1. Introduction and Preliminaries

Partial metric spaces were introduced by Matthews in ^[9] as a part of the study of denotational semantics of dataow networks. In fact, it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation ^{[10, 11, 12, 13, 14]}.

Definition 1. ^[9] A partial metric on a nonempty set is a function such that for all

(pms1)

(pms2)

(pms3)

(pms4)

The pair is called a partial metric space.

If is a partial metric on then the function given by is a metric on Each partial metric p on inroduces a topology on which has as a base the family of open balls for all and

Definition 2. ^[9] Let be a partial metric space, and let be any sequence in and Then

(a) a sequence is convergent to x with respect to if ;

(b) a sequence is a Cauchy sequence in if existsand is finite;

(c) is called complete if for every Cauchy sequence in there exists such that

In 2010, Romaguera proved in [4-Theorem 2.3] that a partial metric space is 0-complete if and only if every -Caristi mapping on has a fixed point. Since then several papers have dealt with fixed point theory for single-valued and multi-valued operators in 0-complete partial metric space (see [1-8]^[1] and references therein).

Definition 3. ^[4] Let be a partial metric space. A sequence in X is called a 0-Cauchy sequence if The space is said to be 0-complete if every 0-Cauchy sequence in converges with respect to to a point such that

Remark 1. ^{[15, 16]} Let be a partial metric space. If as then as for all

Lemma 1. ^[2] Let be a partial metric space and let be a sequence in such that

(1.1)

If is not 0-Cauchy sequence in , then there exists and two sequences and of positive integers such that and the following sequences tend to as

(1.2)

Definition 4. ^[17] Let f and g be self maps of a set X. If for some then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g. The pair f, g of self maps is weakly compatible if they commute at their coincidence points.

Proposition 1. ^[17] Let f and g be weakly compatible self maps of a set X. If f and g have a unique point of coincidence w = fx = gx, then w is the unique common fixed point of f and g.

2. Main Results

Denote by the set of functions satisfying the following conditions:

is continuous nondecreasing;

for all and

Denote by the set of functions satisfying the following conditions:

is a lower semi-continuous functions;

for all and

Theorem 1. Let be a 0-complete partial metric spaces. Suppose mappings satisfy

(2.1)

where and and

(2.2)

for all If the range of g contains the range of f and f (X) or g (X) is a closed subset of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point z and

Proof. First, we prove that f and g have a unique point of coincidence (if it exists). If with and with , we assume c1 6= c2. Using (2.1) and (2.2), we have

which is a contradiction. Thus , that is, Thus, the point of coincidence of f and g is unique (if it exists).

We construct a sequence as follows:

Let Choose a point such that This can be done, as the range of g contains the range of f. Continuing in the same way, having chosen we get such that (say). Therefore, we get the sequence such that for all Consider the two possible cases:

(i) for some

In this case is a point of coincidence and then the proof is finished.

(ii) for every

From (2.1) and (2.2), using properties of functions and , we obtain

which implies that

Then, we have

If then Furthermore, it implies that

which is a contradiction. Therefore, we have

(2.3)

It follows from (2.3) that the sequence is nonincreasing. Therefore,

Letting in inequality

we obtain and Thus

(2.4)

We next prove that is a 0-Cauchy sequence in the space . It is sufficient to show that is a 0-Cauchy sequence. Suppose the opposite. Then using Lemma 1, we see that there exist and two sequences and of positive integers and sequences

(2.5)

all tend to when Using (2.1) and (2.2), we get that

(2.6)

Using (2.4) and (2.5), we obtain

Letting in (2.6), we get that which is a contradiction if

This show that is a 0-Cauchy sequence in the space and is a 0-Cauchy sequence in the space

If is closed in then there exist such that and

Now, putting and in (2.1) and (2.2) we have

(2.7)

Letting in (2.7) and by Remark 1, we obtain

This implies that is, Hence, f and g have a unique point of coincidence. By Proposition 1, f and g have a unique common fixed point.

When is closed set in the proof similar.

Corollary 1. Let be a 0-complete partial metric spaces. Suppose mapping satisfy

(2.8)

where and and

(2.9)

for all Then f has a unique fixed point and

Proof. Taking (the identity mapping of ), along the lines of the proof of Theorem 1, we get the desired results. In view of the analogy, we skip the details of the proof.

Corollary 2. Let be a 0-complete partial metric spaces. Suppose mappings satisfy

(2.10)

where and

(2.11)

for all If the range of g contains the range of f and f (X) or g (X) is a closed subset of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point z and

Proof. To prove the above corollary it suffices to take in Theorem 1.

Corollary 3. Let (X; p) be a 0-complete partial metric spaces. Suppose mapping f : X ! X satisfy

(2.12)

where and

(2.13)

for all Then has a unique fixed point and

Proof. Taking in Corollary 2, we have desired results.

Corollary 4. ^[2] Let be a 0-complete partial metric spaces. Suppose mappings satisfy

(2.14)

where and

(2.15)

for all If the range of g contains the range of f and or is a closed subset of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point z and

Proof. To prove the above corollary it suffices to take in Corollary 2.

Corollary 5. Let be a 0-complete partial metric spaces. Suppose mapping satisfy

(2.16)

where and

(2.17)

for all Then f has a unique fixed point and

Corollary 6. ^[18] Let be a 0-complete partial metric spaces. Suppose mapping and there exist nonnegative constants bi satisfying such that, for each

(2.18)

Then f has a unique fixed point and

Corollary 6 is a simple consequence of Corollary 5.

Corollary 7. Let be a 0-complete partial metric spaces. Suppose mapping satisfy

(2.19)

for each and . Then f has a unique fixed point and

Proof. It follows from Corollary 6.

Conclusion 1. 1. Our theorems and corolaries which include the corresponding results announced in Boyd and Wong ^[19] (1969), Rhoades ^[20] (1977) as special cases fundamentally improve and generalize the results of Ahmad et al. ^[2] (2012) and Radenović ^[18] (2013).

2. Taking and in Corollary 6, we obtain extension of Kannan Theorem on a 0-complete partial metric spaces.

3. Taking and in Corollary 6, we obtain extension of Reich Theorem on a 0-complete partial metric spaces.

4. Taking and in Corollary 6, we obtain extension of Chatterjea Theorem on a 0-complete partial metric spaces.

Now, we give a example which illustrate Theorem 1.

Example 1. Let , and le be defined by for all Then, is a 0-complete partial metric space. Define

Take and for each

We distinguish five cases:

Case 1: If (x = 0 and y = 0) or (x = 0 and y = 1) or (x = 0 and y = 2) or (x = 1 and y = 1) or (x = 1 and y = 2) or (x = 2 and y = 2), we have

where

Case 2: If x = 0 and y = 3, we have

and

Hence,

Case 3: If x = 1 and y = 3, we have

and

Thus,

Case 4: If x = 2 and y = 3, we have

and

Thus,

Case 5: If x = 3 and y = 3, we have

and

Thus,

It is obvious that all the condition of Theorem 1 is satisfied. Therefore, we apply Theorem 1 and f and g have a unique common fixed point, i.e. 0.

The following is a example which illustrate our results and that the generalizations are proper.

Example 2. Let and let be defined by for all Then, is a 0-complete partial metric space, but it is not complete partial metric space. Define by

Then all the conditions of Theorem 1 are satisfied with and and f and g have a unique common fixed point, i.e. 0.

Acknowledgement 1. The authors wish to thank the editor and referees for their helpful comments and suggestions.

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