- Research
- Open access
- Published:
Mixed g-monotone property and quadruple fixed point theorems in partially ordered G-metric spaces using contractions
Fixed Point Theory and Applications volume 2012, Article number: 199 (2012)
Abstract
In this paper, we prove some quadruple coincidence and quadruple common fixed point theorems for and satisfying contractions in partially ordered G-metric spaces. We illustrate our results based on an example of the main theorems. Also, we deduce quadruple coincidence points for mappings satisfying the contraction condition of integral type.
MSC:54H25, 47H10, 54E50.
1 Introduction and preliminaries
In 1992, Dhage introduced a new class of generalized metric spaces called D-metric spaces (see [1]). In a subsequent series of papers, Dhage attempted to develop topological structures in such spaces (see [2–4]). He claimed that D-metrics provide a generalization of ordinary metric functions and went on to present several fixed point results. In [5], in collaboration with Sims we demonstrate that most of the claims concerning the fundamental topological structure of D-metric space are incorrect, we also introduce a valid generalized metric space structure, which we call G-metric spaces. Some other papers dealing with G-metric spaces are those in [6–17]. Recently, there has been growing interest in establishing fixed point theorems in partially ordered complete G-metric spaces with the contractive condition which holds for all points that are related by partial ordering ([18, 19] and [20]).
In [21], coupled fixed point results in partially ordered metric spaces were established. After the publication of this work, several coupled fixed point and coincidence point results have appeared in recent literatures (see, for instance, [19, 22–37] and [38]).
Recently, Vasile Berinde and Marin Borcut [39] extended and generalized the results of [21] to the case of a contractive operator , where X is a complete ordered metric space. They introduced the concept of a tripled fixed point and the mixed monotone property of a mapping . For more details on tripled fixed point results, we refer the reader to [39] and [40].
Very recently, the notion of a fixed point of order was introduced in [30], and later in [41] Erdal Karapinar and Nguyen Van Luong introduced the concept of a quadruple fixed point and the mixed monotone property of a mapping and they presented some new fixed point results. Then, a quadruple fixed point is developed and related fixed points are obtained (see [41–47]).
In this paper, we prove some quadruple coincidence and quadruple common fixed point theorems for and satisfying contractions in partially ordered G-metric spaces. We illustrate our results based on an example of the main theorems. Also, we deduce quadruple coincidence points for mappings satisfying the contraction condition of integral type, we shall recall some mathematical preliminaries.
Definition 1.1 [48]
Let X be a nonempty set, and let be a function satisfying the following properties:
(G1) if ;
(G2) for all with ;
(G3) for all with ;
(G4) (symmetry in all three variables); and
(G5) for all (rectangle inequality).
Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair is called a G-metric space.
Example 1.1 [48]
Let be a usual metric space, and define and on to by
for all . Then and are G-metric spaces.
Definition 1.2 [48]
Let be a G-metric space, and let be a sequence of points of X. A point is said to be the limit of the sequence if , and one says that the sequence is G-convergent to x.
Thus, if in a G-metric space , then for any , there exists such that for all (we mean by N the natural numbers).
Proposition 1.1 [48]
Let be a G-metric space. Then the following are equivalent.
-
(1)
is G-convergent to x.
-
(3)
, as .
-
(4)
, as .
-
(5)
, as .
Definition 1.3 [48]
Let be a G-metric space. A sequence is called G-Cauchy if, given , there is such that for all . That is, as .
Proposition 1.2 [48]
In a G-metric space, , the following are equivalent.
-
(1)
The sequence is G-Cauchy.
-
(2)
For every , there exists such that for all .
Proposition 1.3 [48]
Let and be two G-metric spaces. Then a function is G-continuous at a point if and only if it is G-sequentially continuous at x; that is, whenever is G-convergent to x, we have is G-convergent to .
Definition 1.4 [48]
A G-metric space is called a symmetric G-metric space if for all .
It is clear that any G-metric space where G derives from an underlying metric via or in Example 1.1 is symmetric.
Proposition 1.4 [48]
Let be a G-metric space, then the function is jointly continuous in all three of its variables.
Proposition 1.5 [48]
Every G-metric space induces a metric space defined by
Note that if is symmetric, then
However, if is not symmetric, then it holds by the G-metric properties that
Definition 1.5 [48]
A G-metric space is said to be G-complete (or complete G-metric) if every G-Cauchy sequence in is G-convergent in .
Definition 1.6 Let be a G-metric space. A mapping is said to be continuous if for any G-convergent sequences , , , and converging to x, y, z, and w respectively, is G-convergent to .
Proposition 1.6 [48]
A G-metric space is G-complete if and only if is a complete metric space.
Following Erdal [41], we introduce the following definitions.
Definition 1.7 [41] Let X be a nonempty set and be a given mapping. An element is called a quadruple fixed point of F if
Definition 1.8 [41] Let be a partially ordered set and be a mapping. We say that F has the mixed monotone property if is monotone non-decreasing in x and z and is monotone non-increasing in y and w; that is, for any ,
and
Definition 1.9 [41] Let X be a non-empty set. Then we say that the mappings and are commutative if for all ,
Definition 1.10 [47] Let be a partially ordered set. Let and . The mapping F is said to have the mixed g-monotone property if for any ,
Definition 1.11 [47] Let and . An element is called a quadruple coincidence point of F and g if
is said to be a quadruple point of coincidence of F and g.
Definition 1.12 [47] Let and . An element is called a quadruple common fixed point of F and g if
Let Φ be the set of all functions which satisfy
-
(1)
ϕ is continuous and non-decreasing;
-
(2)
iff ;
-
(3)
, .
And let Ψ denote all functions which satisfy
-
(1)
for all , and
-
(2)
.
For example [26], the functions , , are in Φ and , , are in Ψ.
Remark 1 .
Remark 2 For all , we have .
2 Main results
Theorem 2.1 Let be a partially ordered set and be a G-metric space. Let and be such that F has the mixed g-monotone property. Assume that there exist and such that
for all with , , , and . Suppose , g is continuous and commutes with F. If there exist such that
suppose either
-
(a)
is a complete G-metric space and F is continuous or
-
(b)
is complete and has the following property: then there exist such that
that is, F and g have a quadruple coincidence point.
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n,
Proof Let be such that
Since , then we can choose such that
Taking into account , by continuing this process, we can construct sequences , , and in X such that
We shall show that
For this purpose, we use the mathematical induction. Since , , , and , then by (2.2), we get
that is, (2.4) holds for .
We presume that (2.4) holds for some . As F has the mixed g-monotone property and , , , and , we obtain
and
Thus, (2.4) holds for any . Assume for some ,
then by (2.3), we have , , , and is a quadruple coincidence point of F and g. From now on, assume for any that at least
Since , , , and , then from (2.1) and (2.3), we have
and
From (2.6), (2.7), (2.8), and (2.9) it follows that
but from the property (3) of ϕ, we get ; therefore,
which implies that
Using the fact that ϕ is non-decreasing, we get
Let .
Then the sequence is decreasing; therefore, there is some such that
We will show that . Suppose to the contrary that , taking the limit as of both sides of (1.3) and using the fact that ϕ is continuous and for , we have
a contradiction. Thus, , that is,
We will show that the sequences , , , and are G-Cauchy sequences in .
Suppose to the contrary that at least one of , , , and is not a G-Cauchy sequence, so there exists for which we can find sequences of , of , of , and of with such that
Further, corresponding to , we may choose such that it is the smallest integer satisfying (2.16) and .
Thus,
Using the rectangle inequality and having in mind (2.16) and (2.17), we get
Letting in (2.18) and using , we get
Again by the rectangle inequality and using the fact that , we have
Using the property of ϕ, we have
Since , then , , , and .
Then from (2.1), we have
and similarly,
and
Combining (2.22), (2.23), (2.24), and (2.25) in (2.21), we get
Letting, and using (2.15) and (2.19), we get
a contradiction. This implies that , , , and are G-Cauchy sequences in .
Now suppose that the assumption (a) holds.
Since X is a G-complete metric space, there exist such that
From (2.28) and the continuity of g, we have
From the commutativity of F and g, we have
and
We shall show that , , , and .
By letting in (2.29) → (2.32) and using the continuity of F, we obtain
Similarly, , , and .
Hence, is a coincidence point of F and g.
Now suppose that the assumption (b) holds.
Since , , , and are G-Cauchy sequences in the complete G-metric space , then there exist such that
Since , are non-decreasing and , are non-increasing and since satisfies conditions (i) and (ii), we have
If , , , and for some , then , , , and , which implies that
and
that is, is a quadruple coincidence point of F and g. Then, we suppose that for all . By (2.1), consider now
Taking the limit as in (2.34) and using the property of ϕ, hence we get that . Thus, . Analogously, one finds
Thus, we proved that F and g have a quadruple coincidence point. This completes the proof of Theorem 2.1. □
Corollary 2.1 Let be a partially ordered set and be a G-metric space. Let and be such that F has the mixed g-monotone property. Assume that there exists such that
for all with , , , and . Suppose , g is continuous and commutes with F. If there exist such that
suppose either
-
(a)
is a complete G-metric space and F is continuous or
-
(b)
is complete and has the following property: then there exist such that
that is, F and g have a quadruple coincidence point.
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n,
Proof In Theorem 2.1 taking , we get Corollary 2.1. □
Corollary 2.2 Let be a partially ordered set and be a G-metric space. Let and be such that F has the mixed g-monotone property such that
for all with , , , and . Suppose , g is continuous and commutes with F. If there exist such that
suppose either
-
(a)
is a complete G-metric space and F is continuous or
-
(b)
has the following property: then there exist such that
that is, F and g have a quadruple coincidence point.
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n,
Proof In Corollary 2.1 taking , we get Corollary 2.2. □
Now, we shall prove the existence and uniqueness of a quadruple common fixed point. For a product of a partially ordered set , we define a partial ordering in the following way. For all ,
We say that and are comparable if
Also, we say that is equal to if and only if , , , .
Theorem 2.2 In addition to the hypothesis of Theorem 2.1, suppose that for all , there exists such that is comparable to and . Then F and g have a unique quadruple common fixed point such that , , , and .
Proof The set of quadruple coincidence points of F and g is not empty due to Theorem 2.1. Assume now and are two quadruple coincidence points of F and g, that is,
We shall show that and are equal. By assumption, there exists such that is comparable to .
Define the sequences , , , and such that
for all n. Further, set , , , and , , , and in the same way define the sequences , , , and , , , . Then it is easy to see that
for all . Since is comparable to , then it is easy to show . Recursively, we get that
By (2.39) and (2.1), we have
and
It follows from (2.40)-(2.43) and the property of ϕ that
Thus,
and therefore,
Hence, the sequence is a decreasing sequence; therefore, there exists such that
We shall show that . Suppose to the contrary . Taking the limit as in (2.44), then we have
a contradiction. Thus, , that is,
This yields that
Analogously, we show that
Combining (2.47) and (2.48) yields that and are equal.
Since , , , and , by commutativity of F and g, we have
and
where , , , and . Thus, is a quadruple coincidence point of F and g. Consequently, and are equal. We deduce
Therefore, is a quadruple common fixed point of F and g. Its uniqueness follows easily from (2.1). □
Example 2.1 Let with a usual ordering. Define by . Let and be defined by
Take be given by for all and by for all . Then
-
a.
is a complete ordered G-metric space.
-
b.
For with , , , and , we have the condition (2.1) of Theorem 2.1 satisfied.
-
c.
F and g have the mixed g-monotone property.
Proof To prove (b), given with , , , and . Then
but
Similarly,
and
Therefore,
Hence,
To prove (c), let . To show that is g-monotone non-decreasing in x, let with . Then , and so . Hence, .
Therefore, is g-monotone non-decreasing in x. Similarly, we may show that is g-monotone non-decreasing in z.
To show that is g-monotone non-increasing in y, let with , then . Hence, , so .
Therefore, is g-monotone non-increasing in y. Similarly, we may show that is g-monotone non-increasing in w.
Let . It is obvious that other hypotheses of Theorem 2.1 are satisfied. Thus, by Theorems 2.1 and 2.2, F and g have a unique quadruple common fixed point. Here, is the unique quadruple common fixed point of F and g. □
3 Application
In this part, we use previously obtained results to deduce some quadruple coincidence point results for mappings satisfying a contraction of integral type in a complete G-metric space. We first introduce some notations.
We denote by Γ the set of functions satisfying the following conditions:
-
(i)
α is a Lebesgue integrable mapping on each compact subset of ;
-
(ii)
for all , we have
-
(iii)
α is sub-additive on each , that is,
Let be fixed. Let be a family of N functions that belong to Γ. For all , we denote as follows:
We have the following result.
Theorem 3.1 Let be a partially ordered set and be a complete G-metric space. Suppose and are such that F is continuous and F has the mixed g-monotone property. Assume that there exist and such that
for any , for which , , , and . Also, suppose , g is continuous and commutes with F. If there exist such that , , , and , then there exist such that
that is, F and g have a quadruple coincidence point.
Proof Take and .
Note that the are taken to be sub-additive on each in order to get
Moreover, it is easy to show that is continuous, non-decreasing and verifies iff .
We get that . Also, we can find that . From (3.1), we have
Now, applying Theorem 2.1, we obtain the desired result. □
References
Dhage BC: Generalized metric space and mapping with fixed point. Bull. Calcutta Math. Soc. 1992, 84: 329–336.
Dhage BC: Generalized metric space and topological structure I. An. ştiinţ. Univ. “Al.I. Cuza” Iaşi, Mat. 2000, 46: 3–24.
Dhage BC: On generalized metric spaces and topological structure II. Pure Appl. Math. Sci. 1994, 40: 37–41.
Dhage BC: On continuity of mappings in D-metric spaces. Bull. Calcutta Math. Soc. 1994, 86: 503–508.
Mustafa Z, Sims B: Some remarks concerning D-metric spaces. Proceedings of the International Conferences on Fixed Point Theory and Applications 2003, 189–198. Valencia (Spain), July
Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mappings on complete G -metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 189870
Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed point results in G -metric spaces. Int. J. Math. Math. Sci. 2009., 2009: Article ID 283028
Mustafa Z, Obiedat H: A fixed point theorem of Reich in G -metric spaces. CUBO 2010, 12(01):83–93. 10.4067/S0719-06462010000100008
Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175. doi:10.1155/2009/917175
Mustafa Z, Khandagjy M, Shatanawi W: Fixed point results on complete G -metric spaces. Studia Sci. Math. Hung. 2011, 48(3):304–319. doi:10.1556/SScMath.2011.1170
Mustafa Z, Awawdeh F, Shatanawi W: Fixed point theorem for expansive mappings in G -metric spaces. Int. J. Contemp. Math. Sci. 2010, 5: 49–52.
Obiedat H, Mustafa Z: Fixed point results on a nonsymmetric G -metric spaces. Jordan J. Math. Stat. 2010, 3(2):65–79.
Mustafa Z, Aydi H, Karapinar E: On common fixed points in G -metric spaces using (E.A) property. Comput. Math. Appl. 2012. doi:10.1016/j.camwa.2012.03.051
Mustafa Z: Common fixed points of weakly compatible mappings in G -metric spaces. Appl. Math. Sci. 2012, 6(92):4589–4600.
Mustafa Z: Some new common fixed point theorems under strict contractive conditions in G -metric spaces. J. Appl. Math. 2012., 2012: Article ID 248937. doi:10.1155/2012/248937
Rao KPR, Bhanu Lakshmi K, Mustafa Z: Fixed and related fixed point theorems for three maps in G -metric space. J. Adv. Dtud. Topol. 2012, 3(4):12–19.
Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in G -metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 181650
Shatanawi W: Some fixed point theorems in ordered G -metric spaces and applications. Abstr. Appl. Anal. 2011., 2011: Article ID 126205
Aydi H, Damjanović B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered G -metric spaces. Math. Comput. Model. 2011. doi:10.1016/j.mcm.2011.05.059
Shatanawi W, Samet B: On -weakly contractive condition in partially ordered metric spaces. Comput. Math. Appl. 2011, 62: 3204–3214. 10.1016/j.camwa.2011.08.033
Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
Lakshmikantham V, Ciric L: Couple fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020
Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036
Karapinar E: Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl. 2010, 59: 3656–3668. 10.1016/j.camwa.2010.03.062
Lakshmikantham V, Ćirić Lj: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020
Luong NV, Thuan NX: Coupled fixed points in partially ordered metric spaces and applications. Nonlinear Anal. 2011, 72: 983–992.
Nashine, HK, Shatanawi, W: Coupled common fixed point theorems for a pair of commuting mappings in partially ordered complete metric spaces. Comput. Math. Appl. (to appear)
Sabetghadam F, Masiha HP, Sanatpour AH: Some coupled fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 125426
Samet B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72: 4508–4517. 10.1016/j.na.2010.02.026
Samet B, Vetro C: Coupled fixed point, f -invariant set and fixed point of N -order. Ann. Funct. Anal. 2010, 1(2):46–56.
Samet B, Yazidi H: Coupled fixed point theorems in partially ordered ε -chainable metric spaces. TJMCS 2010, 1(3):142–151.
Sedghi S, Altun I, Shobe N: Coupled fixed point theorems for contractions in fuzzy metric spaces. Nonlinear Anal. 2010, 72: 1298–1304. 10.1016/j.na.2009.08.018
Shatanawi W: Some common coupled fixed point results in cone metric spaces. Int. J. Math. Anal. 2010, 4: 2381–2388.
Shatanawi W: Partially ordered cone metric spaces and coupled fixed point results. Comput. Math. Appl. 2010, 60: 2508–2515. 10.1016/j.camwa.2010.08.074
Shatanawi W, Samet B, Abbas M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. 2012, 55: 680–687. 10.1016/j.mcm.2011.08.042
Shatanawi W: Coupled fixed point theorems in generalized metric spaces. Hacet. J. Math. Stat. 2011, 40(3):441–447.
Shatanawi W, Abbas M, Nazir T: Common coupled coincidence and coupled fixed point results in two generalized metric spaces. Fixed Point Theory Appl. 2011, 2011: 80. 10.1186/1687-1812-2011-80
Shatanawi W: Partially ordered cone metric spaces and coupled fixed point results. Comput. Math. Appl. 2010, 60: 2508–2515. 10.1016/j.camwa.2010.08.074
Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74(15):4889–4897. 10.1016/j.na.2011.03.032
Aydi H, Karapınar E, Postolache M: Tripled coincidence point theorems for weak φ -contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2012, 2012: 44. 10.1186/1687-1812-2012-44
Karapınar E, Luong NV: Quadruple fixed point theorems for nonlinear contractions. Comput. Math. Appl. 2012. doi:10.1016/j.camwa.2012.02.061
Karapınar, E: Quartet fixed point for nonlinear contraction. http://arxiv.org/abs/1106.5472
Karapınar E: Quadruple fixed point theorems for weak ϕ -contractions. 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. 2012, 6(1):74–89.
Karapınar E: A new quartet fixed point theorem for nonlinear contractions. JP J. Fixed Point Theory Appl. 2011, 6(2):119–135.
Karapinar E, Shatanawi W, Mustafa Z: Quadruple fixed point theorems under nonlinear contractive conditions in partially ordered metric spaces. J. Appl. Math. 2012., 2012: Article ID 951912. doi:10.1155/2012/951912
Mustafa Z, Aydi H, Karapinar E: Mixed g -monotone property and quadruple fixed point theorems in partial ordered metric space. Fixed Point Theory Appl. 2012., 2012: Article ID 71. doi:10.1186/1687–1812–2012–71
Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7(2):289–297.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Mustafa, Z. Mixed g-monotone property and quadruple fixed point theorems in partially ordered G-metric spaces using contractions. Fixed Point Theory Appl 2012, 199 (2012). https://doi.org/10.1186/1687-1812-2012-199
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2012-199