Abstract

In 2011, E. Karapinar et al. [12] proved some common fixed point theorems for four weakly compatible self-maps in complete partial metric spaces. In this paper, we extend these theorems using E.A. property and (CLR)-property in complete partial metric spaces.

1. Introduction

Partial metric spaces, introduced by Matthews ^{2, 3}, are a generalization of the notion of the metric space in which in definition of metric the condition is replaced by the condition Matthews discussed some properties of convergence of sequence and proved the fixed point theorems for contractive mapping on partial metric spaces: any mapping of a complete partial metric spaces into itself that satisfies, where the inequality for all has a unique common fixed point. Recently, many authors ^{4, 5, 6, 7, 8, 9, 10, 11, 12} have focused on this subject and generalized some fixed point theorems from the class of metric spaces to the class of partial metric spaces.

The definition of partial metric space was given by Matthews ¹ as follows:

Definition 1.1. ¹ Let be a nonempty set and let satisfy

(PM1)

(PM2)

(PM3)

(PM4)

(1.1)

for all where Then the pair is called a partial metric space (in short PMS) and is called a partial metric on

Let be a PMS. Then, the functions given by

are usual metrics on It is clear that and are equivalent. Each partial metric on generates a topology on with a base of the open p-balls where for all and

E. Karapinar et al. ¹³ proved the following theorem for four weakly compatible mappings in partial metric spaces.

Theorem 1.2. ¹³ Let be a complete PMS. Suppose that are self-mappings on and and are continuous. Suppose also that and are continuing pairs and that

(1.2)

If there exists an and such that

(1.3)

for any in where

(1.4)

Then and have a unique common fixed point in

In 2002, Aamri and Moutawakil ¹ introduced the notion of E.A. property as follows:

Definition 1.3. ¹ Let be a metric space and Then and are said to satisfy E.A. property if there exists a sequence such that

for some in

In 2012, Sintunavarat and Kumam ¹⁴ introduced the notion of property as follows:

Definition 1.4. ¹⁴ Two self-mappings f and g of a metric spaces are said to satisfy property if there exists a sequences in such that,

for some in

2. Main Results

Theorem 2.1. Let be four self-mappings on a complete partial metric space satisfying (1.3), (1.4) and the followings:

(1.5) pairs and are weakly compatible,

(1.6) pair or satisfy E.A. property.

If any one of is a complete subspace of then have a unique common fixed point.

Proof: Let us suppose satisfies the E.A. property. Then, a sequence in such that for some in

Since a sequence in X such that

Hence,

We shall show that

Let, if possible,

From (1.3), we have

Letting limit as we get

(1.7)

where

Thus, from (1.7), we get

which is a contradiction.

Therefore, that is,

Suppose that is a complete subspace of X. Then, for some in

Subsequently, we have

Now, we shall show that

Let, if possible,

From (1.3), we have

Letting limit as we get

(1.8)

where

Thus, from (1.8), we get

a contradiction.

Therefore,

Since, and are weakly compatible, therefore, implies that

Since, there exists such that

Now, we claim that

Let, if possible,

From (1.3), we have

Letting limit as we get

(1.9)

where

Then, from (1.9), we get a contradiction.

Therefore,

Thus, we have

The weak compatibility of and implies that

Now, we claim that, is the common fixed point of and

Suppose that,

From (1.3), we have

(1.10)

where

Thus, from (1.10), we have a contradiction.

Therefore,

Hence, is the common fixed point of and

Similarly, we prove that is the common fixed point of and Since, is the common fixed point of and The proof is similar when is assumed to be a complete subspace of are similar to the cases in which or respectively is complete subspace of since and

Now, we shall prove that the common fixed point is unique.

If possible, let c and d be two common fixed points of and such that

From (1.3), we have

(1.11)

where

Thus, from (1.11), we get a contradiction.

Therefore, and the uniqueness follows.

Theorem 2.2. Let be four self-mappings on a complete partial metric space satisfying (1.3), (1.5) and the followings:

(1.12)

Then, have a unique common fixed point.

Proof: Without loss of generality, assume that and the pair satisfies (CL) property, then there exists a sequence in such that for some in

Since there exists a sequence in such that

Hence,

We shall show that

Let, if possible

From (1.3), we have

Letting limit as we get

(1.13)

where

Thus, from (1.13), we get

a contradiction.

Therefore, that is,

Subsequently, we have

Now, we will show that

Let, if possible,

From (1.3), we have

Letting limit as we get

(1.14)

where,

Thus, from (1.14), we get a contradiction.

Therefore,

Since, the pair is weakly compatible, it follows that

Also, since there exists some y in such that that is,

Now, we show that

Let, if possible,

From (1.3), we have

Letting limit as we get

(1.15)

where,

Thus, from (1.15), we get

a contradiction.

Hence,

Since the pair is weakly compatible, it follows that

Let, if possible,

From (1.3), we have

(1.16)

where,

Thus, from (1.16), we get a contradiction.

Therefore, that is,

Now, we shall show that

Let, if possible,

From (1.3), we have

(1.17)

where,

Thus, from (1.17), we get a contradiction.

Therefore,

Hence, z is the common fixed point and

Now, we shall prove that the common fixed point is unique.

Let be the another common fixed point of and

Let, if possible,

From (1.3), we have

where,

Therefore, we get a contradiction.

Thus, and hence the uniqueness follows.

3. Conclusion

In this paper, some results in complete partial metric spaces are proved using two important properties in fixed point theory, viz., E.A. property and (CLR) property. The results proved are the extended version of the result proved by E. Karapinar et al. for weakly compatible maps.

References