Abstract
In this paper, we establish common fixed point theorems for two weakly compatible self-mappings satisfying the contractive condition or the quasi-contractive condition in the case of a quasi-contractive constant in cone b-metric spaces without the normal cone, where the coefficient s satisfies . The main results generalize, extend and unify several well-known comparable results in the literature.
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1 Introduction and preliminaries
Huang and Zhang [1] introduced the concept of a cone metric space, proved the properties of sequences on cone metric spaces and obtained various fixed point theorems for contractive mappings. The existence of a common fixed point on cone metric spaces was considered recently in [2–5]. Also, Ilic and Rakocevic [6] introduced a quasi-contraction on a cone metric space when the underlying cone was normal. Later on, Kadelburg et al. obtained a few similar results without the normality of the underlying cone, but only in the case of a quasi-contractive constant . However, Gajic [7] proved that result is true for on a cone metric space by a new way, which answered the open question whether the result is true for . Recently, Hussain and Shah [8] introduced cone b-metric spaces, as a generalization of b-metric spaces and cone metric spaces, and established some important topological properties in such spaces. Following Hussain and Shah, Huang and Xu [9] obtained some interesting fixed point results for contractive mappings in cone b-metric spaces. Although Ion Marian [10] proved some common fixed point theorems in complete b-cone metric spaces, the main ways of the proof depend strongly on the nonlinear scalarization function . In the present paper, we will show common fixed point theorems for two weakly compatible self-mappings satisfying the contractive condition or quasi-contractive condition in the case of a quasi-contractive constant in cone b-metric spaces without the assumption of normality, where the coefficient s satisfies . As consequences, our results generalize, extend and unify several well-known comparable results (see, for example, [2–7, 9–13]).
Consistent with Huang and Zhang [1], the following definitions and results will be needed in the sequel.
Let E be a real Banach space and let P be a subset of E. By θ we denote the zero element of E and by intP the interior of P. The subset P is called a cone if and only if:
-
(i)
P is closed, nonempty, and ;
-
(ii)
, , ;
-
(iii)
.
On this basis, we define a partial ordering ⪯ with respect to P by if and only if . We write to indicate that but , while stands for . Write as the norm on E. The cone P is called normal if there is a number such that for all , implies . The least positive number satisfying the above is called the normal constant of P. It is well known that .
In the following, we always suppose that E is a Banach space, P is a cone in E with and ⪯ is a partial ordering with respect to P.
Definition 1.1 [8]
Let X be a nonempty set and let be a given real number. A mapping is said to be cone b-metric if and only if for all the following conditions are satisfied:
-
(i)
with and if and only if ;
-
(ii)
;
-
(iii)
.
The pair is called a cone b-metric space.
Example 1.2 Consider the space () of all real function () such that . Let , , and such that
where are constants. Then is a cone b-metric space with the coefficient .
Remark 1.3 It is obvious that any cone metric space must be a cone b-metric space. Moreover, cone b-metric spaces generalize cone metric spaces, b-metric spaces and metric spaces.
Definition 1.4 [8]
Let be a cone b-metric space, and be a sequence in X. Then
-
(i)
converges to x whenever, for every with , there is a natural number N such that for all . We denote this by or ().
-
(ii)
is a Cauchy sequence whenever, for every with , there is a natural number N such that for all .
-
(iii)
is a complete cone b-metric space if every Cauchy sequence is convergent.
Lemma 1.5 [8]
Let be a cone b-metric space. The following properties are often used while dealing with cone b-metric spaces in which the cone is not necessarily normal.
-
(1)
If and , then ;
-
(2)
If for each , then ;
-
(3)
If for each , then ;
-
(4)
If and , then ;
-
(5)
If , where and , then ;
-
(6)
If , and , then there exists such that for all .
Lemma 1.6 [8]
The limit of a convergent sequence in a cone b-metric space is unique.
Definition 1.7 [2]
The mappings are weakly compatible if for every , holds whenever .
Definition 1.8 [3]
Let f and g be self-maps of a set X. If for some x in X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g.
Lemma 1.9 [3]
Let f and g be weakly compatible self-maps of a set X. If f and g have a unique point of coincidence , then w is the unique common fixed point of f and g.
Definition 1.10 [13]
Let be a cone metric space. A mapping is such that, for some constant and for every , there exists an element
for which is called a g-quasi-contraction.
2 Main results
In this section, we give some common fixed point results for two weakly compatible self-mappings satisfying the contractive condition and quasi-contractive condition in the case of a contractive constant in cone b-metric spaces without the assumption of normality.
Theorem 2.1 Let be a cone b-metric space with the coefficient and let () be constants with . Suppose that the mappings satisfy the condition, for all ,
If the range of g contains the range of f and or is a complete subspace 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 in X.
Proof For an arbitrary , since , there exists an such that . By induction, a sequence can be chosen such that (). If for some natural number , then is a coincidence point of f and g in X. Suppose that for all .
Thus, by (2.1) for any , we have
and
Hence
Since , we have
where . Obviously, .
Thus, setting any positive integers m and n, we have
Since , we notice that as for any . By Lemma 1.5, for any , we can choose such that for all . Thus, for each , for all , . Therefore is a Cauchy sequence in .
If is complete, there exist and such that as and . (If is complete, there exists such that as . Since , we can find such that .)
Now, from (2.1) we show that ,
Similarly,
thus, we have
Since , by the triangular inequality, it follows that
Since is a Cauchy sequence and (), for any , we can choose such that for all ,
and
Thus, for any , for all . Therefore, by Lemma 1.5, we have .
Assume that there exist u, w in X such that .
Since , by Lemma 1.5, we can obtain that , i.e., . Moreover, the mappings f and g are weakly compatible, by Lemma 1.9, we know that q is the unique common fixed point of f and g. □
Example 2.2 Let , , and . Then is a cone b-metric space with the coefficient , but it is not a cone metric space. We consider the functions defined by , . Hence
Here is the unique common fixed point of f and g.
Example 2.3 Let X be the set of Lebesgue measurable functions on such that , , . We define as
for all . Then is a cone b-metric space with the coefficient , but it is not a cone metric space. Considering the functions and (), we have
Clearly, is the unique common fixed point of f and g.
Remark 2.4 Compared with the common fixed point results on cone metric spaces in [2, 3, 5], the common fixed point theorems in complete b-cone metric spaces in [10] and the fixed point results in cone b-metric spaces in [9], Theorem 2.1 is shown to be a proper generalization by Examples 2.2 and 2.3. Furthermore, Theorem 2.1 generalizes and unifies [[9], Theorem 2.1 and 2.3].
Definition 2.5 Let be a cone b-metric space with the coefficient . A mapping is such that, for some constant and for every , there exists an element
for which is called a g-quasi-contraction.
Theorem 2.6 Let be a cone b-metric space with the coefficient and let the mapping be a g-quasi-contraction. If the range of g contains the range of f and or is a complete subspace 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 in X.
Proof For each , set and (). If for some natural number , then is a coincidence point of f and g in X.
Suppose that for all . Now we prove that is a Cauchy sequence. First, we show that
Clearly, we note (2.3) holds when . We assume that (2.3) holds for some (), then we prove that (2.3) holds for all . Because f is a g-quasi-contractive mapping, there exists a real number such that
In order to prove that (2.4) holds, we show that for all , there exists such that
Clearly, (2.5) is true for . Suppose that (2.5) is true for each , that is, for all , there exists such that
Let us prove (2.5) holds for .
By (2.6), we only show that for any , there exists such that
Since f is a g-quasi-contractive mapping, there exists
such that .
By (2.6), we discuss that there exists an element
such that ().
If the above inequality does not hold for , then (2.5) is true for by (2.6).
We continue in the same way, and after steps, we get () such that
Notice that there exist such that . That is,
As , by Lemma 1.5(5), we get a contradiction. From (2.6), (2.5) is true for .
Hence, (2.5) is true for all , which implies that (2.4) holds for .
Next, let us prove that for all ,
Using the triangular inequality, from (2.3) we obtain
Now, we show that is a Cauchy sequence. For all , there exists
such that .
By the contractive condition, there exist but not all
such that
In fact, from (2.8) we have
Let , where
In general, if there exists
then we have
such that ().
As
we can obtain (2.9).
Using the triangular inequality, we get
so we obtain
Since as , by Lemma 1.5, it is easy to see that for any , there exists such that for all ,
So, is a Cauchy sequence in . If is complete, there exist and such that as and .
Now, from (2.2) we get
such that .
We have the following five cases:
-
(1)
;
-
(2)
;
-
(3)
, that is, ;
-
(4)
, that is, ;
-
(5)
.
As , then we obtain that
Since as , for any , there exists such that for all ,
By Lemmas 1.5 and 1.6, we have as and .
Now, if w is another point such that , then
where and
It is obvious that , i.e., . Therefore, q is the unique point of coincidence of f, g in X. Moreover, the mappings f and g are weakly compatible, by Lemma 1.9 we know that q is the unique common fixed point of f and g.
Similarly, if is complete, the above conclusion is also established. □
Example 2.7 Let , and . Define by where such that . It is easy to see that is a cone b-metric space with the coefficient , but it is not a cone metric space. The mappings are defined by and (). The mapping f is a g-quasi-contraction with the constant . Moreover, is the unique common fixed point of f and g.
Remark 2.8 Kadelburg and Radenovi [11] obtained a fixed point result without the normality of the underlying cone, but only in the case of a quasi-contractive constant (see [[11], Theorem 2.2]). However, Ljiljana [7] proved the result is true for on a cone metric space by a new way. Referring to this way, Theorem 2.6 presents a similar common fixed point result in the case of the contractive constant in cone b-metric spaces without the assumption of normality. Moreover, it is obvious that Example 2.7 given above shows that Theorem 2.6 not only improves and generalizes [[11], Theorem 2.2], but also generalizes and unifies [[7], Theorem 3].
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Acknowledgements
The first author thanks Doctor Hao Liu for his help and encouragement. Besides, the authors are extremely grateful to the referees for their useful comments and suggestions which helped to improve this paper. The research is partially supported by the Foundation of Education Ministry, Hubei Province, China (No: D20102502).
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Shi, L., Xu, S. Common fixed point theorems for two weakly compatible self-mappings in cone b-metric spaces. Fixed Point Theory Appl 2013, 120 (2013). https://doi.org/10.1186/1687-1812-2013-120
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DOI: https://doi.org/10.1186/1687-1812-2013-120