On a New Generalization of Banach Contraction Principle with Application

Abstract

The main purpose of the current work is to present firstly a new generalization of Caristi’s fixed point result and secondly the Banach contraction principle. An example and an application is given to show the usability of our results.

1. Introduction and Preliminaries

Metric fixed point theory plays a crucial role in the field of functional analysis. It was first introduced by the great Polish mathematician Banach [1]. Over the years, due to its significance and application in different fields of science, a lot of generalizations have been done in different directions by several authors see, for example, [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] and references therein. Assuredly, the Caristi’s fixed point theorem [18] is the most valuable generalization of this principle.

For any nonempty set $\mathsf{\Lambda }$, set:

$$\mathsf{\Xi }=\{\mathsf{\varrho }:\mathsf{\Lambda }\to \mathbb{R}:\mathsf{\varrho }\mathrm{is}\mathrm{a}\mathrm{lower}\mathrm{semi-continuous}\mathrm{and}\mathrm{bounded}\mathrm{below}\mathrm{function}\}.$$

Theorem 1.

[18] Let $(\mathsf{\Lambda },d)$ be a complete metric space and $\mathsf{\Gamma }:\mathsf{\Lambda }\to \mathsf{\Lambda }$ be a self-map. If there exists $\mathsf{\varrho }\in \mathsf{\Xi }$ such that:

$$d(\eta ,\mathsf{\Gamma }\eta )\le \mathsf{\varrho }\left(\eta \right)-\mathsf{\varrho }\left(\mathsf{\Gamma }\eta \right)$$

for all $\eta \in \mathsf{\Lambda }$. Then Γ has a fixed point.

Recently, Du [19] established a direct proof of Caristi’s fixed point theorem without using Zorn’s lemma. In the next section we introduce a new generalization of Caristi’s fixed point theorem and provide the proof without using Zorn’s lemma.

2. A Generalization of Caristi’s Fixed Point Theorem

Let $\mathsf{\Omega }$ be the collection of functions $\vartheta :\mathbb{R}\to (0,\infty )$ satisfying the following conditions:

(${\mathsf{\Omega }}_1$)

$\vartheta$ is strictly increasing and continuous;

(${\mathsf{\Omega }}_2$)

For every sequence $\left\{{\alpha }_n\right\}\subseteq R^{+}$, ${lim}_{n\to \infty }{\alpha }_n=0$ if and only if ${lim}_{n\to \infty }\vartheta \left({\alpha }_n\right)=1;$

(${\mathsf{\Omega }}_3$)

For every $\alpha ,\beta \in \mathbb{R}$, $\vartheta (\alpha +\beta )\le \vartheta \left(\alpha \right)\vartheta \left(\beta \right);$

Obviously, for a function $\vartheta$, satisfying (${\mathsf{\Omega }}_2$), $\vartheta \left(\alpha \right)=1$ iff $\alpha =0$.

Example 1.

${\vartheta }_1\left(t\right)=1+tanht\in \mathsf{\Omega }$, ${\vartheta }_2\left(t\right)=e^t$,

$${\vartheta }_3\left(t\right)=\left\{\begin{array}{cc}1+ln(1+t),\hfill & ift\in \left[0,\infty \right),\hfill \\ e^t,\hfill & ift\in \left(-\infty ,0\right],\hfill \end{array}\right.$$

are some elements in $\mathsf{\Omega }.$

Theorem 2.

Let $(\mathsf{\Lambda },d)$ be a complete metric space and $\mathsf{\Gamma }:\mathsf{\Lambda }\to \mathsf{\Lambda }$ be a self-map. If there exist $\mathsf{\varrho }\in \mathsf{\Xi }$ and $\vartheta \in \mathsf{\Omega }$ such that:

$$\vartheta \left(d(\eta ,\mathsf{\Gamma }\eta )\right)\le {\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\right(\eta \left)\right)}{\vartheta \left(\mathsf{\varrho }\right(\mathsf{\Gamma }\eta \left)\right)}},$$

(1)

for all $\eta \in \mathsf{\Lambda }$, then Γ has a fixed point.

Proof. 

For any $\eta \in \mathsf{\Lambda }$, define:

$$\mathsf{{\rm Y}}\eta =\{\mu \in \mathsf{\Lambda }:\vartheta \left(d(\eta ,\mu )\right)\le {\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\right(\eta \left)\right)}{\vartheta \left(\mathsf{\varrho }\right(\mu \left)\right)}}\}.$$

Obviously, $\mathsf{{\rm Y}}\eta \ne \varnothing$ for any $\eta \in \mathsf{\Lambda }$, since $\eta \in \mathsf{{\rm Y}}\eta$. Let us firstly show that for any $\mu \in \mathsf{{\rm Y}}\eta$, we have $\mathsf{\varrho }\left(\mu \right)\le \mathsf{\varrho }\left(\eta \right)$ and $\mathsf{{\rm Y}}\mu \subseteq \mathsf{{\rm Y}}\eta$. Suppose $\mu \in \mathsf{{\rm Y}}\eta$. Then,

$$1\le \vartheta \left(d(\eta ,\mu )\right)\le {\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\right(\eta \left)\right)}{\vartheta \left(\mathsf{\varrho }\right(\mu \left)\right)}},$$

(2)

which implies $\vartheta \left(\mathsf{\varrho }\right(\mu \left)\right)\le \vartheta \left(\mathsf{\varrho }\right(\eta \left)\right)$ and since $\vartheta$ is strictly increasing, we get $\mathsf{\varrho }\left(\mu \right)\le \mathsf{\varrho }\left(\eta \right)$. Now let $\zeta \in \mathsf{{\rm Y}}\mu$. Then:

$$1\le \vartheta \left(d(\mu ,\zeta )\right)\le {\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\right(\mu \left)\right)}{\vartheta \left(\mathsf{\varrho }\right(\zeta \left)\right)}}.$$

(3)

From Equation (2) and Equation (3), we get:

$$\begin{array}{cc}\hfill \vartheta \left(d\right(\eta ,\zeta \left)\right)& \le \vartheta \left(d\right(\eta ,\mu )+d(\mu ,\zeta \left)\right)\hfill \\ \hfill & \le \vartheta \left(d\right(\eta ,\mu \left)\right)\vartheta \left(d\right(\mu ,\zeta \left)\right)\hfill \\ \hfill & \le {\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\right(\eta \left)\right)}{\vartheta \left(\mathsf{\varrho }\right(\mu \left)\right)}}{\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\right(\mu \left)\right)}{\vartheta \left(\mathsf{\varrho }\right(\zeta \left)\right)}}\hfill \\ \hfill & ={\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\right(\eta \left)\right)}{\vartheta \left(\mathsf{\varrho }\right(\zeta \left)\right)}}.\hfill \end{array}$$

Therefore, $\zeta \in \mathsf{{\rm Y}}\eta$. Thus, $\mathsf{{\rm Y}}\mu \subseteq \mathsf{{\rm Y}}\eta$. Choose a point ${\eta }_1\in \mathsf{\Lambda }$ and construct a sequence $\left\{{\eta }_n\right\}$ in $\mathsf{\Lambda }$ in the following way: For any ${\eta }_n$ there exists ${\eta }_{n+1}\in \mathsf{{\rm Y}}{\eta }_n$ such that:

$$\mathsf{\varrho }\left({\eta }_{n+1}\right)\le \underset{\zeta \in \mathsf{{\rm Y}}{\eta }_n}{inf}\mathsf{\varrho }\left(\zeta \right)+\frac{1}{n}.$$

Since ${\eta }_{n+1}\in \mathsf{{\rm Y}}{\eta }_n$, we get $\mathsf{\varrho }\left({\eta }_{n+1}\right)\le \mathsf{\varrho }\left({\eta }_n\right)$, for all $n\in \mathbb{N}$. Thus, the sequence $\left\{\mathsf{\varrho }\left({\eta }_n\right)\right\}$ is non-increasing. Since $\mathsf{\varrho }$ is bounded below, there exists $L\in \mathbb{R}$ such that $lim\mathsf{\varrho }\left({\eta }_n\right)=L$. For any $n,m\in \mathbb{N}$ with $n<m$,

$$\begin{array}{cc}\hfill \vartheta \left(d({\eta }_n,{\eta }_m)\right)& \le \vartheta (\sum _{i=n}^{m-1}d({\eta }_i,{\eta }_{i+1}))\hfill \\ \hfill & \le \prod _{i=n}^{m-1}\vartheta \left(d({\eta }_i,{\eta }_{i+1})\right)\hfill \\ \hfill & \le \prod _{i=n}^{m-1}{\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\left({\eta }_i\right)\right)}{\vartheta \left(\mathsf{\varrho }\left({\eta }_{i+1}\right)\right)}}\hfill \\ \hfill & ={\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\left({\eta }_n\right)\right)}{\vartheta \left(\mathsf{\varrho }\left({\eta }_m\right)\right)}}\hfill \\ \hfill & \le {\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\left({\eta }_n\right)\right)}{\vartheta \left(L\right)}}.\hfill \end{array}$$

(4)

Therefore, from continuity of $\vartheta$ and by taking the limit in both sides of Equation (4), we obtain that ${lim}_{n,m\to \infty }\vartheta \left(d({\eta }_n,{\eta }_m)\right)=1$. Therefore, $\vartheta \left({lim}_{n,m\to \infty }d({\eta }_n,{\eta }_m)\right)=1$, which gives us ${lim}_{n,m\to \infty }d({\eta }_n,{\eta }_m)=0$. Thus, we proved that $\left\{{\eta }_n\right\}$ is a Cauchy sequence. Completeness of $\mathsf{\Lambda }$ ensures that there exists $v\in \mathsf{\Lambda }$ such that ${\eta }_n\to v$ as $n\to \infty$. We claim that v is a fixed point of $\mathsf{\Gamma }$. Taking the limit in both sides of Equation (4) as $m\to \infty$, we obtain:

$$\vartheta \left(d({\eta }_n,v)\right)\le \frac{\vartheta \left(\mathsf{\varrho }\left({\eta }_n\right)\right)}{\vartheta \left(\mathsf{\varrho }\right(v\left)\right)}.$$

This gives us $v\in \mathsf{{\rm Y}}{\eta }_n$, for all $n\in \mathbb{N}$ and so $v\in {\cap }_{n=1}^{\infty }\mathsf{{\rm Y}}{\eta }_n$. Also, for any $w\in {\cap }_{n=1}^{\infty }\mathsf{{\rm Y}}{\eta }_n$, we have:

$$\begin{array}{c}\hfill \vartheta \left(d({\eta }_n,w)\right)\le {\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\left({\eta }_n\right)\right)}{\vartheta \left(\mathsf{\varrho }\right(w\left)\right)}}\le {\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\left({\eta }_n\right)\right)}{{inf}_{\zeta \in \mathsf{{\rm Y}}{\eta }_n}\vartheta \left(\mathsf{\varrho }\left(\zeta \right)\right)}}\le {\displaystyle \frac{\vartheta \left(\mathsf{\varrho }\left({\eta }_n\right)\right)\vartheta ({\displaystyle \frac{1}{n}})}{\vartheta \left(\mathsf{\varrho }\left({\eta }_{n+1}\right)\right)}}.\end{array}$$

(5)

Taking the limit in both sides of Equation (5) as $n\to \infty$, we obtain $\vartheta \left(d\right(v,w\left)\right)\le 1$ and so $d(v,w)=0$. Thus, $w=v$. Therefore, ${\cap }_{n=1}^{\infty }\mathsf{{\rm Y}}{\eta }_n=\left\{v\right\}$. On the other hand, $v\in {\cap }_{n=1}^{\infty }\mathsf{{\rm Y}}{\eta }_n$ implies $\mathsf{{\rm Y}}v\subseteq {\cap }_{n=1}^{\infty }\mathsf{{\rm Y}}{\eta }_n=\left\{v\right\}$. Thus $\mathsf{{\rm Y}}v=\left\{v\right\}$. Furthermore, from Equation (1), we have $\mathsf{\Gamma }v\in \mathsf{{\rm Y}}v=\left\{v\right\}$. Therefore, $\mathsf{\Gamma }v=v$. The proof is completed. □

Note that taking $\vartheta \left(t\right)=e^t$, Theorem (2) reduces to Carisi’s fixed point theorem. Thus, Theorem (2) is a generalization of Caristi’s theorem.

Theorem 3.

Let $(\mathsf{\Lambda },d)$ be a complete metric space and $\mathsf{\Gamma }:\mathsf{\Lambda }\to \mathsf{\Lambda }$ be a self-map. If there exist $\mathsf{\varrho }\in \mathsf{\Xi }$ and $\vartheta \in \mathsf{\Omega }$ such that:

$$\vartheta \left(\xi \left(d(\eta ,\mathsf{\Gamma }\eta )\right)\right)\le \frac{\vartheta \left(\mathsf{\varrho }\right(\eta \left)\right)}{\vartheta \left(\mathsf{\varrho }\right(\mathsf{\Gamma }\eta \left)\right)},$$

(6)

for all $\eta \in \mathsf{\Lambda }$, where $\xi :[0,\infty )\to [0,\infty )$ is a continuous, non-decreasing, and concave downward function such that ${\xi }^{-1}\left(\left\{0\right\}\right)=\left\{0\right\}$, then Γ has a fixed point.

Proof. 

Define a function:

$$d^{\prime }(\eta ,\mu )=\xi \left(d(\eta ,\mu )\right)$$

for all $\eta ,\mu \in \mathsf{\Lambda }$. Then it is easy to check that $(\mathsf{\Lambda },d^{\prime })$ is a complete metric space and the conditions of Theorem (2) holds for $(\mathsf{\Lambda },d^{\prime })$. Thus, by Theorem (2), $\mathsf{\Gamma }$ has a fixed point. □

3. A Generalization of Banach’s Fixed Point Theorem

In this section, we introduce a generalization of Banach contraction principle via a different approach from Caristi’s result.

Theorem 4.

Let $(\mathsf{\Lambda },d)$ be a complete metric space and $\mathsf{\Gamma }:\mathsf{\Lambda }\to \mathsf{\Lambda }$ be a continuous self-map. If there exists a function $\mathsf{\varrho }:[0,\infty )\to [0,\infty )$ such that ${lim}_{t\to 0^{+}}\mathsf{\varrho }\left(t\right)=0$, $\mathsf{\varrho }\left(0\right)=0$ and:

$$d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )\le \mathsf{\varrho }\left(d\right(\eta ,\mu \left)\right)-\mathsf{\varrho }\left(d\right(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu \left)\right),$$

(7)

for all $\eta ,\mu \in \mathsf{\Lambda }$, then Γ has a unique fixed point.

Proof. 

Consider an arbitrary element ${\eta }_0\in \mathsf{\Lambda }$. Construct a sequence $\left\{{\eta }_n\right\}$ in $\mathsf{\Lambda }$ with ${\eta }_{n+1}=\mathsf{\Gamma }\left({\eta }_n\right)$, for all $n\in \mathbb{N}\cup \left\{0\right\}$. Using Equation (7) for $\eta ={\eta }_n$ and $\mu ={\eta }_{n+1}$, we have:

$$\begin{array}{cc}\hfill 0\le d({\eta }_n,{\eta }_{n+1})=d(\mathsf{\Gamma }{\eta }_{n-1},\mathsf{\Gamma }{\eta }_n)& \le \mathsf{\varrho }\left(d({\eta }_{n-1},{\eta }_n)\right)-\mathsf{\varrho }\left(d(\mathsf{\Gamma }{\eta }_{n-1},\mathsf{\Gamma }{\eta }_n)\right)\hfill \\ \hfill & =\mathsf{\varrho }\left(d({\eta }_{n-1},{\eta }_n)\right)-\mathsf{\varrho }\left(d({\eta }_n,{\eta }_{n+1})\right).\hfill \end{array}$$

(8)

Thus, the sequence $\left\{\mathsf{\varrho }\left(d({\eta }_n,{\eta }_{n+1})\right)\right\}$ is nonincreasing. Since $\mathsf{\varrho }$ is bounded below, there exists $L\in {\mathbb{R}}^{+}$ such that ${lim}_{n\to \infty }\mathsf{\varrho }\left(d({\eta }_n,{\eta }_{n+1})\right)=L$. For any $n,m\in \mathbb{N}$ with $n<m$,

$$\begin{array}{cc}\hfill d({\eta }_n,{\eta }_m)& \le \sum _{i=n}^{m-1}d({\eta }_i,{\eta }_{i+1})\hfill \\ \hfill & \le \sum _{i=n}^{m-1}(\mathsf{\varrho }\left(d({\eta }_{i-1},{\eta }_i)\right)-\mathsf{\varrho }\left(d({\eta }_i,{\eta }_{i+1})\right))\hfill \\ \hfill & =\mathsf{\varrho }\left(d({\eta }_{n-1},{\eta }_n)\right)-\mathsf{\varrho }\left(d({\eta }_{m-1},{\eta }_m)\right)\hfill \\ \hfill & \le \mathsf{\varrho }\left(d({\eta }_n,{\eta }_{n+1})\right)-L.\hfill \end{array}$$

(9)

Taking the limit in both sides of Equation (9), we obtain ${lim}_{n,m\to \infty }d({\eta }_n,{\eta }_m)=0$. Thus, we proved that $\left\{{\eta }_n\right\}$ is a Cauchy sequence. Completeness of $\mathsf{\Lambda }$ ensures that there exists $\zeta \in \mathsf{\Lambda }$ such that ${\eta }_n\to \zeta$ as $n\to \infty$. We claim that $\zeta$ is a fixed point of $\mathsf{\Gamma }$. We have:

$$\begin{array}{cc}\hfill d(\zeta ,\mathsf{\Gamma }\zeta )=\underset{n\to \infty }{lim}d({\eta }_{n+1},\mathsf{\Gamma }\zeta )=& \underset{n\to \infty }{lim}d(\mathsf{\Gamma }{\eta }_n,\mathsf{\Gamma }\zeta )\hfill \\ \hfill & \le \underset{n\to \infty }{lim}\mathsf{\varrho }\left(d({\eta }_n,\zeta )\right)-\mathsf{\varrho }\left(d(\mathsf{\Gamma }{\eta }_n,\mathsf{\Gamma }\zeta )\right)\hfill \\ \hfill & \le \underset{n\to \infty }{lim}\mathsf{\varrho }\left(d({\eta }_n,\zeta )\right)=0.\hfill \end{array}$$

The proof is completed. □

Remark 1.

Note that Theorem 4 is a generalization of the Banach contraction principle. If $\mathsf{\Gamma }:\mathsf{\Lambda }\to \mathsf{\Lambda }$ is a Banach contraction, there exists $k\in [0,1)$ such that $d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )\le kd(\eta ,\mu )$, for all $\eta ,\mu \in \mathsf{\Lambda }$. Hence:

$$d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )\le kd(\eta ,\mu )\le \frac{k}{1+k-\sqrt{k}}d(\eta ,\mu ),$$

for all $\eta \in \mathsf{\Lambda }$. Consequently,

$$kd(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )+(1-\sqrt{k})d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )\le kd(\eta ,\mu )$$

and so,

$$(1-\sqrt{k})d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )\le kd(\eta ,\mu )-kd(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu ).$$

Therefore,

$$d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )\le \frac{k}{1-\sqrt{k}}d(\eta ,\mu )-\frac{k}{1-\sqrt{k}}d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu ).$$

Taking $\mathsf{\varrho }\left(t\right)=\frac{k}{1-\sqrt{k}}t$, we have $d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )\le \mathsf{\varrho }\left(d\right(\eta ,\mu \left)\right)-\mathsf{\varrho }\left(d\right(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu \left)\right),$ for all $\eta ,\mu \in \mathsf{\Lambda }$.

Choosing $\mathsf{\varrho }\left(t\right)=t{e}^t$, for all $t\ge 0$, we deduce the following corollary.

Corollary 1.

Let $(\mathsf{\Lambda },d)$ be a complete metric space and $\mathsf{\Gamma }:\mathsf{\Lambda }\to \mathsf{\Lambda }$ be a continuous self-map. Let:

$$\frac{d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )(1+e^{d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )})}{d(\eta ,\mu )e^{d(\eta ,\mu )}}\le 1,$$

(10)

for all $\eta ,\mu \in \mathsf{\Lambda }$ with $\eta \ne \mu$. Then Γ has a unique fixed point.

Example 2.

Let $\mathsf{\Lambda }=\{{\kappa }_j=\frac{j(j+1)}{2}:j=1,2,\cdots \}$, $d(\eta ,\mu )=|\eta -\mu |$ and:

$$\mathsf{\Gamma }\eta =\{\begin{array}{cc}{\kappa }_1,\hfill & \eta ={\kappa }_1,\hfill \\ {\kappa }_{j-1},\hfill & \eta ={\kappa }_j,j\ge 2.\hfill \end{array}$$

We need only check the following two cases:

Case 1:$\eta ={\kappa }_j,j\ge 2$ and $\mu ={\kappa }_1$.

$$d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )=|{\kappa }_{j-1}-1|$$

and $d(\eta ,\mu )=|{\kappa }_j-1|$. Then,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )(1+e^{d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )})}{d(\eta ,\mu )e^{d(\eta ,\mu )}}}& ={\displaystyle \frac{({\kappa }_{j-1}-1)(1+e^{{\kappa }_{j-1}-1})}{({\kappa }_j-1)e^{{\kappa }_j-1}}}\hfill \\ \hfill & ={\displaystyle \frac{(\frac{j(j-1)}{2}-1)(1+e^{\frac{j(j-1)}{2}-1})}{(\frac{j(j+1)}{2}-1)e^{\frac{j(j+1)}{2}-1}}}\hfill \\ \hfill & \le {\displaystyle \frac{2(e^{\frac{j(j-1)}{2}-1})}{e^{\frac{j(j+1)}{2}-1}}}\hfill \\ \hfill & <2e^{-j}\le 1.\hfill \end{array}$$

Case 2:$\eta ={\kappa }_j,\mu ={\kappa }_l,j>l$. So,

$$d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )=|{\kappa }_{j-1}-{\kappa }_{l-1}|$$

and $d(\eta ,\mu )=|{\kappa }_j-{\kappa }_l|$. Then,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )(1+e^{d(\mathsf{\Gamma }\eta ,\mathsf{\Gamma }\mu )})}{d(\eta ,\mu )e^{d(\eta ,\mu )}}}& ={\displaystyle \frac{({\kappa }_{j-1}-{\kappa }_{m-1})(1+e^{{\kappa }_{j-1}-{\kappa }_{l-1}})}{({\kappa }_j-{\kappa }_l)e^{{\kappa }_j-{\kappa }_l}}}\hfill \\ \hfill & ={\displaystyle \frac{(\frac{j(j-1)}{2}-\frac{l(l-1)}{2})(1+e^{\frac{j(j-1)}{2}-\frac{l(l-1)}{2}})}{(\frac{j(j+1)}{2}-\frac{l(l+1)}{2})e^{\frac{j(j+1)}{2}-\frac{l(l+1)}{2}}}}\hfill \\ \hfill & \le {\displaystyle \frac{j+l-1}{j+l+1}}{\displaystyle \frac{2(e^{\frac{j(j-1)}{2}-\frac{l(l-1)}{2}})}{e^{\frac{j(j+1)}{2}-\frac{l(l+1)}{2}}}}\hfill \\ \hfill & <2e^{-(j-l)}\le 2e^{-1}\le 1.\hfill \end{array}$$

So, by Corollary 1, Γ has a unique fixed point. Here $\mathsf{\Gamma }{\kappa }_1={\kappa }_1$.

Note that Γ is not a Banach contraction. Since,

$$\begin{array}{cc}\hfill sup{\displaystyle \frac{d(\mathsf{\Gamma }{\kappa }_j,\mathsf{\Gamma }{\kappa }_1)}{d({\kappa }_j,{\kappa }_1)}}& =sup{\displaystyle \frac{{\kappa }_{j-1}-1}{{\kappa }_j-1}}\hfill \\ \hfill & =sup{\displaystyle \frac{{\displaystyle \frac{j(j-1)}{2}}-1}{{\displaystyle \frac{j(j+1)}{2}}-1}}=1.\hfill \end{array}$$

4. Application to Integral Equations

Take $\mathcal{I}=[0,\mathcal{T}]$. Let $\mathsf{\Lambda }=\mathcal{C}(\mathcal{I},\mathbb{R})$ be the set of all real valued continuous functions with domain $\mathcal{I}$. Define:

$$d(\eta ,\mu )=\underset{t\in \mathcal{I}}{sup}\left(\right|\eta \left(t\right)-\mu \left(t\right)\left|\right)=\left|\right|\eta -\mu \left|\right|.$$

Consider the integral equation:

$$\eta \left(t\right)=p\left(t\right)+{\int }_0^{\mathcal{T}}\mathcal{G}(t,s)\mathcal{K}(s,\eta \left(s\right))ds,t\in [0,\mathcal{T}]$$

(11)

Assume that the following conditions hold:

(A)

$p:\mathcal{I}\to \mathbb{R}$ and $\mathcal{K}:\mathcal{I}\times \mathbb{R}\to \mathbb{R}$ are continuous;

(B)

$\mathcal{G}:\mathcal{I}\times \mathcal{I}\to \mathbb{R}$ is continuous and measurable at $s\in \mathcal{I}$ for all $t\in \mathcal{I}$;

(C)

$\mathcal{G}(t,s)\ge 0$ for all $t,s\in \mathcal{I}$ and ${\int }_0^{\mathcal{T}}\mathcal{G}(t,s)ds\le 1$ for all $t\in \mathcal{I}$;

(D)

For each $t\in \mathcal{I}$ and for all $\eta ,\mu \in \mathsf{\Lambda }$.

$$\begin{array}{c}\hfill |\mathcal{K}(t,\eta \left(t\right))-\mathcal{K}(t,\mu \left(t\right))|\le {\displaystyle \frac{-1+\sqrt{1+4{(\eta \left(t\right)-\mu \left(t\right))}^2}}{2}}\end{array}$$

Theorem 5.

Under the assumptions (A)–(D), the integral Equation (7) has a solution in Λ.

Proof. 

Define $\mathsf{{\rm Y}}:\mathsf{\Lambda }\to \mathsf{\Lambda }$ as:

$$\mathsf{{\rm Y}}\eta \left(t\right)=p\left(t\right)+{\int }_0^{\mathcal{T}}\mathcal{G}(t,s)\mathcal{K}(s,\eta \left(s\right))ds,t\in [0,\mathcal{T}].$$

We have:

$$\begin{array}{cc}\hfill & |\mathsf{{\rm Y}}\eta \left(t\right)-\mathsf{{\rm Y}}\mu \left(t\right)|=|{\int }_0^{\mathcal{T}}\mathcal{G}(t,s)(\mathcal{K}(s,\eta \left(s\right))-\mathcal{K}(s,\mu \left(s\right))ds|\hfill \\ \hfill & \le {\int }_0^{\mathcal{T}}\mathcal{G}(t,s)|\mathcal{K}(s,\eta \left(s\right))-\mathcal{K}(s,\mu \left(s\right))|ds\hfill \\ \hfill & \le {\int }_0^{\mathcal{T}}\mathcal{G}(t,s)({\displaystyle \frac{-1+\sqrt{1+4{(\eta \left(s\right)-\mu \left(s\right))}^2}}{2}})ds\hfill \\ \hfill & \le {\int }_0^{\mathcal{T}}\mathcal{G}(t,s)({\displaystyle \frac{-1+\sqrt{1+4\left|\right|\eta -{\mu \left|\right|}^2}}{2}})ds\hfill \\ \hfill & ={\displaystyle \frac{-1+\sqrt{1+4\left|\right|\eta -{\mu \left|\right|}^2}}{2}}\hfill \\ \hfill & ={\displaystyle \frac{-1+\sqrt{1+4{\left[d(\eta ,\mu )\right]}^2}}{2}}\hfill \end{array}$$

for every $t\in [0,1]$. Take sup to find that:

$$\begin{array}{cc}\hfill d(\mathsf{{\rm Y}}\eta ,\mathsf{{\rm Y}}\mu )& =\left|\right|\mathsf{{\rm Y}}\eta -\mathsf{{\rm Y}}\mu \left|\right|\hfill \\ \hfill & \le {\displaystyle \frac{-1+\sqrt{1+4{\left[d(\eta ,\mu )\right]}^2}}{2}}.\hfill \end{array}$$

From the above inequality, we obtain:

$$\begin{array}{c}\hfill {(1+2d(\mathsf{{\rm Y}}\eta ,\mathsf{{\rm Y}}\mu ))}^2\le 1+4{\left[d(\eta ,\mu )\right]}^2.\end{array}$$

This is equivalent to:

$$\begin{array}{c}\hfill {d(\mathsf{{\rm Y}}\eta ,\mathsf{{\rm Y}}\mu ))+\left[d(\mathsf{{\rm Y}}\eta ,\mathsf{{\rm Y}}\mu )\right]}^2\le {\left[d(\eta ,\mu )\right]}^2.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill {d(\mathsf{{\rm Y}}\eta ,\mathsf{{\rm Y}}\mu ))\le \left[d(\eta ,\mu )\right]}^2-{\left[d(\mathsf{{\rm Y}}\eta ,\mathsf{{\rm Y}}\mu )\right]}^2.\end{array}$$

Taking $\mathsf{\varrho }\left(t\right)=t^2$, we get:

$$d(\mathsf{{\rm Y}}\eta ,\mathsf{{\rm Y}}\mu )\le \mathsf{\varrho }\left(d\right(\eta ,\mu \left)\right)-\mathsf{\varrho }\left(d\right(\mathsf{{\rm Y}}\eta ,\mathsf{{\rm Y}}\mu \left)\right),$$

for all $\eta ,\mu \in \mathsf{\Lambda }$, which is Equation (7). Therefore, by Theorem 4, $\mathsf{{\rm Y}}$ has a fixed point. Hence there is a solution for Equation (11). □

5. Conclusions

In this paper, we introduced a new generalization of the Banach contraction principle. The new contraction will be a powerful tool for the existence solution of integral equations, differential equations, and also the fractional integro-differential equations. We think that the multi-valued version of this new contraction can be considered by researchers. The new multi-valued contraction will be a powerful tool for the existence solution of Volterra-integral inclusions.