Abstract
In this paper, we prove a common fixed point theorem for two pairs of non-self-mappings satisfying the generalized contraction condition of Ćirić type in cone metric spaces. Our result generalizes and extends some recent results related to non-self-mappings in the setting of cone metric spaces.
MSC:47H10, 54H25.
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1 Introduction and preliminaries
Recently, Huang and Zhang [1] generalized the concept of a metric space, replacing the set of real numbers by ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. Subsequently, the study of fixed point theorems in such spaces is followed by some other mathematicians; see [2–24]. The study of fixed point theorems for non-self-mappings in metrically convex metric spaces was initiated by Assad and Kirk [25]. Utilizing the induction method of Assad and Kirk [25], many authors like Assad [26], Ćirić [27–36], Ćirić et al. [37–42], Kumam et al. [43], Hadžić [44], Hadžić and Gajić [45], Imdad and Kumar [46], Rhoades [47, 48] have obtained common fixed point in metrically convex spaces. Recently, Ćirić and Ume [49] defined a wide class of multi-valued non-self-mappings which satisfy a generalized contraction condition and proved a fixed point theorem which generalizes the results of Itoh [50] and Khan [51].
Very recently, Radenović and Rhoades [15] extended the fixed point theorem of Imdad and Kumar [46] for a pair of non-self-mappings to non-normal cone metric spaces. Huang et al. [6] proved a fixed point theorem for four non-self-mappings in cone metric spaces which generalizes the result of Radenović and Rhoades [15]. Janković et al. [9] proved new common fixed point results for a pair of non-self-mappings defined on a closed subset of metrically convex cone metric space which is not necessarily normal by adapting Assad-Kirk’s method. Sumitra et al. [20] generalized the fixed point theorems of Ćirić and Ume [49] for a pair of non-self-mappings to non-normal cone metric spaces. In the same time, Sumitra et al.’s [20] results extended the results of Radenović and Rhoades [15] and Janković et al. [9]. The aim of this paper is to prove a common fixed point theorem for two pairs of non-self-mappings on cone metric spaces in which the cone need not be normal and the condition is weaker. This result generalizes the result of Sumitra et al. [20] and Huang et al. [6].
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 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.
Example 1.1 [1]
Let , , , and be such that , where is a constant. Then is a cone 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 is convergent in X. It is well known that converges to if and only if as . It is a Cauchy sequence if and only if ().
Remark 1.1 [52]
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 [16]
-
(1)
If and , then .
Indeed, implies .
-
(2)
If and , then .
Indeed, implies .
-
(3)
If for each then .
If , , and , then there exists an such that for all we have .
If E is a real Banach space with cone P and if where and , then .
Definition 1.3 [2]
Let f and g be self-maps on 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 weakly compatible if they commute at their coincidence point; i.e., if for some , then .
2 Main results
The following theorem is Sumitra et al.’s [20] generalization of Ćirić and Ume’s [49] result in cone metric spaces.
Theorem 2.1 Let be a complete cone metric space, C a nonempty closed subset of X such that for each and there exists a point (the boundary of C) such that
Suppose that are two non-self-mappings satisfying, for all with ,
where , , and α, β, γ are nonnegative real numbers such that
Also assume that
-
(i)
, ,
-
(ii)
implies that ,
-
(iii)
fC is closed in X.
Then the pair has a coincidence point in C. Moreover, if the pair is weakly compatible, then f and g have a unique common fixed point in C.
Remark 2.1 From the proof of this theorem, it is easy to see that condition (2.2) can be weakened to .
The purpose of this paper is to extend the above theorem for two pairs of non-self-mappings in cone metric spaces with a weaker condition.
We state and prove our main result as follows.
Theorem 2.2 Let be a complete cone metric space, C a nonempty closed subset of X such that for each and there exists a point such that
Suppose that are two pairs of non-self-mappings satisfying, for all with ,
where , , and α, β, γ are nonnegative real numbers such that
Also assume that
-
(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 weakly compatible pairs, then F, G, S, and T have a unique common fixed point.
Proof Firstly, we proceed to construct two sequences and in the following way.
Let be arbitrary. Then (due to ) there exists a point such that . Since implies that , one concludes that . 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 (2.3)
where
Clearly, there are infinitely many n such that at least one of the following four cases holds:
-
(1)
If and , then
It implies that .
-
(2)
If and , then
It implies that .
-
(3)
If and , then
It implies that .
-
(4)
If and , then
It implies that .
From (1), (2), (3), (4) it follows that
where by (2.4).
Similarly, if , we have
If , we have
Case 2. If , then and
which in turns yields
and hence
Now, proceeding as in Case 1, we see that (2.5) holds.
If , then . We show that
Using (2.8), we get
By noting that , one can conclude that
and
in view of Case 1.
Thus,
and we proved (2.11).
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 (2.15).
Similarly, if , then , and
From this, we have
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 [25], it can easily be shown by induction that, for , there exists such that
From (2.21) and by the triangle inequality, for we have
From Remark 1.2 and Corollary 1.1(1), it follows that .
Thus, the sequence is a Cauchy sequence. Then, as noted in [45], there exists at least one subsequence, or , which is contained in or , respectively, and one finds its limit . Furthermore, subsequences and both converge to as C is a closed subset of complete cone metric space . We assume that there exists a subsequence for each , then Since TC as well as SC are closed in X and is Cauchy in TC, it converges to a point . Let , then . Similarly, and being a subsequence of a Cauchy sequence, also converges to z as SC is closed. Let , then , where α, β, γ are nonnegative real numbers with .
Using (2.3), one can write
where
Let . Clearly at least one of the following four cases holds for infinitely many n:
-
(1)
If and , then
-
(2)
If and , then
-
(3)
If and , then
-
(4)
If and , then
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 F, T.
Further, since Cauchy sequence converges to and , , there exists such that . Again using (2.3), we get
where
Hence, we get the following cases:
Since , using Remark 1.3 and Corollary 1.1(3), it follows that , therefore, , that is, z is a coincidence point of .
In case FC and GC 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 SC are closed in X, then noting that is a Cauchy sequence in SC, the foregoing arguments establish (IV) and (V).
Suppose now that and are weakly compatible pairs, then
Then, from (2.3),
where
Hence, we get the following cases:
Since , 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 F, G, S, and T.
Uniqueness of the common fixed point follows easily from (2.3). □
The following example shows that in general F, G, S, and T satisfying the hypotheses of Theorem 2.2 need not have a common coincidence justifying the two separate conclusions (IV) and (V).
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 a nonempty interior. Define F, G, S, and as
Since . Clearly, for each and there exists a point such that . Further, , , , and, SC, TC, FC, and GC are closed in X.
Also,
Moreover, for each ,
that is, (2.3) is satisfied with , .
Obviously, and . Notice that two separate coincidence points are not common fixed points as and , which shows the necessity of the weakly compatibility property in Theorem 2.2.
Next, we furnish an illustrate example in support of our result. In doing so, we are essentially inspired by Imdad and Kumar [46].
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 F, G, S, and as
Since . 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 (2.3) is satisfied if we choose , . Moreover, 1 is a point of coincidence as as well as , whereas both pairs and are weakly compatible as and . Also, SC, TC, FC, and GC are closed in X. Thus, all the conditions of Theorem 2.2 are satisfied and 1 is the unique common fixed point of F, G, S, and T. One may note that 1 is also a point of coincidence for both pairs and .
Remark 2.2 1. Setting and in Theorem 2.2, one deduces Theorem 2.1 due to Sumitra et al. [20] with weaker condition.
-
2.
Setting and in Theorem 2.2, we obtain the following result.
Corollary 2.1 Let be a complete cone metric space, and C a nonempty closed subset of X such that for each and there exists a point such that
Suppose that satisfies, for all with ,
where
and α, β, γ are nonnegative real numbers such that and g has the additional property that if, for each , , then g has a unique fixed point in C.
Corollary 2.2 Let be a complete cone metric space, C 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 F, G, S, T, and C satisfy the following conditions:
-
(I)
, , ,
-
(II)
, ,
-
(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 weakly compatible pairs, then F, G, S, and T have a unique common fixed point.
Corollary 2.3 Let be a complete cone metric space, C 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 F, G, S, T, and C satisfy the following conditions:
-
(I)
, , ,
-
(II)
, ,
-
(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 weakly compatible pairs, then F, G, S, and T have a unique common fixed point.
Corollary 2.4 Let be a complete cone metric space, C 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 F, G, S, T, and C satisfy the following conditions:
-
(I)
, , ,
-
(II)
, ,
-
(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 weakly compatible pairs, then F, G, S, and T have a unique common fixed point.
Remark 2.3 Setting and in Corollaries 2.2-2.4, we obtain the following results.
Corollary 2.5 Let be a complete cone metric space, C 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)
,
-
(III)
gC is closed in X.
Then the pair has a coincidence point in C. Moreover, if the pair is weakly compatible, then f and g have a unique common fixed point in C.
Corollary 2.6 Let be a complete cone metric space, C a nonempty closed subset of X such that for each and there exists a point (the boundary of C) such that
Let be such that
for some and for all . Suppose, further, that f, g, and C satisfy the following conditions:
-
(I)
, ,
-
(II)
,
-
(III)
gC is closed in X.
Then the pair has a coincidence point in C. Moreover, if the pair is weakly compatible, then f and g have a unique common fixed point in C.
Corollary 2.7 Let be a complete cone metric space, C a nonempty closed subset of X such that for each and there exists a point (the boundary of C) such that
Let be such that
for some and for all . Suppose, further, that f, g, and C satisfy the following conditions:
-
(I)
, ,
-
(II)
,
-
(III)
gC is closed in X.
Then the pair has a coincidence point in C. Moreover, if the pair is weakly compatible, then f and g have a unique common fixed point in C.
Remark 2.4 Corollaries 2.5-2.7 are the corresponding theorems of Abbas and Jungck from [2] in the case that f, g are non-self-mappings.
There have been a number of papers written about fixed points for non-self-maps. One of the most general papers involving two maps is that in [30]. For cone metric spaces, the four-function analog of this result would have the contractive condition.
Suppose that are two pairs of non-self-mappings satisfying, for all with ,
where , a is a positive real number satisfying and .
Note that if F, G, S, and T satisfy condition (2.28), then F, G, S, and T satisfy condition (2.3), but the implication is not reversible.
Indeed, there are four cases to consider:
-
(1)
If in (2.28), then . So setting , in (2.3), it follows that .
-
(2)
If in (2.28), then . So setting , , in (2.3), it follows that , where .
-
(3)
If in (2.28), then . So setting , , in (2.3), it follows that , where .
-
(4)
If in (2.28), then . So setting , , in (2.3), it follows that , where .
Therefore, in all cases we find that F, G, S, and T satisfy condition (2.28), then F, G, S, and T satisfy condition (2.3).
Now, we give an example to show that condition (2.3) is more general than condition (2.28) above.
Example 2.3 Let , , , , and let be defined by , where is a fixed function, e.g., . Then is a complete cone metric space with a non-normal cone having a nonempty interior. Define F, G, S, and as
Note that for all with ,
Therefore, condition (2.3) is satisfied if we choose , .
Next, we shall see that the inequality (2.28) is not satisfied for all and by taking and .
Indeed, and .
Since and , for all possible cases of w, , and , that is, (2.3) is more general than (2.28).
So, we can obtain the following corollary.
Corollary 2.8 Let be a complete cone metric space, C a nonempty closed subset of X such that for each and there exists a point such that
Suppose that are two pairs of non-self-mappings satisfying, for all with ,
where , a is a positive real number satisfying and . Also assume that
-
(I)
, , ,
-
(II)
, ,
-
(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 weakly compatible pairs, then F, G, S, and T have a unique common fixed point.
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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 was supported by the National Natural Science Foundation of China (11361042 and 11326099) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20142BAB201005) and the Science and Technology Project of Educational Commission of Jiangxi Province, China (GJJ11294).
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Huang, X., Luo, J., Zhu, C. et al. Common fixed point theorem for two pairs of non-self-mappings satisfying generalized Ćirić type contraction condition in cone metric spaces. Fixed Point Theory Appl 2014, 157 (2014). https://doi.org/10.1186/1687-1812-2014-157
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DOI: https://doi.org/10.1186/1687-1812-2014-157