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Fixed point theory of cyclical generalized contractive conditions in partial metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 17 (2013)
Abstract
The purpose of this paper is to study fixed point theorems for a mapping satisfying the cyclical generalized contractive conditions in complete partial metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.
MSC:47H10, 54C60, 54H25, 55M20.
1 Introduction and preliminaries
Throughout this paper, by , we denote the set of all nonnegative real numbers, while ℕ is the set of all natural numbers. Let be a metric space, D be a subset of X and be a map. We say f is contractive if there exists such that for all ,
The well-known Banach fixed point theorem asserts that if , f is contractive and is complete, then f has a unique fixed point in X. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, in 1969, Boyd and Wong [2] introduced the notion of Φ-contraction. A mapping on a metric space is called Φ-contraction if there exists an upper semi-continuous function such that
In 1994, Mattews [3] introduced the following notion of partial metric spaces.
Definition 1 [3] A partial metric on a nonempty set X is a function such that for all ,
(p1) if and only if ;
(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.
Remark 1 It is clear that if , then from (p1) and (p2), . But if , may not be 0.
Each partial metric p on X generates a topology on X which has as a base the family of open p-balls , where for all and . If p is a partial metric on X, then the function given by
is a metric on X.
We recall some definitions of a partial metric space as follows.
Definition 2 [3]
Let be a partial metric space. Then
-
(1)
a sequence in a partial metric space converges to if and only if ;
-
(2)
a sequence in a partial metric space is called a Cauchy sequence if and only if 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 ;
-
(4)
a subset A of a partial metric space is closed if whenever is a sequence in A such that converges to some , then .
Remark 2 The limit in a partial metric space is not unique.
-
(a)
is a Cauchy sequence in a partial metric space if and only if it is a Cauchy sequence in the metric space ;
-
(b)
a partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if .
In 2003, Kirk, Srinivasan and Veeramani [5] introduced the following notion of the cyclic representation.
Definition 3 [5]
Let X be a nonempty set, and be an operator. Then is called a cyclic representation of X with respect to f if
-
(1)
, are nonempty subsets of X;
-
(2)
.
Kirk, Srinivasan and Veeramani [5] also proved the following theorem.
Theorem 1 [5]
Let be a complete metric space, , , be closed nonempty subsets of X and . Suppose that f satisfies the following condition:
where is upper semi-continuous from the right and for . Then f has a fixed point .
Recently, the fixed theorems for an operator defined on a metric space X with a cyclic representation of X with respect to f have appeared in the literature (see, e.g., [6–8]). In 2010, Pǎcurar and Rus [7] introduced the following notion of a cyclic weaker φ-contraction.
Definition 4 [7]
Let be a metric space, , be closed nonempty subsets of X and . An operator is called a cyclic weaker φ-contraction if
-
(1)
is a cyclic representation of X with respect to f;
-
(2)
there exists a continuous, non-decreasing function with for and such that
for any , , , where .
And Pǎcurar and Rus [7] proved the following main theorem.
Theorem 2 [7]
Let be a complete metric space, , be closed nonempty subsets of X and . Suppose that f is a cyclic weaker φ-contraction. Then f has a fixed point .
In the recent years, fixed point theory has developed rapidly on cyclic contraction mappings, see [9–15].
The purpose of this paper is to study fixed point theorems for a mapping satisfying the cyclical generalized contractive conditions in complete partial metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.
2 Fixed point theorems (I)
In the section, we denote by Ψ the class of functions satisfying the following conditions:
() ψ is an increasing and continuous function in each coordinate;
() for , , and .
Next, we denote by Θ the class of functions satisfying the following conditions:
() φ is continuous and non-decreasing;
() for , and .
And we denote by Φ the class of functions satisfying the following conditions:
() ϕ is continuous;
() for , and .
We now state a new notion of cyclic -contractions in partial metric spaces as follows.
Definition 5 Let be a partial metric space, , be nonempty subsets of X and . An operator is called a cyclic -contraction if
-
(1)
is a cyclic representation of Y with respect to f;
-
(2)
for any , , ,
(2.1)
where , , , and .
Theorem 3 Let be a complete partial metric space, , be nonempty closed subsets of X and . Let be a cyclic -contraction. Then f has a unique fixed point .
Proof Given and let for . If there exists such that , then we finished the proof. Suppose that for any . Notice that for any , there exists such that and .
Step 1. We will prove that
Using (2.1), we have
where
If , then
which implies that , and hence . This contradicts our initial assumption.
From the above argument, we have that for each ,
and
And since the sequence is decreasing, it must converge to some . Taking limit as in (2.2) and by the continuity of φ and ϕ, we get
and so we conclude that and . Thus, we have
By (p2), we also have
Since for all , using (2.3) and (2.4), we obtain that
Step 2. We show that is a Cauchy sequence in the metric space . We claim that the following result holds.
Claim For every , there exists such that if with , then .
Suppose the above statement is false. Then there exists such that for any , there are with with satisfying
Now, we let . Then corresponding to use, we can choose in such a way it is the smallest integer with satisfying and . Therefore, and
Letting , we obtain that
On the other hand, we can conclude that
Letting , we obtain that
Since and using (2.4), (2.6) and (2.7), we have that
and
Since and lie in different adjacently labeled sets and for certain , by using the fact that f is a cyclic -contraction, we have
where
Thus, letting , we can conclude that
which implies , that is, . So, we get a contradiction. Therefore, our claim is proved.
In the sequel, we will show that is a Cauchy sequence in the metric space . Let be given. By our claim, there exists such that if with , then
Since , there exists such that
for any .
Let and . Then there exists such that . Therefore, for , and so we have
Thus, is a Cauchy sequence in the metric space .
Step 3. We show that f has a fixed point ν in .
Since Y is closed, the subspace is complete. Then from Lemma 1, we have that is complete. Thus, there exists such that
And it follows from Lemma 1 that we have
On the other hand, since the sequence is a Cauchy sequence in the metric space , we also have
Since , we can deduce that
Since is a cyclic representation of X with respect to f, the sequence has infinite terms in each for . Now, for all , we may take a subsequence of with and also all converge to ν. Using (2.10) and (2.11), we have
By (2.1),
where
Letting , we have
which implies , that is, . So, .
Step 4. Finally, to prove the uniqueness of the fixed point, suppose that μ, ν are fixed points of f. Then using the inequality (2.1), we obtain that
where
So, we also deduce that
which implies that , and hence , that is, . So, we complete the proof. □
The following provides an example for Theorem 3.
Example 1 Let and , , . We define the partial metric p on X by
and define the function by
Now, we let and be
Then f is a cyclic -contraction and 0 is the unique fixed point.
Proof We claim that f is a cyclic -contraction.
-
(1)
Note that , and . Thus, is a cyclic representation of X with respect to f;
-
(2)
For and (or, and ), without loss of generality, we may assume that , then we have
and
Since
we have
On the other hand, for and , without loss of generality, we may assume that , then it is easy to get the above inequality.
Note that Example 1 satisfies all of the hypotheses of Theorem 3, and we get that 0 is the unique fixed point. □
3 Fixed point theorems (II)
In this article, we also recall the notion of a Meir-Keeler function (see [16]). A function is said to be a Meir-Keeler function if for each , there exists such that for with , we have . We now introduce a new notion of a weaker Meir-Keeler function in a partial metric space as follows.
Definition 6 Let be a partial metric space. We call a weaker Meir-Keeler function in X if for each , there exists such that for with , there exists such that .
In the section, we denote by Φ the class of weaker Meir-Keeler functions in a partial metric space in satisfying the following conditions:
() for , ;
() is decreasing;
() for ,
-
(a)
if , then and
-
(b)
if , then .
And we denote by the class Ψ of functions a continuous function satisfying for , .
First, we state a new notion of cyclic -contractions in partial metric spaces as follows.
Definition 7 Let be a partial metric space, , be nonempty subsets of X and . An operator is called a cyclic -contraction if
-
(1)
is a cyclic representation of Y with respect to f;
-
(2)
for any , , ,
(3.1)
where , and .
Theorem 4 Let be a complete partial metric space, , be nonempty closed subsets of X and . Let be a cyclic -contraction. Then f has a unique fixed point .
Proof Given and let , for . If there exists such that , then we finished the proof. Suppose that for any . Notice that for any , there exists such that and . Then by (3.1), we have
Step 1. We will prove that
Since f is a cyclic -contraction, we can conclude that
Since is decreasing, it must converge to some . We claim that . On the contrary, assume that . Then by the definition of a weaker Meir-Keeler function ϕ, there exists such that for with , there exists such that . Since , there exists such that , for all . Thus, we conclude that . So, we get a contradiction. Therefore, , and so we have
By (p2), we also have
Since for all , using (3.2) and (3.3), we obtain that
Step 2. We show that is a Cauchy sequence in the metric space . We claim that the following result holds.
Claim For every , there exists such that if with , then .
Suppose the above statement is false. Then there exists such that for any , there are with with satisfying
Now, we let . Then corresponding to use, we can choose in such a way it is the smallest integer with satisfying and . Therefore, and
Letting , we obtain that
On the other hand, we can conclude that
Letting , we obtain that
Since and using (3.5) and (3.6), we have that
and
Since and lie in different adjacently labeled sets and for certain , by using the fact that f is a cyclic -contraction, we have
Letting , by using the condition of the function ϕ, we obtain that
and consequently, . By the definition of a function ψ, we get which is a contraction. Therefore, our claim is proved.
In the sequel, we will show that is a Cauchy sequence in the metric space . Let be given. By our claim, there exists such that if with , then
Since , there exists such that
for any .
Let and . Then there exists such that . Therefore, for , and so we have
Thus, is a Cauchy sequence in the metric space .
Step 3. We show that f has a fixed point ν in .
Since Y is closed, the subspace is complete. Then from Lemma 1, we have that is complete. Thus, there exists such that
And it follows from Lemma 1 that we have
On the other hand, since the sequence is a Cauchy sequence in the metric space , we also have
Since , we can deduce that
Since is a cyclic representation of X with respect to f, the sequence has infinite terms in each for . Now, for all , we may take a subsequence of with and also all converge to ν. Using (3.9) and (3.10), we have
By (3.1),
Letting , we have
and so .
Step 4. Finally, to prove the uniqueness of the fixed point, let μ be another fixed point of f in . By the cyclic character of f, we have . Since f is a cyclic weaker -contraction, we have
and we can conclude that
which implies . So, we have . We complete the proof. □
The following provides an example for Theorem 4.
Example 2 Let and , , . We define the partial metric p on X by
and define the function by
Now, we let be
Then f is a cyclic -contraction and 0 is the unique fixed point.
By Theorem 4, it is easy to get the following corollary.
Corollary 1 Let be a complete partial metric space, , be nonempty closed subsets of X, and let . Assume that
-
(1)
is a cyclic representation of Y with respect to f;
-
(2)
for any , , ,
where and .
Then f has a unique fixed point .
References
Banach S: Sur les operations dans les ensembles abstraits et leur application aux equations integerales. Fundam. Math. 1922, 3: 133–181.
Boyd DW, Wong SW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S0002-9939-1969-0239559-9
Mattews SG: Partial metric topology. Ann. New York Acad. Sci. 728. Proc. 8th Summer of Conference on General Topology and Applications 1994, 183–197.
Oltra S, Valero O: Banach’s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste 2004, 36: 17–26.
Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory Appl. 2003, 4(1):79–89.
Karapinar E: Fixed point theory for cyclic weaker ϕ -contraction. Appl. Math. Lett. 2011, 24(6):822–825. 10.1016/j.aml.2010.12.016
Pǎcurar M, Rus IA: Fixed point theory for cyclic φ -contractions. Nonlinear Anal. 2010, 72(3–4):2683–2693.
Rus IA: Cyclic representations and fixed points. Ann. “Tiberiu Popoviciu” Sem. Funct. Equ. Approx. Convexity 2005, 3: 171–178.
Agarwal RP, Alghamdi MA, Shahzad N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 40
Di Bari C, Suzuki T, Vetro C: Best proximity for cyclic Meir-Keeler contractions. Nonlinear Anal. 2008, 69: 3790–3794. 10.1016/j.na.2007.10.014
Karpagam S, Agrawal S: Best proximity point theorems for p -cyclic Meir-Keeler contractions. Fixed Point Theory Appl. 2009., 2009: Article ID 197308
Karpagam S, Agrawal S: Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps. Nonlinear Anal. 2010, 74: 1040–1046.
Karapinar E, Erhan IM: Best proximity point on different type contraction. Inf. Sci. Appl. Math. 2011, 5: 342–353.
Karapinar E, Erhan IM: Fixed point theorem for cyclic maps on partial metric spaces. Inf. Sci. Appl. Math. 2012, 6: 239–244.
Karapinar E, Erhan IM: Cyclic contractions and fixed point theorems. Filomat 2012, 26: 777–782.
Meir A, Keeler E: A theorem on contraction mappings. J. Math. Anal. Appl. 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6
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The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.
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Chen, CM. Fixed point theory of cyclical generalized contractive conditions in partial metric spaces. Fixed Point Theory Appl 2013, 17 (2013). https://doi.org/10.1186/1687-1812-2013-17
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DOI: https://doi.org/10.1186/1687-1812-2013-17