New coincidence point and fixed point theorems for essential distances and $e^{0}$-metrics

TL;DR

This paper introduces new fixed point and coincidence point theorems for essential distances and $e^{0}$-metrics, generalizing and improving several classical fixed point results and principles in metric space theory.

Contribution

It presents novel theorems that extend existing fixed point results to essential distances and $e^{0}$-metrics, broadening their applicability.

Findings

Generalized fixed point theorems for essential distances and $e^{0}$-metrics

Improved upon classical fixed point theorems

Unified various fixed point principles in a broader framework

Abstract

In this paper, we establish some new fixed point theorems and coincidence point theorems for essential distances and $e^{0}$-metrics which generalize and improve Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, Nadler's fixed point theorem and Banach contraction principle and many known results in the literature.

Equations

$$\mathcal{H}(A,B)=\text{max}\left\{\underset{x\in B}{\sup }d(x,A)\text{,}\underset{x\in A}{\sup }d(x,B)\right\}$$

$$\mathcal{H}(Tx,Ty)\le \alpha (d(x,y))d(x,y)+Ld(y,Tx)\text{,}$$

$$\kappa (x,C)=\underset{y\in C}{\inf }\kappa (x,y).$$

$${\mathcal{D}}_{\kappa }(A,B)=\text{max}\{{\xi }_{\kappa }(A,B),{\xi }_{\kappa }(B,A)\}\text{,}$$

$$\kappa (y,z)\le \mu (k(x,y))\kappa (x,y)\text{;}$$

Full text

New coincidence point and fixed point theorems for essential distances and $e^{0}$-metrics

**Wei-Shih Du111E-mail address: wsdu@mail.nknu.edu.tw; Tel: +886-7-7172930 ext 6809; Fax: +886-7-6051061.

**

Department of Mathematics, National Kaohsiung Normal University,

Kaohsiung 82444, Taiwan

**Abstract: **In this paper, we establish some new fixed point theorems and coincidence point theorems for essential distances and $e^{0}$-metrics which generalize and improve Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem and Banach contraction principle and many known results in the literature.

2010 Mathematics Subject Classification: 47H10, 54H25.

Key words and phrases: $\mathcal{MT}$-function, $\mathcal{MT}(\lambda)$-function, $\tau$-function, coincidence point, essential distance ($e$-distance), $e^{0}$-metric, Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, Banach contraction principle.

1. Introduction and preliminaries

Let $(X,d)$ be a metric space. Denote by $\mathcal{N}(X)$ the class of all nonempty subsets of $X$, $\mathcal{C}(X)$ the class of all nonempty closed subsets of $X$ and $\mathcal{CB}(X)$ the family of all nonempty closed and bounded subsets of $X$. A function $\mathcal{H}:\mathcal{CB}(X)\times\mathcal{CB}(X)\rightarrow[0,\infty)$ defined by

[TABLE]

is said to be the Hausdorff metric on $\mathcal{CB}(X)$ induced by the metric $d$ on $X$. A point $v$ in $X$ is a fixed point of a mapping $T$ if $v=Tv$ (when $T:X\rightarrow X$ is a single-valued mapping) or $v\in Tv$ (when $T:X\rightarrow\mathcal{N}(X)$ is a multivalued mapping). The set of fixed points of $T$ is denoted by $\mathcal{F}(T)$. Let $g:X\rightarrow X$ be a self-mapping and $T:X\rightarrow\mathcal{N}(X)$ be a multivalued mapping. A point $x$ in $X$ is said to be a coincidence point of $g$ and $T$ if $gx\in Tx$. The set of coincidence point of $g$ and $T$ is denoted by $\mathcal{COP}(g,T)$. Throughout this paper, we denote by $\mathbb{N}$ and $\mathbb{R}$, the set of positive integers and real numbers, respectively.

In 2007, M. Berinde and V. Berinde [2] established the following interesting fixed point theorem which generalizes Mizoguchi-Takahashi’s fixed point theorem [13], Nadler’s fixed point theorem [14] and Banach contraction principle [1].

Theorem 1.1 (M. Berinde and V. Berinde [2]). Let $(X,d)$ be a complete metric space, $T:X\rightarrow\mathcal{CB}(X)$ be a multivalued mapping and $L\geq 0$. Assume that

[TABLE]

for all $x,y\in X$, where $\alpha$ is an $\mathcal{MT}$-function, that is, $\alpha$ is a function from $[0,\infty)$ into $[0,1)$ satisfying $\limsup_{s\rightarrow t+0}\alpha(s)<1$ for all $t\in[0,\infty)$. Then there exists $v\in X$ such that $v\in Tv$.

Let $(X,d)$ be a metric space. Recall that a function $\kappa:X\times X\rightarrow[0,\infty)$ is said to be a $\tau$-function [3, 4, 6, 7, 10, 12], introduced and studied by Lin and Du, if the following conditions hold:

1. ($\tau 1$)

$\kappa(x,z)\leq\kappa(x,y)+\kappa(y,z)$ for all $x,y,z\in X$; 2. ($\tau 2$)

if $x\in X$ and $\{y_{n}\}$ in $X$ with $\lim_{n\rightarrow\infty}y_{n}=y$ such that $\kappa(x,y_{n})\leq M$ for some $M=M(x)>0$, then $\kappa(x,y)\leq M$; 3. ($\tau 3$)

for any sequence $\{x_{n}\}$ in $X$ with $\lim_{n\rightarrow\infty}\sup\{\kappa(x_{n},x_{m}):m>n\}=0$, if there exists a sequence $\{y_{n}\}$ in $X$ such that $\lim_{n\rightarrow\infty}\kappa(x_{n},y_{n})=0$, then $\lim_{n\rightarrow\infty}\rho(x_{n},y_{n})=0$; 4. ($\tau 4$)

for $x,$ $y,$ $z\in X$, $\kappa(x,y)=0$ and $\kappa(x,z)=0$ imply $y=z$.

It is obvious that the metric $d$ is a $w$-distance [3, 4, 6, 7, 9-12] and any $w$-distance is a $\tau$-function, but the converse is not true; see [3, 10, 12] for more detail.

The following result is very crucial in our proofs.

Lemma 1.1 (see [8, Lemma 1.1]). If condition $(\tau 4)$ is weakened to the following condition $(\tau 4)^{\prime}:$

1. $(\tau 4)^{\prime}$

for any $x\in X$ with $\kappa(x,x)=0$, if $\kappa(x,y)=0$ and $\kappa(x,z)=0$, then $y=z$,

then $(\tau 3)$ implies $(\tau 4)^{\prime}$.

In 2016, Du [8] introduced the concept of essential distance as follows.

Definition 1.1 (see [8, Definition 1.2]). Let $(X,d)$ be a metric space. A function $\kappa:X\times X\rightarrow[0,+\infty)$ is called an essential distance (abbreviated as ”$e$-distance”) if conditions $(\tau 1)$, $(\tau 2)$ and $(\tau 3)$ hold.

Remark 1.1. Clearly any $\tau$-function is an $e$-distance. By Lemma 1.1, we know that if $\kappa$ is an $e$-distance, then condition $(\tau 4)^{\prime}$ holds.

The following known result is very crucial in this paper.

Lemma 1.2 (see [4, Lemma 2.1]). Let $(X,d)$ be a metric space and $\kappa:X\times X\rightarrow[0,+\infty)$ be a function. Assume that $\kappa$ satisfies the condition $(\tau 3)$. If a sequence $\{x_{n}\}$ in $X$ with $\lim_{n\rightarrow\infty}\sup\{\kappa(x_{n},x_{m}):m>n\}=0$, then $\{x_{n}\}$ is a Cauchy sequence in $X$.

In 2016, Du introduced the concept of $\mathcal{MT}(\lambda)$-function [7] as follows.

Definition 1.2. Let $\lambda>0$. A function $\mu:$ $[0,+\infty)\rightarrow$ $[0,\lambda)$ is said to be an $\mathcal{MT}(\lambda)$-function [7] if $\limsup\limits_{s\rightarrow t^{+}}\mu(s)<\lambda$ for all $t\in[0,\infty)$. As usual, we simply write ”$\mathcal{MT}$-function (see [3-6])” instead of ”$\mathcal{MT}(1)$-function”.

In [7], Du established the following useful characterizations of $\mathcal{MT}(\lambda)$-functions.

Theorem 1.2 (see [7, Theorem 2.4]). Let $\lambda>0$ and let $\mu:$ $[0,+\infty)\rightarrow$ $[0,\lambda)$ be a function. Then the following statements are equivalent.

1. (1)

$\mu$* is an $\mathcal{MT}(\lambda)$-function.* 2. (2)

$\lambda^{-1}\mu$* is an $\mathcal{MT}$-function.* 3. (3)

For each $t\in[0,\infty)$, there exist $\xi_{t}^{(1)}\in[0,\lambda)$ and $\epsilon_{t}^{(1)}>0$ such that $\mu(s)\leq\xi_{t}^{(1)}$ for all $s\in(t,t+\epsilon_{t}^{(1)})$. 4. (4)

For each $t\in[0,\infty)$, there exist $\xi_{t}^{(2)}\in[0,\lambda)$ and $\epsilon_{t}^{(2)}>0$ such that $\mu(s)\leq\xi_{t}^{(2)}$ for all $s\in[t,t+\epsilon_{t}^{(2)}]$. 5. (5)

For each $t\in[0,\infty)$, there exist $\xi_{t}^{(3)}\in[0,\lambda)$ and $\epsilon_{t}^{(3)}>0$ such that $\mu(s)\leq\xi_{t}^{(3)}$ for all $s\in(t,t+\epsilon_{t}^{(3)}]$. 6. (6)

For each $t\in[0,\infty)$, there exist $\xi_{t}^{(4)}\in[0,\lambda)$ and $\epsilon_{t}^{(4)}>0$ such that $\mu(s)\leq\xi_{t}^{(4)}$ for all $s\in[t,t+\epsilon_{t}^{(4)})$. 7. (7)

For any nonincreasing sequence $\{x_{n}\}_{n\in\mathbb{N}}$ in $[0,\infty)$, we have $0\leq\sup\limits_{n\in\mathbb{N}}\mu(x_{n})<\lambda$. 8. (8)

For any strictly decreasing sequence $\{x_{n}\}_{n\in\mathbb{N}}$ in $[0,\infty)$, we have $0\leq\sup\limits_{n\in\mathbb{N}}\mu(x_{n})<\lambda$. 9. (9)

*For any eventually nonincreasing sequence $\{x_{n}\}_{n\in\mathbb{N}}$ *(i.e. there exists $\ell\in\mathbb{N}$ such that $x_{n+1}\leq x_{n}$ for all $n\in\mathbb{N}$ with $n\geq\ell$) in $[0,\infty)$, we have $0\leq\sup\limits_{n\in\mathbb{N}}\mu(x_{n})<\lambda$. 10. (10)

*For any eventually strictly decreasing sequence $\{x_{n}\}_{n\in\mathbb{N}}$ *(i.e. there exists $\ell\in\mathbb{N}$ such that $x_{n+1}<x_{n}$ for all $n\in\mathbb{N}$ with $n\geq\ell$) in $[0,\infty)$, we have $0\leq\sup\limits_{n\in\mathbb{N}}\mu(x_{n})<\lambda$.

Let $\kappa$ be an $e$-function. For each $x\in X$ and $C\subseteq X$, denote by

[TABLE]

Lemma 1.3 (see [3, Lemma 1.2]. Let $C$ be a closed subset of a metric space $(X,d)$ and $\kappa$ be a function satisfying the condition $(\tau 3)$. Suppose that there exists $z\in X$ such that $\kappa(z,z)=0$. Then $\kappa(z,C)=0$ if and only if $z\in C$.

Now, we introduce the concepts of $e^{0}$-distance and $e^{0}$-metric.

Definition 1.3. Let $(X,d)$ be a metric space. A function $\kappa:X\times X\rightarrow[0,\infty)$ is called an $e^{0}$-distance if it is an $e$-distance on $X$ with $\kappa(x,x)=0$ for all $x\in X$.

Definition 1.4. Let $(X,d)$ be a metric space and $\kappa$ be an $e^{0}$-distance. For any $A$, $B\in\mathcal{CB}(X)$, define a function $\mathcal{D}_{\kappa}:\mathcal{CB}(X)\times\mathcal{CB}(X)\rightarrow[0,+\infty)$ by

[TABLE]

where $\xi_{\kappa}(A,B)=\sup_{x\in A}\kappa(x,B)$, then $\mathcal{D}_{\kappa}$ is said to be the $e^{0}$-$metric$ on $\mathcal{CB}(X)$ induced by $\kappa$.

Clearly, any Hausdorff metric is an $e^{0}$-metric, but the reverse is not true. It is not difficult to prove the following theorem.

Theorem 1.3. Let $(X,d)$ be a metric space and $\mathcal{D}_{\kappa}$ be an $e^{0}$-metric defined as in Def. 1.4 on $\mathcal{CB}(X)$ induced by an $e^{0}$-distance $\kappa$. Then for $A$, $B$, $C\in\mathcal{CB}(X)$, the following hold:

1. (i)

$\xi_{\kappa}(A,B)=0$* $\Longleftrightarrow$ $A\subseteq B$;* 2. (ii)

$\xi_{\kappa}(A,B)\leq\xi_{\kappa}(A,C)+\xi_{\kappa}(C,B)$; 3. (iii)

Every $e^{0}$-metric $\mathcal{D}_{\kappa}$ is a metric on $\mathcal{CB}(X)$.

In this paper, some new fixed point theorems and coincidence point theorems for essential distances and $e^{0}$-metrics are established. Our new results generalize and improve Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, Banach contraction principle and many known results in the literature.

2. New fixed point and coincidence point theorems

In this section, we first establish the following new fixed point theorem for $e^{0}$-distances which extends and generalizes Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem and Banach contraction principle.

Theorem 2.1. Let $(X,d)$ be a complete metric space, $\kappa$ be a $e^{0}$-distance and $T:X\rightarrow\mathcal{C}(X)$ be a multivalued mapping. Suppose that

1. $(S1)$

there exists an $\mathcal{MT}$-function $\mu:$ $[0,\infty)\rightarrow$ $[0,1)$ such that for each $x\in X$, if $y\in Tx$ with $y\neq x$ then there exists $z\in Ty$ such that

[TABLE] 2. $(S2)$

$T$* further satisfies one of the following conditions:*

1. (H1)

$T$* is closed;* 2. (H2)

the function $f:X\rightarrow[0,\infty)$ defined by $f(x)=\kappa(x,Tx)$ is l.s.c.; 3. (H3)

the function $g:X\rightarrow[0,\infty)$ defined by $g(x)=d(x,Tx)$ is l.s.c.; 4. (H4)

for any sequence $\{x_{n}\}$ in $X$ with $x_{n+1}\in Tx_{n}$, $n\in\mathbb{N}$ and $\lim_{n\rightarrow\infty}x_{n}=v$, we have $\lim_{n\rightarrow\infty}\kappa(x_{n},Tv)=0$; 5. (H5)

$\inf\{\kappa(x,z)+\kappa(x,Tx):$* $x\in X\}>0$ for every $z\notin\mathcal{F}(T)$.*

Then $\mathcal{F}(T)\neq\emptyset$.

Proof. Since $\kappa$ is a $e^{0}$-distance, by Lemma 1.1, we know that for $x,y\in X$, $\kappa(x,y)=0$* $\Longleftrightarrow$ $x=y$*. Following a similar argument as the proof of [6, Lemma 3.1], we can prove the conclusion. $\Box$

The following result is an immediate consequence of Theorem 2.1.

Theorem 2.2. Let $(X,d)$ be a complete metric space, $\kappa$ be a $e^{0}$-distance, $T:X\rightarrow C(X)$ be a multivalued mapping and $\mu:$ $[0,\infty)\rightarrow$ $[0,1)$ be an $\mathcal{MT}$-function. Suppose that the condition $(S2)$ as in Theorem 2.1 holds and further assume that

1. $(S3)$

for each $x\in X$, $\kappa(y,Ty)\leq\mu(\kappa(x,y))\kappa(x,y)$ for all $y\in Tx$.

Then $\mathcal{F}(T)\neq\emptyset$.

By applying Theorem 2.2, we establishe the following existence theorem of coincidence point and fixed point.

Theorem 2.3. Let $(X,d)$ be a complete metric space, $\kappa$ be a $e^{0}$-distance, $T:X\rightarrow\mathcal{C}(X)$ be a multivalued mapping, $\varphi:X\rightarrow X$ be a continuous self-mapping, $\mu:$ $[0,\infty)\rightarrow$ $[0,1)$ be an $\mathcal{MT}$-function and $L\geq 0$. Suppose that the condition $(S2)$ as in Theorem 2.1 holds and further assume

1. $(S4)$

$Tx$* is $\varphi$-invariant (i.e. $\varphi(Tx)\subseteq Tx$) for each $x\in X$;* 2. $(S5)$

$\kappa(y,Ty)\leq\mu(\kappa(x,y))\kappa(x,y)+L\kappa(\varphi y,Tx)$* for all $x,y\in X$.*

Then $\mathcal{COP}(\varphi,T)\cap\mathcal{F}(T)\neq\emptyset$.

The following existence theorem of coincidence point and fixed point for $e^{0}$-distances and $e^{0}$-metrics is an immediate consequence of Theorem 2.3.

Theorem 2.4. Let $(X,d)$ be a complete metric space, $\kappa$ be a $e^{0}$-distance, $\mathcal{D}_{\kappa}$ be a $e^{0}$-metric on $\mathcal{CB}(X)$, $T:X\rightarrow\mathcal{CB}(X)$ be a multivalued mapping, $\varphi:X\rightarrow X$ be a continuous self-mapping, $\mu:$ $[0,\infty)\rightarrow$ $[0,1)$ be an $\mathcal{MT}$-function and $L\geq 0$. Suppose that the conditions $(S2)$ and $(S4)$ hold and further assume

1. $(S6)$

$\mathcal{D}_{\kappa}(Tx,Ty)\leq\mu(\kappa(x,y))\kappa(x,y)+L\kappa(\varphi y,Tx)$* for all $x,y\in X$.*

Then $\mathcal{COP}(\varphi,T)\cap\mathcal{F}(T)\neq\emptyset$.

Remark 2.1. Theorems 2.1-2.4 all improve and generalize Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, Banach contraction principle and some results in [3, 5, 6] and references therein.

Acknowledgments: The author is supported by Grant No. MOST 107-2115-M-017-004-MY2 of the Ministry of Science and Technology of the Republic of China.

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