Nonlinear Relation-Theoretic Suzuki-Generalized Ćirić-Type Contractions and Application to Fractal Spaces

Abstract

In this article, we introduce the idea of relation-theoretic Suzuki-generalized nonlinear contractions and utilized the same to prove some fixed point results in an ℜ-complete partial metric space. Our newly established results are sharpened versions of earlier existing results in the literature. Indeed, we give an application to construct multivalued fractals using a newly introduced contraction in the iterated function space.

1. Introduction

In metric fixed point theory, the first result on the existence and uniqueness of fixed points was established by Banach [1] in 1922. The Banach contraction principle (or, in short, BCP) is referred to as the classical result of fixed point theory, which is still a crucial one in nonlinear functional analysis. Later, numerous researchers generalized and extended the BCP in different directions, namely by enlarging the ambient space (cf. [2,3,4]), enhancing the number of involved mappings and weakening underlying contraction conditions (cf. [5,6,7]). Under the class of contractive conditions, the first important generalization of contraction is $\varphi$-contraction or nonlinear contractions, i.e., A self-mapping T defined on a metric space $(\mathcal{M},d)$ is called a nonlinear contraction under $\varphi$ (or, in short, $\varphi$-contraction) if $d(T\varsigma ,T\wp )\le \varphi \left(d\right(\varsigma ,\wp \left)\right)\mathrm{for}\mathrm{all}\varsigma ,\wp \in \mathcal{M}$, where $\varphi :[0,+\infty )\to [0,+\infty )$ satisfying $\varphi \left(\varrho \right)<\varrho \mathrm{for}\mathrm{each}\varrho >0$ is a control function. In 1968, Browder [8] employed the notion of $\varphi$-contraction to generalize the BCP, where $\varphi$ is right-continuous and increasing. Furthermore, Boyd–Wong [9] and Matkowski [10], extended the notion of $\varphi$-contraction by slightly altering the characteristics of the control function $\varphi$.

In 1968, Kannan [11] modified the contractive condition of Banach [1] by introducing a different contractive condition. In this continuation in 1971, Riech [12] extended and generalized the Banach and Kannan contractive conditions. Ćirić [13] enhanced the Riech [12] results by using a new contractive terminology, known as generalized contraction.

Recently, in 2018, Pant [14] furnished some fixed point results in a metric space under Suzuki-type generalized $\varphi$-contraction ( $\varphi$ is of Boyd and Wong type), which is stated as follows:

Definition 1 

([14]). Let $(\mathcal{M},d)$ be a metric space. A self-mapping $T:\mathcal{M}\to \mathcal{M}$ is called a Suzuki-type generalized ϕ-contractive if for all $\varsigma ,\wp \in \mathcal{M}$

$$\frac{1}{2}d(\varsigma ,T\varsigma )\le d(\varsigma ,\wp )\Rightarrow d(T\varsigma ,T\wp )\le \varphi \left(m(\varsigma ,\wp )\right),$$

where $m(\varsigma ,\wp )=max\left\{d(\varsigma ,\wp ),d(\varsigma ,T\varsigma ),d(\wp ,T\wp )\right\}$ and $\varphi :[0,+\infty )\to [0,+\infty )$ is a function such that $\varphi \left(\varrho \right)<\varrho$ and ${lim\; sup}_{s\to {\varrho }^{+}}\varphi \left(s\right)<\varrho$ for all $\varrho >0$.

Alam and Imdad [15] established a substantial generalization of the Banach contraction principle in 2015 under arbitrary binary relation. Soon after, various relation-theoretic results were proposed by several researchers (cf. [16]). Most recently, Arif and Imdad [17] proved fixed point results by utilizing a Suzuki-generalized Ćirić–Boyd and Wong-type nonlinear contraction employing locally T-transitive binary relation.

Under the class of ambient space, partial metric space is one of the extensions of metric spaces and was first proposed by Mathews [18] in 1994. In this generalization, the distance between the self-point need not be zero, and a modified triangle inequality is also included. Following that, Matthews established the partial metric version of the Banach fixed point theorem [1].

A symmetric closure of any binary relation was used by Samet and Turinici [19] to derive a fixed point theorem for nonlinear contraction. The auxiliary function stated by Samet and Turinici is as follows:

Let $\mathsf{\Phi }$ be the family of all mappings $\varphi :[0,+\infty )\to [0,+\infty )$ satisfying the following properties

$\left({\varphi }_1\right):\varphi \mathrm{is}\mathrm{increasing}$;

$\left({\varphi }_2\right):{\sum }_{n=1}^{+\infty }{\varphi }^n\left(\varrho \right)<+\infty \mathrm{for}\mathrm{each}\varrho >0.$

The aim of this article is to prove the existence and uniqueness of fixed point results for Suzuki-generalized Ćirić-type $\varphi$-contraction (under the class $\mathsf{\Phi }$, as described above) in partial metric space endowed with arbitrary binary relation. An example is provided to demonstrate our newly proved results. Finally, we furnish an application to construct multivalued fractals using the Suzuki-generalized Ćirić $\varphi$-contraction.

2. Preliminaries of Partial Metric Space

From now onward, we use the abbreviation PMS for partial metric space, NES($\ne \varnothing$) for nonempty set, $\mathbb{N}$ for the set of all natural numbers and ${\mathbb{N}}_0$ for the set of all whole numbers, i.e., ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$.

Mathews [18] defined the idea of PMS as follows.

Definition 2 

([18]). Let $\mathcal{M}$ be a NES($\ne \varnothing$) and $d_p:\mathcal{M}\times \mathcal{M}\to [0,+\infty )$ a mapping satisfying the following conditions:

1. 

$\varsigma =\wp ⟺d_p(\varsigma ,\varsigma )=d_p(\varsigma ,\wp )=d_p(\wp ,\wp )$;

2. 

$d_p(\varsigma ,\varsigma )\le d_p(\varsigma ,\wp )$;

3. 

$d_p(\varsigma ,\wp )=d_p(\wp ,\varsigma )$;

4. 

$d_p(\varsigma ,\wp )\le d_p(\varsigma ,\xi )+d_p(\xi ,\wp )-d_p(\xi ,\xi ),\mathrm{for}\mathrm{all}\varsigma ,\wp ,\xi \in \mathcal{M}$.

Then, $d_p$ is known as the partial metric, and the pair $(\mathcal{M},d_p)$ is called PMS.

Let $d_p$ be the partial metric on $\mathcal{M}$. Then, the mapping $d:\mathcal{M}\times \mathcal{M}\to [0,+\infty )$ defined by

$$d(\varsigma ,\wp )=2d_p(\varsigma ,\wp )-d_p(\varsigma ,\varsigma )-d_p(\wp ,\wp ),forall\varsigma ,\wp \in \mathcal{M},$$

is a metric on $\mathcal{M}$ and, hence, $(\mathcal{M},d)$ is a metric space.

Definition 3 

([18]). Let $(\mathcal{M},d_p)$ be a PMS. Then,

(i) a sequence $\left\{{\varsigma }_n\right\}$ is convergent to a point $\varsigma \in \mathcal{M}$ if $\underset{n\to +\infty }{lim}{d}_p({\varsigma }_n,\varsigma )=d_p(\varsigma ,\varsigma ),$

(ii) a sequence $\left\{{\varsigma }_n\right\}$ is Cauchy if $\underset{m,n\to +\infty }{lim}{d}_p({\varsigma }_m,{\varsigma }_n)$ exists and finite and

(iii) $(\mathcal{M},d_p)$ is said to be complete if every Cauchy sequence $\left\{{\varsigma }_n\right\}$ in $\mathcal{M}$ converges to a point $\varsigma \in \mathcal{M}$ and $d_p(\varsigma ,\varsigma )=\underset{n,m\to +\infty }{lim}{d}_p({\varsigma }_n,{\varsigma }_m)$.

Lemma 1 

([18]). Let $(\mathcal{M},d_p)$ be a PMS. Then,

(i) a sequence $\left\{{\varsigma }_n\right\}$ is Cauchy in $(\mathcal{M},d_p)$ if and only if it is Cauchy in $(\mathcal{M},d)$.

(ii) $(\mathcal{M},d_p)$ is complete if $(\mathcal{M},d)$ is complete, and

$$\underset{n\to +\infty }{lim}d({\varsigma }_n,\varsigma )=0⟺d_p(\varsigma ,\varsigma )=\underset{n\to +\infty }{lim}{d}_p({\varsigma }_n,\varsigma )=\underset{n,m\to +\infty }{lim}{d}_p({\varsigma }_n,{\varsigma }_m).$$

3. Relation-Theoretic Definitions and Results

Definition 4 

([20]). Let $\mathcal{M}$ be a NES($\ne \varnothing$). A subset ℜ of $\mathcal{M}\times \mathcal{M}$ is called a binary relation on $\mathcal{M}$. If $(\varsigma ,\wp )\in \Re$, then we say that “ς is related to ℘” or “ς relates to ℘" under ℜ. The subsets, $\mathcal{M}\times \mathcal{M}$ and ∅ of $\mathcal{M}\times \mathcal{M}$ are called the universal relation and empty relation, respectively.

Definition 5 

([19,20,21,22,23,24]). A binary relation ℜ defined on $\mathcal{M}$(NES($\ne \varnothing$)) is called:

1. 

“Amorphous”if ℜ has no specific property;

2. 

“Reflexive”if $(\varsigma ,\varsigma )\in \Re \mathrm{for}\mathrm{all}\varsigma \in \mathcal{M}$;

3. 

“Symmetric”if $(\varsigma ,\wp )\in \Re$ implies $(y,x)\in \Re$;

4. 

“Antisymmetric”if $(\varsigma ,\wp )\in \Re$ and $(\wp ,\varsigma )\in \Re$ implies $\varsigma =\wp$;

5. 

“Transitive”if $(\varsigma ,\wp )\in \Re$ and $(\wp ,\xi )\in \Re$ implies $(\varsigma ,\xi )\in \Re$;

6. 

A “partial order”if ℜ is “reflexive”, “antisymmetric”and “transitive”.

Definition 6 

([15,20,25]). Let $\mathcal{M}$ be a NES($\ne \varnothing$) equipped with a binary relation ℜ, then for $\varsigma ,\wp \in \mathcal{M}$:

1. 

“Inverse relation” ${\Re }^{-1}=\{(\varsigma ,\wp )\in {\mathcal{M}}^2:(\wp ,\varsigma )\in \Re \}$ and “symmetric closure” ${\Re }^s:=\Re \cup {\Re }^{-1}$;

2. 

ς and ℘ are “ℜ-comparative” if either $(\varsigma ,\wp )\in \Re$ or $(\wp ,\varsigma )\in \Re$. It is denoted by $[\varsigma ,\wp ]\in \Re ;$

3. 

A sequence $\left\{{\varsigma }_n\right\}\subset \mathcal{M}$ is termed as “ℜ-preserving” if

$$({\varsigma }_n,{\varsigma }_{n+1})\in \Re \mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0,$$

4. 

A path in ℜ from ς to ℘ is a finite sequence $\{{\xi }_0,{\xi }_1,\dots ,{\xi }_k\}\subseteq \mathcal{M}$ which satisfies the following conditions:

$\left(i\right)$${\xi }_0=\varsigma$ and ${\xi }_k=\wp$;

$\left(ii\right)$$({\xi }_i,{\xi }_{i+1})\in \Re$ for all $i\in \{0,1,2,3,\dots ,k-1\}$;

5. 

A subset D of $\mathcal{M}$ is called “ℜ-connected” if each pair of elements of D has a path in ℜ.

Definition 7 

([15]). Let $\mathcal{M}$ be an NES($\ne \varnothing$) and T a self-mapping on $\mathcal{M}$. A binary relation ℜ defined on $\mathcal{M}$ is called T-closed if for any $\varsigma ,\wp \in \mathcal{M}$:

$(\varsigma ,\wp )\in \Re \Rightarrow (T\varsigma ,T\wp )\in \Re .$

Proposition 1 

([15]). ${\Re }^s$ is T-closed, if ℜ is T-closed.

Proposition 2 

([25]). ℜ is $T^n$-closed, if ℜ is T-closed.

Definition 8 

([26]). Let $(\mathcal{M},d_p)$ be a PMS endowed with a binary relation ℜ. A mapping $T:\mathcal{M}\to \mathcal{M}$ is said to be “ℜ-continuous” at an element $\varsigma \in \mathcal{M}$ if for any “ℜ-preserving” sequence $\left\{{\varsigma }_n\right\}\subset \mathcal{M}$ such that $\left\{{\varsigma }_n\right\}$ is convergent to ς, we have $T\left({\varsigma }_n\right)$ is convergent to $T\left(\varsigma \right)$. Moreover, T is called “ℜ-continuous” if it is “ℜ-continuous” at each point of $\mathcal{M}$.

Definition 9 

([26]). Given a PMS $(\mathcal{M},d_p)$, a binary relation ℜ defined on $\mathcal{M}$ is called $d_p$-self-closed if for an “ℜ-preserving” sequence $\left\{{\varsigma }_n\right\}\subset \mathcal{M}$ converging to $\varsigma \in \mathcal{M}$, there exists a subsequence $\left\{{\varsigma }_{n_k}\right\}$ of $\left\{{\varsigma }_n\right\}$ such that $[{\varsigma }_{n_k},\varsigma ]\in \Re$ for all $k\in \mathbb{N}$.

Definition 10 

([27]). Let ℜ be a binary relation defined on an $\mathcal{M}$(NES($\ne \varnothing$)). We say that $(\mathcal{M},d_p)$ is “ℜ-complete” if every “ℜ-preserving” Cauchy sequence in $\mathcal{M}$ converges.

Inspired by Turinici [28,29], Alam and Imdad [25] introduced the following notions by localizing the transitivity condition.

Definition 11 

([25]). Let $\mathcal{M}$ be a NES($\ne \varnothing$) equipped with a binary relation ℜ and T a self-mapping on $\mathcal{M}$. Then:

1. 

ℜ on $\mathcal{M}$ is called “locally transitive” if for each (effectively) “ℜ-preserving” sequence $\left\{{\varsigma }_n\right\}\subset \mathcal{M}$ (with range $E=\{{\varsigma }_n:n\in \mathbb{N}\})$, such that ${\Re |}_E$ is “transitive”;

2. 

ℜ on $\mathcal{M}$ is called “locally T-transitive” if for each (effectively) “ℜ-preserving” sequence $\left\{{\varsigma }_n\right\}\subset T\left(\mathcal{M}\right)$ (with range $E=\{{\varsigma }_n:n\in \mathbb{N}\})$, such that ${\Re |}_E$ is “transitive”.

Proposition 3 

([25]). Let $\mathcal{M}$ be a NES($\ne \varnothing$), ℜ a binary relation on $\mathcal{M}$ and T a self-mapping on $\mathcal{M}$. Then:

(i) 

ℜ is “T-transitive”⇔${\Re |}_{T\left(\mathcal{M}\right)}$ is “transitive”;

(ii) 

ℜ is “locally T-transitive”⇔${\Re |}_{T\left(\mathcal{M}\right)}$ is “locally transitive”;

(iii) 

ℜ is “transitive”⇒ℜ is “locally transitive”⇒ℜ is “locally T-transitive”;

(iv) 

ℜ is “transitive”⇒ℜ is “T-transitive”⇒ℜ is “locally T-transitive”.

Lemma 2 

([19]). Let $\varphi \in \mathsf{\Phi }$. Then, for all $\varrho >0$, we have $\varphi \left(\varrho \right)<\varrho$.

Given a binary relation ℜ and a self-mapping T on an NES($\ne \varnothing$), $\mathcal{M}$, we use the following notations:

(i) 

$F\left(T\right)$: = the set of all fixed points of T;

(ii) 

$\mathcal{M}(T,\Re ):=\{\varsigma \in \mathcal{M}:(\varsigma ,T\varsigma )\in \Re \}$;

(iii) 

${\rm Y}(\varsigma ,\wp ,\Re ):\mathrm{the}\mathrm{class}\mathrm{of}\mathrm{all}\mathrm{paths}\mathrm{in}\Re \mathrm{from}\varsigma \mathrm{to}\wp ;$

(iv) 

$N(\varsigma ,\wp ):=max\left\{d_p(\varsigma ,\wp ),d_p(\varsigma ,T\varsigma ),d_p(\wp ,T\wp )\right\}$;

(v) 

$M(\varsigma ,\wp ):=max\left\{d_p(\varsigma ,\wp ),d_p(\varsigma ,T\varsigma ),d_p(\wp ,T\wp ),\frac{1}{2}\{d_p(\varsigma ,T\wp )+d_p(\varsigma ,T\varsigma )\}\right\}$.

Remark 1. 

Observe that $N(\varsigma ,\wp )\le M(\varsigma ,\wp )$$(\mathit{for}\mathit{all}\varsigma ,\wp \in \mathcal{M})$.

In view of the symmetry of metric $d_p$, the following conclusion is immediate.

Proposition 4. 

If $(\mathcal{M},d_p)$ is a PMS, ℜ is a binary relation on $\mathcal{M}$, T a self-mapping on $\mathcal{M}$ and $\varphi \in \mathsf{\Phi }$, then the following contractivity conditions are equivalent:

$$\begin{array}{c}\hfill \left(I\right)\frac{1}{2}{d}_p(\varsigma ,T\varsigma )\le d_p(\varsigma ,\wp )⟹d_p(T\varsigma ,T\wp )\le \varphi \left(M(\varsigma ,\wp )\right)\mathit{for}\mathit{all}\varsigma ,\wp \in \mathcal{M}with(\varsigma ,\wp )\in \Re ,\\ \hfill \left(II\right)\frac{1}{2}{d}_p(\varsigma ,T\varsigma )\le d_p(\varsigma ,\wp )⟹d_p(T\varsigma ,T\wp )\le \varphi \left(M(\varsigma ,\wp )\right)\mathit{for}\mathit{all}\varsigma ,\wp \in \mathcal{M}with[\varsigma ,\wp ]\in \Re .\end{array}$$

4. Main Results

Now, we are equipped to prove some fixed point results in relational partial metric space under the Suzuki-generalized Ćirić $\varphi$-contraction.

Theorem 1. 

Let $(\mathcal{M},d_p)$ be a PMS endowed with a binary relation ℜ and T a self-mapping on $\mathcal{M}$. Suppose that the following conditions hold:

(a) 

$(\mathcal{M},d_p)$ is ℜ-complete;

(b) 

$\mathcal{M}(T,\Re )$ is non-empty;

(c) 

ℜ is T-closed;

(d) 

if there exists $\varphi \in \mathsf{\Phi }$ such that

$$\frac{1}{2}{d}_p(\varsigma ,T\varsigma )\le d_p(\varsigma ,\wp )⟹d_p(T\varsigma ,T\wp )\le \varphi \left(M(\varsigma ,\wp )\right)\mathit{for}\mathit{all}\varsigma ,\wp \in \mathcal{M}with(\varsigma ,\wp )\in \Re ;$$

(e) 

either T is ℜ-continuous or ℜ is $d_p$-self-closed.

Then, T has a fixed point.

Proof. 

By condition $\left(b\right)$, choose ${\varsigma }_0\in \mathcal{M}$, now we can construct a Picard sequence $\left\{{\varsigma }_n\right\}$ with initial point ${\varsigma }_0$ such that

$${\varsigma }_{n+1}=T^n\left({\varsigma }_0\right)=T\left({\varsigma }_n\right)\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0.$$

(1)

Since $({\varsigma }_0,T{\varsigma }_0)\in \Re$ and ℜ is T-closed, inductively, we have

$$(T{\varsigma }_0,T^2{\varsigma }_0),(T^2{\varsigma }_0,T^3{\varsigma }_0),(T^3{\varsigma }_0,T^4{\varsigma }_0),\dots ,(T^n{\varsigma }_0,T^{n+1}{\varsigma }_0),\dots \in \Re .$$

Consequently, we have $({\varsigma }_n,{\varsigma }_{n+1})\in \Re$ $\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0$. Hence, $\left\{{\varsigma }_n\right\}$ is a ℜ-preserving sequence. For, if $d_p({\varsigma }_{n_0+1},{\varsigma }_{n_0})=0$ for some $n_0\in {\mathbb{N}}_0$, then, we have $T\left({\varsigma }_{n_0}\right)={\varsigma }_{n_0}$ so that ${\varsigma }_{n_0}$ is a fixed point of T.

Otherwise, if $d_p({\varsigma }_{n+1},{\varsigma }_n)>0$$\mathrm{for}\mathrm{all}n\in {\mathbb{N}}_0,$ implies $\frac{1}{2}{d}_p({\varsigma }_n,{\varsigma }_{n+1})<d_p({\varsigma }_n,{\varsigma }_{n+1}),$ then applying the contractivity condition $\left(d\right)$ to (1), using the triangle inequality of $d_p$ and increasing property of $\varphi$, we deduce, for all $n\in {\mathbb{N}}_0$,

$$\begin{array}{cc}\hfill & d_p({\varsigma }_{n+1},{\varsigma }_n)\hfill \\ \hfill & =d_p(T{\varsigma }_n,T{\varsigma }_{n-1})\hfill \\ \hfill & \le \varphi \left(M({\varsigma }_n,{\varsigma }_{n-1})\right)\hfill \\ \hfill & =\varphi \left(max\right\{d_p({\varsigma }_n,{\varsigma }_{n-1}),d_p({\varsigma }_n,T{\varsigma }_n),d_p({\varsigma }_{n-1},T{\varsigma }_{n-1}),\frac{1}{2}\{d_p({\varsigma }_n,T{\varsigma }_{n-1})+d_p({\varsigma }_{n-1},T{\varsigma }_n)\}\hfill \\ \hfill & =\varphi \left(max\right\{d_p({\varsigma }_n,{\varsigma }_{n-1}),d_p({\varsigma }_n,{\varsigma }_{n+1}),d_p({\varsigma }_{n-1},{\varsigma }_n),\frac{1}{2}\{d_p({\varsigma }_{n-1},{\varsigma }_n)+d_p({\varsigma }_n,{\varsigma }_{n+1})\}\hfill \\ \hfill & =\varphi \left(max\{d_p({\varsigma }_n,{\varsigma }_{n-1}),d_p({\varsigma }_n,{\varsigma }_{n+1})\}\right)\hfill \end{array}$$

so that

$$\begin{array}{ccc}\hfill d_p({\varsigma }_{n+1},{\varsigma }_n)& \le & \varphi \left(max\{d_p({\varsigma }_n,{\varsigma }_{n-1}),d_p({\varsigma }_n,{\varsigma }_{n+1})\}\right).\hfill \end{array}$$

(2)

In the case if $max\{d_p({\varsigma }_n,{\varsigma }_{n-1}),d_p({\varsigma }_n,{\varsigma }_{n+1})\}=d_p({\varsigma }_n,{\varsigma }_{n+1})$ and using the property of $\varphi$, then by (2), we obtain $d_p({\varsigma }_{n+1},{\varsigma }_n)<d_p({\varsigma }_n,{\varsigma }_{n+1})$, which is a contradiction, and hence (2) reduces to

$$\begin{array}{ccc}\hfill d_p({\varsigma }_{n+1},{\varsigma }_n)& \le & \varphi \left(d_p({\varsigma }_n,{\varsigma }_{n-1})\right).\hfill \end{array}$$

(3)

Thus, by mathematical induction, we obtain

$$\begin{array}{ccc}\hfill d_p({\varsigma }_{n+1},{\varsigma }_n)& \le & {\varphi }^n\left(d_p({\varsigma }_1,{\varsigma }_0)\right).\hfill \end{array}$$

(4)

Now, we show that $\left\{{\varsigma }_n\right\}$ is a Cauchy sequence. Then, for $m>n$ and triangle inequality, we obtain

$$\begin{array}{cc}\hfill d_p({\varsigma }_n,{\varsigma }_m)& \le d_p({\varsigma }_n,{\varsigma }_{n+1})+d_p({\varsigma }_{n+1},{\varsigma }_{n+2})+\cdots +d_p({\varsigma }_{m-1},{\varsigma }_m)\hfill \\ \hfill & -d_p({\varsigma }_{n+1},{\varsigma }_{n+1})-d_p({\varsigma }_{n+2},{\varsigma }_{n+2})-\cdots -d_p({\varsigma }_{m-1},{\varsigma }_{m-1})\hfill \\ \hfill & \le d_p({\varsigma }_n,{\varsigma }_{n+1})+d_p({\varsigma }_{n+1},{\varsigma }_{n+2})+\cdots +d_p({\varsigma }_{m-1},{\varsigma }_m)\hfill \\ \hfill & \le {\varphi }^n\left(d_p({\varsigma }_1,{\varsigma }_0)\right)+{\varphi }^{n+1}\left(d_p({\varsigma }_1,{\varsigma }_0)\right)+\cdots +{\varphi }^m\left(d_p({\varsigma }_1,{\varsigma }_0)\right)\hfill \\ \hfill & \le \sum _{k=n}^m{\varphi }^k\left(d_p({\varsigma }_1,{\varsigma }_0)\right)\hfill \\ \hfill & \le \sum _{k\ge n}^{}{\varphi }^k\left(d_p({\varsigma }_1,{\varsigma }_0)\right)\hfill \end{array}$$

as $n\to \infty$, by the condition $\left({\varphi }_2\right)$, we obtain $d_p({\varsigma }_n,{\varsigma }_m)\to 0$ as $n,m\to \infty$. Thus, $\underset{n,m\to \infty }{lim}{d}_p({\varsigma }_n,{\varsigma }_m)$$=0$, which amounts to say that $\left\{{\varsigma }_n\right\}$ is a Cauchy sequence in $(\mathcal{M},d_p)$. Since $({\varsigma }_n,{\varsigma }_{n+1})\in \Re$ for all $n\in {\mathbb{N}}_0$, and we have $\underset{n,m\to \infty }{lim}{d}_p({\varsigma }_m,{\varsigma }_n)=0$, owing to Lemma 1, $\left\{{\varsigma }_n\right\}$ is a Cauchy sequence in both $(\mathcal{M},d_p)$ and $(\mathcal{M},d)$. Since $(\mathcal{M},d_p)$ is ℜ-complete, so is $(\mathcal{M},d)$. Now, the ℜ-continuity of T implies that

$\varsigma =\underset{n\to \infty }{lim}{\varsigma }_{n+1}=\underset{n\to \infty }{lim}T{\varsigma }_n=T\varsigma .$

Hence, $\varsigma$ is a fixed point of T.

Alternately, assume that ℜ is $d_p$-self-closed. As $\left\{{\varsigma }_n\right\}$ is ℜ-preserving such that ${\varsigma }_n\stackrel{d_p}{⟶}\xi$, the $d_p$-self-closedness of ℜ guarantees the existence of a subsequence $\left\{{\varsigma }_{n_k}\right\}$ of $\left\{{\varsigma }_n\right\}$ with $[{\varsigma }_{n_k},\xi ]\in \Re (\mathrm{for}\mathrm{all}k\in {\mathbb{N}}_0)$.

Now, we claim that (for all $k\in {\mathbb{N}}_0$),

$$\begin{array}{c}\hfill \frac{1}{2}{d}_p({\varsigma }_{n_k},{\varsigma }_{n_k+1})\le d_p({\varsigma }_{n_k},\xi )\mathrm{or}\frac{1}{2}{d}_p({\varsigma }_{n_k+1},{\varsigma }_{n_k+2})\le d_p({\varsigma }_{n_k+1},\xi ).\end{array}$$

(5)

Arguing by contradiction, we assume that (for some $k_1\in {\mathbb{N}}_0$)

$$\begin{array}{c}\hfill \frac{1}{2}{d}_p({\varsigma }_{n_{k_1}},{\varsigma }_{n_{k_1}+1})>d_p({\varsigma }_{n_{k_1}},\xi )\mathrm{and}\frac{1}{2}{d}_p({\varsigma }_{n_{k_1}+1},{\varsigma }_{n_{k_1}+2})>d_p({\varsigma }_{n_{k_1}+1},\xi )\end{array}$$

Applying the triangle inequality of partial metric, we obtain

$$\begin{array}{ccc}\hfill d_p({\varsigma }_{n_{k_1}},{\varsigma }_{n_{k_1}+1})& \le & d_p({\varsigma }_{n_{k_1}},\xi )+d_p(\xi ,{\varsigma }_{n_{k_1}+1})-d_p(\xi ,\xi )\hfill \\ & \le & d_p({\varsigma }_{n_{k_1}},\xi )+d_p(\xi ,{\varsigma }_{n_{k_1}+1})\hfill \\ & <& \frac{1}{2}{d}_p({\varsigma }_{n_{k_1}},{\varsigma }_{n_{k_1}+1})+\frac{1}{2}{d}_p({\varsigma }_{n_{k_1}+1},{\varsigma }_{n_{k_1}+2})\hfill \\ & <& \frac{1}{2}{d}_p({\varsigma }_{n_{k_1}},{\varsigma }_{n_{k_1}+1})+\frac{1}{2}{d}_p({\varsigma }_{n_{k_1}},{\varsigma }_{n_{k_1}+1})\hfill \\ & <& \frac{1}{2}\{d_p({\varsigma }_{n_{k_1}},{\varsigma }_{n_{k_1}+1})+d_p({\varsigma }_{n_{k_1}},{\varsigma }_{n_{k_1+1}})\}=d_p({\varsigma }_{n_{k_1}},{\varsigma }_{n_{k_1}+1}),\hfill \end{array}$$

a contradiction. Therefore, (5) for all $k\in {\mathbb{N}}_0$ holds immediately.

On using assumption $\left(d\right)$, (5), $[{\varsigma }_{n_k},\xi ]\in \Re$ and Proposition 4, we have

$$d_p({\varsigma }_{n_{k+1}},T\xi )=d_p(T{\varsigma }_{n_k},T\xi )\le \varphi \left(M\left({\varsigma }_{n_k},\xi \right)\right).$$

(6)

If $M\left({\varsigma }_{n_k},\xi \right)=d_p(T\xi ,\xi )=\alpha$, then we have

$d_p({\varsigma }_{n_{k+1}},T\xi )=d_p(T{\varsigma }_{n_k},T\xi )\le \varphi \left(M\left({\varsigma }_{n_k},\xi \right)\right)\le \varphi \left(d_p(\xi ,T\xi )\right)\le \varphi \left(\alpha \right).$

Taking $k\to \infty$, we obtain

$$\begin{array}{ccc}\hfill d_p(T\xi ,\xi )& \le & \varphi \left(\alpha \right)\hfill \\ \hfill \alpha & \le & \varphi \left(\alpha \right)<\alpha ,\hfill \end{array}$$

which is a contradiction. Otherwise, if

$M({\varsigma }_{n_k},\xi )=max\left\{d_p({\varsigma }_{n_k},\xi ),d_p({\varsigma }_{n_k},{\varsigma }_{n_{k+1}}),\frac{1}{2}\left[d_p({\varsigma }_{n_k},T\xi )+d_p(\xi ,{\varsigma }_{n_k+1})\right]\right\},$

then due to the fact ${\varsigma }_n\stackrel{d_p}{\to }\xi$, there always exists $N=N\left(\alpha \right)$ such that

$M({\varsigma }_{n_k},\xi )\le \frac{3}{4}\alpha \mathrm{for}\mathrm{all}k\ge N.$

As $\varphi$ is increasing, we have

$$\varphi \left(M\left({\varsigma }_{n_k},\xi \right)\right)\le \varphi \left(\frac{3}{4}\alpha \right)\mathrm{for}\mathrm{all}k\ge N.$$

(7)

Employing (6) and (7), we have,

$d_p({\varsigma }_{n_{k+1}},T\xi )=d_p(T{\varsigma }_{n_k},T\xi )\le \varphi \left(M\left({\varsigma }_{n_k},\xi \right)\right)\le \varphi \left(\frac{3}{4}\alpha \right)\mathrm{for}\mathrm{all}k\ge N.$

On taking $k\to \infty$ and using Lemma 2, we obtain

$\alpha =\varphi \left(\frac{3}{4}\alpha \right)<\frac{3}{4}\alpha <\alpha ,$

which is a contradiction. Hence, $\alpha =0$, so that

$d_p(\xi ,T\xi )=\alpha =0\Rightarrow T\xi =\xi .$

Hence, $\xi$ is a fixed point of T. □

Theorem 2. 

In addition to all hypotheses of Theorem 1, if we have that

(f) ${\rm Y}\left(\varsigma ,\wp ,{\Re }^s\right)$ is non-empty,

then T has a unique fixed point.

Proof. 

To prove uniqueness, choose $\varsigma ,\wp \in F\left(T\right)$. Then,

$$T\varsigma =\varsigma \mathrm{and}T\wp =\wp .$$

(8)

As ${\rm Y}\left(\varsigma ,\wp ,{\Re }^s\right)$ is non-empty, there exists a path $\left(say,\{{\xi }_0,{\xi }_1,\dots ,{\xi }_k\}\right)$ of some finite length k in ${\Re }^s$ from $\varsigma$ to ℘, such that

$$\varsigma ={\xi }_0,\wp ={\xi }_k,[{\xi }_i,{\xi }_{i+1}]\in \Re \mathrm{for}\mathrm{each}i(0\le i\le k-1).$$

(9)

Moreover, ℜ is T-closed, then by Proposition 4, we have

$$[T^n{\xi }_i,T^n{\xi }_{i+1}]\in \Re .$$

(10)

So, we see that the sequence $\left\{T^n{\xi }_i\right\}$ is an “ℜ-preserving” sequence. We can say that $(\varsigma ={\xi }_0,\wp ={\xi }_k)\in \Re$. Again, we see that

$\frac{1}{2}{d}_p(T^n{\xi }_i,T^n{\xi }_i)\le d_p(T^n{\xi }_i,T^n{\xi }_i)\le d_p(T^n{\xi }_i,T^n{\xi }_{i+1})$

$\frac{1}{2}{d}_p(T^n{\xi }_i,T^n{\xi }_i)\le d_p(T^n{\xi }_i,T^n{\xi }_{i+1}).$

So, by (8)–(10), triangle inequality, condition $\left(d\right)$ and Proposition 4, we have

$$\begin{array}{cc}\hfill d_p(\varsigma ,\wp )& =d_p(T^n{\xi }_0,T^n{\xi }_k)\hfill \\ \hfill & =d_p(T^n{\xi }_0,T^n{\xi }_1)+d_p(T^n{\xi }_1,T^n{\xi }_2)+\cdots +d_p(T^n{\xi }_{k-1},T^n{\xi }_k)\hfill \\ \hfill & -d_p(T^n{\xi }_1,T^n{\xi }_1)-d_p(T^n{\xi }_2,T^n{\xi }_2)-\cdots -d_p(T^n{\xi }_{k-1},T^n{\xi }_{k-1})\hfill \\ \hfill & \le d_p(T^n{\xi }_0,T^n{\xi }_1)+d_p(T^n{\xi }_1,T^n{\xi }_2)+\cdots +d_p(T^n{\xi }_{k-1},T^n{\xi }_k)\hfill \\ \hfill & \le \sum _{i=0}^{k-1}{d}_p(T^n{\xi }_i,T^n{\xi }_{i+1})\hfill \\ \hfill & \le \sum _{i=0}^{k-1}{\varphi }^n\left(d_p({\xi }_i,{\xi }_{i+1})\right)\hfill \\ \hfill & <\sum _{i=0}^{}{\varphi }^n\left(d_p({\xi }_i,{\xi }_{i+1})\right)\hfill \\ \hfill & \to 0\mathrm{as}n\to \infty .\hfill \end{array}$$

So, we have $d_p(\varsigma ,\wp )=0$. Owing to the use of a partial metric, we obtain $\varsigma =\wp$. □

Theorem 3. 

In addition to the hypotheses of Theorem 1, if we have additional conditions,

($f_1$) $T\left(\mathcal{M}\right)$ is “ℜ-connected”,

($f_2$) ℜ is “locally T-transitive”and

($f_3$) $F\left(T\right)\subset \mathcal{M}(T,\Re )$.

Then, T has a unique fixed point.

Proof. 

Let $\varsigma ,\wp \in F\left(T\right)$. Since, $T\left(\mathcal{M}\right)$ is “ℜ-connected”, there is a finite path $\{{\xi }_0,{\xi }_1,\dots ,{\xi }_{k-1}\}$ such that

$\varsigma ={\xi }_0,\wp ={\xi }_{k-1},{\xi }_i\in T\left(\mathcal{M}\right)\mathrm{and}({\xi }_i,{\xi }_{i+1})\in \Re \mathrm{for}0\le i\le k-1$.

Now, we construct a sequence as follows

${\varsigma }_1=\varsigma ={\xi }_0,{\varsigma }_2={\xi }_1,\dots ,{\varsigma }_k={\xi }_{k-1}=\wp \mathrm{and}{\varsigma }_n=\wp \mathrm{for}n\ge k+1$.

As ℘ is a fixed point of T such that $T\wp =\wp$, then by $\left(f_3\right)$, we obtain

$\wp \in \mathcal{M}(T,\Re )\mathrm{implies}(\wp ,T\wp )\in \Re \Rightarrow (\wp ,\wp )\in \Re .$

So, we see that the sequence $\left\{{\varsigma }_n\right\}$ is a “ℜ-preserving”sequence. Making use of $\left(f_2\right)$, we can say that $(\varsigma ,\wp )\in \Re$. Again, we see that

$\frac{1}{2}{d}_p(\varsigma ,\varsigma )\le d_p(\varsigma ,\varsigma )\le d_p(\varsigma ,\wp )$

$\frac{1}{2}{d}_p(\varsigma ,\varsigma )\le d_p(\varsigma ,\wp ).$

Applying the contractive condition $\left(d\right)$ of Theorem 1, we obtain,

$$\begin{array}{cc}\hfill d_p(\varsigma ,\wp )=d_p(T\varsigma ,T\wp )& \le \varphi \left(M(\varsigma ,\wp )\right)\hfill \\ \hfill & \le \varphi \left(max\left\{d_p(\varsigma ,\wp ),d_p(\varsigma ,T\varsigma ),d_p(\wp ,T\wp ),\frac{1}{2}\left[d_p(\varsigma ,T\wp )+d_p(\wp ,T\varsigma )\right]\right\}\right)\hfill \\ \hfill & \le \varphi \left(max\left\{d_p(\varsigma ,\wp ),d_p(\varsigma ,\varsigma ),d_p(\wp ,\wp )\right\}\right)\hfill \\ \hfill & \le \varphi \left(p(\varsigma ,\wp )\right)\hfill \\ \hfill & <d_p(\varsigma ,\wp )\hfill \end{array}$$

which is a contradiction. Hence, $d_p(\varsigma ,\wp )=0\Rightarrow \varsigma =\wp .$

□

Corollary 1. 

Theorem 3 remains true if “locally T-transitivity”is replaced by any one of the following conditions:

1. 

ℜ is “transitive”;

2. 

ℜ is “T-transitive”.

Corollary 2. 

If we replace the conditions $\left(f_1\right)$ and $\left(f_2\right)$ of Theorem 3 by any one of the following conditions

$\left(f^{\prime }\right)$$T\left(\mathcal{M}\right)$ is “ℜ-connected”and ℜ is “T-transitive”, and

$\left(f^{″}\right)$$T\left(\mathcal{M}\right)$ is “ℜ-connected”and ℜ is “transitive”.

Then, T has a unique fixed point.

Theorem 4. 

In addition to hypotheses of Theorem 1, if we have completeness of ℜ, then T has a unique fixed point.

Proof. 

Let $\varsigma ,\wp \in F\left(T\right)$, then we have to show $\varsigma =\wp$. Since ℜ is complete, then $(\varsigma ,\wp )\in \Re$ and, also, we have $\frac{1}{2}{d}_p(\varsigma ,\varsigma (=T\varsigma ))<d_p(\varsigma ,\wp )$. Applying the contractivity condition $\left(d\right)$ of Theorem 1, we obtain

$$\begin{array}{cc}\hfill d_p(\varsigma ,\wp )=d_p(T\varsigma ,T\wp )& \le \varphi \left(M(\varsigma ,\wp )\right)\hfill \\ \hfill & \le \varphi \left(max\left\{d_p(\varsigma ,\wp ),d_p(\varsigma ,T\varsigma ),d_p(\wp ,T\wp ),\frac{1}{2}\left[d_p(\varsigma ,T\wp )+d_p(\wp ,T\varsigma )\right]\right\}\right)\hfill \\ \hfill & \le \varphi \left(max\left\{d_p(\varsigma ,\wp ),d_p(\varsigma ,\varsigma ),d_p(\wp ,\wp )\right\}\right)\hfill \\ \hfill & \le \varphi \left(d_p(\varsigma ,\wp )\right)\hfill \\ \hfill & <d_p(\varsigma ,\wp ),\hfill \end{array}$$

which is a contradiction. Hence $d_p(\varsigma ,\wp )=0$ implies that $\varsigma =\wp$.

□

Under the universal relation (i.e., $\Re =\mathcal{M}\times \mathcal{M})$, Theorems 1 and 4 remain sharpened versions of the following results (in the form of Remarks) in the context of the PMS and Suzuki condition.

Remark 2. 

If we replace $\varphi \left(t\right)=\alpha t\mathrm{for}\alpha \in [0,1)$ and the setting $M(\varsigma ,\wp )$ to be $N(\varsigma ,\wp )$, we obtain a sharpened version of the Ćirić fixed point theorem [13].

Remark 3. 

Under the setting of $\varphi \left(t\right)=\beta t$ $(\beta \in [0,1\left)\right)$ and $M(\varsigma ,\wp )=d_p(\varsigma ,\wp )$ in Theorem 1, we obtain the sharpened version of the Banach contraction principle [1].

Remark 4. 

Under the setting of $\varphi \left(t\right)=\beta \left(2t\right)$ $(\beta \in [0,\frac{1}{2}))$ and $M(\varsigma ,\wp )=\frac{1}{2}\left(d_p(\varsigma ,T\wp )+d_p\right.$$\left.(\wp ,T\varsigma )\right)$ in Theorem 1, we obtain a version of Theorem 1, which remains an improved version of the Chatterjea fixed point theorem (see [30]).

Remark 5. 

If we replace $M(\varsigma ,\wp )$ by the condition $N(\varsigma ,\wp )=\{d_p(\varsigma ,\wp ),\frac{1}{2}[d_p(\varsigma ,T\varsigma )+d_p(\wp ,T\wp )]\}$, $\frac{1}{2}\left[d_p(\varsigma ,T\wp )+d_p(\wp ,T\varsigma )\right]\}$, we obtain a consequences of Theorem 1, which remains an improved version of Theorem 1.17 contained in Ahmadullah et al. [31].

Example 1. 

Let $\mathcal{M}=[0,1)\cap \mathbb{Q}$ endowed with a partial metric $d_p(\varsigma ,\wp )=max\{\varsigma ,\wp \}$ and a binary relation $\Re =\{(0,0),(0,\frac{1}{4}),(\frac{1}{4},0),(\frac{1}{4},\frac{1}{4}),(\frac{1}{2},0)\}.$ We see that ℜ is not “transitive”, but it is “T-transitive”. By Proposition 3, it is “locally T-transitive”. Define a self-mapping T on $\mathcal{M}$ by

$$T\left(\varsigma \right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\varsigma \in [0,\frac{1}{4}]\hfill \\ \frac{1}{4},\hfill & \mathrm{if}\varsigma \in (\frac{1}{4},1).\hfill \end{array}\right.$$

Clearly, we see that ℜ is “T-closed”, T is “ℜ-continuous” and $(\mathcal{M},d_p)$ is “ℜ-complete”. Consider a function $\varphi :[0,+\infty )\to [0,+\infty )$ by $\varphi \left(t\right)=\frac{t}{4}$, then ϕ is increasing and ${\sum }_{n=1}^{+\infty }{\varphi }^n\left(t\right)<+\infty$, and hence a member of the family Φ. The contractive condition of Theorem 1 can be easily verified for all $(\varsigma ,\wp )\in \Re$. For any ℜ-preserving sequence $\left\{{\varsigma }_n\right\}$ such that ${\varsigma }_n\stackrel{d_p}{\to }\varsigma$. As $({\varsigma }_n,{\varsigma }_{n+1})\in \Re$, for all $n\in \mathbb{N}$, there always exists $N\in \mathbb{N}$ such that ${\varsigma }_n=\varsigma \in \{0,\frac{1}{4}\}$ for all $n\ge N$. Hence, ℜ is $d_p$-self-closed. Thus, all the conditions of Theorems 1 and 4 are satisfied and, hence, T has a unique fixed point (i.e., $\varsigma =0$). This example cannot be satisfied by Theorem 1 (due to Pant [14]) because the space $(\mathcal{M},d)$ is not complete (due to Lemma 1 and $(\mathcal{M},d_p)$ is not complete) and ℜ is not a partial order.

5. Application to Fractal Space

Let $(\mathcal{M},d_p)$ be a PMS and $\mathcal{C}\left(\mathcal{M}\right)$ be the collection of all non-empty compact subsets of $\mathcal{M}$.

Define, $r_{d_p}(E,D):=inf\left\{d_p(\wp ,\varrho ):\wp \in E,\varrho \in D\right\}$.

${\rho }_{d_p}(E,D):=sup\left\{r_{d_p}(\wp ,D):\wp \in E\right\}=\underset{\wp \in E}{sup}\underset{\varrho \in D}{inf}{d}_p(\wp ,\varrho )$.

So, then ${\rho }_{d_p}(D,E):=sup\left\{r_{d_p}(\varrho ,E):\varrho \in D\right\}=\underset{\varrho \in D}{sup}\underset{\wp \in E}{inf}{d}_p(\varrho ,\wp )$.

Then, the Hausdorff metric induced by $d_p$ is defined by

$${\mathcal{H}}_{d_p}(E,D)=max\left\{\underset{\wp \in E}{sup}{r}_{d_p}(\wp ,D),\underset{\varrho \in D}{sup}{r}_{d_p}(\varrho ,E)\right\}=max\left\{{\rho }_{d_p}(E,D),{\rho }_{d_p}(D,E)\right\}\mathrm{for}\mathrm{all}E,D\in \mathcal{C}\left(\mathcal{M}\right)$$

where $d_p(\wp ,D)=\underset{\varrho \in D}{inf}{d}_p(\wp ,\varrho ).$

Hutchinson [32] and Barnsley [33] developed a brilliant method of defining and constructing fractals as compact invariant subsets of an abstract complete metric space with reference to the union of contractions $T_i(i=1,2,\dots ,n).$ Hutchinson introduced that the operator

$$\mathcal{F}=T_1\left(E\right)\cup T_2\left(E\right)\cup \cdots \cup T_n\left(E\right),E\in \mathcal{M},$$

is a contraction with reference to the Hausdorff distance. Thus, the contraction mapping principle can be applied to the iteration of Hutchinson operator $\mathcal{F}$. Consequently, whichever initial image is chosen to begin the iteration under the Iterated Function System (IFS), such as $E_0$, the generated sequence

$$E_{k+1}=\mathcal{F}\left(E_k\right)k=0,1,\dots ,$$

will tend towards a distinguish image, the attractor $E_{\infty }$ of the IFS. Moreover, this image is invariant, i.e., $\mathcal{F}\left(E_{\infty }\right)=E_{\infty }$.

Now, we construct a result in the form of Lemma 3.2 of [34] for the PMS under the Suzuki-generalized Ćirić $\varphi$-contractions.

Lemma 3. 

Let $(\mathcal{M},d_p)$ be a PMS and $T:\mathcal{M}\to \mathcal{M}$ a continuous Suzuki-generalized Ćirić ϕ-contractive mapping with reference to operator $\mathcal{F}$, i.e., $\frac{1}{2}{\mathcal{H}}_{d_p}(E,T\left(E\right))\le {\mathcal{H}}_{d_p}(E,D)\Rightarrow {\mathcal{H}}_{d_p}({\mathcal{F}}_T\left(E\right),$${\mathcal{F}}_T\left(D\right))\le \varphi \left({\mathcal{M}}_T(E,D)\right)\mathrm{for}\mathrm{all}E,D\in \mathcal{C}\left(\mathcal{M}\right)$ where, $M_T(E,D)=max\left\{{\mathcal{H}}_{d_p}\right(E,$$D),$${\mathcal{H}}_{d_p}$$(E,T(E\left)\right),$${\mathcal{H}}_{d_p}(D,T\left(D\right)),$$\frac{1}{2}[{\mathcal{H}}_{d_p}(E,T\left(D\right))+{\mathcal{H}}_{d_p}(D,T\left(E\right))]\}$. Then, ${\mathcal{F}}_T:\mathcal{C}\left(\mathcal{M}\right)\to \mathcal{C}\left(\mathcal{M}\right)$ is also a Suzuki-generalized Ćirić ϕ-contractive mapping, $\mathrm{and}{\mathcal{F}}_T\left(W\right)=T\left(W\right)\mathrm{for}\mathrm{all}W\in \mathcal{C}\left(\mathcal{M}\right)$.

Proof. 

Let $E,D\in \mathcal{C}\left(\mathcal{M}\right)$ and any point ${\wp }_0\in E$. Using the compactness of E, there always exists ${\varrho }_{{\wp }_0}\in D$ such that $\underset{\varrho \in D}{inf}{d}_p({\wp }_0,\varrho )=d_p({\wp }_0,{\varrho }_{{\wp }_0})$. Then, we have

$$\underset{\varrho \in D}{inf}\varphi \left(d_p({\wp }_0,\varrho )\right)\le \varphi \left(d_p({\wp }_0,{\varrho }_{{\wp }_0})\right)=\varphi \left(\underset{\varrho \in D}{inf}{d}_p({\wp }_0,\varrho )\right).$$

Because $\varphi :[0,+\infty )\to [0,+\infty )$ is increasing, it follows that

$$\varphi \left(\underset{\varrho \in D}{inf}{d}_p({\wp }_0,\varrho )\right)\le \varphi \left(\underset{\wp \in E}{sup}\underset{\varrho \in D}{inf}{d}_p(\wp ,\varrho )\right)\le \varphi \left({\mathcal{H}}_{d_p}(E,D)\right).$$

Since ${\wp }_0$ was arbitrary, then $\underset{\wp \in E}{sup}\varphi \left(\underset{\varrho \in D}{inf}{d}_p(\wp ,\varrho )\right)\le \varphi \left({\mathcal{H}}_{d_p}(E,D)\right)$.

Then, for all $\wp \in E$ and $\varrho \in D$, we have

$\underset{\wp \in E}{sup}\underset{\varrho \in D}{inf}\varphi \left(d_p(\wp ,\varrho )\right)\le \underset{\wp \in E}{sup}\varphi \left(\underset{\varrho \in D}{inf}{d}_p(\wp ,\varrho )\right)\le \varphi \left(M_T(E,D)\right).$

$\underset{\varrho \in D}{sup}\underset{\wp \in E}{inf}\varphi \left(d_p(\wp ,\varrho )\right)\le \underset{\varrho \in D}{sup}\varphi \left(\underset{\wp \in E}{inf}{d}_p(\wp ,\varrho )\right)\le \varphi \left(M_T(E,D)\right).$

Furthermore, for all $\wp \in E$ and $\varrho \in D$

$\frac{1}{2}{d}_p(\wp ,T\wp )\le d_p(\wp ,\varrho )\Rightarrow \frac{1}{2}{\rho }_{d_p}(E,T\left(E\right))\le \frac{1}{2}{\mathcal{H}}_{d_p}(E,T\left(E\right))\le {\mathcal{H}}_{d_p}(E,D).$

and

$\frac{1}{2}{d}_p(\varrho ,T\varrho )\le d_p(\wp ,\varrho )\Rightarrow \frac{1}{2}{\rho }_{d_p}(D,T\left(D\right))\le \frac{1}{2}{\mathcal{H}}_{d_p}(D,T\left(D\right))\le {\mathcal{H}}_{d_p}(E,D).$

Next, we have

${\rho }_{d_p}\left({\mathcal{F}}_T\left(E\right),{\mathcal{F}}_T\left(D\right)\right)=\underset{T\wp \in \mathcal{F}\left(E\right)}{sup}\underset{T\varrho \in \mathcal{F}\left(D\right)}{inf}{d}_p(T\wp ,T\varrho )=\underset{\wp \in E}{sup}\underset{\varrho \in D}{inf}{d}_p(T\wp ,T\varrho ).$

Since T is Suzuki-generalized Ćirić $\varphi$-contractive mapping, we have

$\frac{1}{2}{\rho }_{d_p}(E,T\left(E\right))\le \frac{1}{2}{\mathcal{H}}_{d_p}(E,T\left(E\right))$, then the above inequality reduces to

${\rho }_{d_p}\left({\mathcal{F}}_T\left(E\right),{\mathcal{F}}_T\left(D\right)\right)=\underset{\wp \in E}{sup}\underset{\varrho \in D}{inf}{d}_p(T\wp ,T\varrho )\le \varphi \left(M_T(E,D)\right).$

Similarly, $\frac{1}{2}{\rho }_{d_p}(D,T\left(D\right))\le \frac{1}{2}{\mathcal{H}}_{d_p}(D,T\left(D\right))$ we always have,

${\rho }_{d_p}\left({\mathcal{F}}_T\left(D\right),{\mathcal{F}}_T\left(E\right)\right)=\underset{\varrho \in D}{sup}\underset{\wp \in E}{inf}{d}_p(T\wp ,T\varrho )\le \varphi \left(M_T(E,D)\right).$

Since the Hausdorff metric is symmetric, i.e., ${\mathcal{H}}_{d_p}(E,D)={\mathcal{H}}_{d_p}(D,E)$, then we obtain

$${\mathcal{H}}_{d_p}\left({\mathcal{F}}_T\left(E\right),{\mathcal{F}}_T\left(D\right)\right)=max\left\{{\rho }_{d_p}\left({\mathcal{F}}_T\left(E\right),{\mathcal{F}}_T\left(D\right)\right),{\rho }_{d_p}\left({\mathcal{F}}_T\left(D\right),{\mathcal{F}}_T\left(E\right)\right)\right\},$$

hence, we always have

$$\frac{1}{2}{\mathcal{H}}_{d_p}(E,T\left(E\right))\le {\mathcal{H}}_{d_p}(E,D)\Rightarrow {\mathcal{H}}_{d_p}\left({\mathcal{F}}_T\left(E\right),{\mathcal{F}}_T\left(D\right)\right)\le \varphi \left(M_T(E,D)\right)$$

for all $E,D\in \mathcal{C}\left(\mathcal{M}\right)$. Therefore, ${\mathcal{F}}_T$ is a Suzuki-generalized Ćirić $\varphi$-contractive mapping. □

Lemma 4. 

Let $(\mathcal{M},d_p)$ be a complete PMS. Then, $(\mathcal{C}\left(\mathcal{M}\right),{\mathcal{H}}_{d_p})$ is also a complete PMS.

Lemma 5. 

Let $(\mathcal{M},d_p)$ be a PMS and $T_n:\mathcal{C}\left(\mathcal{M}\right)\to \mathcal{C}\left(\mathcal{M}\right)(n=1,2,\dots ,p)$ continuous Suzuki-generalized Ćirić ϕ-contractive mapping, i.e., for all $E,D\in \mathcal{C}\left(\mathcal{M}\right),$

$\frac{1}{2}{\mathcal{H}}_{d_p}(E,T_n\left(E\right))\le {\mathcal{H}}_{d_p}(E,D)\Rightarrow {\mathcal{H}}_{d_p}(T_n\left(E\right),T_n\left(D\right))\le {\varphi }_n\left(M_{T_n}(E,D)\right).$

Define $T:\mathcal{C}\left(\mathcal{M}\right)\to \mathcal{C}\left(\mathcal{M}\right)$ by $T\left(E\right)=T_1\left(E\right)\cup T_2\left(E\right)\cup \cdots \cup T_p\left(E\right)={\cup }_{n=1}^p{T}_n\left(E\right)$ for each $E\in \mathcal{C}\left(\mathcal{M}\right)$. Then, T also satisfies

$\frac{1}{2}{\mathcal{H}}_{d_p}(E,T\left(E\right))\le {\mathcal{H}}_{d_p}(E,D)\Rightarrow {\mathcal{H}}_{d_p}\left(T\left(E\right),T\left(D\right)\right)\le \lambda \left(M_T(E,D)\right)$

for all $E,D\in \mathcal{C}\left(\mathcal{M}\right)$, where $\lambda =max\{{\varphi }_n:n=1,2,\dots ,p\}$.

Proof. 

We prove the above Lemma by the principle of mathematical induction. For $n=1$, the result is obvious. For $n=2$, we have

$$\begin{array}{cc}\hfill {\mathcal{H}}_{d_p}(T\left(E\right),T\left(D\right))& ={\mathcal{H}}_{d_p}(T_1\left(E\right)\cup T_2\left(E\right),T_1\left(D\right)\cup T_2\left(D\right))\hfill \\ \hfill & \le max\left\{{\mathcal{H}}_{d_p}(T_1\left(E\right),T_1\left(D\right)),{\mathcal{H}}_{d_p}(T_2\left(E\right),T_2\left(D\right))\right\}.\hfill \end{array}$$

Since each $T_1$ and $T_2$ are Suzuki-generalized Ćirić $\varphi$-contractive, that is

$\frac{1}{2}{\mathcal{H}}_{d_p}(E,T_1\left(E\right))\le {\mathcal{H}}_{d_p}(E,D)\Rightarrow {\mathcal{H}}_{d_p}(T_1\left(E\right),T_1\left(D\right))\le {\varphi }_1\left(M_{T_1}(E,D)\right)$

$\frac{1}{2}{\mathcal{H}}_{d_p}(E,T_2\left(E\right))\le {\mathcal{H}}_{d_p}(E,D)\Rightarrow {\mathcal{H}}_{d_p}(T_2\left(E\right),T_2\left(D\right))\le {\varphi }_2\left(M_{T_2}(E,D)\right)$,

then we have

$$\begin{array}{cc}\hfill {\mathcal{H}}_{d_p}(T\left(E\right),T\left(D\right))& \le max\left\{{\varphi }_1\left(M_{T_1}(E,D)\right),{\varphi }_2\left(M_{T_2}(E,D)\right)\right\}\hfill \\ \hfill & =\lambda \left(max\right\{{\mathcal{H}}_{d_p}(E,D),{\mathcal{H}}_{d_p}(E,T_1\left(E\right)\cup T_2\left(E\right)),{\mathcal{H}}_{d_p}(D,T_1\left(D\right)\cup T_2\left(D\right)),\hfill \\ \hfill & \frac{1}{2}\left[{\mathcal{H}}_{d_p}(E,T_1\left(D\right)\cup T_2\left(D\right))+{\mathcal{H}}_{d_p}(D,T_1\left(E\right)\cup T_2\left(E\right))\right]\left\}\right)\hfill \\ \hfill & =\lambda \left(max\left\{{\mathcal{H}}_{d_p}(E,D),{\mathcal{H}}_{d_p}(E,T\left(E\right)),{\mathcal{H}}_{d_p}(D,T\left(D\right))\right\}\right)\hfill \\ \hfill & =\lambda \left(M_T(E,D)\right),\hfill \end{array}$$

where $\lambda =max\{{\varphi }_1,{\varphi }_2\}$.

□

As a consequence of Theorem 1, Lemmas 3 and 5, we have the following result in fractal spaces.

Theorem 5. 

Let $(\mathcal{M},d_p)$ be a complete PMS and $T_n:\mathcal{C}\left(\mathcal{M}\right)\to \mathcal{C}\left(\mathcal{M}\right)$ continuous Suzuki-generalized Ćirić ϕ-contractive mapping such that one of $T_n\left(E\right)\subseteq E$. Then, the transformation $T:\mathcal{C}\left(\mathcal{M}\right)\to \mathcal{C}\left(\mathcal{M}\right)$ defined by $T\left(E\right)={\cup }_{n=1}^p{T}_n\left(E\right)$ for each $E\in \mathcal{C}\left(\mathcal{M}\right)$ satisfies the following condition

$$\frac{1}{2}{\mathcal{H}}_{d_p}(E,T\left(E\right))\le {\mathcal{H}}_{d_p}(E,D)\Rightarrow {\mathcal{H}}_{d_p}\left(T\left(E\right),T\left(D\right)\right)\le \lambda \left(M(E,D)\right),$$

for all $E,D\in \mathcal{C}\left(\mathcal{M}\right)$, where $\lambda =max\{{\varphi }_n:n=1,2,\dots ,p\}$.

Moreover,

(1) T has a unique fixed point E in $\mathcal{C}\left(\mathcal{M}\right);and$

(2) $\underset{n\to \infty }{lim}{T}^n\left(D\right)=E\mathrm{for}\mathrm{all}D\in \mathcal{C}\left(\mathcal{M}\right)$.

Proof. 

Define a binary relation as

$$\Re =\left\{(E,D)\in \mathcal{C}\left(\mathcal{M}\right)\times \mathcal{C}\left(\mathcal{M}\right)\mathrm{such}\mathrm{that}E\subseteq D\right\}.$$

Then, T is well-defined and ℜ on $\mathcal{C}\left(\mathcal{M}\right)$ is T-closed. Since $T_n\left(E\right)\subseteq E$ (for some ‘n’), then $T\left(E\right)={\cup }_{n=1}^p{T}_n\left(E\right)\subseteq E$ implies that $(E,T(E\left)\right)\in \Re$, which amounts to say that $\mathcal{M}(T,\Re )$ is non-empty. In view of Lemma 5 we can say that the mapping T satisfies Suzuki-generalized Ćirić $\varphi$-contractive mapping for any $(E,D)\in \Re$. Moreover, T is ℜ-continuous being union of continuous maps. Then, by Theorem 1, we can say that T has a fixed point. Moreover, by use of Theorem 3.1 of Si Ri [34], we can say that T has a fixed point E in $\mathcal{C}\left(\mathcal{M}\right)$ and $\underset{n\to \infty }{lim}{T}^n\left(D\right)=E\mathrm{for}\mathrm{all}D\in \mathcal{C}\left(\mathcal{M}\right)$. □

Remark 6. 

If we see the results of Rhoades [35] in the setting of PMS, Theorems 1 and 5 generalize corresponding results in [32,34].

6. Conclusions

In the Theorems 1–3, we furnished the existence and uniqueness of the fixed points for the Suzuki-generalized Ćirić $\varphi$-contraction in the ℜ-complete partial metric spaces. Moreover, we derived some known results (as corollaries). An example is constructed to demonstrate the worth of our newly proved results. Moreover, an application is provided for multivalued fractals using the Suzuki-generalized Ćirić $\varphi$-contraction. One can prove these results in different spaces, such as metric, b-metric, quasi-metric space, etc., or for different classes of control functions, such as Matkowski and the subclass of Byod and Wong besides employing a weaker form of transitivity.