Existence of Fixed Points of Suzuki-Type Contractions of Quasi-Metric Spaces

Abstract

In order to generalize classical Banach contraction principle in the setup of quasi-metric spaces, we introduce Suzuki-type contractions of quasi-metric spaces and prove some fixed point results. Further, we suggest a correction in the definition of another class of quasi-metrics known as $\Delta$-symmetric quasi-metrics satisfying a weighted symmetry property. We discuss equivalence of various types of completeness of $\Delta$-symmetric quasi-metric spaces. At the end, we consider the existence of fixed points of generalized Suzuki-type contractions of $\Delta$-symmetric quasi-metric spaces. Some examples have been furnished to make sure that generalizations we obtain are the proper ones.

1. Introduction

Numerous disciplines of pure and applied mathematics, including nonlinear functional analysis, and the numerical solutions of differential equations make substantial use of fixed point theory ([1]). Banach contraction principle (BCP), which is a key finding in metric fixed point theory provides a useful method for solving functional equations. Due to its usefulness, this theorem has been expanded and generalized over time to encompass various spaces and mappings satisfying distinct contraction conditions.

Suzuki [2] and Kikkawa and Suzuki [3] generalized BCP by introducing Suzuki-type contractions of metric spaces. Note that Suzuki-type contractions characterize the completeness of underlying metric spaces. Moreover, Suzuki-type contractions are in the form of implication which allow many discontinuous mappings of metric spaces to be Suzuki-type contractions, while Banach contractions are necessarily continuous mappings of metric spaces. Suzuki-type contractions have a wide scope of applications, for instance, in delay differential equations (see [4]). Eroglu et al. [5] discussed fixed point results for rational type contractions in quasi-metric spaces. Romaguera [6] discussed fixed point results for generalized Ćirić’s contractions of quasi-metric spaces. Ghasab et al. [7] presented some new fixed point results in $\mathcal{F}$-quasi-metric spaces and an application.

Kannan [8] presented a fixed point theorem for contractions known as Kannan-type contractions. Unlike Banach contractions, Kannan-type contractions are not necessarily continuous. Moreover, Kannan-type contractions characterize the completeness of underlying metric spaces (compare [9]). Note that Banach contractions do not characterize metric completeness (see [10]). Further, Banach and Kannan-type contractions are independent of each other. For more on Kannan-type contractions, we refer the reader to [11,12].

Another important class of contractions is due to Suzuki [2] known as Suzuki-type contractions. These contractions characterize metric completeness and are not necessarily continuous as well. Suzuki-type contractions properly generalize Banach contractions.

Quasi-metric spaces, by relaxing the symmetry property present in metric spaces, give rise to asymmetric functional analysis. The topology of quasi-metric space, in general, is $T_0.$ Further, these spaces have various applications, for instance, in fractal theory [13], software engineering [14], denotational semantics and complexity analysis [15], and many more in the literature. Dağ et al. [16] have generalized the classical Banach contraction principle in the setup of quasi-metric spaces with some illustrative examples. In [17], they have obtained Kannan-type fixed point theorems in quasi-metric spaces along with quasi-metric completeness characterization.

Recently, Romaguera’s investigations revealed that unlike the Banach contraction principle and Kannan-type fixed point theorem, a basic Suzuki-type fixed point theorem cannot be transported as it is in the setup of quasi-metric spaces (see for details [18] (Example 3)). Romaguera proposed a new type of Suzuki-type contraction by tweaking the original Suzuki type contraction a little bit and obtained a variant of Suzuki type fixed point theorems for mappings of quasi-metric spaces. Romaguera introduced a $2-$basic contraction of the Suzuki type and proved fixed point results in the setup of Smyth’s complete quasi-metric space. Note that $2-$basic contraction of the Suzuki type does not reduce to the original Suzuki type contraction whenever quasi-metric is replaced by metric.

In this manuscript, in the context of quasi-metric spaces, we further investigate the Suzuki-type fixed point theorems. We proposed a new type of Suzuki-type contraction, which is reduced to the original Suzuki-type contraction when quasi-metric is replaced with metric.

As we know from [18] (Example 3), Suzuki-type contractions cannot be transported in quasi-metric spaces, so the transportation of generalization of such contractions is out of the question. There is a special type of quasi-metric space known as $\Delta$-symmetric quasi-metric spaces (see [19]). These spaces are Hausdorff quasi-metric spaces. In this paper, we prove the existence and uniqueness of fixed points of generalized Suzuki-type contractions of $\Delta$-symmetric quasi-metric spaces. We present a remark about the possible range for $\Delta$ that is being used in $\Delta$-symmetric quasi-metric spaces and suggest a correction in the definition of these spaces given in [19]. Moreover, we discuss, bicompleteness, Smyth, and $d-$sequential completeness of these spaces as well. We present a situation (an example) where the mapping is generalized Suzuki type contraction, but it is neither Suzuki type contraction nor $2-$basic contraction of Suzuki type.

For the purposes of this paper, the symbols $\mathbb{R}$, $\mathbb{N}$, and $\mathbb{Z}$ will represent the sets of real numbers, natural numbers, and integer numbers, respectively. Also, $\left(\Im ,d\right)$ represents the quasi-metric space, and $\left(\Im ,d^s\right)$ represents the metric space induced by quasi-metric d otherwise mentioned.

Definition 1.

Let $d:\Im \times \Im \to \mathbb{R}$ be a function on a non-empty set ℑ. Consider for all $\ell ,\hslash ,\wp \in \Im ,$ the following conditions:

(d₁) 

$d(\ell ,\hslash )\ge 0,$

(d₂) 

$d(\ell ,\hslash )=0$ iff $\ell =\hslash ,$

(d₃) 

$d(\ell ,\hslash )=d(\hslash ,\ell ),$

(d₄) 

$d(\ell ,\hslash )\le d(\ell ,\wp )+d(\wp ,\hslash ),$

(d₅) 

$d\left(\ell ,\hslash \right)=d\left(\hslash ,\ell \right)=0$ iff $\ell =\hslash .$

The function d is a metric on ℑ if it satisfies the conditions (d${}_1$) through (d${}_4$), and a quasi-metric on ℑ if d satisfies (d${}_1$), (d${}_4$), and (d${}_5$).

For a positive real number $\Delta >0$, then quasi-metric space $(\Im ,d)$ is said to be $\Delta$-symmetric if

$$d(\ell ,\hslash )\le \Delta d(\hslash ,\ell ),$$

for all $\ell ,\hslash \in \Im .$ It is obvious that the $\Delta$-symmetric quasi-metric space $(\Im ,d)$ constitutes a metric space if $\Delta =1$. See [19] for more details and examples of $\Delta$-symmetric quasi-metric spaces.

Remark 1.

Let $(\Im ,d)$ be a Δ-symmetric quasi-metric space with at least two distinct points. Recently, in [19] Δ-symmetric quasi-metric is defined for $\Delta >0$ but

$$d\left(\ell ,\hslash \right)\le \Delta d\left(\hslash ,\ell \right)\le {\Delta }^{2}d\left(\ell ,\hslash \right)$$

for all $\ell ,\hslash \in \Im$ such that $\ell \ne \hslash$ implies $1-{\Delta }^2\le 0$ which means that $1\le {\Delta }^2$, then we have either $\Delta \le -1$ or $\Delta \ge 1$. Hence $\Delta \ge 1$ (as $\Delta >0$) for Δ-symmetric quasi-metric spaces with at least two distinct points.

If $(\Im ,d)$ is a quasi-metric space, then the mapping $d_{-1}$ is defined as

$$d_{-1}(\ell ,\hslash )=d(\hslash ,\ell ),$$

is another quasi-metric on ℑ. It is named a conjugate of d, and the function

$$d^s\left(\ell ,\hslash \right)=max\left\{d\left(\ell ,\hslash \right),d_{-1}\left(\ell ,\hslash \right)\right\}$$

is a metric on ℑ. Every quasi-metric d on ℑ produces a topology ${\tau }_d$, which is $T_0$ (in general) with the class of open balls

$$\left\{B_d\left(\ell ,\epsilon \right):\ell \in \Im ,\epsilon >0\right\}$$

as its basis, where

$$B_d\left(\ell ,\epsilon \right)=\left\{\hslash \in \Im :d\left(\ell ,\hslash \right)<\epsilon \right\}$$

for all $\ell \in \Im$ and $\epsilon >0.$ If the topology ${\tau }_d$ on ℑ is $T_1$$\left(\mathrm{or}{T}_2\right)$, then quasi-metric space $\left(\Im ,d\right)$ is said to be $T_1$$\left(\mathrm{or}{T}_2\right)$. Note that $\left(\Im ,d\right)$ is $T_1$ iff

$$d(\ell ,\hslash )=0\Rightarrow \ell =\hslash$$

for all $\ell ,\hslash \in \Im$. A sequence ${\left({\ell }_k\right)}_{k\in \mathbb{N}}$ in $\left(\Im ,d\right)$ is convergent to $z_0\in \Im$ in ${\tau }_d$ (${\tau }_{d_{-1}}$) if for every $\epsilon >0$ there is a $n_0\in \mathbb{N}$ such that

$$d\left(z_0,{\ell }_k\right)<\epsilon (d\left({\ell }_k,z_0\right)<\epsilon )$$

for all $k\ge n_0.$ A sequence ${\left({\ell }_k\right)}_{k\in \mathbb{N}}$ in $\left(\Im ,d\right)$ is said to be a left (right) $\tilde{K}$-Cauchy sequence if for any $\epsilon >0$ there exists an $n_0\in \mathbb{N}$ such that

$$d\left({\ell }_k,{\ell }_j\right)<\epsilon (d\left({\ell }_j,{\ell }_k\right)<\epsilon ),$$

for any $j\ge k\ge n_0$. A quasi-metric space $\left(\Im ,d\right)$ is

• left (right) $\tilde{K}$-sequentially complete if each left (right) $\tilde{K}$-Cauchy sequence converges in the topology ${\tau }_d,$
• d-sequentially complete if any Cauchy sequence in $(\Im ,d^s)$ converges in topology ${\tau }_d$,
• bicomplete if the metric space $\left(\Im ,d^s\right)$ is complete, and
• Smyth is complete if every left $\tilde{K}$-Cauchy sequence in $(\Im ,d)$ converges in the topology ${\tau }_{d^s}$.

Note that in $\Delta$-symmetric quasi-metric spaces $\left(\Im ,d\right),$ every left (right) $\tilde{K}$-Cauchy sequence is a right (left) K-Cauchy sequence. If a sequence ${\left({\ell }_k\right)}_{k\in \mathbb{N}}$ in $\Delta$-symmetric quasi-metric space $(\Im ,d)$ is left (right) $\tilde{K}$-Cauchy, then we refer to it as $\Delta$-Cauchy in $\left(\Im ,d\right).$ Smyth completeness implies bicompleteness, but the converse does not hold in general. Both bicompleteness and left (right) $\tilde{K}$-sequentially completeness imply d-sequentially completeness but converses do not hold in general. For more details on above mentioned notions, we refer the readers to [16,18], and the references therein.

Throughout this paper, otherwise stated, $\lambda$ denotes a decreasing function $\lambda :\left[0,1\right)\to \left({\displaystyle \frac{1}{2}},1\right]$ defined as

$$\lambda \left(\alpha \right)=\frac{1}{1+\alpha }.$$

2. Fixed Points of Suzuki-Type d $\left(d_{-1}\right)$-Contractions of Quasi-Metric Spaces

In this section, we start with the following definition.

Definition 2.

For a quasi-metric space $\left(\Im ,d\right)$ and a mapping $\mathsf{\Psi }:\Im \to \Im ,$

1. 

if there is a $\alpha \in \left[0,1\right)$ such that

$$\begin{array}{ccc}\hfill d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)& \le & \alpha d\left(\ell ,\hslash \right)\mathrm{and}\hfill \\ \hfill d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)& \le & \alpha d\left(\hslash ,\ell \right)\hfill \end{array}$$

for all $\ell ,\hslash \in \Im ,$ then Ψ is called d -contraction and $d_{-1}$-contraction on $(\Im ,d),$ respectively (see [16]),

2. 

if there is a $\alpha \in \left[0,1\right)$ such that

$$\lambda \left(\alpha \right)d\left(\ell ,\mathsf{\Psi }\ell \right)\le d^s\left(\ell ,\hslash \right)\Rightarrow d^s\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)\le \alpha d\left(\ell ,\hslash \right)$$

(1)

for all $\ell ,\hslash \in \Im ,$ then Ψ is called Suzuki-type d-contraction on $(\Im ,d)$,

3. 

if there is a $\alpha \in \left[0,1\right)$ such that

$$\lambda \left(\alpha \right)d\left(\ell ,\mathsf{\Psi }\ell \right)\le d^s\left(\ell ,\hslash \right)\Rightarrow d^s\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)\le \alpha d\left(\hslash ,\ell \right)$$

(2)

for all $\ell ,\hslash \in \Im ,$ then Ψ is called Suzuki-type $d_{-1}$-contraction on $(\Im ,d)$,

4. 

if there is a $\alpha \in \left[0,1\right)$ such that

$$\lambda \left(\alpha \right)d\left(\ell ,\mathsf{\Psi }\ell \right)\le d\left(\ell ,\hslash \right)\Rightarrow d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)\le \alpha d\left(\ell ,\hslash \right)$$

(3)

for all $\ell ,\hslash \in \Im ,$ then Ψ is called Suzuki type contraction on $(\Im ,d)$,

5. 

if there is a $\alpha \in \left(0,1\right)$ such that

$$min\left\{d\left(\ell ,\mathsf{\Psi }\ell \right),d\left(\hslash ,\mathsf{\Psi }\hslash \right)\right\}\le 2d\left(\ell ,\hslash \right)\Rightarrow d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)\le \alpha d\left(\ell ,\hslash \right)$$

(4)

for all $\ell ,\hslash \in \Im ,$ then Ψ is called $2-$basic contraction of Suzuki type (see [18]).

The following theorem is a direct consequence of [2] (Theorem 2).

Theorem 1.

Let $(\Im ,d)$ be a complete metric space and $\mathsf{\Psi }:\Im \to \Im$. Assume that there exists $\alpha \in \left[0,1\right)$ such that

$$d\left(\ell ,\mathsf{\Psi }\ell \right)\le 2d\left(\ell ,\hslash \right)\Rightarrow d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)\le \alpha d\left(\ell ,\hslash \right)$$

for all $\ell ,\hslash \in \Im .$ Then there exists a unique fixed point z of Ψ. Moreover

$$\underset{k\to \infty }{lim}{\mathsf{\Psi }}^k\ell =z$$

for all $\ell \in \Im .$

Remark 2.

Theorem 1 does not hold for quasi-metric spaces (see [18] (Example 3)).

The following Theorem is a direct consequence of [3] (Theorem 2).

Theorem 2.

Let $\left(\Im ,d\right)$ be a complete metric space and $\mathsf{\Psi }:\Im \to \Im$ a Suzuki-type contraction. Then Ψ has a unique fixed point.

The following example (compare with [18] (Example 3)) shows that the above Theorem cannot be transported as it is in quasi-metric space.

Example 1.

Let $\Im =\mathbb{N}\cup \left\{\infty \right\}$ and d be a quasi-metric on ℑ given as

$$\begin{array}{c}d\left(\ell ,\ell \right)=0\mathit{for}\mathit{all}\ell \in \Im ,\hfill \\ d\left(p,\infty \right)=0\mathit{for}\mathit{all}p\in \mathbb{N},\hfill \\ d\left(\infty ,p\right)=\frac{1}{p}\mathit{for}\mathit{all}p\in \mathbb{N},\hfill \\ d\left(p,q\right)=\frac{1}{q}\mathit{for}\mathit{all}p,q\in \mathbb{N}\mathrm{with}p\ne q.\hfill \end{array}$$

We conclude that $\left(\Im ,d\right)$ is Smyth complete since any non-eventually constant left-$\tilde{K}$ Cauchy sequence in $\left(\Im ,d\right)$ converges to ∞ in $\left(\Im ,d^s\right).$ Let $\mathsf{\Psi }:\Im \to \Im$ be a mapping defined as

$$\mathsf{\Psi }\ell =\left\{\begin{array}{c}1,\mathrm{if}\ell =\infty ,\hfill \\ 2\ell ,\mathrm{if}\ell \in \mathbb{N}.\hfill \end{array}\right.$$

Clearly Ψ has no fixed point. But it fulfills the condition (3) for $\alpha ={\displaystyle \frac{1}{2}}.$

• If $\ell =p$ and $\hslash =q$ with $p\ne q,$ we have

  $$d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)=d\left(2p,2q\right)=\frac{1}{2q}=\frac{1}{2}d\left(p,q\right)=\frac{1}{2}d\left(\ell ,\hslash \right).$$
• If $\ell =\infty$ and $\hslash =p,$$p\in \mathbb{N},$ we have

  $$d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)=d\left(1,2p\right)=\frac{1}{2p}=\frac{1}{2}d\left(\infty ,p\right)=\frac{1}{2}d\left(\ell ,\hslash \right).$$
• If $\ell =p,$$p\in \mathbb{N},$ and $\hslash =\infty ,$$p\in \mathbb{N},$ we have

$$\lambda \left(\alpha \right)d\left(\ell ,\mathsf{\Psi }\ell \right)\le d\left(\ell ,\mathsf{\Psi }\ell \right)=d\left(p,2p\right)=\frac{1}{2p}>0=d\left(p,\infty \right)=d\left(\ell ,\hslash \right).$$

Thus we have shown that Ψ is a Suzuki-type contraction of quasi-metric space $\left(\Im ,d\right)$ without a fixed point.

Romaguera [18] presented the following theorem.

Theorem 3.

[18] Every $2-$basic contraction $\mathsf{\Psi }:\Im \to \Im$ of a Smyth’s complete $T_1$ quasi-metric space $\left(\Im ,d\right)$ has a unique fixed point.

We present the following proposition, which we will use to prove some fixed point theorems for Suzuki-type d$\left(d_{-1}\right)$-contractions.

Proposition 1.

Let $(\Im ,d)$ be a quasi-metric space. If Ψ is a Suzuki-type d($d_{-1}$)-contraction, then the following holds:

(a) 

Ψ is a Suzuki-type contraction on $\left(\Im ,d^s\right)$.

(b) 

For any ${\ell }_0\in \Im$, the sequence ${\left({\mathsf{\Psi }}^k{\ell }_0\right)}_{k\in \mathbb{N}}$ is a Cauchy sequence in $\left(\Im ,d^s\right)$.

Proof. 

(a) Assuming that $\mathsf{\Psi }$ is a Suzuki-type d-contraction on $(\Im ,d)$, there exists $\alpha \in \left(0,1\right)$ such that

$$\lambda \left(\alpha \right)d\left(\ell ,\mathsf{\Psi }\ell \right)\le d^s\left(\ell ,\hslash \right)\Rightarrow d^s\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)\le \alpha d\left(\ell ,\hslash \right),$$

(5)

for all $\ell ,\hslash \in \Im .$ Since

$$\lambda \left(\alpha \right)d^s\left(\ell ,\mathsf{\Psi }\ell \right)\le d^s\left(\ell ,\hslash \right)\Rightarrow \lambda \left(\alpha \right)d\left(\ell ,\mathsf{\Psi }\ell \right)\le d^s\left(\ell ,\hslash \right),$$

therefore by (5), we get

$$d^s\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)\le \alpha d\left(\ell ,\hslash \right)\le \alpha d^s\left(\ell ,\hslash \right).$$

(6)

Hence,

$$\lambda \left(\alpha \right)d^s\left(\ell ,\mathsf{\Psi }\ell \right)\le d^s\left(\ell ,\hslash \right)\Rightarrow d^s\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)\le \alpha d^s\left(\ell ,\hslash \right)$$

for all $\ell ,\hslash \in \Im$. In a similar way, we prove that if $\mathsf{\Psi }$ is a Suzuki-type $d_{-1}$-contraction on $(\Im ,d)$, then $\mathsf{\Psi }$ is a Suzuki-type contraction on $(\Im ,d^s)$.

(b) Since $\mathsf{\Psi }$ is a Suzuki-type contraction on $\left(\Im ,d^s\right)$ from (a), so from [3], $\left({\mathsf{\Psi }}^k{\ell }_0\right)$ for $k\in \mathbb{N}$ is a Cauchy sequence in $\left(\Im ,d^s\right).$ □

Theorem 4.

Suppose $\left(\Im ,d\right)$ is bicomplete then any Suzuki-type d($d_{-1}$)-contraction on $(\Im ,d)$ has a unique fixed point.

Proof. 

From Proposition 1 (a), if $\mathsf{\Psi }$ is a Suzuki-type d ($d_{-1}$)-contraction on $\left(\Im ,d\right)$, then $\mathsf{\Psi }$ is a Suzuki-type contraction on $\left(\Im ,d^s\right).$ As $\left(\Im ,d\right)$ is bicomplete, $\left(\Im ,d^s\right)$ is complete. Hence, from [2,3], we obtain that $\mathsf{\Psi }$ has a unique fixed point. □

3. Fixed Points and $\Delta$-Symmetric Quasi-Metric Spaces

We start this section with a lemma showing that bicompleteness implies Smyth completeness in $\Delta$-symmetric quasi-metric spaces.

Lemma 1.

Every Δ-symmetric quasi-metric space $(\Im ,d)$ is bicomplete if it is Smyth complete.

Proof. 

Let $(\Im ,d)$ be bicomplete $\Delta$-symmetric quasi-metric space. Let ${\left({\ell }_k\right)}_{k\in \mathbb{N}}$ be a left $\tilde{K}$-Cauchy sequence in $(\Im ,d).$ As d is $\Delta$-symmetric, so ${\left({\ell }_k\right)}_{k\in \mathbb{N}}$ is the right $\tilde{K}$-Cauchy sequence in $(\Im ,d).$ Hence ${\left({\ell }_k\right)}_{k\in \mathbb{N}}$ is a Cauchy sequence in $(\Im ,d^s)$ which is complete. Consequently ${\left({\ell }_k\right)}_{k\in \mathbb{N}}$ converges in $(\Im ,d^s).$ This implies that $(\Im ,d)$ is Smyth’s complete quasi-metric. The Converse part is trivial and left for the reader. □

The following proposition shows that $\Delta$-symmetric quasi-metric spaces are $T_1.$ In fact, they are more than that.

Proposition 2.

Every Δ-symmetric quasi-metric space $\left(\Im ,d\right)$ is $T_1$ quasi-metric space.

Proof. 

Suppose that for $\ell ,\hslash \in \Im ,$

$$d\left(\ell ,\hslash \right)=0$$

implies that

$$d\left(\hslash ,\ell \right)\le \Delta d\left(\ell ,\hslash \right)=0.$$

Hence $\ell =\hslash ,$ that is, $\left(\Im ,d\right)$ is $T_1$ quasi-metric space. □

Proposition 3.

The topology of Δ-symmetric quasi-metric spaces $\left(\Im ,d\right)$ is Hausdorff.

Proof. 

As $\Delta$-symmetric quasi-metric space is $T_1$ and ${\tau }_d$ convergence implies ${\tau }_{d_{-1}}$ convergence, so from [13] (Proposition 2.1), the topology of $\Delta$-symmetric quasi-metric space is Hausdorff. □

Proposition 4.

A Δ-symmetric quasi-metric space $\left(\Im ,d\right)$ is d ($d_{-1}$)-sequentially complete if $\left(\Im ,d\right)$ is bicomplete.

Proof. 

Suppose $\left(\Im ,d\right)$ is a d-sequentially complete $\Delta$-symmetric quasi-metric space, then the Cauchy sequence ${\left({\ell }_k\right)}_{k\in \mathbb{N}}$ in metric space $\left(\Im ,d^s\right)$ converges in the topology ${\tau }_d.$ By $\Delta$-symmetric condition, ${\left({\ell }_k\right)}_{k\in \mathbb{N}}$ is also converged in the topology ${\tau }_{d_{-1}}.$ This implies ${\left({\ell }_k\right)}_{k\in \mathbb{N}}$ converges in $\left(\Im ,d^s\right).$ Hence, $\left(\Im ,d\right)$ is bicomplete. Converse follows directly from the definition of bicompleteness. On similar lines, the proof follows for $d_{-1}$-sequentially complete $\Delta$-symmetric quasi-metric space. □

Theorem 5.

Every Suzuki-type d ($d_{-1}$)-contraction on a d-sequentially complete Δ-symmetric quasi-metric space $\left(\Im ,d\right)$ has a unique fixed point.

Proof. 

Suppose $\left(\Im ,d\right)$ is d-sequentially complete $\Delta$-symmetric quasi-metric space and$\mathsf{\Psi }$ is a Suzuki-type d-contraction. As $\left(\Im ,d\right)$ is $\Delta$-symmetric quasi-metric space, by Proposition 4, $\left(\Im ,d\right)$ is bicomplete. So by Theorem 4, $\left(\Im ,d\right)$ has a unique fixed point. □

Consider the following definition.

Definition 3.

Let $\left(\Im ,d\right)$ be a quasi-metric space and $\mathsf{\Psi }:\Im \to \Im$ a mapping. If there is a $\alpha \in \left[0,1\right)$ such that

$$\lambda \left(\alpha \right)d\left(\ell ,\mathsf{\Psi }\ell \right)\le d^s\left(\ell ,\hslash \right)\Rightarrow d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)\le \alpha U_{\mathsf{\Psi }}\left(\ell ,\hslash \right),$$

(7)

where,

$$U_{\mathsf{\Psi }}\left(\ell ,\hslash \right)=max\left\{d\left(\ell ,\hslash \right),d\left(\ell ,\mathsf{\Psi }\ell \right),d\left(\hslash ,\mathsf{\Psi }\hslash \right),d\left(\mathsf{\Psi }\ell ,{\mathsf{\Psi }}^2\ell \right),\frac{d\left({\mathsf{\Psi }}^2\ell ,\mathsf{\Psi }\hslash \right)}{2},{\displaystyle \frac{d\left(\hslash ,\mathsf{\Psi }\ell \right)+d(\ell ,\mathsf{\Psi }\hslash )}{2}}\right\}.$$

(8)

for all $\ell ,\hslash \in \Im ,$ then Ψ is called generalized Suzuki-type (GST) contraction on $(\Im ,d)$.

Now we present fixed point results for GST contractions of $\Delta$-symmetric quasi-metric spaces.

Theorem 6.

Let $\left(\Im ,d\right)$ be a d-sequentially complete Δ-symmetric quasi-metric space. Then every GST contraction on $(\Im ,d)$ has a unique fixed point.

Proof. 

Let $\mathsf{\Psi }$ be a GST contraction on $(\Im ,d)$ and ${\alpha }_1$ be a real number such that $0\le \alpha <{\alpha }_1<1.$ For a fixed ${\varrho }_1\in \Im ,$ define an iterative sequence as ${\varrho }_{k+1}=\mathsf{\Psi }{\varrho }_k$ for all $k\in \mathbb{N}.$ If there exists a $k\in \mathbb{N}$ such that

$${\varrho }_k={\varrho }_{k+1}=\mathsf{\Psi }{\varrho }_k,$$

then ${\varrho }_k$ is a fixed point of $\mathsf{\Psi }$. So assume that ${\varrho }_k\ne {\varrho }_{k+1}$ for all $k\in \mathbb{N}.$ Since $\lambda \left(\alpha \right)\le 1,$ therefore

$$\lambda \left(\alpha \right)d\left({\varrho }_{k+1},\mathsf{\Psi }{\ell }_k\right)\le d\left({\varrho }_k,\mathsf{\Psi }{\varrho }_k\right)=d\left({\varrho }_k,{\varrho }_{k+1}\right),$$

holds for all $k\in \mathbb{N}.$ By the given hypothesis,

$$\begin{array}{c}d\left({\varrho }_{k+1},{\varrho }_{k+2}\right)=d\left(\mathsf{\Psi }{\varrho }_k,\mathsf{\Psi }{\varrho }_{k+1}\right)\hfill \\ \le \alpha U_{\mathsf{\Psi }}\left({\varrho }_k,{\varrho }_{k+1}\right)\hfill \\ ={\alpha }_{1}max\left\{\begin{array}{c}d\left({\varrho }_k,{\varrho }_{k+1}\right),d\left({\varrho }_k,\mathsf{\Psi }{\varrho }_k\right),d\left({\varrho }_{k+1},\mathsf{\Psi }{\varrho }_{k+1}\right),d\left(\mathsf{\Psi }{\varrho }_k,{\mathsf{\Psi }}^2{\varrho }_k\right),\hfill \\ {\displaystyle \frac{d\left({\mathsf{\Psi }}^2{\varrho }_k,\mathsf{\Psi }{\varrho }_{k+1}\right)}{2}},{\displaystyle \frac{d\left({\varrho }_{k+1},\mathsf{\Psi }{\varrho }_k\right)+d({\varrho }_k,\mathsf{\Psi }{\varrho }_{k+1})}{2}}\hfill \end{array}\right\}\hfill \\ \le {\alpha }_{1}max\left\{d\left({\varrho }_k,{\varrho }_{k+1}\right),d\left({\varrho }_{k+1},{\varrho }_{k+2}\right),{\displaystyle \frac{d\left({\varrho }_k,{\varrho }_{k+2}\right)}{2}}\right\}\hfill \\ \le {\alpha }_{1}max\left\{d\left({\varrho }_k,{\varrho }_{k+1}\right),d\left({\varrho }_{k+1},{\varrho }_{k+2}\right)\right\}.\hfill \end{array}$$

Thus

$$d\left({\varrho }_{k+1},{\varrho }_{k+2}\right)\le {\alpha }_{1}max\left\{d\left({\varrho }_k,{\varrho }_{k+1}\right),d\left({\varrho }_{k+1},{\varrho }_{k+2}\right)\right\}.$$

If

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

then

$$d\left({\varrho }_{k+1},{\varrho }_{k+2}\right)\le {\alpha }_{1}d\left({\varrho }_{k+1},{\varrho }_{k+2}\right)<d\left({\varrho }_{k+1},{\varrho }_{k+2}\right),$$

a contradiction. Consequently, we obtain $d\left({\varrho }_{k+1},{\varrho }_{k+2}\right)\le {\alpha }_{1}d\left({\varrho }_k,{\varrho }_{k+1}\right).$

• That is, $d\left({\varrho }_{k+1},{\varrho }_{k+2}\right)\le {\alpha }_1^{k}d\left({\varrho }_1,{\varrho }_2\right)$ for all $k\in \mathbb{N}.$ This implies $\left\{{\varrho }_k\right\}$ is a $\Delta$-Cauchy sequence in $\left(\Im ,d\right).$ Since $\left(\Im ,d\right)$ is d-sequential complete, there exists a $z_0\in \Im$ such that

$$\underset{k\to \infty }{lim}d\left({\varrho }_k,z_0\right)=0.$$

(9)

Now, we claim that

$$d\left(z_0,\mathsf{\Psi }\ell \right)\le \alpha max\left\{d\left(z_0,\ell \right),d\left(\ell ,z_0\right),d\left(\ell ,\mathsf{\Psi }\ell \right)\right\},$$

(10)

for all $\ell \ne z_0.$ By (9), there exists an $n_0\in \mathbb{N}$ such that

$$d\left(z_0,{\varrho }_k\right)<\frac{1}{3\Delta }d\left(z_0,\ell \right)$$

for all $k\ge n_0,$ which implies

$$\begin{array}{ccc}\hfill \lambda \left(\alpha \right)d\left({\varrho }_k,\mathsf{\Psi }{\varrho }_k\right)& \le & d\left({\varrho }_k,\mathsf{\Psi }{\varrho }_k\right)=d\left({\varrho }_k,{\varrho }_{k+1}\right)\hfill \\ & \le & d\left({\varrho }_k,z_0\right)+d\left(z_0,{\varrho }_{k+1}\right)\hfill \\ & \le & \Delta d\left(z_0,{\varrho }_k\right)+d\left(z_0,{\varrho }_{k+1}\right)\hfill \\ & \le & \Delta \frac{1}{3\Delta }d\left(z_0,\ell \right)+\frac{1}{3\Delta }d\left(z_0,\ell \right)\hfill \\ & =& \frac{2}{3}d\left(z_0,\ell \right)\hfill \\ & =& d\left(z_0,\ell \right)-\frac{1}{3}d\left(z_0,\ell \right)\hfill \\ & \le & d\left(z_0,\ell \right)-\Delta d\left(z_0,{\varrho }_k\right)\hfill \\ & \le & d\left(z_0,\ell \right)-d\left(z_0,{\varrho }_k\right)\le d\left({\varrho }_k,\ell \right).\hfill \end{array}$$

Thus,

$$\lambda \left(\alpha \right)d\left({\varrho }_k,\mathsf{\Psi }{\varrho }_k\right)\le d^s\left({\varrho }_k,\ell \right)$$

holds for all $k\ge n_0.$ From (8), we have

$$\begin{array}{ccc}\hfill d\left({\varrho }_{k+1},\mathsf{\Psi }\ell \right)& =& d\left(\mathsf{\Psi }{\varrho }_k,\mathsf{\Psi }\ell \right)\le \alpha U_{\mathsf{\Psi }}\left({\varrho }_k,\ell \right)\hfill \\ & \le & \alpha max\left\{\begin{array}{c}d\left({\varrho }_k,\ell \right),d\left({\varrho }_k,\mathsf{\Psi }{\varrho }_k\right),d\left(\ell ,\mathsf{\Psi }\ell \right),d\left(\mathsf{\Psi }{\varrho }_k,{\mathsf{\Psi }}^2{\varrho }_k\right),\hfill \\ {\displaystyle \frac{d\left({\mathsf{\Psi }}^2{\varrho }_k,\mathsf{\Psi }\ell \right)}{2}},{\displaystyle \frac{d\left(\ell ,\mathsf{\Psi }{\varrho }_k\right)+d({\varrho }_k,\mathsf{\Psi }\ell )}{2}}\hfill \end{array}\right\}\hfill \\ & =& \alpha max\left\{\begin{array}{c}d\left({\varrho }_k,\ell \right),d\left({\varrho }_k,{\varrho }_{k+1}\right),d\left(\ell ,\mathsf{\Psi }\ell \right),d\left({\varrho }_{k+1},{\varrho }_{k+2}\right),\hfill \\ {\displaystyle \frac{d\left({\varrho }_{k+2},\mathsf{\Psi }\ell \right)}{2}},{\displaystyle \frac{d\left(\ell ,{\varrho }_{k+1}\right)+d({\varrho }_k,\mathsf{\Psi }\ell )}{2}}\hfill \end{array}\right\}.\hfill \end{array}$$

On taking limit as $k\to \infty$, we have

$$\begin{array}{ccc}\hfill d\left(z_0,\mathsf{\Psi }\ell \right)& \le & \alpha max\left\{d\left(z_0,\ell \right),d\left(\ell ,\mathsf{\Psi }\ell \right),\frac{d\left(z_0,\mathsf{\Psi }\ell \right)}{2},\frac{d\left(\ell ,z_0\right)+d\left(z_0,\mathsf{\Psi }\ell \right)}{2}\right\}\hfill \\ & \le & \alpha max\left\{d\left(z_0,\ell \right),d\left(\ell ,\mathsf{\Psi }\ell \right),\frac{d\left(z_0,\ell \right)+d\left(\ell ,\mathsf{\Psi }\ell \right)}{2},\frac{d\left(\ell ,z_0\right)+d\left(z_0,\mathsf{\Psi }\ell \right)}{2}\right\}\hfill \\ & =& \alpha max\left\{d\left(z_0,\ell \right),d\left(\ell ,\mathsf{\Psi }\ell \right),\frac{d\left(\ell ,z_0\right)+d\left(z_0,\mathsf{\Psi }\ell \right)}{2}\right\}\hfill \end{array}$$

holds for all $\ell \ne z_0.$ If

$$max\left\{d\left(z_0,\ell \right),d\left(\ell ,\mathsf{\Psi }\ell \right),{\displaystyle \frac{d\left(z_0,\mathsf{\Psi }\ell \right)+d\left(\ell ,z_0\right)}{2}}\right\}=\frac{d\left(z_0,\mathsf{\Psi }\ell \right)+d\left(\ell ,z_0\right)}{2},$$

then

$$d\left(z_0,\mathsf{\Psi }\ell \right)\le \alpha \left({\displaystyle \frac{d\left(z_0,\mathsf{\Psi }\ell \right)+d\left(\ell ,z_0\right)}{2}}\right)$$

implies that

$$d\left(z_0,\mathsf{\Psi }\ell \right)\le \left(\frac{\alpha }{2-\alpha }\right)d\left(\ell ,z_0\right)<\alpha d\left(\ell ,z_0\right)\le \alpha max\left\{d\left(z_0,\ell \right),d\left(\ell ,z_0\right),d\left(\ell ,\mathsf{\Psi }\ell \right)\right\}$$

for all $\ell \ne z_0.$ Hence, (10) holds. Now, we prove $z_0=\mathsf{\Psi }{z}_0.$ Assume on the contrary that

$$x\ne \mathsf{\Psi }x$$

(11)

for all $x\in \Im .$ Set $\ell =\mathsf{\Psi }{z}_0.$ In view of (11), $\ell \ne z_0.$ Since

$$\lambda \left(\alpha \right)d\left(z_0,\mathsf{\Psi }{z}_0\right)\le d\left(z_0,\mathsf{\Psi }{z}_0\right)=d\left(z_0,\ell \right),$$

then, by (8), we have

$$\begin{array}{ccc}\hfill d\left(\ell ,\mathsf{\Psi }\ell \right)& =& d\left(\mathsf{\Psi }{z}_0,\mathsf{\Psi }\ell \right)\hfill \\ & \le & \alpha max\left\{\begin{array}{c}d\left(z_0,\ell \right),d\left(z_0,\mathsf{\Psi }{z}_0\right),d\left(\ell ,\mathsf{\Psi }\ell \right),d\left(\mathsf{\Psi }{z}_0,{\mathsf{\Psi }}^2{z}_0\right),\hfill \\ {\displaystyle \frac{d\left({\mathsf{\Psi }}^2{z}_0,\mathsf{\Psi }\ell \right)}{2}},{\displaystyle \frac{d\left(\ell ,\mathsf{\Psi }{z}_0\right)+d(z_0,\mathsf{\Psi }\ell )}{2}}\hfill \end{array}\right\}\hfill \\ & \le & \alpha max\left\{d\left(z_0,\ell \right),d\left(\ell ,\mathsf{\Psi }\ell \right),\frac{d\left(z_0,\mathsf{\Psi }\ell \right)}{2}\right\}\hfill \\ & =& \alpha max\left\{d\left(z_0,\ell \right),d\left(\ell ,\mathsf{\Psi }\ell \right),\frac{d\left(z_0,\ell \right)+d\left(\ell ,\mathsf{\Psi }\ell \right)}{2}\right\}\hfill \\ & \le & \alpha max\left\{d\left(z_0,\ell \right),d\left(\ell ,\mathsf{\Psi }\ell \right)\right\},\hfill \end{array}$$

that is,

$$d\left(\ell ,\mathsf{\Psi }\ell \right)\le \alpha max\left\{d\left(z_0,\ell \right),d\left(\ell ,\mathsf{\Psi }\ell \right)\right\}.$$

If

$$max\left\{d\left(z_0,\ell \right),d\left(\ell ,\mathsf{\Psi }\ell \right)\right\}=d\left(\ell ,\mathsf{\Psi }\ell \right),$$

then

$$d\left(\ell ,\mathsf{\Psi }\ell \right)=d\left(\mathsf{\Psi }{z}_0,\mathsf{\Psi }\ell \right)\le \alpha d\left(\ell ,\mathsf{\Psi }\ell \right)<d\left(\ell ,\mathsf{\Psi }\ell \right),$$

which is a contradiction. Hence, we get

$$d\left(\ell ,\mathsf{\Psi }\ell \right)\le \alpha d\left(z_0,\ell \right)$$

along with (10) implies

$$d\left(z_0,\mathsf{\Psi }\ell \right)\le \alpha max\left\{d\left(z_0,\ell \right),d\left(\ell ,z_0\right)\right\}.$$

Therefore, we obtain

$$\begin{array}{ccc}\hfill d\left(\ell ,\mathsf{\Psi }\ell \right)& \le & d\left(\ell ,z_0\right)+d\left(z_0,\mathsf{\Psi }\ell \right)\hfill \\ & \le & d\left(\ell ,z_0\right)+\alpha max\left\{d\left(z_0,\ell \right),d\left(\ell ,z_0\right)\right\}\hfill \\ & \le & d^s\left(\ell ,z_0\right)+\alpha d^s\left(\ell ,z_0\right)\hfill \\ & =& \left(1+\alpha \right)d^s\left(\ell ,z_0\right).\hfill \end{array}$$

Thus,

$$\frac{1}{\left(1+\alpha \right)}d\left(\ell ,\mathsf{\Psi }\ell \right)\le d^s\left(\ell ,z_0\right).$$

By (8), we have

$$\begin{array}{ccc}\hfill d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }{z}_0\right)& \le & \alpha max{U}_{\mathsf{\Psi }}\left(\ell ,z_0\right)\hfill \\ & \le & \alpha max\left\{\begin{array}{c}d\left(\ell ,z_0\right),d\left(\ell ,\mathsf{\Psi }\ell \right),d\left(z_0,\mathsf{\Psi }{z}_0\right),d\left(\mathsf{\Psi }\ell ,{\mathsf{\Psi }}^2\ell \right),\hfill \\ {\displaystyle \frac{d\left({\mathsf{\Psi }}^2\ell ,\mathsf{\Psi }{z}_0\right)}{2}},{\displaystyle \frac{d\left(z_0,\mathsf{\Psi }\ell \right)+d(\ell ,\mathsf{\Psi }{z}_0)}{2}}\hfill \end{array}\right\},\hfill \end{array}$$

holds for all $\ell \ne z_0\in \Im .$ Put $\ell ={\varrho }_k$ in above inequality

$$\begin{array}{ccc}\hfill d\left({\varrho }_{k+1},\mathsf{\Psi }{z}_0\right)& =& d\left(\mathsf{\Psi }{\varrho }_k,\mathsf{\Psi }{z}_0\right)\hfill \\ & \le & \alpha max\left\{\begin{array}{c}d\left({\varrho }_k,z_0\right),d\left({\varrho }_k,\mathsf{\Psi }{\varrho }_k\right),d\left(z_0,\mathsf{\Psi }{z}_0\right),d\left(\mathsf{\Psi }{\varrho }_k,{\mathsf{\Psi }}^2{\varrho }_k\right),\hfill \\ {\displaystyle \frac{d\left({\mathsf{\Psi }}^2{\varrho }_k,\mathsf{\Psi }{z}_0\right)}{2}},{\displaystyle \frac{d\left(z_0,\mathsf{\Psi }{\varrho }_k\right)+d({\varrho }_k,\mathsf{\Psi }{z}_0)}{2}}\hfill \end{array}\right\}\hfill \\ & \le & \alpha max\left\{\begin{array}{c}d\left({\varrho }_k,z_0\right),d\left({\varrho }_k,{\varrho }_{k+1}\right),d\left(z_0,\mathsf{\Psi }{z}_0\right),d\left({\varrho }_{k+1},{\varrho }_{k+2}\right),\hfill \\ {\displaystyle \frac{d\left({\varrho }_{k+2},\mathsf{\Psi }{z}_0\right)}{2}},{\displaystyle \frac{d\left(z_0,{\varrho }_{k+1}\right)+d({\varrho }_k,\mathsf{\Psi }{z}_0)}{2}}\hfill \end{array}\right\}\hfill \end{array}$$

holds for all $k\in \mathbb{N}.$ Taking the limit as $k\to \infty$ on both sides of the above inequality, we have

$$d\left(z_0,\mathsf{\Psi }{z}_0\right)\le \alpha d\left(z_0,\mathsf{\Psi }{z}_0\right)<d\left(z_0,\mathsf{\Psi }{z}_0\right),$$

a contradiction. Hence, $z_0=\mathsf{\Psi }{z}_0$. That is, $\mathsf{\Psi }$ has a fixed point. Now, we prove the uniqueness. For the proof by contradiction suppose that $\mathsf{\Psi }$ has two distinct fixed points say $z_0,$ and $z_1.$ Since

$$\lambda \left(\alpha \right)d\left(z_0,\mathsf{\Psi }{z}_0\right)\le d\left(z_0,\mathsf{\Psi }{z}_0\right)=d\left(z_0,z_0\right)=0\le d\left(z_0,z_1\right),$$

therefore we get

$$\begin{array}{ccc}\hfill d\left(z_0,z_1\right)& =& d\left(\mathsf{\Psi }{z}_0,\mathsf{\Psi }{z}_1\right)\hfill \\ & & \le kmax\left\{\begin{array}{c}d\left(z_0,z_1\right),d\left(z_0,\mathsf{\Psi }{z}_0\right),d\left(z_1,\mathsf{\Psi }{z}_1\right),d\left(\mathsf{\Psi }{z}_0,{\mathsf{\Psi }}^2{z}_0\right),\hfill \\ {\displaystyle \frac{d\left({\mathsf{\Psi }}^2{z}_0,\mathsf{\Psi }{z}_1\right)}{2}},{\displaystyle \frac{d\left(z_1,\mathsf{\Psi }{z}_0\right)+d(z_0,\mathsf{\Psi }{z}_1)}{2}}\hfill \end{array}\right\}\hfill \\ & =& kd\left(z_0,z_1\right)<d\left(z_0,z_1\right),\hfill \end{array}$$

a contradiction. This proves the uniqueness of fixed points and completes the proof as well. □

Now we give an example to show that (GST) contractions are proper generalization of d-contractions and Suzuki-type d-contractions in the context of $\Delta$-symmetric quasi-metric space.

Example 2.

Let $\Im =\left\{\ell ,\hslash ,\wp \right\}$ and mapping $d:\Im \times \Im \to \left[0,+\infty \right)$ defined as

$$\begin{array}{ccc}& & d\left(\ell ,\hslash \right)=1,d\left(\hslash ,\ell \right)=1.1,d\left(\wp ,\ell \right)=1.4,d\left(\ell ,\wp \right)=2,\hfill \\ & & d\left(\hslash ,\wp \right)=1.3,d\left(\wp ,\hslash \right)=1.25,d\left(\ell ,\ell \right)=0\hfill \end{array}$$

for all $\ell \in \Im .$ Note that $(\Im ,d)$ is a Δ-symmetric quasi-metric space for any $\Delta \ge {\displaystyle \frac{1.4}{1.1}}.$ Consider a mapping $\mathsf{\Psi }:\Im \to \Im$ defined as

$$\mathsf{\Psi }x=\left\{\begin{array}{cc}\hslash ,\hfill & \mathit{if}x\in \{\hslash ,\wp \},\hfill \\ \wp ,\hfill & \mathit{if}x=\ell .\hfill \end{array}\right.$$

As

$$\lambda \left(\alpha \right)d\left(\hslash ,\mathsf{\Psi }\hslash \right)=\lambda \left(\alpha \right)d\left(\hslash ,\hslash \right)=0\le 1.1=d^s\left(\hslash ,\ell \right).$$

Note that

$$d\left(\mathsf{\Psi }\hslash ,\mathsf{\Psi }\ell \right)=d^s\left(\mathsf{\Psi }\hslash ,\mathsf{\Psi }\ell \right)=d^s\left(\hslash ,\wp \right)=d\left(\hslash ,\wp \right)=1.3\nleqq 1.1\alpha =\alpha d\left(\hslash ,\ell \right)$$

for any $\alpha \in \left[0,1\right).$ Hence Ψ is neither a d-contraction nor a Suzuki-type d-contraction. Moreover

$$min\left\{d\left(\ell ,\mathsf{\Psi }\ell \right),d\left(\hslash ,\mathsf{\Psi }\hslash \right)\right\}=min\left\{d\left(\ell ,\wp \right),d\left(\hslash ,\hslash \right)\right\}=0\le 2d\left(\ell ,\hslash \right)$$

and

$$d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)=d\left(\wp ,\hslash \right)=1.25\nleqq \alpha =\alpha d\left(\ell ,\hslash \right)$$

for any $\alpha \in \left(0,1\right)$, implies that Ψ is not a $2-$basic contraction on $\left(\Im ,d\right)$. Now, we show that Ψ is a GST contraction of $(\Im ,d).$ Now,

$$d\left(\mathsf{\Psi }\hslash ,\mathsf{\Psi }\ell \right)=d\left(\hslash ,\wp \right)=1.3$$

and

$$\begin{array}{ccc}\hfill U_{\mathsf{\Psi }}\left(\hslash ,\ell \right)& =& max\left\{\begin{array}{c}d\left(\hslash ,\ell \right),d\left(\hslash ,\mathsf{\Psi }\hslash \right),d\left(\ell ,\mathsf{\Psi }\ell \right),d\left(\mathsf{\Psi }\hslash ,{\mathsf{\Psi }}^2\hslash \right),\hfill \\ {\displaystyle \frac{d\left({\mathsf{\Psi }}^2\hslash ,\mathsf{\Psi }\ell \right)}{2}},{\displaystyle \frac{d\left(\ell ,\mathsf{\Psi }\hslash \right)+d(\hslash ,\mathsf{\Psi }\ell )}{2}}\hfill \end{array}\right\}\hfill \\ & =& max\left\{\begin{array}{c}d\left(\hslash ,\ell \right),d\left(\hslash ,\hslash \right),d\left(\ell ,\wp \right),d\left(\hslash ,\hslash \right),\hfill \\ {\displaystyle \frac{d\left(\hslash ,\wp \right)}{2}},{\displaystyle \frac{d\left(\ell ,\hslash \right)+d\left(\hslash ,\wp \right)}{2}}\hfill \end{array}\right\}\hfill \\ & =& max\left\{1.1,0,2,0,\frac{1.3}{2},{\displaystyle \frac{1+1.3}{2}}\right\}=2.\hfill \end{array}$$

Thus

$$d\left(\mathsf{\Psi }\hslash ,\mathsf{\Psi }\ell \right)\le \alpha U_{\mathsf{\Psi }}\left(\hslash ,\ell \right)$$

holds for $\alpha =0.7.$ Note that

$$\lambda \left(\alpha \right)d\left(\ell ,\mathsf{\Psi }\ell \right)=\lambda \left(\alpha \right)d\left(\ell ,\wp \right)=1.17\nleqq 1.1=d^s\left(\ell ,\hslash \right)$$

for $\alpha =0.7.$ Note that

$$d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\wp \right)=d\left(\wp ,\hslash \right)=1.25$$

and

$$\begin{array}{ccc}\hfill U_{\mathsf{\Psi }}\left(\ell ,\wp \right)& =& max\left\{\begin{array}{c}d\left(\ell ,\wp \right),d\left(\ell ,\mathsf{\Psi }\ell \right),d\left(\wp ,\mathsf{\Psi }\wp \right),d\left(\mathsf{\Psi }\ell ,{\mathsf{\Psi }}^2\ell \right),\hfill \\ {\displaystyle \frac{d\left({\mathsf{\Psi }}^2\ell ,\mathsf{\Psi }\wp \right)}{2}},{\displaystyle \frac{d\left(\wp ,\mathsf{\Psi }\ell \right)+d(\ell ,\mathsf{\Psi }\wp )}{2}}\hfill \end{array}\right\}\hfill \\ & =& max\left\{\begin{array}{c}d\left(\ell ,\wp \right),d\left(\ell ,\wp \right),d\left(\wp ,\hslash \right),d\left(\wp ,\hslash \right),\hfill \\ {\displaystyle \frac{d\left(\hslash ,\hslash \right)}{2}},{\displaystyle \frac{d\left(\wp ,\wp \right)+d\left(\ell ,\hslash \right)}{2}}\hfill \end{array}\right\}\hfill \\ & =& max\left\{2,2,1.25,1.25,0,\frac{1}{2}\right\}=2.\hfill \end{array}$$

Thus

$$d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\wp \right)\le \alpha U_{\mathsf{\Psi }}\left(\ell ,\wp \right)$$

holds for $\alpha =0.7.$ Now

$$d\left(\mathsf{\Psi }\wp ,\mathsf{\Psi }\ell \right)=d\left(\hslash ,\wp \right)=1.3$$

and

$$\begin{array}{ccc}\hfill U_{\mathsf{\Psi }}\left(\wp ,\ell \right)& =& max\left\{\begin{array}{c}d\left(\wp ,\ell \right),d\left(\wp ,\mathsf{\Psi }\wp \right),d\left(\ell ,\mathsf{\Psi }\ell \right),d\left(\mathsf{\Psi }\wp ,{\mathsf{\Psi }}^2\wp \right),\hfill \\ {\displaystyle \frac{d\left({\mathsf{\Psi }}^2\wp ,\mathsf{\Psi }\ell \right)}{2}},{\displaystyle \frac{d\left(\ell ,\mathsf{\Psi }\wp \right)+d(\wp ,\mathsf{\Psi }\ell )}{2}}\hfill \end{array}\right\}\hfill \\ & =& max\left\{\begin{array}{c}d\left(\wp ,\ell \right),d\left(\wp ,\hslash \right),d\left(\ell ,\wp \right),d\left(\hslash ,\hslash \right),\hfill \\ {\displaystyle \frac{d\left(\hslash ,\wp \right)}{2}},{\displaystyle \frac{d\left(\ell ,\hslash \right)+d\left(\wp ,\wp \right)}{2}}\hfill \end{array}\right\}\hfill \\ & =& max\left\{1.4,1.25,2,0,\frac{1.3}{2},\frac{1}{2}\right\}=2.\hfill \end{array}$$

Thus

$$d\left(\mathsf{\Psi }\wp ,\mathsf{\Psi }\ell \right)\le \alpha U_{\mathsf{\Psi }}\left(\wp ,\ell \right)$$

holds for $\alpha =0.7.$ As

$$d\left(\mathsf{\Psi }\hslash ,\mathsf{\Psi }\wp \right)=d\left(\hslash ,\hslash \right)=0\mathit{and}d\left(\mathsf{\Psi }\wp ,\mathsf{\Psi }\hslash \right)=d\left(\hslash ,\hslash \right)=0$$

so

$$d\left(\mathsf{\Psi }\hslash ,\mathsf{\Psi }\wp \right)\le \alpha U_{\mathsf{\Psi }}\left(\hslash ,\wp \right)\mathit{and}d\left(\mathsf{\Psi }\wp ,\mathsf{\Psi }\hslash \right)\le \alpha U_{\mathsf{\Psi }}\left(\wp ,\hslash \right)$$

holds for $\alpha =0.7.$ Hence, Ψ is a (GST) contraction for $\alpha =0.7.$

Corollary 1.

Let $\left(\Im ,d\right)$ be a d-sequential complete Δ-symmetric quasi-metric space and $\mathsf{\Psi }:\Im \to \Im$ a mapping. If there is an $\alpha \in \left[0,1\right)$ such that

$$\begin{array}{c}\lambda \left(\alpha \right)d\left(\ell ,\mathsf{\Psi }\ell \right)\le d^s\left(\ell ,\hslash \right)\hfill \\ \mathit{implies}\hfill \\ d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)\le \alpha max\left\{d\left(\ell ,\hslash \right),d\left(\ell ,\mathsf{\Psi }\ell \right),d\left(\hslash ,\mathsf{\Psi }\hslash \right),{\displaystyle \frac{d\left(\hslash ,\mathsf{\Psi }\ell \right)+d(\ell ,\mathsf{\Psi }\hslash )}{2}}\right\}\hfill \end{array}$$

for all $\ell ,\hslash \in \Im .$ Then Ψ has a unique fixed point.

The following corollary generalizes Theorems 1, 2 and [16] (Theorem 5).

Corollary 2.

Let $\left(\Im ,d\right)$ be a d-sequential complete Δ-symmetric quasi-metric space and $\mathsf{\Psi }:\Im \to \Im$ a mapping. If there is an $\alpha \in \left[0,1\right)$ such that

$$\lambda \left(\alpha \right)d\left(\ell ,\mathsf{\Psi }\ell \right)\le d^s\left(\ell ,\hslash \right)\Rightarrow d\left(\mathsf{\Psi }\ell ,\mathsf{\Psi }\hslash \right)\le \alpha d\left(\ell ,\hslash \right)$$

or all $\ell ,\hslash \in \Im .$ Then Ψ has a unique fixed point.

4. Conclusions

In this paper, we obtained fixed point results for Suzuki-type contractions of quasi-metric spaces. We discussed the equivalence of various types of completeness notions of quasi-metric spaces, including Smyth completeness. Note that Smyth completeness has various applications in computer science and engineering, for instance, in denotational semantics and complexity analysis (see [15]); further quasi metrics have applications in software engineering [14] and fractal theory [13]. Results obtained here in this paper can further be explored in connection with these applications.