{x}},\xi\_{\mathbb{s}\_k}^{\mathbf{z}}\right) \stackrel{>}{\geq} \varepsilon, \quad \text{and} \quad Q\_{\mathcal{W}}\left(\xi\_{t\_k-1}^{\mathbf{x}},\xi\_{\mathbb{s}\_k}^{\mathbf{x}}\right) \stackrel{<}{\leq} \varepsilon. \tag{9}$$

From (9) and (*q*5), it follows that:

$$\begin{array}{l} \varepsilon \le Q\_{\mathfrak{m}}\left(\mathfrak{f}\_{t\_{k}}^{\mathfrak{x}}, \mathfrak{f}\_{s\_{k}}^{\mathfrak{x}}\right) = Q\_{\max\left\{\mathfrak{m}, \mathfrak{m}\right\}}\left(\mathfrak{f}\_{t\_{k}}^{\mathfrak{x}}, \mathfrak{f}\_{s\_{k}}^{\mathfrak{x}}\right) \\ \quad \le Q\_{\mathfrak{m}}\left(\mathfrak{f}\_{t\_{k}}^{\mathfrak{x}}, \mathfrak{f}\_{t\_{k}-1}^{\mathfrak{x}}\right) + Q\_{\mathfrak{m}}\left(\mathfrak{f}\_{t\_{k}-1}^{\mathfrak{x}}, \mathfrak{f}\_{s\_{k}}^{\mathfrak{x}}\right) \\ \quad < \varepsilon + Q\_{\mathfrak{m}}\left(\mathfrak{f}\_{t\_{k}}^{\mathfrak{x}}, \mathfrak{f}\_{t\_{k}-1}^{\mathfrak{x}}\right) . \end{array} \tag{10}$$

On taking the limit as *k* → ∞ in the above relation, we obtain that:

$$\lim\_{k \to \infty} Q\_{\mathfrak{m}}\left(\xi\_{t\_{k'}}, \xi\_{s\_k}\right) = \varepsilon. \tag{11}$$

Also, from (9) and (*q*5), it follows that:

*Qm ξtk*+1, *ξsk*+<sup>1</sup> = *Q*max{*<sup>m</sup>*,*<sup>m</sup>*} *ξtk*+1, *ξsk*+<sup>1</sup> ≤ *Qm ξtk*+1, *ξtk* + *Qm ξtk* , *ξsk*+<sup>1</sup> = *Qm ξtk*+1, *ξtk* + *Q*max{*<sup>m</sup>*,*<sup>m</sup>*} *ξtk* , *ξsk*+<sup>1</sup> ≤ *Qm ξtk* , *ξtk*−<sup>1</sup> + *Qm ξtk*−1, *ξsk*+<sup>1</sup> + *Qm ξtk*+1, *ξtk* = *Qm ξtk* , *ξtk*−<sup>1</sup> + *Qm ξtk*+1, *ξtk* + *Q*max{*<sup>m</sup>*,*<sup>m</sup>*} *ξtk*−1, *ξsk*+<sup>1</sup> ≤ *Qm ξtk*−1, *ξsk* + *Qm ξsk* , *ξsk*+<sup>1</sup> +*Qm ξtk* , *ξtk*−<sup>1</sup> + *Qm ξtk*+1, *ξtk* < *ε* + *Qm ξsk* , *ξsk*+<sup>1</sup> + *Qm ξtk* , *ξtk*−<sup>1</sup> +*Qm ξtk*+1, *ξtk* . (12)

Next, we claim that:

If:

$$\begin{aligned} \frac{1}{2}Q\_{\mathfrak{m}}\left(\mathfrak{z}\_{t\_{k'}}^{\mathfrak{x}}, V^{\mathfrak{x}}\_{\mathfrak{t}\_{k}}\right) & \leq Q\_{\mathfrak{m}}\left(\mathfrak{z}\_{t\_{k'}}^{\mathfrak{x}}, \mathfrak{z}\_{s\_{k}}^{\mathfrak{x}}\right), \\\\ \frac{1}{2}Q\_{\mathfrak{m}}\left(\mathfrak{z}\_{t\_{k'}}^{\mathfrak{x}}, V^{\mathfrak{x}}\_{\mathfrak{t}\_{k}}\right) & > Q\_{\mathfrak{m}}\left(\mathfrak{z}\_{t\_{k'}}^{\mathfrak{x}}, \mathfrak{z}\_{s\_{k}}\right) \\\\ \frac{1}{2}Q\_{\mathfrak{m}}\left(\mathfrak{z}\_{t\_{k'}}^{\mathfrak{x}}, \mathfrak{z}\_{t\_{k}+1}\right) & > Q\_{\mathfrak{m}}\left(\mathfrak{z}\_{t\_{k'}}^{\mathfrak{x}}, \mathfrak{z}\_{s\_{k}}\right), \end{aligned} \tag{13}$$

.

then letting *k* → ∞ in (13), from (11) and (8), we have that 0 > *ε* is a contradiction. Hence,

$$\frac{1}{2}Q\_m\left(\mathfrak{f}\_{t\_{k'}}^\times V \mathfrak{f}\_{\mathfrak{f}\_k}^\times\right) \le Q\_m\left(\mathfrak{f}\_{t\_{k'}}^\times \mathfrak{f}\_{\mathfrak{s}\_k}^\times\right)$$

From the generalized Suzuki-simulation-type contractive mapping, we get:

$$\begin{split} \mathbb{C}\_{\mathbb{G}} &\leq \mathbb{Z}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(V^{\mathbb{Z}}\_{\mathbb{G},t\_{k}},V^{\mathbb{Z}}\_{\mathbb{G},s\_{k}}\right)\right),\,\mathrm{\uprho}\left(\Sigma\left(P\left(\mathbb{Z}\_{t\_{k}},\mathbb{Z}\_{s\_{k}}\right)\right)\right)\right) \\ &= \mathbb{Z}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{t\_{k}+1},\mathbb{Z}\_{s\_{k}+1}\right)\right),\,\mathrm{\uprho}\left(\Sigma\left(P\left(\mathbb{Z}\_{t\_{k}},\mathbb{Z}\_{s\_{k}}\right)\right)\right)\right), \end{split} \tag{14}$$

where:

$$\begin{split} P\left(\mathbf{\tilde{\xi}}\_{l\_{k'}}\mathbf{\tilde{\xi}}\_{\mathbf{\tilde{s}}\_{k}}\right) &= \max\left\{ Q\_{\mathcal{m}}\left(\mathbf{\tilde{\xi}}\_{l\_{k'}}\mathbf{\tilde{\xi}}\_{\mathbf{\tilde{s}}\_{k}}\right), Q\_{\mathcal{m}}\left(\mathbf{\tilde{\xi}}\_{l\_{k'}}, V\mathbf{\tilde{\xi}}\_{\mathbf{\tilde{t}}\_{k}}\right), Q\_{\mathcal{m}}\left(\mathbf{\tilde{\xi}}\_{\mathbf{\tilde{s}}\_{k'}}, V\mathbf{\tilde{\xi}}\_{\mathbf{\tilde{s}}\_{k}}\right) \right\} \\ &= \max\left\{ Q\_{\mathcal{m}}\left(\mathbf{\tilde{\xi}}\_{l\_{k'}}\mathbf{\tilde{\xi}}\_{\mathbf{\tilde{s}}\_{k}}\right), Q\_{\mathcal{m}}\left(\mathbf{\tilde{\xi}}\_{l\_{k'}}\mathbf{\tilde{\xi}}\_{\mathbf{\tilde{t}}\_{k}+1}\right), Q\_{\mathcal{m}}\left(\mathbf{\tilde{\xi}}\_{\mathbf{\tilde{s}}\_{k'}}\mathbf{\tilde{\xi}}\_{\mathbf{\tilde{s}}\_{k}+1}\right) \right\}. \end{split} \tag{15}$$

Taking the limit *k* → ∞ using (8), (11), and (12) in (14) and (15), we obtain:

$$C\_G \le \mathcal{J}\left(\Sigma\left(\varepsilon\right), \varrho\left(\Sigma\left(\varepsilon\right)\right)\right) < G\left(\wp\left(\Sigma\left(\varepsilon\right)\right), \Sigma\left(\varepsilon\right)\right).$$

From Definition 5, we get:

$$
\Sigma\left(\mathfrak{e}\right) < \mathfrak{p}\left(\Sigma\left(\mathfrak{e}\right)\right) < \Sigma\left(\mathfrak{e}\right).
$$

It follows that Σ (*ε*) < Σ (*ε*), a contradiction. Hence, {*ξk*} is a left *Q*-*<sup>K</sup>*-Cauchy sequence. As *SQ* is a *Q*-Smyth-complete non-Archimedean quasi modular metric space, there exists *u* ∈ *SQ* such that:

$$\lim\_{k \to \infty} Q\_{\mathcal{W}}^{\;\;\;E} \left( \S\_k, u \right) = 0.$$

Thus, we have:

$$\lim\_{k \to \infty} Q\_m\left(\mathcal{J}\_{k'}\mu\right) = 0 \qquad \text{and} \qquad \lim\_{k \to \infty} Q\_m\left(\mu, \mathcal{J}\_k\right) = 0.$$

Now, we show that *u* is a fixed point of *V*. Assume that *Qm* (*Vu*, *u*) > 0. We claim that for each *k* ≥ 0, the following holds:

$$\frac{1}{2}\mathcal{Q}\_{\mathfrak{m}}\left(\mathcal{\xi}\_{k\prime}V\mathcal{\xi}\_{k}\right) \le \mathcal{Q}\_{\mathfrak{m}}\left(\mathcal{\xi}\_{k\prime}\mathfrak{u}\right).$$

On the contrary, suppose that:

$$\frac{1}{2}Q\_{\mathfrak{m}}\left(\mathfrak{f}\_{k\prime}^{\mathfrak{x}}V^{\mathfrak{x}}\_{\xi k}\right) > Q\_{\mathfrak{m}}\left(\mathfrak{f}\_{k\prime}^{\mathfrak{x}}\mathfrak{u}\right) = \frac{1}{2}Q\_{\mathfrak{m}}\left(\mathfrak{f}^{\mathfrak{x}}\_{k\prime}\mathfrak{f}^{\mathfrak{x}}\_{k+1}\right) > Q\_{\mathfrak{m}}\left(\mathfrak{f}^{\mathfrak{x}}\_{k\prime}\mathfrak{u}\right).\tag{16}$$

Taking the limit as *k* → ∞ in (16), we obtain 0 > 0, a contradiction. Thus, the claim is true, and so, from the generalized Suzuki-simulation-type contractive mapping, we get:

$$\begin{split} \mathbb{C}\_{G} &\leq \mathbb{f}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(V\_{\mathfrak{}^{\mathfrak{K}}},Vu\right)\right), \mathfrak{q}\left(\Sigma\left(P\left(\mathbb{Z}\_{k},u\right)\right)\right)\right) \\ &= \mathbb{f}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{k+1},Vu\right)\right), \mathfrak{q}\left(\Sigma\left(P\left(\mathbb{Z}\_{k},u\right)\right)\right)\right) \\ &< G\left(\mathfrak{q}\left(\Sigma\left(P\left(\mathbb{Z}\_{k},u\right)\right)\right), \Sigma\left(Q\_{\mathfrak{m}}\left(\mathbb{Z}\_{k+1},Vu\right)\right)\right). \end{split} \tag{17}$$

By Definition 5,

$$
\Sigma\left(Q\_{\mathfrak{m}}\left(\mathfrak{J}\_{k+1}, V\mathfrak{u}\right)\right) < \mathfrak{q}\left(\Sigma\left(P\left(\mathfrak{J}\_{k}, \mathfrak{u}\right)\right)\right),
\tag{18}
$$

 .

where:

$$\begin{split} P\left(\xi\_{k'}\boldsymbol{u}\right) &= \max\left\{Q\_{\mathfrak{m}}\left(\xi\_{k'}\boldsymbol{u}\right), Q\_{\mathfrak{m}}\left(\xi\_{k'}\,V\_{\xi\_{k}}^{\mathfrak{x}}\right), Q\_{\mathfrak{m}}\left(\boldsymbol{u},\boldsymbol{V}\boldsymbol{u}\right)\right\} \\ &= \max\left\{Q\_{\mathfrak{m}}\left(\xi\_{k'}\boldsymbol{u}\right), Q\_{\mathfrak{m}}\left(\xi\_{k'}\,\xi\_{k+1}\right), Q\_{\mathfrak{m}}\left(\boldsymbol{u},\boldsymbol{V}\boldsymbol{u}\right)\right\}. \end{split} \tag{19}$$

Letting *k* → ∞ in (17)–(19), we have:

$$
\Sigma\left(Q\_{\mathfrak{m}}\left(\mathfrak{u}, V\mathfrak{u}\right)\right) \prec \varphi\left(\Sigma\left(Q\_{\mathfrak{m}}\left(\mathfrak{u}, V\mathfrak{u}\right)\right)\right) \prec \Sigma\left(Q\_{\mathfrak{m}}\left(\mathfrak{u}, V\mathfrak{u}\right)\right).
$$

That is, Σ(*Qm* (*<sup>u</sup>*,*Vu*)) < Σ(*Qm* (*<sup>u</sup>*,*Vu*)), a contradiction. Thus, *u* is a fixed point of *V*. Suppose that there is an another fixed point *u*<sup>∗</sup> of *V* such that *Vu*∗ = *u*<sup>∗</sup> and *u* = *<sup>u</sup>*<sup>∗</sup>. Then, *Qm* (*Vu*,*Vu*<sup>∗</sup>) = *Qm* (*u*, *u*<sup>∗</sup>) > 0, and:

$$0 = \frac{1}{2} \mathcal{Q}\_m \left( \mu\_\prime \, V \mu \right) \le \mathcal{Q}\_m \left( \mu\_\prime \, \mu^\* \right).$$

By the generalized Suzuki-simulation-type contractive mapping, we have:

$$\begin{split} \mathbb{C}\_{G} &\leq \mathbb{Z}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(Vu, Vu^{\*}\right)\right), \varrho\left(\Sigma\left(P\left(u, u^{\*}\right)\right)\right)\right) \\ &= \mathbb{Z}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(u, u^{\*}\right)\right), \varrho\left(\Sigma\left(P\left(u, u^{\*}\right)\right)\right)\right) \\ &< G\left(\varrho\left(\Sigma\left(P\left(u, u^{\*}\right)\right)\right), \Sigma\left(Q\_{\mathfrak{m}}\left(u, u^{\*}\right)\right)\right). \end{split} \tag{20}$$

From the property of *G*,

$$
\Sigma\left(Q\_m\left(u, u^\*\right)\right) < \left.\wp\left(\Sigma\left(P\left(u, u^\*\right)\right)\right)\right|,\tag{21}
$$

where:

$$P\left(\mu,\mu^\*\right) = \max\left\{Q\_{\mathfrak{m}}\left(\mu,\mu^\*\right), Q\_{\mathfrak{m}}\left(\mu,V\mathfrak{u}\right), Q\_{\mathfrak{m}}\left(\mu^\*,V\mathfrak{u}^\*\right)\right\} = Q\_{\mathfrak{m}}\left(\mu,\mu^\*\right). \tag{22}$$

From (20)–(22), we attain the following ordering:

$$
\Sigma\left(Q\_{\mathfrak{m}}\left(\mu,\mu^\*\right)\right) < \wp\left(\Sigma\left(Q\_{\mathfrak{m}}\left(\mu,\mu^\*\right)\right)\right) < \Sigma\left(Q\_{\mathfrak{m}}\left(\mu,\mu^\*\right)\right),
$$

which is a contradiction. Hence, *u* is a unique fixed point of *V*.

Now, we give some corollaries that are directly acquired from our main results. **Corollary 1.** *Let SQ be a Q-Smyth-complete non-Archimedean quasi modular metric space and V* : *SQ* → *SQ be a mapping. If there exists* Σ ∈ Θ ˜ *, ϕ* ∈ Φ*, and ζ* ∈ *TZ such that:*

$$\frac{1}{2}Q\_{\mathfrak{m}}\left(\mathfrak{J}\_{\prime}V\_{\!\!/\!}\right) \le Q\_{\mathfrak{m}}\left(\mathfrak{J}\_{\prime}\ell\right) \qquad \text{implies}\_{\prime}$$

$$\mathcal{Z}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(V\_{\mathcal{I}},V\ell\right)\right),\mathfrak{p}\left(\Sigma\left(Q\_{\mathfrak{m}}\left(\mathfrak{j},\ell\right)\right)\right)\right) \geq C\_{G,\ell}$$

*for all j*, ∈ *SQ, then V has a unique fixed point.*

**Corollary 2.** *Let SQ be a Q-Smyth-complete non-Archimedean quasi modular metric space and V* : *SQ* → *SQ be a mapping. If there exists* Σ ∈ Θ ˜ *, ϕ* ∈ Φ*, and ζ* ∈ *TZ such that:*

$$\zeta\left(\Sigma\left(Q\_{m}\left(V\mathfrak{j},V\ell\right)\right),\mathfrak{p}\left(\Sigma\left(P\left(\mathfrak{j},\ell\right)\right)\right)\right) \ge C\_G$$

*where:*

$$P\left(\mathfrak{j},\ell\right) = \max\left\{Q\_{\mathfrak{m}}\left(\mathfrak{j},\ell\right), Q\_{\mathfrak{m}}\left(\mathfrak{j},V\mathfrak{j}\right), Q\_{\mathfrak{m}}\left(\ell,V\ell\right)\right\},$$

*for all j*, ∈ *SQ, then V has a unique fixed point.*

**Corollary 3.** *Let SQ be a Q-Smyth-complete non-Archimedean quasi modular metric space and V* : *SQ* → *SQ be a mapping. If there exists* Σ ∈ Θ ˜ *and ϕ* ∈ Φ *such that:*

$$\frac{1}{2}\mathcal{Q}\_{\mathfrak{m}}\left(\mathcal{J}\_{\mathfrak{l}}\,V\_{\mathcal{I}}\right) \le \mathcal{Q}\_{\mathfrak{m}}\left(\mathcal{J}\_{\mathfrak{l}}\,\ell\right) \qquad \text{implies},$$

$$\Sigma\left(\mathcal{Q}\_{\mathfrak{m}}\left(V\_{\mathcal{I}}\,V\ell\right)\right) \le \mathcal{q}\left(\Sigma\left(P\left(\mathcal{J}\_{\mathfrak{l}}\ell\right)\right)\right)$$

*where:*

$$P\left(\mathfrak{j},\ell\right) = \max\left\{Q\_{\mathfrak{m}}\left(\mathfrak{j},\ell\right), Q\_{\mathfrak{m}}\left(\mathfrak{j},V\mathfrak{j}\right), Q\_{\mathfrak{m}}\left(\ell,V\ell\right)\right\},$$

*for all j*, ∈ *SQ, then V has a unique fixed point.*

**Corollary 4.** *Let SQ be a Q-Smyth-complete non-Archimedean quasi modular metric space and V* : *SQ* → *SQ be a mapping. If there exists* Σ ∈ Θ ˜ *and ϕ* ∈ Φ *such that:*

$$
\Sigma\left(Q\_{\mathfrak{m}}\left(V\_{\mathfrak{l}}, V\ell\right)\right) \le \wp\left(\Sigma\left(P\left(\mathfrak{l}, \ell\right)\right)\right),
$$

*where:*

> *P* (*j*, ) = max {*Qm* (*j*, ), *Qm* (*j*, *<sup>V</sup>j*), *Qm* (, *V*)} ,

*for all j*, ∈ *SQ, then V has a unique fixed point.*

**Corollary 5.** *Let SQ be a Q-Smyth-complete non-Archimedean quasi modular metric space and V* : *SQ* → *SQ be a mapping. If there exists* Σ ∈ Θ ˜ *and ϕ* ∈ Φ *such that:*

$$
\Sigma\left(Q\_m\left(V\_{\mathcal{I}}, V\ell\right)\right) \le q\left(\Sigma\left(Q\_m\left(\mathcal{I}\_{\mathcal{I}}\ell\right)\right)\right),
$$

*for all j*, ∈ *SQ, then V has a unique fixed point.*

#### **4. Application to a Graph Structure**

Let *SQ* be a non-Archimedean quasi modular metric space and Δ = {(*j*, *j*) : *j* ∈ *SQ*} denote the diagonal of *SQ* × *SQ*. Let *H* be a directed graph such that the set *C*(*H*) of its vertices coincides with *SQ* and *B*(*H*) is the set of edges of the graph such that Δ ⊆ *<sup>B</sup>*(*H*). *H* is determined with the pair (*C*(*H*), *<sup>B</sup>*(*H*)).

If *j* and are vertices of *H*, then a path in *H* from *j* to of length *p* ∈ N is a finite sequence {*jp*} of vertices such that *j* = *j*0, ..., *jp* = *η* and (*ji*−1, *ji*) ∈ *B*(*H*) for *i* ∈ {1, 2, ..., *p*}.

Recall that *H* is connected if there is a path between any two vertices, and it is weakly connected if *H* is connected, where *H* defines the undirected graph obtained from *H* by ignoring the direction of edges. Define by *H*−<sup>1</sup> the graph obtained from *H* by reversing the direction of edges. Thus,

$$B\left(H^{-1}\right) = \left\{ (\mathfrak{J}, \ell) \in \mathcal{S}\_{\mathcal{Q}} \times \mathcal{S}\_{\mathcal{Q}} : (\ell, \mathfrak{j}) \in B\left(H\right) \right\}.$$

It is more convenient to treat *H* as a directed graph for which the set of its edges is symmetric, under this convention; we have that:

$$B(\tilde{H}) = B(H) \cup B(H^{-1}).$$

Let *Hj* be the component of *H* consisting of all the edges and vertices that are contained in some way in *H* starting at *j*. We denote the relation (*R*) in the following way:

We have *j*(*R*) if and only if, there is a path in *H* from *j* to , for *j*, ∈ *<sup>C</sup>*(*H*).

If *H* is such that *B*(*H*) is symmetric, then for *j* ∈ *<sup>C</sup>*(*H*), the equivalence class [*j*]*G* in *V*(*G*) described by the relation (*R*) is *<sup>C</sup>*(*<sup>H</sup>j*).

Let *SQ* be a non-Archimedean quasi modular metric space endowed with a graph *H* and *h*¯ : *SQ* → *SQ*. We set:

$$\mathcal{S}\_{\hbar} = \left\{ \mathfrak{f} \in \mathcal{S}\_{Q} : (\mathfrak{f}, \hbar \mathfrak{f}) \in B\left(H\right) \right\}.$$

**Definition 13** ([20])**.** (*<sup>S</sup>*, *d*) *is a metric space, and h*¯ : *S* → *S is a mapping. Then, h*¯ *is called a Banach H-contraction if the following hold:*

*B*1. *h preserves edges of H, i.e., for all* ¯ *j*, ∈ *S*,

$$(\mathfrak{h}, \ell) \in B\left(H\right) \qquad \Rightarrow \qquad (\mathfrak{h}\mathfrak{h}, \mathfrak{h}\ell) \in B\left(H\right)\,.$$

*B*2. *there exists δ* ∈ (0, 1) *such that:*

$$d\left(\hbar j\_\prime \hbar \ell\right) \le \delta d\left(j\_\prime \ell\right)$$

*for all* (*j*, ) ∈ *<sup>B</sup>*(*H*)*.*

After that, many fixed point researchers investigated fixed point results improving the Jachymski fixed point theorems in [17,21–23].

Now, motivated by [24–26], we generate a new contraction and obtain fixed point results using a graph structure.

**Definition 14.** *Let SQ be a non-Archimedean quasi modular metric space and h*¯ : *SQ* → *SQ be a mapping. Then, we say that h*¯ *is a generalized Suzuki-simulation-H-type contractive mapping if the following conditions hold:*

*H*1. *h preserves edges of G;* ¯ *H*2. *there exists* Σ ∈ Θ ˜ *, ϕ* ∈ Φ *and ζ* ∈ *TZ such that:*

$$\begin{aligned} \,\_2^1Q\_{\mathfrak{m}}\left(\mathfrak{h}, \mathfrak{h}\mathfrak{h}\right) &\leq Q\_{\mathfrak{m}}\left(\mathfrak{f}, \ell\right) \quad \text{implies},\\ \,\_2^\zeta\left(\Sigma\left(Q\_{\mathfrak{m}}\left(\mathfrak{h}\mathfrak{h}, \mathfrak{h}\ell\right)\right), \mathfrak{g}\left(\Sigma\left(P\left(\mathfrak{f}, \ell\right)\right)\right)\right) &\geq C\_{\mathsf{G}, \ell} \end{aligned} \tag{23}$$

*where*

$$P\left(\mathfrak{j},\ell\right) = \max\left\{Q\_{\mathfrak{m}}\left(\mathfrak{j},\ell\right), Q\_{\mathfrak{m}}\left(\mathfrak{j},\mathfrak{h}\mathfrak{j}\right), Q\_{\mathfrak{m}}\left(\ell,\mathfrak{h}\ell\right)\right\}$$

*for all j*, ∈ *B*(*H*) *and for all m* > 0*.* **Remark 3.** *Let SQ be a non-Archimedean quasi modular metric space with a graph H and h*¯ : *SQ* → *SQ be a generalized Suzuki-simulation-H-type contractive mapping. If there exists j*0 ∈ *SQ such that h*¯ *j*0 ∈ [*j*0]*H*˜ *, then:*


**Theorem 6.** *Let SQ be a Q-Smyth-complete non-Archimedean quasi modular metric space with a graph H and h*¯ : *SQ* → *SQ be a mapping.*



*Then, h has a unique fixed point.* ¯

**Proof.** Define a sequence {*jk*} in *SQ* by:

$$
\eta\_{k+1} = \hbar \eta\_{k\prime} \tag{24}
$$

for all *k* ∈ N. Let *j*0 be a given point in *Sh*¯ ; thus, (*j*0, ¯*hj*0) = (*j*0, *j*1) ∈ *B* (*H*). Because *h*¯ preserves the edges of *H*,

> (*j*0, *j*1) ∈ *B* (*H*) ⇒ (*h*¯ *j*0, ¯*hj*1) ∈ *B* (*H*).

Continuing this way, we get:

$$(\mathfrak{j}\_{k\prime Jk+1}) \in \mathcal{B}\left(H\right).$$

Then from Theorem 5, we ge<sup>t</sup> that {*jk*} is a left *Q*-*<sup>K</sup>*-Cauchy sequence in *SQ*. By the *Q*-Smythcompleteness of *SQ*, there exists *u* ∈ *SQ* such that:

$$\lim\_{k \to \infty} Q\_m^E \left( y\_k, \mu \right) = 0. \tag{25}$$

Thus, we have:

$$\lim\_{k \to \infty} Q\_{\mathfrak{m}}\left(\jmath\_k, \mathfrak{u}\right) = 0 \quad \text{and} \quad \lim\_{k \to \infty} Q\_{\mathfrak{m}}\left(\mathfrak{u}, \jmath\_k\right) = 0. \tag{26}$$

Now, we show that *u* is a fixed point of ¯*h*. Using (*iv*), we ge<sup>t</sup> (*jks*, *u*) ∈ *B* (*H*). We claim that:

$$\frac{1}{2}Q\_m\left(\jmath\_{k\_{s'}}\hbar\jmath\_{k\_s}\right) \le Q\_m\left(\jmath\_{k\_{s'}}\mu\right).\tag{27}$$

If

$$\frac{1}{2}Q\_{\mathfrak{m}}\left(\jmath\_{k\_{s'}}\hbar\jmath\_{k\_s}\right) > Q\_{\mathfrak{m}}\left(\jmath\_{k\_{s'}}\hbar\right) = \frac{1}{2}Q\_{\mathfrak{m}}\left(\jmath\_{k\_{s'}}\jmath\_{k\_s+1}\right) > Q\_{\mathfrak{m}}\left(\jmath\_{k\_{s'}}\hbar\right) \tag{28}$$

and taking the limit *s* → ∞ in (28), we ge<sup>t</sup> 0 > 0, a contradiction. Hence, the claim is true. Since *h*¯ is a generalized Suzuki-simulation-*H* ˜ -type contractive mapping, we have:

$$\mathbb{C}\_{G} \le \mathbb{Z}\left(\Sigma\left(Q\_{m}\left(\hbar\_{\mathcal{I}k\_{s}}, \hbar u\right)\right), \varrho\left(\Sigma\left(P\left(\mathcal{I}\_{k\_{s}}, u\right)\right)\right)\right)$$

$$\le \mathbb{Z}\left(\Sigma\left(Q\_{m}\left(\hbar\_{\mathcal{I}k\_{s}}, \hbar u\right)\right), \varrho\left(\Sigma\left(P\left(\mathcal{I}\_{k\_{s}}, u\right)\right)\right)\right) \tag{29}$$

$$\le G\left(\varrho\left(\Sigma\left(P\left(\mathcal{I}\_{k\_{s}}, u\right)\right)\right), \Sigma\left(Q\_{m}\left(\hbar\_{\mathcal{I}k\_{s}}, \hbar u\right)\right)\right),$$

from Definition 5, we get:

$$
\Sigma\left(Q\_{\mathfrak{m}}\left(\hbar j\_{k\_{\mathfrak{s}}}, \hbar u\right)\right), \varphi\left(\Sigma\left(P\left(j\_{k\_{\mathfrak{s}}}, \mathfrak{u}\right)\right)\right),
\tag{30}
$$

where:

$$\begin{split} P\left(j\_{k\_{\rm s}},\boldsymbol{\mu}\right) &= \max\left\{ Q\_{\rm m}\left(j\_{k\_{\rm s}},\boldsymbol{\mu}\right), Q\_{\rm m}\left(j\_{k\_{\rm s}},\hbar j\_{k\_{\rm s}}\right), \, Q\_{\rm m}\left(\boldsymbol{\mu},\hbar \boldsymbol{\mu}\right) \right\} \\ &= \max\left\{ Q\_{\rm m}\left(j\_{k\_{\rm s}},\boldsymbol{\mu}\right), Q\_{\rm m}\left(j\_{k\_{\rm s}},j\_{k\_{\rm s}+1}\right), \, Q\_{\rm m}\left(\boldsymbol{\mu},\hbar \boldsymbol{\mu}\right) \right\}. \end{split} \tag{31}$$

Taking the limit as *s* → ∞ in (29)–(31), we get:

$$
\Sigma\left(Q\_m\left(\mathfrak{u}, h\mathfrak{u}\right)\right) < \left\|\left(\Sigma\left(Q\_m\left(\mathfrak{u}, h\mathfrak{u}\right)\right)\right) < \Sigma\left(Q\_m\left(\mathfrak{u}, h\mathfrak{u}\right)\right)\right\|
$$

It follows that Σ (*Qm* (*<sup>u</sup>*, *hu*)) < Σ (*Qm* (*<sup>u</sup>*, *hu*)), a contradiction. Therefore, we ge<sup>t</sup> *Qm* (*<sup>u</sup>*, ¯*hu*) = 0, that is *u* = *hu*¯ since *Q* is regular.

Next, we show that *u* is a unique fixed point of *h*¯. On the contrary, we suppose that *u*<sup>∗</sup> is another fixed point of *h*¯, i.e., *u*<sup>∗</sup> = *hu*¯ ∗ and *u* = *<sup>u</sup>*<sup>∗</sup>. Then, there exist *σ* ∈ *SQ* such that (*<sup>u</sup>*, *σ*) ∈ *B*(*H*) and (*<sup>σ</sup>*, *u*<sup>∗</sup>) ∈ *<sup>B</sup>*(*H*). Using (*iii*), we ge<sup>t</sup> that (*<sup>u</sup>*, *u*<sup>∗</sup>) ∈ *B*(*H*˜ ). Furthermore,

$$0 = \frac{1}{2} Q\_{\mathfrak{m}}\left(\mathfrak{u}, h\mathfrak{u}\right) < Q\_{\mathfrak{m}}\left(\mathfrak{u}, \mathfrak{u}^\*\right). \tag{32}$$

From the generalized Suzuki-Simulation-*H* ˜ -type contractive mapping we have:

$$\begin{split} \mathsf{C}\_{G} &\leq \mathsf{f}\left(\Sigma\left(Q\_{\mathsf{m}}\left(\mathsf{hu},\mathsf{hu}^{\*}\right)\right), \mathsf{q}\left(\Sigma\left(P\left(\mathsf{u},\mathsf{u}^{\*}\right)\right)\right)\right) \\ &\leq \mathsf{f}\left(\Sigma\left(Q\_{\mathsf{m}}\left(\mathsf{u},\mathsf{u}^{\*}\right)\right), \mathsf{q}\left(\Sigma\left(P\left(\mathsf{u},\mathsf{u}^{\*}\right)\right)\right)\right) \\ &\leq \mathsf{G}\left(\mathsf{q}\left(\Sigma\left(P\left(\mathsf{u},\mathsf{u}^{\*}\right)\right)\right), \Sigma\left(Q\_{\mathsf{m}}\left(\mathsf{hu},\mathsf{hu}^{\*}\right)\right)\right). \end{split} \tag{33}$$

Using Definition 5, we get:

$$
\Sigma\left(Q\_m\left(\mu,\mu^\*\right)\right) < \wp\left(\Sigma\left(P\left(\mu,\mu^\*\right)\right)\right) \tag{34}
$$

where:

$$\begin{split} P\left(\boldsymbol{\mu},\boldsymbol{\mu}^\*\right) &= \max\left\{ Q\_{\mathfrak{M}}\left(\boldsymbol{\mu},\boldsymbol{\mu}^\*\right), Q\_{\mathfrak{M}}\left(\boldsymbol{\mu},\hbar\boldsymbol{\mu}\right), Q\_{\mathfrak{M}}\left(\boldsymbol{\mu}^\*,\hbar\boldsymbol{\mu}^\*\right) \right\} \\ &= \max\left\{ Q\_{\mathfrak{M}}\left(\boldsymbol{\mu},\boldsymbol{\mu}^\*\right), 0 \right\} = Q\_{\mathfrak{M}}\left(\boldsymbol{\mu},\boldsymbol{\mu}^\*\right). \end{split} \tag{35}$$

From (33)–(35), it follows that:

$$
\Sigma\left(Q\_m\left(\mathfrak{u},\mathfrak{u}^\*\right)\right) < \left.\wp\left(\Sigma\left(Q\_m\left(\mathfrak{u},\mathfrak{u}^\*\right)\right)\right) < \Sigma\left(Q\_m\left(\mathfrak{u},\mathfrak{u}^\*\right)\right).
$$

This is an incorrect statement. Hence, *u* = *<sup>u</sup>*<sup>∗</sup>.

## **5. Conclusions**

First, motivated by [4,10,15], we established a new contractive mapping, which is called the generalized Suzuki-simulation-type contractive mapping. Second, in [19], we constituted a new quasi metric space, which is named the non-Archimedean quasi modular metric space, and so using this, we attained fixed point theorems via generalized Suzuki-simulation-type contractive mapping. Finally, we acquired graphical fixed point results in non-Archimedean quasi modular metric spaces.

**Author Contributions:** The authours contributed equally in writing this article. Authours read and approved the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors are grateful to the editor and reviewers for their careful reviews and useful comments.

**Conflicts of Interest:** The authors declare no conflict of interest.
