Abstract

In this paper, we obtain some fixed point theorems of expanding mappings in G-metric spaces. And the existence and uniqueness of the fixed point and common fixed point of some expansive mapping in the complete G-metric space are discussed. The results not only directly improve and generalize some fixed point results in G-metric spaces, but also expand and complement some previous results in the papers by Asadi, et al. [1] and Lei et al. [2].

Mathematics Subject Classification (2010): 47H09; Secondary 47H10, 47J20

1. Introduction and Preliminaries

Fixed point theory plays a basic role in applications of many branches mathematics. Finding the fixed point of contractive mappings becomes the center of strong research activity ^{3, 4}. In 2007, Mustafa and Sims ³ introduced the notion of G-metric and investigated the topology of such spaces. Mustafa ⁵ provided many examples of G-metric spaces and developed some of their properties. Samet et al. ⁶ and Jleli and Samet ⁷ reported that some published results can be considered as a straight consequence of the existence theorem in the setting of the usual metric spaces. Asadi et al. ¹ stated and proved some fixed point theorems in the framework of G-metric space. At the same time, the authors of those papers established some fixed point results for expansive mappings.

The object of this paper is to get some fixed point results in the complete G-metric space and some of the results are different from ¹.

First, we recollect some necessary definitions and results in this direction. The notion of G-metric spaces is defined as follows.

Definition 1.1. (See ³) A G-metric space is a pair where is a nonempty set and is a function such that, for all , the following conditions are fulfilled:

(G₁) if ;

(G₂) for all with ;

(G₃) for all with ;

(G₄) (symmetry in all three variables);

(G₅) (rectangle inequality).

Then the function is called generalized metric or, more specifically, a -metric in and the pair is called a -metric space.

Remark 1.2. Throughout this paper we denote the set of all positive real numbers and the set of all natural numbers.

For a better understanding of the subject, we give the following example of -metric.

Example 1.3. If is a non-empty subset of , then the function , given by

is a G-metric on X.

Example 1.4. Let be the interval of nonnegative real numbers and let be defined by:

Then is a complete -metric on .

Definition 1.5. (See ³) Let be a -metric space, let be a sequence of points of , a point is said to be the limit of the sequence if and one say that the sequence is -convergent to . That is, for any , there exists such that for all . We call is the limit of the sequence and write or .

Proposition 1.6. (See ³) Let be a -metric space. The following are equivalent:

(1) is -convergent to

(2) as

(3) as

(4) as

Definition 1.7. (See ³) Let be a -metric space. Sequence is called a -Cauchy sequence if, for any there exists such that for all that is as

Proposition 1.8. (See ³) Let be a -metric space. Then the following are equivalent:

(1) The sequence is -Cauchy;

(2) For any there exists such that for all .

Definition 1.9. (See ³) A -metric space is called -complete if every -Cauchy sequence is -convergent in .

The following fixed point theorem for a contractive mapping on G-metric space has proved in ⁵.

Theorem 1.10. (See ⁵) Let be a complete -metric space and be a mapping satisfying the following condition for all :

(1.1)

where Then has a unique fixed point.

Theorem 1.11. Let be a complete -metric space and be a mapping satisfying the following condition for all :

(1.2)

where . Then has a unique fixed point.

Remark 1.12. We notice that condition (1.1) implies condition (1.2). The converse is true only if For detail see ⁵.

Lemma 1.13. (See ⁵) By the rectangle inequality together with the symmetry , we have

(1.3)

Definition 1.14. A (c)-comparison function is a non-decreasing function such that there exist and a convergent series of nonnegative terms verifying

Let denote the family of all (c)-comparison functions. Consider the family

Lemma 1.15. (See ⁵) Let be a sequence in a -metric space and assume that there exist a function and such that, at least, one of the following conditions holds:

(a) for all ;

(b) for all .

Then is a Cauchy sequence in .

If we take for all , where , then and the above result can be stated as follows.

Lemma 1.16. (See ⁵) Let be a sequence in a -metric space and assume that there exist a constant and such that , at least, one of the following conditions holds:

(a) for all ;

(b) for all .

Then is a Cauchy sequence in .

Definition 1.17. (See ⁸) A mapping from a -metric space into itself is said to be:

Ÿ expansive of type if there exists such that

(1.4)

Ÿ expansive of type II if there exists such that

(1.5)

2. Main Results

In this section, we start our work by proving the following theorem:

Theorem 2.1. Let be a complete -metric space and be a onto mapping. Suppose that there exists such that

(2.1)

Then has a unique fixed point.

Proof. Let be arbitrary. Since is onto, then there exists such that By continuing this process, we get for all If there exists some such that , then is a fixed point of Now assume that for all For (2.1) with and , we have

which implies that

where . Form Lemma 1.16, is a Cauchy sequence. Since, is complete, there exists such that . As is onto, there exists such that . From (2.1) with and we have that, for all ,

Taking the limit as in the above inequality we get,

that is, Then, is a fixed point of because We shall show that is the unique fixed point of Suppose, on the contrary, that there exists another fixed point such that If then From (2.1) and we have that

which is a contradiction. Hence, is the unique fixed point of .

Remark 2.2. Condition (2.1) was inspired by (1.4) in the definition 1.17. If z = y, then condition (2.1) implies condition (1.4), and if x = y, then condition (2.1) implies condition (1.5).

Example 2.3. Let be the interval of nonnegative real numbers and let the complete -metric on defined by

Defined by for all Then, all the hypotheses of Theorem 2.1 hold. In fact,

and

Therefore,

for all Then has a unique fixed point on which is

Based on theorem 2.1, the following result considered two nonnegative real numbers and four nonnegative real numbers can be proved.

Theorem 2.4. Let be a complete -metric space and be a onto mapping. Suppose that there exist nonnegative real numbers a, b, with such that, for all

(2.2)

Then has a unique fixed point.

Proof. Let be arbitrary. Since is onto, then there exists such that By continuing this process, we can find a sequence such that for all . If there exists some such that then is a fixed point of Now assume that for all . For (2.2) with and we have that, for all ,

which implies that

where . Using that is,

where . Then we have,

(2.3)

From Lemma 1.13 we get,

Then by (2.3), we have

Moreover, for all we have by rectangle inequality that

and so, as Thus, is -Cauchy sequence. Due to is complete, there exists such that is -convergent to Since is onto, there exists such that Form (2.2) with and we have that, for all

Taking the limit as in the above inequality we get,

That is, Then So, is a fixed point of because To prove uniqueness, suppose that is another fixed point of such that . If , again by (2.2), we get

which is a contradiction. Hence . Therefore, has a unique fixed point.

Theorem 2.5. Let be a complete -metric space and let be a onto mapping. Assume that there exist nonnegative real numbers a, b, c and with and such that, for all ,

(2.4)

Then has a unique fixed point.

Proof. Let since is onto, then there exists such that . Continuing in this way, we get a sequence such that for all If there exists some such that then is a fixed point of because . On the contrary case, assume that for all By taking and in the (2.4), we have that, for all ,

which implies that

and so,

(2.5)

where . Proceeding in this way, we get

(2.6)

From Lemma 1.13 we get,

Then by (2.6), we have

Moreover, for all we have by rectangle inequality that

So, as and is -Cauchy sequence. Due to the completeness of there exists such that is -convergent to As is onto, there exists such that . Form (2.4) with and we have that, for all

which implies that

Taking the limit as in the above inequality we get,

So, , then . Therefore, is a fixed point of because Suppose there is another fixed point of such that If , again by (2.4), we get

Then

That is which is a contradiction because of Hence Therefore, has a unique fixed point.

Corollary 2.6. Let be a complete -metric space and let be a onto mapping. Assume that there exist nonnegative real numbers a, b, c and with and , such that, for all ,

Then has a unique fixed point.

Proof. From the previous theorem, we see that has a unique fixed point (say ), that is, . But , so $Tu$ is another fixed point for and by uniqueness . Therefore, has a unique fixed point.

Theorem 2.7. Let be a symmetric complete -metric space and let be two continuous onto mappings. Suppose that there exist nonnegative real numbers a, b, c, d, e with and , such that, for all ,

(2.7)

Then and have a common fixed point; Specially, if , then and have a unique common fixed point.

Proof. Suppose is an arbitrary point in . Since S, T are onto, there exist such that , Continuing this process, we can define by , , for all . By (2.7), we have

Apply to the symmetric of , we have

which implies that

(2.8)

Similarly, it can be shown that

which implies that

(2.9)

Let From and , , we know and . Thus, let , then . So, from (2.8) and (2.9), for all , we get

Hence, for it follows that

Moreover, for all we have by rectangle inequality that

So, , as and is -Cauchy sequence. Due to the completeness of , there exists such that as It's equivalent to , as Since S, T are continuous, then we have and that is, Therefore, is a common fixed point of and

If , assume that is another common fixed point of and , then we hav

so , that is, . Therefore, when , and have a unique common fixed point.

Remark 2.8. Theorem 2.7 of this paper extends Theorem 1 of ² from metric spaces to G-metric spaces, but we add to the continuity of the mappings.

Corollary 2.9. Let be a symmetric complete -metric space and let be two continuous onto mappings. Suppose that there exist nonnegative real numbers with and such that, for all ,

Then S and T have a common fixed point.

Corollary 2.10. Let be a complete -metric space and let be two onto mappings. Suppose that there exists such that, for all ,

Then S and T have a unique common fixed point.

Corollary 2.11. Let be a complete -metric space and let is onto mappings. Suppose that there exist p, q are positive integers and such that, for all ,

Then has a unique common fixed point.

Proof. Let , . Since is an onto mapping, then , are onto mappings, the conditions of Corollary 2.10 are satisfied.

Acknowledgements

This work is supported by the Humanity and Social Science Planning (Youth) Foundation of Ministry of Education of China (Grand No. 14YJAZH095, 16YJC630004), the National Natural Science Foundation of China (Grand No. 61374081), the Natural Science Foundation of Guangdong Province (2015A030313485) and the Guangzhou Science and Technology Project (Grant No.201707010494).

References