# On Fixed Point Theorems for Multivalued Contractions

- Zeqing Liu
^{1}, - Wei Sun
^{1}, - ShinMin Kang
^{2}Email author and - JeongSheok Ume
^{3}

**2010**:870980

https://doi.org/10.1155/2010/870980

© Zeqing Liu et al. 2010

**Received: **13 August 2010

**Accepted: **15 November 2010

**Published: **1 December 2010

## Abstract

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

Such a mapping
is called *a generalized Hausdorff metric in*
*induced by*
.

Throughout this paper, we assume that , , and denote the sets of all real numbers, nonnegative real numbers, and positive integers, respectively.

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 [7] 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 [7]).

In 1989, Mizoguchi and Takahashi [6] generalized the Nadler fixed point theorem and got a fixed point theorem for the multivalued contraction as follows.

Theorem 1.2 (see [6]).

In 2006, Feng and Liu [3] obtained a new extension of the Nadler fixed point theorem and proved the following fixed point theorem.

Theorem 1.3 (see [3]).

then has a fixed point in provided a function , is lower semicontinuous.

In 2007, Klim and Wardowski [5] improved the result of Feng and Liu and proved the following results.

Theorem 1.4 (see [5]).

Let be a complete metric space, and let be a multivalued mapping from into . Assume that

(a)the mapping , defined by , , is lower semi-continuous;

Theorem 1.5 (see [5]).

Let be a complete metric space, and let be a mapping from into . Assume that

(a)the mapping , defined by , , is lower semi-continuous;

(b)there exists a function satisfying

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 [2]).

Let be a complete metric space, and let be a mapping from into . Assume that

(a)the mapping , defined by , , is lower semi-continuous;

(b)there exists a function , , satisfying

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 [2], Feng and Liu [3], and Klim and Wardowski [5] and are different from those results of Mizoguchi and Takahashi [6] and Nadler Jr [7].

Next we recall and introduce the following result in [4] and some notions and terminologies.

Lemma 1.7 (see [4]).

where
and
, for all
. The function
is said to be
*-orbitally lower semicontinuous at*
if
is an arbitrary orbit of
with
impling that
.

## 2. Main Results

In this section, we establish three fixed point theorems for three new multivalued contractions in complete metric spaces.

Theorem 2.1.

Then,

(a1) for each there exists an orbit of and such that ;

(a2) is a fixed point of in if and only if the function is -orbitally lower semi-continuous at .

Proof.

which is a contradiction. Thus, ; that is, (2.8) holds.

which implies that is a Cauchy sequence because . It follows from completeness of that there is some such that .

which means that because is closed.

that is, is -orbitally lower semi-continuous in . This completes the proof.

Theorem 2.2.

Then,

(a1) for each there exists an orbit of and such that ;

(a2) is a fixed point of in if and only if the function is -orbitally lower semi-continuous at .

Proof.

which together with (2.20) and (2.21) ensures that (2.1) and (2.2) hold with and . Thus, Theorem 2.2 follows from Theorem 2.1. This completes the proof.

Theorem 2.3.

and one of and is nondecreasing. Then,

(a1) for each there exists an orbit of and such that ;

(a2) is a fixed point of in if and only if the function is -orbitally lower semi-continuous at .

Proof.

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.

Remark 3.1.

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.

Example 3.2.

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 .

In fact, for any , we consider two possible cases as follows.

Case 1.

Case 2.

That is, the conditions of Theorem 1.3 do not hold.

for any with . That is, the conditions of Theorems 1.1 and 1.2 do not hold.

Remark 3.3.

If and for all , then Theorem 2.1 changes into a result, which is an extension of Theorem 1.6. The following example demonstrates that Theorem 2.1 generalizes substantially Theorem 1.6.

Example 3.4.

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.

Case 1.

Case 2.

Suppose that . As in the proof of Case 1 stated first, we conclude similarly the conclusion of Case 1 stated after then. Therefore, the assumptions of Theorem 1.6 do not hold.

Remark 3.5.

The following example is an application of Theorem 2.2.

Example 3.6.

Therefore, all assumptions of Theorem 2.2 are satisfied. It follows from Theorem 2.2 that has a fixed point in .

Remark 3.7.

If and for all , then Theorem 2.3 comes down to a result, which extends Theorem 1.5. The following example shows that Theorem 2.3 is a genuine generalization of Theorem 1.5.

Example 3.8.

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 .

Remark 3.9.

In case and for all , then Theorem 2.3 reduces to a result, which extends Theorem 1.4. The following example reveals that Theorem 2.3 generalizes properly Theorem 1.4.

Example 3.10.

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.

Case 1.

Case 2.

Therefore, the assumptions of Theorem 1.4 are not satisfied.

## Declarations

### Acknowledgment

This work was supported by the Korea Research Foundation (KRF) Grant funded by the Korea government (MEST) (2009-0073655).

## Authors’ Affiliations

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