#### Abstract

We introduce generalized -contractive mappings of integral type in the context of generalized metric spaces. The results of this paper generalize and improve several results on the topic in literature.

#### 1. Introduction and Preliminaries

In fixed point theory, one of the interesting research trends is to investigate the existence and uniqueness of certain mappings in the various abstract spaces. As a result of this approach, the notion of metric has been extended in several ways to get distinct abstract spaces. Among all, we mention the concept of generalized metric space that was introduced by Branciari [1] in 2001. The notion of generalized metric can be considered as a natural extension of the concept of a metric since it is obtained by replacing the the triangle inequality condition by a weaker condition, namely, quadrilateral inequality. Branciari [1] proved Banach’s fixed point theorem in such a space. For more details, the reader can refer to [2–21].

At this point, we emphasize why the generalized metric space is interesting. Although the definitions of metric and generalized metric are very close to each other, the topology of the corresponding spaces is very different. In particular, a generalized metric may or may not be continuous. Furthermore, a convergent sequence in generalized metric spaces need not be Cauchy. Besides them, we cannot guarantee that a generalized metric space is Hausdorff, and hence the uniqueness of limits cannot be provided easily.

On the other hand, a notion of -admissible mappings was defined by Samet et al. [22]. By using this notion, the authors introduced contractive mappings and investigated the existence and uniqueness of a fixed point of such mappings in the context of metric space. Their results have attracted several authors since they are very interesting and that several existing fixed point theorems listed as consequences of the main result of this paper [22]. The approaches used in this paper have been extended and improved by a number of authors to get similar results in different settings; see, for example, [13, 15, 23–26].

The aim of this paper is to examine the existence and uniqueness of fixed points of -admissible mappings of integral type in the setting of generalized metric spaces. We also underline that the phrase “a generalized metric” has been used for distinct notions since all such concepts generalize the notion of metric. For this reason, when we mention a “generalized metric” we mean the distance function introduced by Branciari [1]. It is evident that any metric space is a generalized metric space but the converse is not true [1].

For the sake of completeness, we recall some basic definitions and notations and fundamental results that will be used in the sequel.

and denote the set of positive integers and the set of nonnegative reals, respectively. Let be the family of functions satisfying the following conditions:(i) is upper semicontinuous;(ii) converges to as for all ;(iii), for any .

In the following, we recall the notion of a generalized metric space.

*Definition 1 (see [1]). *Let be a nonempty set and let as satisfy the following conditions for all and all distinct each of which is different from and . Consider
Then, the map is called a generalized metric and abbreviated as GMS. Here, the pair is called a generalized metric space.

In the above definition, if satisfies only (GMS1) and (GMS2), then it is called a semimetric (see, e.g., [27]).

The concepts of convergence, Cauchy sequence, completeness, and continuity on a GMS are defined below.

*Definition 2. *(1)A sequence in a GMS is GMS convergent to a limit if and only if as .(2)A sequence in a GMS is GMS Cauchy if and only if for every there exists positive integer such that for all .(3)A GMS is called complete if every GMS Cauchy sequence in is GMS convergent.(4)A mapping is continuous if for any sequence in for which , we have .

The following assumption was suggested by Wilson [27] to replace the triangle inequality with the weakened condition.(W):for each pair of (distinct) points , there is a number such that for every

Proposition 3 (see [28]). *In a semimetric space, the assumption is equivalent to the assertion that limits are unique.*

Proposition 4 (see [28]). *Suppose that is a Cauchy sequence in a GMS with , where . Then for all . In particular, the sequence does not converge to if .*

The following concepts were defined by Samet et al. [22].

*Definition 5 (see [22]). *For a nonempty set , let and be mappings. We say that is -admissible if for all , one has

In what follows we recall the notion of a contractive mapping.

*Definition 6 (see [22]). *Let be a metric space and let be a given mapping. One says that is a contractive mapping if there exist two functions and a certain such that

Notice that any contractive mapping, that is a mapping satisfying the Banach contraction, is a contractive mapping with for all and , .

Inspired by the results of Samet et al. [22], Karapınar [13] gave the analog of the notion of a contractive mapping in the context of generalized metric spaces as follows.

*Definition 7. *Let be a generalized metric space and let be a given mapping. One says that is a contractive mapping if there exist two functions and a certain such that

Let be a generalized metric space. A sequence is called regular if is a sequence in such that for all and as ; then for all .

Karapınar [13] also stated the following fixed point theorems.

Theorem 8. *Let be a complete generalized metric space and let be a contractive mapping. Suppose that *(i)* is -admissible;*(ii)*there exists such that and ;*(iii)*either is continuous or is regular.**
Then there exists a such that .*

For the uniqueness, an additional condition was considered.():for all , one has , where denotes the set of fixed points of .

Theorem 9. *Adding condition to the hypotheses of Theorem 8, one obtains that is the unique fixed point of .*

As an alternative condition for the uniqueness of a fixed point of a contractive mapping, one will consider the following hypothesis.(H):for all , there exists such that and .

Theorem 10. *Adding conditions and to the hypotheses of Theorem 8, one obtains that is the unique fixed point of .*

Corollary 11. *Adding condition to the hypotheses of Theorem 8 and assuming that is Hausdorff, one obtains that is the unique fixed point of .*

#### 2. Main Results

In this section, we will present our main results. For this purpose, we first define the following class of functions: such that is nonnegative, Lebesgue integrable and satisfies

*Definition 12 (see [29]). *One says that is an integral subadditive if for each , one has

One denotes by the class of all integral subadditive functions .

*Example 13 (see [29]). *Let for all , for all , and for all . Then , where .

In what follows we introduce notions of generalized -contractive type mappings of integral type I and type II.

*Definition 14. *Let be a generalized metric space and let be a given mapping. One says that is generalized -contractive type mappings of integral type I if there exist two functions and such that for each where and

*Definition 15. *Let be a generalized metric space and let be a given mapping. One says that is generalized -contractive type mappings of integral type II if there exist two functions and such that for each where and

Now, we state our first fixed point result.

Theorem 16. *Let be a complete generalized metric space and let be a generalized -contractive type mappings of integral type I. Suppose that *(i)* is -admissible;*(ii)*there exists such that and ;*(iii)* is continuous.**
Then there exists a such that .*

*Proof. *Regarding assumption (ii), we guarantee that there exists a point such that and . Starting this initial value , we define an iterative sequence in as follows:
Notice that if for some , then the proof is completed in this case. Indeed, we have . As a consequence of this observation, throughout the proof, we assume that
It is evident that
since is -admissible. Recursively, we find that
By repeating the same arguments, used above, we also derive that
From the previous inequalities, we conclude that
We divide the proofs into 4 steps.*Step 1.* We show that
By taking (8) and (15) into account, we obtain that for all , where
If we have for some , then inequality (19) turns into
by regarding the property (iii) of the auxiliary function . This is a contradiction. Consequently, we have for all and (19) becomes
This yields that
by recalling the property (iii) of the auxiliary function . Due to (22), we find that
By property of again, we deduce that and hence *Step 2.* We show that
Combining (8) and (17), we conclude that
for all , where
By (23), we have where and . Thus, inequality (28) can be considered as
On the other hand, by (23)
Therefore,
Then, the sequence is monotone nonincreasing, so it converges to some . Assume that . Now, by (18)
Taking in (31) which is a contradiction; that is, (27) is proved.*Step 3.* We will prove that
We argue by contradiction. Suppose that for some with . Since for each , so without loss of generality, assume that Consider now where
If , then from (37) we get that
If , inequality (37) becomes
Due to a property of , inequalities (39) and (40) together yield that
respectively. In each case, there is a contradiction.*Step 4.* We will prove that is a Cauchy sequence; that is,
The cases and are proved, respectively, by (18) and (27). Now, take arbitrary. It is sufficient to examine two cases.*Case (I)*. Suppose that where . Then, by using step 3 and the quadrilateral inequality together with (24), we find *Case (II)*. Suppose that where . Again, by applying the quadrilateral inequality and step 3 together with (24), we find
By combining expressions (44) and (45), we have
Hence, we have
We conclude that is a Cauchy sequence in . Since is complete, there exists such that
Since is continuous, we obtain from (48) that that is, . Taking Proposition 4 into account, we conclude that ; that is, is a fixed point of .

The following result is deduced from Theorem 16 due to the obvious inequality .

Theorem 17. *Let be a complete generalized metric space and let be generalized -contractive type mappings of integral type II. Suppose that *(i)* is -admissible;*(ii)*there exists such that and ;*(iii)* is continuous.**
Then there exists a such that .*

Theorem 16 remains true if we replace the continuity hypothesis by the following property.

If is a sequence in such that for all and as , then there exists a subsequence of such that for all .

This statement is given as follows.

Theorem 18. *Let be a complete generalized metric space and let be generalized -contractive type mappings of integral type I. Suppose that *(i)* is -admissible;*(ii)*there exists such that and ;*(iii)*if is a sequence in such that for all and as , then there exists a subsequence of such that for all .**
Then, there exists such that .*

*Proof. *Following the lines in the proof of Theorem 8, we deduce that the sequence defined by for all is Cauchy and converges to some . In view of Proposition 4,
By using the method of* reductio ad absurdum*, we will show that . Suppose, on the contrary, that ; that is, . From (15) and condition (iii), there exists a subsequence of such that for all .

By applying (8), we find that where
By (18) and (50), we obtain Since is upper semicontinuous, by letting in (51) we derive that
This is a contradiction. Hence, we obtain that is a fixed point of ; that is, .

In the following, the hypothesis of upper semicontinuity of is not required. Similar to Theorem 18, for the generalized contractive mappings of type II, we have the following.

Theorem 19. *Let be a complete generalized metric space and let be generalized -contractive type mappings of integral type II. Suppose that *(i)* is -admissible;*(ii)*there exists such that and ;*(iii)*if is a sequence in such that for all and as , then there exists a subsequence of such that for all .**
Then, there exists such that .*

*Proof. *Following the proof of Theorem 17 (which is the same as Theorem 16), we know that the sequence defined by for all is Cauchy and converges to some . Similarly, in view of Proposition 4,
We will show that . Suppose, on the contrary, that . From (15) and condition (iii), there exists a subsequence of such that for all . By applying (10), for all , we get that where
Letting in (56), we have
From (58), for large enough, we have , which implies that
Thus, from (56) and (58), we have which is a contradiction. Hence, we obtain that is a fixed point of ; that is, .

Theorem 20. *Adding condition to the hypotheses of Theorem 16 (resp., Theorem 18), one obtains that is the unique fixed point of .*

*Proof. *By using the method of* reductio ad absurdum*, we will show that is the unique fixed point of . Let be another fixed point of with . By hypothesis ,
Now, due to (8), we have which is a contradiction. Hence, .

Theorem 21. *Adding condition to the hypotheses of Theorem 17 (resp., Theorem 19), one obtains that is the unique fixed point of .*

*Proof. *As in Theorem 20, we use the method of* reductio ad absurdum* to show that is the unique fixed point of . Suppose, on the contrary, that is another fixed point of with . It is evident that .

Now, due to (10), we have which is a contradiction. Hence, .

For the uniqueness of a fixed point of a generalized contractive mapping, we will consider the following hypotheses suggested in [11].(H1):for all , there exists in such that and .(H2):let . If there exists in such that and , then

Theorem 22. *Adding conditions , , and to the hypotheses of Theorem 16 (resp., Theorem 18), one obtains that is the unique fixed point of .*

*Proof. *We will show that is the unique fixed point of , by using the method of* reductio ad absurdum*. Let be another fixed point of with ; that is, . Due to , there exists such that
Since is -admissible, from (65), we have
Define the sequence in by for all and . From (66), for all , we have where
By , we get
Iteratively, by using inequality (67), we get that for all . Letting in the above inequality, we obtain and hence
Similarly, one can show that
Regarding , there exists such that for all and hence From (71) and (73), by passing , it follows that , which is a contradiction. Thus, we proved that is the unique fixed point of .

Theorem 23. *Adding conditions , , and to the hypotheses of Theorem 17 (resp., Theorem 19), one obtains that is the unique fixed point of .*

*Proof. *Suppose that is another fixed point of and . From , there exists such that
Since is -admissible, from (76), we have
Define the sequence in by for all and . From (77), for all , we have where
By , we get
Iteratively, by using inequality (78), we get that for all . Letting in the above inequality, we obtain and hence
Analogously, one can show that
Similarly, regarding together with (83) and (84), it follows that . Thus we proved that is the unique fixed point of .

It is known that Hausdorffness property implies the uniqueness of the limit, so the condition in Theorem 22 (resp., Theorem 23) can be replaced by Hausdorff property. Then, the proof of the following result is clear and hence it is omitted.

Corollary 24. *Adding conditions and to the hypotheses of Theorem 16 (resp., Theorems 18, 17, and 19) and assuming that is Hausdorff, one obtains that is the unique fixed point of .*

#### 3. Consequences

In what follows we introduce the notion of -contractive type mappings of integral type.

*Definition 25 (Karapınar, [14]). *Let be a generalized metric space and let be a given mapping. One says that is an -contractive mapping of integral type if there exist two functions and such that for each where .

Now, we state the following fixed point theorem.

Theorem 26 (Karapınar, [14]). *Let be a complete generalized metric space and let be an contractive mapping of integral type. Suppose that *(i)* is -admissible;*(ii)*there exists such that and ;*(iii)*either is continuous or is regular.**
Then there exists a such that .*

*Proof. *The proof is verbatim of the proofs of Theorems 16 and 18.

Theorem 27 (Karapınar, [14]). *Adding condition to the hypotheses of Theorem 26, one obtains that is the unique fixed point of .*

*Proof. *The proof is verbatim of the proofs of Theorem 20.

*Remark 28. *The uniqueness condition in Theorem 27 can be replaced with alternative criteria , , and as in Theorems 22 and 23.

Corollary 29. *Let be a complete generalized metric space and let be a continuous mapping. Suppose that there exists a function such that **for all , where and **
Then has a unique fixed point.*

*Proof. *Let be the mapping defined by , for all . Then is an -contraction mapping of integral type I. It is clear that all conditions of Theorem 20 are satisfied. Hence, has a unique fixed point.

Corollary 30. *Let be a complete generalized metric space and let be a continuous mapping. Suppose that there exists a function such that **for all , where and **
Then has a unique fixed point.*

*Proof. *As in the corollary, it is sufficient to define such that , for all . Then, evidently, is an -contraction mapping of integral type II. Hence, all conditions of Theorem 21 are fulfilled. So, has a unique fixed point.

The following fixed point theorems follow immediately from Corollary 29 by taking , where .

Corollary 31. *Let be a complete generalized metric space and let be a continuous mapping. Suppose that there exists a constant such that**for all , where and **
Then has a unique fixed point.*

By taking , where , in Corollary 30, we derive the following result.

Corollary 32. *Let be a complete generalized metric space and let be a continuous mapping. Suppose that there exists a constant such that **for all , where and **
Then has a unique fixed point.*

Corollary 33 (cf. [11]). *Let be a complete generalized metric space and let be a continuous mapping. Suppose that there exists a function such that **for all . Then has a unique fixed point.*

*Proof. *Let be the mapping defined by , for all . Then is an -contraction mapping. It is evident that all conditions of Theorem 8 are satisfied. Hence, has a unique fixed point.

The following fixed point theorems follow immediately from Corollary 33 by taking , where .

Corollary 34 (see e.g. [11]). *Let be a complete generalized metric space and let be a continuous mapping. Suppose that there exists a constant such that **for all . Then has a unique fixed point.*

Now, we will show that many existing results in the literature can be deduced easily from our obtained results. The following theorems are the main results of Aydi et al. [11].

Theorem 35 (Aydi et al. [11]). *Let be a complete generalized metric space and let be a generalized contractive mapping of type I. Suppose that *(i)* is -admissible;*(ii)*there exists such that and ;*(iii)*either is continuous or if is a sequence in such that for all and as , then there exists a subsequence of such that for all .**
Then there exists a such that .*

*Proof. *It is sufficient to take in Theorems 16 and 18.

Theorem 36 (Aydi et al. [11]). *Let be a complete generalized metric space and let be a generalized contractive mapping of type II. Suppose that *(i)* is -admissible;*(ii)*there exists such that and ;*(iii)*either is continuous or if is a sequence in such that for all and as , then there exists a subsequence of such that for all .**
Then there exists a such that .*

*Proof. *If we take in Theorems 17 and 19, then the proof follows immediately.

Theorem 37 (Aydi et al. [11]). *Adding condition to the hypotheses of Theorem 35 (resp., Theorem 36), one obtains that is the unique fixed point of .*

*Proof. *Let in Theorems 20 and 21.

*Remark 38. *Notice that all consequences and corollaries of Aydi et al. [11] can be added here since their main results are corollaries of the main results of this paper. To avoid the repetition, we do not want to state them here but we underline this fact.

*Example 39. *Let and . We define the distance function as follows:
It is clear that is a generalized metric space. Notice also that is not a metric since
We define as . Furthermore, let be defined as and . Now, we define as follows:
Hence, all conditions of Theorem 20 are satisfied and is a unique fixed point of .

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The author thanks the anonymous referees for their remarkable comments, suggestions, and ideas that helped the author to improve this paper.