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Some common fixed point theorems for a family of non-self mappings in cone metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 144 (2013)
Abstract
Some common fixed point theorems for a family of non-self mappings defined on a closed subset of a metrically convex cone metric space (over the cone which is not necessarily normal) are obtained which generalize earlier results due to Imdad et al. and Janković et al.
MSC:47H10, 54H25.
1 Introduction and preliminaries
The existing literature of fixed point theory contains many results enunciating fixed point theorems for self-mappings in metric and Banach spaces. Recently, Huang and Zhang [1] have replaced the real numbers by ordering Banach space and defining cone metric space. They have proved some fixed point theorems of contractive mappings on cone metric spaces. The study of fixed point theorems in such spaces is followed by some other mathematicians; see [2–17]. However, fixed point theorems for non-self mappings are not frequently discussed and so they form a natural subject for further investigation. The study of fixed point theorems for non-self mappings in metrically convex metric spaces was initiated by Assad and Kirk [18]. Recently, Janković et al. [10] obtained a fixed point theorem for two non-self mappings in cone metric spaces. Motivated by Janković et al. [10], we prove some common fixed point theorems for a family of non-self mappings on cone metric spaces in which the cone need not be normal.
Consistent with Huang and Zhang [1], the following definitions and results will be needed in the sequel.
Let E be a real Banach space. A subset P of E is called a cone if and only if:
-
(a)
P is closed, nonempty and ;
-
(b)
, , implies ;
-
(c)
.
Given a cone , we define a partial ordering ⪯ with respect to P by if and only if . A cone P is called normal if there is a number such that for all ,
The least positive number K satisfying the above inequality is called the normal constant of P, while stands for (interior of P).
Definition 1.1 [1]
Let X be a nonempty set. Suppose that the mapping satisfies:
-
(d1)
for all and if and only if ;
-
(d2)
for all ;
-
(d3)
for all .
Then d is called a cone metric on X and is called a cone metric space.
The concept of a cone metric space is more general than that of a metric space.
Definition 1.2 [1]
Let be a cone metric space. We say that is:
-
(e)
a Cauchy sequence if for every with , there is an N such that for all , ;
-
(f)
a convergent sequence if for every with , there is an N such that for all , for some fixed .
A cone metric space X is said to be complete if for every Cauchy sequence in X, it is convergent in X. It is known that converges to if and only if as . It is a Cauchy sequence if and only if ().
Remark 1.1 [19]
Let E be an ordered Banach (normed) space. Then c is an interior point of P, if and only if is a neighborhood of θ.
Corollary 1.1 [9]
(1) If and , then .
Indeed, implies .
(2) If and , then .
Indeed, implies .
(3) If for each , then .
Remark 1.2 [11]
If , and , then there exists an such that for all , we have .
Remark 1.3 [11]
If E is a real Banach space with cone P and if , where and , then .
We find it convenient to introduce the following definition.
Definition 1.3 [11]
Let be a complete cone metric space, let C be a nonempty closed subset of X, and let be non-self mappings. Denote, for ,
Then f is called a generalized -contractive mapping of C into X if, for some , there exists such that for all ,
Definition 1.4 [2]
Let f and g be self-maps of a set X (i.e., ). 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. Self-maps f and g are said to be coincidentally commuting if they commute at their coincidence point; i.e., if for some , then .
2 Main results
Recently, Janković et al. [10] proved some fixed point theorems for a pair of non-self mappings defined on a nonempty closed subset of complete metrically convex cone metric spaces with new contractive conditions.
Theorem 2.1 [10]
Let be a complete cone metric space, let C be a nonempty closed subset of X such that for each and there exists a point (the boundary of C) such that
Suppose that are such that f is a generalized -contractive mapping of C into X, and
-
(i)
, ,
-
(ii)
,
-
(iii)
gC is closed in X.
Then the pair has a coincidence point. Moreover, if are coincidentally commuting, then f and g have a unique common fixed point.
The purpose of this paper is to extend the above theorem for a family of non-self mappings in cone metric spaces. We begin with the following definition.
Definition 2.1 Let be a complete cone metric space, let C be a nonempty closed subset of X, and let be non-self mappings. Denote, for ,
where , for some . Then is called a pair of generalized -contractive mappings of C into X if for some there exists such that for all with ,
Notice that by setting , and in (2.1), one deduces a slightly generalized form of (1.1).
We state and prove our main result as follows.
Theorem 2.2 Let be a complete cone metric space, let C be a nonempty closed subset of X such that for each and there exists a point such that
Suppose that are such that is a pair of generalized -contractive mappings of C into X for all , (), and
-
(I)
, , ,
-
(II)
implies that , implies that ,
-
(III)
SC and TC (or and ) are closed in X.
Then
-
(IV)
has a point of coincidence,
-
(V)
has a point of coincidence.
Moreover, if and are coincidentally commuting pairs, then , S and T have a unique common fixed point.
Proof Let be arbitrary. Then (due to ) there exists a point such that . Since , from (I) and (II), we have . Thus, there exists such that . Since , there exists a point such that
Suppose . Then , which implies that there exists a point such that . Otherwise, if , then there exists a point such that
Since , there exists a point with so that
Let be such that . Thus, repeating the foregoing arguments, one obtains two sequences and such that
-
(a)
, ,
-
(b)
implies that or implies that and
-
(c)
implies that or implies that and
We denote
Note that , as if , then and one infers that , which implies that . Hence . Similarly, one can argue that .
Now, we distinguish the following three cases.
Case 1. If , then from (2.2)
where
Clearly, there are infinitely many n such that at least one of the following three cases holds:
-
(1)
;
-
(2)
implies that ;
-
(3)
implies that .
From (1), (2), (3) it follows that
Similarly, if , we have
If , we have
Case 2. If , then and
which in turn yields
and hence
Now, proceeding as in Case 1, we have that (2.3) holds.
If , then . We show that
Using (2.6), we get
By noting that , one can conclude that
and
in view of Case 1.
Thus,
and we proved (2.9).
Case 3. If , then . We show that
Since , then
From this, we get
By noting that , one can conclude that
and
in view of Case 1.
Thus,
and we proved (2.13).
Similarly, if , then , and
From this, we have
This implies that .
By noting that , one can conclude that
in view of Case 1.
Thus, in all Cases 1-3, there exists such that
and there exists such that
Following the procedure of Assad and Kirk [18], it can easily be shown by induction that, for , there exists such that
From (2.19) and by the triangle inequality, for , we have
From Remark 1.2 and Corollary 1.1(1), .
Thus, the sequence is a Cauchy sequence. Then, as noted in [20], there exists at least one subsequence or which is contained in or , respectively, having as a limit point z. Furthermore, subsequences and both converge to as C is a closed subset of a complete cone metric space . We assume that there exists a subsequence for each , and TC as well as SC are closed in X. Since is a Cauchy sequence in TC, it converges to a point . Let , then . Similarly, being a subsequence of the Cauchy sequence also converges to z as SC is closed. Using (2.2), one can write
where
for any odd integer and even integer .
Let . Clearly at least one of the following four cases holds for infinitely many n.
-
(1)
;
-
(2)
;
-
(3)
-
(4)
In all cases, we obtain for each . Using Corollary 1.1(3), it follows that or . Thus, , that is, z is a coincidence point of , T for any odd integer .
Further, since the Cauchy sequence converges to and , , there exists such that . Again, using (2.2), we get
where
for any odd integer and even integer .
Hence, we get the following cases:
Using Remark 1.3 and Corollary 1.1(3), it follows that ; therefore, , that is, z is a coincidence point of for any even integer .
In case and are closed in X, then or . The analogous arguments establish (IV) and (V). If we assume that there exists a subsequence with TC as well as SC closed in X, then noting that is a Cauchy sequence in SC, foregoing arguments establish (IV) and (V).
Suppose now that and are coincidentally commuting pairs, then
Then, from (2.2),
where
Hence, we get the following cases:
Using Remark 1.3 and Corollary 1.1(3), it follows that . Thus, .
Similarly, we can prove . Therefore , that is, z is a common fixed point of , S and T.
The uniqueness of the common fixed point follows easily from (2.2). □
Example 2.1 Let , , , and defined by , where is a fixed function, e.g., . Then is a complete cone metric space with a non-normal cone having the nonempty interior. Define , , S and as
Since . Clearly, for each and , there exists a point such that . Further, , , , and SC, TC, and are closed in X.
Also,
Moreover, for each ,
that is, (2.2) is satisfied with .
Evidentially, and . Notice that two separate coincidence points are not common fixed points as and , which shows the necessity of coincidentally commuting property in Theorem 2.2.
Next, we furnish an illustrative example in support of our result. In doing so, we are essentially inspired by Imdad and Kumar [21].
Example 2.2 Let , , , and defined by , where is a fixed function, e.g., . Then is a complete cone metric space with a non-normal cone having the nonempty interior. Define , , S and as
Note that . Clearly, for each and , there exists a point such that . Further, , and .
Also,
Moreover, if and , then
Next, if , then
Finally, if , then
Therefore, condition (2.2) is satisfied if we choose . Moreover, 1 is a point of coincidence as as well as , whereas both the pairs and are weakly compatible as and . Also, SC, TC, and are closed in X. Thus, all the conditions of Theorem 2.2 are satisfied and 1 is the unique common fixed point of , , S and T. One may note that 1 is also a point of coincidence for both the pairs and .
Remark 2.1 Setting and in Theorem 2.2, we obtain the following result.
Corollary 2.1 Let be a complete cone metric space, let C be a nonempty closed subset of X such that for each and there exists a point such that
Suppose that are such that is a pair of generalized -contractive mappings of C into X, and
-
(I)
, , ,
-
(II)
implies that , implies that ,
-
(III)
SC and TC (or FC and GC) are closed in X.
Then
-
(IV)
has a point of coincidence,
-
(V)
has a point of coincidence.
Moreover, if and are coincidentally commuting pairs, then F, G, S and T have a unique common fixed point.
Remark 2.2 1. Theorem 2.2 in [10] is a special case of Theorem 2.2 with , and .
2. Setting and (the identity mapping on X) in Theorem 2.2, we obtain the following result.
Corollary 2.2 Let be a complete cone metric space, and let C be a nonempty closed subset of X such that for each and there exists a point such that
Suppose that satisfies the condition
where
for all , and f has the additional property that for each , . Then f has a unique fixed point.
Remark 2.3 The following definition is a special case of Definition 2.1 when is a metric space. But when is a cone metric space, which is not a metric space, this is not true. Indeed, there may exist such that the vectors , and are incomparable. For the same reason Theorems 2.2 and 2.3 (given below) are incomparable.
Definition 2.2 Let be a complete cone metric space, let C be a nonempty closed subset of X, and let be non-self mappings. Denote, for ,
where , for some . Then is called a pair of generalized -contractive mappings of C into X if for some there exists such that for all with ,
Our next result is the following.
Theorem 2.3 Let be a complete cone metric space, let C be a nonempty closed subset of X such that for each and there exists a point such that
Suppose that are such that is a pair of generalized -contractive mappings of C into X for all , (), and
-
(I)
, , ,
-
(II)
implies that , implies that ,
-
(III)
SC and TC (or and ) are closed in X.
Then
-
(IV)
has a point of coincidence,
-
(V)
has a point of coincidence.
Moreover, if and are coincidentally commuting pairs, then , S and T have a unique common fixed point.
The proof of this theorem is very similar to the proof of Theorem 2.2 and it is omitted.
Remark 2.4 Setting and in Theorem 2.3, we obtain the following result.
Corollary 2.3 Let be a complete cone metric space, let C be a nonempty closed subset of X such that for each and there exists a point such that
Suppose that are such that is a pair of generalized -contractive mappings of C into X, and
-
(I)
, , ,
-
(II)
implies that , implies that ,
-
(III)
SC and TC (or FC and GC) are closed in X.
Then
-
(IV)
has a point of coincidence,
-
(V)
has a point of coincidence.
Moreover, if and are coincidentally commuting pairs, then F, G, S and T have a unique common fixed point.
We now list some corollaries of Theorems 2.2 and 2.3.
Corollary 2.4 Let be a complete cone metric space, let C be a nonempty closed subset of X such that for each and there exists a point such that
Let be such that
for some and for all , (), with .
Suppose, further, that , S, T and C satisfy the following conditions:
-
(I)
, , ,
-
(II)
implies that , implies that ,
-
(III)
SC and TC (or and ) are closed in X.
Then
-
(IV)
has a point of coincidence,
-
(V)
has a point of coincidence.
Moreover, if and are coincidentally commuting pairs, then , S and T have a unique common fixed point.
Corollary 2.5 Let be a complete cone metric space, let C be a nonempty closed subset of X such that for each and there exists a point such that
Let be such that
for some and for all , (), with .
Suppose, further, that , S, T and C satisfy the following conditions:
-
(I)
, , ,
-
(II)
implies that , implies that ,
-
(III)
SC and TC (or and ) are closed in X.
Then
-
(IV)
has a point of coincidence,
-
(V)
has a point of coincidence.
Moreover, if and are coincidentally commuting pairs, then , S and T have a unique common fixed point.
Corollary 2.6 Let be a complete cone metric space, let C be a nonempty closed subset of X such that for each and there exists a point such that
Let be such that
for some and for all , (), with .
Suppose, further, that , S, T and C satisfy the following conditions:
-
(I)
, , ,
-
(II)
implies that , implies that ,
-
(III)
SC and TC (or and ) are closed in X.
Then
-
(IV)
has a point of coincidence,
-
(V)
has a point of coincidence.
Moreover, if and are coincidentally commuting pairs, then , S and T have a unique common fixed point.
Remark 2.5 Setting and in Corollaries 2.4-2.6, we obtain the following result.
Corollary 2.7 Let be a complete cone metric space, let C be a nonempty closed subset of X such that for each and there exists a point such that
Let be such that
for some and for all . Suppose, further, that f, g and C satisfy the following conditions:
-
(I)
, ,
-
(II)
implies that ,
-
(III)
gC is closed in X.
Then there exists a coincidence point z of f, g in C. Moreover, if are coincidentally commuting, then z is the unique common fixed point of f and g.
Corollary 2.8 Let be a complete cone metric space, let C be a nonempty closed subset of X such that for each and there exists a point such that
Let be such that
for some and for all . Suppose, further, that f, g and C satisfy the following conditions:
-
(I)
, ,
-
(II)
implies that ,
-
(III)
gC is closed in X.
Then there exists a coincidence point z of f, g in C. Moreover, if are coincidentally commuting, then z is the unique common fixed point of f and g.
Corollary 2.9 Let be a complete cone metric space, let C be a nonempty closed subset of X such that for each and there exists a point such that
Let be such that
for some and for all . Suppose, further, that f, g and C satisfy the following conditions:
-
(I)
, ,
-
(II)
implies that ,
-
(III)
gC is closed in X.
Then there exists a coincidence point z of f, g in C. Moreover, if are coincidentally commuting, then z is the unique common fixed point of f and g.
Remark 2.6 Corollaries 2.7-2.9 are the corresponding theorems of Abbas and Jungck from [2] in the case that f, g are non-self mappings.
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Acknowledgements
The authors would like to express their sincere appreciation to the referees for their very helpful suggestions and kind comments. Project is supported by the National Natural Science Foundation of China (11071108) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20114BAB201003) and the Science and Technology Project of Educational Commission of Jiangxi Province, China (GJJ11346).
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Huang, X., Zhu, C., Wen, X. et al. Some common fixed point theorems for a family of non-self mappings in cone metric spaces. Fixed Point Theory Appl 2013, 144 (2013). https://doi.org/10.1186/1687-1812-2013-144
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DOI: https://doi.org/10.1186/1687-1812-2013-144