# Sehgal Type Contractions on b-Metric Space

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## Abstract

**:**

## 1. Introduction and Preliminaries

**Theorem**

**1.**

**Example**

**1.**

## 2. Main Results

**Definition**

**1.**

**Theorem**

**2.**

**Proof.**

- (a)
- For $p={p}_{2k-1}$ and $q={p}_{2k}$, the inequality (2) becomes:$$\begin{array}{cc}\hfill 0<d({p}_{2k},{p}_{2k+1})& =d({O}^{{n}_{k}}{p}_{2k-1},{R}^{{m}_{k}}{p}_{2k})\le c\left[d({p}_{2k-1},{p}_{2k})+\left|d({p}_{2k-1},{O}^{{n}_{k}}{p}_{2k-1})-d({p}_{2k},{R}^{{m}_{k}}{p}_{2k})\right|\right]\hfill \\ & =c\xb7\left[d({p}_{2k-1},{p}_{2k})+\left|d({p}_{2k-1},{p}_{2k})-d({p}_{2k},{p}_{2k+1})\right|\right].\hfill \end{array}$$If $max\left\{d({p}_{2k-1},{p}_{2k}),d({p}_{2k},{p}_{2k+1})\right\}=d({p}_{2k},{p}_{2k+1})$, then from (6), we obtain a contradiction:$$0<d({p}_{2k},{p}_{2k+1})=d({O}^{{n}_{k}}{p}_{2k-1},{R}^{{m}_{k}}{p}_{2k})\le c\xb7d({p}_{2k},{p}_{2k+1})<d({p}_{2k},{p}_{2k+1}),$$$$d({p}_{2k},{p}_{2k+1})\le c\xb7\left(2d({p}_{2k-1},{p}_{2k})-d({p}_{2k},{p}_{2k+1})\right),$$$$d({p}_{2k},{p}_{2k+1})\le \phantom{\rule{4pt}{0ex}}\frac{2c}{1+c}d({p}_{2k-1},{p}_{2k}),\text{}\mathrm{for}\text{}\mathrm{all}\text{}k\ge 1.$$
- (b)
- For $p={p}_{2k+1}$, $q={p}_{2k}$, we have:$$\begin{array}{cc}\hfill 0<d({p}_{2k+2},{p}_{2k+1})& =d({O}^{{n}_{k+1}}{p}_{2k+1},{R}^{{m}_{k}}{p}_{2k})\hfill \\ & \le c\xb7\left[d({p}_{2k+1},{p}_{2k})+\left|d({p}_{2k+1},{O}^{{n}_{k+1}}{p}_{2k+1})-d({p}_{2k},{R}^{{m}_{k}}{p}_{2k})\right|\right]\hfill \\ & =c\xb7\left(d({p}_{2k+1},{p}_{2k})+\left|d({p}_{2k+1},{p}_{2k+2})-d({p}_{2k},{p}_{2k+1})\right|\right).\hfill \end{array}$$If $max\left\{d({p}_{2k+1},{p}_{2k+2}),d({p}_{2k},{p}_{2k+1})\right\}=d({p}_{2k+1},{p}_{2k+2})$, then (10) turns into:$$d({p}_{2k+1},{p}_{2k+2})\le cd({p}_{2k+1},{p}_{2k+2})<d({p}_{2k+1},{p}_{2k+2}),$$$$0<d({p}_{2k+2},{p}_{2k+1})\le c\left[2d({p}_{2k},{p}_{2k+1})-d({p}_{2k+1},{p}_{2k+2})\right]$$$$d({p}_{2k+1},{p}_{2k+2})\le \frac{2c}{1+c}d({p}_{2k},{p}_{2k+1}),\text{}\mathrm{for}\text{}\mathrm{all}\text{}k\ge 1.$$

**Example**

**2.**

**Example**

**3.**

**Corollary**

**1.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Example**

**4.**

**Corollary**

**3.**

**Corollary**

**4.**

**Proof.**

## 3. Application to Nonlinear Fredholm Integral Equation

**Theorem**

**4.**

- 1.
- Define$${O}^{n(p)}p(t)={\int}_{0}^{t}{\mathrm{Y}}_{1}(t,s,p(s))ds+\varrho (t),$$$${O}^{n(q)}q(t)={\int}_{0}^{t}{\mathrm{Y}}_{2}(t,s,q(s))ds+\varrho (t).$$
- 2.
- Suppose there exists $\tau >0$ and a non-negative constant z, where $0<z<\frac{1}{s}$ such that:$$|{\mathrm{Y}}_{1}(t,s,p(s))-{\mathrm{Y}}_{2}{(t,s,q(s))|}^{2}\le z\tau {|p(s)-q(s)|}^{2}.$$

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Alqahtani, B.; Fulga, A.; Karapınar, E.
Sehgal Type Contractions on *b*-Metric Space. *Symmetry* **2018**, *10*, 560.
https://doi.org/10.3390/sym10110560

**AMA Style**

Alqahtani B, Fulga A, Karapınar E.
Sehgal Type Contractions on *b*-Metric Space. *Symmetry*. 2018; 10(11):560.
https://doi.org/10.3390/sym10110560

**Chicago/Turabian Style**

Alqahtani, Badr, Andreea Fulga, and Erdal Karapınar.
2018. "Sehgal Type Contractions on *b*-Metric Space" *Symmetry* 10, no. 11: 560.
https://doi.org/10.3390/sym10110560