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Common fixed points of ordered g-contractions in partially ordered metric spaces
Fixed Point Theory and Applications volume 2014, Article number: 28 (2014)
Abstract
The concept of ordered g-contraction is introduced, and some fixed and common fixed point theorems for g-nondecreasing ordered g-contraction mapping in partially ordered metric spaces are proved. We also show the uniqueness of the common fixed point in the case of an ordered g-contraction mapping. The theorems presented are generalizations of very recent fixed point theorems due to Golubović et al. (Fixed Point Theory Appl. 2012:20, 2012).
MSC:47H10, 47N10.
1 Introduction
Thenomark]The word “contraction” occurs throughout the article, where “contradiction” might be intended. See for instance below Eq. (21). Please check. Banach fixed point theorem for contraction mappings has been extended in many directions (cf. [1–48]). Very recently, Golubović et al. [49] presented some new results for ordered quasicontractions and ordered g-quasicontractions in partially ordered metric spaces.
Recall that if is a partially ordered set and is such that, for , implies , then a mapping f is said to be nondecreasing. The main result of Golubović et al. [49] is the following common fixed point theorem.
Theorem 1.1 (See [49], Theorem 1)
Let be a partially ordered metric space and let be two self-maps on X satisfying the following conditions:
-
(i)
;
-
(ii)
gX is complete;
-
(iii)
f is g-nondecreasing;
-
(iv)
f is an ordered g-quasicontraction;
-
(v)
there exists such that ;
-
(vi)
if is a nondecreasing sequence that converges to some , then for each and .
Then f and g have a coincidence point, i.e., there exists such that . If, in addition,
-
(vii)
f and g are weakly compatible [50, 51], i.e., implies , for each , then they have a common fixed point.
An open problem is to find sufficient conditions for the uniqueness of the common fixed point in the case of an ordered g-quasicontraction in Theorem 1.1.
In Section 2 of this article, we introduce ordered g-contractions in partially ordered metric spaces and prove the respective (common) fixed point results, which generalizes the results of Theorem 1.1.
In Section 3 of this article, a theorem on the uniqueness of a common fixed point is obtained when for all , there exists such that fa is comparable to fx and fu, in addition to the hypotheses in Theorem 2.1 of Section 2. Our result is an answer to finding sufficient conditions for the uniqueness of the common fixed point in the case of ordered g-contractions in Theorem 1.1. Finally, two examples show that the comparability is a sufficient condition for the uniqueness of common fixed point in the case of ordered g-contractions, so our results are extensions of known ones.
2 Common fixed points of ordered g-contractions
We start this section with the following definitions. Consider a partially ordered set and two mappings and such that .
Definition 2.1 (See [1])
We shall say that the mapping f is g-nondecreasing (resp., g-nonincreasing) if
(resp., ) holds for each .
Definition 2.2 (See [49])
We shall say that the mapping f is an ordered g-quasicontraction if there exists such that for each satisfying , the inequality
holds, where
Definition 2.3 We shall say that the mapping f is an ordered g-contraction if there is a continuous and nondecreasing function with and if there exists , the inequality
holds for all for which .
It is obviously that if , then ordered g-contraction reduces to ordered g-quasicontraction.
For arbitrary one can construct a so-called Jungck sequence in the following way: denote ; there exists such that ; now and there exists such that and the procedure can be continued.
Theorem 2.1 Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Let be two self-maps on X satisfying the following conditions:
-
(i)
;
-
(ii)
is closed;
-
(iii)
f is a g-nondecreasing mapping;
-
(iv)
f is an ordered g-contraction;
-
(v)
there exists an with ;
-
(vi)
is a nondecreasing sequence with in , then , , ∀n hold.
Then f and g have a coincidence point. Further, if f and g are weakly compatible, then f and g have a common fixed point.
Proof Let be such that . Since , we can choose such that . Again from , we can choose such that . Continuing this process we can choose a sequence in X such that
Since and , we have . Then by (1),
Thus, by (3), . Again by (1),
that is, . Continuing this process, we obtain
Now, we will claim that is a Cauchy sequence. In what follows, we will suppose that for all n, since if for some n, by (3),
that is, f and g have a coincidence at , and so we have finished the proof. Thus we assume that for all n. We will show that
From (3) and (6), it follows that for all . Then apply the contractivity condition (2) with and ,
Thus by (3),
We divide the proof of (8) into three cases in the following:
-
(I)
If , from (10), then
(11)
Since ψ is nondecreasing, . By virtue of , it follows that . Thus (8) holds.
(II) If , from (10), then
Since ψ is nondecreasing, . By virtue of , , and it is a contraction with the assumption that for all n!
-
(III)
If , from (10) and the triangle inequality, we have
(13)
Since ψ is nondecreasing,
Then it follows that
Thus (8) trivially holds when . Hence
Taking into account the previous considerations, we proved that (8) holds. From (8), it follows that the sequence of real non-negative numbers is monotone nonincreasing. Therefore, there exists some such that
Next we will prove that . We suppose that . By the triangle inequality,
Hence, by (8),
Taking the upper limit as , we get
Set
Then it follows that . Now, taking the upper limit on the both sides of (10) and being continuous, we get
From (15) and (19),
If , from (21), it yields . Since ψ is nondecreasing, then . When , then , it is a contradiction! When , then , it is a contradiction! When , then , it yields . Since and , it is also a contradiction!
If , then from (21), it yields . Since ψ is nondecreasing, then . By virtue of , it yields σ. It is a contradiction with the assumption that !
Taking into account the previous consideration, . Therefore, we proved that
Now, we prove that is a Cauchy sequence. Suppose, to the contrary, that is not a Cauchy sequence. Then there exist an and two sequences of integers , , with
We may also assume
by choosing to be the smallest number which satisfies , and (23) holds. From (23), (24), and by the triangle inequality,
Hence, by (22),
Since from (3) and (6), we have , from (2) and (3) with and , we get
Denote . Then we have
where the first equality holds, since ψ is nondecreasing, and for all . Again, since ψ is nondecreasing, by (28), it follows that
Therefore, since
we have
By the triangle inequality, (23), and (24),
From the equality of (31) and the last inequality of (31), it yields
Similarly, we obtain
And it follows that
Adding the two inequalities above,
From (32) and (34), we have
Thus from (22) and (35), we get
Letting in (30), then by (22), (26), and (36), we get, as ψ is continuous,
it is a contradiction! Thus our assumption (23) is wrong. Therefore, is a Cauchy sequence. Since by (3), we have and is closed, there exists such that
Now we show that z is a coincidence point of f and g. Since from condition (vi) and (39), we have for all n, then by the triangle inequality and (2), we have
So letting , and ψ being continuous, we have
Since ψ is nondecreasing, then . Since , it follows that . Hence . Thus we proved that f and g have a coincidence point.
Suppose now that f and g commute at z. Set . Since f and g are weakly compatible,
Since from condition (vi), we have and as and , from (2), we have
Since ψ is nondecreasing, , i.e., . Again from , , that is, . Therefore,
Thus, we have proved that f and g have a common fixed point. The proof is completed. □
If we replace some conditions in Theorem 2.1, then we can obtain the following conclusions. Note that the way followed in Theorem 2.2 is different from that in the proof of Theorem 2.1. In fact, we can use the way in Theorem 2.2 to prove the conclusions in Theorem 2.1. Similarly, we can also use the way in Theorem 2.1 to prove Theorem 2.2. Here, our aim is to show two different methods of proof. Comparing Theorem 2.1 with Theorem 2.2, we can find that the conclusions cover Theorem 2.2; in other words, the condition of Theorem 2.2 is more extensive than that in Theorem 2.1. Now, let us treat the following theorem.
Theorem 2.2 Let the conditions of Theorem 2.1 be satisfied, except that (iii), (v) and (vi) are, respectively, replaced by
(iii′) f is a g-nonincreasing mapping;
(v′) there exists such that and are comparable;
(vi′) if is a sequence in which has comparable adjacent terms and that converges to some , then there exists a subsequence of having all the terms comparable with gz and gz is comparable with . Then all the conclusions of Theorem 2.1 hold.
Proof Regardless of whether or (condition (v′)), Lemma 1 of [49] implies that two arbitrary adjacent terms of the Jungck sequence are comparable. This is again sufficient to imply that is a Cauchy sequence. In the following, we assume the other case to prove the conclusions of Theorem 2.2.
Let be such that , where it is different from in Theorem 2.1. Since , we can choose such that . Again from , we can choose such that . Continuing this process, we can choose a sequence in X such that
Since and , we have . Then by condition (iii′), f is a g-nonincreasing mapping,
Thus, by (43), it follows that . Again by condition (iii′),
that is, . Continuing this process, we obtain the result that two arbitrary adjacent terms of the Jungck sequence are comparable.
Let . We will show that is a Cauchy sequence. To prove our claim, we follow the arguments of Das and Naik [12] again. Fix and . If , then is also a Cauchy sequence. Thus our claims holds. Now we suppose that . Now for i, j with , by (2), we have
where the third equality holds, since ψ is nondecreasing, and for all . Since ψ is nondecreasing,
Now for some i, j with , . If , by (2) and (46), then we have
where the inequality (47) holds as . Then from (47) and , we have . It is a contradiction with the assumption that ! Thus,
Also, by (46) and (48), we have
Using the triangle inequality, by (46), (48), and (49), we obtain
and so
As a result, we have
where the first inequality holds by the expression (49) and the last inequality holds by (51). Now let ; there exists an integer such that
For , we have
Therefore, is a Cauchy sequence. Since is closed, it converges to some .
By condition (vi′), there exists a subsequence , , having all the terms comparable with gz. Hence, we can apply the contractivity condition to obtain
So letting , and as ψ is continuous, we have
Since ψ is nondecreasing, . Since , it follows that . Hence . Thus we proved that f and g have a coincidence point.
Suppose now that f and g commute at z. Set . Since f and g are weakly compatible,
Since from condition (vi′), we have and as and , from (2), we have
Since ψ is nondecreasing, , i.e., . Again from , we have , that is, . Therefore,
Thus, we have proved that f and g have a common fixed point. The proof is completed. □
Corollary 2.1 (a) Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Let be a nondecreasing self-mapping such that for some
for all for which . Suppose also that either
-
(i)
is a nondecreasing sequence with in X, then , ∀n holds, or
-
(ii)
f is continuous.
If there exists an with , then f has a fixed point.
(b) The same holds if f is nonincreasing; there exists comparable with and (i) is replaced by
(i′) if a sequence converging to some has every two adjacent terms comparable, then there exists a subsequence having each term comparable with x.
Proof (a) If (i) holds, then take and (I= the identity mapping) in Theorem 2.1.
If (ii) holds, then from (3) with , we get
(b) Let u be the limit of the Picard sequence and let be a subsequence having all the terms comparable with u. Then we can apply the contractivity condition in the (a) term to obtain
Letting , we have
It follows that . Therefore, .
Note also that instead of the completeness of X, its f-orbitally completeness is sufficient to obtain the conclusion of the corollary. The proof is completed. □
3 Uniqueness of common fixed point of f and g
The following theorem gives the sufficient condition for the uniqueness of the common fixed point of f and g in the case of ordered g-contractions in partially ordered metric spaces.
Theorem 3.1 In addition to the hypotheses of Theorem 2.1, suppose that for all , there exists such that
Then f and g have a unique common fixed point x such that
Proof Since a set of common fixed points of f and g is not empty due to Theorem 2.1, assume now that x and u are two common fixed points of f and g, i.e.,
We claim that .
By the assumption, there exists such that fa is comparable to fx and fu. Define a sequence such that and
Further, set and and in the same way, define and such that
Since is comparable to , without loss of generality, we assume that , i.e., ; then it is easy to show
Since f is g-nondecreasing, we obtain . Since , it follows that , i.e., . Recursively, we get
By (66), we have
By the proof of Theorem 2.1, we find that is a convergent sequence, and there exists such that . Letting in (67), we obtain
Therefore, it yields
Hence
Similarly, we can also show that
Apply the contractivity condition, we obtain
Therefore, it yields
Hence
Combining (68) with (69), we obtain . It follows that
The proof is completed. □
Remark 3.1 Theorem 3.1 can be considered as an answer to Theorem 3 in [49]. We find the sufficient conditions for the uniqueness of the common fixed point in the case of an ordered g-quasicontraction. In this paper, condition (vi) in Theorem 2.1 is weaker than that ordered g-quasicontraction in [49]. When (I= the identity mapping), our condition (vi) reduces to an ordered g-quasicontraction in [49].
Example 3.1 Let , let if and only if and , and let d be the Euclidean metric. We define the functions as follows:
Let be given by
The only comparable pairs of points in X are for and then the contractivity condition (2) reduces to , and condition (iv) of Theorem 2.1 is trivially fulfilled. The other conditions of Theorem 2.1 are also satisfied. It is obvious that for and , is not comparable to , i.e., comparability in Theorem 3.1 is not satisfied. In fact, f and g have two common fixed points and , since
Example 3.2 Let with the usual metric , for all . Let and be given by
Let be given by
It is easy to check that all the conditions of Theorem 2.1 are satisfied. We have
and this holds when and , i.e, , i.e., . This means that the contractivity condition (2) holds when .
In addition, , there exists such that is comparable to and . So, all the conditions of Theorem 3.1 are satisfied.
By applying Theorem 3.1, we conclude that f and g has a unique common fixed point. In fact, f and g has only one common fixed point. It is .
Example 3.3 Let be the closed interval with usual metric and let and be mappings defined as follows:
Let x, y in X be arbitrary. We say that if . For any such that , we have
Since and , f is nondecreasing at , then . By the definition of ψ, we have
and
Since for all , it follows that
Thus we have
where the last inequality holds whenever . Therefore, f and g satisfy (2). Also it is easy to see that the mappings possess all properties in Definition 2.3, as well as hypotheses (v), (vi), and (vii) in Theorem 2.1. Thus we can apply Theorem 2.1 and Theorem 3.1.
On the other hand, for and each the contractive condition in Theorem 1 and Theorem 2 of Golubović, Kadelburg and Radenović [49]:
where and
is not satisfied. Indeed,
Thus, for any fixed λ; , we have, for and each with ,
Thus, f does not satisfy the contractive condition in Definition 2.2. Therefore, the theorems of Jungck and Hussain [52], Al-Thagafi and Shahzad [53] and Das and Naik [54] also cannot be applied.
Author’s contributions
The author read and approved the final manuscript.
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Acknowledgements
This work is partially supported by National Natural Science Foundation of China (Grant No. 11126336), the Scientific Research Fund of Sichuan Provincial Education Department (14ZB0208), Scientific Research Fund of Sichuan University of Science and Engineering (2012KY08).
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Liu, Xl. Common fixed points of ordered g-contractions in partially ordered metric spaces. Fixed Point Theory Appl 2014, 28 (2014). https://doi.org/10.1186/1687-1812-2014-28
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DOI: https://doi.org/10.1186/1687-1812-2014-28