Fixed Points for Multivalued Mappings and the Metric Completeness
© S. Dhompongsa and H. Yingtaweesittikul. 2009
Received: 24 December 2008
Accepted: 6 May 2009
Published: 14 May 2009
We consider the equivalence of the existence of fixed points of single-valued mappings and multivalued mappings for some classes of mappings by proving some equivalence theorems for the completeness of metric spaces.
The Banach contraction principle  states that for a complete metric space , every contraction on , that is, for some , for all , has a (unique) fixed point.
Connell  gave an example of a noncomplete metric space on which every contraction on has a fixed point. Thus contractions cannot characterize the metric completeness of
Theorem 1.1 (see [3, Kannan]).
Subrahmanyam  proved that Kannan mappings can be used to characterize the completeness of the metric. That is, a metric space is complete if and only if every Kannan mapping on has a fixed point.
Theorem 1.2 (see ).
In 2008, Kikkawa and Suzuki  partially extended Theorem 1.2 to multivalued mappings.
Theorem 1.3 (see ).
Moţ and Petruşel  proved the following theorem which is a generalization of Kikkawa and Suzuki Theorem.
Theorem 1.4 (see ).
In this paper, we will characterize the completeness of a metric space by the existence of fixed points for both single-valued and multivalued mappings. We first aim to extend, in Section 3, the Suzuki's result (Theorem 1.2) to more general classes of mappings. We then consider multivalued mappings in Section 4. We also show in this section that the converse of Theorem 1.4 is true.
The next theorem plays important roles in this paper.
Theorem 2.1 (see cf. ).
3. Completeness and Single-valued Mappings
Theorem 3.1 (see ).
Theorem 3.2 (see ).
Thus (3.8) holds and now we find a contradiction in each of the following cases.
In fact the following theorem shows that the converse of Theorem 3.3 is valid.
(iii) (i). We know that every Kannan mapping belongs to the class of mappings in (iii). Thus is complete by Subrahmanyam .
4. Completeness and Multivalued Mappings
Inspired by Theorem 1.2 and Theorem 1.3, we prove the following theorem for a larger class of mappings under some certain assumptions.
Observe that Theorem 4.1 is not covered by Theorem 3.4 when considering as single-valued mappings.
Proof of Theorem 4.1.
Therefore (4.6) holds.
The converse of Theorem 1.4 is also valid by following the same proof of Theorem 1.2. Assuming that is not complete, we find a fixed point free mapping satisfying the condition in Theorem 1.4. Following the same proof of Theorem 1.2 by replacing by where , we obtain for all and is fixed point free. We now verify the condition in Theorem 1.4 for
Therefore (4.13) holds, and the proof of the converse of Theorem 1.4 is complete.
has a fixed point.
5. Caristi Set-Valued Mappings
In 2008, Ćirić  proved the following fixed point theorems.
Theorem 5.1 ().
Theorem 5.2 ().
We give a simple proof of each of these theorems.
Proof of Theorem 5.1.
Proof of Theorem 5.2.
The authors would like to thank the Thailand Research Fund (grant BRG4780016) for its support.
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