Picard Sequence and Fixed Point Results on -Metric Spaces

Abstract

We obtain some fixed point results for single-valued and multivalued mappings in the setting of a -metric space. These results are generalizations of the analogous ones recently proved by Khojasteh, Abbas, and Costache.

1. Introduction

It is well known that the contraction mapping principle, formulated and proved in the Ph.D. dissertation of Banach in 1920, which was published in 1922 [1], is one of the most important theorems in classical functional analysis. It is widely considered as the source of metric fixed point theory. Also its significance lies in its vast applicability in a number of branches of mathematics. The study of fixed points of mappings satisfying a certain metrical contractive condition attracted many researchers; see, for example, [2–10].

The concept of -metric space appeared in some works, such as Bakhtin [11] and Czerwik [12, 13]. In dealing with the fixed point theory for single-valued and multivalued mappings in -metric spaces, seven years later, Khamsi [14] and Khamsi and Hussain [15] reintroduced such spaces under the name of metric-type spaces. For some results of fixed and common fixed point in the setting of -metric spaces, see [3, 14–18].

In the first part of this paper we consider a class of Picard sequences with a property of regularity and prove that these sequences are Cauchy. Then we use this result for establishing existence of fixed points for single-valued mappings. The last part contains a fixed point result for multivalued mappings on complete -metric spaces. Clearly, these results are a generalization of the results recently obtained from Khojasteh et al. in [5].

2. Preliminaries

The aim of this section is to collect some relevant definitions and results for our further use. Let be the set of positive integers, the set of real numbers, and the set of nonnegative real numbers.

Definition 1. Let be a nonempty set and let be a given real number. A function is said to be a -metric if and only if for all the following conditions are satisfied: (1) if and only if ;(2);(3).A triplet is called a -metric space.

We observe that a -metric space is a metric space if . So the notions of convergent sequence, Cauchy sequence, and complete space are defined as in metric spaces.

Next, we give some examples of -metric spaces.

Example 2. Let and defined by , for each . Clearly, is a -metric space.

Example 3. Let and let . The function defined by is a -metric with constant and so is a -metric space.
We note that if are two nonnegative real numbers, then This implies that
The following results are useful for some of the proofs in the paper.

Lemma 4. Let be a -metric space and let be a sequence in . If and , then .

Lemma 5. Let be a -metric space and let . Then

From Lemma 5, we deduce the following lemma.

Lemma 6. Let be a sequence in a -metric space such that for some , , and each . Then is a Cauchy sequence in .

Let be a -metric space and let be the collection of all nonempty closed bounded subsets of . For , define where with Note that is called the Hausdorff -metric induced by the -metric .

We recall the following properties from Czerwik et al. [12, 13, 19].

Lemma 7. Let be a -metric space. For any and any , we have the following:(i), for any ;(ii);(iii), for any ;(iv);(v);(vi);(vii).

Lemma 8. Let be a -metric space and with . Then for each and for all there exists such that .

Lemma 9. Let be a -metric space. For and , we have where denotes the closure of the set .

3. Picard Sequence in -Metric Spaces

Let be a -metric space, let , and let be a given mapping. The sequence with for all is called a Picard sequence of initial point . In this section we consider a class of Picard sequences which are Cauchy.

Proposition 10. Let be a -metric space and let be a given mapping. Assume that a Picard sequence of initial point satisfies one of the following conditions: or where is a positive real number. Then is a Cauchy sequence.

Proof. Let be an arbitrary point, and let be a Picard sequence of initial point . If for some , then is a fixed point of and so is a Cauchy sequence.
Assume that (10) holds for the sequence . If for all , from (10), we deduce that the sequence is decreasing. Thus there exists a nonnegative real number such that . Then we claim that . If , on taking limit as on both sides of (10), we obtain which is a contradiction. It follows that .
Now, we prove that is a Cauchy sequence. Let . Since , then there exists such that This implies that and by Lemma 6 we deduce again that is a Cauchy sequence.
The proof that is a Cauchy sequence if (11) holds is the same.

As a consequence of the previous result, we establish the following result of existence of fixed points.

Theorem 11. Let be a complete -metric space and let be a mapping such that for all . Then(i) has at least one fixed point ;(ii)every Picard sequence of initial point converges to a fixed point of ;(iii)if are two distinct fixed points of , then .

Proof. Let be an arbitrary point and let be a Picard sequence of initial point . If for some , then is a fixed point of . If for all , using the contractive condition (15) with and , we get that is, condition (10) holds for the sequence . Then, by Proposition 10, is a Cauchy sequence. Since is a complete -metric space, the sequence converges to some . Now, we prove that is a fixed point of . Using (15) with and , we obtain
Moreover, from as , we deduce that
On taking , as , on both sides of (17), by (19) we get This implies that ; that is, and hence is a fixed point of . Thus (i) and (ii) hold.
If , with , is another fixed point of , then using (15) with and , we get and hence ; that is, (iii) holds.

Remark 12. From Theorem 11, we obtain Theorem 1 of [5] if ; that is, if is a metric space.

In the following result we consider a weak contractive condition.

Theorem 13. Let be a complete -metric space and let be a mapping such that for all , where is a nonnegative real number. Then (i) has at least one fixed point ;(ii)every Picard sequence of initial point converges to a fixed point of ;(iii)if are two distinct fixed points of , then .

Proof. Let be an arbitrary point and let be a Picard sequence of initial point . If for some , then is a fixed point of . If for all , using the contractive condition (22) with and , we get that is, condition (10) holds for the sequence . Then, by Proposition 10, is a Cauchy sequence. Since is a complete -metric space, the sequence converges to some . Now, we prove that is a fixed point for . Using (22) with and , we obtain
On taking as on both sides of (24), by (19) we get This implies that ; that is, and hence is a fixed point of . Thus (i) and (ii) hold.
If , with , is another fixed point of , then using (22) with and , we get and hence ; that is, (iii) holds.

Next examples illustrate Theorem 11.

Example 14. Let and defined by Clearly, is a complete -metric space. Consider the mapping defined by Now, we have Then Thus all the hypotheses of Theorem 11 are satisfied. In this case has two fixed points and . Clearly, is not a contraction.

Example 15. Let and defined by Define by . Clearly, is a complete -metric space. For all with , we have Then all the hypotheses of Theorem 11 are satisfied. In this case has infinite fixed points. Clearly, is not a contraction.

4. Fixed Points for Multivalued Mappings in -Metric Spaces

In this section, we give a result of existence of fixed point for a class of multivalued mappings.

Theorem 16. Let be a complete -metric space and let be a multivalued mapping such that for all , where is a positive real number. Then has a fixed point .

Proof. Let be arbitrary and such that . Clearly, if or , we deduce that is a fixed point of and so we can conclude the proof. Now, we assume that and and hence . This implies that . Now, we set By Lemma 8 one can choose such that Then This implies Now, we suppose to have chosen such that , , and We set Again, by Lemma 8, one can choose such that Then From this, we get If , then is a fixed point of and the proof is finished. If , itering this procedure we construct a sequence such that , , and (42) holds for all . Then, by Proposition 10, is a Cauchy sequence. Since is a complete -metric space, the sequence converges to some . Now, we prove that is a fixed point of . Using (33) with and , we obtain
On taking limit as on both sides, we get . As is closed, by Lemma 9, we obtain that ; that is, is a fixed point of .

Proceeding as in the proof of Theorem 16, we establish the following theorem.

Theorem 17. Let be a complete -metric space and let be a multivalued mapping such that for all , where is a positive real number and a nonnegative real number. Then has a fixed point .

Example 18. Let and defined by Define by for all . Clearly, is a complete -metric space. For all with , we have Then all the hypotheses of Theorem 16 are satisfied with . In this case has infinite fixed points. Clearly, is not a multivalued contraction.

Conflict of Interests

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

Acknowledgments

The authors are grateful to the referees for their helpful comments.