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.

Definition 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 Define 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

Definition 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 finite integral) on each compact subset of such that for each then f has a unique fixed 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 fixed 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 fixed 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 fixed 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. Define by

Then the condition of Theorem 3.1 holds. In fact,

and

and so,

That is, condition of Theorem 3.1 holds with

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