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Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 54 (2013)
Abstract
In this paper, we introduce the notion of a generalized -weak contraction and we prove some common fixed point results for self-mappings T and S and some fixed point results for a single mapping T by using a -comparison function and a comparison function in the sense of Berinde in a partial metric space. Also, we introduce an example to support the useability of our results.
MSC:47H10, 54H25.
1 Introduction and preliminaries
The contraction principle of Banach is one of the most important results in nonlinear analysis. After Banach established his existence and uniqueness result, many authors established important fixed point theorems in the literature. For the development of our research, in this article, the article by Matthews [1] is the background.
In 1994, in his elegant article [1], Matthews introduced the notion of a partial metric space and proved the contraction principle of Banach in this new framework. After then, many fixed point theorems in partial metric spaces have been given by several authors (for example, please see [2–24]).
Following Matthews [1], the notion of a partial metric space is given as follows.
Definition 1.1 [1]
A partial metric on a nonempty set X is a function such that for all :
-
(p1) ,
-
(p2) ,
-
(p3) ,
-
(p4) .
A partial metric space is a pair such that X is a nonempty set and p is a partial metric on X.
It is clear that each partial metric p on X generates a topology on X. The set , where for all and , forms the base of .
It is remarkable that if p is a partial metric on X, then the functions
and
are ordinary equivalent metrics on X.
Definition 1.2 [1]
Let be a partial metric space. Then:
-
(1)
A sequence in a partial metric space converges to a point if and only if .
-
(2)
A sequence in a partial metric space is called a Cauchy sequence if there exists (and is finite) .
-
(3)
A partial metric space is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that .
The following lemma is crucial in proving our main results.
Lemma 1.1 [1]
Let be a partial metric space.
-
(1)
is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space .
-
(2)
A partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if
The definition of a 0-complete partial metric space is given by Romaguera [19] as follows.
Definition 1.3 [19]
A sequence in a partial metric space is called 0-Cauchy if . We say that is 0-complete if every 0-Cauchy sequence in X converges, with respect to , to a point such that .
We need the following useful lemma in the proof of our main result.
Lemma 1.2 [2]
Assume that as in a partial metric space such that . Then for every .
In [25], Berinde introduced the nonlinear type weak contraction using a comparison function. A map is called a comparison function if it satisfies:
-
(i)
Ï• is monotone increasing,
-
(ii)
for all .
If Ï• satisfies (i) and
-
(iii)
converges for all ,
then Ï• is said to be a -comparison function.
It is an easy matter to see that if Ï• is a comparison function or a -comparison function, then for all and .
Berinde [26, 27] initiated the concept of weak contraction mappings, the concept of almost contraction mappings and the concepts of -weak contractions. Berinde [25–32] studied many interesting fixed point theorems for weak contraction mappings, almost contraction mappings and -weak contraction mappings in metric spaces. We have to recall the following definition.
Definition 1.4 [25]
A single-valued mapping is called a Ćirić strong almost contraction if there exist a constant and some such that
for all .
For some theorems of almost contractive mappings in the sense of Berinde on metric spaces, we refer the reader to [33–42].
Very recently, Ishak Altun and Özlem Acar initiated the notions of a -weak contraction and a -weak contraction in partial metric spaces as follows.
Definition 1.5 [43]
Let be a partial metric space. A map T is called a -weak contraction if there exist a and some such that
Because of the symmetry of the distance, the -weak contraction condition implicitly includes the following dual one:
Thus by (1.3) and (1.4), the -weak contraction condition can be replaced by the following condition:
Definition 1.6 [43]
Let be a partial metric space. A map T is called -weak contraction if there exist a comparison function Ï• and some such that
As above, because of the symmetry of the distance, the -weak contraction condition implicitly includes the following dual one:
Thus, by (1.6) and (1.7), the -weak contraction condition can be replaced by the following condition:
Altun and Acar [43] proved the following interesting theorems.
Theorem 1.1 [43]
Let be a 0-complete partial metric space and be a -weak contraction mapping with a -comparison function, then T has a fixed point.
Theorem 1.2 [43]
Let be a 0-complete partial metric space and be a -weak contraction mapping. Suppose T also satisfies the following condition: There exist a comparison function and some such that
for all . Then T has a unique fixed point.
In this paper, we introduce the notion of a generalized -weak contraction mapping and a generalized -weak contraction mapping in partial metric spaces. Then after, we prove some fixed point results for two mappings S and T and some fixed point results for a single mapping T. Our results generalize Theorems 1.1 and 1.2.
2 The main result
We start our work by introducing the following two concepts.
Definition 2.1 Let be a partial metric space and be two mappings. The pair is called a generalized -weak contraction if there exist and some such that
for all .
Definition 2.2 Let be a partial metric space and be two mappings. Then the pair is called a generalized -weak contraction if there exist a control function Ï• and some such that
for all .
Now, we give and prove our first result.
Theorem 2.1 Let be a 0-complete partial metric space and be two mappings such that the pair is a generalized -weak contraction. If Ï• is a -comparison function, then T and S have a common fixed point.
Proof Choose . Put . Again, put . Continuing this process, we construct a sequence in X such that and . Suppose for some . Without loss of generality, we assume for some . Thus . Now, by (2.2), we have
Since for all , we conclude that . By (p1) and (p2) of the definition of partial metric spaces, we have . So, . Therefore and hence is a fixed point of T and S. Thus, we may assume that for all . Given . If n is even, then for some . By (2.2), we have
Using (p4) of the definition of partial metric spaces and the definition of , we arrive at
If , then (2.3) yields a contradiction. Thus, and hence
If n is odd, then for some . By similar arguments as above, we can show that
By (2.4) and (2.5), we have
By repeating (2.6) n-times, we get . For with , we have
Since Ï• is -comparison, we have converges and hence
So, . Thus is a 0-Cauchy sequence in X. Since X is 0-complete, there exists such that with . So,
Now, we prove that and . Since and , then by Lemma 1.2 we get
and
By using (2.2), we have
On letting in the above inequality and using (2.7) and (2.8), we get that . Since for all , we conclude that . By using (p1) and (p2) of the definition of partial metric spaces, we get that . By similar arguments as above, we may show that . □
The common fixed point of S and T in Theorem 2.1 is unique if we replaced by in (2.2). So, we have the following result.
Theorem 2.2 Let be a 0-complete partial metric space and be two mappings such that
for all . If Ï• is a -comparison function, then the common fixed point of T and S is unique.
Proof The existence of the common fixed point of T and S follows from Theorem 2.1. To prove the uniqueness of the common fixed point of T and S, we let u, v be two common fixed points of T and S. Then and . Now, we will show that . By (2.10), we have
Since for all , we conclude that . By (p1) and (p2) of the definition of partial metric spaces, we get that . □
Taking in Theorems 2.1 and 2.2, we have the following results.
Corollary 2.1 Let be a 0-complete partial metric space and be a mapping such that
for all . If Ï• is a -comparison function, then T has a fixed point.
Corollary 2.2 Let be a 0-complete partial metric space and be a mapping such that
for all . If Ï• is a -comparison function, then T has a unique fixed point.
By the aid of Lemma 2.1 of Ref. [44], we have the following consequence results of Corollaries 2.1 and 2.2.
Corollary 2.3 Let be a partial metric space and be two mappings such that
for all . Also, suppose that
-
(1)
.
-
(2)
SX is a 0-complete subspace of the partial metric space X.
If Ï• is a -comparison function, then T and S have a coincidence point. Moreover, the point of coincidence of T and S is unique.
Corollary 2.4 Let be a partial metric space and be two mappings such that
for all . Also, suppose that
-
(1)
.
-
(2)
SX is a 0-complete subspace of the partial metric space X.
If Ï• is a -comparison function, then the point of coincidence of T and S is unique; that is, if and , then .
By taking , in Corollaries 2.1 and 2.2, we have the following results.
Corollary 2.5 Let be a 0-complete partial metric space and be a mapping such that
for all . If , then T has a fixed point.
Corollary 2.6 Let be a 0-complete partial metric space and be a mapping such that
for all . If , then T has a unique fixed point.
By the aid of Lemma 2.1 of Ref. [44], we have the following consequence results of Corollaries 2.5 and 2.6.
Corollary 2.7 Let be a partial metric space and be two mappings such that
for all . Also, suppose that
-
(1)
.
-
(2)
SX is a 0-complete subspace of the partial metric space X.
If , then T and S have a coincidence point.
Corollary 2.8 Let be a partial metric space and be two mappings such that
for all . Also, suppose that
-
(1)
.
-
(2)
SX is a 0-complete subspace of the partial metric space X.
If , then the point of coincidence of T and S is unique; that is, if and , then .
The -comparison function in Theorems 2.1 and 2.2 can be replaced by a comparison function if we formulated the contractive condition to a suitable form. For this instance, we have the following result.
Theorem 2.3 Let be a 0-complete partial metric space and be a mapping such that
for all . If Ï• is a comparison function, then T has a unique fixed point.
Proof Choose . Put . Again, put such that . Continuing the same process, we can construct a sequence in X such that . If for some , then by the definition of partial metric spaces, we have , that is, is a fixed point of T. Thus, we assume that for all . By (2.11), we have
If
then
a contradiction. Thus,
and hence
Repeating (2.12) n times, we get that
Now, we will prove that is a Cauchy sequence in the partial metric space . For this, given , since and , there exists such that for all . Now, given with . Claim: for all . We prove our claim by induction on m. Since , then
The last inequality proves our claim for . Assume that our claim holds for . To prove our claim for , we have
By (2.11), we have
If , then by (2.13) we have
If , then by (2.13) we have
If , then by (2.13) we have
Thus is a 0-Cauchy sequence in X. Since X is 0-complete, then converges, with respect to , to a point z for some such that
Now, assume that . By using (p4) of the definition of partial metric spaces and (2.11), we have
Since
and , we can choose such that
for all . Thus (2.15) becomes
for all . On letting in the above inequality and using 2.14, we get that , a contradiction. Thus . By using (p1) and (p2) of the definition of a partial metric space, we get that ; that is, z is a fixed point of T. To prove that the fixed point of T is unique, we assume that u and v are fixed points of T. Thus, we have and . By (2.11), we have
Since for all , we have . By (p1) and (p2), we have . □
By the aid of Lemma 2.1 of Ref. [44], we have the following consequence result of Theorem 2.3.
Corollary 2.9 Let be a partial metric space and be two mappings such that for some , we have
for all . Also, suppose that
-
(1)
.
-
(2)
SX is a 0-complete subspace of the partial metric space X.
If Ï• is a comparison function, then T and S have a coincidence point. Moreover, the point of coincidence of T and S is unique.
The uniqueness of a common fixed point of T and S in Theorem 2.1 can be proved under an additional contractive condition based on a comparison function .
Corollary 2.10 Let be a 0-complete partial metric space and be two mappings. Assume there exists a -comparison function Ï• such that the pair is a generalized -weak contraction. Also, suppose that there exist a comparison function and such that
for all . Then T and S have a unique common fixed point.
Proof The existence of the common fixed point of T and S follows from Theorem 2.1. To prove the uniqueness of the fixed point, we assume that u and v are two fixed points of T and S. Then by (2.16), we have
Since for all , we get and hence . □
Taking in Corollary 2.10, we have the following result.
Corollary 2.11 Let be a 0-complete partial metric space and be a mapping. Assume there exists a -comparison function Ï• such that
for all . Also, suppose that there exist a comparison function and such that
for all . Then T has a unique fixed point.
By the aid of Lemma 2.1 of [44], we have the following result.
Corollary 2.12 Let be a partial metric space and be two mappings. Suppose there exist a -comparison function Ï• and such that
for all . Also, assume that there exist a comparison function and such that
for all . Moreover, assume that
-
(1)
.
-
(2)
SX is a 0-complete subspace of the partial metric space X.
Then the point of coincidence of T and S is unique.
Now, we introduce an example satisfying the hypotheses of Theorem 2.3 to support the useability of our results.
Example 2.1 Let . Define a partial metric by the formula . Also, define by and the comparison function by . Then, we have
-
(1)
is a 0-complete partial metric space.
-
(2)
For any , the inequality
holds for all .
-
(3)
There are no -comparison function Ï• and such that the inequality
holds for all .
Proof To prove (2), given . Without loss of generality, we may assume that . Thus, we have
To prove (3), we assume that there exist a -comparison function Ï• and some such that
holds for all .
Thus,
holds for all . Hence
holds for all . Therefore, for , we have
By induction on n, we can show that
holds for all . Since diverges, we have diverges. So, ϕ is not a -comparison function. □
Remark Example 2.1 satisfies the hypotheses of Theorem 2.3 and does not satisfy the hypotheses of Theorem 3 and Theorem 4 of [43].
Remarks
3 Conclusion
In this paper, we introduced the notion of a generalized -weak contraction. In the first part of this paper, we utilized our definition to derive a common fixed point of two self-mappings T and S under a -comparison function Ï•. Also, we used Lemma 2.1 of Ref. [44] to derive a common fixed point of two self-mappings T and S. In the second part of this paper, we generalized the main result (Theorem 3) of [43] by proving Theorem 2.3 under a comparison function. Also, we utilized Lemma 2.1 of Ref. [44] to derive a common coincidence point of two self-mappings T and S. Finally, we closed our paper by introducing Example 2.1 which satisfies the hypotheses of our result Theorem 2.3 and does not satisfy the hypotheses of Theorems 3, 4 of [43].
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Shatanawi, W., Postolache, M. Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces. Fixed Point Theory Appl 2013, 54 (2013). https://doi.org/10.1186/1687-1812-2013-54
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DOI: https://doi.org/10.1186/1687-1812-2013-54