Journal of Function Spaces
Volume 2015 (2015), Article ID 709391, 6 pages
http://dx.doi.org/10.1155/2015/709391

Research Article

Caristi’s Fixed Point Theorem and Subrahmanyam’s Fixed Point Theorem in -Generalized Metric Spaces

Badriah Alamri,¹ Tomonari Suzuki,1,2 and Liaqat Ali Khan¹

¹Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
²Department of Basic Sciences, Faculty of Engineering, Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan

Received 20 January 2015; Accepted 26 February 2015

Academic Editor: Hemant Kumar Nashine

Copyright © 2015 Badriah Alamri 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.

Abstract.

We discuss the completeness of -generalized metric spaces in the sense of Branciari. We also prove generalizations of Subrahmanyam’s and Caristi’s fixed point theorem.

1. Introduction

In 2000, Branciari introduced the following very interesting concept.

Definition 1 (Branciari [1]). Let be a set, let be a function from into , and let . Then, is said to be a -generalized metric space if the following hold.

) iff for any .() for any .() for any such that are all different.

Remark 2. Every metric space is a -generalized metric space.

Branciari in [1] also proved a generalization of the Banach contraction principle; however, the proof is not correct; see [2–4]. When proving theorems on -generalized metric spaces , we may have to be careful because does not necessarily have the compatible topology; see [4]. See also [5, 6].

In this paper, we first discuss the completeness of -generalized metric spaces. We next prove generalizations of Subrahmanyam’s and Caristi’s fixed point theorem.

2. Preliminaries

Throughout this paper we denote by the set of all positive integers and by the set of all real numbers.

In this section, we give some preliminaries.

As we mentioned in Section 1, -generalized metric spaces do not necessarily have the compatible topology. So we have to define something connected with convergence.

Definition 3. Let be a -generalized metric space.

(i)A sequence in is said to converge to iff .(ii)A sequence in is said to converge only to iff holds and does not hold for .(iii)A mapping on is said to be sequentially continuous iff converges to whenever converges to .(iv)A function from into is said to be sequentially lower semicontinuous iff whenever converges to .

We introduce new concepts.

Definition 4. Let be a -generalized metric space and let .

(i)A sequence in is said to be -Cauchy iff

holds.(ii) is -complete iff every -Cauchy sequence converges.

Remark 5. We sometimes write “Cauchy” instead of “-Cauchy” and “complete” instead of “-complete.”

The following is obvious.

Proposition 6. Let be a -generalized metric space and let such that is divisible by . Then, the following hold.

(i)Every -Cauchy sequence is -Cauchy.(ii)If is -complete, then is -complete.

The following are partially converse to Proposition 6(i).

Proposition 7. Let be a -generalized metric space where is odd. Let be a -Cauchy sequence such that are all different. Then, is Cauchy.

Proof. In the case where , the conclusion clearly holds. So we assume . Fix . Then, there exists such that

for any and with . Fix and with . We first show

for . It is obvious that (3) holds when . We assume that (3) holds for some with . Then, we have by ()

Hence, (3) holds when . Therefore, (3) holds for every , which implies

for any , , and with . Using this, we have

for any , , and with . So is Cauchy.

Proposition 8. Let be a -generalized metric space where is even. Let be a -Cauchy sequence such that are all different. Then, is -Cauchy.

Proof. Fix . Then, there exists such that

for any and with . Fix and with . Then, as in the proof of Proposition 7, by induction, we can show

for . Therefore, we obtain

for any . So is -Cauchy.

Lemma 9. Let be a -generalized metric space and let be a sequence in such that are all different and . Then, is -Cauchy.

Remark 10. Example 1 in [7] tells that there exists some sequence in a -generalized metric space such that and is not Cauchy.

Proof. Fix . Then, there exists such that . Fix with . We will show

by induction. It is obvious that (10) holds for . We assume that (10) holds for some . Then, by (), we have

Therefore, (10) holds for . By induction, (10) holds for any . Hence,

holds. Therefore, is -Cauchy.

Lemma 11. Let be a -generalized metric space and let be a sequence in such that are all different, , and converges to some . Then, converges only to .

Proof. Arguing by contradiction, we assume that converges to which differs from . Since are all different, and for sufficiently large . By (), we have

where we define . By (), we obtain . This is a contradiction.

Lemma 12. Let be a -generalized metric space satisfying either of the following:

(i) is odd and is complete;(ii) is even and is -complete.

Let be a sequence in such that are all different and . Then, there exists such that converges only to .

Proof. By Lemma 9, is -Cauchy. By Propositions 7 and 8, the following hold.

(i)If is odd, then is Cauchy.(ii)If is even, then is -Cauchy.

From the assumption on the completeness of , converges to some point . By Lemma 11, converges only to .

3. Fixed Point Theorems

The following is a generalization of Subrahmanyam’s fixed point theorem [8]; see [9–11].

Theorem 13. Let be as in Lemma 12. Let be a sequentially continuous mapping on . Assume that there exists satisfying

for all . Then, for any , converges only to a fixed point of .

Proof. Define a sequence in by for . We prove the conclusion, dividing the following three cases.

(i)There exists such that .(ii) for all and there exist such that and .(iii) are all different.

In the first case, is a fixed point of . By (), converges only to . In the second case, since , we have

which implies . This is a contradiction. Thus, the second case cannot be possible. In the third case, we have

So by Lemma 12, there exists such that converges only to . We note that also converges only to . Since is sequentially continuous, we obtain .

A function from into is proper iff is nonempty.

The following is a generalization of Caristi’s fixed point theorem [12, 13].

Theorem 14. Let be as in Lemma 12. Let be a mapping on . Let be a proper, sequentially lower semicontinuous function from into bounded from below. Assume that

for all . Then, has a fixed point.

Remark 15. This theorem is connected with Theorem 2 in [14]. See Section 4.

Proof. In the case where , Theorem 14 becomes the original Caristi fixed point theorem. So we assume . Arguing by contradiction, we assume that does not have a fixed point. Then, we note for every with . By induction, we define a sequence in satisfying the following:

Fix with . We assume that is defined for some . Then, we put

Since , is nonempty. Since , we can define satisfying (18). By induction, we have defined . We note that are all different because for any . Since is bounded from below, converges. We have

So by Lemma 12, there exists such that converges only to . Since is sequentially lower semicontinuous, we have

We note that for any . Since , for any . Fix with . Then, for any , we have

As tends to infinity, we obtain and hence . Then, we have

This is a contradiction.

4. Counterexample

Kirk and Shahzad in [7] gave a counterexample to Theorem 2 in [14]. In this section, we give another example.

Lemma 16 (see [4]). Let be a bounded metric space and let be a real number satisfying

Let and be two subsets of with . Define a function from into by

Then, is a -generalized metric space.

Remark 17. We assume in Lemma 4 in [4]. However, we do not use this assumption in the proof.

Example 18. Let and define a metric on as usual. Define two subsets and of by

Define a function from into as in Lemma 16 with . Define a mapping on by

And define a function from into by

Then, is a complete, -generalized metric space, is sequentially continuous with respect to , and (17) holds. However, does not have a fixed point.

Proof. By Lemma 16, is a -generalized metric space. We will show that is complete. Let be a Cauchy sequence in . We consider the following three cases.

(i), where is the number of the elements of .(ii) for some .(iii) and for any .

In the first case, since , holds for sufficiently large . Thus, converges to . In the second case, since

holds for sufficiently large . Thus, converges to . We consider the third case. Since is Cauchy, there exists such that

for any with . We note that because of the definition of . Without loss of generality, we may assume that . There exists such that and for with . Then, clearly holds. Also, there exists such that and . Then, and clearly hold. We obtain . This is a contradiction. Therefore, the third case cannot be possible. We have shown that is complete. We next show that is sequentially continuous. Let a sequence in converge to some . Then, from the definition of , there exists such that for with . This fact implies that is sequentially continuous. For any , holds. So (17) is satisfied. However, it is clear that does not have a fixed point.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant no. 35–130–35–HiCi. The authors, therefore, acknowledge technical and financial support of KAU. The second author is supported in part by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science.

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