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Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph^{ †}

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## Abstract

**:**

^{s}) contractions; directed graph; R-Weakly graph preserving; rational contractions; R-weakly α-admissible

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (1)
- F is continuous,
- (2)
- $F(p,q)\le p$,
- (3)
- $F(p,q)=p\to $$p=0$ or $q=0$

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- for each $x,y\in X$, $(x,y)\in E(G)\to $ for each $u\in Tx$ there exists $v\in {P}_{Ty}(u)$ such that $(u,v)\in E(G)$.

**Lemma**

**1.**

## 3. Main Results

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Example**

**1.**

#### 3.1. R-Weakly Graph Preserving and R-Weakly $\alpha $-Admissible Mappings

**Definition**

**9.**

- (9.1)
- For ${u}_{G}\in S{x}_{G}$, there exists ${v}_{G}\in {R}_{T{y}_{G}}({u}_{G})$ such that $({u}_{G},{v}_{G})\in E(G)$
- (9.2)
- For ${u}_{G}\in T{x}_{G}$, there exists ${v}_{G}\in {R}_{S{y}_{G}}({u}_{G})$ such that $({u}_{G},{v}_{G})\in E(G)$

**Remark**

**1.**

**Remark**

**2.**

**Example**

**2.**

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

**Definition**

**13.**

- (13.1a)
- For $x\in Su$, we can find $y\in {R}_{Tv}(x)$ such that $\alpha (x,y)\ge s$.
- (13.1b)
- For $x\in Tu$, we can find $y\in {R}_{Sv}(x)$ such that $\alpha (x,y)\ge s$.

**Remark**

**3.**

**Example**

**3.**

#### 3.2. Common Fixed Point Theorems in b-Metric Space Endowed with a Graph

**Definition**

**14.**

- (14.1)
- there exists $F\in \mathcal{C}$, $\psi \in {\mathsf{\Psi}}^{*}$ and $\gamma >1$ such that

- $\psi (\gamma sH(S{x}_{G},T{y}_{G}))\le F(\psi (M({x}_{G},{y}_{G})),M({x}_{G},{y}_{G}))$ and
- $\psi (\gamma sH(T{x}_{G},S{y}_{G}))\le F(\psi (M({y}_{G},{x}_{G})),M({y}_{G},{x}_{G}))$

**Theorem**

**1.**

- (1.1)
- For some arbitrary ${{x}_{G}}_{0}\in {X}_{G}$ there exists ${{x}_{G}}_{1}\in T{{x}_{G}}_{0}\bigcup S{{x}_{G}}_{0}$ such that $({{x}_{G}}_{0},{{x}_{G}}_{1})\in E(G)$,
- (1.2)
- S and T are pairwise R-weakly graph preserving,
- (1.3)
- $(S,T)\in (\mathcal{C},{\mathsf{\Psi}}^{*},G,\gamma s)$ for some $F\in \mathcal{C}$ and $\psi \in {\mathsf{\Psi}}^{*}$.

**Proof.**

**Definition**

**15.**

- (15.1)
- there exists $F\in \mathcal{C}$, $\psi \in {\mathsf{\Psi}}^{*}$ and $\gamma >1$ such that

- $\psi (\gamma H(S{x}_{G},T{y}_{G}))\le F(\psi (M({x}_{G},{y}_{G})),M({x}_{G},{y}_{G}))$ and
- $\psi (\gamma H(T{x}_{G},S{y}_{G}))\le F(\psi (M({y}_{G},{x}_{G})),M({y}_{G},{x}_{G}))$

**Theorem**

**2.**

- (2.1)
- $(S,T)\in (\mathcal{C},{\mathsf{\Psi}}^{*},G,\gamma )$ for some $F\in \mathcal{C}$, $\psi \in {\mathsf{\Psi}}^{*}and\gamma >1$.

**Proof.**

**Definition**

**16.**

- (16.1)
- there exists $F\in \mathcal{C}$, $\psi \in {\mathsf{\Psi}}^{*}$ such that

- $\psi (H(S{x}_{G},T{y}_{G}))\le F(\psi (M({x}_{G},{y}_{G})),M({x}_{G},{y}_{G}))$ and
- $\psi (H(T{x}_{G},S{y}_{G}))\le F(\psi (M({y}_{G},{x}_{G})),M({y}_{G},{x}_{G}))$

**Theorem**

**3.**

- (3.1)
- G satisfies transitivity property,
- (3.2)
- $(S,T)\in (\mathcal{C},{\mathsf{\Psi}}^{*},G)$ for some $F\in \mathcal{C}$, $\psi \in {\mathsf{\Psi}}^{*}$.

**Proof.**

**Corollary**

**1.**

- (1.1)
- For all ${x}_{G},{y}_{G}\in {X}_{G}$ with $({x}_{G},{y}_{G})\in E(G)$

- $\psi (\gamma sH(S{x}_{G},T{y}_{G}))\le \psi (M({x}_{G},{y}_{G}))-\varphi (M({x}_{G},{y}_{G}))$ and
- $\psi (\gamma sH(T{x}_{G},S{y}_{G}))\le \psi (M({y}_{G},{x}_{G}))-\varphi (M({y}_{G},{x}_{G}))$

**Proof.**

**Corollary**

**2.**

- (2.1)
- For all ${x}_{G},{y}_{G}\in {X}_{G}$ with $({x}_{G},{y}_{G})\in E(G)$

- $H(S{x}_{G},T{y}_{G})\le \theta (M({x}_{G},{y}_{G}))M({x}_{G},{y}_{G})$ and
- $H(T{x}_{G},S{y}_{G})\le \theta (M({y}_{G},{x}_{G}))M({y}_{G},{x}_{G})$

**Proof.**

**Example**

**4.**

- For $({x}_{G},{y}_{G})=(\frac{1}{{2}^{n}},0)\in E(G)$, we have $S{x}_{G}=\{\frac{1}{{2}^{n+1}},0\},S{y}_{G}=\{0\},T{x}_{G}=\{\frac{1}{{2}^{n+2}},0\},\phantom{\rule{0.277778em}{0ex}}T{y}_{G}=\{0\},\phantom{\rule{0.277778em}{0ex}}{d}_{G}({x}_{G},{y}_{G})=\frac{1}{{2}^{2n}},\phantom{\rule{0.277778em}{0ex}}H(S{x}_{G},T{y}_{G})=\frac{1}{{2}^{2n+2}},\psi (\gamma H(S{x}_{G},T{y}_{G}))=\frac{3}{{2}^{2n+2}}+1,M({x}_{G},{y}_{G})=\frac{1}{{2}^{2n}},\psi (M({x}_{G},{y}_{G}))=\frac{1}{{2}^{2n-1}}+1,\varphi (M({x}_{G},{y}_{G}))=\frac{1}{{2}^{2n+2}}$ and$$\begin{array}{cc}\hfill \psi (\gamma H(S{x}_{G},T{y}_{G}))& =\frac{3}{{2}^{2n+2}}+1<\frac{7}{{2}^{2n+2}}+1\hfill \\ & =\frac{1}{{2}^{2n-1}}+1-\frac{1}{{2}^{2n+2}}=\psi (M({x}_{G},{y}_{G}))-\varphi (M({x}_{G},{y}_{G})).\hfill \end{array}$$
- For $({x}_{G},{y}_{G})=(0,0)\in E(G)$, we have $S{x}_{G}=\{0\}=S{y}_{G}=T{x}_{G}=T{y}_{G},0={d}_{G}({x}_{G},{y}_{G})=H(S{x}_{G},T{y}_{G})=M({x}_{G},{y}_{G}),\psi (H(S{x}_{G},T{y}_{G}))=\psi (M({x}_{G},{y}_{G}))=1,\varphi (M({x}_{G},{y}_{G}))=0$ and$$\begin{array}{c}\hfill \psi (H(S{x}_{G},T{y}_{G}))=1=\psi (M({x}_{G},{y}_{G}))-\varphi (M({x}_{G},{y}_{G})).\end{array}$$
- In addition, for $({x}_{G},{y}_{G})=(\frac{1}{{2}^{n}},\frac{1}{{2}^{n+1}})\in E(G)$, we have $S{x}_{G}=\{\frac{1}{{2}^{n+1}},0\}$, $S{y}_{G}=\{\frac{1}{{2}^{n+2}},0\}$, $T{x}_{G}=\{\frac{1}{{2}^{n+2}},0\}$, $T{y}_{G}=\{\frac{1}{{2}^{n+3}},0\}$. Simple calculations shows that ${d}_{G}({x}_{G},{y}_{G})=\frac{1}{{2}^{2n+2}}$, $H(S{x}_{G},T{y}_{G})=\frac{9}{{2}^{2n+6}}$, $\psi (\gamma H(S{x}_{G},T{y}_{G}))=\frac{27}{{2}^{2n+6}}+1$, $M({x}_{G},{y}_{G})=\frac{1}{{2}^{2n+2}}$, $\psi (M({x}_{G},{y}_{G}))=\frac{1}{{2}^{2n+1}}+1$, $\varphi (M({x}_{G},{y}_{G}))=\frac{1}{{2}^{2n+4}}$ and$$\begin{array}{cc}\hfill \psi (\gamma H(S{x}_{G},T{y}_{G}))& =\frac{27}{{2}^{2n+6}}+1<\frac{28}{{2}^{2n+6}}+1\hfill \\ & <\frac{1}{{2}^{2n+1}}+1-\frac{1}{{2}^{2n+4}}=\psi (M({x}_{G},{y}_{G}))-\varphi (M({x}_{G},{y}_{G}))\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}n\in \mathbb{N}.\hfill \end{array}$$

**Definition**

**17.**

- (17.1)
- there exists $F\in \mathcal{C}$, $\psi \in {\mathsf{\Psi}}^{*}$ and $\gamma >1$ such that

- $\psi (\gamma sH(S{x}_{G},T{y}_{G}))\le F(\psi ({M}^{RC}({x}_{G},{y}_{G})),{M}^{RC}({x}_{G},{y}_{G}))$ and
- $\psi (\gamma sH(T{x}_{G},S{y}_{G}))\le F(\psi ({M}^{RC}({y}_{G},{x}_{G})),{M}^{RC}({x}_{G},{y}_{G}))$

**Theorem**

**4.**

- (4.1)
- $(S,T)\in (\mathcal{C},{\mathsf{\Psi}}^{*},G)$ rational contractions.

**Proof.**

**Definition**

**18.**

- (18.1)
- there exists $F\in \mathcal{C}$, $\psi \in {\mathsf{\Psi}}^{*}$ and $\gamma >1$ such that

- $\psi (\gamma H(S{x}_{G},T{y}_{G}))\le F(\psi ({M}^{RC}({x}_{G},{y}_{G})),{M}^{RC}({x}_{G},{y}_{G}))$ and
- $\psi (\gamma H(T{x}_{G},S{y}_{G}))\le F(\psi ({M}^{RC}({y}_{G},{x}_{G})),{M}^{RC}({y}_{G},{x}_{G}))$

**Theorem**

**5.**

- (5.1)
- $(S,T)\in (\mathcal{C},{\mathsf{\Psi}}^{*},G,\gamma )$ rational contractions class.

#### 3.3. Common Fixed Point Theorems for R-Weakly $\alpha $-Admissible Mappings in a b-Metric Space

**Theorem**

**6.**

- (6.1)
- There exists ${x}_{0},{x}_{1}\in X$ such that ${x}_{1}\in T{x}_{0}\bigcup S{x}_{0}$ and $\alpha ({x}_{0},{x}_{1})\ge s$,
- (6.2)
- The pair $(S,T)$ is R-weakly α-admissible of type S,
- (6.3)
- for some $F\in \mathcal{C}$, $\psi \in {\mathsf{\Psi}}^{*}$ and for all $u,v\in X$ with $\alpha (u,v)\ge s$

- $\psi (\gamma sH(Su,Tv))\le F(\psi ({M}^{\alpha}(u,v)),{M}^{\alpha}(u,v))$ and
- $\psi (\gamma sH(Tu,Sv))\le F(\psi ({M}^{\alpha}(v,u)),{M}^{\alpha}(v,u))$

**Proof.**

**Theorem**

**7.**

- (7.1)
- for some $F\in \mathcal{C}$, $\psi \in {\mathsf{\Psi}}^{*}$ and for all $u,v\in X$ with $\alpha (u,v)\ge s$

- $\psi (\gamma H(Su,Tv))\le F(\psi ({M}^{\alpha}(u,v)),{M}^{\alpha}(u,v))$ and
- $\psi (\gamma H(Tu,Sv))\le F(\psi ({M}^{\alpha}(v,u)),{M}^{\alpha}(v,u))$

**Theorem**

**8.**

- (8.1)
- α is a triangular function, that is if $\alpha (u,v)\ge s$ and $\alpha (v,w)\ge s$ then $\alpha (u,w)\ge s$,
- (8.2)
- for some $F\in \mathcal{C}$, $\psi \in {\mathsf{\Psi}}^{*}$ and for all $u,v\in X$ with $\alpha (u,v)\ge s$

- $\psi (H(Su,Tv))\le F(\psi ({M}^{\alpha}(u,v)),{M}^{\alpha}(u,v))$ and
- $\psi (H(Tu,Sv))\le F(\psi ({M}^{\alpha}(v,u)),{M}^{\alpha}(v,u))$

**Theorem**

**9.**

- (9.1)

- $\psi (\gamma sH(Su,Tv))\le F(\psi ({M}^{RC}(u,v)),{M}^{\alpha}(u,v))$ and
- $\psi (\gamma sH(Tu,Sv))\le F(\psi ({M}^{RC}(v,u)),{M}^{\alpha}(v,u))$

**Theorem**

**10.**

- (10.1)

- $\psi (\gamma H(Su,Tv))\le F(\psi ({M}^{RC}(u,v)),{M}^{\alpha}(u,v))$ and
- $\psi (\gamma H(Tu,Sv))\le F(\psi ({M}^{RC}(v,u)),{M}^{\alpha}(v,u))$

**Corollary**

**3.**

- (3.1)
- For all $u,v\in X$ with $\alpha (u,v)\ge s$

- $H(Su,Tv)\le \theta ({M}^{RC}(u,v)){M}^{RC}(u,v)$ and
- $H(Tu,Sv)\le \theta ({M}^{RC}(v,u)){M}^{RC}(v,u)$

**Proof.**

## 4. Discussions

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

**Remark**

**8.**

**Remark**

**9.**

**Remark**

**10.**

**Remark**

**11.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

George, R.; Nabwey, H.A.; Ramaswamy, R.; Radenović, S.
Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph. *Mathematics* **2019**, *7*, 754.
https://doi.org/10.3390/math7080754

**AMA Style**

George R, Nabwey HA, Ramaswamy R, Radenović S.
Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph. *Mathematics*. 2019; 7(8):754.
https://doi.org/10.3390/math7080754

**Chicago/Turabian Style**

George, Reny, Hossam A. Nabwey, Rajagopalan Ramaswamy, and Stojan Radenović.
2019. "Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph" *Mathematics* 7, no. 8: 754.
https://doi.org/10.3390/math7080754