Abstract
In this paper, we establish common coupled fixed point and coupled fixed point theorems in cone b-metric spaces. The presented theorems extend and generalize several well-known comparable results in literature. We supply some examples to elucidate our obtained results.
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1 Introduction and preliminaries
In [1], Bakhtin introduced b-metric spaces (or metric-type spaces) as a generalization of metric spaces. He evidenced the contraction mapping principle in b-metric spaces that generalized the famous Banach contraction principle in metric spaces. From that time on, manifold papers have treated fixed point theory or the variational principle for single-valued and multi-valued operators in b-metric spaces (see [2–7] and the references therein).
Ordered normed spaces, cones and topical functions have applications in applied mathematics, for instance, in using Newton’s approximation method [8–12] and optimization theory [13, 14]. In the mid-twentieth century [9], k-metric and k-normed spaces were established (see also [11, 12]) by replacing an ordered Banach space instead of the set of real numbers, as the codomain for a metric. Due to defining convergent and Cauchy sequences in terms of interior points of the underlying cone, Huang and Zhang [15] re-introduced such spaces under the name of cone metric space. Even though they used only normal cones, nonnormal cones can be used as well in such a way but by taking into the consideration that the Sandwich theorem and continuity of the metric may not hold. Some fixed point theorems for contractive-type mappings in cone metric spaces have been substantiated; for more details, see [16–25].
As a generalization of b-metric spaces and cone metric spaces, Hussain and Shah [26] announced cone b-metric spaces, which was in 2011. They built up some topological properties in such spaces and upgraded some latest results about KKM mappings in the setting of a cone b-metric space. Hussain and Shah [26] have done initial work that stimulated many authors to prove fixed point theorems, as well as common fixed point theorems for two or more mappings on cone b-metric spaces (see [27–30] and the references therein).
The following will be needed in the sequel.
Let E be a real Banach space and let θ denote the zero element in E. A cone P is a subset of E such that:
-
1.
P is nonempty set closed and .
-
2.
If a, b are nonnegative real numbers and , then .
-
3.
and implies .
For any cone , the partial ordering ⪯ with respect to P is defined by if and only if . The notation of ≺ stands for but . Also, we used to indicate that , where intP denotes the interior of P. A cone P is called normal if there exists a number K such that
for all . The least positive number K satisfying the above condition is called the normal constant of P. Throughout this paper, we do not impose the normality condition for the cones, but the only assumption is that the cone P is solid, that is, .
Definition 1.1 [26]
Let X be a nonempty set and E be a real Banach space equipped with the partial ordering ⪯ with respect to the cone P. A vector-valued function is said to be a cone b-metric function on X with the constant if the following conditions are satisfied:
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1.
for all and if and only if ,
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2.
for all ,
-
3.
for all .
Then the pair is called a cone b-metric space (or a cone metric-type space); we will use the first mentioned term.
Observe that if then the ordinary triangle inequality in a cone metric space is satisfied; however, it does not hold true when . Thus the class of tvs-cone b-metric spaces is effectively larger than that of the ordinary cone metric spaces. That is, every cone metric space is a cone b-metric space, but the converse need not be true. The following examples illustrate the above remarks.
Example 1.2 [29]
Let , , . Define by for all , , and , . Then is a complete cone b-metric space, but the triangle inequality is not satisfied. Indeed, we have that . It is not hard to verify that .
The following example is a modification of Example 3 from [31].
Example 1.3 Let , and . Define by
Then is a cone b-metric space with the coefficient . But it is not a cone metric space since the triangle inequality is not satisfied. Indeed, .
Definition 1.4 [26]
Let be a cone b-metric space, and let be a sequence in X and .
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1.
For all with , if there exists a positive integer N such that for all , then is said to be convergent and x is the limit of . We denote this by .
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2.
For all with , if there exists a positive integer N such that for all , then is called a Cauchy sequence in X.
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3.
A cone metric space is called complete if every Cauchy sequence in X is convergent.
The following lemma is helpful in proving our results.
Lemma 1.5 [24]
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1.
If E is a real Banach space with a cone P and , where and , then .
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2.
If , and , then there exists a positive integer N such that for all .
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3.
If and , then .
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4.
If for each , then .
Recall the following definitions.
Definition 1.6 [32]
An element is said to be a coupled fixed point of the mapping if and .
Definition 1.7 [33]
An element is called
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1.
a coupled coincidence point of mappings and if and , and is called a coupled point of coincidence;
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2.
a common coupled fixed point of mappings and if and .
Definition 1.8 [25]
The mappings and are called w-compatible if whenever and .
2 Coupled coincidence point results
In this section, we prove some coupled coincidence point results in cone b-metric spaces.
Theorem 2.1 Let be a cone b-metric space with the coefficient relative to a solid cone P. Let and be two mappings and suppose that there exist nonnegative constants , , with and such that the following contractive condition holds for all :
If and is a complete subspace of X, then F and g have a coupled coincidence point .
Proof Choose . Set , , this can be done because . Continuing this process, we obtain two sequences , such that and . Then we have
So that
Hence,
Similarly, we can prove that
Put
Adding inequalities (2.1) and (2.2), one can assert that
On the other hand, we have
So that
Hence,
Similarly,
Adding inequalities (2.4) and (2.5), one can assert that
Finally, from (2.3) and (2.6), we have
that is,
where .
Consequently, we have
Let . It follows that
and
Now, (2.7) and imply that
According to Lemma 1.5(2), and for any with , there exists such that for any , . Furthermore, from (2.8) and for any , Lemma 1.5(3) shows that
which implies that
and
Hence, by Definition 1.4(2), and are Cauchy sequences in . Since is complete, there exist and such that and as .
On the other hand,
Hence,
Similarly,
Put
Adding the above inequalities, we get
Then
where , and . Since and as , then by Definition 1.4(1) and for , there exists such that for all , , , and . Hence,
Now, according to Lemma 1.5(4), it follows that , that is, , which implies that and . Hence, and . Therefore is a coupled coincidence point of F and g. □
Remark 2.2 Theorem 2.1 extends and generalizes Theorem 2.4 of Abbas et al. [25] to cone b-metric spaces.
Corollary 2.3 Let be a cone b-metric space with the coefficient relative to a solid cone P. Let and be two mappings and suppose that there exist nonnegative constants k, l with such that the following contractive condition holds for all :
If and is a complete subspace of X, then F and g have a coupled coincidence point .
Corollary 2.4 Let be a cone b-metric space with the coefficient relative to a solid cone P. Let and be two mappings and suppose that there exist nonnegative constants k, l with such that the following contractive condition holds for all :
If and is a complete subspace of X, then F and g have a coupled coincidence point .
Corollary 2.5 Let be a cone b-metric space with the coefficient relative to a solid cone P. Let and be two mappings and suppose that there exist nonnegative constants k, l with such that the following contractive condition holds for all :
If and is a complete subspace of X, then F and g have a coupled coincidence point .
Remark 2.6 All of the coupled coincidence point results may be proved for a partially ordered cone b-metric space by inserting well-known conditions like
-
(1)
F has the mixed g-monotone property and g is b-continuous;
and either
-
(a)
F is b-continuous, or
-
(b)
is regular.
3 Common coupled fixed point results
The conditions of Theorem 2.1 are not enough to prove the existence of a common coupled fixed point for the mappings F and g. By restricting to w-compatibility for F and g, we obtain the following theorem.
Theorem 3.1 In addition to the hypotheses of Theorem 2.1, if F and g are w-compatible, then F and g have a unique common coupled fixed point. Moreover, a common coupled fixed point of F and g is of the form for some .
Proof From Theorem 2.1, F and g have a coupled coincidence point . Then is a coupled point of coincidence of F and g such that and . First, we will show that the coupled point of coincidence is unique. Suppose that F and g have another coupled point of coincidence such that and , where . Then we have
Hence,
By a similar way, we can show that
By adding inequalities (3.1) and (3.2), we get
Since , Lemma 1.5(1) shows that . But and . Hence, and , that is,
which implies the uniqueness of the coupled point of coincidence of F and g, that is, . By a similar way, we can prove that
In view of (3.3) and (3.4), one can assert that
That is, the unique coupled point of coincidence of F and g is .
Now, let . Since F and g are w-compatible, then we have
Then is a coupled point of coincidence, and also we have is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that . Therefore . Hence, is the unique common coupled fixed point of F and g. This completes the proof. □
Now, we present one example to illustrate our results.
Example 3.2 Let and with , and let . It is well known that this cone is solid but it is not normal. Define a cone b-metric by . Then is a complete cone b-metric space with the coefficient . Let us define and as and for all . Now we obtain that
where , , , . Note that , and is a complete subspace of X. Hence, the conditions of Theorem 2.1 are satisfied, that is, F and g have a coupled coincidence point . Also, F and g are w-compatible at . Hence, Theorem 3.1 shows that is the unique common coupled fixed point of F and g.
Finally, we have the following result (immediate consequence of Theorems 2.1 and 3.1).
Theorem 3.3 Let be a complete cone b-metric space with the coefficient relative to a solid cone P. Let be a mapping and suppose that there exist nonnegative constants , , with and such that the following contractive condition holds for all :
Then F has a coupled fixed point . Moreover, the coupled fixed point is unique and of the form for some .
As consequences of Theorem 3.3, we have the following results which are the extension of main results of Sabetghadam et al. [34] to cone b-metric spaces.
Corollary 3.4 Let be a complete cone b-metric space with the coefficient relative to a solid cone P. Let be a mapping and suppose that there exist nonnegative constants k, l with such that the following contractive condition holds for all :
Then F has a coupled fixed point . Moreover, the coupled fixed point is unique and of the form for some .
Corollary 3.5 Let be a complete cone b-metric space with the coefficient relative to a solid cone P. Let be a mapping and suppose that there exist nonnegative constants k, l with such that the following contractive condition holds for all :
Then F has a coupled fixed point . Moreover, the coupled fixed point is unique and of the form for some .
Corollary 3.6 Let be a complete cone b-metric space with the coefficient relative to a solid cone P. Let be a mapping and suppose that there exist nonnegative constants k, l with such that the following contractive condition holds for all :
Then F has a coupled fixed point . Moreover, the coupled fixed point is unique and of the form for some .
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The authors would like to acknowledge the financial support received from Faculty of Science and Technology, Universiti Kebangsaan Malaysia. The authors thank the referee for his/her careful reading of the manuscript and useful suggestions.
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Fadail, Z.M., Ahmad, A.G.B. Coupled coincidence point and common coupled fixed point results in cone b-metric spaces. Fixed Point Theory Appl 2013, 177 (2013). https://doi.org/10.1186/1687-1812-2013-177
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DOI: https://doi.org/10.1186/1687-1812-2013-177