Modified Suzuki-Simulation Type Contractive Mapping in Non-Archimedean Quasi Modular Metric Spaces and Application to Graph Theory

Abstract

In this paper, we establish generalized Suzuki-simulation-type contractive mapping and prove fixed point theorems on non-Archimedean quasi modular metric spaces. As an application, we acquire graphic-type results.

1. Introduction

In the sequel, the letter ${\mathbb{R}}_{+}$ will denote the set of all nonnegative real numbers.

Let S be a nonempty set and $V:S\to S$ be given mappings. A point $𝚥\in S$ is said to be:

i.

a fixed point of V if and only if $V𝚥=𝚥$;

ii.

a common fixed point of V and Z if and only if $V𝚥=Z𝚥=𝚥$.

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

Definition 1

([1]). Let $\zeta :{\left[0,\infty \right)}^2\to \mathbb{R}$ be a mapping. The mapping ζ is named a simulation function satisfying the following conditions:

ζ₁.

$\zeta \left(0,0\right)=0$,

ζ₂.

$\zeta \left(a,b\right)<a-b$, for all $a,b>0$,

ζ₃.

if $\left\{a_k\right\}$ and $\left\{b_k\right\}$ are sequences in ${\mathbb{R}}_{+}$ such that $\underset{k\to \infty }{lim}{a}_k=\underset{k\to \infty }{lim}{b}_k=l,$ $l\in {\mathbb{R}}_{+}$. Thus,

$$\underset{k\to \infty }{limsup\zeta \left(a_k,b_k\right)}<0.$$

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:

i.

$\zeta \left(a,b\right)<a-b$, for all $a,b>0$,

ii.

if $\left\{a_k\right\}$ and $\left\{b_k\right\}$ are sequences in ${\mathbb{R}}_{+}$ such that $\underset{k\to \infty }{lim}{a}_k=\underset{k\to \infty }{lim}{b}_k>0,$ and $a_k<b_k$, then $\underset{k\to \infty }{limsup\zeta \left(a_k,b_k\right)}<0.$

For examples and related results on simulation functions, one may refer to [1,2,3,4,5,6,7,8].

Radenovic and Chandok generalized the simulation function combining the C-class function as follows.

Definition 3

([4]). A mapping $G:{\left[0,\infty \right)}^2\to \mathbb{R}$ is named a C-class function if it is continuous and satisfies the following conditions:

i.

$G\left(a,b\right)\le a$,

ii.

$G\left(a,b\right)=a$ implies that either $a=0$ or $b=0$, for all $a,b\in \left[0,\infty \right).$

Definition 4

([4]). A $C_G$-simulation function is a mapping $\zeta :{\left[0,\infty \right)}^2\to \mathbb{R}$ satisfying the following conditions:

i.

$\zeta \left(a,b\right)<G\left(a,b\right)$ for all $a,b>0$, where $G:{\left[0,\infty \right)}^2\to \mathbb{R}$ is a C-class function,

ii.

if $\left\{a_k\right\}$ and $\left\{b_k\right\}$ are sequences in $\left(0,\infty \right)$ such that $\underset{k\to \infty }{lim}{b}_k=\underset{k\to \infty }{lim}{a}_k>0,$ and $b_k<a_k$, then $\underset{k\to \infty }{limsup\zeta \left(a_k,b_k\right)}<C_G.$

Definition 5

([4]). A mapping $G:{\left[0,\infty \right)}^2\to \mathbb{R}$ has the property $C_G$, if there exists a $C_G\ge 0$ such that:

i.

$G\left(a,b\right)>C_G$ implies $a>b$,

ii.

$G\left(a,a\right)\le C_G$ for all $a\in \left[0,\infty \right).$

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 $(S,d)$ be a compact metric space and $V:S\to S$ be a mapping. Suppose that, for all $𝚥,\ell \in S$ with $𝚥\ne \ell$,

$$\frac{1}{2}d\left(𝚥,V𝚥\right)<d\left(𝚥,\ell \right)\Rightarrow d\left(V𝚥,V\ell \right)<d\left(𝚥,\ell \right).$$

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 $(S,\delta )$ be a partial metric space and $f,g,V,Z:S\to S$ be mappings satisfying:

$$\frac{1}{2}min\left\{\delta \left(f𝚥,V𝚥\right),\delta \left(g\ell ,Z\ell \right)\right\}\le \ell \left(f𝚥,g\ell \right)\Rightarrow \varphi \left(V𝚥,Z\ell \right)\le \alpha \left(M\left(𝚥,\ell \right)\right)-\beta \left(M\left(𝚥,\ell \right)\right),$$

for all $𝚥,\ell \in S,$ where $\varphi ,\alpha ,\beta :\left[0,\infty \right)\to \left[0,\infty \right)$ are such that ϕ is an altering distance function, α is continuous, and β is lower-semi continuous $\alpha \left(0\right)=\beta \left(0\right)=0$ and $\varphi \left(t\right)-\alpha \left(t\right)+\beta \left(t\right)>0,$ for all $t>0$ and:

$$M\left(𝚥,\ell \right)=max\left\{\delta \left(f𝚥,g\ell \right),\delta \left(f𝚥,V\ell \right),\delta \left(g\ell ,Z\ell \right),\frac{\delta \left(f𝚥,Z\ell \right)+\delta \left(g\ell ,V𝚥\right)}{2}\right\},$$

i.

$V\left(S\right)\subseteq g\left(S\right),Z\left(S\right)\subseteq f\left(S\right)$;

ii.

either $f\left(S\right)$ or $g\left(S\right)$ is a complete subspace of S;

iii.

the pairs $(f,V)$ and $(g,Z)$ are weakly compatible.

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

Jleli and Samet [12] introduced a $\mathsf{\Sigma }$-contraction and established fixed point results in generalized metric spaces. Jleli and Samet [12] also introduced a class of $\mathsf{\Theta }$ such that $\mathsf{\Sigma }:\left(0,\infty \right)\to \left(1,\infty \right)$ of all functions, providing the following conditions:

Σ₁.

$\mathsf{\Sigma }$ is nondecreasing;

Σ₂.

for any sequence $\left\{a_k\right\}$ in $\left(0,\infty \right)$, $\underset{k\to \infty }{lim}\mathsf{\Sigma }\left(a_k\right)=1$ if and only if $\underset{k\to \infty }{lim}{a}_k=0;$

Σ₃.

there exist $r\in \left(0,1\right)$ and $l\in \left(0,\infty \right)$ such that $\underset{k\to 0^{+}}{lim}\frac{\mathsf{\Sigma }\left(k\right)-1}{k^r}=l.$

Theorem 3

([12]). Let $(S,d)$ be a complete generalized metric space and $V:S\to S$ be a mapping. Suppose that there exist $\mathsf{\Sigma }\in \mathsf{\Theta }$ and $\gamma \in \left(0,1\right)$ such that:

$$d\left(V𝚥,V\ell \right)\ne 0\Rightarrow \mathsf{\Sigma }\left(d\left(V𝚥,V\ell \right)\right)\le {\left[\mathsf{\Sigma }\left(d\left(𝚥,\ell \right)\right)\right]}^{\gamma },$$

for all $𝚥,\ell \in S.$ Then, V has a unique fixed point.

After that, many authors generalized such a contraction in different spaces [13,14,15,16,17].

Liu et al. [15] modified the class of function $\mathsf{\Theta }$ exchanging conditions. The class of functions $\tilde{\mathsf{\Theta }}$ was defined by the set of $\mathsf{\Sigma }:\left(0,\infty \right)\to \left(1,\infty \right)$ satisfying the following conditions:

• ${\tilde{\mathsf{\Sigma }}}_1.$ $\mathsf{\Sigma }$ is non-decreasing and continuous,
• ${\tilde{\mathsf{\Sigma }}}_2.$ $\underset{k\in \left(0,\infty \right)}{inf}\mathsf{\Sigma }\left(k\right)=1.$

Lemma 1

([15]). Let $\mathsf{\Sigma }:\left(0,\infty \right)\to \left(1,\infty \right)$ be a non-decreasing and continuous function with $\underset{k\in \left(0,\infty \right)}{inf}\mathsf{\Sigma }\left(k\right)=1$ and $\left\{a_k\right\}$ be a sequence in $\left(0,\infty \right)$. Then, the following condition holds:

$$\underset{k\to \infty }{lim}\mathsf{\Sigma }\left(a_k\right)=1\iff \underset{k\to \infty }{lim}{a}_k=0.$$

Zheng et al. [18] denoted new set functions $\mathsf{\Phi }$ satisfying the following conditions:

Φ₁.

$\phi :\left[1,\infty \right)\to \left[1,\infty \right)$ is nondecreasing,

Φ₂.

for each $k>0,$ $\underset{n\to \infty }{lim}{\phi }^n\left(k\right)=1,$

Φ₃.

$\phi$ is continuous on $\left[1,\infty \right).$

Lemma 2

([18]). If $\phi \in \mathsf{\Phi },$ then $\phi \left(1\right)=1$ and $\phi \left(t\right)<t$ for each $t>1.$

Definition 6

([18]). Let $(S,d)$ be a metric space and $V:S\to S$ be a mapping. V is said to be a $\mathsf{\Sigma }-\phi$-contraction if there exist $\mathsf{\Sigma }\in \mathsf{\Theta }$ and $\phi \in \mathsf{\Phi }$ such that for any $𝚥,\ell \in S,$

$$\mathsf{\Sigma }\left(d\left(V𝚥,V\ell \right)\right)\le \phi \left(\mathsf{\Sigma }\left(N\left(𝚥,\ell \right)\right)\right),$$

where:

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

Theorem 4

([18]). Let $(S,d)$ be a complete metric space and $V:S\to S$ be a $\mathsf{\Sigma }-\phi$-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 $Q:\left(0,\infty \right)\times S\times S\to \left[0,\infty \right]$ is called a quasi modular metric on S if the following hold:

q₁.

$\xi =\eta$ if and only if $Q_m\left(\xi ,\eta \right)=0$ for all $m>0$;

q₂.

$Q_{m+n}\left(\xi ,\eta \right)\le Q_m\left(\xi ,\nu \right)+Q_n\left(\nu ,\eta \right)$ for all $m,n>0$ and $\xi ,\eta ,\nu \in S$.

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

q_1^′.

$Q_m\left(\xi ,\xi \right)=0$ for all $m>0$ and $\xi \in S,$

instead of $\left(q_1\right)$, 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 $\left(q_1\right)$ is satisfied:

q₃.

$\xi =\eta$ if and only if $Q_m\left(\xi ,\eta \right)=0$ for some $m>0.$

Again, Q is called a convex if for $m,n>0$ and $\xi ,\eta ,\nu \in S$, the inequality holds:

q₄.

$Q_{m+n}\left(\xi ,\eta \right)\le \frac{m}{m+n}{Q}_m\left(\xi ,\nu \right)+\frac{n}{m+n}{Q}_n\left(\nu ,\eta \right).$

Definition 8

([19]). In Definition 7, if we replace $\left(q_2\right)$ by:

q₅.

$Q_{max\left\{m,n\right\}}\left(\xi ,\eta \right)\le Q_m\left(\xi ,\nu \right)+Q_n\left(\nu ,\eta \right)$

for all $m,n>0$ and $\xi ,\eta ,\nu \in S$, then $S_Q$ is called a non-Archimedean quasi modular metric space.

Note that the function $Q_{max\left\{m,n\right\}}$ is more general than the function $Q_{m+n}\left(\xi ,\eta \right)$, so every non-Archimedean quasi modular metric space is a quasi modular metric space.

Example 1

([19]). Let $S=\left[0,\infty \right)$ and Q be defined by:

$$Q_m\left(\xi ,\eta \right)=\left\{\begin{array}{c}\frac{\xi -\eta }{m}\mathrm{if}\xi \ge \eta \hfill \\ 1\mathrm{if}\xi <\eta .\hfill \end{array}\right.$$

Then, $S_Q$ is a quasi modular metric space with $m=\frac{1}{3}$ and $n=\frac{2}{3}$, but is not modular metric space since $Q_m\left(0,1\right)=1$ and $Q_m\left(1,0\right)=\frac{1}{3}$.

Remark 1

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

i.

For a quasi modular metric Q on S, the conjugate quasi modular metric $Q^{-1}$ on S of Q is defined by $Q_m^{-1}\left(\xi ,\eta \right)=Q_m\left(\eta ,\xi \right).$

ii.

If Q is a $T_0$-quasi pseudo modular metric on S, then the function $Q^E$ defined by $Q^E=Q^{-1}\vee Q$, that is $Q_m^E\left(\xi ,\eta \right)=max\left\{Q_m\left(\xi ,\eta \right),Q_m\left(\eta ,\xi \right)\right\}$, defines a modular metric space.

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

Definition 9

([19]). A sequence $\left\{{\xi }_p\right\}$ in $S_Q$ converges to ξ and is called:

a.

Q-convergent or left convergent if ${\xi }_p\to \xi \iff Q_m\left(\xi ,{\xi }_p\right)\to 0.$

b.

$Q^{-1}$-convergent or right convergent if ${\xi }_p\to \xi \iff Q_m\left({\xi }_p,\xi \right)\to 0.$

c.

$Q^E$-convergent if $Q_m\left(\xi ,{\xi }_p\right)\to 0$ and $Q_m\left({\xi }_p,\xi \right)\to 0.$

Definition 10

([19]). A sequence $\left\{{\xi }_p\right\}$ in a quasi modular metric space $S_Q$ is called:

d.

left (right) Q-K-Cauchy if for every $\epsilon >0$, there exists $p_{\epsilon }\in N$ such that $Q_m\left({\xi }_r,{\xi }_p\right)<\epsilon$ for all $p,r\in N$ with $p_{\epsilon }\le r\le p\left(p_{\epsilon }\le p\le r\right)$ and for all $m>0$.

e.

$Q^E$-Cauchy if for every $\epsilon >0$, there exists $p_{\epsilon }\in N$ such that $Q_m\left({\xi }_p,{\xi }_r\right)<\epsilon$ for all $p,r\in N$ with $p,r\ge p_{\epsilon }$.

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 $Q^E$-Cauchy if and only if it is left and right Q-K-Cauchy.

Definition 11

([19]). A quasi modular metric space $S_Q$ is called:

i.

left Q-K-complete if every left Q-K-Cauchy is Q-convergent.

ii.

Q-Smyth-complete if every left Q-K-Cauchy sequence is $Q^E$-convergent.

3. Common Fixed Point Results

In the sequel, Q is regular and convex and $T_Z$ denotes the family of all $C_G$-simulation functions $\zeta :{\left[0,\infty \right)}^2\to \mathbb{R}$.

Definition 12.

Let $S_Q$ be a non-Archimedean quasi modular metric space and $V:S_Q\to S_Q$ be a mapping. We say that V is a generalized Suzuki-simulation-type contractive mapping if there exist $\mathsf{\Sigma }\in \tilde{\mathsf{\Theta }}$, $\phi \in \mathsf{\Phi }$ and $\zeta \in T_Z$ such that:

$$\begin{array}{c}\frac{1}{2}{Q}_m\left(\xi ,V\xi \right)\le Q_m\left(\xi ,\eta \right)\mathrm{implies}\hfill \\ \zeta \left(\mathsf{\Sigma }\left(Q_m\left(V\xi ,V\eta \right)\right),\phi \left(\mathsf{\Sigma }\left(P\left(\xi ,\eta \right)\right)\right)\right)\ge C_G\hfill \end{array}$$

(1)

where:

$$P\left(\xi ,\eta \right)=max\left\{Q_m\left(\xi ,\eta \right),Q_m\left(\xi ,V\xi \right),Q_m\left(\eta ,V\eta \right)\right\}$$

for all $\xi ,\eta \in S_Q$.

Theorem 5.

Let $S_Q$ 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 $\left\{{\xi }_k\right\}$ in $S_Q$ by:

$$\begin{array}{c}{\xi }_{k+1}=V{\xi }_k,\hfill \end{array}$$

(2)

for all $k\in \mathbb{N}$. If there exists an $k_0$ such that ${\xi }_{k_0}={\xi }_{k_0+1},$ then $z={\xi }_{k_0}$ becomes a fixed point of V. Consequently, we shall assume that ${\xi }_k\ne {\xi }_{k+1}$ for all $k\in \mathbb{N}$. Therefore, we have:

$$Q_m\left({\xi }_k,{\xi }_{k+1}\right)>0,\mathrm{for}\mathrm{all}n=0,1,2\dots .$$

(3)

Hence, we have:

$$\frac{1}{2}{Q}_m\left({\xi }_k,V{\xi }_k\right)<Q_m\left({\xi }_k,V{\xi }_k\right)=Q_m\left({\xi }_k,{\xi }_{k+1}\right)\mathrm{implies},$$

$$\begin{array}{c}{C}_G\le \zeta \left(\mathsf{\Sigma }\left(Q_m\left(V{\xi }_k,V{\xi }_{k+1}\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left({\xi }_k,{\xi }_{k+1}\right)\right)\right)\right)\hfill \\ =\zeta \left(\mathsf{\Sigma }\left(Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left({\xi }_k,{\xi }_{k+1}\right)\right)\right)\right)\hfill \\ <G\left(\phi \left(\mathsf{\Sigma }\left(P\left({\xi }_k,{\xi }_{k+1}\right)\right)\right),\mathsf{\Sigma }\left(Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)\right)\right),\hfill \end{array}$$

(4)

by Definition 5, we get that:

$$\mathsf{\Sigma }\left(Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)\right)<\phi \left(\mathsf{\Sigma }\left(P\left({\xi }_k,{\xi }_{k+1}\right)\right)\right),$$

(5)

where:

$$\begin{array}{c}P\left({\xi }_k,{\xi }_{k+1}\right)=max\left\{Q_m\left({\xi }_k,{\xi }_{k+1}\right),Q_m\left({\xi }_k,V{\xi }_k\right),Q_m\left({\xi }_{k+1},V{\xi }_{k+1}\right)\right\}\hfill \\ =max\left\{Q_m\left({\xi }_k,{\xi }_{k+1}\right),Q_m\left({\xi }_k,{\xi }_{k+1}\right),Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)\right\}\hfill \\ =max\left\{Q_m\left({\xi }_k,{\xi }_{k+1}\right),Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)\right\}.\hfill \end{array}$$

(6)

If:

$$max\left\{Q_m\left({\xi }_k,{\xi }_{k+1}\right),Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)\right\}=Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)$$

for some $k\in \mathbb{N}$, then it follows from (5) and Lemma 2 that we get:

$$\mathsf{\Sigma }\left(Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)\right)<\phi \left(\mathsf{\Sigma }\left(Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)\right)\right)<\mathsf{\Sigma }\left(Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)\right)$$

which is a contradiction. Therefore, we have:

$$P\left({\xi }_k,{\xi }_{k+1}\right)=Q_m\left({\xi }_k,{\xi }_{k+1}\right)$$

for each $k\in \mathbb{N}$. Also, by (5), we have

$$\mathsf{\Sigma }\left(Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)\right)<\phi \left(\mathsf{\Sigma }\left(Q_m\left({\xi }_k,{\xi }_{k+1}\right)\right)\right).$$

Repeating this step, we conclude that:

$$\begin{array}{c}\mathsf{\Sigma }\left(Q_m\left({\xi }_{k+1},{\xi }_{k+2}\right)\right)<\phi \left(\mathsf{\Sigma }\left(Q_m\left({\xi }_k,{\xi }_{k+1}\right)\right)\right)\hfill \\ <{\phi }^2\left(\mathsf{\Sigma }\left(Q_m\left({\xi }_{k-1},{\xi }_k\right)\right)\right)\hfill \\ ⋮\hfill \\ <{\phi }^k\left(\mathsf{\Sigma }\left(Q_m\left({\xi }_1,{\xi }_2\right)\right)\right),\hfill \end{array}$$

for all $k\in \mathbb{N}$. Taking the limit $k\to \infty$ above, by the definition of $\phi$ and property ${\mathsf{\Theta }}_2$, we have:

$$\underset{k\to \infty }{lim}{\phi }^k\left(Q_m\left({\xi }_1,{\xi }_2\right)\right)=1.$$

(7)

Thus, from Lemma 1, it follows that:

$$\underset{k\to \infty }{lim}{Q}_m\left({\xi }_{k+1},{\xi }_{k+2}\right)=0,$$

(8)

for all $k\in \mathbb{N}$. Now, we show that $\left\{{\xi }_k\right\}$ is a left Q-K-Cauchy sequence. Assume the contrary. There exists $\epsilon >0$ such that we can find two subsequences $\left\{t_k\right\}$ and $\left\{s_k\right\}$ of positive integers satisfying the following inequalities:

$$Q_m\left({\xi }_{t_k},{\xi }_{s_k}\right)\ge \epsilon ,\mathrm{and}{Q}_m\left({\xi }_{t_k-1},{\xi }_{s_k}\right)<\epsilon .$$

(9)

From (9) and ($q_5$), it follows that:

$$\begin{array}{cc}\epsilon \hfill & \le Q_m\left({\xi }_{t_k},{\xi }_{s_k}\right)=Q_{max\left\{m,m\right\}}\left({\xi }_{t_k},{\xi }_{s_k}\right)\hfill \\ & \le Q_m\left({\xi }_{t_k},{\xi }_{t_k-1}\right)+Q_m\left({\xi }_{t_k-1},{\xi }_{s_k}\right)\hfill \\ & <\epsilon +Q_m\left({\xi }_{t_k},{\xi }_{t_k-1}\right).\hfill \end{array}$$

(10)

On taking the limit as $k\to \infty$ in the above relation, we obtain that:

$$\underset{k\to \infty }{lim}{Q}_m\left({\xi }_{t_k},{\xi }_{s_k}\right)=\epsilon .$$

(11)

Also, from (9) and ($q_5$), it follows that:

$$\begin{array}{cc}{Q}_m\left({\xi }_{t_k+1},{\xi }_{s_k+1}\right)=\hfill & Q_{max\left\{m,m\right\}}\left({\xi }_{t_k+1},{\xi }_{s_k+1}\right)\hfill \\ & \le Q_m\left({\xi }_{t_k+1},{\xi }_{t_k}\right)+Q_m\left({\xi }_{t_k},{\xi }_{s_k+1}\right)\hfill \\ & =Q_m\left({\xi }_{t_k+1},{\xi }_{t_k}\right)+Q_{max\left\{m,m\right\}}\left({\xi }_{t_k},{\xi }_{s_k+1}\right)\hfill \\ & \le Q_m\left({\xi }_{t_k},{\xi }_{t_k-1}\right)+Q_m\left({\xi }_{t_k-1},{\xi }_{s_k+1}\right)+Q_m\left({\xi }_{t_k+1},{\xi }_{t_k}\right)\hfill \\ & =Q_m\left({\xi }_{t_k},{\xi }_{t_k-1}\right)+Q_m\left({\xi }_{t_k+1},{\xi }_{t_k}\right)+Q_{max\left\{m,m\right\}}\left({\xi }_{t_k-1},{\xi }_{s_k+1}\right)\hfill \\ & \le Q_m\left({\xi }_{t_k-1},{\xi }_{s_k}\right)+Q_m\left({\xi }_{s_k},{\xi }_{s_k+1}\right)\hfill \\ & +Q_m\left({\xi }_{t_k},{\xi }_{t_k-1}\right)+Q_m\left({\xi }_{t_k+1},{\xi }_{t_k}\right)\hfill \\ & <\epsilon +Q_m\left({\xi }_{s_k},{\xi }_{s_k+1}\right)+Q_m\left({\xi }_{t_k},{\xi }_{t_k-1}\right)\hfill \\ & +Q_m\left({\xi }_{t_k+1},{\xi }_{t_k}\right).\hfill \end{array}$$

(12)

Next, we claim that:

$$\frac{1}{2}{Q}_m\left({\xi }_{t_k},V{\xi }_{t_k}\right)\le Q_m\left({\xi }_{t_k},{\xi }_{s_k}\right).$$

If:

$$\begin{array}{c}\frac{1}{2}{Q}_m\left({\xi }_{t_k},V{\xi }_{t_k}\right)>Q_m\left({\xi }_{t_k},{\xi }_{s_k}\right)\hfill \\ =\frac{1}{2}{Q}_m\left({\xi }_{t_k},{\xi }_{t_k+1}\right)>Q_m\left({\xi }_{t_k},{\xi }_{s_k}\right),\hfill \end{array}$$

(13)

then letting $k\to \infty$ in (13), from (11) and (8), we have that $0>\epsilon$ is a contradiction. Hence,

$$\frac{1}{2}{Q}_m\left({\xi }_{t_k},V{\xi }_{t_k}\right)\le Q_m\left({\xi }_{t_k},{\xi }_{s_k}\right).$$

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

$$\begin{array}{c}{C}_G\le \zeta \left(\mathsf{\Sigma }\left(Q_m\left(V{\xi }_{t_k},V{\xi }_{s_k}\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left({\xi }_{t_k},{\xi }_{s_k}\right)\right)\right)\right)\hfill \\ =\zeta \left(\mathsf{\Sigma }\left(Q_m\left({\xi }_{t_k+1},{\xi }_{s_k+1}\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left({\xi }_{t_k},{\xi }_{s_k}\right)\right)\right)\right),\hfill \end{array}$$

(14)

where:

$$\begin{array}{c}P\left({\xi }_{t_k},{\xi }_{s_k}\right)=max\left\{Q_m\left({\xi }_{t_k},{\xi }_{s_k}\right),Q_m\left({\xi }_{t_k},V{\xi }_{t_k}\right),\left.Q_m\left({\xi }_{s_k},V{\xi }_{s_k}\right)\right\}\right.\hfill \\ =max\left\{Q_m\left({\xi }_{t_k},{\xi }_{s_k}\right),Q_m\left({\xi }_{t_k},{\xi }_{t_k+1}\right),\left.Q_m\left({\xi }_{s_k},{\xi }_{s_k+1}\right)\right\}\right..\hfill \end{array}$$

(15)

Taking the limit $k\to \infty$ using (8), (11), and (12) in (14) and (15), we obtain:

$$C_G\le \zeta \left(\mathsf{\Sigma }\left(\epsilon \right),\phi \left(\mathsf{\Sigma }\left(\epsilon \right)\right)\right)<G\left(\phi \left(\mathsf{\Sigma }\left(\epsilon \right)\right),\mathsf{\Sigma }\left(\epsilon \right)\right).$$

From Definition 5, we get:

$$\mathsf{\Sigma }\left(\epsilon \right)<\phi \left(\mathsf{\Sigma }\left(\epsilon \right)\right)<\mathsf{\Sigma }\left(\epsilon \right).$$

It follows that $\mathsf{\Sigma }\left(\epsilon \right)<\mathsf{\Sigma }\left(\epsilon \right),$ a contradiction. Hence, $\left\{{\xi }_k\right\}$ is a left Q-K-Cauchy sequence. As $S_Q$ is a Q-Smyth-complete non-Archimedean quasi modular metric space, there exists $u\in S_Q$ such that:

$$\underset{k\to \infty }{lim}{Q_m}^E\left({\xi }_k,u\right)=0.$$

Thus, we have:

$$\underset{k\to \infty }{lim}{Q}_m\left({\xi }_k,u\right)=0\mathrm{and}\underset{k\to \infty }{lim}{Q}_m\left(u,{\xi }_k\right)=0.$$

Now, we show that u is a fixed point of V. Assume that $Q_m\left(Vu,u\right)>0.$ We claim that for each $k\ge 0$, the following holds:

$$\frac{1}{2}{Q}_m\left({\xi }_k,V{\xi }_k\right)\le Q_m\left({\xi }_k,u\right).$$

On the contrary, suppose that:

$$\frac{1}{2}{Q}_m\left({\xi }_k,V{\xi }_k\right)>Q_m\left({\xi }_k,u\right)=\frac{1}{2}{Q}_m\left({\xi }_k,{\xi }_{k+1}\right)>Q_m\left({\xi }_k,u\right).$$

(16)

Taking the limit as $k\to \infty$ 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:

$$\begin{array}{c}{C}_G\le \zeta \left(\mathsf{\Sigma }\left(Q_m\left(V{\xi }_k,Vu\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left({\xi }_k,u\right)\right)\right)\right)\hfill \\ =\zeta \left(\mathsf{\Sigma }\left(Q_m\left({\xi }_{k+1},Vu\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left({\xi }_k,u\right)\right)\right)\right)\hfill \\ <G\left(\phi \left(\mathsf{\Sigma }\left(P\left({\xi }_k,u\right)\right)\right),\mathsf{\Sigma }\left(Q_m\left({\xi }_{k+1},Vu\right)\right)\right).\hfill \end{array}$$

(17)

By Definition 5,

$$\mathsf{\Sigma }\left(Q_m\left({\xi }_{k+1},Vu\right)\right)<\phi \left(\mathsf{\Sigma }\left(P\left({\xi }_k,u\right)\right)\right),$$

(18)

where:

$$\begin{array}{cc}P\left({\xi }_k,u\right)\hfill & =max\left\{Q_m\left({\xi }_k,u\right),Q_m\left({\xi }_k,V{\xi }_k\right),Q_m\left(u,Vu\right)\right\}\hfill \\ & =max\left\{Q_m\left({\xi }_k,u\right),Q_m\left({\xi }_k,{\xi }_{k+1}\right),Q_m\left(u,Vu\right)\right\}.\hfill \end{array}$$

(19)

Letting $k\to \infty$ in (17)–(19), we have:

$$\mathsf{\Sigma }\left(Q_m\left(u,Vu\right)\right)<\phi \left(\mathsf{\Sigma }\left(Q_m\left(u,Vu\right)\right)\right)<\mathsf{\Sigma }\left(Q_m\left(u,Vu\right)\right).$$

That is, $\mathsf{\Sigma }\left(Q_m\left(u,Vu\right)\right)<\mathsf{\Sigma }\left(Q_m\left(u,Vu\right)\right)$, a contradiction. Thus, u is a fixed point of V. Suppose that there is an another fixed point $u^{\ast }$ of V such that $V{u}^{\ast }=u^{\ast }$ and $u\ne u^{\ast }.$ Then, $Q_m\left(Vu,V{u}^{\ast }\right)=Q_m\left(u,u^{\ast }\right)>0,$ and:

$$0=\frac{1}{2}{Q}_m\left(u,Vu\right)\le Q_m\left(u,u^{\ast }\right).$$

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

$$\begin{array}{c}{C}_G\le \zeta \left(\mathsf{\Sigma }\left(Q_m\left(Vu,V{u}^{\ast }\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left(u,u^{\ast }\right)\right)\right)\right)\hfill \\ =\zeta \left(\mathsf{\Sigma }\left(Q_m\left(u,u^{\ast }\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left(u,u^{\ast }\right)\right)\right)\right)\hfill \\ <G\left(\phi \left(\mathsf{\Sigma }\left(P\left(u,u^{\ast }\right)\right)\right),\mathsf{\Sigma }\left(Q_m\left(u,u^{\ast }\right)\right)\right).\hfill \end{array}$$

(20)

From the property of G,

$$\mathsf{\Sigma }\left(Q_m\left(u,u^{\ast }\right)\right)<\phi \left(\mathsf{\Sigma }\left(P\left(u,u^{\ast }\right)\right)\right),$$

(21)

where:

$$P\left(u,u^{\ast }\right)=max\left\{Q_m\left(u,u^{\ast }\right),Q_m\left(u,Vu\right),Q_m\left(u^{\ast },V{u}^{\ast }\right)\right\}=Q_m\left(u,u^{\ast }\right).$$

(22)

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

$$\mathsf{\Sigma }\left(Q_m\left(u,u^{\ast }\right)\right)<\phi \left(\mathsf{\Sigma }\left(Q_m\left(u,u^{\ast }\right)\right)\right)<\mathsf{\Sigma }\left(Q_m\left(u,u^{\ast }\right)\right),$$

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 $S_Q$ be a Q-Smyth-complete non-Archimedean quasi modular metric space and $V:S_Q\to S_Q$ be a mapping. If there exists $\mathsf{\Sigma }\in \tilde{\mathsf{\Theta }}$, $\phi \in \mathsf{\Phi }$, and $\zeta \in T_Z$ such that:

$$\frac{1}{2}{Q}_m\left(𝚥,V𝚥\right)\le Q_m\left(𝚥,\ell \right)\mathrm{implies},$$

$$\zeta \left(\mathsf{\Sigma }\left(Q_m\left(V𝚥,V\ell \right)\right),\phi \left(\mathsf{\Sigma }\left(Q_m\left(𝚥,\ell \right)\right)\right)\right)\ge C_G,$$

for all $𝚥,\ell \in S_Q$, then V has a unique fixed point.

Corollary 2.

Let $S_Q$ be a Q-Smyth-complete non-Archimedean quasi modular metric space and $V:S_Q\to S_Q$ be a mapping. If there exists $\mathsf{\Sigma }\in \tilde{\mathsf{\Theta }}$, $\phi \in \mathsf{\Phi }$, and $\zeta \in T_Z$ such that:

$$\zeta \left(\mathsf{\Sigma }\left(Q_m\left(V𝚥,V\ell \right)\right),\phi \left(\mathsf{\Sigma }\left(P\left(𝚥,\ell \right)\right)\right)\right)\ge C_G$$

where:

$$P\left(𝚥,\ell \right)=max\left\{Q_m\left(𝚥,\ell \right),Q_m\left(𝚥,V𝚥\right),Q_m\left(\ell ,V\ell \right)\right\},$$

for all $𝚥,\ell \in S_Q$, then V has a unique fixed point.

Corollary 3.

Let $S_Q$ be a Q-Smyth-complete non-Archimedean quasi modular metric space and $V:S_Q\to S_Q$ be a mapping. If there exists $\mathsf{\Sigma }\in \tilde{\mathsf{\Theta }}$ and $\phi \in \mathsf{\Phi }$ such that:

$$\frac{1}{2}{Q}_m\left(𝚥,V𝚥\right)\le Q_m\left(𝚥,\ell \right)\mathrm{implies},$$

$$\mathsf{\Sigma }\left(Q_m\left(V𝚥,V\ell \right)\right)\le \phi \left(\mathsf{\Sigma }\left(P\left(𝚥,\ell \right)\right)\right)$$

where:

$$P\left(𝚥,\ell \right)=max\left\{Q_m\left(𝚥,\ell \right),Q_m\left(𝚥,V𝚥\right),Q_m\left(\ell ,V\ell \right)\right\},$$

for all $𝚥,\ell \in S_Q$, then V has a unique fixed point.

Corollary 4.

Let $S_Q$ be a Q-Smyth-complete non-Archimedean quasi modular metric space and $V:S_Q\to S_Q$ be a mapping. If there exists $\mathsf{\Sigma }\in \tilde{\mathsf{\Theta }}$ and $\phi \in \mathsf{\Phi }$ such that:

$$\mathsf{\Sigma }\left(Q_m\left(V𝚥,V\ell \right)\right)\le \phi \left(\mathsf{\Sigma }\left(P\left(𝚥,\ell \right)\right)\right)$$

where:

$$P\left(𝚥,\ell \right)=max\left\{Q_m\left(𝚥,\ell \right),Q_m\left(𝚥,V𝚥\right),Q_m\left(\ell ,V\ell \right)\right\},$$

for all $𝚥,\ell \in S_Q$, then V has a unique fixed point.

Corollary 5.

Let $S_Q$ be a Q-Smyth-complete non-Archimedean quasi modular metric space and $V:S_Q\to S_Q$ be a mapping. If there exists $\mathsf{\Sigma }\in \tilde{\mathsf{\Theta }}$ and $\phi \in \mathsf{\Phi }$ such that:

$$\mathsf{\Sigma }\left(Q_m\left(V𝚥,V\ell \right)\right)\le \phi \left(\mathsf{\Sigma }\left(Q_m\left(𝚥,\ell \right)\right)\right),$$

for all $𝚥,\ell \in S_Q$, then V has a unique fixed point.

4. Application to a Graph Structure

Let $S_Q$ be a non-Archimedean quasi modular metric space and $∆=\{(𝚥,𝚥):𝚥\in S_Q\}$ denote the diagonal of $S_Q\times S_Q.$ Let H be a directed graph such that the set $C\left(H\right)$ of its vertices coincides with $S_Q$ and $B\left(H\right)$ is the set of edges of the graph such that $∆\subseteq B\left(H\right)$. H is determined with the pair $\left(C\right(H),B(H\left)\right)$.

If 𝚥 and ℓ are vertices of H, then a path in H from 𝚥 to ℓ of length $p\in \mathbb{N}$ is a finite sequence $\left\{{𝚥}_p\right\}$ of vertices such that $𝚥={𝚥}_0,\dots ,{𝚥}_p=\eta$ and $({𝚥}_{i-1},{𝚥}_i)\in B\left(H\right)$ for $i\in \{1,2,\dots ,p\}$.

Recall that H is connected if there is a path between any two vertices, and it is weakly connected if $\tilde{H}$ is connected, where $\tilde{H}$ defines the undirected graph obtained from H by ignoring the direction of edges. Define by $H^{-1}$ the graph obtained from H by reversing the direction of edges. Thus,

$$B\left(H^{-1}\right)=\left\{\left(𝚥,\ell \right)\in S_Q\times S_Q:\left(\ell ,𝚥\right)\in B\left(H\right)\right\}.$$

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

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

Let $H_{𝚥}$ be the component of H consisting of all the edges and vertices that are contained in some way in H starting at $𝚥.$ We denote the relation $\left(R\right)$ in the following way:

We have $𝚥\left(R\right)\ell$ if and only if, there is a path in H from 𝚥 to $\ell ,$ for $𝚥,\ell \in C\left(H\right).$

If H is such that $B\left(H\right)$ is symmetric, then for $𝚥\in C\left(H\right)$, the equivalence class ${[𝚥]}_G$ in $V\left(G\right)$ described by the relation $\left(R\right)$ is $C\left(H_{𝚥}\right).$

Let $S_Q$ be a non-Archimedean quasi modular metric space endowed with a graph H and $\hslash :S_Q\to S_Q$. We set:

$$S_{\hslash }=\left\{𝚥\in S_Q:\left(𝚥,\hslash 𝚥\right)\in B\left(H\right)\right\}.$$

Definition 13

([20]). $(S,d)$ is a metric space, and $\hslash :S\to S$ is a mapping. Then, ℏ is called a Banach H-contraction if the following hold:

B₁.

ℏ preserves edges of H, i.e., for all $𝚥,\ell \in S,$

$$\left(𝚥,\ell \right)\in B\left(H\right)\Rightarrow \left(\hslash 𝚥,\hslash \ell \right)\in B\left(H\right),$$

B₂.

there exists $\delta \in \left(0,1\right)$ such that:

$$d\left(\hslash 𝚥,\hslash \ell \right)\le \delta d\left(𝚥,\ell \right)$$

for all $(𝚥,\ell )\in B\left(H\right)$.

After that, many fixed point researchers investigated fixed point results improving the Jachymski fixed point theorems in [17,21,22,23].

Now, motivated by [24,25,26], we generate a new contraction and obtain fixed point results using a graph structure.

Definition 14.

Let $S_Q$ be a non-Archimedean quasi modular metric space and $\hslash :S_Q\to S_Q$ be a mapping. Then, we say that ℏ is a generalized Suzuki-simulation-H-type contractive mapping if the following conditions hold:

H₁.

ℏ preserves edges of G;

H₂.

there exists $\mathsf{\Sigma }\in \tilde{\mathsf{\Theta }}$, $\phi \in \mathsf{\Phi }$ and $\zeta \in T_Z$ such that:

$$\begin{array}{c}\frac{1}{2}{Q}_m\left(𝚥,\hslash 𝚥\right)\le Q_m\left(𝚥,\ell \right)\mathrm{implies},\hfill \\ \zeta \left(\mathsf{\Sigma }\left(Q_m\left(\hslash 𝚥,\hslash \ell \right)\right),\phi \left(\mathsf{\Sigma }\left(P\left(𝚥,\ell \right)\right)\right)\right)\ge C_G,\hfill \end{array}$$

(23)

where

$$P\left(𝚥,\ell \right)=max\left\{Q_m\left(𝚥,\ell \right),Q_m\left(𝚥,\hslash 𝚥\right),Q_m\left(\ell ,\hslash \ell \right)\right\}$$

for all $𝚥,\ell \in B\left(H\right)$ and for all $m>0$.

Remark 3.

Let $S_Q$ be a non-Archimedean quasi modular metric space with a graph H and $\hslash :S_Q\to S_Q$ be a generalized Suzuki-simulation-H-type contractive mapping. If there exists ${𝚥}_0\in S_Q$ such that $\hslash {𝚥}_0\in {\left[{𝚥}_0\right]}_{\tilde{H}}$, then:

R₁.

ℏ is both a generalized Suzuki-simulation-$H^{-1}$-type contractive mapping and a generalized Suzuki- Simulation-$\tilde{H}$-type contractive mapping.

R₂.

${\left[{𝚥}_0\right]}_{\tilde{H}}$ is ℏ-invariant, and $\hslash \left|{}_{{\left[{𝚥}_0\right]}_{\tilde{H}}}\right.$ is a generalized Suzuki-simulation-${\tilde{H}}_{{𝚥}_0}$-type contractive mapping.

Theorem 6.

Let $S_Q$ be a Q-Smyth-complete non-Archimedean quasi modular metric space with a graph H and $\hslash :S_Q\to S_Q$ be a mapping.

i.

there exists ${𝚥}_0\in S_{\hslash }$;

ii.

ℏ is the generalized Suzuki-simulation-$\tilde{H}$-type contractive mapping;

iii.

H is weakly connected;

iv.

if $\left\{{𝚥}_k\right\}$ is a sequence in $S_Q$ such that $\underset{k\to \infty }{lim}{Q_m}^E\left({𝚥}_k,u\right)=0$ and $\left({𝚥}_k,{𝚥}_{k+1}\right)\in B\left(H\right)$, then there exists a subsequence $\left\{{𝚥}_{k_s}\right\}$ of $\left\{{𝚥}_k\right\}$ such that $\left({𝚥}_{k_s},u\right)\in B\left(H\right)$.

Then, ℏ has a unique fixed point.

Proof. 

Define a sequence $\left\{{𝚥}_k\right\}$ in $S_Q$ by:

$$\begin{array}{c}{𝚥}_{k+1}=\hslash {𝚥}_k,\hfill \end{array}$$

(24)

for all $k\in \mathbb{N}$. Let ${𝚥}_0$ be a given point in $S_{\hslash }$; thus, $\left({𝚥}_0,\hslash {𝚥}_0\right)=\left({𝚥}_0,{𝚥}_1\right)\in B\left(H\right).$ Because ℏ preserves the edges of H,

$$\left({𝚥}_0,{𝚥}_1\right)\in B\left(H\right)\Rightarrow \left(\hslash {𝚥}_0,\hslash {𝚥}_1\right)\in B\left(H\right).$$

Continuing this way, we get:

$$\left({𝚥}_k,{𝚥}_{k+1}\right)\in B\left(H\right).$$

Then from Theorem 5, we get that $\left\{{𝚥}_k\right\}$ is a left Q-K-Cauchy sequence in $S_Q$. By the Q-Smyth- completeness of $S_Q$, there exists $u\in S_Q$ such that:

$$\underset{k\to \infty }{lim}{Q}_m^E\left({𝚥}_k,u\right)=0.$$

(25)

Thus, we have:

$$\underset{k\to \infty }{lim}{Q}_m\left({𝚥}_k,u\right)=0\mathrm{and}\underset{k\to \infty }{lim}{Q}_m\left(u,{𝚥}_k\right)=0.$$

(26)

Now, we show that u is a fixed point of ℏ. Using $\left(iv\right)$, we get $\left({𝚥}_{k_s},u\right)\in B\left(H\right)$. We claim that:

$$\frac{1}{2}{Q}_m\left({𝚥}_{k_s},\hslash {𝚥}_{k_s}\right)\le Q_m\left({𝚥}_{k_s},u\right).$$

(27)

If

$$\begin{array}{c}\frac{1}{2}{Q}_m\left({𝚥}_{k_s},\hslash {𝚥}_{k_s}\right)>Q_m\left({𝚥}_{k_s},u\right)=\frac{1}{2}{Q}_m\left({𝚥}_{k_s},{𝚥}_{k_s+1}\right)>Q_m\left({𝚥}_{k_s},u\right)\hfill \end{array}$$

(28)

and taking the limit $s\to \infty$ in (28), we get $0>0,$ a contradiction. Hence, the claim is true. Since ℏ is a generalized Suzuki-simulation-$\tilde{H}$-type contractive mapping, we have:

$$\begin{array}{c}{C}_G\le \zeta \left(\mathsf{\Sigma }\left(Q_m\left(\hslash {𝚥}_{k_s},\hslash u\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left({𝚥}_{k_s},u\right)\right)\right)\right)\hfill \\ \le \zeta \left(\mathsf{\Sigma }\left(Q_m\left(\hslash {𝚥}_{k_s},\hslash u\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left({𝚥}_{k_s},u\right)\right)\right)\right)\hfill \\ \le G\left(\phi \left(\mathsf{\Sigma }\left(P\left({𝚥}_{k_s},u\right)\right)\right),\mathsf{\Sigma }\left(Q_m\left(h{𝚥}_{k_s},hu\right)\right)\right),\hfill \end{array}$$

(29)

from Definition 5, we get:

$$\mathsf{\Sigma }\left(Q_m\left(\hslash {𝚥}_{k_s},\hslash u\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left({𝚥}_{k_s},u\right)\right)\right),$$

(30)

where:

$$\begin{array}{cc}P\left({𝚥}_{k_s},u\right)\hfill & =max\left\{Q_m\left({𝚥}_{k_s},u\right),Q_m\left({𝚥}_{k_s},\hslash {𝚥}_{k_s}\right),\right.\left.Q_m\left(u,\hslash u\right)\right\}\hfill \\ & =max\left\{Q_m\left({𝚥}_{k_s},u\right),Q_m\left({𝚥}_{k_s},{𝚥}_{k_s+1}\right),\right.\left.Q_m\left(u,\hslash u\right)\right\}.\hfill \end{array}$$

(31)

Taking the limit as $s\to \infty$ in (29)–(31), we get:

$$\mathsf{\Sigma }\left(Q_m\left(u,hu\right)\right)<\phi \left(\mathsf{\Sigma }\left(Q_m\left(u,hu\right)\right)\right)<\mathsf{\Sigma }\left(Q_m\left(u,hu\right)\right).$$

It follows that $\mathsf{\Sigma }\left(Q_m\left(u,hu\right)\right)<\mathsf{\Sigma }\left(Q_m\left(u,hu\right)\right),$ a contradiction. Therefore, we get $Q_m\left(u,\hslash u\right)=0$, that is $u=\hslash u$ since Q is regular.

Next, we show that u is a unique fixed point of ℏ. On the contrary, we suppose that $u^{\ast }$ is another fixed point of ℏ, i.e., $u^{\ast }=\hslash u^{\ast }$ and $u\ne u^{\ast }.$ Then, there exist $\sigma \in S_Q$ such that $(u,\sigma )\in B\left(H\right)$ and $(\sigma ,u^{\ast })\in B\left(H\right)$. Using $\left(iii\right)$, we get that $(u,u^{\ast })\in B(\tilde{H}).$ Furthermore,

$$0=\frac{1}{2}{Q}_m\left(u,hu\right)<Q_m\left(u,u^{\ast }\right).$$

(32)

From the generalized Suzuki-Simulation-$\tilde{H}$-type contractive mapping we have:

$$\begin{array}{c}{C}_G\le \zeta \left(\mathsf{\Sigma }\left(Q_m\left(hu,h{u}^{\ast }\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left(u,u^{\ast }\right)\right)\right)\right)\hfill \\ \le \zeta \left(\mathsf{\Sigma }\left(Q_m\left(u,u^{\ast }\right)\right),\phi \left(\mathsf{\Sigma }\left(P\left(u,u^{\ast }\right)\right)\right)\right)\hfill \\ \le G\left(\phi \left(\mathsf{\Sigma }\left(P\left(u,u^{\ast }\right)\right)\right),\mathsf{\Sigma }\left(Q_m\left(hu,h{u}^{\ast }\right)\right)\right).\hfill \end{array}$$

(33)

Using Definition 5, we get:

$$\mathsf{\Sigma }\left(Q_m\left(u,u^{\ast }\right)\right)<\phi \left(\mathsf{\Sigma }\left(P\left(u,u^{\ast }\right)\right)\right)$$

(34)

where:

$$\begin{array}{cc}P\left(u,u^{\ast }\right)\hfill & =max\left\{Q_m\left(u,u^{\ast }\right),Q_m\left(u,\hslash u\right),\right.\left.Q_m\left(u^{\ast },\hslash u^{\ast }\right)\right\}\hfill \\ & =max\left\{Q_m\left(u,u^{\ast }\right),\right.\left.0\right\}=Q_m\left(u,u^{\ast }\right).\hfill \end{array}$$

(35)

From (33)–(35), it follows that:

$$\mathsf{\Sigma }\left(Q_m\left(u,u^{\ast }\right)\right)<\phi \left(\mathsf{\Sigma }\left(Q_m\left(u,u^{\ast }\right)\right)\right)<\mathsf{\Sigma }\left(Q_m\left(u,u^{\ast }\right)\right).$$

This is an incorrect statement. Hence, $u=u^{\ast }.$ □

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.