Caristi’s Fixed Point Theorem in Cone Metric Space

Lael, Fatemeh; Saleem, Naeem; George, Reny

doi:https://doi.org/10.1155/2022/7523333

Journal of Function Spaces

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Abstract Introduction and Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Fixed Point Theory and Applications for Function Spaces 2021

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Research Article | Open Access

Volume 2022 | Article ID 7523333 | https://doi.org/10.1155/2022/7523333

Caristi’s Fixed Point Theorem in Cone Metric Space

Fatemeh Lael,¹Naeem Saleem,²and Reny George³

Academic Editor: Richard I. Avery

Received25 Jan 2022

Revised18 Feb 2022

Accepted21 Feb 2022

Published17 Mar 2022

Abstract

In this paper, we provide a short, comprehensive, and brief proof for Caristi-Kirk fixed point result for single and set-valued mappings in cone metric spaces. In addition, we partially addressed an open problem in which Caristi-Kirk fixed point result in cone metric spaces reduces to a classical result in metric spaces and provided a brief justification for a partial positive answer to this open problem using Caristi-Kirk fixed point theorem on uniform space. The proofs given to Caristi-Kirk’s result vary and use different techniques.

1. Introduction and Preliminaries

Caristi-Kirk’s fixed point theorem in [1] states that if is a complete metric space and is a lower semicontinuous mapping from into the nonnegative real numbers, then any mapping satisfying has a fixed point.

Several researchers generalized the Caristi-Kirk’s fixed point theorem in various directions, for details see [2–9].

Angelov [10] provided an extension of the Caristi-Kirk theorem to -separated uniform spaces, the uniform space is known as -separated if each convergent sequence in has a unique limit. As we know that every uniform space is generated by a family of pseudometrics , where is an indexing set. Also, a sequence is known as a Cauchy sequence, if for each , we have , and a sequence is convergent and converges to , if for each , we have . Thus, a uniform space is called complete if every Cauchy sequence is convergent in .

In this regard, Angelov [10] generalizes the Caristi-Kirk fixed point theorem on uniform space, which stated as:

Theorem 1 [10]. Let be a -separated complete uniform space which is generated by a family of pseudometrics , where is an indexing set. Let be a mapping and be a family of lower semicontinuous functionals. Suppose that the following inequality holds for each , where and is a surjective mapping. Then, has a fixed point in .

The following theorem is a Banach fixed point theorem on uniform space, which stated as:

Theorem 2 [11]. Let be a -separated complete uniform space which is generated by a family of pseudometrics , where is an indexing set. Suppose that is a mapping which is satisfying for each and . Then, has a unique fixed point in .

In 2007, Huang et al. [12] introduced the concept of cone metric space and proved some well-known fixed point results. The authors extended fixed point results proved for cone metric spaces which was just a simple reformulation of classical results presented in metric spaces. The obtained results are generalizations from classical results to cone metric spaces, for details see [13–16].

In this paper, we aim to reformulate Caristi-Kirk’s fixed point theorem for single and set-valued mappings in cone metric space and obtained a detailed answer to a question posed by Khamsi and Wojciechowski in ([17], Theorem 3-1) “whether the vectorial version and the classical version of Caristi-Kirk’s fixed point theorem are equivalent.” To address this particular answer, we defined a uniform space by considering cone metric space, and then, we addressed Theorem 1 in cone metric spaces and uniform spaces. Our proof is shorter, comprehensive, and easier than proof provided until now and our results generalize the existing results due to Khamsi and Wojciechowski in [17].

2. Cone Metric Version of Caristi-Kirk’s Theorem

Suppose that is a nonempty closed convex cone of a real Banach space such that where is the null vector, and .

In addition, induces a partial order on which is defined as if and only if and we write if and only if .

A convex subset is a base of if and and is the topological dual space of and is known as dual cone of . The dual cone of a cone in a Banach space has a weak-compact base . A set is called bounded from above (below) if there exists such that for all , (). A cone is called regular if every nondecreasing (decreasing) sequence which is bounded from above (below) is convergent in norm. The cone is called normal if there is a number , such that we have for all ,

The least positive number satisfying this inequality is called a normal constant of .

The following lemma will be used in proving our main results.

Lemma 3 [18, 19]. The weak-compact base satisfies: (1)any element if and only if , for all (2)any element if and only if , for all

Definition 4 [12]. Let be a nonempty set. Consider a mapping is satisfied as follows: (1) for all and if and only if (2) for all (3) for all Then, is called a cone metric on and is named a cone metric space.

Definition 5 [12]. Let be a sequence in a cone metric space and some . A sequence is as follows: (1)a -Cauchy sequence if for every , there exists , such that for all (2)a -convergent and -converges to if for every there exists , such that for all which is denoted as

Definition 6 [12]. A cone metric space is complete if every Cauchy sequence is -convergent in .

Definition 7 [20]. Let be a cone metric space. A mapping is considered as a cone lower semicontinuous mapping at if for any , there exists a natural number such that for all , where is a sequence in and . If , then , is a metric space and , so is a lower semicontinuous mapping at , if for any , there exists such that for any , we have where is a convergent sequence and converges to in a metric space .

The following theorem is a cone metric version of the Caristi-Kirk’s theorem with the some extra normal cone condition.

Theorem 8 [21]. Let be a cone metric space with normal and regular cone of a Banach space such that implies for some . Also, satisfies , for every . Suppose that the mapping satisfying the following condition: for all . Then, has a fixed point.

Since then, some studies have focused on extending and improving the cone metric version of the Caristi-Kirk’s fixed point theorem in many ways. In [16, 17], authors proved the Caristi-Kirk’s fixed point theorem but the authors supposed that the cone is normal which is a strict condition and researchers did not accept it as a good condition. Further, the results are proved for regular and normal cone in [21].

In this paper, none of these conditions was considered for the cone. Now, we will omit the stronger conditions, the normality, and regularity of the cone in our main results and we will prove this result under the weaker condition as compared to the result proved in the literature under strict conditions.

3. Main Results

The following lemmas are handy tools that are used in the sequel.

Lemma 9. Let be a cone metric space and . Also suppose that is a uniform space which is generated by a family of pseudometrics . Then, is -separated.

Proof. On contrary suppose that the sequence has two different limits, i.e., and in the uniform space . Then, according to definition, for each pseudometric , we have and . In addition, by the third property of the cone metric, we have , (by lemma 3) we have , for each . When , we have that for each , . Thus, . Thus, , i.e., .

Lemma 10. Let be a cone metric space and . Also suppose that is a uniform space which is generated by a family of pseudometrics . Then, is a complete uniform space if and only if is a -complete cone metric space.

Proof. First, we suppose that is a complete uniform space. Let be a -Cauchy sequence in the cone metric space . Then, for each , where , there exists such that for each , we have . Using Lemma 3 part , for each , we have and For any and , there is a such that . For , inequality (8) implies that . Using the definition, is a Cauchy sequence in a complete uniform space. Therefore, is convergent and converges to which belongs to the uniform space . Then, for each and , there exists , such that for each , Now, we demonstrate that the sequence is -convergent. On the contrary, suppose that there is some , such that for each there is such that . From lemma 3 part (8), for each , , and which is a contradiction, if .
On contrary suppose that is a -complete space. Let be a Cauchy sequence in a uniform space , for each and , there is , such that for each , Now, we show that is a Cauchy sequence. On the contrary, suppose that is not a Cauchy sequence. Then, there is such that for each , there are such that . Thus, by using lemma 3 for each , we have and , which is a contradiction, as . Therefore, is a Cauchy sequence, and accordingly, it is a -convergent and converges to some (from definition 5) for each where , there is such that for each , we have , (using lemma 3) for each , we have and For each and , there is such that thus (11) implies that for , we have Thus, is convergent to in the uniform space .

Lemma 11. Let be a cone metric space and . Then, is a cone lower semicontinuous mapping.

Proof. Let and be a sequence in such that . There exists such that for all . Then, , for all . Thus, is a cone lower semicontinuous mapping.

Lemma 12. Let be a cone metric space, be a cone lower semicontinuous mapping and . Then, is a lower semicontinuous function.

Proof. Let be fixed. For any there exists such that , ( is linear) and is cone lower semicontinuous and . Thus, there exists such that for all , and so for all . This relation indicates the lower semicontinuity of .

As is shown in [14], all fixed point results in cone metric spaces obtained recently, in which the assumption that the underlying cone is normal and with the nonempty interior is present, can be reduced to the corresponding results in metric spaces. On the other hand, when we deal with nonnormal cones, this is not possible.

Theorem 13 is a cone metric version of Caristi-Kirk’s theorem without extra conditions normality and regularity which are always put in cone metric theorems, so our results are original. To prove this theorem, we show that the cone metric space is uniform too; then, it will be proved by applying Theorem 1. We know that a -separated uniform space is metrizable if its uniformity can be defined by a countable family of pseudometrics. Indeed, such uniformity can be defined by a single pseudometric, which is necessarily a metric. This implies that a cone metric version of Caristi-Kirk’s theorem may be derived from the classical one if which is defined in Section 2 is countable. This is a partial answer to the open question mentioned before.

Theorem 13. Let be a -complete cone metric space and be a cone lower semicontinuous mapping. Suppose that the self-mapping satisfying the following condition: for all . Then, has a fixed point.

Proof. We provide the conditions of Theorem 1 to conclude that has a fixed point. It is easily shown that is a family of pseudometrics on and will be a uniform space with the topology generated by these pseudometrics. By Lemma 9, the uniform space is -separated. Using lemma 10, is a complete cone metric space, since is a complete uniform space. By lemma 12, is a lower semicontinuous mapping. Further, lemma 3 and assumption imply that for each , . By considering as an identity mapping, all assumptions are satisfied.

In Theorem 13, the regularity of the cone, which is an essential condition in [21] is omitted. So, our theorem is a real generalization of Theorem 8.

For example, we cannot even conclude from Theorem 8 that the identity mapping has a fixed point but it is possible by Theorem 8. The following example is presented in the support of the theorem 13.

Example 14. Consider the Banach space with its cone . It is not difficult to see that is complete and is normal with nonempty interior. Let be a subset of consisting of all which are nondecreasing and converging to with , for all . Define as, for every . It is not hard to check that is a -complete space. Now, define the mapping by and is the inclusion mapping. It is clear that is cone lower semicontinuous and satisfies , since . But is not regular because the sequence that , for each , is nondecreasing and bounded from above but it is not convergent.

Thus, one of the conditions of Theorem 8 is not satisfied, although , for all . But Theorem 13 implies that has a fixed point.

Remark 15. In example 14, one of conditions of Theorem 8 is not satisfied, as is not regular although , for all . In the example 14, all the conditions of the Theorem 13 are satisfied, and hence, the underlying mapping has a fixed point. This shows that Theorem 13 is a real and proper generalization of Theorem 8.

The following example shows that the cone lower semicontinuity of is essential in Theorem 13 and may not be dropped.

Example 16. Let be a family of the sequences and is defined as same as the cone defined in Example 14 and the cone metric is for every . Define the mapping and in the following way: and . Obviously but is not cone lower semicontinuous map because does imply , and therefore, one of conditions of Theorem 13 is not satisfied. It is clear that has no fixed point because .

In the next theorem, we give a short proof for a set-valued version of Caristi-Kirk’s fixed point theorem in cone metric space. An element is considered as a fixed point of set-valued mapping if .

Theorem 17. Let be a -complete cone metric space, be a cone lower semicontinuous mapping and there exists for a set-valued mapping such that for each . Then, has a fixed point.

Proof. By assumption, for each , the set is nonempty. Using the axiom of choice, there is a single-valued mapping such that , for each . Theorem 13 is applied for to find a fixed point (say) of . Since , we have .

Additionally, Khamsi (2010) proved the Theorem 18, which is the cone metric version of the Banach fixed point theorem. In Theorem 19, we improve it by removing the stronger condition of “normal cone.”

Theorem 18 [16]. Let be a -complete cone metric space over the Banach space with the cone which is normal. Suppose that for some , the mapping satisfies for all . Then, has a unique fixed point.

Theorem 19. Let be a -complete cone metric space and for some , the mapping satisfying for all . Then, has a unique fixed point.

Proof. We provide the conditions of Theorem 2 to conclude that has a fixed point. We know that is a family of pseudometrics on and will be a uniform space with the topology generated by these pseudometrics. By lemma 9, this uniform space is -separated. Using lemma 10, is a complete uniform space since is a complete cone metric space. In addition, lemma 3 and assumption imply that for each , . Thus, Theorem 2 implies that has a fixed point.

Remark 20. It is worth noting that Theorem 19 is a generalization of the Theorem 18. We used cone metric space with a nonnormal cone in our main results. Therefore, our theorems are the strict generalizations of the results which are proved in [16, 17, 21].

Theorem 21. Let be a -complete cone metric space, and be a set-valued mapping that for each and, there exists such that

Then, has a fixed point.

Proof. It is direct consequence of the Theorem 19.

In this article, we provided a brief proof for the Caristi-Kirk’s fixed point result for single and set-valued mappings in cone metric spaces. Also, we partially addressed an open problem in which Caristi-Kirk’s fixed point resulted in cone metric spaces. We improved the already existing results on Caristi-Kirk’s fixed point in cone metric spaces by improving and removing the extra and strict conditions on the underlying spaces and mappings as well. Further, we provided a brief justification as a partial positive answer to this open problem using Caristi-Kirk’s fixed point theorem on uniform space. We further provided a short proof in the cone metric version of the Banach fixed point theorem by using a short and comprehensive approach.

Data Availability

No data were used.

Conflicts of Interest

The authors declare no conflict of interest.

Acknowledgments

This research is supported by the Deanship of Scientific Research, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia.

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Copyright © 2022 Fatemeh Lael et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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