A Mizoguchi–Takahashi Type Fixed Point Theorem in Complete Extended b-Metric Spaces

Abstract

In this paper, we prove a new fixed point theorem for a multi-valued mapping from a complete extended b-metric space U into the non empty closed and bounded subsets of U, which generalizes Nadler’s fixed point theorem. We also establish some fixed point results, which generalize our first result. Furthermore, we establish Mizoguchi–Takahashi’s type fixed point theorem for a multi-valued mapping from a complete extended b-metric space U into the non empty closed and bounded subsets of U that improves many existing results in the literature.

1. Introduction

Throughout this paper, $(\mathbf{U},d_{\varphi })$ is an extended b-metric space. We denote by $\mathcal{CL}\left(\mathbf{U}\right)$ the set of all subsets of U that are non empty and closed, by $\mathcal{CLB}\left(\mathbf{U}\right)$ the set of all subsets of U that are non empty closed and bounded and by $\mathcal{K}\left(\mathbf{U}\right)$ the set of all subsets of U that are non empty compacts.

An element $u^{\prime }\in \mathcal{U}$ is called a fixed point of a multi-valued map $F:\mathbf{U}\to \mathcal{CLB}\left(\mathbf{U}\right)$ if $u^{\prime }\in F{u}^{\prime }.$ An orbit for a mapping $F:\mathbf{U}\to \mathcal{CLB}\left(\mathbf{U}\right)$ at a point $u_0\in \mathbf{U}$ denoted by $O\left(F\right)$ is a sequence ${\left\{u_n\right\}}_{n=0}^{\infty }$ in $\mathbf{U}$ such that $u_{n+1}\in F{u}_n.$ A mapping $f:\mathbf{U}\to \mathbb{R}$ is said to be F-orbitally lower semi-continuous if for any sequence ${\left\{u_n\right\}}_{n=0}^{\infty }$ in $O\left(F\right)$ and $u\in \mathbf{U}$, $u_n\to u$ implies $f\left(u\right)\le {lim}_{n\to \infty }inff\left(u_n\right)$.

Define a function $f:\mathbf{U}\to \mathbb{R}$ as $f\left(u\right)=d_{\varphi }(u,Fu).$ For a constant $q\in (0,1)$, define the set $I_q^u\subset \mathbf{U}$ as

$$I_q^u=\{v\in Fu|q{d}_{\varphi }(u,v)\le d_{\varphi }(u,Fu)\}.$$

The Pompeiu–Hausdorff distance measuring the distance between the subsets of a metric space was initiated by D. Pompeiu in [1]. The fixed point theory of set-valued contractions was initiated by Nadler [2], but later many authors extrapolated it multi directionally (see [3,4]).

Theorem 1

(Reich [5]). Let $(\mathbf{U},d)$ be a complete metric space and let $F:\mathbf{U}\to \mathcal{K}\left(\mathbf{U}\right)$. Assume that there exists a map $\eta :[0,\infty )\to [0,1)$ such that

$$lim\underset{s\to t^{+}}{sup}\eta \left(s\right)<1,\mathit{for}\mathit{all}t\in (0,\infty ),$$

and

$$\mathbf{H}(Fu,Fv)\le \eta \left(d\right(u,v\left)\right)d(u,v),\mathit{for}\mathit{all}u,v\in \mathbf{U}.$$

Then F has a fixed point.

In [5] Reich raised the question if the above theorem is also true for $F:\mathbf{U}\to \mathcal{CLB}\left(\mathbf{U}\right)$. In [6], Mizoguchi and Takahashi gave supportive solution to the conjecture of [5] under the hypothesis $lim{sup}_{s\to t^{+}}\eta \left(s\right)<1,\mathit{for}\mathit{all}t\in [0,\infty ).$ In particular, they proved the following result:

Theorem 2

(Mizoguchi, Takahashi [6]). Let $(\mathbf{U},d)$ be a complete metric space and let $F:\mathbf{U}\to \mathcal{CLB}\left(\mathbf{U}\right)$. Assume that there exists a map $\eta :[0,\infty )\to [0,1)$ such that

$$lim\underset{s\to t^{+}}{sup}\eta \left(s\right)<1,\mathit{for}\mathit{all}t\in [0,\infty ),$$

and

$$\mathbf{H}(Fu,Fv)\le \eta \left(d\right(u,v\left)\right)d(u,v),\mathit{for}\mathit{all}u,v\in \mathbf{U},u\ne v.$$

Then F has a fixed point.

In [7], Feng and Liu extended Nadler’s fixed point theorem, other than the direction of Reich and Takahashi. They proved a theorem as follows:

Theorem 3

(Feng, Liu [7]). Let $(\mathbf{U},d)$ be a complete metric space and let $F:\mathbf{U}\to \mathcal{CLB}\left(\mathbf{U}\right)$. Assume that:

(i) 

The map $f:\mathbf{U}\to \mathbb{R}$ defined by $f\left(u\right)=d(u,Fu)$, $u\in \mathbf{U}$, is lower semi-continuous;

(ii) 

There exist $p,q\in (0,1)$, $p<q$ such that for all $u\in \mathbf{U}$ there exists $v\in \{v\in Fu|qd(u,v)\le d(u,Fu)\}$ satisfying

$$d(v,Fv)\le pd(u,v).$$

Then F has a fixed point.

Hicks and Rhodes [8] and Klim and Wardowski [9] proved the following results:

Theorem 4

([8]). Let $(\mathbf{U},d)$ be a complete metric space and let $g:\mathbf{U}\to \mathbf{U}$, $0\le h<1.$ Suppose there exists q such that

$$d(gv,g^{2}v)\le hd(v,gv),foreveryy\in \{x,gx,g^{2}x,\dots \}.$$

Then

(i) 

${lim}_n{g}^{n}x=q$ exists;

(ii) 

$d(g^{n}x,q)\le \frac{h^n}{1-h}d(x,gx)$;

(iii) 

q is a fixed point of g iff $G\left(x\right)=d(x,gx)$ is g-orbitally lower semi-continuous at $q.$

Theorem 5

([9]). Let $(\mathbf{U},d)$ be a complete metric space and let $F:\mathbf{U}\to \mathcal{K}\left(\mathbf{U}\right)$. Assume that the following conditions hold:

(i) 

The map $f:\mathbf{U}\to \mathbb{R}$ defined by $f\left(u\right)=d(u,Fu)$, $u\in \mathbf{U}$, is lower semi-continuous;

(ii) 

There exists a map $\eta :[0,\infty )\to [0,1)$ such that

$$lim\underset{s\to t^{+}}{sup}\eta \left(s\right)<1,\mathit{for}\mathit{all}t\in (0,\infty ),$$

and for all $u\in \mathbf{U}$ there exists $v\in \{v\in Fu:d(u,v)\le d(u,Fu\left)\right\}$ satisfying

$$d(v,Fv)\le \eta \left(d\right(u,v\left)\right)d(u,v).$$

Then F has a fixed point.

In 2007, Kamran [10] logically presented Mizoguchi–Takahashi’s type fixed point theorem, that simply generalizes Theorems 4 and 5.

The idea of generalizing metric spaces into b-metric spaces was initiated from the works of Bakhtin [11], Bourbaki [12], and Czerwik [13,14]. In [15], the notion of b-metric space was generalized further by introducing the concept of extended b-metric spaces (see also [16,17,18]) as follows:

Definition 1

([15]). Let $\mathbf{U}$ be a non empty 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:

(i) 

$d_{\varphi }(u_1,u_2)=0$ if and only if $u_1=u_2,$

(ii) 

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

(iii) 

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

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

Remark 1

([15]). Every b-metric space is an extended b-metric space with a constant function $\varphi (x_1,x_2)=s$, for $s\ge 1$, but its converse is not true in general.

Example 1.

Let $\mathbf{U}=\{u\in \mathbb{R}:u\ge 1\}$. Define $d_{\varphi }:\mathbf{U}\times \mathbf{U}\to [0,\infty )$ and $\varphi :\mathbf{U}\times \mathbf{U}\to [1,\infty )$ as follows:

$$d_{\varphi }(u_1,u_2)={(u_1-u_2)}^2,\varphi (u_1,u_2)=1+u_1+u_2,$$

for all $u_1,u_2\in \mathbf{U}.$ Then $(\mathbf{U},d_{\varphi })$ is an extended b-metric space.

For more examples and recent results see [19]. Also, in [20] Muhammad Usman Ali et al. established fixed point results for new F-contractions of Hardy–Rogers type in the setting of b-metric space and proved the existence theorem for Volterra-type integral inclusion. Their results generalized many existence results in the literature. Finally in [21], authors introduced the notion of a generalized Pompeiu–Hausdorff metric induced by the extended b-metric as follows:

Definition 2.

([21]) 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 $\mathbf{A},\mathbf{B}\in \mathcal{CLB}\left(\mathbf{U}\right)$, where $\mathcal{CLB}\left(\mathbf{U}\right)$ denotes the family of all non empty closed and bounded subsets of $\mathbf{U}$, the Hausdorff–Pompieu metric on $\mathcal{CLB}\left(\mathbf{U}\right)$ induced by $d_{\varphi }$ is defined by

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

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

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

Theorem 6.

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

In this paper, we extend Nadler’s fixed point theorem for the extended b-metric space. Moreover, we improve Mizoguchi–Takahashi’s type fixed point theorem (Theorem 1.2, [10]) for the extended b-metric space when F is a multi-valued mapping from $\mathbf{U}$ to $\mathcal{CLB}\left(\mathbf{U}\right)$. Our results generalize Theorems 4 and 5 in the setting of extended b-metric spaces which in turn generalize many existing results including Theorems 1–3.

2. Main Results

We start with the following lemma.

Lemma 1.

Let $\mathbf{X},\mathbf{Y}\in \mathcal{CLB}\left(\mathbf{U}\right)$, then for every $\eta >0$ and $y\in \mathbf{Y}$ there exists $x\in \mathbf{X}$ such that

$$d_{\varphi }(x,y)\le {\mathbf{H}}_{\mathsf{\Phi }}(\mathbf{X},\mathbf{Y})+\eta .$$

Proof. 

By definition of the Hausdorff metric, for $\mathbf{X},\mathbf{Y}\in \mathcal{CLB}\left(\mathbf{U}\right)$ and for any $y\in \mathbf{Y}$, we have

$$d_{\varphi }(\mathbf{X},y)\le {\mathbf{H}}_{\mathsf{\Phi }}(\mathbf{X},\mathbf{Y}).$$

By the definition of an infimum, we can let ${\left\{x_n\right\}}_{n=0}^{\infty }$ be a sequence in $\mathbf{X}$ such that

$$d_{\varphi }(y,x_n)<d_{\varphi }(y,\mathbf{X})+\eta ,where\eta >0.$$

(1)

We know that $\mathbf{X}$ is closed and bounded, so there exists $x\in \mathbf{X}$ such that $x_n\to x$. Therefore by (1), we have

$$\begin{array}{c}\hfill d_{\varphi }(x,y)<d_{\varphi }(\mathbf{X},y)+\eta \le {\mathbf{H}}_{\mathsf{\Phi }}(\mathbf{X},\mathbf{Y})+\eta .\end{array}$$

 ☐

Theorem 7.

Let $(\mathbf{U},d_{\varphi })$ be a complete extended b-metric space. If $F:\mathbf{U}\to \mathcal{CLB}\left(\mathbf{U}\right)$ satisfies the inequality

$${\mathbf{H}}_{\mathsf{\Phi }}(Fu,Fv)\le \eta d_{\varphi }(u,v),\mathit{for}\mathit{all}u,v\in \mathbf{U},$$

(2)

where $\eta \in [0,1)$ is a real constant such that ${lim}_{n,m\to \infty }\eta \varphi (u_n,u_m)<1,$ then F has a fixed point.

Proof. 

Let us consider $\eta >0.$ Let $u_0\in \mathbf{U}$ and choose $u_1\in F{u}_0$. Since $F{u}_0,F{u}_1\in \mathcal{CLB}\mathbf{U})$ and $u_1\in F{u}_0,$ then by Lemma 1, there exists $u_2\in F{u}_1$ such that

$$d_{\varphi }(u_1,u_2)\le {\mathbf{H}}_{\mathsf{\Phi }}(F{u}_0,F{u}_1)+\eta .$$

Now since $F{u}_1,F{u}_2\in \mathcal{CLB}\mathbf{U})$ and $u_2\in F{u}_1$, there is a point $u_3\in F{u}_2$ such that

$$d_{\varphi }(u_2,u_3)\le {\mathbf{H}}_{\mathsf{\Phi }}(F{u}_1,F{u}_2)+{\eta }^2.$$

Continuing in this fashion, we obtain a sequence ${\left\{u_n\right\}}_{n=0}^{\infty }$ of elements of $\mathbf{U}$ such that $u_{n+1}\in F{u}_n$ and

$$d_{\varphi }(u_n,u_{n+1})\le {\mathbf{H}}_{\mathsf{\Phi }}(F{u}_{n-1},F{u}_n)+{\eta }^n,\mathrm{for}\mathrm{all}n\ge 1.$$

By (2), we note that

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_{n+1})& \le & \eta d_{\varphi }(u_{n-1},u_n)+{\eta }^n\hfill \\ & \le & \eta (\eta d_{\varphi }(u_{n-2},u_{n-1})+{\eta }^{n-1})+{\eta }^n\hfill \\ & \le & {\eta }^2{d}_{\varphi }(u_{n-2},u_{n-1})+2{\eta }^n.\hfill \end{array}$$

Continuing in this way, we have

$$d_{\varphi }(u_n,u_{n+1})\le {\eta }^n{d}_{\varphi }(u_0,u_1)+n{\eta }^n,\mathrm{for}\mathrm{all}n\ge 1.$$

(3)

By the triangle inequality and (3) for $m>n,$ we have

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_m)& \le & \varphi (u_n,u_m)[{\eta }^n{d}_{\varphi }(u_0,u_1)+n{\eta }^n]+\varphi (u_n,u_m)\varphi (u_{n+1},u_m)[{\eta }^{n+1}{d}_{\varphi }(u_0,u_1)\hfill \\ & & +(n+1){\eta }^{n+1}]+\dots \dots +\varphi (u_n,u_m)\varphi (u_{n+1},u_m)\dots \varphi (u_{m-1},u_m)\hfill \\ & & [{\eta }^{m-1}{d}_{\varphi }(u_0,u_1)+(m-1){\eta }^{m-1}],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_m)& \le & d_{\varphi }(u_0,u_1)[\varphi (u_n,u_m){\eta }^n+\varphi (u_n,u_m)\varphi (u_{n+1},u_m){\eta }^{n+1}+\dots +\varphi (u_n,u_m)\hfill \\ & & \varphi (u_{n+1},u_m)\dots \varphi (u_{m-1},u_m){\eta }^{m-1}]+[\varphi (u_n,u_m)n{\eta }^n+\varphi (u_n,u_m)\hfill \\ & & \varphi (u_{n+1},u_m)(n+1){\eta }^{n+1}+\dots +\varphi (u_n,u_m)\varphi (u_{n+1},u_m)\dots \hfill \\ & & \varphi (u_{m-1},u_m)(m-1){\eta }^{m-1}],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_m)& \le & d_{\varphi }(u_0,u_1)[\varphi (u_1,u_m)\varphi (u_2,u_m)\dots \varphi (u_n,u_m){\eta }^n+\varphi (u_1,u_m)\varphi (u_2,u_m)\dots \hfill \\ & & \varphi (u_{n+1},u_m){\eta }^{n+1}+\dots +\varphi (u_1,u_m)\varphi (u_2,u_m)\dots \varphi (u_n,u_m)\varphi (u_{n+1},u_m)\dots \hfill \\ & & \varphi (u_{m-1},u_m){\eta }^{m-1}]+[\varphi (u_1,u_m)\varphi (u_2,u_m)\dots \varphi (u_n,u_m)n{\eta }^n+\varphi (u_1,u_m)\hfill \\ & & \varphi (u_2,u_m)\dots \varphi (u_{n+1},u_m)(n+1){\eta }^{n+1}+\dots +\varphi (u_1,u_m)\varphi (u_2,u_m)\dots \hfill \\ & & \varphi (u_n,u_m)\varphi (u_{n+1},u_m)\dots \varphi (u_{m-1},u_m)(m-1){\eta }^{m-1}].\hfill \end{array}$$

Since ${lim}_{n,m\to \infty }\varphi (u_{n+1},u_m)\eta <1$, the series

$$\sum _{n=1}^{\infty }{\eta }^n\prod _{i=1}^n\varphi (u_i,u_m)\mathrm{and}\sum _{n=1}^{\infty }n{\eta }^n\prod _{i=1}^n\varphi (u_i,u_m)$$

converges by the ratio test for each $m\in \mathbb{N}.$ Let

$$S=\sum _{n=1}^{\infty }{\eta }^n\prod _{i=1}^n\varphi (u_i,u_m),S_n=\sum _{j=1}^n{\eta }^j\prod _{i=1}^j\varphi (u_i,u_m),$$

and

$$S^{{}^{\prime }}=\sum _{n=1}^{\infty }n{\eta }^n\prod _{i=1}^n\varphi (u_i,u_m),S_n^{{}^{\prime }}=\sum _{j=1}^{n}j{\eta }^j\prod _{i=1}^j\varphi (u_i,u_m).$$

Thus for $m>n$, the above inequality implies

$$d_{\varphi }(u_n,u_m)\le d_{\varphi }(u_0,u_1)[S_{m-1}-S_n]+[S_{m-1}^{{}^{\prime }}-S_n^{{}^{\prime }}].$$

By letting $n\to \infty$, we conclude that ${\left\{u_n\right\}}_{n=1}^{\infty }$ is a Cauchy sequence. Since $\mathbf{U}$ is complete, there exists $u\in \mathbf{U}$ such that ${lim}_{n\to \infty }{u}_n=u$ (so ${lim}_{n\to \infty }{u}_{n+1}=u$). Now by the triangle inequality

$$\begin{array}{ccc}\hfill d_{\varphi }(Fu,u)& \le & \varphi (Fu,u)[d_{\varphi }(Fu,u_n)+d_{\varphi }(u_n,u)]\hfill \\ & \le & \varphi (Fu,u)[\eta d_{\varphi }(u,u_{n-1})+d_{\varphi }(u_n,u)].\hfill \end{array}$$

This implies that

$$d_{\varphi }(Fu,u)\le 0asn\to \infty .$$

$$d_{\varphi }(Fu,u)=0.$$

Hence u is a fixed point of $F.$  ☐

Theorem 8.

Let us consider a multi-valued mapping $F:\mathbf{U}\to \mathcal{CLB}\left(\mathbf{U}\right)$, where $(\mathbf{U},d_{\varphi })$ is a complete extended b-metric space. Furthermore, let us consider that the following two conditions hold:

(i) 

The map $f:\mathbf{U}\to \mathbb{R}$ defined by $f\left(u\right)=d_{\varphi }(u,Fu)$, $u\in \mathbf{U}$, is lower semi-continuous;

(ii) 

There exist $p,q\in (0,1)$, $p<q$ such that for all $u\in \mathbf{U}$ there exists $v\in I_q^u$ satisfying

$$d_{\varphi }(v,Fv)\le p{d}_{\varphi }(u,v).$$

Moreover ${lim}_{n,m\to \infty }\alpha \varphi (u_n,u_m)<1,$ for all $\alpha \in (0,1)$. Then F has a fixed point in $\mathbf{U}.$

Proof. 

As $Fu\in \mathcal{CLB}\left(\mathbf{U}\right)$ for any $u\in \mathbf{U}$, $I_q^u$ is non void for any constant $q\in (0,1)$. For some arbitrary point $u_0\in \mathbf{U}$, there exists $u_1\in I_q^{u_0}$ such that

$$d_{\varphi }(u_1,F{u}_1)\le p{d}_{\varphi }(u_0,u_1).$$

And, for $u_1\in \mathbf{U}$, there exists $u_2\in I_q^{u_1}$ satisfying

$$d_{\varphi }(u_2,F{u}_2)\le p{d}_{\varphi }(u_1,u_2).$$

Continuing in this fashion, we can get an iterative sequence ${\left\{u_n\right\}}_{u=0}^{\infty }$, where $u_{n+1}\in I_q^{u_n}$ and

$$\begin{array}{c}\hfill d_{\varphi }(u_{n+1},F{u}_{n+1})\le p{d}_{\varphi }(u_n,u_{n+1}),n=0,1,2,\cdots .\end{array}$$

Now we will prove that ${\left\{u_n\right\}}_{n=0}^{\infty }$ is a Cauchy sequence. On the one hand,

$$d_{\varphi }(u_{n+1},F{u}_{n+1})\le p{d}_{\varphi }(u_n,u_{n+1}),n=0,1,2,\cdots .$$

(4)

On the other hand, $u_{n+1}\in I_q^{u_n}$ implies

$$q{d}_{\varphi }(u_n,u_{n+1})\le d_{\varphi }(u_n,F{u}_n),n=0,1,2,\cdots .$$

By the above two equations, we have

$$d_{\varphi }(u_{n+1},u_{n+2})\le \frac{p}{q}{d}_{\varphi }(u_n,u_{n+1}),n=0,1,2,\cdots ,$$

(5)

$$\begin{array}{c}\hfill d_{\varphi }(u_{n+1},F{u}_{n+1})\le \frac{p}{q}{d}_{\varphi }(u_n,F{u}_n),n=0,1,2,\cdots .\end{array}$$

By inequality (5), it is easy to prove that

$$\begin{array}{c}\hfill d_{\varphi }(u_n,u_{n+1})\le \frac{p^n}{q^n}{d}_{\varphi }(u_0,u_1),n=0,1,2,\cdots ,\end{array}$$

$$d_{\varphi }(u_n,F{u}_n)\le \frac{p^n}{q^n}{d}_{\varphi }(u_0,F{u}_0),n=0,1,2,\cdots .$$

(6)

Let $\alpha =\frac{p}{q}$. Since $p<q$ we have $\alpha =\frac{p}{q}<1$. By taking $n\to \infty$ in (6), we obtain

$$\underset{n\to \infty }{lim}{d}_{\varphi }(u_n,F{u}_n)=0.$$

(7)

By the triangle inequality and (6), for $m,n\in \mathbb{N}$, $m>n$

$$\begin{array}{c}\hfill d_{\varphi }(u_n,u_m)\le \varphi (u_n,u_m)[d_{\varphi }(u_n,u_{n+1})+d_{\varphi }(u_{n+1},u_m)],\end{array}$$

$$\begin{array}{c}\hfill d_{\varphi }(u_n,u_m)\le \varphi (u_n,u_m)d_{\varphi }(u_n,u_{n+1})+\varphi (u_n,u_m)\varphi (u_{n+1},u_m)[d_{\varphi }(u_{n+1},u_{n+2})+d_{\varphi }(u_{n+2},u_m)],\end{array}$$

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_m)& \le & \varphi (u_n,u_m)d_{\varphi }(u_n,u_{n+1})+\varphi (u_n,u_m)\varphi (u_{n+1},u_m)d_{\varphi }\left(u_{n+1}\right)+\cdots \cdots \hfill \\ & & +\varphi (u_n,u_m)\varphi (u_{n+1},u_m)\dots \varphi (u_{m-1},u_m)d_{\varphi }(u_{m-1},u_m),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_m)& \le & \varphi (u_n,u_m){\alpha }^n{d}_{\varphi }(u_0,u_1)+\varphi (u_n,u_m)\varphi (u_{n+1},u_m){\alpha }^{n+1}{d}_{\varphi }(u_0,u_1)+\cdots \cdots \hfill \\ & & +\varphi (u_n,u_m)\varphi (u_{n+1},u_m)\dots \varphi (u_{m-1},u_m){\alpha }^{m-1}{d}_{\varphi }(u_0,u_1),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_m)& \le & d_{\varphi }(u_0,u_1)[\varphi (u_1,u_m)\varphi (u_2,u_m)\dots \varphi (u_n,u_m){\alpha }^n+\varphi (u_1,u_m)\varphi (u_2,u_m)\dots \hfill \\ & & \varphi (u_{n+1},u_m){\alpha }^{n+1}+\dots +\varphi (u_1,u_m)\varphi (u_2,u_m)\dots \varphi (u_n,u_m)\varphi (u_{n+1},u_m)\hfill \\ & & \dots \varphi (u_{m-1},u_m){\alpha }^{m-1}].\hfill \end{array}$$

Since $\alpha <1$ so ${lim}_{n,m\to \infty }\alpha \varphi (u_n,u_m)<1.$ Therefore the series ${\sum }_{n=1}^{\infty }{\alpha }^n{\prod }_{i=1}^n\varphi (u_i,u_m)$ converges by ratio test for all $m\in \mathbb{N}.$ Let

$$S=\sum _{n=1}^{\infty }{\alpha }^n\prod _{i=1}^n\varphi (u_i,u_m),\mathrm{and}{S}_n=\sum _{j=1}^n{\alpha }^j\prod _{i=1}^j\varphi (u_i,u_m).$$

Thus for $m>n$ the above inequality implies

$$d_{\varphi }(u_n,u_m)\le d_{\varphi }(u_0,u_1)[S_{m-1}-S_n].$$

By taking $n\to \infty$, we conclude that ${\left\{u_n\right\}}_{n=0}^{\infty }$ is a Cauchy sequence. As $\mathbf{U}$ is complete, there exists $u\in \mathbf{U}$ such that ${lim}_{n\to \infty }{u}_n=u.$

On the other hand as f is lower semi-continuous, so from (7) we have

$$0\le f\left(u\right)\le \underset{n\to \infty }{lim}inff\left(u_n\right)=0.$$

Hence $f\left(u\right)=d_{\varphi }(u,Fu)=0$. Finally, by the closeness of $Fu$, we have $u\in Fu$.  ☐

Theorem 9.

Let us consider a multi-valued mapping $F:\mathbf{U}\to \mathcal{CLB}\left(\mathbf{U}\right)$, where $(\mathbf{U},d_{\varphi })$ is a complete extended b-metric space. Furthermore, let us consider that the following two conditions hold:

(i) 

The map $f:\mathbf{U}\to \mathbb{R}$ defined by $f\left(u\right)=d_{\varphi }(u,Fu)$, $u\in \mathbf{U}$, is lower semi-continuous;

(ii) 

There exist $q\in (0,1)$ and $\eta :[0,\infty )\to [0,q)$ such that

$$lim\underset{s\to t^{+}}{sup}\eta \left(s\right)<q,\mathit{for}\mathit{all}t\in [0,\infty )$$

(8)

and for all $u\in \mathbf{U}$, there exists $v\in I_q^u$ satisfying

$$d_{\varphi }(v,Fv)\le \eta \left(d_{\varphi }(u,v)\right)d_{\varphi }(u,v),\mathit{for}\mathit{all}u\in \mathbf{U}andv\in Fu.$$

(9)

Moreover ${lim}_{n,m\to \infty }\alpha \varphi (u_n,u_m)<1,$ for all $\alpha \in (0,1)$. Then F has a fixed point in $\mathbf{U}.$

Proof. 

Let us assume that F has no fixed point, so $d_{\varphi }(u,Fu)>0$ for each $u\in \mathbf{U}$. Since $Fu\in \mathcal{CLB}\left(\mathbf{U}\right),$ for any $u\in \mathbf{U}$, $I_q^u$ is non void for any constant $q\in (0,1).$ If $v=u$ then $u\in Fu$, which is a contradiction. Hence for all $q\in (0,1)$ and $u\in \mathbf{U}$, there exist $v\in Tu$ with $u\ne v$ such that

$$q{d}_{\varphi }(u,v)\le d_{\varphi }(u,Fu).$$

(10)

Let us take an arbitrary point $u_0\in \mathbf{U}$. By (10) and $\left(ii\right)$, there exists $u_1\in F{u}_0$ with $u_1\ne u_0$, satisfying

$$q{d}_{\varphi }(u_0,u_1)\le d_{\varphi }(u_0,F{u}_0),$$

(11)

and

$$d_{\varphi }(u_1,F{u}_1)\le \eta \left(d_{\varphi }(u_0,u_1)\right)d_{\varphi }(u_0,u_1),\eta (d_{\varphi }(u_0,u_1)<q.$$

(12)

From (11) and (12), we have

$$\begin{array}{ccc}\hfill d_{\varphi }(u_0,F{u}_0)-d_{\varphi }(u_1,F{u}_1)& \ge & q{d}_{\varphi }(u_0,u_1)-\eta \left(d_{\varphi }(u_0,u_1)\right)d_{\varphi }(u_0,u_1)\hfill \\ & \ge & [q-\eta \left(d_{\varphi }(u_0,u_1)\right)]d_{\varphi }(u_0,u_1)>0.\hfill \end{array}$$

Further, for $u_1$, there exists $u_2\in F{u}_1$, $u_2\ne u_1$, such that

$$q{d}_{\varphi }(u_1,u_2)\le d_{\varphi }(u_1,F{u}_1),$$

(13)

and

$$d_{\varphi }(u_2,F{u}_2)\le \eta \left(d_{\varphi }(u_1,u_2)\right)d_{\varphi }(u_1,u_2),\eta (d_{\varphi }(u_1,u_2)<q.$$

(14)

By (13) and (14), we have

$$\begin{array}{ccc}\hfill d_{\varphi }(u_1,F{u}_1)-d_{\varphi }(u_2,F{u}_2)& \ge & q{d}_{\varphi }(u_1,u_2)-\eta \left(d_{\varphi }(u_1,u_2)\right)d_{\varphi }(u_1,u_2)\hfill \\ & \ge & [q-\eta \left(d_{\varphi }(u_1,u_2)\right)]d_{\varphi }(u_1,u_2)>0.\hfill \end{array}$$

Furthermore from (12) and (13)

$$d_{\varphi }(u_1,u_2)\le \frac{1}{q}{d}_{\varphi }(u_1,F{u}_1)\le \frac{1}{q}\eta \left(d_{\varphi }(u_0,u_1)\right)d_{\varphi }(u_0,u_1)<d_{\varphi }(u_0,u_1).$$

Continuing in this fashion, for $u_n$, $n>1$, there exists $u_{n+1}\in F{u}_n$, $u_{n+1}\ne u_n$ satisfying

$$q{d}_{\varphi }(u_n,u_{n+1})\le d_{\varphi }(u_n,F{u}_n),$$

(15)

and

$$d_{\varphi }(u_{n+1},F{u}_{n+1})\le \eta \left(d_{\varphi }(u_n,u_{n+1})\right)d_{\varphi }(u_n,u_{n+1}),\eta (d_{\varphi }(u_n,u_{n+1})<q.$$

(16)

From (15) and (16), we have

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,F{u}_n)-d_{\varphi }(u_{n+1},F{u}_{n+1})& \ge & q{d}_{\varphi }(u_n,u_{n+1})-\eta \left(d_{\varphi }(u_n,u_{n+1})\right)d_{\varphi }(u_n,u_{n+1})\hfill \\ & \ge & [q-\eta \left(d_{\varphi }(u_n,u_{n+1})\right)]d_{\varphi }(u_n,u_{n+1})>0\hfill \end{array}$$

and

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

(17)

From above both equations, it follows that the sequences $\left\{d_{\varphi }(u_n,F{u}_n)\right\}$ and $\left\{d_{\varphi }(u_n,u_{n+1})\right\}$ are decreasing, and hence convergent. Now from (8), there exists $q^{{}^{\prime }}\in [0,q)$ such that ${lim}_{n\to \infty }sup\eta \left(d_{\varphi }(u_n,u_{n+1})\right)=q^{{}^{\prime }}.$ Therefore for any $q_0\in (q^{{}^{\prime }},q)$, there exists $n_0\in \mathbb{N}$ such that

$$\eta \left(d_{\varphi }(u_n,u_{n+1})\right)<q_0,\mathrm{for}\mathrm{all}n>n_0$$

(18)

Consequently from (15) and (16), we have

$$d_{\varphi }(u_n,u_{n+1})<\alpha d_{\varphi }(u_{n-1},u_n),$$

(19)

where $\alpha =\frac{q_0}{q}$ and $n>n_0.$ Furthermore, from (15)–(17), for $n>n_0$, we have

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,F{u}_n)& \le & \eta d_{\varphi }(u_{n-1},u_n)\le \frac{\eta \left(d_{\varphi }(u_{n-1},u_n)\right)}{q}{d}_{\varphi }(u_{n-1},F{u}_{n-1})\hfill \\ & \le & \dots \le \frac{(\eta \left(d_{\varphi }(u_{n-1},u_n)\right)\dots \eta \left(d_{\varphi }(u_0,u_1)\right)}{q^n}{d}_{\varphi }(u_0,F{u}_0)\hfill \\ & =& \frac{\eta \left(d_{\varphi }(u_{n-1},u_n)\right)\dots \eta \left(d_{\varphi }(u_{n_0+1},u_{n_0+2})\right)}{q^{n-n_0}}\hfill \\ & & \times \frac{\eta \left(d_{\varphi }(u_{n_0},u_{n_0+1})\right)\dots \eta \left(d_{\varphi }(u_0,u_1)\right)}{q^{n_0}}{d}_{\varphi }(u_0,F{u}_0)\hfill \\ & <& {\left(\frac{q_0}{q}\right)}^{n-n_0}\frac{\eta \left(d_{\varphi }(u_{n_0},u_{n_0+1})\right)\dots \eta \left(d_{\varphi }(u_0,u_1)\right)}{q^{n_0}}{d}_{\varphi }(u_0,F{u}_0).\hfill \end{array}$$

Since $q_0<q$, clearly ${lim}_{n\to \infty }{\left(\frac{q_0}{q}\right)}^{n-n_0}=0.$ This gives

$$\underset{n\to \infty }{lim}{d}_{\varphi }(u_n,F{u}_n)=0.$$

Let $m>n>n_0$, from the triangle inequality and (19), we have

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_m)& \le & \varphi (u_n,u_m)d_{\varphi }(u_n,u_{n+1})+\varphi (u_n,u_m)\varphi (u_{n+1},u_m)d_{\varphi }\left(u_{n+1}\right)+\cdots \cdots \hfill \\ & & +\varphi (u_n,u_m)\varphi (u_{n+1},u_m)\dots \varphi (u_{m-1},u_m)d_{\varphi }(u_{m-1},u_m),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_m)& \le & \varphi (u_n,u_m){\alpha }^n{d}_{\varphi }(u_0,u_1)+\varphi (u_n,u_m)\varphi (u_{n+1},u_m){\alpha }^{n+1}{d}_{\varphi }(u_0,u_1)+\cdots \cdots \hfill \\ & & +\varphi (u_n,u_m)\varphi (u_{n+1},u_m)\dots \varphi (u_{m-1},u_m){\alpha }^{m-1}{d}_{\varphi }(u_0,u_1).\hfill \end{array}$$

By using the analogous procedure as in Theorem 8, there exists a Cauchy sequence ${\left\{u_n\right\}}_{n=0}^{\infty }$ such that $u_{n+1}\in F{u}_n$, $u_{n+1}\ne u_n$. As $\mathbf{U}$ is complete, therefore there exists $u\in \mathbf{U}$ such that $u_n\to u.$ By $\left(i\right)$, we obtain

$$0\le d_{\varphi }(u,Fu)\le \underset{n\to \infty }{lim}inf{d}_{\varphi }(u_n,F{u}_n)=0.$$

By the closedness of $Fu$, we have $u\in Fu$, which contradicts our assumption that F has no fixed point.  ☐

Corollary 1.

Let $F:\mathbf{U}\to \mathcal{K}\left(\mathbf{U}\right)$ be a multi-valued mapping, where $(\mathbf{U},d_{\varphi })$ is a complete extended b-metric space. Furthermore, let us consider that the following conditions hold:

(i) 

The map $f:\mathbf{U}\to \mathbb{R}$ defined by $f\left(u\right)=d_{\varphi }(u,Fu)$, $u\in \mathbf{U}$, is lower semi-continuous;

(ii) 

There exists $\eta :[0,\infty )\to [0,1)$ such that

$$lim\underset{s\to t^{+}}{sup}\eta \left(s\right)<1,\mathit{for}\mathit{all}t\in [0,\infty ),$$

and for all $u\in \mathbf{U}$, there exists $v\in I_1^u$ satisfying

$$d_{\varphi }(v,Fv)\le \eta \left(d_{\varphi }(u,v)\right)d_{\varphi }(u,v),\mathit{for}\mathit{all}u\in \mathbf{U}andv\in Fu.$$

Moreover ${lim}_{n,m\to \infty }\alpha \varphi (u_n,u_m)<1,$ for all $\alpha \in (0,1)$. Then F has a fixed point in $\mathbf{U}.$

Proof. 

Let us assume that F has no fixed point, so $d_{\varphi }(u,Fu)>0$ for any $u\in \mathbf{U}$. Since $Fu\in \mathcal{K}\left(\mathbf{U}\right)$ for any $u\in \mathbf{U}$, $I_1^u$ is non empty. If $v=u$ then $u\in Fu$, which is a contradiction. Hence for all $u\in \mathbf{U}$, there exists $v\in Fu$ with $u\ne v$ such that

$$d_{\varphi }(u,v)\le d_{\varphi }(u,Fu).$$

(20)

Let us consider an arbitrary point $u_0\in \mathbf{U}$. From (20), by using the analogous procedure as in Theorem 9, we obtain the existence of a Cauchy sequence ${\left\{u_n\right\}}_{n=0}^{\infty }$ such that $u_{n+1}\in F{u}_n$, $u_{n+1}\ne u_n$, satisfying

$$d_{\varphi }(u_n,u_{n+1})=d_{\varphi }(u_n,F{u}_n)$$

and

$$d_{\varphi }(u_n,F{u}_n)\le \eta \left(d_{\varphi }(u_{n-1},u_n)\right)d_{\varphi }(u_{n-1},u_n),\eta \left(d_{\varphi }(u_{n-1},u_n)\right)<1.$$

Since $\mathbf{U}$ is complete, there exists $u\in \mathbf{U}$ such that $u_n\to u.$ By (i), we obtain

$$0\le d_{\varphi }(u,Fu)\le \underset{n\to \infty }{lim}inf{d}_{\varphi }(u_n,F{u}_n)=0.$$

By the closedness of $Fu$, we have $u\in Fu$, which contradicts our assumption that F has no fixed point.  ☐

Lemma 2.

Let $(\mathbf{U},d_{\varphi })$ be an extended b-metric space. Then for any $u\in \mathbf{U}$ and $\alpha >1$, there exists an element $x\in \mathbf{X}$, where $\mathbf{X}\in \mathcal{CLB}\left(\mathbf{U}\right)$ such that

$$d_{\varphi }(u,x)\le \alpha d_{\varphi }(u,\mathbf{X}).$$

(21)

Proof. 

Let us suppose that $d_{\varphi }(u,\mathbf{X})=0$ then $u\in \mathbf{X}$, since $\mathbf{X}$ is a closed subset of $\mathbf{U}$. Further, let us suppose that $x=u$, so (21) holds. Now, suppose that $d_{\varphi }(u,\mathbf{X})>0$ and choose

$$ϵ=(\alpha -1)d_{\varphi }(u,\mathbf{X}).$$

(22)

Then using the definition of $d_{\varphi }(u,\mathbf{X})$, there exists $x\in \mathbf{X}$ such that

$$d_{\varphi }(u,x)\le d_{\varphi }(u,\mathbf{X})+ϵ,\mathrm{where}ϵ>0.$$

(23)

By putting (22) in (23), we get

$$d_{\varphi }(u,x)\le \alpha d_{\varphi }(u,\mathbf{X}).$$

 ☐

Theorem 10.

Let $(\mathbf{U},d_{\varphi })$ be a complete extended b-metric space and $F:\mathbf{U}\to \mathcal{CLB}\left(\mathbf{U}\right)$ be a multi-valued mapping satisfying

$$d_{\varphi }(v,Fv)\le \eta \left(d_{\varphi }(u,v)\right)d_{\varphi }(u,v),forallu\in \mathbf{U}andv\in Fu,$$

(24)

where $\eta :(0,\infty )\to [0,1)$ such that

$$lim\underset{s\to t^{+}}{sup}\eta \left(s\right)<1,\mathit{for}\mathit{all}t\in [0,\infty ).$$

(25)

Moreover, let us suppose that ${lim}_{n,m\to \infty }\alpha \varphi (u_n,u_m)<1,$ for all $\alpha \in (0,1)$. Then

(i) 

There exists an orbit ${\left\{u_n\right\}}_{n=0}^{\infty }$ of F for each $u_0\in \mathbf{U}$ such that ${lim}_{n\to \infty }{u}_n=u$ for $u\in \mathbf{U}$;

(ii) 

u is a fixed point of F, if and only if the function $f\left(u\right)=d_{\varphi }(u,Fu)$ is F-orbitally lower semi-continuous at $u.$

Proof. 

Let us assume $u_0\in \mathbf{U}$ and choose $u_1\in F{u}_0$, since $F{u}_0\ne 0.$ If $u_0=u_1$, then $u_0$ is a fixed point of $F.$ Let $u_0\ne u_1$, by taking $\alpha =\frac{1}{\sqrt{\eta \left(d_{\varphi }(u_0,u_1)\right)}}$, it follows from Lemma 2 that there exists $u_2\in F{u}_1$ such that

$$d_{\varphi }(u_1,u_2)\le \frac{1}{\sqrt{\eta \left(d_{\varphi }(u_0,u_1)\right)}}{d}_{\varphi }(u_1,F{u}_1).$$

Continuing in this fashion, we produce a sequence ${\left\{u_n\right\}}_{n=1}^{\infty }$ of points in $\mathbf{U}$ such that $u_{n+1}\in F{u}_n$ and

$$d_{\varphi }(u_n,u_{n+1})\le \frac{1}{\sqrt{\eta \left(d_{\varphi }(u_{n-1},u_n)\right)}}{d}_{\varphi }(u_n,F{u}_n).$$

(26)

Now assume that $u_{n-1}\ne u_n$, for otherwise $u_{n-1}$ is fixed point of $F.$ Using (24), it follows from (26) that

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_{n+1})& \le & \sqrt{\eta \left(d_{\varphi }(u_{n-1},u_n)\right)}{d}_{\varphi }(u_{n-1},u_n)\hfill \\ & <& d_{\varphi }(u_{n-1},u_n).\hfill \end{array}$$

(27)

Hence $\left\{d_{\varphi }(u_n,u_{n+1})\right\}$ is a decreasing sequence, so it is converges to some non-negative real number. Let a be the limit of $\left\{d_{\varphi }(u_n,u_{n+1})\right\}$. Clearly, $a=0$, for otherwise by taking limits in (27), we obtain $a\le \sqrt{c}a$, where $c=lim{sup}_{s\to a^{+}}\eta \left(s\right).$ From (27), we have

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_{n+1})& \le & \sqrt{\eta \left(d_{\varphi }(u_{n-1},u_n)\right)}\sqrt{\eta \left(d_{\varphi }(u_{n-2},u_{n-1})\right)}{d}_{\varphi }(u_{n-2},u_{n-1})\dots \hfill \\ & & \dots \le \sqrt{\eta \left(d_{\varphi }(u_{n-1},u_n)\right)}\dots \sqrt{\eta \left(d_{\varphi }(u_0,u_1)\right)}]d_{\varphi }(u_0,u_1).\hfill \end{array}$$

From (25), we can choose $\delta >0$ and $\alpha \in (0,1)$ such that

$$\eta \left(t\right)<{\alpha }^2,\mathrm{for}t\in (0,\delta ).$$

Let N be such that $d_{\varphi }(u_{n-1},u_n)<\delta$ for $n\ge N.$ From (27), we have

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_{n+1})& \le & \alpha d_{\varphi }(u_{n-1},u_n)\le \dots \hfill \\ & \le & {\alpha }^{n-N+1}{d}_{\varphi }(u_{N-1},u_n).\hfill \end{array}$$

Hence from the inequality (27), we get

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_{n+1})& \le & {\alpha }^{n-N+1}[\sqrt{\eta \left(d_{\varphi }(u_{N-2},u_{N-1})\right)}\dots \sqrt{\eta \left(d_{\varphi }(u_0,u_1)\right)}]d_{\varphi }(u_0,u_1)\hfill \\ & & <{\alpha }^{n-N+1}{d}_{\varphi }(u_0,u_1).\hfill \end{array}$$

(28)

Therefore from the triangle inequality and (28) for any $m\in \mathbb{N}$ with $m>n$, we have

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_{n+m})& \le & \varphi (u_n,u_{n+m})d_{\varphi }(u_n,u_{n+1})+\varphi (u_n,u_{n+m})\varphi (u_{n+1},u_{n+m})d_{\varphi }(u_{n+1},u_{n+2})+\hfill \\ & & \cdots \cdots +\varphi (u_n,u_{n+m})\varphi (u_{n+1},u_{n+m})\dots \varphi (u_{n+m-1},u_{n+m})\hfill \\ & & d_{\varphi }(u_{n+m-1},u_{n+m}),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_{n+m})& \le & {\alpha }^{n-N+1}[\varphi (u_n,u_{n+m})+{\alpha }^2\varphi (u_n,u_{n+m})\varphi (u_{n+1},u_{n+m})+\cdots \cdots +{\alpha }^{m-n-1}\hfill \\ & & \varphi (u_n,u_{n+m})\varphi (u_{n+1},u_{n+m})\dots \varphi (u_{n+m-1},u_{n+m})]d_{\varphi }(u_0,u_1),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,u_{n+m})& \le & {\alpha }^{n-N+1}[\varphi (u_1,u_{n+m})\varphi (u_2,u_{n+m})\dots \varphi (u_n,u_{n+m})+\varphi (u_1,u_{n+m})\hfill \\ & & \varphi (u_2,u_{n+m})\dots \varphi (u_{n+m-1},u_{n+m})]d_{\varphi }(u_0,u_1).\hfill \end{array}$$

Since ${lim}_{n,m\to \infty }\varphi (u_n,u_m)\alpha <1$, the series ${\sum }_{j=1}^{\infty }{\alpha }^j{\prod }_{i=1}^j\varphi (u_j,u_{n+m})$ converges by the ratio test for each $m\in \mathbb{N}.$ Let

$$S=\sum _{j=1}^{\infty }{\alpha }^j\prod _{i=1}^j\varphi (u_i,u_{n+m}),S_n=\sum _{j=1}^n{\alpha }^j\prod _{i=1}^j\varphi (u_i,u_{n+m}).$$

Thus for $m\in \mathbb{N}$ with $m>n$, the above inequality implies

$$d_{\varphi }(u_n,u_{n+m})\le {\alpha }^{n-N+1}[S_{m-1}-S_n].$$

By letting $n\to \infty$, we conclude that ${\left\{u_n\right\}}_{n=1}^{\infty }$ is a Cauchy sequence in $\mathbf{U}.$ As $\mathbf{U}$ is complete, there exists $u\in \mathbf{U}$ such that ${lim}_{n\to \infty }{u}_n=u.$ Since $u_n\in F{u}_{n-1}$, it follows from (24) that

$$\begin{array}{ccc}\hfill d_{\varphi }(u_n,F{u}_n)& \le & \eta \left(d_{\varphi }(u_{n-1},u_n)\right)d_{\varphi }(u_{n-1},u_n)\hfill \\ & & <d_{\varphi }(u_{n-1},u_n).\hfill \end{array}$$

Letting $n\to \infty$, from the above inequality we have

$$\underset{n\to \infty }{lim}{d}_{\varphi }(u_n,F{u}_n)=0.$$

Suppose $f\left(u\right)=d_{\varphi }(u,Fu)$ is F orbitally semi-continuous at u,

$$d_{\varphi }(u,Fu)=f\left(u\right)\le \underset{n\to \infty }{lim}inff\left(u_n\right)=\underset{n\to \infty }{lim}inf{d}_{\varphi }(u_n,F{u}_n)=0.$$

Hence $u\in Fu$, since $Fu$ is closed. Conversely let us suppose that u is a fixed point of F ($u\in Fu$), then $f\left(u\right)=0\le {lim}_{n\to \infty }inff\left(u_n\right).$ Hence f is F orbitally semi-continuous at $u.$  ☐

Remark 2.

Theorem 10 improves Theorem 1, since F may take values in $\mathcal{CLB}\left(\mathbf{U}\right).$ Since $d_{\varphi }(v,Fv)\le H(Fu,Fv)$ for $v\in Fu$. We have the following corollary.

Corollary 2.

Let $(\mathbf{U},d_{\varphi })$ be a complete extended b-metric space and $F:\mathbf{U}\to \mathcal{CLB}\left(\mathbf{U}\right)$ be such that

$${\mathbf{H}}_{\mathsf{\Phi }}(Fu,Fv)\le \eta \left(d_{\varphi }(u,v)\right)d_{\varphi }(u,v),\mathit{for}\mathrm{each}u\in \mathbf{U}andv\in Fu,$$

where $\eta :(0,\infty )\to (0,1]$ is such that

$$lim\underset{s\to t^{+}}{sup}\eta \left(s\right)<1,\mathit{for}\mathit{all}t\in [0,\infty ).$$

Then

(i) 

there exist an orbit ${\left\{u_n\right\}}_{n=0}^{\infty }$ of F for each $u_0\in \mathbf{U}$ and $u\in \mathbf{U}$ such that ${lim}_{n\to \infty }{u}_n=u;$

(ii) 

u is a fixed point of F, if and only if the function $f\left(u\right)=d_{\varphi }(u,Fu)$ is F-orbitally lower semi-continuous at $u.$

Remark 3.

Theorem 7 extends Nadler’s fixed point theorem when $\mathbf{U}$ is the extended b-metric space.

Remark 4.

Theorem 8 is a generalization of 7. The following example shows that generalization.

Example 2.

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

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

In a sense of Theorem 7, clearly F is not contractive, in fact

$${\mathbf{H}}_{\mathsf{\Phi }}\left(F\left(\frac{1}{2^n}\right),F\left(0\right)\right)=\frac{1}{2}\ge \frac{1}{2^{2n}}=d_{\varphi }(u_1,u_2),forn=1,2,3,\dots .$$

On the other way,

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

Hence f is continuous, so it is clearly lower semi-continuous. Furthermore there exists $v\in I_{0.7}^u$ for any $u\in \mathbf{U}$ such that

$$d_{\varphi }(v,F\left(v\right))=\frac{1}{4}{d}_{\varphi }(u,v).$$

Thus the existence of a fixed point follows from Theorem 8. Hence Theorem 8 is a generalization of Theorem 7.

Remark 5.

Theorem 9 is an extension of Theorem 8. In fact, let us consider a constant map $\eta =c,$ where $0<c<q.$ Thus the hypotheses of Theorem 9 are fulfilled. On the other hand, there exists a map which fulfills the hypotheses of Theorem 9, but does not fulfill the hypotheses of Theorem 8. See the following example:

Example 3.

Let $\mathbf{U}=[0,1]$ and $d_{\varphi }:\mathbf{U}\times \mathbf{U}\to [0,\infty )$ be a mapping defined as $d_{\varphi }(u_1,u_2)={(u_1-u_2)}^2$, for $u_1,u_2\in \mathbf{U}$, where $\varphi :\mathbf{U}\times \mathbf{U}\to [1,\infty )$ is a mapping defined by $\varphi (u_1,u_2)=u_1+u_2+2.$ Then $(\mathbf{U},d_{\varphi })$ is a complete extended b-metric space. Let $F:\mathbf{U}\to \mathcal{CLB}\left(\mathbf{U}\right)$ be such that

$$F\left(u\right)=\left\{\begin{array}{cc}\left\{\frac{1}{2^{u^2}}\right\},\hfill & \mathrm{u}\in [0,\frac{15}{32})\cup (\frac{15}{32},1],\hfill \\ \{\frac{17}{96},\frac{1}{4}\},\hfill & \mathrm{u}=\frac{15}{32}.\hfill \end{array}\right.$$

Let $q=\frac{3}{4}$ and let $\eta :[0,\infty )\to [0,q)$ be of the form

$$\eta \left(t\right)=\left\{\begin{array}{cc}\frac{3}{2}t,\hfill & \mathit{for}\mathrm{t}\in [0,\frac{7}{24})\cup (\frac{7}{24},\frac{1}{2}),\hfill \\ \frac{425}{768},\hfill & \mathit{for}\mathrm{t}=\frac{7}{24},\hfill \\ \frac{1}{2},\hfill & \mathit{for}\mathrm{t}=[\frac{1}{2},\infty ).\hfill \end{array}\right.$$

Since

$$f\left(u\right)=\left\{\begin{array}{cc}{(u-\frac{1}{2}{u}^2)}^2,\hfill & \mathit{for}\mathrm{u}\in [0,\frac{15}{32})\cup (\frac{15}{32},1],\hfill \\ \frac{49}{1024},\hfill & \mathit{for}\mathrm{u}=\frac{15}{32}.\hfill \end{array}\right.$$

Obviously f is a lower semi-continuous. Further, for any $u\in [0,\frac{15}{32})\cup (\frac{15}{32},1]$ and $v=\frac{1}{2}{u}^2$, we have

$$q{d}_{\varphi }(u,v)\le d_{\varphi }(u,Fu),$$

and

$$d_{\varphi }(v,Fv)\le \eta \left(d_{\varphi }(u,v)\right)d_{\varphi }(u,v).$$

Of course these both inequalities hold for $u=\frac{15}{32}$ and $v=\frac{17}{96}.$ Hence all the hypotheses of Theorem 9 are satisfied and the fixed point of F is $\left\{0\right\}.$ Next let us suppose that, if $q\in (0,\frac{3}{4}]$ and $p\in (0,1)$ is such that $p<q$, then, for $u=1$, we have $v=1/2$ and consequently

$$d_{\varphi }\left(\frac{1}{2},F\left(\frac{1}{2}\right)\right)>p{d}_{\varphi }\left(1,\frac{1}{2}\right).$$

If $q\in (3/4,1)$ and $p\in (0,1)$ is such that $p<q$, then for $u=\frac{15}{32}$, we have $Fu=\{\frac{17}{96},\frac{1}{4}\}$. Thus, in the case $v=\frac{17}{96}$, we obtain

$$q{d}_{\varphi }\left(\frac{15}{32},\frac{17}{96}\right)>d_{\varphi }\left(\frac{15}{32},F\left(\frac{15}{32}\right)\right),$$

and, in the case $v=\frac{1}{4}$, we have

$$d_{\varphi }\left(\frac{1}{4},F\left(\frac{1}{4}\right)\right)>p{d}_{\varphi }\left(\frac{15}{32},\frac{1}{4}\right).$$

Hence hypotheses of Theorem 8 are not fulfilled.

Remark 6.

Theorem 10 is an extension of (Theorem 2.1, [10]) for the case when F is a multi-valued mapping from $\mathbf{U}$ to $\mathcal{CLB}\left(\mathbf{U}\right)$ and hence generalizes Theorems 4 and 5 and also the results of [2,5,7,22].