- Research Article
- Open Access

# Fixed Point Results for Multivalued Maps in Cone Metric Spaces

- Abdul Latif
^{1}Email author and - FawziaY Shaddad
^{1}

**2010**:941371

https://doi.org/10.1155/2010/941371

© The Author(s). 2010

**Received:**4 February 2010**Accepted:**16 April 2010**Published:**23 May 2010

## Abstract

We prove some fixed point theorems for multivalued maps in cone metric spaces. We improve and extend a number of known fixed point results including the corresponding recent fixed point results of Feng and Liu (1996) and Chifu and Petrusel (1997). The remarks and example provide improvement in the mentioned results.

## Keywords

- Banach Space
- Fixed Point Theorem
- Lower Semicontinuity
- Normal Constant
- Normal Cone

## 1. Introduction

The well-known Banach contraction principle and its several generalizations in the setting of metric spaces play a central role for solving many problems of nonlinear analysis. For example, see [1–5]. Using the concept of the Hausdorff metric, Nadler [6] obtained a multivalued version of the Banach contraction principle. Without using the concept of the Hausdorff metric, recently, Feng and Liu [7] obtained a new fixed point theorem for nonlinear contractions in metric spaces, extending Nadler's result. Recently, Chifu and Petrusel obtained a fixed point result [18, Theorem 2.1] which contains [7, Theorem 3.1].

In 1980, Rzepecki [8] introduced a generalized metric by replacing the set of real numbers with normal cone of the Banach space. In 1987, Lin [9] introduced the notion of -metric spaces by replacing the set of real numbers with cone in the metric function. Zabrejko [10] studied new revised version of the fixed point theory in -metric and -normed linear spaces. Most recently, Huang and Zhang [11] announced the notion of cone metric spaces, replacing the set of real numbers by an ordered Banach spaces. They proved some basic properties of convergence of sequences and also obtained various fixed point theorems for contractive single-valued maps in such spaces. For more fixed point results in cone metric spaces, see [12–17].

In this paper, first we prove a useful lemma in the setting of cone metric spaces. Then, we prove some results on the existence of fixed points for multivalued maps in cone metric spaces. Consequently, our results improve and extend a number of known fixed point results including the corresponding recent main fixed point results of Chifu and Petrusel [18, Theorems 2.1 and 2.5].

## 2. Preliminaries

Let
be a real Banach space and
a subset of
.
is called a *cone* if and only if

(i) is closed, nonempty, and

(ii) , and imply that ;

(iii) and imply that .

For a given cone , we define partial ordering on with respect to by the following: for , we say that if and only if Also,we write if int where int denotes the interior of

The least positive number satisfying the above inequality is called the normal constant of ; for details see ([3, 11]).

In the sequel, is a real Banach space, is a cone in , and is partial ordering with respect to .

Definition 2.1.

Let be a nonempty set. Suppose that the map satisfies

(i) for all and if and only if ;

(ii) for all ;

(iii) for all

Then
is called a *cone metric* on
and
is called a*cone metric space*([11]).

Example 2.2 (see [11]).

Let and defined by , where is a constant. Then is a cone metric space.

Example 2.3 (see [16]).

Let , a metric space, and defined by . Then is a cone metric space.

Clearly, the above examples show that class of cone metric spaces contains the class of metric spaces.

Now, we recall some basic definitions of sequences in cone metric spaces (see, [11, 17].

Let be a cone metric space and a sequence in . Then

(i) converges to whenever for every with there is a natural number such that for all we denote this by or ;

(ii) is a Cauchy sequence whenever for every with there is a natural number such that for all ;

(iii) is said to be complete space if every Cauchy sequence in is convergent in ;

(iv)A set is said to be closed if for any sequence converges to we have ;

(v)A map
is called *lower semicontinuous* if for any sequence
such that
we have
.

Lemma 2.4 (see [11]).

Let be a cone metric space, and let be a normal cone with normal constant . Let be any sequence in . Then

(a) converges to if and only if as ;

(b) is a Cauchy sequence if and only if as

Let be a cone metric space. We denote as a collection of nonempty subsets of , and as a collection of nonempty closed subsets of . An element is called a fixed point of a multivalued map if . Denote

The set is closed [16, Lemma 2.3].

## 3. The Results

First, we prove our key lemma.

Lemma 3.1.

then is a Cauchy sequence.

Proof.

taking limit as , we get Thus is a Cauchy sequence.

Applying Lemma 3.1, we prove the following result.

Theorem 3.2.

Let be a complete cone metric space, a normal cone with normal constant , and Suppose that the following hold for arbitrary but fixed and with :

(i)there exist constants with such that for each and for any there exist and satisfying

;

the function defined by is lower semicontinuous.

Then

Proof.

From (3.21), it follows that there exists a sequence such that , and thus as Hence, Since is closed, we get Thus,

Remark 3.3.

Our Theorem 3.2 extends the main fixed point result of Chifu and Petrusel [18, Theorem 2.1] to the setting of cone metric spaces, and thus the result of Feng and Liu [7, Theorem 2.1] follows from our Theorem 3.2 as well. Theorem 3.2 also extends some results from [2, 5, 6].

Another fixed point result is the following.

Theorem 3.4.

Let be a complete cone metric space, a normal cone with normal constant , and Suppose that the following hold for arbitrary but fixed and with

there exist with such that for each and for any there exist and satisfying

;

the function defined by is lower semicontinuous.

Then

Proof.

The rest of the proof runs as the proof of Theorem 3.2, and hence we get .

Remark 3.5.

Theorem 3.4 extends the fixed point result of Chifu and Petrusel [18, Theorem 2.5] to cone metric spaces.

Most recently, Asadi et al. [13, Lemma 2.1] proved the closedness of the set in complete cone metric spaces without the normality assumption. In the following remark, we obtain the same conclusion without normality and completeness assumptions.

Remark 3.6.

Let be a cone metric space, and let be any multivalued map. If the function defined by is lower semicontinuous, then the set is closed.

So, there exists a sequence such that Hence

Example 3.7.

Therefore, all the assumptions of Theorem 3.2 are satisfied, and note that

## Declarations

### Acknowledgments

The authors are grateful to Professor Sh. Rezapour for providing a copy of the paper [16]. Also, the authors are thankful to the referees for their valuable suggestions to improve this paper. Finally, the first author thanks the Deanship of Scientific Research, King Abdulaziz University for the research Grant no. 3-62/429.

## Authors’ Affiliations

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