Fixed Point Results for Multivalued Maps in Cone Metric Spaces

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.

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 cone is called normal if there is a constant such that for all

(2.1)

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 acone 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

For and one denotes

(2.2)

For with one denotes

(2.3)

The set is closed [16, Lemma  2.3].

3. The Results

First, we prove our key lemma.

Lemma 3.1.

Let be a cone metric space and let be a normal cone with normal constant . If there exist a sequence in and a real number such that for every ,

(3.1)

then is a Cauchy sequence.

Proof.

Let be an arbitrary but fixed. Note that

(3.2)

Now, for all

(3.3)

Since is a normal cone with the normal constant , we have

(3.4)

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

(3.5)

;

the function defined by is lower semicontinuous.

Then

Proof.

Since and it follows from (i) and (ii) that there exist and satisfying

(3.6)

(3.7)

Note that

(3.8)

and thus . From (3.6) and (3.7) it follows that

(3.9)

Now, since and there exist and such that

(3.10)

(3.11)

Using (3.9), (3.10) (3.11) we obtain

(3.12)

From (3.7), (3.8) and (3.10) it follows that

(3.13)

Note that

(3.14)

and so Continuing this process, we obtain and such that and satisfying

(3.15)

and we get

(3.16)

Thus by Lemma 3.1, is a Cauchy sequence in the closed set Due to the completeness of , there exists such that Note that

(3.17)

and thus

(3.18)

From (3.18) and the fact the cone is normal, we have

(3.19)

and thus as ; it follows that is convergent to . Since there exists a sequence such that and Now by the convergence of the sequence and by assumption (iii) we obtain

(3.20)

Thus

(3.21)

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

(3.22)

;

the function defined by is lower semicontinuous.

Then

Proof.

Since and there exist and satisfying

(3.23)

(3.24)

From (3.23) and (3.24) we have

(3.25)

Using (3.23) and (ii), we get

(3.26)

and so Therefore, there exist and satisfying

(3.27)

(3.28)

Using (3.25), (3.27), and (3.28), we get

(3.29)

Now, using (3.23), (3.24), (3.27), and (ii), we have

(3.30)

Note that

(3.31)

Thus, Continuing this process, we get and such that and satisfying

(3.32)

Thus, we get

(3.33)

Now, for , we have

(3.34)

where Since is normal, we have

(3.35)

and hence is a Cauchy sequence. Due to the completeness of there exists such that . Also, note that

(3.36)

and thus,

(3.37)

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.

Indeed, let be such that as Clearly, because Using the lower semicontinuity of the function , we get

(3.38)

Thus

(3.39)

So, there exists a sequence such that Hence

Example 3.7.

Let a Banach space with the maximum norm, and a normal cone. Define by

(3.40)

Then the pair is a complete cone metric space. Now, define the map by

(3.41)

Note that the map

(3.42)

is lower semicontinuous. Now, if we take we get

(3.43)

Now, for the case and we obtain

(3.44)

Now, taking and , we get

(3.45)

Now, for the case we have

(3.46)

And also, for this case we get

(3.47)

Further, for we have

(3.48)

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