Rectangular Metric-Like Type Spaces and Related Fixed Points

Abstract

In this paper, we introduce the concept of a rectangular metric-like space, along with its topology, and we prove some fixed point theorems for different contraction types. We also introduce the concept of modified metric-like spaces and we prove some topological and convergence properties under the symmetric convergence. Some examples are given to illustrate the new introduced metric type spaces.

1. Introduction

The generalization of Banach contraction principle, which has many applications in different branches of science and engineering, depends on either generalizing the metric type space or the contractive type mapping (see [1] and the references therein). The generalization of a metric space is based on reducing or modifying the metric axioms; for example, we cite quasi-metrics, partial metrics, metrics, metrics, rectangular metrics, metrics. For more details, see [2–15]. Note that losing or weakening some of the metric axioms causes the loss of some topological properties, hence bringing obstacles in proving some fixed point theorems. These obstacles force researchers to develop new techniques in the development of fixed point theory in order to resolve more real concrete applications. In this article, we restrict ourselves on developing metric-like spaces by introducing modified metric-like spaces, rectangular metric-like spaces, and rectangular modified metric-like spaces. We shall prove some fixed point theorems in rectangular metric-like spaces. Examples will be given to support the given concepts. The notion of symmetric convergence will be also studied in the setting of modified metric-like spaces.

Definition 1 ([13] (partial metric space)). Let be a nonempty set. A mapping is said to be a partial metric on , if, for any , it satisfies the following conditions: x = y if and only if ; p; p; p. In this case, the pair is called a partial metric (PM) space.

Definition 2 ([9] (rectangular (or Branciari) metric space)). Let be a nonempty set. A mapping is said to be a rectangular metric on if, for any and all distinct points , it satisfies the following conditions: if and only if ;;. In this case, the pair is called a rectangular metric (RM) space.

In [15], the notion of a rectangular metric space was extended to rectangular partial metric spaces as follows.

Definition 3 ([15] (rectangular partial metric space)). Let be a nonempty set. A mapping is said to be a rectangular metric on if, for any and all distinct points , it satisfies the following conditions: if and only if ;;;. In this case, the pair is called a rectangular partial metric (RPM) space.

It is clear that every rectangular metric space is a rectangular partial metric space, but the converse is not true.

Example 4 (see [15]). Let and . Define the mapping by Then is a rectangular partial metric space, but it is not a rectangular metric space, because, for any , we have

For convergence, completeness, and examples of RM, PM, and RPM spaces, we refer to [9, 13, 15]. See also the papers [16, 17].

Definition 5 (see [11]). Let be a nonempty set. A mapping is said to be a metric-like on if, for any , it satisfies the following conditions: implies ;;. In this case, the pair is called a metric-like space (-space).

Every metric-like space is a topological space whose topology is generated by the base consisting of the open balls Note the difference between the balls and the balls , which is due to the absence of the smallness of the self distance condition from the metric-like. Also, since the self distance need not be zero in metric-like spaces, then convergence and completeness in metric-like spaces still resemble the case of partial metric spaces. Indeed, a sequence in a metric-like space converges to a point if and only if and the sequence is called Cauchy if exists and is finite. The metric-like space is called complete if for each Cauchy sequence there exists such that

Remark 6. Metric-like spaces lose some topological and convergence properties as known for metric spaces. We state the following. For example, limits are not unique in -spaces. Take and let for any . Then, clearly the constant sequence converges to both and . Notice that . However, if and such that , then . Letting , we conclude that , and hence .

2. Main Results

Upon Remark 6 above, we define the following modified metric-like () space.

Definition 7 (modified metric-like space). Let be a nonempty set. A mapping is said to be a modified metric-like on , if for any , it satisfies the following conditions: implies ;;. The pair is called a modified metric-like space (-space).

It is clear that every partial metric space is an space and every space is an space.

Example 8. Let and define the mapping such that for all , for all , and . It is clear that the conditions and in Definition 5 are satisfied. We need to verify . If , then we have . Also, if , then . Finally, if , . Therefore, is a -space, but it is not a space, because .

As in the case of metric-like spaces, the open balls in modified metric-like spaces are given as Also, note that a sequence in a modified metric-like space converges to a point if and only if and the sequence is called Cauchy if exists and is finite.

Definition 9 (symmetrical convergence in modified metric-like spaces). We shall say that a sequence in a modified metric-like space is symmetrically convergent to if, for every , there exists such that, for each , we have Equivalently,

We shall denote for symmetrical convergence. It is clear that symmetrical convergence implies -convergence. Our first result is as follows.

Theorem 10. Let be an -space. Then one has the following: (1)If , then is -Cauchy.(2)If is -Cauchy and has a subsequence such that , then .(3)If and are -Cauchy sequences, then exists.(4)If and , then .

Proof. (1)Assume . Then . By (), for each we have  and  Let . Then . Hence , and so is -Cauchy.(2)Let be a -Cauchy and have a subsequence such that . Then  Since is Cauchy, there exists such that . It is clear that as well. On the other hand, by (), we have  and  Therefore,  If we take , then . It follows that , and hence .(3)Assume that and are -Cauchy sequences in . Then there exist such that and . It is sufficient to prove that the sequence is Cauchy in . By (), for each , we have  and  From what is mentioned, it follows that  Let , then  Hence , and so is Cauchy in .(4)Assume that and . Then  and  Now, for each , we have  and  Finally, letting yields that  and thus .

Now, we introduce the concepts of rectangular metric-like and rectangular modified metric-like spaces.

Definition 11. Let be a nonempty set and be a function. If the following conditions are satisfied for all in , (1);(2);(3), for all distinct , then the pair is called a rectangular metric-like (RML) space.

Definition 12. Let be a nonempty set and be a function. If the following conditions are satisfied for all in , (1);(2);(3), for all distinct , then the pair is called a rectangular modified metric-like (RMML) space.

Example 13. Let and define the mapping by It is clear that conditions and in Definition 11 are satisfied. We need to verify the last condition. For all distinct , we have , for all . Therefore, is a RML-space, but it is not a RMML-space. Indeed, . Moreover, the space is not a rectangular partial metric, because the condition of Definition 2 does not hold.

Example 14. Let be a RML-space. Consider for . Then the space is a RMML-space.

Example 15. Let and define the mapping by . Then is a RMML-space. Indeed, for any and distinct , we have

Definition 16. (1)A sequence is called convergent (resp., convergent) in a rectangular metric-like space (resp., a rectangular modified metric-like space ), if there exists such that  (resp. ).(2)A sequence is called Cauchy if and only if  (resp., ) exists and is finite.(3)A rectangular metric-like space is called complete (resp., complete) if, for every Cauchy (resp., Cauchy) sequence in , there exists , such that  (resp., ).

Remark 17. The convergence defined in Definition 16 is the convergence obtained in the sense of the topology generated by the open balls , , . This convergence is weaker than the symmetric convergence discussed before.

Definition 18 (continuity of maps). Let and be two metric type (such as ML, MML, PM, RPM, RML, RMML) spaces. The mapping is said (sequentially) continuous at if and only if , whenever is a sequence in such that .

Our second main result concerns an existence and uniqueness theorem on rectangular metric-like spaces.

Theorem 19. Let be a complete rectangular metric-like space and be a self mapping on If there exists such thatthen has a unique fixed point in such that

Proof. Let in be arbitrary. Using (23), we havefor all . We distinguish two cases.
Case 1. Let for some integers . For example, take . We have . Choose and . Then that is, is a periodic point of . By (23) and (24), we have Since , we get , so ; that is, is a fixed point of .
Case 2. Suppose that for all integers .
We rewrite (24) asSimilarly, by (23), we haveNow, if is odd, then consider with . By (23) and (27), we have On the other hand, if is even, then consider with . Again, by (23), (27), and (28), We deduce from all cases thatThe right-hand side tends to 0 as , so the sequence is Cauchy in the complete rectangular metric-like space . Due to Definition 16, there exists some such that In view of (31), we getWe shall prove that . Mention that we are still in case 2, that is, for all integers . Now, we distinguish three subcases.
Subcase 1. If, for all , , the rectangular inequality implies that Taking limit as and using (27) and (33), we get ; that is,
Subcase 2. If there exists an integer such that , due to case 2, for all . Similarly, for all . We reach subcase 1, so is a fixed point of .
Subcase 3. If there exists an integer such that , again, necessarily and for all . Similarly, we get .
We deduce that is a fixed point of . To show the uniqueness of the fixed point , assume that has another fixed point . By (23), which holds unless , so .

Next, we present the following example.

Example 20. Let and define the mapping by It is not difficult to see that is a complete rectangular metric-like space. Let be a self mapping on defined by and Note that satisfies the contraction of Theorem 19 with , and is the unique fixed point of

Our third main result is as follows.

Theorem 21. Let be a complete rectangular metric-like space and be a self mapping on If there exists such thatthen has a unique fixed point in such that

Proof. Fix in . If for some , then is a fixed point of . The proof is completed. From now on, we assume that , so for all By (37), we have for all . If, for some , , then which is a contradiction. Thus,Here, the sequence is nonincreasing. As the proof of Theorem 19, we distinguish two cases.
Case 1. Let for some integers (take ). Let , where . Again, using (40), Since , we get , so is a fixed point of .
Case 2. Suppose that for all integers .
We rewrite (40) asSimilarly, by (37) and (42), we have Now, if is odd, then consider with . By (37) and (42), we have On the other hand, if is even, then consider with . Again, by (37), (42), We deduce from all cases thatThe right-hand side tends to 0 as , so the sequence is Cauchy in the complete rectangular metric-like space . Due to Definition 16, there exists some such that, in view of (46),We shall prove that . Without loss of generality, we may assume that, for all , . By the rectangular inequality, we have Taking limit as and using (42) and (47), we get , that is,
Now, let be a fixed point of . From (37), It is true unless , so is the unique fixed point of .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare thast they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RG-DES-2017-01-17.