Some Fixed Point Theorems of Integral Type Contraction in Cone Metric Spaces

Abstract

We define a new concept of integral with respect to a cone. Moreover, certain fixed point theorems in those spaces are proved. Finally, an extension of Meir-Keeler fixed point in cone metric space is proved.

1. Introduction

In 2007, Huang and Zhang in [1] introduced cone metric space by substituting an ordered Banach space for the real numbers and proved some fixed point theorems in this space. Many authors study this subject and many fixed point theorems are proved; see [2–5]. In this paper, the concept of integral in this space is introduced and a fixed point theorem is proved. In order to do this, we recall some definitions, examples, and lemmas from [1, 4] as follows.

Let be a real Banach space. A subset of is called a cone if and only if the following hold:

(i) is closed, nonempty, and ,

(ii), , and imply that

(iii) and imply that

Given a cone we define a partial ordering with respect to by if and only if We will write to indicate that but , while will stand for int where int  denotes the interior of The cone is called normal if there is a number such that implies for all The least positive number satisfying above is called the normal constant [1].

The cone is called regular if every increasing sequence which is bounded from above is convergent. That is, if is a sequence such that for some , then there is such that . Equivalently, the cone is regular if and only if every decreasing sequence which is bounded from below is convergent [1]. Also every regular cone is normal [4]. In addition, there are some nonnormal cones.

Example 1.1.

Suppose with the norm and consider the cone : . For all , set and Then and Since is not normal constant of Therefore, is non-normal cone.

From now on, we suppose that is a real Banach space, is a cone in with and is partial ordering with respect to . Let be a nonempty set. As it has been defined in [1], a function is called a cone metric on if it satisfies the following conditions:

(i) for all and if and only if

(ii), for all

(iii), for all

Then is called a cone metric space.

Example 1.2.

Suppose is a metric space and is defined by Then is a cone metric space and the normal constant of is equal to

Definition 1.3.

Let be a cone metric space. Let be a sequence in and If for any with there is such that for all , then is said to be convergent to and is the limit of . We denote this by

(1.1)

Definition 1.4.

Let be a cone metric space and be a sequence in If for any with , there is such that for all then is called a Cauchy sequence in

Definition 1.5.

Let be a cone metric space, if every Cauchy sequence is convergent in then is called a complete cone metric space.

Definition 1.6.

Let be a cone metric space. Let be a self-map on If for all sequence in

(1.2)

then is called continuous on

The following lemmas are useful for us to prove the main result.

Lemma 1.7.

Let be a cone metric space and a normal cone with normal constant Let be a sequence in Then converges to if and only if

(1.3)

Lemma 1.8.

Let be a cone metric space and a normal cone with normal constant Let be a sequence in . Then is a Cauchy sequence if and only if

(1.4)

Lemma 1.9.

Let be a cone metric space and a sequence in If is convergent, then it is a Cauchy sequence.

Lemma 1.10.

Let be a cone metric space and be a normal cone with normal constant . Let and be two sequences in and . Then

(1.5)

The following example is a cone metric space.

Example 1.11.

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

Theorem 1.12.

Let be a complete cone metric space and a normal cone with normal constant Suppose the mapping satisfies the contractive condition

(1.6)

for all where is a constant. Then has a unique fixed point Also, for all the sequence converges to

2. Certain Integral Type Contraction Mapping in Cone Metric Space

In 2002, Branciari in [6] introduced a general contractive condition of integral type as follows.

Theorem 2.1.

Let be a complete metric space, , and is a mapping such that for all

(2.1)

where is nonnegative and Lebesgue-integrable mapping which is summable (i.e., with finite integral) on each compact subset of such that for each , , then has a unique fixed point , such that for each

In this section we define a new concept of integral with respect to a cone and introduce the Branciari's result in cone metric spaces.

Definition 2.2.

Suppose that is a normal cone in . Let and . We define

(2.2)

Definition 2.3.

The set is called a partition for if and only if the sets are pairwise disjoint and

Definition 2.4.

For each partition of and each increasing function we define cone lower summation and cone upper summation as

(2.3)

respectively.

Definition 2.5.

Suppose that is a normal cone in . is called an integrable function on with respect to cone or to simplicity, Cone integrable function, if and only if for all partition of

(2.4)

where must be unique.

We show the common value by

(2.5)

We denote the set of all cone integrable function by .

Lemma 2.6.

• (1) If , then for (2)  for and .

Proof.

• (1) Suppose that and are partitions for and respectively. That is,

  

  (2.6)

Let    is a partition for Therefore one can write

(2.7)

So

(2.8)

• (2) Suppose is an partition for , that is

  

  (2.9)

Then

(2.10)

Thus

(2.11)

Definition 2.7.

The function is called subadditive cone integrable function if and only if for all

(2.12)

Example 2.8.

Let , , and for all Then for all

(2.13)

Since then . Therefore

(2.14)

This shows that is an example of subadditive cone integrable function.

Theorem 2.9.

Let be a complete cone metric space and a normal cone. Suppose that is a nonvanishing map and a subadditive cone integrable on each such that for each , . If is a map such that, for all

(2.15)

for some then has a unique fixed point in

Proof.

Let Choose We have

(2.16)

Since thus

(2.17)

If then and this is a contradiction, so

(2.18)

We now show that is a Cauchy sequence. Due to this, we show that

(2.19)

By triangle inequality

(2.20)

and by sub-additivity of we get

(2.21)

Thus

(2.22)

This means that is a Cauchy sequence and since is a complete cone metric space, thus is convergent to Finally, since

(2.23)

thus This means that If are two distinct fixed points of then

(2.24)

which is a contradiction. Thus has a unique fixed point

Lemma 2.10.

Let and Suppose that is defined by where is a constant. Suppose that is defined by where are two Riemann-integrable functions. Then

(2.25)

Proof.

Let be a partition of set such that and , then (by Definitions 2.4 and 2.5)

(2.26)

Thus

(2.27)

Example 2.11.

Let and Suppose for some constant Firstly, is a complete cone metric space. Secondly, if and are defined by

(2.28)

respectively, then

(2.29)

In order to obtain inequality (2.29), set and where Hence

(2.30)

Suppose for all and . Thus . By Lemma 2.10

(2.31)

Since thus

(2.32)

It means that

(2.33)

On the other side, Branciari in [6] shows that

(2.34)

for all . Therefore

(2.35)

Thus inequalities (2.33) and (2.35) imply that

(2.36)

or in other words

(2.37)

Thus by Theorem 2.9, has a fixed point. But, on the other hand,

(2.38)

and this means that does not satisfy in Theorem 1.12.

3. Extension of Meir-Keeler Contraction in Cone Metric Space

In 2006, Suzuki in [7] proved that the integral type contraction (see [6]) is a special case of Meir-Keeler contraction (see [8]). Haghi and Rezapour in [5] extended Meir-Keeler contraction in cone metric space as follows.

Theorem 3.1 (see[5]).

Let be a complete regular cone metric space and has the property (KMC) on that is, for all , there exists such that

(3.1)

for all . Then has a unique fixed point.

An extension of Theorem 3.1 is as follows.

Theorem 3.2.

Let be a complete regular cone metric space and a mapping on . Suppose that there exists a function from into itself satisfying the following:

and for all ,

is nondecreasing and continuous function. Moreover, its inverse is continuous,

for all there exists such that for all

(3.2)

for all

(3.3)

Then has a unique fixed point.

Proof.

First, note that for all with Since exists, thus for all with Now Let Set for all If, there is a natural such that then and so has a fixed point. If for all then Hence, according to regularity of there exists such that We claim If then according to there is such that for all with Choose such that and take the natural number such that for all We obtain

(3.4)

Thus

(3.5)

So, Since has the property    for all . This is a contradiction because for all Thus

(3.6)

Now, we show that is a Cauchy sequence. If this is not, then there is a such that for all natural number there are so that the relation does not hold. Since has continuous inverse thus there exists such that for all natural number there are so that the relation does not hold. For each there exists such that for all with Choose a natural number such that for all Also, take so that the relation does not hold. Then yields

(3.7)

Hence, Similarly, Thus

(3.8)

which is a contradiction. Therefore is a Cauchy sequence. Since is a complete cone metric space, there is such that . Since , for all with , thus for each , there is a natural number such that for all , Since thus for all . It means that In the other side, and the limit point is unique in cone metric spaces. Thus has at least one fixed point. Now, if are two distinct fixed points for then

(3.9)

which is a contradiction. Therefore has a unique fixed point.

Remark 3.3.

Set , then Theorem 3.1 is a direct result of Theorem 3.2.

Let be a nonvanishing map and a subadditive cone integrable on each such that for each , If then satisfies all conditions of Theorem 3.2. In other words, Theorem 2.9 is a direct result of Theorem 3.2.