# Fixed Point Results for a Selected Class of Multi-Valued Mappings under (θ,ℛ)-Contractions with an Application

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## Abstract

**:**

## 1. Introduction

- To exhibit the utility, we provide some illustrative examples.
- We obtain some relation-theoretic existence and uniqueness results for single-valued mappings.
- As consequences of our results, we deduce some corollaries in the setting of ordered-metric spaces.
- We show the applicability of our newly obtained results by investigating the existence and uniqueness of a positive solution for Volterra type integral equation under some suitable conditions.

## 2. Preliminaries

- (Θ
_{1}) - $\theta $ is nondecreasing;
- (Θ
_{2}) - for each sequence {${\beta}_{n}$}$\subset (0,\infty )$, ${lim}_{n\to \infty}\theta \left({\beta}_{n}\right)=1$$\u27fa{lim}_{n\to \infty}{\beta}_{n}={0}^{+}$;
- (Θ
_{3}) - there exist $r\in (0,1)$ and $\gamma \in (0,\infty ]$ such that ${lim}_{\beta \to {0}^{+}}\frac{\theta \left(\beta \right)-1}{{\beta}^{r}}=\gamma $; and
- (Θ
_{4}) - $\theta $ is continuous.

- ${\Theta}_{1,2,3,4}$ denotes the set of all functions $\theta $ satisfying $\left({\Theta}_{1}\right)-\left({\Theta}_{4}\right)$.
- ${\Theta}_{1,2,3}$ denotes the set of all functions $\theta $ satisfying $\left({\Theta}_{1}\right)-\left({\Theta}_{3}\right)$.
- ${\Theta}_{1,2,4}$ denotes the set of all functions $\theta $ satisfying $\left({\Theta}_{1}\right),\left({\Theta}_{2}\right)$, and $\left({\Theta}_{4}\right)$.
- ${\Theta}_{2,3}$ denotes the set of all functions $\theta $ satisfying $\left({\Theta}_{2}\right)$ and $\left({\Theta}_{3}\right)$.
- ${\Theta}_{2,4}$ denotes the set of all functions $\theta $ satisfying $\left({\Theta}_{2}\right)$ and $\left({\Theta}_{4}\right)$.

**Example**

**1.**

**Example**

**2.**

**Definition**

**1**

**Theorem**

**1**

**Definition**

**2**

**Theorem**

**2**

- (${\Theta}_{4}^{\prime}$)
- $\theta (infB)=inf\theta \left(B\right),$$\forall B\subset (0,\infty )$ with $infB>0.$

**Theorem**

**3**

**Definition**

**3**

**Definition**

**4**

**Definition**

**5**

**Definition**

**6**

**Definition**

**7**

**Definition**

**8**

**Definition**

**9**

**Definition**

**10**

- (i)
- $(U,V)\in {\mathcal{R}}_{1}$ if $(u,v)\in \mathcal{R}$, for all $u\in U$ and $v\in V$.
- (ii)
- $(U,V)\in {\mathcal{R}}_{2}$ if, for each $u\in U$, there exists $v\in V$ such that $(u,v)\in \mathcal{R}$.

**Remark**

**1.**

**Definition**

**11**

- (i)
- monotone of Type (I) if$$u,v\in M,\phantom{\rule{4pt}{0ex}}(u,v)\in \mathcal{R}\phantom{\rule{4pt}{0ex}}\mathit{implies}\phantom{\rule{4.pt}{0ex}}\mathit{that}\phantom{\rule{4pt}{0ex}}(Su,Sv)\in {\mathcal{R}}_{1};$$and
- (ii)
- monotone of Type (II) if$$u,v\in M,\phantom{\rule{4pt}{0ex}}(u,v)\in \mathcal{R}\phantom{\rule{4pt}{0ex}}\mathit{implies}\phantom{\rule{4.pt}{0ex}}\mathit{that}\phantom{\rule{4pt}{0ex}}(Su,Sv)\in {\mathcal{R}}_{2}.$$

**Remark**

**2.**

**Definition**

**12.**

## 3. Main Results

**Definition**

**13.**

**Remark**

**3.**

**Remark**

**4.**

**Theorem**

**4.**

- (a)
- S is monotone of Type $\left(I\right)$;
- (b)
- there exists ${u}_{0}\in M$ such that $(\left\{{u}_{0}\right\},S{u}_{0})\in {\mathcal{R}}_{2}$;
- (c)
- S is multi-valued $(\theta ,\mathcal{R})$-contraction with $\theta \in {\Theta}_{1,2,3}$;
- (d)
- M is $\mathcal{R}$-complete; and
- (e)
- one of the following holds:
- (e′)
- S is ${\mathcal{R}}_{\mathcal{H}}$-continuous, or
- (e″)
- $\mathcal{R}$ is ρ-self-closed.

**Proof.**

**Case 1:**When $\gamma <\infty $. Take $A=\frac{\gamma}{2}>0$; then, by the definition of the limit, there exists ${n}_{0}\in \mathbb{N}$ such that

**Case 2:**When $\gamma =\infty $. Let ${A}^{*}>0$ be any positive real number. Then, by the definition of limit, there exists ${n}_{1}\in \mathbb{N}$ such that

**Remark**

**5.**

**Example**

**3.**

**Theorem**

**5.**

- (a)
- S is monotone of Type $\left(I\right)$;
- (b)
- there exists ${u}_{0}\in M$ such that $(\left\{{u}_{0}\right\},S{u}_{0})\in {\mathcal{R}}_{2}$;
- (c)
- S is multi-valued $(\theta ,\mathcal{R})$-contraction with $\theta \in {\Theta}_{1,2,4}$;
- (d)
- M is $\mathcal{R}$-complete; and
- (e)
- one of the following holds:
- (e′)
- either S is ${\mathcal{R}}_{\mathcal{H}}$-continuous; or
- (e″)
- $\mathcal{R}$ is ρ-self-closed.

**Proof.**

**Example**

**4.**

**Case 1:**When $i=1or2$ and $j>3$. In this case, we get

**Case 2:**When $j>i>2$. We have

**Remark**

**6.**

**Corollary**

**1.**

- (a)
- $\mathcal{R}$ is S-closed;
- (b)
- there exists ${u}_{0}\in M$ such that $({u}_{0},S{u}_{0})\in \mathcal{R}$;
- (c)
- S is $(\theta ,\mathcal{R})$-contraction with $\theta \in {\Theta}_{2,3}$;
- (d)
- M is $\mathcal{R}$-complete; and
- (e)
- one of the following holds:
- (e′)
- S is $\mathcal{R}$-continuous; or
- (e″)
- $\mathcal{R}$ is ρ-self-closed.

**Corollary**

**2.**

- (a)
- $\mathcal{R}$ is S-closed;
- (b)
- there exists ${u}_{0}\in M$ such that $({u}_{0},S{u}_{0})\in \mathcal{R}$;
- (c)
- S is $(\theta ,\mathcal{R})$-contraction with $\theta \in {\Theta}_{2,4}$;
- (d)
- M is $\mathcal{R}$-complete; and
- (e)
- one of the following holds:
- (e′)
- S is $\mathcal{R}$-continuous; or
- (e″)
- $\mathcal{R}$ is ρ-self-closed.

**Remark**

**7.**

**Theorem**

**6.**

**Proof.**

**Remark**

**8.**

## 4. Some Consequences in Ordered Metric Spaces

**Definition**

**14.**

**Definition**

**15.**

**Corollary**

**3.**

- (a)
- S is monotone of Type $\left(I\right)$;
- (b)
- there exists ${u}_{0}\in M$ such that $\left\{{u}_{0}\right\}{\prec}_{2}S{u}_{0}$;
- (c)
- S is multi-valued $(\theta ,\u2aaf)$-contraction with $\theta \in {\Theta}_{1,2,3}$;
- (d)
- M is ⪯-complete; and
- (e)
- one of the following holds:
- (e′)
- S is ${\u2aaf}_{\mathcal{H}}$-continuous; or
- (e″)
- $(M,\rho ,\u2aaf)$ is regular.

**Corollary**

**4.**

- (a)
- S is monotone of Type $\left(I\right)$;
- (b)
- there exists ${u}_{0}\in M$ such that $\left\{{u}_{0}\right\}{\prec}_{2}S{u}_{0}$;
- (c)
- S is multi-valued $(\theta ,\u2aaf)$-contraction with $\theta \in {\Theta}_{1,2,4}$;
- (d)
- M is ⪯-complete; and
- (e)
- one of the following holds:
- (e′)
- S is ${\u2aaf}_{\mathcal{H}}$-continuous; or
- (e″)
- $(M,\rho ,\u2aaf)$ is regular.

**Corollary**

**5.**

- (a)
- S is non-decreasing;
- (b)
- there exists ${u}_{0}\in M$ such that ${u}_{0}\u2aafS{u}_{0}$;
- (c)
- S is $(\theta ,\u2aaf)$-contraction with $\theta \in {\Theta}_{2,3}$;
- (d)
- M is ⪯-complete; and
- (e)
- one of the following holds:
- (e′)
- S is ⪯-continuous; or
- (e″)
- $(M,\rho ,\u2aaf)$ is regular.

**Corollary**

**6.**

- (a)
- S is non-decreasing;
- (b)
- there exists ${u}_{0}\in M$ such that $({u}_{0},S{u}_{0})\in \mathcal{R}$;
- (c)
- S is $(\theta ,\mathcal{R})$-contraction with $\theta \in {\Theta}_{2,4}$;
- (d)
- M is ⪯-complete; and
- (e)
- one of the following holds:
- (e′)
- S is ⪯-continuous; or
- (e″)
- $(M,\rho ,\u2aaf)$ is regular.

**Corollary**

**7.**

## 5. Application to Integral Equation

**Theorem**

**7.**

- (a
_{1}) - $g({t}_{1},{r}_{1},u)>0$, for all $u>0$ and ${t}_{1},{r}_{1}\in I$; and
- (a
_{2}) - g is non-decreasing in the third variable and there exists $h>0$ such that$$|g(t,r,u)-g(t,r,v)|\le \frac{\left|u\right(t)-v(t\left)\right|}{h\parallel u-v\parallel +1},$$$\forall \phantom{\rule{4pt}{0ex}}t,r\in I$ and $u,v>0$ with $uv\ge (u\vee v)$, where $u\vee v=u\phantom{\rule{4pt}{0ex}}or\phantom{\rule{4pt}{0ex}}v$.

**Proof.**

**Theorem**

**8.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Hasanuzzaman, M.; Sessa, S.; Imdad, M.; Alfaqih, W.M. Fixed Point Results for a Selected Class of Multi-Valued Mappings under (*θ*,*ℛ*)-Contractions with an Application. *Mathematics* **2020**, *8*, 695.
https://doi.org/10.3390/math8050695

**AMA Style**

Hasanuzzaman M, Sessa S, Imdad M, Alfaqih WM. Fixed Point Results for a Selected Class of Multi-Valued Mappings under (*θ*,*ℛ*)-Contractions with an Application. *Mathematics*. 2020; 8(5):695.
https://doi.org/10.3390/math8050695

**Chicago/Turabian Style**

Hasanuzzaman, Md, Salvatore Sessa, Mohammad Imdad, and Waleed M. Alfaqih. 2020. "Fixed Point Results for a Selected Class of Multi-Valued Mappings under (*θ*,*ℛ*)-Contractions with an Application" *Mathematics* 8, no. 5: 695.
https://doi.org/10.3390/math8050695