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Common fixed points of a generalized ordered g-quasicontraction in partially ordered metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 53 (2013)
Abstract
The concept of a generalized ordered g-quasicontraction is introduced, and some fixed and common fixed point theorems for a g-nondecreasing generalized ordered g-quasicontraction mapping in partially ordered complete metric spaces are proved. We also show the uniqueness of the common fixed point in the case of a generalized ordered g-quasicontraction mapping. Finally, we prove fixed point theorems for mappings satisfying the so-called weak contractive conditions in the setting of a partially ordered metric space. Presented theorems 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
The Banach fixed point theorem for contraction mappings has been extended in many directions (cf. [1–15]). Very recently Golubović et al. [16] presented some new results for ordered quasicontractions and 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 non-decreasing. The main result of Golubović et al. [16] is the following fixed point theorem.
Theorem 1.1 (See [16], 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 [17, 18], 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 generalized ordered g-quasicontractions in partially ordered metric spaces and prove the respective (common) fixed point theorems which generalize the results of Theorem 1.1.
In Section 3 of this article, the uniqueness of a common fixed point theorem 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 results are an answer to finding sufficient conditions for the uniqueness of a common fixed point in the case of an ordered g-quasicontraction in Theorem 1.1. Finally, two examples show that the comparability is a sufficient condition for the uniqueness of a common fixed point in the case of an ordered g-quasicontraction, so our results are extensions of known ones.
In Section 4 of this article, we consider weak contractive conditions in the setting of a partially ordered metric space and prove respective common fixed point theorems.
2 Common fixed points of a generalized ordered g-quasicontraction
We start this section with the following definitions. Consider a partially ordered set and two mappings and such that .
Definition 2.1 (See [19])
We will say that the mapping f is g-nondecreasing (resp., g-nonincreasing) if
(resp., ) holds for each .
Definition 2.2 (See [16])
We will 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 will say that the mapping f is a generalized ordered g-quasicontraction if there is a continuous and non-decreasing function with for each , for and there exists
for all for which ;
It is obvious that if , then a generalized ordered g-quasicontraction reduces to an ordered g-quasicontraction.
For arbitrary , one can construct the 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 a generalized ordered g-quasicontraction;
-
(v)
there exists an with ;
-
(vi)
is a non-decreasing 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 so that . Again from , we can choose such that . Continuing this process, we can construct a Jungck 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
Let . We will claim that is a Cauchy sequence. To prove our claim, we follow the arguments of Das and Naik [20]. Fix and . If , then , which yields that is a constant sequence and also a Cauchy sequence. Then our claims holds. Thus we suppose that . Now, for i, j with , by (2), we have
and so
Now, for some i, j with , . If by (2) and (7), then we have
It follows that , as , then . It is a contradiction! Thus,
Also, by (7) and (9), we have
Using the triangle inequality, by (7), (9) and (10), we obtain that
and so
As a result, we have
Now let , there exists an integer such that
For , we have
Since as , then . Therefore, is a Cauchy sequence.
Since by (3) we have and is closed, then there exists such that
Now we show that z is a coincidence point of f and g. Since from condition (iv) and (9) we have for all n, then by the triangle inequality and (2), we have that
So, letting yields . Hence , hence , which yields . Thus we have proved that f and g have a coincidence point.
Suppose now that f and g commute at z. Set . Then
Since from (vi) we have that and as and , from (2) we have that
Hence, , that is, . Therefore,
Thus, we have proved that f and g have a common fixed point. □
Accordingly, we can also obtain the results similar to Theorem 2 in [16].
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 [16] implies that the adjacent terms of the Jungck sequence are comparable. This is again sufficient to imply that is a Cauchy sequence. Hence, it converges to some .
By (vi′), there exists a subsequence , , having all the terms comparable with gz. Hence, we can apply the contractive condition to obtain
Letting , it yields that , then . Thus . It follows that . The rest of conclusions follow in the same way as in Theorem 2.1. □
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-map such that for some
for all for which . Suppose also that either
-
(i)
is a non-decreasing sequence with in X, then , ∀n hold, 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 (16) 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 to obtain
Letting , we have that
It follows that . Thus as . Therefore, .
Note also that instead of the completeness of X, its f-orbitally completeness is sufficient to obtain the conclusion of the corollary. □
3 Uniqueness of a common fixed point of f and g
The following theorem gives the sufficient condition for the uniqueness of a common fixed point of f and g.
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.
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 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 fx () is comparable to fa () and f is g-nondecreasing, it is easy to show
Recursively, we can get that
By (27), we have that
By the proof of Theorem 2.1, we obtain that is a convergent sequence, and there exists such that . Letting in (28) and ψ is continuous, we can obtain that
Therefore, we obtain
Since as , then and hence
Similarly, we can show that
Therefore, we obtain
Since as , then and hence
Thus, from (29) and (30), we have . It follows that
It means that x is the unique common fixed point of f and g. □
Remark 3.1 Theorem 3.1 can be considered as an answer to Theorem 3 in [16]. 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 the ordered g-quasicontraction in [16]. When (I= the identity mapping), our condition (vi) reduces to the ordered g-quasicontraction in [16].
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
Obviously, for and , but is not comparable to . However, 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
for all . Let be given by
It is easy to check that all the conditions of Theorem 2.1 are satisfied.
It holds when and , i.e., , i.e., .
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 have a unique common fixed point. In fact, f and g have only one common fixed point. It is .
4 Weak ordered contractions
We denote by Ψ the set of functions satisfying the following hypotheses:
() ψ is continuous and nondecreasing,
() if and only if .
We denote by Φ the set of functions satisfying the following hypotheses:
() for all ,
() if and only if .
Let be a metric space and let . In the article [16] (in the setting of partially ordered metric spaces), the authors obtained contractive conditions of the form
where
We will use here the following more general contractive condition:
We begin with the following result.
Theorem 4.1 Let be a partially ordered metric space and let f and g be self-mappings of X satisfying the following conditions:
-
(i)
;
-
(ii)
is complete;
-
(iii)
f is g-nondecreasing;
-
(iv)
f and g satisfy the following condition:
(35)for all such that , where , and
(36)Suppose that, in addition,
-
(v)
is nondecreasing;
-
(vi)
for each ;
-
(vii)
;
-
(viii)
there exists such that ;
-
(ix)
if is a nondecreasing sequence that converges to some , then for each and .
Then f and g have a coincidence point. If, in addition,
-
(x)
f and g are weakly compatible, then they have a common fixed point.
Further, if
-
(xi)
for arbitrary , there exists such that is comparable to fv and fw, then f and g have a unique common fixed point.
Proof As in the proof of Theorem 2.1, we can construct a nondecreasing Jungck sequence with
for all . Denote
We will prove that the Jungck sequence is bounded, that is,
for some . Let be any fixed positive integer and let for some i, j with . We will show that
Since , , and , then from (35) we have
where
Since , then
So, from (v),
Hence from (41) we obtain (40).
Note that , and so from (40),
Now we will show that if , then , that is,
Suppose, to the contrary, that . Then and hence we conclude that
This contradicts (42). Therefore, and so we have proved (43).
We will prove that the Jungck sequence is bounded. From (43) it follows that for some . By the triangle inequality,
Now, from () and (), we get
Since and as , from (35) we have
where
Clearly, . Thus by (v), we get
Now, by (44),
Hence
Since , the sequence is nondecreasing, and so there exists its limit , which is finite or infinite. Suppose that . Then (vii) implies that the left-hand side of (45) becomes unbounded when n tends to infinity, but the right-hand side is bounded, a contradiction. Therefore, . Thus we have proved (39).
Now we show that is a Cauchy sequence. For all , set similarly as in (38),
Clearly, and so . Therefore, is the monotone decreasing sequence of finite nonnegative numbers and converges to some .
We will prove that . Let and . Since , from (35),
where
Since , we conclude that , and so by (v), we get
Since and ψ is continuous, we have . Thus, taking the limit in (46) when , we get
Suppose that . Since as , then from (), we have . Therefore, taking the limits as in (47) and using the continuity of ψ, we get
a contradiction. Therefore, and so we have proved that
Hence we conclude that is a Cauchy sequence.
Since , by the assumption (ii) that is complete, there is some such that
We show that . Suppose, to the contrary, that . Condition (ix) implies that and we can apply the contractive condition (35) to obtain
where
By the triangle inequality,
Now, from () and (),
Hence from (48) we have
Since , for large enough n, we have
If , then from (49)
Letting n tend to infinity and using the continuity of ψ, we get
Hence , a contradiction with () and the assumption .
Similarly, if , then from (48)
Letting n tend to infinity and having in mind that , we obtain
and hence we get
a contradiction with (). Thus our assumption is wrong. Therefore, . Hence , that is, z is a coincidence point of f and g.
If the condition (x) is fulfilled, put . We will show that w is a common fixed point of f and g. Since and f and g are weakly compatible, we obtain, by the definition of weak compatibility, that . Thus we have . Using the condition (ix) that , we can apply the contractive condition (35) to obtain
where
Thus
Hence , and so by (), . Hence . Therefore
Thus we showed that w is a common fixed point of f and g.
Suppose now that the condition (xi) is fulfilled. Since a set of common fixed points of f and g is not empty, assume that w and v are two common fixed points of f and g, i.e.,
We claim that .
By assumption, there exists such that is comparable to fw and fv. Define a sequence such that
Further, set and and, in the same way, define and such that
From (50) and (52), we have and . Since is comparable to fw and fv, and f is g-nondecreasing, it is easy to show
Recursively, we can get that
By (35), we have that
Similarly as in the proof of Theorem 2.1, we can prove that is a convergent sequence. Thus there exists such that . Since also , for large enough n, we have
Thus from (55), for large enough n,
Letting in (56), by () we get
Hence we obtain
Then by (), and hence
Similarly, we can show that
and hence we obtain
Therefore, from (57) and (58), we have . It follows that
It means that w is the unique common fixed point of f and g. □
Corollary 4.1 Let be a complete partially ordered metric space and let f be a self-mapping of X satisfying the following condition:
for all such that , where
and . Suppose that, in addition, is non-decreasing, , there exists such that and if is a nondecreasing sequence such that it converges to some , then . Then f has a unique fixed point.
Proof Taking and in the proof of Theorem 4.1, we obtain Corollary 4.1. □
Remark 4.1 Theorem 4.1 extends Theorem 1 due to Berinde [21], Theorems 2.1 and 2.5 due to Beg and Abbas [22] and Theorem 3.1 due to Song [23].
We present an example to show that our result is a real generalization of the recent result of Golubović et al. [16] as well as of the existing results in the literature.
Example 4.1 Let be the closed interval with the 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 for all , it follows that
Thus we have
Therefore, f and g satisfy (35). Also, it is easy to see that the mappings and possess all properties (), () and (), () respectively, as well as hypotheses (v), (vi) and (vii) in Theorem 4.1. Thus we can apply our Theorem 4.1 and Corollary 4.1.
On the other hand, for and each , the contractive condition in Theorems 1 and 2 of Golubović et al. [16]:
where and
is not satisfied. Indeed,
Thus, for any fixed λ; , we have, for and each with ,
Thus, f does not satisfy (60). Therefore, the theorems of Jungck and Hussain [24], Al-Thagafi and Shahzad [25] and Das and Naik [26] also cannot be applied.
References
Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87(1):109–116. 10.1080/00036810701556151
Agarwal RP, O’Regan D, Sambandham M: Random and deterministic fixed point theory for generalized contractive maps. Appl. Anal. 2004, 83(7):711–725. 10.1080/00036810410001657206
Ahmad B, Nieto JJ: The monotone iterative technique for three-point second-order integrodifferential boundary value problems with p -Laplacian. Bound. Value Probl. 2007., 2007: Article ID 57481
Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20(2):458–464. 10.1090/S0002-9939-1969-0239559-9
Cabada A, Nieto JJ: Fixed points and approximate solutions for nonlinear operator equations. J. Comput. Appl. Math. 2000, 113(1–2):17–25. 10.1016/S0377-0427(99)00240-X
Ćirić LB: Generalized contractions and fixed-point theorems. Publ. Inst. Math. (Belgr.) 1971, 12(26):19–26.
Ćirić LB: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 1974, 45(2):267–273.
Ćirić LB: Fixed points of weakly contraction mappings. Publ. Inst. Math. (Belgr.) 1976, 20(34):79–84.
Ćirić LB: Coincidence and fixed points for maps on topological spaces. Topol. Appl. 2007, 154(17):3100–3106. 10.1016/j.topol.2007.08.004
Ćirić LB, Ume JS: Nonlinear quasi-contractions on metric spaces. Prakt. Akad. Athēnōn 2001, 76(A):132–141.
Ćirić LB: Common fixed points of nonlinear contractions. Acta Math. Hung. 1998, 80(1–2):31–38.
Drici Z, McRae FA, Vasundhara Devi J: Fixed-point theorems in partially ordered metric spaces for operators with PPF dependence. Nonlinear Anal., Theory Methods Appl. 2007, 67(2):641–647. 10.1016/j.na.2006.06.022
Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal., Theory Methods Appl. 2006, 65(7):1379–1393. 10.1016/j.na.2005.10.017
Gajić L, Rakočević V: Quasicontraction nonself-mappings on convex metric spaces and common fixed point theorems. Fixed Point Theory Appl. 2005, 3: 365–375.
Hussain N: Common fixed points in best approximation for Banach operator pairs with Ćrić type I -contractions. J. Math. Anal. Appl. 2008, 338(2):1351–1363. 10.1016/j.jmaa.2007.06.008
Golubović Z, Kadelburg Z, Radenović S: Common fixed points of ordered g -quasicontractions and weak contractions in ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 20
Jungck G: Commuting mappings and fixed points. Am. Math. Mon. 1976, 83: 261–263. doi:10.2307/2318216 10.2307/2318216
Jungck G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 1986, 9: 771–779. doi:10.1155/S0161171286000935 10.1155/S0161171286000935
Ćirić LB, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 131294. doi:10.1155/2008/131294
Das KM, Naik KV: Common fixed point theorems for commuting maps on a metric space. Proc. Am. Math. Soc. 1979, 77: 369–373.
Berinde V: A common fixed point theorem for quasi contractive type mappings. Ann. Univ. Sci. Bp. 2003, 46: 81–90.
Beg I, Abbas M: Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition. Fixed Point Theory Appl. 2006., 2006: Article ID 74503
Song Y: Coincidence points for noncommuting f -weakly contractive mappings. Int. J. Comput. Appl. Math. 2007, 2: 51–57.
Jungck G, Hussain N: Compatible maps and invariant approximations. J. Math. Anal. Appl. 2007, 325: 1003–1012. 10.1016/j.jmaa.2006.02.058
Al-Thagafi MA, Shahzad N: Banach operator pairs, common fixed points, invariant approximations and ∗-nonexpansive multimaps. Nonlinear Anal. 2008, 69: 2733–2739. 10.1016/j.na.2007.08.047
Das KM, Naik KV: Common fixed point theorems for commuting maps on a metric space. Proc. Am. Math. Soc. 1979, 77: 369–373.
Acknowledgements
Siniša Ješić was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, Project grant number 174032. Xiaolan Liu was supported by Scientific Research Fund of Sichuan Provincial Education Department (12ZA098), Scientific Research Fund of Sichuan University of Science and Engineering (2012KY08), and Scientific Research Fund of School of Science SUSE (10LXYB03).
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Liu, X., Ješić, S. Common fixed points of a generalized ordered g-quasicontraction in partially ordered metric spaces. Fixed Point Theory Appl 2013, 53 (2013). https://doi.org/10.1186/1687-1812-2013-53
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DOI: https://doi.org/10.1186/1687-1812-2013-53