On Some New Fixed Point Results in Complete Extended b-Metric Spaces

Abstract

In this paper, we specified a method that generalizes a number of fixed point results for single and multi-valued mappings in the structure of extended b-metric spaces. Our results extend several existing ones including the results of Aleksic et al. for single-valued mappings and the results of Nadler and Miculescu et al. for multi-valued mappings. Moreover, an example is given at the end to show the superiority of our results.

1. Introduction and Preliminaries

Banach contraction principle [1] is a fundamental tool for providing the existence of solutions for many mathematical problems involving differential equations and integral equations. A mapping $T:\mathbf{U}\to \mathbf{U}$ on a metric space $(\mathbf{U},d)$ is called a contraction mapping, if there exists $\eta <1$ such that for all $u,v\in \mathbf{U},$

$$d(Tu,Tv)\le \eta d(u,v).$$

(1)

If the metric space is complete and T satisfies inequality (1), then T has a unique fixed point. Clearly, inequality (1) implies continuity of $T.$ Naturally, a question arises as to whether we can find contractive conditions which will imply the existence of fixed points in a complete metric space, but will not imply continuity. In [2], Kannan derived the following result, which answers the said question. Let $T:\mathbf{U}\to \mathbf{U}$ be a mapping on a complete metric space $(\mathbf{U},d)$, which satisfies inequality:

$$d(Tu,Tv)\le \eta \left[d\right(u,Tu)+d(v,Tv\left)\right],$$

(2)

where $\eta \in [0,\frac{1}{2})$ and $u,v\in \mathbf{U}.$ The mapping satisfying inequality (2) is called a Kannan type mapping. There are number of generalizations of the contraction principle of Banach both for single-valued and multi-valued mappings, see ([3,4,5,6,7,8,9,10,11,12,13]). Chatterjea in [14] established the following alike co ntractive condition. Let $(\mathbf{U},d)$ be a complete metric space. A mapping $T:\mathbf{U}\to \mathbf{U}$ has a unique fixed point, if it satisfies the following inequality:

$$d(Tu,Tv)\le \eta \left[d\right(u,Tv)+d(v,Tu\left)\right].$$

(3)

where $\eta \in [0,\frac{1}{2})$ and $u,v\in \mathbf{U}.$ The mapping satisfying inequality (3) is called a Chatterjea type mapping.

Due to the problem of the convergence of measurable functions with respect to a measure, Bakhtin [15], Bourbaki [16], and Czerwik [17,18] introduced the concept of b-metric spaces by weakening the triangle inequality of the metric space as follows:

Definition 1

([17]). Let $\mathbf{U}$ be a set and $s\ge 1$ a real number. A function $d:\mathbf{U}\times \mathbf{U}\to [0,\infty )$ is called a b-metric space, if it satisfies the following axioms for all $u_1,u_2,u_3\in \mathbf{U}:$

(1)

$d(u_1,u_2)=0$ if and only if $u_1=u_2;$

(2)

$d(u_1,u_2)=d(u_2,u_1);$

(3)

$d(u_1,u_3)\le s[d(u_1,u_2)+d(u_2,u_3)]$.

The pair $(\mathbf{U},d)$ is called a b-metric space.

Clearly, every metric space is a b-metric space with $s=1$, but its converse is not true in general. After that, a number of research papers have been established that generalized the Banach fixed point result in the framework of b-metric spaces. In [19], Kir and Kiziltunc introduced the following results, which generalized Kannan and Chatterjea type mappings in b-metric spaces. Let $T:\mathbf{U}\to \mathbf{U}$ be a mapping on a complete b-metric space $(\mathbf{U},d)$, which satisfies inequality:

$$d(Tu,Tv)\le \eta \left[d\right(u,Tu)+d(v,Tv\left)\right].$$

(4)

where $s\eta \in [0,\frac{1}{2})$ and $u,v\in \mathbf{U}.$ Then T has a unique fixed point.

Let $(\mathbf{U},d)$ be a complete b-metric space. A mapping $T:\mathbf{U}\to \mathbf{U}$ has a unique fixed point in $\mathbf{U}$, if it satisfies the following inequality:

$$d(Tu,Tv)\le \eta \left[d\right(u,Tv)+d(v,Tu\left)\right],$$

(5)

for all $u,v\in \mathbf{U},$ where $\eta \in [0,\frac{1}{2}).$ In [20], the given below results, which generalized Equation (4) for ${\kappa }_1={\kappa }_2={\kappa }_3=0$ and (5) for ${\kappa }_1={\kappa }_4=0$ and ${\kappa }_2={\kappa }_3$, have been derived.

Theorem 1

([20]). Let $(\mathbf{U},d)$ be a complete b-metric space with constant $s\ge 1$. If $T:\mathbf{U}\to \mathbf{U}$ satisfies the inequality:

$$d(Tu,Tv)\le {\kappa }_{1}d(u,v)+{\kappa }_{2}d(u,Tu)+{\kappa }_{3}d(v,Tv)+{\kappa }_4[d(v,Tu)+d(u,Tv)],$$

(6)

where,

$${\kappa }_1+2s{\kappa }_2+{\kappa }_3+2s{\kappa }_4<1,$$

then T has a unique fixed point.

Theorem 2

([20]). Let $(\mathbf{U},d)$ be a complete b-metric space with constant $s\ge 1$. If $T:\mathbf{U}\to \mathbf{U}$ satisfies the inequality:

$$d(Tu,Tv)\le {\kappa }_1{d}_{\varphi }(u,v)+{\kappa }_2[d_{\varphi }(u,Tu)+d_{\varphi }(v,Tv)],$$

(7)

for all $u,v\in \mathbf{U},$ where ${\kappa }_1,{\kappa }_2\in [0,\frac{1}{3})$, then T has a unique fixed point.

In [21], Koleva and Zlatanov proved the following result, which generalizes Chatterjea’s type mappings in b-metric spaces and do not involve the b-metric constant.

Theorem 3

([21]). Let $(\mathbf{U},d)$ be a complete b-metric space and d be a continuous function. If $T:\mathbf{U}\to \mathbf{U}$ is a Chatterjea’s mapping, i.e., it satisfies inequality (3) such that ${\displaystyle \underset{n\in \mathbb{N}}{sup}\left\{d(T^{n}u,u)\right\}<\infty }$ holds for every $u\in \mathbf{U}.$ Then:

(i)

There exists a unique fixed point of T, say ξ;

(ii)

For any $u_0\in \mathbf{U}$, the sequence ${\left\{u_n\right\}}_{n=1}^{\infty }$ converges to ξ, where $u_{n+1}=T^n{u}_n$, $n=0,1,2,\dots ;$

(iii)

There holds the priori error estimate.

$$d(\xi ,T^{m}u)\le {\left(\frac{\eta }{1-\eta }\right)}^m\underset{j\in \mathbb{N}}{sup}\left\{d(T^{j}u,u)\right\},$$

where $\eta \in [0,\frac{1}{2}).$

Ilchev and Zlatanov in [22] proved the following result generalizing Theorem 3 for ${\kappa }_1=0.$

Theorem 4

([22]). Let $(\mathbf{U},d)$ be a complete b-metric space and d be a continuous function. If,

(1)

$T:\mathbf{U}\to \mathbf{U}$ is a Reich mapping, i.e., there exist ${\kappa }_1,{\kappa }_2\ge 0$, such that ${\kappa }_1+2{\kappa }_2<1,$ so that the inequality

$$d(Tu,Tv)\le {\kappa }_1{d}_{\varphi }(u,v)+{\kappa }_2[d(u,Tv)+d(v,Tu)],$$

(8)

holds for every $u,v\in \mathbf{U};$

(2)

the inequality ${\displaystyle \underset{n\in \mathbb{N}}{sup}\left\{d(T^{n}u,u)\right\}<\infty }$ holds for every $u\in \mathbf{U},$

then:

(i)

There exists a unique fixed point of T, say ξ;

(ii)

For any $u_0\in \mathbf{U}$, the sequence ${\left\{u_n\right\}}_{n=1}^{\infty }$ converges to ξ, where $u_{n+1}=T^n{u}_n$, $n=0,1,2,\dots ;$

(iii)

There holds the priori error estimate.

$$d(\xi ,T^{m}u)\le {\left(\frac{{\kappa }_1+{\kappa }_2}{1-{\kappa }_2}\right)}^m\underset{j\in \mathbb{N}}{sup}\left\{d(T^{j}u,u)\right\}.$$

In [23], the author introduced the following results, which improve Theorems 1 and 2 of [20].

Theorem 5

([23]). Let $(\mathbf{U},d)$ be a complete b-metric space with a constant $s\ge 1$. If $T:\mathbf{U}\to \mathbf{U}$ satisfies the inequality:

$$d(Tu,Tv)\le {\kappa }_{1}d(u,v)+{\kappa }_{2}d(u,Tu)+{\kappa }_{3}d(v,Tv)+{\kappa }_4[d(v,Tu)+d(u,Tv)],$$

(9)

where ${\kappa }_i\ge 0,$ for $i=1,2,3,4$ and

$${\kappa }_1+{\kappa }_2+{\kappa }_3+2s{\kappa }_4<1,$$

then T has a unique fixed point.

Theorem 6

([23]). Let $(\mathbf{U},d)$ be a complete b-metric space with a constant $s\ge 1$. If $T:\mathbf{U}\to \mathbf{U}$ satisfies the inequality:

$$d(Tu,Tv)\le {\kappa }_{1}d(u,v)+{\kappa }_2[d(u,Tu)+d(v,Tv)],$$

(10)

for all $u,v\in \mathbf{U},$ where ${\kappa }_1,{\kappa }_2\in [0,\frac{1}{3})$ such that ${\kappa }_2<min\{\frac{1}{3},\frac{1}{s}\},$ then T has a unique fixed point.

If $s=1$, then $(\mathbf{U},d)$ is a metric space and condition (9) implies:

$$d(Tu,Tv)\le kmax\{d(u,v),d(u,Tu),d(v,Tv),\frac{d(v,Tu)+d(u,Tv)}{2}\},$$

(11)

where ${\kappa }_1+{\kappa }_2+{\kappa }_3+2{\kappa }_4<1.$ With Equation (11), we recover the well-known result for generalized Ciric’s contraction mapping in the metric space and obtain a unique fixed point.

In 1969, Nadler [24] generalized the single-valued Banach contraction principle into a multi-valued contraction principle. This mapping has been carried out for a complete metric space $(\mathbf{U},d)$ by using subsets of $\mathbf{U}$ that are nonempty closed and bounded. There are number of generalizations for Nadler’s fixed point theorem (see [25,26,27]). In [28], the author introduced the given below quasi-contraction mapping and proved an existence and uniqueness fixed point theorem.

A mapping $T:\mathbf{U}\to \mathbf{U}$ on a metric space $(\mathbf{U},d)$ is called a quasi-contraction, if there exists $q<1$ such that for all $u,v\in \mathbf{U},$

$$d(Tu,Tv)\le qmax\left\{d\right(u,v),d(u,Tu),d(v,Tv),d(u,Tv),d(v,Tu\left)\right\}.$$

Amini-Harandi in [29] introduced the concept of q-multi-valued quasi-contractions and derived a fixed point theorem, which generalized Ciric’s theorem [28].

A multi-valued map $T:\mathbf{U}\to \mathcal{CB}\left(\mathbf{U}\right)$ on a metric space $(\mathbf{U},d)$ is called a q-multi-valued quasi-contraction, if there exists $q<1$ such that for all $u,v\in \mathbf{U},$

$$d(Tu,Tv)\le qmax\left\{d\right(u,v),d(u,Tu),d(v,Tv),d(u,Tv),d(v,Tu\left)\right\},$$

where $\mathcal{CB}\left(\mathbf{U}\right)$ denotes the non-empty closed and bounded subsets of $\mathbf{U}.$ In [30], Aydi et al. established the following result, which generalized Theorem 2.2 from [29] and Ciric’s result [28].

Theorem 7

([30]). Let $(\mathbf{U},d)$ be a complete b-metric space. Suppose that T is a q-multi-valued quasi-contraction and $q<\frac{1}{s^2+s}$, then T has a fixed point in $\mathbf{U}.$

In 2017, Kamran et al. generalized the structure of a b-metric space and called it, an extended b-metric space. Thereafter, a number of research articles have appeared, which generalize the contraction principle of Banach in extended b-metric spaces for both single and multi-valued mappings (see [31,32,33,34,35,36,37]). In this paper, we illustrate a method (see Lemma 3), to generalize a number of fixed point results of single-valued and multi-valued mappings in the structure of extended b-metric spaces.

Definition 2

([38]). Let $\mathbf{U}$ be a nonempty set and $\varphi :\mathbf{U}\times \mathbf{U}\to [1,\infty ).$ A function $d_{\varphi }:\mathbf{U}\times \mathbf{U}\to [0,\infty )$ is called an extended b-metric, if for all $u_1,u_2,u_3\in \mathbf{U}$, it satisfies:

$\left(d_1\right)$

$d_{\varphi }(u_1,u_2)=0$ iff $u_1=u_2;$

$\left(d_2\right)$

$d_{\varphi }(u_1,u_2)=d_{\varphi }(u_2,u_1);$

$\left(d_3\right)$

$d_{\varphi }(u_1,u_3)\le \varphi (u_1,u_3)[d_{\varphi }(u_1,u_2)+d_{\varphi }(u_2,u_3)]$.

The pair $(\mathbf{U},d_{\varphi })$ is called an extended b-metric space.

Example 1.

Let $\mathbf{U}=[0,\infty ).$ Define $d_{\varphi }:\mathbf{U}\times \mathbf{U}\to [0,\infty )$ by:

$$d_{\varphi }(u,v)=\left\{\begin{array}{cc}0,\hfill & ifu=v;\hfill \\ 3,\hfill & ifuorv\in \{1,2\},u\ne v;\hfill \\ 5,\hfill & ifu\ne v\in \{1,2\};\hfill \\ 1,\hfill & otherwise.\hfill \end{array}\right.$$

Then $(\mathbf{U},d_{\varphi })$ is an extended b-metric space, where $\varphi :\mathbf{U}\times \mathbf{U}\to [1,\infty )$ is defined by:

$$\varphi (u,v)=u+v+1,$$

for all $u,v\in \mathbf{U}.$

Remark 1.

Every b-metric space is an extended b-metric space with constant function $\varphi (u_1,u_2)=s$, for $s\ge 1$, but its converse is not true in general.

Definition 3

([35]). Let $(\mathbf{U},d_{\varphi })$ be an extended b-metric space, where $\varphi :\mathbf{U}\times \mathbf{U}\to [1,\infty )$ is bounded. Then for all $\mathbf{A},\mathbf{B}\in \mathcal{CB}\left(\mathbf{U}\right)$, where $\mathcal{CB}\left(\mathbf{U}\right)$ denotes the family of all nonempty closed and bounded subsets of $\mathbf{U}$, the Hausdorff–Pompieu metric on $\mathcal{CB}\left(\mathbf{U}\right)$ induced by $d_{\varphi }$ is defined by:

$$H_{\mathsf{\Phi }}(\mathbf{A},\mathbf{B})=max\{\underset{a\in \mathbf{A}}{sup}{d}_{\varphi }(a,\mathbf{B}),\underset{b\in \mathbf{B}}{sup}{d}_{\varphi }(b,\mathbf{A})\},$$

where for every $a\in \mathbf{A}$, $d_{\varphi }(a,\mathbf{B})=inf\{d_{\varphi }(a,b):b\in \mathbf{B}\}$ and $\mathsf{\Phi }:\mathcal{CB}\left(\mathbf{U}\right)\times \mathcal{CB}\left(\mathbf{U}\right)\to [1,\infty )$ is such that:

$$\mathsf{\Phi }(\mathbf{A},\mathbf{B})=sup\left\{\varphi \right(a,b):a\in \mathbf{A},b\in \mathbf{B}\}.$$

Theorem 8

([31]). Let $(\mathbf{U},d_{\varphi })$ be an extended b-metric space. Then $(\mathcal{CB}\left(\mathbf{U}\right),H_{\mathsf{\Phi }})$ is an extended Hausdorff–Pompieu b-metric space.

Lemma 1

([39]). Every sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ of elements from an extended b-metric space $(\mathbf{U},d_{\varphi })$, having the property that for every $n\in \mathbb{N}$, there exists $\gamma \in [0,1)$ such that:

$$d_{\varphi }(u_{n+1},u_n)\le \gamma d_{\varphi }(u_n,u_{n-1})$$

(12)

where for each $u_0\in \mathbf{U}$, ${lim}_{n,m\to \infty }\varphi (u_n,u_m)<\frac{1}{\gamma }$. Then ${\left\{u_n\right\}}_{n=0}^{\infty }$ is a Cauchy sequence.

Definition 4.

Let $\mathbf{U}$ be any set and $T:\mathbf{U}\to \mathcal{CB}\left(\mathbf{U}\right)$ be a multi-valued map. For any point $u_0\in \mathbf{U}$, the sequence ${\left\{u_n\right\}}_{n=0}^{\infty }$ given by:

$$u_{n+1}\in T{u}_n,n=0,1,2,\dots$$

(13)

is called an iterative sequence with initial point $u_0.$

2. Main Results

Definition 5.

Let $(\mathbf{U},d_{\varphi })$ be an extended b-metric space. A function $T:\mathbf{U}\to \mathcal{CB}\left(\mathbf{U}\right)$ is called continuous, if for every sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ and ${\left\{v_n\right\}}_{n\in \mathbb{N}}$ belongs to $\mathbf{U}$ and $u,v\in \mathbf{U}$ such that ${lim}_{n\to \infty }{u}_n=u$, ${lim}_{n\to \infty }{v}_n=v$ and $v_n\in T{u}_n.$ We have $v\in Tu.$

Definition 6.

An extended b-metric space $(\mathbf{U},d_{\varphi })$ is called ∗-continuous, if for every $A\in \mathcal{CB}\left(\mathbf{U}\right)$, ${\left\{u_n\right\}}_{n\in \mathbb{N}}\in \mathbf{U}$ and $u\in \mathbf{U}$ such that ${lim}_{n\to \infty }{u}_n=u$. We have ${lim}_{n\to \infty }{d}_{\varphi }(u_n,A)=d_{\varphi }(u,A).$

Remark 2.

Note that ∗- continuity of $d_{\varphi }$ is stronger than continuity of $d_{\varphi }$ in first variable.

Lemma 2.

For every sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ of elements from an extended b-metric space $(\mathbf{U},d_{\varphi })$, the inequality

$$d_{\varphi }(u_0,u_k)\le \sum _{i=0}^{k-1}{d}_{\varphi }(u_i,u_{i+1})\prod _{l=0}^i\varphi (u_l,u_k),$$

(14)

is valid for every $k\in \mathbb{N}.$

Proof. 

From the triangle inequality for $k>0$, we haveL

$$\begin{array}{ccc}\hfill d_{\varphi }(u_0,u_k)& \le & \varphi (u_0,u_k)d_{\varphi }(u_0,u_1)+\varphi (u_0,u_k)\varphi (u_1,u_k)d_{\varphi }(u_1,u_2)\hfill \\ & & +\cdots +\varphi (u_0,u_k)\varphi (u_1,u_k)\dots \varphi (u_{k-1},u_k)d_{\varphi }(u_{k-1},u_k).\hfill \end{array}$$

This implies that:

$$d_{\varphi }(u_0,u_k)\le \sum _{i=0}^{k-1}{d}_{\varphi }(u_i,u_{i+1})\prod _{l=0}^i\varphi (u_l,u_k).$$

□

Lemma 3.

Every sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ of elements from an extended b-metric space $(\mathbf{U},d_{\varphi })$, having the property that there exists $\gamma \in [0,1)$ such that:

$$d_{\varphi }(u_{n+1},u_n)\le \gamma d_{\varphi }(u_n,u_{n-1})$$

(15)

for every $n\in \mathbb{N}$ is Cauchy.

Proof. 

First, by successively applying (15), we get:

$$d_{\varphi }(u_n,u_{n+1})\le {\gamma }^n{d}_{\varphi }(u_0,u_1),$$

(16)

for every $n\in \mathbb{N}.$ Then by the Lemma 3, for all $m,k\in \mathbb{N}$, we have:

$$\begin{array}{c}\hfill d_{\varphi }(u_m,u_{m+k})\le \sum _{n=m}^{m+k-1}{d}_{\varphi }(u_n,u_{n+1})\prod _{l=0}^n\varphi (u_l,u_{m+k})\end{array}$$

$$\begin{array}{c}\hfill d_{\varphi }(u_m,u_{m+k})\le d_{\varphi }(u_0,u_1)\sum _{n=m}^{m+k-1}{\gamma }^n\prod _{l=0}^n\varphi (u_l,u_{m+k})\end{array}$$

$$\begin{array}{c}\hfill d_{\varphi }(u_m,u_{m+k})\le d_{\varphi }(u_0,u_1)\sum _{n=0}^{k-1}{\gamma }^{n+m}\prod _{l=0}^{n+m}\varphi (u_l,u_{m+k})\end{array}$$

$$\begin{array}{c}\hfill d_{\varphi }(u_m,u_{m+k})\le {\gamma }^m{d}_{\varphi }(u_0,u_1)\sum _{n=0}^{k-1}{\gamma }^n\prod _{l=0}^{n+m}\varphi (u_l,u_{m+k})\end{array}$$

$$d_{\varphi }(u_m,u_{m+k})\le {\gamma }^m{d}_{\varphi }(u_0,u_1)\sum _{n=0}^{k-1}{\gamma }^{{log}_{\gamma }{\prod }_{l=0}^{n+m}\varphi (u_l,u_{m+k})+n}.$$

(17)

Now let us take two cases for ${log}_{\gamma }{\prod }_{l=0}^{n+m}\varphi (u_l,u_{m+k})+n.$

Case 1:

If ${\prod }_{l=0}^{n+m}\varphi (u_l,u_{m+k})$ is finite, let us say $M,$ then ${lim}_{n\to \infty }{log}_{\gamma }M+n=\infty .$ Hence the series ${\sum }_{n=0}^{k-1}{\gamma }^{{log}_{\gamma }M+n}$ is convergent.

Case 2:

If ${\prod }_{l=0}^{n+m}\varphi (u_l,u_{m+k})$ is infinite, then ${lim}_{n\to \infty }{log}_{\gamma }{\prod }_{l=0}^{n+m}\varphi (u_l,u_{m+k})=\infty ,$ so there exist $n_0\in \mathbb{N}$ such that ${log}_{\gamma }{\prod }_{l=0}^{n+m}\varphi (u_l,u_{m+k})>M$, i.e.,

$${\gamma }^{{log}_{\gamma }{\prod }_{l=0}^{n+m}\varphi (u_l,u_{m+k})+n}\le {\gamma }^M·{\gamma }^n,foreachn\in \mathbb{N},n\ge n_0.$$

Hence the series ${\sum }_{n=0}^{k-1}{\gamma }^{{log}_{\gamma }{\prod }_{l=0}^{n+m}\varphi (u_l,u_{m+k})+n}$ is convergent. In both cases denoting by S the sum of this series, we come to the conclusion that:

$$d_{\varphi }(u_m,u_{m+k})\le {\gamma }^m{d}_{\varphi }(u_0,u_1)S,$$

for all $m,k\in \mathbb{N}.$ Consequently, as ${\displaystyle \underset{m\to \infty }{lim}{\gamma }^m=0,}$ we conclude that ${\left\{u_m\right\}}_{m\in \mathbb{N}}$ is a Cauchy sequence. □

Remark 3.

Lemma 3 shows that the condition on ϕ in Lemma 1 corresponding to that for each $u_0\in \mathbf{U}$, ${\displaystyle \underset{n,m\to \infty }{lim}\varphi (u_n,u_m)<\frac{1}{\gamma }}$, can be avoided. Therefore, Lemma 3 generalizes Lemma 1, which is the basis of the results from [36].

Lemma 4.

Let $\mathbf{A},\mathbf{B}\in \mathcal{CB}\left(\mathbf{U}\right)$, then for every $\eta >0$ and $b\in \mathbf{B}$ there exists $a\in \mathbf{A}$ such that:

$$d_{\varphi }(a,b)\le H_{\mathsf{\Phi }}(\mathbf{A},\mathbf{B})+\eta .$$

(18)

Proof. 

By definition of Hausdorff metric, for $\mathbf{A},\mathbf{B}\in \mathcal{CB}\left(\mathbf{U}\right)$ and for any $b\in \mathbf{Y}$, we have:

$$d_{\varphi }(\mathbf{A},b)\le H_{\mathsf{\Phi }}(\mathbf{A},\mathbf{B}).$$

By the definition of infimum, we can let $\left\{a_n\right\}$ be a sequence in $\mathbf{A}$ such that:

$$d_{\varphi }(b,a_n)<d_{\varphi }(b,\mathbf{A})+\eta ,where\eta >0.$$

(19)

We know that $\mathbf{A}$ is closed and bounded, so there exists $a\in \mathbf{A}$ such that $a_n\to a$. Therefore, by (19), we have:

$$\begin{array}{c}\hfill d_{\varphi }(a,b)<d_{\varphi }(\mathbf{A},b)+\eta \le H_{\mathsf{\Phi }}(\mathbf{A},\mathbf{B})+\eta .\end{array}$$

□

Theorem 9.

Let $(\mathbf{U},d_{\varphi })$ be a complete extended b-metric space with $\varphi :\mathbf{U}\times \mathbf{U}\to [1,\infty )$. If $T:\mathbf{U}\to \mathbf{U}$ satisfies the inequality:

$$d_{\varphi }(Tu,Tv)\le {\kappa }_1{d}_{\varphi }(u,v)+{\kappa }_2{d}_{\varphi }(u,Tu)+{\kappa }_3{d}_{\varphi }(v,Tv)+{\kappa }_4[d_{\varphi }(v,Tu)+d_{\varphi }(u,Tv)],$$

(20)

where ${\kappa }_i\ge 0$, for $i=1,\dots ,4$ and for each $u_0\in \mathbf{U}$,

$${\displaystyle {\kappa }_1+{\kappa }_2+{\kappa }_3+2{\kappa }_4\underset{n,m\to \infty }{lim}\varphi (u_n,u_m)<1,}$$

then T has a fixed point.

Proof. 

Let us choose an arbitrary $u_0\in \mathbf{U}$ and define the iterative sequence ${\left\{u_n\right\}}_{n=0}^{\infty }$ by $u_n=T{u}_{n-1}=T^{n-1}{u}_0$ for all $n\ge 1.$ If $u_n=u_{n-1},$ then $u_n$ is a fixed point of T and the proof holds. So we suppose $u_n\ne u_{n-1}$, ∀ $n\ge 1.$ Then from Equation (20), we have:

$$\begin{array}{ccc}\hfill d_{\varphi }(T{u}_n,T{u}_{n-1})& \le & {\kappa }_1{d}_{\varphi }(u_n,u_{n-1})+{\kappa }_2{d}_{\varphi }(u_n,T{u}_n)+{\kappa }_3{d}_{\varphi }(u_{n-1},T{u}_{n-1})\hfill \\ & & +{\kappa }_4[d_{\varphi }(u_{n-1},T{u}_n)+d_{\varphi }(u_n,T{u}_{n-1})].\hfill \end{array}$$

From the triangle inequality, we get:

$$\begin{array}{ccc}\hfill d_{\varphi }(T{u}_n,T{u}_{n-1})& \le & {\kappa }_1{d}_{\varphi }(u_n,u_{n-1})+{\kappa }_2{d}_{\varphi }(u_n,T{u}_n)+{\kappa }_3{d}_{\varphi }(u_{n-1},T{u}_{n-1})\hfill \\ & & +{\kappa }_4\varphi (u_{n-1},u_{n+1})[d_{\varphi }(u_{n-1},u_n)+d_{\varphi }(u_n,u_{n+1})].\hfill \end{array}$$

This implies that:

$$\begin{array}{ccc}\hfill d_{\varphi }(u_{n+1},u_n)& \le & ({\kappa }_1+{\kappa }_3+{\kappa }_4\varphi (u_{n-1},u_{n+1}))d_{\varphi }(u_n,u_{n-1})\hfill \\ & & +({\kappa }_2+{\kappa }_4\varphi (u_{n-1},u_{n+1}))d_{\varphi }(u_n,u_{n+1}).\hfill \end{array}$$

(21)

Similarly,

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_{n+1})& \le & ({\kappa }_1+{\kappa }_2+{\kappa }_4\varphi (u_{n-1},u_{n+1}))d_{\varphi }(u_n,u_{n-1})\hfill \\ & & +({\kappa }_3+{\kappa }_4\varphi (u_{n-1},u_{n+1}))d_{\varphi }(u_n,u_{n+1}).\hfill \end{array}$$

(22)

By adding Equations (21) and (22), we get:

$$d_{\varphi }(u_{n+1},u_n)\le \eta d_{\varphi }(u_n,u_{n-1}).$$

(23)

where,

$$\eta =\frac{2{\kappa }_1+{\kappa }_2+{\kappa }_3+2{\kappa }_4\varphi (u_{n-1},u_{n+1})}{2-{\kappa }_2-{\kappa }_3-2{\kappa }_4\varphi (u_{n-1},u_{n+1})}.$$

Since ${\kappa }_1+{\kappa }_2+{\kappa }_3+2{\kappa }_4{lim}_{n,m\to \infty }\varphi (u_n,u_m)<1$, multiply by 2,

$$2{\kappa }_1+2{\kappa }_2+2{\kappa }_3+4{\kappa }_4\underset{n,m\to \infty }{lim}\varphi (u_n,u_m)<2,$$

$$2{\kappa }_1+2{\kappa }_2+2{\kappa }_3+(2{\kappa }_4\underset{n,m\to \infty }{lim}\varphi (u_n,u_m)+2{\kappa }_4\underset{n,m\to \infty }{lim}\varphi (u_n,u_m))<2.$$

This implies that:

$$2{\kappa }_1+{\kappa }_2+{\kappa }_3+2{\kappa }_4\underset{n,m\to \infty }{lim}\varphi (u_n,u_m)<2-{\kappa }_2-{\kappa }_3-2{\kappa }_4\underset{n,m\to \infty }{lim}\varphi (u_n,u_m).$$

⇒ $\eta <1.$ Hence from Lemma 3, ${\left\{u_n\right\}}_{n=0}^{\infty }$ is a Cauchy sequence. As $\mathbf{U}$ is complete, therefore there exists $u\in \mathbf{U}$ such that ${\displaystyle \underset{n\to \infty }{lim}{u}_n=u.}$ Next, we will show that u is a fixed point of T. From the triangle inequality and Equation (20), we have:

$$\begin{array}{ccc}\hfill d_{\varphi }(u,Tu)& \le & \varphi (u,Tu)[d_{\varphi }(u,u_{n+1})+d_{\varphi }(u_{n+1},Tu)]\hfill \\ & \le & \varphi (u,Tu)[d_{\varphi }(u,u_{n+1})+{\kappa }_1{d}_{\varphi }(u_n,u)+{\kappa }_2{d}_{\varphi }(u_n,u_{n+1})\hfill \\ & & +{\kappa }_3{d}_{\varphi }(u,Tu)+{\kappa }_4[d_{\varphi }(u_n,Tu)+d_{\varphi }(u,u_{n+1})]\hfill \\ & \le & \varphi (u,Tu)[d_{\varphi }(u,u_{n+1})+{\kappa }_1{d}_{\varphi }(u_n,u)+{\kappa }_2{d}_{\varphi }(u_n,u_{n+1})\hfill \\ & & +{\kappa }_3{d}_{\varphi }(u,Tu)+{\kappa }_4{d}_{\varphi }(u,u_{n+1})+{\kappa }_4\varphi (u_n,Tu)\hfill \\ & & [d_{\varphi }(u_n,u)+d_{\varphi }(u,Tu)]\hfill \\ & \le & \varphi (u,Tu)[(1+{\kappa }_4)d_{\varphi }(u,u_{n+1})+({\kappa }_1+{\kappa }_4\varphi (u_n,Tu))d_{\varphi }(u,u_n)\hfill \\ & & {\kappa }_2{d}_{\varphi }(u_n,u_{n+1})+({\kappa }_3+{\kappa }_4\varphi (u_n,Tu))d_{\varphi }(u,Tu)]..\hfill \end{array}$$

So,

$$\begin{array}{cc}\hfill (1-{\kappa }_3-{\kappa }_4\varphi (u_n,Tu))d_{\varphi }(u,Tu)\le & \varphi (u,Tu)[(1+{\kappa }_4)d_{\varphi }(u,u_{n+1})+({\kappa }_1+{\kappa }_4\varphi (u_n,Tu))d_{\varphi }(u,u_n)\hfill \\ & +{\kappa }_2{d}_{\varphi }(u_n,u_{n+1})].\hfill \end{array}$$

(24)

Similarly,

$$\begin{array}{cc}\hfill (1-{\kappa }_2-{\kappa }_4\varphi (u_n,Tu))d_{\varphi }(u,Tu)\le & \varphi (u,Tu)[(1+{\kappa }_4)d_{\varphi }(u,u_{n+1})+({\kappa }_1+{\kappa }_4\varphi (u_n,Tu))d_{\varphi }(u,u_n)\hfill \\ & +{\kappa }_3{d}_{\varphi }(u_n,u_{n+1})].\hfill \end{array}$$

(25)

By adding Equations (24) and (25), we have:

$$\begin{array}{cc}\hfill (2-{\kappa }_2-{\kappa }_3-2{\kappa }_4\varphi (u_n,Tu))d_{\varphi }(u,Tu)& \le \varphi (u,Tu)[2(1+{\kappa }_4)d_{\varphi }(u,u_{n+1})+2({\kappa }_1+{\kappa }_4\varphi (u_n,Tu))d_{\varphi }(u,u_n)\hfill \\ & +({\kappa }_2+{\kappa }_3)d_{\varphi }(u_n,u_{n+1})]\to 0,\hfill \end{array}$$

as $n\to \infty .$ This implies that:

$$(2-{\kappa }_2-{\kappa }_3-2{\kappa }_4\varphi (u_n,Tu))d_{\varphi }(u,Tu)\le 0.$$

Since $(2-{\kappa }_2-{\kappa }_3-2{\kappa }_4\varphi (u_n,Tu))>0$, we get $d_{\varphi }(u,Tu)=0,$ i.e., $Tu=u.$ Now, we show that u is the unique fixed point of T. Assume that $u^{\prime }$ is another fixed point of T, then we have $T{u}^{\prime }=u^{\prime }$. Also,

$$\begin{array}{ccc}\hfill d_{\varphi }(u,u^{\prime })& =& d_{\varphi }(Tu,T{u}^{\prime })\hfill \\ & \le & {\kappa }_1{d}_{\varphi }(u,u^{\prime })+{\kappa }_2{d}_{\varphi }(u,T{u}^{\prime })+{\kappa }_3{d}_{\varphi }(u^{\prime },Tu)+{\kappa }_4[d_{\varphi }(u,T{u}^{\prime })+d_{\varphi }(u^{\prime },Tu)\hfill \\ & \le & {\kappa }_1{d}_{\varphi }(u,u^{\prime })+{\kappa }_2{d}_{\varphi }(u,u^{\prime })+{\kappa }_3{d}_{\varphi }(u^{\prime },u)+{\kappa }_4[d_{\varphi }(u,u^{\prime })+d_{\varphi }(u^{\prime },u)\hfill \\ & \le & ({\kappa }_1+2{\kappa }_4)d_{\varphi }(u,u^{\prime }).\hfill \end{array}$$

This implies that:

$$(1-{\kappa }_1-2{\kappa }_4)d_{\varphi }(u,u^{\prime })\le 0.$$

As ${\displaystyle {\kappa }_1+{\kappa }_2+{\kappa }_3+2{\kappa }_4\le {\kappa }_1+{\kappa }_2+{\kappa }_3+2{\kappa }_4\underset{n,m\to \infty }{lim}\varphi (u_n,u_m)<1.}$ Therefore $(1-{\kappa }_1-2{\kappa }_4)>0$, and $d_{\varphi }(u,u^{\prime })=0,$ i.e., $u=u^{\prime }.$ Hence T has a unique fixed point in $\mathbf{U}.$ □

Remark 4.

From the symmetry of the distance function $d_{\varphi }$, it is easy to prove similar to that done in [4,22] that ${\kappa }_2={\kappa }_3$. Thus the inequality (20) is equivalent to the following inequality:

$$d_{\varphi }(Tu,Tv)\le {\kappa }_1{d}_{\varphi }(u,v)+{\kappa }_2[d_{\varphi }(u,Tu)+d_{\varphi }(v,Tv)]+{\kappa }_4[d_{\varphi }(v,Tu)+d_{\varphi }(u,Tv)],$$

(26)

where ${\kappa }_1,{\kappa }_2,{\kappa }_4\ge 0$ such that ${\kappa }_1+2{\kappa }_2+2{\kappa }_4{lim}_{n,m\to \infty }\varphi (u_n,u_m)<1.$ If ${\kappa }_1={\kappa }_2=0$ and ${\kappa }_4\in [0,\frac{1}{2})$ in inequality (26), we obtain generalization of Chatterjea’s map [14] in extended b-metric space.

Remark 5.

Theorem 9 generalizes and improves Theorem 1.5 of [23] and therefore Theorem 2.1 of [20]. Moreover, Theorem 9 generalizes and improves Theorem 3.7 from [40], that is, Theorem 2.19 from [41].

Theorem 10.

Let $(\mathbf{U},d_{\varphi })$ be a complete extended b-metric space with $\varphi :\mathbf{U}\times \mathbf{U}\to [1,\infty )$. If $T:\mathbf{U}\to \mathbf{U}$ satisfies the inequality:

$$d_{\varphi }(Tu,Tv)\le {\kappa }_1{d}_{\varphi }(u,v)+{\kappa }_2[d_{\varphi }(u,Tu)+d_{\varphi }(v,Tv)],$$

(27)

for each $u,v\in \mathbf{U}$, where ${\kappa }_1,{\kappa }_2\in [0,\frac{1}{3})$. Moreover for each $u_0\in \mathbf{U}$,

$$\underset{n,m\to \infty }{lim}\varphi (u_n,u_m){\kappa }_2<1,$$

then T has a unique fixed point.

Proof. 

Let us choose an arbitrary $u_0\in \mathbf{U}$ and define the iterative sequence ${\left\{u_n\right\}}_{n=0}^{\infty }$ by $u_n=T{u}_{n-1}=T^{n-1}{u}_0$ for all $n\ge 1.$ If $u_n=u_{n-1},$ then $u_n$ is a fixed point of T and the proof holds. So we suppose $u_n\ne u_{n-1}$, ∀ $n\ge 1.$ Then from Equation (27), we have:

$$\begin{array}{c}\hfill d_{\varphi }(T{u}_n,T{u}_{n-1})\le {\kappa }_1{d}_{\varphi }(u_n,u_{n-1})+{\kappa }_2[d_{\varphi }(u_{n-1},T{u}_{n-1})+d_{\varphi }(u_n,T{u}_n)].\end{array}$$

So,

$$\begin{array}{c}\hfill (1-{\kappa }_2)d_{\varphi }(u_{n+1},u_n)\le ({\kappa }_1+{\kappa }_4)d_{\varphi }(u_n,u_{n-1}).\end{array}$$

$$\begin{array}{c}\hfill d_{\varphi }(u_n,u_{n+1})\le \frac{{\kappa }_1+{\kappa }_4}{1-{\kappa }_4}{d}_{\varphi }(u_n,u_{n-1}).\end{array}$$

This implies that:

$$d_{\varphi }(u_{n+1},u_n)\le \eta d_{\varphi }(u_n,u_{n-1}).$$

(28)

where,

$$\eta =\frac{{\kappa }_1+{\kappa }_4}{1-{\kappa }_4}.$$

Since ${\kappa }_1,{\kappa }_2\in [0,\frac{1}{3})$, so $\eta <1$, from Lemma 3, ${\left\{u_n\right\}}_{n=0}^{\infty }$ is a Cauchy sequence. As $\mathbf{U}$ is complete, therefore there exists $u\in \mathbf{U}$ such that ${lim}_{n\to \infty }{u}_n=u.$ Next, we will show that u is a fixed point of T in $\mathbf{U}.$ From the triangle inequality and Equation (27), we have:

$$\begin{array}{ccc}\hfill d_{\varphi }(u,Tu)& \le & \varphi (u,Tu)[d_{\varphi }(u,u_{n+1})+d_{\varphi }(u_{n+1},Tu)]\hfill \\ & \le & \varphi (u,Tu)[d_{\varphi }(u,u_{n+1})+{\kappa }_1{d}_{\varphi }(u_n,u)+{\kappa }_2[d_{\varphi }(u_n,u_{n+1})+d_{\varphi }(u,Tu)]..\hfill \end{array}$$

So,

$$(1-{\kappa }_2\varphi (u,Tu))d_{\varphi }(u,Tu)\le 0,$$

as $n\to \infty .$ Since ${lim}_{n,m\to \infty }\varphi (u_n,u_m){\kappa }_2<1$, we get $(1-{\kappa }_2\varphi (u,Tu))>0$, and so $d_{\varphi }(u,Tu)=0,$ i.e., $Tu=u.$ We will show that u is the unique fixed point of T. Assume that $u^{\prime }$ is another fixed point of T, then we have $T{u}^{\prime }=u^{\prime }$. Again,

$$\begin{array}{ccc}\hfill d_{\varphi }(u,u^{\prime })& =& d_{\varphi }(Tu,T{u}^{\prime })\hfill \\ & \le & {\kappa }_1{d}_{\varphi }(u,u^{\prime })+{\kappa }_2[d_{\varphi }(u,Tu)+d_{\varphi }(u^{\prime },T{u}^{\prime })]\hfill \\ & & +{\kappa }_1{d}_{\varphi }(u,u^{\prime })<d_{\varphi }(u,u^{\prime }),\hfill \end{array}$$

which is a contradiction. Hence T has a unique fixed point in $\mathbf{U}.$ □

Remark 6.

Theorem 10 generalizes Theorem 1.2 of [20].

For $u,v\in \mathbf{U}$ and $c,d\in [0,1]$, we will use the following notation:

$$N_{c_1,c_2}(u,v)=max\{d_{\varphi }(u,v),c_1{d}_{\varphi }(u,Tu),c_1{d}_{\varphi }(v,Tv),\frac{c_2}{2}(d_{\varphi }(u,Tv)+d_{\varphi }(v,Tu))\}.$$

Theorem 11.

Let $(\mathbf{U},d_{\varphi })$ be an extended b-metric space. Let $T:\mathbf{U}\to \mathcal{CB}\left(\mathbf{U}\right)$ be a multi-valued mapping having the property that there exist $c_1,c_2\in [0,1]$ and $\eta \in [0,1)$ such that:

(i)

For each $u_0\in \mathbf{U}$, ${lim}_{n,m\to \infty }\eta c_2\varphi (u_n,u_m)<1$, here $u_n=T^n{u}_0,$

($ii$)

$H_{\mathsf{\Phi }}(Tu,Tv)\le \eta N_{c_1,c_2}(u,v)$ for all $u,v\in \mathbf{U}.$

Then for every $u_0\in \mathbf{U}$, there exist $\gamma \in [0,1)$ and a sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ of iterates from $\mathbf{U}$ such that for every $n\in \mathbb{N},$

$$d_{\varphi }(u_n,u_{n+1})\le \gamma d_{\varphi }(u_{n-1},u_n).$$

(29)

Proof. 

Let us choose an arbitrary $u_0\in \mathbf{U}$ and $u_1\in T{u}_0.$ Consider:

$$\gamma =max\{\eta ,\frac{\eta c_2\varphi (u_{n-1},u_{n+1})}{2-\eta c_2\varphi (u_{n-1},u_{n+1})}\}.$$

Clearly, $\gamma <1$. If $u_1=u_0$, then for every $n\in \mathbb{N}$, the sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ given by $u_n=u_0$ satisfies Equation(29). Since:

$$\begin{array}{ccc}\hfill d_{\varphi }(u_1,T{u}_1))& \le & d_{\varphi }(T{u}_0,T{u}_1)\le H_{\mathsf{\Phi }}(T{u}_0,T{u}_1)\hfill \\ & \le & \eta N_{c_1,c_2}(u_0,u_1).\hfill \end{array}$$

there exists $u_2\in T{u}_1$ such that $d_{\varphi }(u_1,u_2)\le \eta N_{c_1,c_2}(u_0,u_1).$ If $u_2=u_1$, then for every $n\in \mathbb{N}$, $n\ge 1$, the sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ given by $u_n=u_1$ satisfies Equation (29). By repeating this process, we obtain a sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ of elements from $\mathbf{U}$ such that $u_{n+1}\in T{u}_n$ and $0<d_{\varphi }(u_n,u_{n+1})\le \eta N_{c_1,c_2}(u_{n-1},u_n)$ for every $n\in \mathbb{N}$, $n\ge 1.$ Then we have:

$$\begin{array}{ccc}\hfill 0& <& d_{\varphi }(u_n,u_{n+1})\le \eta N_{c_1,c_2}(u_{n-1},u_n)\hfill \\ & \le & \eta max\{d_{\varphi }(u_{n-1},u_n),c_1{d}_{\varphi }(u_{n-1},T{u}_{n-1}),c_1{d}_{\varphi }(u_n,T{u}_n),\frac{c_2}{2}\hfill \\ & & (d_{\varphi }(u_{n-1},T{u}_n)+d_{\varphi }(u_n,T{u}_{n-1}))\}\hfill \\ & \le & \eta max\{d_{\varphi }(u_{n-1},u_n),c_1{d}_{\varphi }(u_{n-1},u_n),c_1{d}_{\varphi }(u_n,u_{n+1}),\frac{c_2}{2}\left(d_{\varphi }(u_{n-1},u_{n+1})\right)\}\hfill \end{array}$$

(30)

$$\begin{array}{cc}\le & \eta max\{d_{\varphi }(u_{n-1},u_n),c_1{d}_{\varphi }(u_{n-1},u_n),c_1{d}_{\varphi }(u_n,u_{n+1}),\frac{c_2\varphi (u_{n-1},u_{n+1})}{2}\hfill \\ & (d_{\varphi }(u_{n-1},u_n)+d_{\varphi }(u_n,u_{n+1}))\},\hfill \end{array}$$

(31)

for every $n\in \mathbb{N}.$ If we take:

$$\begin{array}{c}max\{d_{\varphi }(u_{n-1},u_n),c_1{d}_{\varphi }(u_{n-1},u_n),c_1{d}_{\varphi }(u_n,u_{n+1}),\frac{c_2\varphi (u_{n-1},u_{n+1})}{2}\hfill \\ (d_{\varphi }(u_{n-1},u_n)+d_{\varphi }(u_n,u_{n+1}))\}=c_1{d}_{\varphi }(u_n,u_{n+1}),\hfill \end{array}$$

then from Equations (30) and (31), $0<d(u_n,u_{n+1})\le \eta c_1{d}_{\varphi }(u_n,u_{n+1})<\eta d_{\varphi }(u_n,u_{n+1}).$ As $\eta <1$, so we obtain the contradiction. Therefore, we have:

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_{n+1})& \le & \eta N_{c_1,c_2}(u_{n-1},u_n)\hfill \\ & \le & \eta max\{d_{\varphi }(u_{n-1},u_n),\frac{c_2\varphi (u_{n-1},u_{n+1})}{2}(d_{\varphi }(u_{n-1},u_n)+d_{\varphi }(u_n,u_{n+1}))\}.\hfill \end{array}$$

Consequently, $d_{\varphi }(u_n,u_{n+1})\le \eta d_{\varphi }(u_{n-1},u_n)$ or

$$d_{\varphi }(u_n,u_{n+1})\le \frac{\eta c_2\varphi (u_{n-1},u_{n+1})}{2}(d_{\varphi }(u_{n-1},u_n)+d_{\varphi }(u_n,u_{n+1})).$$

This implies that $d_{\varphi }(u_n,u_{n+1})\le \eta d_{\varphi }(u_{n-1},u_n)$ or

$$d_{\varphi }(u_n,u_{n+1})\le \frac{\eta c_2\varphi (u_{n-1},u_{n+1})}{2-\eta c_2\varphi (u_{n-1},u_{n+1})}{d}_{\varphi }(u_{n-1},u_n),$$

for every $n\in \mathbb{N}.$ Thus,

$$d_{\varphi }(u_n,u_{n+1})\le max\{\eta ,\frac{\eta c_2\varphi (u_{n-1},u_{n+1})}{2-\eta c_2\varphi (u_{n-1},u_{n+1})}\}{d}_{\varphi }(u_{n-1},u_n),$$

i.e.,

$$d_{\varphi }(u_n,u_{n+1})\le \gamma d_{\varphi }(u_{n-1},u_n).$$

Thus, the sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ satisfies Equation(29). Hence from Lemma 3, we conclude that ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ is Cauchy sequence. □

Theorem 12.

Let $(\mathbf{U},d_{\varphi })$ be a complete extended b-metric space. Let $T:\mathbf{U}\to \mathcal{CB}\left(\mathbf{U}\right)$ be a multi-valued mapping having the property that there exist $c_1,c_2\in [0,1]$ and $\eta \in [0,1)$ such that:

(i)

For each $u_0\in \mathbf{U}$, ${lim}_{n,m\to \infty }\eta c_2\varphi (u_n,u_m)<1$, here $u_n=T^n{u}_0,$

($ii$)

$H_{\mathsf{\Phi }}(Tu,Tv)\le \eta N_{c_1,c_2}(u,v)$ for all $u,v\in \mathbf{U},$

($iii$)

T is continuous.

Then T has a fixed point in $\mathbf{U}.$

Proof. 

From Theorem 11, by taking in account condition $\left(i\right)$ and $\left(ii\right)$, we conclude that ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ is a Cauchy sequence such that:

$$u_{n+1}\in T{u}_n,$$

(32)

for every $n\in \mathbb{N}.$ As $\mathbf{U}$ is complete, so there exists $u\in \mathbf{U}$ such that ${lim}_{n\to \infty }{u}_n=u.$ From inequality (3), by the continuity of T, it follows that:

$$u_{n+1}=T{u}_n\to Tu,asn\to \infty .$$

Therefore, $u\in Tu.$ Hence T has a fixed point in $\mathbf{U}.$ □

Theorem 13.

Let $(\mathbf{U},d_{\varphi })$ be a complete extended b-metric space. Let $T:\mathbf{U}\to \mathcal{CB}\left(\mathbf{U}\right)$ be a multi-valued mapping having the property that there exist $c_1,c_2\in [0,1]$ and $\eta \in [0,1)$ such that:

(i)

For each $u_0\in \mathbf{U}$${lim}_{n,m\to \infty }\eta c_2\varphi (u_n,u_m)<1$, here $u_n=T^n{u}_0,$

($ii$)

$H_{\mathsf{\Phi }}(Tu,Tv)\le \eta N_{c_1,c_2}(u,v)$ for all $u,v\in \mathbf{U},$

($iii$)

T is ∗-continuous.

Then T has a fixed point in $\mathbf{U}.$ □

Proof. 

From Theorem 3, by taking in account condition $\left(i\right)$ and $\left(ii\right)$, we conclude that ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ is a Cauchy sequence such that:

$$u_{n+1}\in T{u}_n,$$

(33)

for every $n\in \mathbb{N}.$ As $\mathbf{U}$ is complete, so there exists $u\in \mathbf{U}$ such that ${lim}_{n\to \infty }{u}_n=u.$ Then we have:

$$\begin{array}{ccc}\hfill d_{\varphi }(u_{n+1},Tu)& =& d_{\varphi }(T{u}_n,Tu)\le H_{\mathsf{\Phi }}(T{u}_n,Tu)\le \eta N_{c_1,c_2}(u_n,u)\le \eta max\{d_{\varphi }(u_n,u),c_1\hfill \\ & & d_{\varphi }(u_n,T{u}_n),c_1{d}_{\varphi }(u,Tu),\frac{c_2}{2}(d_{\varphi }(u_n,Tu)+d_{\varphi }(u,T{u}_n))\}\le \eta max\{\hfill \\ & & d_{\varphi }(u_n,u),c_1{d}_{\varphi }(u_n,u_{n+1}),c_1{d}_{\varphi }(u,Tu),\frac{c_2}{2}(d_{\varphi }(u_n,Tu)+d_{\varphi }(u,T{u}_n))\}\hfill \end{array}$$

(34)

$$\begin{array}{cc}\le & \eta max\{d_{\varphi }(u_n,u),c_1{d}_{\varphi }(u_n,u_{n+1}),c_1{d}_{\varphi }(u,Tu),\frac{c_2}{2}(\varphi (u_n,Tu)\hfill \\ & (d_{\varphi }(u_n,u)+d_{\varphi }(u,Tu)))+d_{\varphi }(u,u_{n+1})\},\hfill \end{array}$$

(35)

for every $n\in \mathbb{N}.$ Since ${\displaystyle \underset{n\to \infty }{lim}{u}_n=u}$, ${\displaystyle \underset{n\to \infty }{lim}{d}_{\varphi }(u_n,u_{n+1})=0}$. Then ${\displaystyle \underset{n\to \infty }{lim}{d}_{\varphi }(u_{n+1},Tu)=d_{\varphi }(u,Tu)}$. Therefore, by taking limit $n\to \infty$ in Equations (34) and (35), we obtain:

$$\begin{array}{ccc}\hfill d_{\varphi }(u,Tu)& \le & \eta N_{c_1,c_2}(u_n,u)\hfill \\ & \le & \eta max\{0,c_1{d}_{\varphi }(u,Tu),\frac{{\displaystyle c_2\underset{n\to \infty }{lim}\varphi (u_n,Tu)}}{2}{d}_{\varphi }(u,Tu)\}\hfill \\ & \le & max\{\eta c_1,\eta \frac{{\displaystyle \eta c_2\underset{n\to \infty }{lim}\varphi (u_n,Tu)}}{2}\}{d}_{\varphi }(u,Tu).\hfill \end{array}$$

As $max\{\eta c_1,\eta \frac{{\displaystyle \eta c_2\underset{n\to \infty }{lim}\varphi (u_n,Tu)}}{2}\}<1,$ so from above inequality $d_{\varphi }(u,Tu)<d_{\varphi }(u,Tu),$ which is impossible, therefore $d_{\varphi }(u,Tu)=0$ i.e., $u\in Tu$. Hence T has a fixed point in $\mathbf{U}.$ □

Theorem 14.

A multi-valued mapping $T:\mathbf{U}\to \mathcal{CB}\left(\mathbf{U}\right)$ has a fixed point in a complete extended b-metric space $(\mathbf{U},d_{\varphi })$, if it satisfies the following two axioms:

(i)

There exist $c_1,c_2\in [0,1]$ and $\eta \in [0,1)$ such that $H_{\mathsf{\Phi }}(Tu,Tv)\le \eta N_{c_1,c_2}(u,v)$ for all $u,v\in \mathbf{U},$

($ii$)

For each $u_0\in \mathbf{U}$, ${\displaystyle max\{\eta c_1\underset{n,m\to \infty }{lim}\varphi (u_n,u_m),\eta c_2\underset{n,m\to \infty }{lim}\varphi (u_n,u_m)\}<1,}$ here $u_n=T^n{u}_0.$

Proof. 

From Theorem 11, by taking in account condition $\left(i\right)$ and $\left(ii\right)$, we conclude that ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ is a Cauchy sequence such that:

$$u_{n+1}\in T{u}_n,$$

(36)

for every $n\in \mathbb{N}.$ As $\mathbf{U}$ is complete, so there exists $u\in \mathbf{U}$ such that ${\displaystyle \underset{n\to \infty }{lim}{u}_n=u.}$ Then for every $n\in \mathbb{N}$, we have:

$$\begin{array}{cc}& d_{\varphi }(u_{n+1},Tu)=d_{\varphi }(T{u}_n,Tu)\le H_{\mathsf{\Phi }}(T{u}_n,Tu)\le \eta N_{c_1,c_2}(u_n,u)\hfill \\ \le & \eta max\{d_{\varphi }(u_n,u),c_1{d}_{\varphi }(u_n,T{u}_n),c_1{d}_{\varphi }(u,Tu),\frac{c_2}{2}(d_{\varphi }(u_n,Tu)+d_{\varphi }(u,T{u}_n))\}\hfill \\ \le & \eta max\{d_{\varphi }(u_n,u),c_1{d}_{\varphi }(u_n,u_{n+1}),c_1{d}_{\varphi }(u,Tu),\frac{c_2}{2}(d_{\varphi }(u_n,Tu)+d_{\varphi }(u,T{u}_n))\}\hfill \end{array}$$

(37)

$$\begin{array}{cc}\le & \eta max\{d_{\varphi }(u_n,u),c_1{d}_{\varphi }(u_n,u_{n+1}),c_1{d}_{\varphi }(u,Tu),\frac{c_2}{2}(\varphi (u_n,Tu)\hfill \\ & (d_{\varphi }(u_n,u)+d_{\varphi }(u,Tu)))+d_{\varphi }(u,u_{n+1})\}.\hfill \end{array}$$

(38)

Now, we will take two cases:

Case (i):

If ${\displaystyle d_{\varphi }(u,Tu)\le \underset{n\to \infty }{lim}sup{d}_{\varphi }(u_n,Tu),}$ then there exists a subsequence ${\left\{u_{n_l}\right\}}_{n\in \mathbb{N}}$ of $\left\{u_n\right\}$ such that ${\displaystyle d_{\varphi }(u,Tu)\le \underset{l\to \infty }{lim}{d}_{\varphi }(u_{n_l+1},Tu),}$ so for each $ϵ>0$, ∃$l_{ϵ}\in \mathbb{N}$ such that for every $l\in \mathbb{N}$, $l\ge l_{ϵ}$, we have:

$$\begin{array}{cc}& d_{\varphi }(u,Tu)-ϵ\le d_{\varphi }(u_{n_l+1},Tu)\hfill \\ \le & \eta max\{d_{\varphi }(u_{n_l},u),c_1{d}_{\varphi }(u_{n_l},u_{n_l+1}),c_1{d}_{\varphi }(u,Tu),\frac{c_2}{2}\hfill \\ & (d_{\varphi }(u_{n_l},Tu)+d_{\varphi }(u,u_{n_l+1}))\}\hfill \end{array}$$

(39)

$$\begin{array}{cc}\le & \eta max\{d_{\varphi }(u_{n_l},u),c_1{d}_{\varphi }(u_{n_l},u_{n_l+1}),c_1{d}_{\varphi }(u,Tu),\frac{c_2}{2}\hfill \\ & (\varphi (u_{n_l},Tu)(d_{\varphi }(u_{n_l},u)+d_{\varphi }(u,Tu))+d_{\varphi }(u,u_{n_l+1})\}.\hfill \end{array}$$

(40)

Since ${\displaystyle \underset{l\to \infty }{lim}{u}_{n_l}=u}$, ${\displaystyle \underset{l\to \infty }{lim}{d}_{\varphi }(u_{n_l},u_{n_l+1})=0}$. Therefore, by taking limit $l\to \infty$ in Equations (39) and (40), we obtain:

$$\begin{array}{cc}\hfill d_{\varphi }(u,Tu)-ϵ& \le \eta max\{0,c_1{d}_{\varphi }(u,Tu),\frac{{\displaystyle c_2\underset{l\to \infty }{lim}\varphi (u_{n_l},Tu)}}{2}{d}_{\varphi }(u,Tu)\}\\ & \le \eta max\{c_1,\eta \frac{{\displaystyle c_2\underset{l\to \infty }{lim}\varphi (u_{n_l},Tu)}}{2}\}{d}_{\varphi }(u,Tu),\end{array}$$

for every $ϵ>0.$ Thus,

$$d_{\varphi }(u,Tu)\le max\{\eta c_1,\eta \frac{{\displaystyle \eta c_2\underset{l\to \infty }{lim}\varphi (u_{n_l},Tu)}}{2}\}{d}_{\varphi }(u,Tu).$$

As $max\{\eta c_1,\eta \frac{{\displaystyle \eta c_2\underset{l\to \infty }{lim}\varphi (u_{n_l},Tu)}}{2}\}<1,$ so from above inequality $d_{\varphi }(u,Tu)<d_{\varphi }(u,Tu),$ which is impossible, therefore $d_{\varphi }(u,Tu)=0$, i.e., $u\in Tu$. Hence T has a fixed point in $\mathbf{U}.$

Case (ii):

If ${\displaystyle d_{\varphi }(u,Tu)>\underset{n\to \infty }{lim}sup{d}_{\varphi }(u_n,Tu),}$ then there exists $N_0\in \mathbb{N}$ such that for every $n\ge N_0$, we have

$$d_{\varphi }(u_{n_l},Tu)\le d_{\varphi }(u,Tu).$$

From the triangle inequality, $d_{\varphi }(u,Tu)\le \varphi (u,Tu)(d_{\varphi }(u,u_{n+1})+d_{\varphi }(u_{n+1},Tu))$, we obtain:

$$\begin{array}{c}{d}_{\varphi }(u,Tu)-\varphi (u,Tu)(d_{\varphi }(u,u_{n+1})\le \varphi (u,Tu)d_{\varphi }(u_{n+1},Tu)\hfill \\ \le \varphi (u,Tu)\eta max\{d_{\varphi }(u_n,u),c_1{d}_{\varphi }(u_n,u_{n+1}),c_1{d}_{\varphi }(u,Tu),\frac{c_2}{2}(d_{\varphi }(u_n,Tu)+d_{\varphi }(u,u_{n+1}))\}\hfill \end{array}$$

(41)

$$\begin{array}{c}\le \eta max\{d_{\varphi }(u_n,u),c_1{d}_{\varphi }(u_n,u_{n+1}),c_1{d}_{\varphi }(u,Tu),\frac{c_2}{2}(\varphi (u_n,Tu)\hfill \\ (d_{\varphi }(u_n,u)+d_{\varphi }(u,Tu)))+d_{\varphi }(u,u_{n+1})\}.\hfill \end{array}$$

(42)

Since ${\displaystyle \underset{n\to \infty }{lim}{u}_n=u}$, ${\displaystyle \underset{n\to \infty }{lim}{d}_{\varphi }(u_n,u_{n+1})=0}$. Therefore by taking limit $n\to \infty$ in Equations (41) and (42), we obtain:

$$\begin{array}{c}{d}_{\varphi }(u,Tu)-\varphi (u,Tu)d_{\varphi }(u,u_{n+1})\le \hfill \\ \varphi (u,Tu)\eta max\{0,c_1{d}_{\varphi }(u,Tu),\frac{{\displaystyle c_2\underset{n\to \infty }{lim}\varphi (u_n,Tu)}}{2}{d}_{\varphi }(u,Tu)\hfill \\ \le \varphi (u,Tu)max\{\eta c_1,\eta \frac{{\displaystyle \eta c_2\underset{n\to \infty }{lim}\varphi (u_n,Tu)}}{2}\}{d}_{\varphi }(u,Tu),\hfill \end{array}$$

(43)

from condition $\left(ii\right)$, since $\varphi (u,Tu)max\{\eta c_1,\eta \frac{{\displaystyle \eta c_2\underset{n\to \infty }{lim}\varphi (u_n,Tu)}}{2}\}<1,$ so from Equation (43), $d_{\varphi }(u,Tu)<d_{\varphi }(u,Tu),$ which is impossible, therefore $d_{\varphi }(u,Tu)=0$, i.e., $u\in Tu$. Hence T has a fixed point in $\mathbf{U}.$

□

Remark 7.

(i)

For $c_1,c_2=0$ in Theorem 12, we obtain Nadler’s contraction principle for multi valued-mappings, i.e., Theorem 5 from [24].

(ii)

Theorem 14 generalizes Theorems 12 and 13;

(ii)

Theorem 14 generalizes Theorem 3.3 from [42], which generalizes Theorem 7 of [30]. Also, Theorem 7, which is a generalization of Theorem 2.2 from [29], improves Theorem 3.3 from [43], Corollary 3.3 from [5], and Theorem 1 from [28].

Example 2.

Let $\mathbf{U}=\{\frac{1}{2},\frac{1}{4},\dots ,\frac{1}{2^n},\dots \}\cup \{0,1\}$, $d_{\varphi }(u_1,u_2)={(u_1-u_2)}^2$, for $u_1,u_2\in \mathbf{U}$, where $\varphi :\mathbf{U}\times \mathbf{U}\to [1,\infty )$ define by $\varphi (u_1,u_2)=u_1+u_2+1.$ Then $\mathbf{U}$ is a complete extended b-metric space. Define mapping $T:\mathbf{U}\to \mathcal{CB}\left(\mathbf{U}\right)$ as

$$Tu=\left\{\begin{array}{cc}\left\{\frac{1}{2^{n+1}}\right\},\hfill & \mathrm{u}=\frac{1}{2^{\mathrm{n}}},\mathrm{n}=0,1,2,\dots \hfill \\ u,\hfill & \mathrm{u}=0.\hfill \end{array}\right.$$

Hence T is continuous. Since $N_{c_1,c_2}(\frac{1}{2^n},0)=\frac{1}{2^{2n}},$ for all $c_1,c_2\in [0,1]$, we get:

$$H_{\mathsf{\Phi }}\left(T\left(\frac{1}{2^n}\right),T\left(0\right)\right)=\frac{1}{2^{2n+2}}\le \frac{1}{2^{2n+1}}\le \frac{1}{2}{N}_{c_1,c_2}\left(\frac{1}{2^n},0\right),$$

where $\eta =\frac{1}{2}$. Also for each $u_0\in \mathbf{U}$, ${\displaystyle \underset{n,m\to \infty }{lim}\eta c_2\varphi (u_n,u_m)<1}$. Clearly, it satisfies all the conditions of Theorem 12, and so there exists a fixed point.

Example 3.

Let $\mathbf{U}=[0,\infty )$. Define $d_{\varphi }(u_1,u_2)={(u_1-u_2)}^2$, for $u_1,u_2\in \mathbf{U}$, where $\varphi :\mathbf{U}\times \mathbf{U}\to [1,\infty )$, where $\varphi (u_1,u_2)=u_1+u_2+2.$ Then $\mathbf{U}$ is a complete extended b-metric space. Define mapping $T:\mathbf{U}\to \mathcal{CB}\left(\mathbf{U}\right)$ as $Tu=\left\{\frac{8}{9}u\right\}$ for every $u\in \mathbf{U}.$ Note that Theorem 14 is applicable by taking $c_1=c_2=0$ and $\eta =\frac{8}{9}.$