On Fixed Point Theorems for Multivalued Contractions
© Zeqing Liu et al. 2010
Received: 13 August 2010
Accepted: 15 November 2010
Published: 1 December 2010
Three concepts of multivalued contractions in complete metric spaces are introduced, and the conditions guaranteeing the existence of fixed points for the multivalued contractions are established. The results obtained in this paper extend genuinely a few fixed point theorems due to Ćirić (2009) Feng and Liu (2006) and Klim and Wardowski (2007). Five examples are given to explain our results.
1. Introduction and Preliminaries
The existence of fixed points for various multivalued contractive mappings had been studied by many authors under different conditions. For details, we refer the reader to [1–7] and the references therein. In 1969, Nadler Jr  extended the famous Banach Contraction Principle from single-valued mapping to multivalued mapping and proved the following fixed point theorem for the multivalued contraction.
Theorem 1.1 (see ).
In 1989, Mizoguchi and Takahashi  generalized the Nadler fixed point theorem and got a fixed point theorem for the multivalued contraction as follows.
Theorem 1.2 (see ).
In 2006, Feng and Liu  obtained a new extension of the Nadler fixed point theorem and proved the following fixed point theorem.
Theorem 1.3 (see ).
In 2007, Klim and Wardowski  improved the result of Feng and Liu and proved the following results.
Theorem 1.4 (see ).
Theorem 1.5 (see ).
In 2008 and 2009, Ćirić [1, 2] introduced new multivalued nonlinear contractions and established a few nice fixed point theorems for the multivalued nonlinear contractions, one of which is as follows.
Theorem 1.6 (see ).
The aim of this paper is both to introduce three new multivalued contractions in complete metric spaces and to prove the existence of fixed points for the multivalued contractions under weaker conditions than the ones in [2, 3, 5]. Five nontrivial examples are given to show that the results presented in this paper generalize substantially and unify the corresponding fixed point theorems of Ćirić , Feng and Liu , and Klim and Wardowski  and are different from those results of Mizoguchi and Takahashi  and Nadler Jr .
Next we recall and introduce the following result in  and some notions and terminologies.
Lemma 1.7 (see ).
2. Main Results
In this section, we establish three fixed point theorems for three new multivalued contractions in complete metric spaces.
The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof.
3. Remarks and Examples
In this section, we construct five examples to illustrate the superiority and applications of the results presented in this paper.
In case and for all , where and are constants in with , then Theorem 2.1 reduces to a result, which is a generalization of Theorem 1.3. The following example reveals that Theorem 2.1 extends both essentially Theorem 1.3 and is different from Theorems 1.1 and 1.2.
That is, the conditions of Theorem 2.1 are fulfilled. It follows from Theorem 2.1 that has a fixed point in . However, we cannot invoke each of Theorems 1.1–1.3 to show that the mapping has a fixed point in .
That is, the conditions of Theorem 1.3 do not hold.
That is, the conditions of Theorem 2.1 are fulfilled. It follows from Theorem 2.1 that has a fixed point in . However, Theorem 1.6 is inapplicable in ensuring the existence of fixed points for the mapping in because there does not exist and satisfying the assumptions of Theorem 1.6. In fact, for any and , we consider the following two possible cases.
The following example is an application of Theorem 2.2.
It is easy to verify that the assumptions of Theorem 2.3 are satisfied. Consequently, Theorem 2.3 guarantees that has a fixed point in . But does not satisfy the conditions of Theorem 1.5 because is not compact for all .
That is, the conditions of Theorem 2.3 are fulfilled. It follows from Theorem 2.3 that has a fixed point in . However, we cannot use Theorem 1.4 to show that the mapping has a fixed point in since there does not exist and satisfying the assumptions in Theorem 1.4. In fact, for any and , we consider two possible cases as follows.
Therefore, the assumptions of Theorem 1.4 are not satisfied.
This work was supported by the Korea Research Foundation (KRF) Grant funded by the Korea government (MEST) (2009-0073655).
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