**Turkish Journal of Analysis and Number Theory**

## Fixed Point Theorems for Expansive Mappings in G-metric Spaces

**Rahim Shah**^{1,}, **Akbar Zada**^{1}

^{1}Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan

Abstract | |

1. | Introduction |

2. | Preliminaries |

3. | Main Results |

4. | Integral Type Contraction for Expansive Mappings |

5. | Example |

References |

### Abstract

In this paper we prove some fixed point theorems for contractive as well as for expansive mappings in G-metric space by using integral type contraction. Finally, we present an example.

**Keywords:** G-metric space, fixed point, integral type contractive mapping, expansive mapping

Received October 08, 2016; Revised January 05, 2017; Accepted February 11, 2017

**Copyright**© 2017 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Rahim Shah, Akbar Zada. Fixed Point Theorems for Expansive Mappings in G-metric Spaces.
*Turkish Journal of Analysis and Number Theory*. Vol. 5, No. 2, 2017, pp 57-62. http://pubs.sciepub.com/tjant/5/2/3

- Shah, Rahim, and Akbar Zada. "Fixed Point Theorems for Expansive Mappings in G-metric Spaces."
*Turkish Journal of Analysis and Number Theory*5.2 (2017): 57-62.

- Shah, R. , & Zada, A. (2017). Fixed Point Theorems for Expansive Mappings in G-metric Spaces.
*Turkish Journal of Analysis and Number Theory*,*5*(2), 57-62.

- Shah, Rahim, and Akbar Zada. "Fixed Point Theorems for Expansive Mappings in G-metric Spaces."
*Turkish Journal of Analysis and Number Theory*5, no. 2 (2017): 57-62.

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### 1. Introduction

In 2006, Mustafa and Sims ^{[10]}, introduced the concept of G-metric spaces and presented some fixed point theorems in G-metric spaces. Mehdi et al. ^{[9]} gave new approach to G-metric spaces and proved some fixed point theorems in G-metric spaces. Further we note that many researchers have studied G-metric spaces see, [1-21]^{[1]}. Moreover, In 2002, A. Branciari ^{[8]} introduced the concept of integral type contractive mappings in complete metric spaces and study the existence of fixed points for mappings which is defined on complete metric space satisfies integral type contraction. Recently A. Zada et al. ^{[22]}, presented the concept of integral type contraction with respect to G-metric spaces and proved some coupled coincidence fixed point results for two pairs in such spaces, by using the notion of integral type contractive mappings given by A. Branciari ^{[8]}. In section 3, we presented some fixed point theorems of integral type contractive mapping in setting of G-metric spaces. Moreover in section 4, we proved some fixed point theorems of integral type contraction for expansive mapping. Also we give suitable example that support our main results.

### 2. Preliminaries

Consistent with Mustafa & Sims ^{[10]} and Branciari ^{[8]}. The following definitions and results will be needed in this paper.

**De****fi****nition 2.1** ^{[10]} *Let Y be a non-empty set and** ** **is a function that satisfies the following conditions:*

*(1)** *

*(2)** *

*(3)** *

*(4)** ** **symmetry in all three variables,*

*(5)** ** **(rectangle inequality) **for all** ** *

*Then the function G is called a generalized metric and the pair *(*Y*,*G*)* is called** **a G-metric space.*

**Example 2.2 **^{[10]} *Let** ** De**fi**ne G on** ** **by*

*and extend G to** ** **by using the symmetry in the variables. Then it** **is clear that** ** **is a G-metric space*.

**Definition 2.3** ^{[10]} *Let** ** **be a G-metric space and** ** **a sequence of points of** ** **A point** ** **is said to be the limit of the sequence** ** **if** ** **and we say that the sequence** ** **is G-convergent to a*.

**Proposition 2.4** ^{[10]} *Let** ** **be a G-metric space. Then the following are equivalent:*

(1) *is G-convergent to a*.

(2) *as** *

(3) * **as** *

(2) *as*

**Definition 2.5** ^{[10]} *Let** ** **be a G-metric space. A sequence** ** **is called G-Cauchy if for every** ** **there is** ** **such that** ** **for all** ** **that is** ** **as** *

**Proposition 2.6** ^{[10]} *Let** ** **be a G-metric space. Then the following** **are equivalent:*

(1) The sequence is *G*-Cauchy.

(2) For every there is such that for all

**De****fi****nition 2.7** ^{[10]} *A G-metric space** ** **is called **G-complete if every** **G-Cauchy sequence in** ** **is **G-convergent in** *

**Lemma 2.8** ^{[11]} *By the rectangle inequality *(5) *together with the symmetry *(4),* we have*

In 2002, Branciari in ^{[8]} introduced a general contractive condition of integral type as follows.

**Theorem 2.9** ^{[8]} *Let** ** **be a complete metric space,** ** **and** ** **is a mapping such that for all** *

*where** ** **is nonnegative and Lebesgue-integrable mapping which is summable **(**i.e., with **fi**nite integral**) **on each compact subset of** ** **such that for each** ** ** **then **f **has a unique **fi**xed** **point** ** **such that for each** ** *

**Lemma 2.10*** By rectangle inequality* (5) *together with the symmetry* (4) *of definition 2.1, we have*

In this paper by using the notion of integral type given by Branciari in ^{[8]}, we presented some fixed point theorems in G-metric space.

### 3. Main Results

In this section, we prove some fixed point theorems in generalize metric space by using integral type contractive mappings. Our first main result is follow as,

Theorem 3.1* **Let** ** **be a G-metric space. Suppose** ** **be a** **mapping satisfy the following condition for all** *

(3.1) |

*where** ** **and** ** **is a Lebesgue integrable mapping which is summable, non-negative and such that for each** ** ** **Then H has a unique **fi**xed point in Y*.

**Proof**. Choose and define by Notice that if for some then obviously has a fixed point. Thus, we suppose that

that is, we have

continuing this process, we get

Moreover, for all we have

So,

Thus

This means that is G-Cauchy sequence. Due to completeness of there exists such that an is convergent to

Suppose that then

Taking limit as and using the fact that function is continuous, then

This contradiction implies that

For uniqueness, let such that and use Lemma 2.10, then

which implies that *l *= *p*. The proof is completed.

**Corollary 3.2** Let be a G-metric space. Suppose be a mapping satisfy the following condition for all

where and is a Lebesgue integrable mapping which is summable, non-negative and such that for each Then has a unique fixed point in

**Theorem 3.3** *Let** ** **be a complete G-metric space. Suppose** ** **be a mapping satisfy the following condition for all** ** **where** *

*And** ** **is a Lebesgue integrable mapping which is summable,** **non-negative and such that for each** ** ** **Then** ** **has a** **unique **fi**xed point in** *

**Proof.** Choose We construct sequence in the following way:

Notice that if for some then obviously has a fixed point. Thus, we suppose that

That is, we have

From above condition, with and we have

implies that

So,

where Then

for all From definition of G-metric space, we know that

with also by Lemma 2.10, we know that

Then by above inequality, we have

Moreover, for all we have

So,

Thus

This means that is G-Cauchy sequence. Due to completeness of , there exists such that is convergent to From the above condition of theorem, with and we have

Then

Taking limit as of above inequality, we have

Now, if then *H* has a fixed point. Hence, we suppose that Therefore, by definition of G-metric space, we get

which implies that

i.e., The proof is completed.

### 4. Integral Type Contraction for Expansive Mappings

In this section of our paper, we prove some fixed point theorems for expansive mapping of integral type contraction in G-metric spaces.

**Theorem 4.1** *Let** ** **be a complete G-metric space. Suppose** ** **be an onto mapping satisfy the following condition for all** ** **where** ** **holds*

*And** ** **is a Lebesgue integrable mapping which is summable, non-negative and such that for each** ** ** **Then H has a unique fixed point in **Y**.*

**Proof.** Choose as H is onto map, then there exists such that If we continue this process, we can get for all In case for some then clearly is a fixed point of Next, we suppose that for all *n*. From above condition of this result, with and we have

which implies that

where Then we have

By Lemma (2.10), we get

If we follow the lines of the proof of result 3.1, we derive that is a G-Cauchy sequence. Since is complete, there exists such that as Consequently, since is onto, so there exists such that From above condition of this theorem, with and we have

Taking limit as in above inequality, we get

That is, Then

For uniqueness, let such that and By above condition of result, we get

which arise contradiction, Hence *r* = *s*. The proof is completed.

**Theorem 4.2** *Let** ** **be a complete G-metric space. Suppose** ** **be an onto mapping satisfy the following condition for all** ** **where** *

(4.1) |

*And** ** **is a Lebesgue integrable mapping **which is summable,** **non-negative and such that for each** ** ** **Then **H **has a** **unique **fi**xed point in **Y*.

**Proof**. Choose as is onto map, then there exists such that If we continue this process, we can get for all In case for some then clearly is a fixed point of Next, we suppose that for all n. From (4.1), with and we have

implies that

and so,

where By the proof of Theorem 3.1, we can show that is a G-Cauchy sequence. Since is complete, it exists such that as Consequently, since is onto, so there exists such that From 4.1, with and we have

On taking limit in above inequality, we have That is, For uniqueness, let such that and

Now, by using condition 4.1, we have

which is contradiction. Hence . The proof is completed.

### 5. Example

In this section, we present an example, which indicates that how our results can be applied to different problems.

**Example 5.1*** **Let** ** **and*

*be a G-metric space on Y. De**fi**ne** ** **by** *

*Then the condition of Theorem 3.1 holds. In fact,*

*and*

*and so*,

That is, condition of Theorem 3.1 holds with

### References

[1] | R. Agarwal, E, Karapinar, Remarks on some coupled fixed point theorems in G-metric spaces, Fixed Point Theory Appl., 2013, 2 (2013). | ||

In article | View Article | ||

[2] | H. Aydi, M. Postolache, W. Shatanawi, Coupled fixed point results for (ϕ-ψ)-weakly contractive mappings in ordered G-metric spaces, Comput. Math. Appl., vol. 63(1), 2012, pp. 298-309. | ||

In article | View Article | ||

[3] | H. Aydi, W. Shatanawi, C. Vetro, On generalized weakly G-contraction mappings in G-metric spaces, Comput. Math. Appl., vol. 62, 2011, pp. 4222-4229. | ||

In article | View Article | ||

[4] | H. Aydi, B. Damjanovic, B. Samet, W. Shatanawi, Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces, Math. Comput. Model., vol. 54, 2011, pp. 2443-2450. | ||

In article | View Article | ||

[5] | H. Aydi, E. Karapinar, W. Shatanawi, Tripled fixed point results in generalized metric spaces, J. Appl. Math., 2012, Article ID 314279. | ||

In article | View Article | ||

[6] | H. Aydi, E. Karapinar, Z. Mustafa. On common fixed points in G-metric spaces using (E.A) property, Comput. Math. Appl., vol. 64(6), 2012, pp. 1944-1956. | ||

In article | View Article | ||

[7] | H. Aydi, E. Karapinar, W. Shatanawi, Tripled common fixed point results for generalized contractions in ordered generalized metric spaces, Fixed Point Theory Appl., 2012, 101 (2012). | ||

In article | View Article | ||

[8] | A. Branciari. A fixed point theorem for mappings satisfying a general contractive condition of integral type. International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 9, 2002, pp. 531-536. | ||

In article | View Article | ||

[9] | A. Mehdi, K. Erdal, S. Peyman, A new approach to G-metric and related fixed point theorems, J. Ineq. Apps., 2013, 2013:454. | ||

In article | |||

[10] | Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., vol. 7 (2), 2006, pp. 289-297. | ||

In article | |||

[11] | Z. Mustafa, A new structure for generalized metric spaces with applications to fixed point theory, PhD thesis, The University of Newcastle, Australia, 2005. | ||

In article | |||

[12] | Z. Mustafa, M. Khandaqji, W. Shatanawi, Fixed point results on complete G-metric spaces, Studia Sci. Math. Hung., vol. 48, 2011, pp. 304-319. | ||

In article | View Article | ||

[13] | Z. Mustafa, H. Aydi, E. Karapinar, Mixed g-monotone property and quadruple fixed point theorems in partially ordered metric space, Fixed Point Theory Appl., 2012, 71. | ||

In article | View Article | ||

[14] | Z. Mustafa, Common fixed points of weakly compatible mappings in G-metric spaces, Appl. Math. Sci., vol. 6(92), 2012, pp. 4589-4600. | ||

In article | |||

[15] | Z. Mustafa, Some new common fixed point theorems under strict contractive conditions in G-metric spaces, J. Appl. Math., 2012, Article ID 248937. | ||

In article | View Article | ||

[16] | Z. Mustafa, Mixed g-monotone property and quadruple fixed point theorems in partially ordered G-metric spaces using (ϕ-ψ) contractions, Fixed Point Theory Appl., 2012, 199. | ||

In article | View Article | ||

[17] | Z. Mustafa, W. Shatanawi, M. Bataineh, Existence of fixed point results in G-metric spaces, Int. J. Math. Math. Sci., 2009, Article ID 283028 (2009). | ||

In article | View Article | ||

[18] | R. Shah, A. Zada and I. Khan, Some fixed point theorems of integral type contraction in cone b-metric spaces, Turkish. J. Ana. Num. Theor., vol. 3(6), 2105, 165-169. | ||

In article | |||

[19] | R. Shah et al., New common coupled fixed point results of integral type contraction in generalized metric spaces, J. Ana. Num. Theor., vol. 4(2), 2106, 1-8. | ||

In article | |||

[20] | K. P. R. Rao, L. Bhanu, Z. Mustafa, Fixed and related fixed point theorems for three maps in G-metric space, J. Adv. Stud. Topol., vol. 3(4), 2012, pp. 12-19. | ||

In article | View Article | ||

[21] | W. Shatanawi, Z. Mustafa, On coupled random fixed point results in partially ordered metric spaces, Mat. Vesn., vol. 64, 2012, pp. 139-146. | ||

In article | |||

[22] | A. Zada, R. Shah, T. Li, Integral Type Contraction and Coupled Coincidence Fixed Point Theorems for Two Pairs in G-metric Spaces, Hacet. J. Math. Stat., vol. 45(5), 2016, pp. 1475-1484. | ||

In article | View Article | ||