An Extended -Metric-Type Space and Related Fixed Point Theorems with an Application to Nonlinear Integral Equations

Abstract

In this paper, the concept of sequential -metric spaces has been introduced as a generalization of usual metric spaces, -metric spaces and specially of -metric spaces. Several topological properties of such spaces have been discussed here. In view of this notion, we prove fixed point theorems for some classes of contractive mappings over such spaces. Supporting examples have been given in order to examine the validity of the underlying space and in respect to our proven fixed point theorems.

1. Introduction and Preliminaries

In the last few decades, several generalizations of usual metric structure have been made by the researchers working in the area of fixed point theory. Different topological structured spaces play vital roles for obtaining fixed point theorems using several contractive or expansive or non-expansive type mappings. There are many interesting two-variable metric-type spaces such as -metric space [1, 2], rectangular metric space [3], extended -metric space [4], -metric space [5], JS-metric space [6], modular metric space [7], multiplicative metric space [8], bipolar metric space [9, 10], cone metric space [11], and -algebra-valued metric space [12]. Also, for considering and analyzing more generalizations of the concept of metric spaces, one can consider the following works dealing with (double) controlled metric spaces and generalized -metric spaces [13–16]. Due to the presence of such interesting spaces and various types of applications of fixed point theorems therein, fixed point theory gains attention in the mathematical community specially to the new researchers working on functional analysis. In the context of various metric-type spaces which are the combination of the above-mentioned spaces, several authors have proved different types of fixed point theorems therein (see [17, 18]). Now, we give definitions of some generalized spaces which are relevant to our research work.

Definition 1. -metric space (see [1, 2]). Let be a nonempty set and be a real number satisfying A function is a -metric on if (1) if and only if ,(2) for all ,(3) for all .The space is called a -metric space.

Let be a nonempty set and be a mapping. For any , let us define the set

Definition 2. JS-metric space (see [6]). Let be a mapping which satisfies (1) implies ,(2)for every , we have ,(3)if and , then , for some .The pair is called a generalized metric space, usually known as a -metric space.

Definition 3. -metric space (see [5]). Let be a nonempty set. A function is said to be -metric if there exists a strictly increasing continuous function with for all and such that for all (1) if and only if ,(2),(3).The pair is called a -metric space.

For various examples of the above spaces, one can see the research papers in the reference section of this manuscript regarding such notions and also the references cited in these papers.

Now, we are ready to prove our main results. We define a new metric-type structure, which is the main concept of this paper.

2. Introduction to Sequential -Metric Space

In this section, we introduce a new type of extended -metric spaces. To develop such a notion, first, we define , where is a given mapping.

Definition 4. Let be a nonempty set. A mapping is said to be a sequential -metric if for all (a) implies ,(b),(c), where and is a strictly increasing continuous function with for all with for .The triplet is called a sequential -metric space. We express a sequential -metric space simply as .

Example 1. Let and the metric be defined by

Then, clearly, implies and for all . Now, we show that satisfies condition (iii) of Definition 4.

For , . Let . If all but finitely many terms of are then we are done. So, suppose that only have finitely many 1’s. Without loss of generality, we can exclude such 1’s and then we get . Therefore, for all .

Hence, is a sequential -metric on for for all and for all .

Proposition 5. If is a -metric space (see Definition 2) then is also a sequential -metric on .

Proof. If is a -metric space then clearly satisfies the first two conditions of Definition 4. We just show that also satisfies the third condition of Definition 4.

Since is a -metric then for all and for any sequence , we have , where .

Then, if we choose for all with , then we have for all and . Therefore, is also a sequential -metric space.

Remark 6. . (i) We know that any metric space, metric space [1, 2], dislocated metric space, and modular metric space with the Fatou property [7] are -metric spaces; therefore, these spaces are also sequential -metric spaces. (ii) Any extended -metric space or -metric space is clearly a sequential -metric space.

Proposition 7. Let -metric space with coefficient . Let , where is a strictly increasing continuous function with for all and . Then, is a sequential -metric for for all .

Proof. Here, we show that satisfies all the conditions of Definition 4. (a) gives . Then implies (b) holds trivially(c)For all we have, where Now, for all . Then, .
Therefore, . This proves our proposition.

Definition 8. Let be a sequential -metric space. Also let be a sequence in and (i) is said to be convergent and converges to if ,(ii) is said to be Cauchy if ,(iii) is called complete if any Cauchy sequence in is convergent.

Definition 9. Let and be two sequential -metric spaces. A mapping is called continuous at a point if for any there exists such that for any whenever is said to be continuous on if is continuous at each point of

Proposition 10. In a sequential -metric space if a sequence is convergent then it converges to a unique element in .

Proof. Suppose be such that and as . Then, we have, implying that , i.e., .

Proposition 11. Let be a sequential -metric space and converges to some then .

Proof. Since converges to , so . Therefore, we have which implies .

Proposition 12. Let be a Cauchy sequence in a sequential -metric space such that is continuous. If has a convergent subsequence which converges to , then also converges to .

Proof. From condition (c) of Definition 4, we have which implies that for all .
Due to the Cauchyness of , it follows that and thus as which implies that as , since is continuous. Hence, converges to .

Proposition 13. In a sequential -metric space , if a self-mapping is continuous at then for any sequence .

Proof. Let be given. Since is continuous at , then for any there exists such that implies
As converges to , so for , there exists such that for all Therefore, for any , and thus as

Some observations regarding sequential -metric spaces are as follows: (1)In a metric space, a convergent sequence is always Cauchy, but it is not true in a sequential -metric space. In Example 1, the sequence converges to , but whenever (2)In a metric space, if and are two sequences converging to and , respectively, then as . In particular, if then as . But this does not always hold in a sequential -metric space. In Example 1, let us consider two sequences and in . Then, both of these two sequences converge to , but as (3)A -metric is always a sequential -metric space, but the converse is not true in general. The metric defined in Example 1 is not a -metric for any . Otherwise, for all with , which at once implies that for all with , arrives at a contradiction

3. Cantor’s Intersection-Like Theorem on Sequential -Metric Spaces

Let be a sequential -metric space with supporting function . Define for all and .

Remark 14. One can easily check that the collection for any, there exists such that forms a topology on .

Definition 15. A set is said to be closed if there exists an open set such that , where denotes the complement of in .

Proposition 16. Let be a sequential -metric space and be closed. Let such that as . Then, .

Proof. Suppose that . Then, . Since , so there exists such that . Again, since as , so for there exists such that for all . That is, for all , which is a contradiction. Hence, .

Proposition 17. Let be a complete sequential -metric space and be closed. Then, is also complete.

Proof. Let be a Cauchy sequence in . Then, is convergent in , since is complete. Let . Then, by Proposition 16, it follows that . Consequently, is complete.

Definition 18. In a sequential -metric space , for , we define

Theorem 19. Let be a complete sequential -metric space and be a decreasing sequence of nonempty closed subsets of such that as . Then, the intersection contains exactly one point.

Proof. Let be chosen as arbitrary for all Since is decreasing, we have for all

Now, for any with , we have , Let be given. Then, there exists some such that , since as From this, it follows that whenever Therefore, is Cauchy in . By the completeness of , there exists some such that converges to Since and is closed for each , using Proposition 16, we have

Next, we prove the uniqueness of the point Let be another point, then . As , there exists such that for all , a contradiction. Hence, and this completes the proof of our theorem.

4. Some Fixed Point Theorems

Theorem 20. Let be a complete sequential -metric space and be a mapping so that (i) for all and for some (ii)there exists such that Then, has at least one fixed point in . Moreover, if and are two fixed points of in with then .

Proof. Let us define , for all . Clearly, for all .

Then, for all and for all , implies, for all ,

Denote for all . For , we have

Therefore, is a Cauchy sequence in . Due to the completeness of , is convergent and let .

Now, as . Therefore, as . Hence, by Proposition 10, it follows that , i.e., is a fixed point of .

Now, if and are two fixed points of in with , then we have which gives implies .

Theorem 21. Let be a complete sequential -metric space and such that (i) for all and for some ,(ii)there exists such that

Then, the Picard iterating sequence , for all , converges to some . If and for all , then is a fixed point of . Moreover, if is a fixed point of in such that and then .

Proof. For all and for all ,

This implies that for all . Then, proceeding in a similar way as in Theorem 20, it can be easily shown that is a Cauchy sequence in , and by the completeness of , there exists some such that

Now, for all , which implies that . Then,

If then , a contradiction. Hence , i.e., is a fixed point of .

Now, if is a fixed point of in with and , then we have , as , implying that .

Theorem 22. Let be a complete sequential -metric space and be a mapping satisfying the following conditions: (i) for all and for some (ii)there exists such that

Then, the Picard iterating sequence , for all , converges to some . If then is a fixed point of . Also, if is a fixed point of in such that then .

Proof. by similar argument as in Theorem 20, is a Cauchy sequence in , and by completeness of , it converges to an element say .

Now, for all , , which implies that and hence . Therefore, , and consequently, .

If is a fixed point of in with , then we have which implies that is .

Example 2. Consider and for all . Then, forms a sequential -metric on with the function for all . (i)Define by for all . Then, satisfies all the conditions of Theorem 20 for and clearly has a unique fixed point (ii)Define, . Then, satisfies all the conditions of Theorem 21 for and clearly has a unique fixed point

5. An Application to Nonlinear Integral Equations

In this section, we discuss about the existence of solutions for nonlinear integral equations as an application of Theorem 20.

Let be the set of all real valued continuous functions on and be defined by

Then, is a complete sequential -metric space with for all Now, let us consider the integral equation where , and are continuous functions.

Theorem 23. Assume that the following hypotheses are satisfied: (i)for all , we have where , ,(ii), where is such that Then, integral equation (12) has a unique solution .

Proof. Let us define by

Then, by conditions and , for all , we get where .

Therefore, Thus, for and for all Hence, the conditions of Theorem 20 are satisfied, and therefore, has a unique fixed point in , provided for some , i.e., nonlinear integral equation (12) has a unique solution in

Now, we give a numerical example in support of Theorem 23.

Example 3. Let us consider the complete sequential -metric space defined in Theorem 23 for , , , and the nonlinear integral equation given by

Then, for all , we have

Therefore, satisfies the contractive condition of Theorem 23 for , and also, we see that for defined by for all , for all Hence, all the conditions of Theorem 20 are satisfied and therefore (17) has a unique solution in .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

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

Acknowledgments

The first and second authors acknowledge the financial support awarded by the Council of Scientific and Industrial Research, New Delhi, India, through research fellowship for carrying out research work leading to the preparation of this manuscript.