## Advances on Multivalued Operators and Related Fixed Point Problems

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# Some Almost Generalized -Contractions in -Metric Spaces

**Academic Editor:**Chi-Ming Chen

#### Abstract

In this paper, we introduce some almost generalized -contractions in the setting of -metric spaces. We prove some fixed points results for such contractions. The presented theorems improve and extend some known results in the literature. An example is also presented.

#### 1. Introduction and Preliminaries

In 2006, a new structure of a generalized metric space was introduced by Mustafa and Sims [1] as an appropriate notion of a generalized metric space called a -metric space. Fixed point theory in this space was initiated in [2]. Particularly, Banach contraction mapping principle was established in this work. Since then the fixed point theory in -metric spaces has been studied and developed by many authors, see [1â€“29].

The following definitions and results will be needed in the sequel.

*Definition 1 (see [1]). *Let be a nonempty set, be a function satisfying the following properties:(*G*1) if ,(*G*2) for all with ,(*G*3) for all with ,(*G*4) (symmetry in all three variables),(*G*5) for all (rectangle inequality).Then the function is called a generalized metric, or, more specially, a -metric on , and the pair is called a -metric space.

Every -metric on defines a metric on by

*Example 2. *Let be a metric space. The function , defined by
or
for all , is a -metric on .

*Definition 3 (see [1]). *Let be a -metric space, and let be a sequence of points of ; therefore, we say that is -convergent to if ; that is, for any , there exists such that , for all . We call the limit of the sequence and write or .

Proposition 4 (see [1]). *Let be a -metric space. The following are equivalent:*(1)* is **-convergent to **,*(2)* as **,*(3)* as **.*

*Definition 5 (see [1]). *Let be a -metric space. A sequence is called a -Cauchy sequence if, for any , there is such that for all ; that is, as .

Proposition 6 (see [1]). *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 7 (see [1]). *A -metric space is called -complete if every -Cauchy sequence is -convergent in .

*Definition 8. *Let be a -metric space. A mapping is said to be -continuous if for any -convergent sequence to , then is -convergent to .

Now, let denote the set of functions such that if and only if . We denote by and the subsets of such that There are a lot of fixed point theorems for different type contractions in the literature. In particular, Berinde [30â€“32] introduced the concept of an almost contraction in metric spaces and studied many interesting fixed point theorems for a Ä†iriÄ‡ strong almost contraction. For other fixed point results on generalized almost contractions, see [33â€“37]. In this paper, we introduce some almost generalized -contractions in the setting of -metric spaces, and we establish some fixed points results for such contractions.

#### 2. Main Results

Let be a -metric space. First, we consider the following expressions: for all .

Our first result for almost generalized -contractions is the following.

Theorem 9. *Let be a complete -metric space. Let be a self-mapping. Suppose there exist , , and such that for all ,
**
Then has a unique fixed point; say .*

*Proof. *Let , and define a sequence in such that
If for some , then , and the proof is completed. Thus, we may assume that for all . By (6) we have
where
From (8) and (9), we get
If
then by (10) we have
Thus , and hence . Therefore, which is a contradiction. So,
Therefore, we get
and (10) becomes
Thus, by (14), the sequence is monotone nonincreasing. It follows that as for some . Next we claim that . On taking limit as in (15), we obtain
Hence and we get . Hence
Next, we show that is a -Cauchy sequence. On contrary, assume that is not a -Cauchy sequence. Then, there is an for which we can find subsequences of with such that
Using (18) and (*G*5), we have
Taking limit as and using (17), we have
By (*G*5), we get
On taking limit as in the above inequality and using (17) and (20), we obtain
Again, by (*G*5), we have
On taking limit as in the above inequality and using (17) and (22), we obtain
We have also, by (*G*5),
On taking limit as in the above inequality and using (17) and (*G*3), we obtain
We have also, by (*G*3) and (*G*5),
Now, on taking limit as in (27), and using (17), (20), and (26), we obtain
Furthermore, by (*G*3) and (*G*5), we get
On taking limit as in the above inequality, and using (17), (20), (22), and (28), we obtain
Now, we have
Letting , and using (17), (20), (22), (24), (28), and (30), and the properties of and , we have
Thus and hence , a contradiction. Thus is a -Cauchy sequence in .

Now, since is -complete, there are such that is -convergent to ; that is
By (6), we get
where
Letting in (35), we get
On letting in (34), and using the properties of and and (36), we obtain
Therefore and hence . Thus is a fixed point of .

Now our purpose is to check that such point is unique. Suppose that there are two fixed points of ; say such that . By (6), we have
where
Similarly, we can prove that
where
If
By (38) and (39), we get
a contradiction. Then
By (40) and (41), we get
a contradiction. Thus, , and hence the fixed point of is unique.

As consequence of Theorem 9, we present the following corollaries.

Corollary 10. *Let be a complete -metric space. Let be a self mapping. Suppose there exist and such that for all **
Then has a unique fixed point; say . *

*Proof. *It suffices to take and in Theorem 9.

Corollary 11. *Let be a complete -metric space. Let be a self mapping. Suppose there exist and such that for all **
Then has a unique fixed point; say . *

*Proof. *Note that in Corollary 10.

*Remark 12. *Corollary 11 is a generalization of Mustafaâ€™s result [2].

Our second main result is given as follows.

Theorem 13. *Let be a complete -metric space. Let be a self mapping. Suppose there exist ,â€‰â€‰ and such that for all :
**
where
**
for all . Then has a unique fixed point; say .*

*Proof. *Let be an arbitrary point in and define a sequence in such that
If for some , then and the proof is completed. Thus, we may assume that for all . By (48), we have
where
From (51) and (52), we get
If
then by (53) we have
Thus and hence . Therefore, , which is a contradiction. So,
Therefore, we get
and (53) becomes
Thus, by (57), the sequence is monotone nonincreasing. It follows that as for some . Next we claim that . On taking limit as in (58), we obtain
Hence and we get . Then
Next, we show that is a -Cauchy sequence. On contrary, assume that is not a -Cauchy sequence. Then, there is an for which we can find subsequences of with such that
Using (61) and (*G*5), we have
Taking limit as and using (60), we have
By (*G*5), and using (60) and (63), we obtain, as stated in the proof of Theorem 9,
Again, by (*G*5), and using (60) and (64), similarly as in the proof of Theorem 9, we obtain
Furthermore, by (*G*3) and (*G*5), we get
By the proof of Theorem 9, we have stated that
So, on taking limit as in (66), and using (60), (64), and (67), we obtain
Furthermore, by (*G*3) and (*G*5), we get
On taking limit as in above inequalities, and using (60) and (63), we have
Also, by (*G*3) and (*G*5), we get
On taking limit as in above inequalities, and using (60), (63), and (70), we obtain
Now, we have