Abstract
We establish coincidence and common fixedpoint theorems for Gisotone mappings in partially ordered metric spaces, which include the corresponding coupled, tripled and quadruple fixedpoint theorems as special cases. Our proofs are simpler and essentially different from the ones devoted to coupled, tripled and quadruple fixedpoint problems that appeared in the last years.
MSC: 47H10, 54H25.
Keywords:
Gisotone mapping; coincidence point; common fixed point; partially ordered metric space1 Introduction
The Banach contraction principle is the most celebrated fixedpoint theorem. There are great number of generalizations of the Banach contraction principle. A very recent trend in metrical fixedpoint theory, initiated by Ran and Reurings [1], and continued by Nieto and Lopez [2,3], Bhaskar and Lakshmikantham [4] and many other authors, is to consider a partial order on the ambient metric space and to transfer a part of the contractive property of the nonlinear operators into its monotonicity properties. This approach turned out to be very productive; see, for example, [19], and the obtained results found important applications to the existence of solutions for matrix equations or ordinary differential equations and integral equations, see [14,9] and reference therein.
In 2006, Bhaskar and Lakshmikantham [4] introduced the notion of coupled fixed point and proved some fixedpoint theorems under certain conditions. Later, Lakshmikantham and Ćirić [8] extended these results by defining the mixedgmonotone property, coupled coincidence point and coupled common fixed point. On the other hand, Berinde and Borcut [6] introduced the concept of tripled fixed point and proved some related theorems. Later, Borcut and Berinde [10] extended these results by defining the mixedgmonotone property, tripled coincidence point and tripled common fixed point. These results were then extended and generalized by several authors in the last five years; see [5,6,1123] and reference therein. Recently, Karapınar [24] introduced the notion of quadruple fixed point and proved some related fixedpoint theorems in partially ordered metric space (see also [2427]). Berzig and Samet [28] extended and generalized the mentioned fixedpoint results to higher dimensions. However, they used permutations of variables and distinguished between the first and the last variables. Very recently, Roldan et al.[29] extend the mentioned previous results for nonlinear mappings of any number of arguments, not necessarily permuted or ordered, in the framework of partially ordered complete metric spaces. We remind the reader of the following fact: in order to guarantee the existence of coupled (tripled or quadruple) coincidence point, the authors constructed two (three or four) Cauchy sequences using the properties of mixed monotone mappings and contractive conditions. It is not easy to prove that two (three or four) sequences are simultaneous Cauchy sequences. Then we spontaneously wonder the following questions:
Question 1.1 Can we obtain more general fixedpoint theorems including the corresponding coupled, tripled and quadruple fixedpoint theorems as three special cases?
Question 1.2 Can we provide a new method for approximating coupled, tripled and quadruple fixed points?
In this work, motivated and inspired by the above results, we establish more general fixedpoint theorems including the coupled, tripled and quadruple fixedpoint theorems as three special cases. Furthermore, we provide affirmative answers to Questions 1.1 and 1.2. The main results extend and improve the recent corresponding results in the literature. Our works bring at least two new features to coupled, tripled and quadruple fixedpoint theory. First, we provide a new method for approximating coupled, tripled and quadruple fixed points. Second, our proofs are simpler and essentially different from the ones devoted to coupled, tripled and quadruple fixedpoint problems that appeared in the last years.
2 Preliminaries
For simplicity, we denote from now on by , where and X is a nonempty set. Let n be a positive integer, will denote the function . If elements x, y of a partially ordered set are comparable (i.e. or holds) we will write .
Let Φ denote the set of all functions , which satisfies
Definition 2.1[12]
Let be a partially ordered set and d be a metric on R. We say that is regular if the following conditions hold:
(i) if a nondecreasing sequence is such that , then for all n,
(ii) if a nonincreasing sequence is such that , then for all n.
Definition 2.2[8]
Let be a partially ordered set and and . We say F has the mixed gmonotone property if F is monotone gnondecreasing in its first argument and is monotone gnonincreasing in its second argument, that is, for any ,
and
Definition 2.3[10]
Let be a partially ordered set and two mappings , . We say that F has the mixed gmonotone property if is gmonotone nondecreasing in x, it is gmonotone nonincreasing in y and it is gmonotone nondecreasing in z, that is, for any ,
and
Note that if g is the identity mapping, then Definitions 2.2 and 2.3 reduce to Definition 1.1 in [4] and Definition 4 in [6] of mixed monotone property, respectively.
Definition 2.4[29]
Let be a mapping. We say that F has the mixed monotone property if is monotone nondecreasing in x and z, and it is monotone nonincreasing in y and w, that is, for any
Some authors introduced the concept of coincidence point in different ways and with different names. Let and be two mappings.
(i) a coupled coincidence point [8] if , and ,
(ii) a tripled coincidence point [10] if , , and ,
(iii) a coupled common fixed point [8] if , and ,
(iv) a tripled common fixed point [10] if , , and ,
(v) a quadruple fixed point [24] if , , , and ,
(vi) a Φcoincidence point [29] if , for all i.
Similarly, note that if g is the identity mapping, then coupled coincidence point, tripled coincidence point and Φcoincidence point reduce to coupled fixed point (GnanaBhaskar and Lakshmikantham [4]), tripled fixed point (Berinde and Borcut [6]) and Φfixed point[29], respectively.
Definition 2.6 We say that the mappings and are commutative
(i) if , for all [8],
(ii) if , for all [10].
Let be a partially ordered set and d be a metric on X. We endow the product space with the following partial order: for ,
which will be denoted in the sequel, for convenience, by ≤, also. Obviously, is a partially ordered set. The mapping , given by
where , defines a metric on . It is easy to see that
In order to prove our main results, we need the following lemma.
Lemma 2.7Letbe a partially ordered set anddbe a metric onX. Ifis regular, thenis regular.
Proof Without loss of generality, we assume that the sequence is nondecreasing with (), where . From (2.3), we have
Now suppose that . As the sequence is nondecreasing and (2.1), we have the sequences are nondecreasing. From (2.4), the regularity of and using Definition 2.1, we have
Suppose that . Since the sequence is nondecreasing and (2.1), we have the sequences are nonincreasing. From (2.4), the regularity of and using Definition 2.1, we have
By (2.1), (2.5) and (2.6), we have for all n. By analogy, we show that if a nonincreasing sequence is such that , then for all n. Therefore, is regular. □
3 Main results
We now state and prove the main results of this paper.
Definition 3.1 We say that the mappings and are commutative if for all .
Definition 3.2 Let be a partially ordered set and , . We say that T is a Gisotone mapping if, for any
Definition 3.3 An element is called a coincidence point of the mappings and if . Furthermore, if , then we say that Y is a common fixed point of T and G.
Theorem 3.4Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letandbe aGisotone mapping for which there existssuch that for all, with,
whereis defined via (2.2). Supposeand also suppose either
(a) Tis continuous, Gis continuous and commutes withTor
If there existssuch that, thenTandGhave a coincidence point.
Proof Since , it follows that there exists such that . In general, there exists such that , . We denote and
Obviously, if for some , then there is nothing to prove. So, we may assume that for all . Since , without loss of generality, we assume that (the case is similar), that is, . Assume that , that is, . Since T is a Gisotone mapping, we get
which shows that for all . This actually means that the sequence is nondecreasing. Since , from (3.1) and (i_{φ}) we have
for all . Hence, the sequence given by is monotone decreasing and bounded below. Therefore, there exists some such that . We shall prove that . Assume that . Then by letting in (3.3) and (ii_{φ}) we have
which is a contradiction. Thus,
We claim that is a Cauchy sequence. Indeed, if it is false, then there exist and the sequences and of such that is the minimal in the sense that and . Therefore, .
Using the triangle inequality, we obtain
Letting in the above inequality and using (3.4), we get
Since , we have and hence . Now, by (3.1), we have
Observe that
Letting in the above inequality and using (3.4)(3.5), we have
where , which is a contradiction. Hence, the sequence is a Cauchy sequence in the metric space . On the other hand, since is a complete metric space, thus the metric space is complete. Therefore, there exists such that , that is, .
Now suppose that the assumption (a) holds. By the continuity of G, we have . On the other hand, by the commutativity of T and G, we have
By (3.6) and the continuity of T, we have
which shows that is a coincidence point of T and G.
Suppose that the assumption (b) holds. Using Lemma 2.7, we have is regular. Since is nondecreasing sequence that converges to , in view of Definition 2.1, we have for all n. Since is closed and by (3.2), we obtain that there exists for which
Then from (3.1), we have
for all . Letting in the above inequality, we have , which implies that . Therefore, is a coincidence point of T and G. □
Remark 3.5 Different kinds of contractive conditions are studied and we use a distinct methodology to prove Theorem 3.4. The authors proved that any number of sequences are simultaneous Cauchy sequence in [29]. However, we only need to proof that one sequence is a Cauchy sequence.
Taking in Theorem 3.4, we can obtain the following result immediately.
Corollary 3.6Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letandbe aGisotone mapping for which there existssuch that for all, with,
Supposeand also suppose either
(a) Tis continuous, Gis continuous and commutes withTor
If there existssuch that, thenTandGhave a coincidence point.
Now, we will show that Theorem 3.4 allow us to derive coupled, tripled and quadruple fixedpoint theorems for mixed monotone mappings in partially ordered metric space.
Taking , and for in Theorem 3.4, we can obtain the following result.
Corollary 3.7Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letandbe a mixedgmonotone mapping for which there existssuch that for allwith, ,
Supposeand also suppose either
(a) Fis continuous, gis continuous and commutes withFor
or
then there existsuch thatand, that is, Fandghave a couple coincidence point.
Proof For simplicity, we denote , and for all . We endow the product space with the following partial order:
Consider the function defined by
Obviously, and are two particular cases of and defined by (2.1) and (2.2), respectively. Now consider the operators and defined by
and
We claim that T is a Gisotone mapping. Indeed, suppose that , . By (3.10) and (3.13), we have and . Since F is gmixed monotone, we have
From (3.10), (3.12) and (3.14), we have
Similarly, we can obtain that for any , . By (3.8)(3.10), we have there exists such that .
From (3.11) and (3.12), we have
and
for any , . It follows from (3.7) that
Now suppose that the assumption (a) holds. By the continuity of g, we have G is continuous. From (3.12), (3.13) and using the commutativity of F and g, we have, for any
which implies that G commutes with T. It is easy to see that T is continuous. Indeed, by (3.11), we obtain that () if and only if and (). Since F is continuous, we have and (), for any (). Therefore, we have
Suppose that the assumption (b) holds. It is easy to see that is closed.
All the hypothesis of Theorem 3.4 () are satisfied, and so we deduce the existence of a coincidence point of T and G. From (3.12) and (3.13), there exists such that and , that is, is a coupled coincidence point of F and g. □
Remark 3.8 Note that in the case of the condition (b) satisfied in Corollary 3.7, we omit the control conditions: g is continuous and commutes with F, which are needed in the proof of Theorem 2.1 in [8] and Theorem 3 in [7].
Taking , and for in Theorem 3.4, we can obtain the following result by the similar argument as we did in the proof of Corollary 3.7.
Corollary 3.9Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letandsuch thatFhas the mixedgmonotone property and. Assume there is a functionsuch that
for anyfor which, and. Suppose either
(a) Fis continuous, gis continuous and commutes withFor
or
that is, Fandghave a tripled coincidence point.
Similarly, taking , and G is the identity mapping on for in Theorem 3.4, we can obtain the following result.
Corollary 3.10Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letsuch thatFhas the mixed monotone property. Assume there is a functionsuch that
for anyfor which, , and. Suppose either
(a) Fis continuous or
or
that is, Fhave a quadruple fixed point.
Theorem 3.11In addition to the hypothesis of Theorem 3.4, suppose that for everythere existssuch thatis comparable toand to. Also, assume thatφis nondecreasing. LetGcommute withTif the assumption (b) holds. ThenTandGhave a unique common fixed point, that is, there exists a unique pointsuch that.
Proof From Theorem 3.4, the set of coincidence points of T and G is nonempty. Assume that and are two coincidence points of T and G. We shall prove that . Put and choose so that . Then, similarly to the proof of Theorem 3.4, we obtain the sequence defined as follows: , . Since and are comparable, without loss of generality, we assume that . Since T is a Gisotone mapping, we have
Recursively, we get that , . Thus, by the contractive condition (3.1), one gets
Thus, by the above inequality, we get
where . Since φ is nondecreasing, it follows that
From the definition of Φ, we get , for each . Then, we have . Thus,
Similarly, we obtain that
Combining (3.15) and (3.16) yields that . Since , by the commutativity of T and G, we have
Denote . By (3.17), we have , that is is a coincidence point of T and G. Thus, we have . Therefore, is a common fixed point of T and G.
To prove the uniqueness, assume is another common fixed point of T and G. Then we have
□
Corollary 3.12In addition to the hypothesis of Corollary 3.7, suppose that for everythere existssuch thatis comparable toand to. Also, assume thatφis nondecreasing. Letgcommute withFif the assumption (b) holds. ThenFandghave a unique coupled common fixed point, that is, there exists a unique pointsuch that
Proof Similarly to the proof of Corollary 3.7, we can obtain all conditions of Theorem 3.4 () are satisfied. In addition, by the commutativity of g and F, we have G commutes with T. For simplicity, we denote , and . By (3.12), we have
By hypothesis, there exists such that is comparable to and to . Hence, there is no doubt that all conditions of Theorem 3.11 are satisfied (). Therefore, there exists a unique point such that . That is, and . □
By the similar argument as we did in the proof of Corollary 3.12, we deduce the following corollary from Theorem 3.11 ().
Corollary 3.13In addition to the hypothesis of Corollary 3.9, suppose that for allandin, there existsinsuch thatis comparable toand. Also, assume thatφis nondecreasing. Letgcommute withFif the assumption (b) holds. ThenFandghave a unique tripled common fixed point, that is,
Competing interests
The author declares that she has no competing interests.
Author’s contributions
SW completed the paper herself. The author read and approved the final manuscript.
Acknowledgements
The author thanks the editor and the referees for their useful comments and suggestions. This study was supported by the Natural Science Foundation of Yancheng Teachers University under Grant (12YCKL001) and UNSF of Jiangsu province, China (09KJD110005).
References

Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc.. 132(5), 1435–1443 (2004). Publisher Full Text

Nieto, JJ, RodriguezLopez, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 22(3), 223–239 (2005). Publisher Full Text

Nieto, JJ, RodriguezLopez, R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser.. 23(12), 2205–2212 (2007). Publisher Full Text

Bhaskar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal.. 65(7), 1379–1393 (2006). Publisher Full Text

Agarwal, RP, ElGebeily, MA, O’Regan, D: Generalized contractions in partially ordered metric spaces. Appl. Anal.. 87, 1–8 (2008). Publisher Full Text

Berinde, V, Borcut, M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal.. 74, 4889–4897 (2011). Publisher Full Text

Berinde, V: Coupled coincidence point theorems for mixed point monotone nonlinear operators. Comput. Math. Appl.. 64, 1770–1777 (2012). Publisher Full Text

Lakshmikantham, V, Ćirić, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal.. 70, 4341–4349 (2009). Publisher Full Text

Luong, NV, Thuan, NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal.. 74, 983–992 (2011). Publisher Full Text

Borcut, M, Berinde, V: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput.. 218(10), 5929–5936 (2012). Publisher Full Text

Abbas, M, Khan, MA, Radenović, S: Common coupled fixed point theorems in cone metric space for ωcompatible mappings. Appl. Math. Comput.. 217, 195–203 (2010). Publisher Full Text

Aydi, H, Karapınar, E, Postolache, M: Tripled coincidence theorems for weak φcontractions in partially ordered metric spaces. Fixed Point Theory Appl.. 2012, (2012) Article ID 44. doi:10.1186/16871812201244

Aydi, H, Karapınar, E: Tripled fixed points in ordered metric spaces. Bull. Math. Anal. Appl.. 4(1), 197–207 (2012)

Aydi, H, Karapınar, E, Radenovic, S: Tripled coincidence fixed point results for BoydWong and Matkowski type contractions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. (2012) doi:10.1007/s1339801200773

Aydi, H, Vetro, C, Karapınar, E: MeirKeeler type contractions for tripled fixed points. Acta Math. Sci.. 32(6), 2119–2130 (2012)

Aydi, H, Karapınar, E: New MeirKeeler type tripled fixed point theorems on ordered partial metric spaces. Math. Probl. Eng.. 2012, (2012) Article ID 409872

Aydi, H, Karapınar, E, Shatnawi, W: Tripled fixed point results in generalized metric spaces. J. Appl. Math.. 2012, (2012) Article ID 314279

Borcut, M: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput.. 218, 7339–7346 (2012). Publisher Full Text

Choudhury, BS, Kundu, A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal.. 73, 2524–2531 (2010). Publisher Full Text

Samet, B: Coupled fixed point theorems for a generalized MeirKeeler contraction in partially ordered metric space. Nonlinear Anal.. 74(12), 4508–4517 (2010)

Shatanawi, W: Partially ordered cone metric spaces and couple fixed point results. Comput. Math. Appl.. 60, 2508–2515 (2010). Publisher Full Text

Roldan, A, MartinezMoreno, J, Roldan, C: Tripled fixed point theorem in fuzzy metric spaces and applications. Fixed Point Theory Appl.. 2013, (2013) Article ID 29. doi:10.1186/16871812201329

Karapınar, E: Coupled fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl.. 59(12), 3656–3668 (2010). Publisher Full Text

Karapınar, E: Quartet fixed point for nonlinear contraction. http://arxiv.org/abs/1106.5472

Karapınar, E: Quadruple fixed point theorems for weak ϕcontractions. ISRN Math. Anal.. 2011, (2011) Article ID 989423

Karapınar, E, Berinde, V: Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces. Banach J. Math. Anal.. 6, 74–89 (2012)

Karapınar, E: A new quartet fixed point theorem for nonlinear contractions. JP J. Fixed Point Theory Appl.. 6, 119–135 (2011)

Berzig, M, Samet, B: An extension of coupled fixed point’s concept in higher dimension and applications. Comput. Math. Appl.. 63(8), 1319–1334 (2012). Publisher Full Text

Roldan, A, MartinezMoreno, J, Roldan, C: Multidimensional fixed point theorems in partially ordered metric spaces. J. Math. Anal. Appl.. 396, 536–545 (2012). PubMed Abstract  Publisher Full Text  PubMed Central Full Text