Abstract

In this paper, we introduce the notion of generalized -contractions which enlarge the class of ℒ-contractions initiated by Cho in 2018. Thereafter, we also, define the notion of -contractions. Utilizing our newly introduced notions, we establish some new fixed-point theorems in the setting of complete Branciari’s metric spaces, without using the Hausdorff assumption. Moreover, some examples and applications to boundary value problems of the fourth-order differential equations are given to exhibit the utility of the obtained results.

1. Introduction

In 2000, Branciari [1] initiated the concept of a generalized metric by replacing the natural triangle inequality of a metric with a relatively more general inequality termed as rectangular (or quadrilateral) inequality which involves four points instead of three. In the literature, such a metric is known as the Branciari metric. Branciari [1] assumed that each of the Branciari metric space becomes a Hausdorff topological space, and the Branciari metric is a continuous function in each coordinate. Sarma et al. [2] and Samet [3] showed that these assumptions were not correct. Meanwhile, several authors obtained various fixed-point results in the Branciari metric space with the assumption that the space is a Hausdorff and (or) the Branciari metric is continuous. However, it was shown (e.g. [4, 5]) that in general, neither the Hausdorff topological property nor the continuity of the metric is required in the proofs. For a recent update or extension of the Branciari metric spaces, we refer to [6].

On the other hand, in 2014, Jleli and Samet [7] introduced the concept of -contractions and proved some fixed-point results in the setting of the Branciari metric spaces. Very recently, Cho [8] introduced the notion of -contractions which unify several concepts of contractions in the existing literature including -contractions in the setting of the Branciari metric spaces. For more notions and results in such spaces, we refer the reader to [9–17].

This paper is aimed at introducing two types of contraction mappings, namely, generalized -contractions and -contractions. For both types of contractions, we prove separate fixed-point theorems in the setting of the complete Branciari metric spaces. The obtained results extend, generalize, and improve some results of the existing literature. Moreover, applications to boundary value problems of the fourth-order are given to exhibit the utility of the obtained results.

2. Preliminaries

The following definitions and basic results are needed in the sequel.

Definition 1 [1]. Let be a nonempty set and a mapping such that for all and all : (BMS1), if and only if (BMS2)(BMS3).

The metric is called a Branciari metric, and the pair () is called a Branciari metric space.

The next example shows that the topologies of the Branciari metric spaces and the usual metric spaces are different. In particular, we have the following: (i)The Branciari metric need not be continuous in both variables(ii)The Branciari metric space is not necessarily Hausdorff(iii)An open ball need not be an open set(iv)A convergent sequence is not necessarily a Cauchy sequence.

Example 2 [2]. Let , where and . Define as Then is a complete Branciari metric space. However, it is easy to see that (i) although , and hence the function is not continuous(ii)There does not exist such that , and hence the respective topology is not a Hausdorff(iii); however, there does not exist such that , and hence an open ball need not be an open set(iv)The sequence converges to both 0 and 2, and it is not a Cauchy sequence.

Definition 3 [7]. Let be a Branciari metric space. A mapping is said to be a -contraction if there exist and such that (for all ) where is the family of all functions : which satisfy the following conditions:
() is nondecreasing
() For each sequence ,
() There exists and such that .

Theorem 4 [7]. Let be a complete Branciari metric space and a -contraction mapping. Then has a unique fixed point.

Imdad et al. [18] observed that this theorem can be proved without the condition (). Also, Ahmad et al. [19] replaced the condition () by the following one:

is continuous.

Remark 5. It is known that every -contraction mapping is continuous.

In the sequel, we adopt the following notations: (i) is the class of all functions which satisfy -(ii) is the class of all functions which satisfy , , and (iii) is the class of all functions which satisfy -.

In 2018, Cho [8] introduced the notion of -contractions which unify several concepts of contractions in the existing literature including -contractions as under one unifying concept:

Definition 6 [8]. Let be a Branciari metric space. A mapping is said to be an -contraction with respect to if there exists such that (for all ) where is the class of all functions which satisfy the following conditions :
()
() for all
() If and are two sequences in with , such that , then .

Example 7 [8]. Let be two functions defined as under: (a), for all , where (b), for all , where is a lower semicontinuous and nondecreasing function with Then .

Example 8. Let be a function defined by where are upper semicontinuous from the right such that , for all . Then .

Based on the above definition, the author in [8] proves the following theorem.

Theorem 9 [8]. Let be a complete Branciari metric space and an -contraction mapping. Then has a unique fixed point.

Remark 10. It is known that every -contraction mapping is continuous.

Remark 11. Let be three sequences of real numbers such that , , and . Then
(i)
(ii) .

The following lemmas are useful in the sequel.

Lemma 12 [20]. Let be a Cauchy sequence in a Branciari metric space such that whenever . Then can converge to at most one point.

Lemma 13 [21]. Let be a Cauchy sequence in a Branciari metric space such that for some . Then, , for all . In particular, does not converge to if .

3. Fixed-Point Results for Generalized -Contractions

We begin this section by introducing the concept of generalized -contractions followed by the main result of this section as follows:

Definition 14. Let be a Branciari metric space and . Then is said to be a generalized -contraction with respect to if there exist and constant such that (for all ) where

Theorem 15. Let be a complete Branciari metric space and . If is a continuous generalized -contraction mapping with respect to , then it has a unique fixed point.

Proof. Let be an arbitrary element of and define a Picard sequence by , for all . If , for some , then is a fixed point of and the proof is finished. Now, assume that , for all . Using the contractive condition (5) and , we have which implies that

Notice that

Hence, inequality (7) becomes which implies (in view of ) that

Therefore, the sequence is decreasing and bounded below by 0. This insures the existence of a number such that . Assume that , then it follows from that

Setting and . In view of (9), (11), and , we have and , for all . Therefore, applying the condition , we deduce which is a contradiction, and hence we must have

Now, assume that , for some . Then, also . Using (9), we get which is a contradiction. Therefore, we conclude that , for all .

Next, we claim that the sequence is a Cauchy sequence in . On the contrary, assume that it is not Cauchy, then there exists an for which we can find two subsequences and of such that , for all and

Suppose that is the least integer exceeding satisfying inequality (15). Then, we have

Using (15), (16), and the rectangular inequality, we get

On taking the limit as and making use of (13), we obtain

Employing the rectangular inequality once again, we get

On letting and using (13) as well as (18), we get

Now, using (5) and , we obtain where

Consequently, we deduce that

Let and . Then in view of Remark 11 and (23), we have and . So, on using , we obtain which is a contradiction. Therefore, must be a Cauchy sequence in . Since is complete, then there exists such that , that is,

As is continuous, then we get that (due to (25)) that is, . Using Lemma 13, we conclude that , that is, is a fixed point of .

Finally, we show that the fixed point of the mapping is unique. On the contrary, assume that there are two fixed points such that . From (5), we have where

This implies that which is a contradiction. Then has a unique fixed point.

Next, we furnish the following illustrative example which shows that Theorem 15 is a genuine extension of Theorems 4 and 9.

Example 16. Let and define a mapping as (i)(ii)(iii)(iv) and ,

Then it is easy to check that is a complete Branciari metric space which is not a metric space since

Consider the mapping defined by

Notice that is neither a -contraction nor an -contraction. Indeed, for and , we have

So, in view of and for any and , we obtain and

Therefore, Theorems 4 and 9 cannot be used here, but Theorem 15 is applicable. In fact, we have and hence, for all ,

Now, we show that is a generalized -contraction with respect to , where , for all , , and .

Define a function by

For all with , we have two cases:

Case 1. If or , then we have Hence,

Case 2. If or , then we have Thus,

Therefore, all the hypotheses of Theorem 15 are satisfied, and hence has a unique fixed point (namely ).

Taking in the contractive condition (5), Theorem 15 reduces to the following fixed-point result.

Corollary 17 [8]. Let be a complete Branciari metric space and . If is an -contraction with respect to , then has a unique fixed point.

Next, we present the following results, which seem new to the existing literature.

Corollary 18. Let be a complete Branciari metric space and let be continuous mapping. Suppose that there exist , and such that (for all ) where Then has a unique fixed point.

Proof. Observe that is a generalized -contraction with respect to . Then, the result follows immediately from Theorem 15.

Corollary 19. Let be a Branciari metric space and let be continuous mapping such that (for all ) where and is nondecreasing and lower semicontinuous such that . Then has a unique fixed point.

Proof. Let , for all . From (42), we have for all with .

Now, define , for all , where is nondecreasing and lower semicontinuous such that .

From (43), we have

Taking and using (44), we have

Therefore, all the requirements of Theorem 15 are satisfied and hence has a unique fixed point.

4. Fixed-Point Results for -Contractions

Before presenting our main result of this section, we give the following definition.

Definition 20. Let be a Branciari metric space and . Then is said to be an -contraction with respect to if there exists such that (for all ) where

Now, we are ready to state and prove the main result of this section.

Theorem 21. Let be a complete Branciari metric space and an -contraction with respect to . Then has a unique fixed point.

Proof. Let be an arbitrary point of . Define a sequence by If there exists such that , then is a fixed point of and hence the proof is done. Assume that , for all .

Using the contractive condition (46) and we have

Consequently, we obtain that where

If , then inequality (49) becomes which is a contradiction. Hence, we must have , for all . Therefore, inequality (49) becomes which implies from that

Thus, the sequence is decreasing and bounded below by 0, so there exists such that . Suppose that , then it follows from that

Taking and , for all . It is clear from (52), (54), and that and . Hence, using we get which is a contradiction. Therefore, , i.e., we have

Now, let us assume that , for some . Then, we have . Using (52), we get which is a contradiction. This concludes that , for all .

Next, we claim that the sequence is a Cauchy sequence in . On the contrary, assume that it is not Cauchy, then we can find two subsequences and of such that is the smallest index for which

By using a similar argument as in the proof of Theorem 15, we obtain

Now, using (46) and , we have which implies that where .

Making use of (56), (59), and Remark 11, we deduce that

Now, let and , for all . In view of (59), (61), (62), and , we have , for all and . Therefore, using we obtain

This contradiction ensures that the sequence is a Cauchy sequence in . As is complete, then there exists such that

Without loss of generality, we can assume that and , for all . Suppose that , it follows from (46) and that where which implies that

From Remark 11 and Lemma 13, we have

Let and , for all . Then it follows from that which is a contradiction. Therefore, we conclude that , that is, is a fixed point of . Finally, we show that the fixed point of the mapping is unique. Assume that and are two distinct fixed points in . Then .

Using (46) and , we deduce that where which implies that which is a contradiction. Therefore, has a unique fixed point.

Remark 22. In Theorem 15, the mapping must be continuous, while in Theorem 21 need not be continuous.

To support Theorem 21, we give an illustrative example. Precisely, we show that Theorem 21 can be used to cover this example while Theorems 4, 9, and 15 are not applicable.

Example 23. Let , where and . Define a mapping as follows: (i)(ii)(iii)(iv), , , and(v) if or , , or , .

Observe that is not a metric on , because the triangle inequality is not satisfied on . To insure this, we have

It is easy to check that is a complete Branciari metric space. Let be defined as

Since is not continuous at , then by Remarks 10 and 5, is neither a -contraction nor an -contraction, and hence Theorems 4 and 9 cannot be applied here.

Observe that is an -contraction with respect to , where and , such that

Indeed, for and , we have and

Hence, all the hypotheses of Theorem 21 are satisfied, and the unique fixed point of is .

Notice that due to Remark 22, Theorem 15 cannot be applied here.

The corollaries that follow are deduced as consequences of Theorem 21.

Corollary 24. Let be a complete Branciari metric space and . Suppose that there exist and such that (for all ) where . Then has a unique fixed point.

Proof. Observe that is an -contraction with respect to . Then, the result follows immediately from Theorem 21.

Remark 25. Corollary 25 is a generalization of Theorem 4 in [17] without assuming condition .

Corollary 26. Let be a Branciari metric space and . Assume that (for all ) where and is nondecreasing and lower semicontinuous such that . Then has a unique fixed point.

Proof. Let , for all . From (77), we have for all with .

Now, define , for all , where is nondecreasing and lower semicontinuous such that .

From (78), we have

Taking and using (79), we have

Therefore, all the requirements of Theorem 21 are satisfied, and hence has a unique fixed point.

5. An Application to Fourth-Order Differential Equation

In this section, we discuss application of the fixed-point theorems obtained in the previous sections in solving the following boundary value problem of a fourth-order differential equation: where is a continuous function. Let be the space of all continuous functions defined on . Define a metric by

It is known that is a complete Branciari distance space. The green function associated to (81) is defined by

Now, we prove the following result on the solution of the boundary value problem (81).

Theorem 27. Assume that the following conditions are satisfied: (i) is a continuous function(ii)There exists such that, for all and where is defined by

Then (81) has a unique solution in .

Proof. Observe that is a solution of (81) if and only if is a solution of the integral equation

In view of condition . For all with and for all , we have where . As , for all , , we obtain or

Observe that as . Therefore, for all we obtain where , , and . Thus, all the hypotheses of Theorem 21 are satisfied, and hence has a unique fixed point in which is a solution of (81).

Theorem 28. Assume that the following conditions are satisfied: (i) is a continuous function(ii)There exists and such that for all and where and is defined by

Then (81) has a unique solution in .

Proof. The proof can be done using similar arguments of the proof of Theorem 28 and applying Theorem 15.

Data Availability

No data were used in this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All the authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The third author thanks Prince Sultan University for funding this research through the group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) (group number RG-DES-2017-01-17).