On coupled fixed point theorems on partially ordered G-metric spaces

In this manuscript, we extend, generalize and enrich some recent coupled fixed point theorems in the framework of partially ordered G-metric spaces in a way that is essentially more natural.

MSC: 46N40, 47H10, 54H25, 46T99.

In [1] Aydi et al. established coupled coincidence and coupled common fixed point results for a mixed g-monotone mapping satisfying nonlinear contractions in partially ordered G-metric spaces. These results generalize those of Choudhury and Maity [2].

Here we generalize, improve, enrich and extend the above mentioned coupled fixed point results of Aydi et al.

Throughout this paper, let denote the set of nonnegative integers, and be the set of positive integers.

Definition 1.1 (See [3])

Let X be a non-empty set, and be a function satisfying the following properties:

(G1) if ,

(G2) for all with ,

(G3) for all with ,

(G4) (symmetry in all three variables),

(G5) for all (rectangle inequality).

Then the function G is called a generalized metric or, more specially, a G-metric on X, and the pair is called a G-metric space.

Every G-metric on X defines a metric on X by

Example 1.2 Let be a metric space. The function , defined by

or

for all , is a G-metric on X.

Definition 1.3 (See [3])

Let be a G-metric space, and let be a sequence of points of X. We say that is G-convergent to if , that is, for any , there exists such that , for all . We call x the limit of the sequence and write or .

Proposition 1.4 (See [3])

Letbe aG-metric space. The following are equivalent:

(1) isG-convergent tox,

(2) as,

(3) as,

(4) as.

Definition 1.5 (See [3])

Let be a G-metric space. A sequence is called a G-Cauchy sequence if, for any , there is such that for all , that is, as .

Proposition 1.6 (See [3])

Letbe aG-metric space. Then the following are equivalent:

(1) the sequenceisG-Cauchy,

(2) for any, there existssuch that, for all.

Proposition 1.7 (See [3])

Letbe aG-metric space. A mappingisG-continuous atif and only if it isG-sequentially continuous at, that is, wheneverisG-convergent to, the sequenceisG-convergent to.

Definition 1.8 (See [3])

A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .

Definition 1.9 (See [2])

Let be a G-metric space. A mapping is said to be continuous if for any two G-convergent sequences and converging to x, y respectively, is G-convergent to .

Let be a partially ordered set and be a G-metric space, be a mapping. A partially ordered G-metric space, , is called g-ordered complete if for each convergent sequence , the following conditions hold:

() if is a non-increasing sequence in X such that implies , ,

() if is a non-decreasing sequence in X such that implies , .

Moreover, a partially ordered G-metric space, , is called ordered complete when g is equal to identity mapping in the above conditions () and ().

Definition 1.10 (See [4])

An element is said to be a coupled fixed point of the mapping if

Definition 1.11 (See [5])

An element is called a coupled coincidence point of a mapping and if

Moreover, is called a common coupled coincidence point of F and g if

Definition 1.12 Let and be mappings. The mappings F and g are said to commute if

Definition 1.13 (See [4])

Let be a partially ordered set and be a mapping. Then F is said to have mixed monotone property if is monotone non-decreasing in x and is monotone non-increasing in y, that is, for any ,

and

Definition 1.14 (See [5])

Let be a partially ordered set and and be two mappings. Then F is said to have mixed g-monotone property if is monotone g-non-decreasing in x and is monotone g-non-increasing in y, that is, for any ,

and

Let Φ denote the set of functions satisfying

(a) ,

(b) for all ,

(c) for all .

Lemma 1.15 (See [5])

Let. For all, we have.

Aydi et al.[1] proved the following theorems.

Theorem 1.16Letbe a partially ordered set andGbe aG-metric onXsuch thatis a completeG-metric space. Suppose that there exist, andsuch that

for allwithand. Suppose also thatFis continuous and has the mixedg-monotone property, andgis continuous and commutes withF. If there existsuch thatand, thenFandghave a coupled coincidence point, that is, there existssuch thatand.

Theorem 1.17Letbe a partially ordered set andGbe aG-metric onXsuch thatis regular. Suppose that there existand mappingsandsuch that

for allwithand. Suppose also thatis complete, Fhas the mixedg-monotone property and. If there existsuch thatand, thenFandghave a coupled coincidence point.

In this manuscript, we generalize, improve, enrich and extend the above coupled fixed point results. We also state some examples to illustrate our results. This paper can be considered as a continuation of the remarkable works of Berinde [6,7].

We begin with an example to illustrate the weakness of Theorem 1.16 and Theorem 1.17 above.

Example 2.1 Let . Define by

for all . Then is a G-metric space. Define a map by and by for all . Suppose

and

It is clear that there is no that provides the statement (1.4) of Theorem 1.16.

Notice that is the unique common coincidence point of F and g. In fact, .

For some coupled fixed point and coupled coincidence point theorems, we refer the reader to [8-34].

We now state our first result which successively guarantees a coupled fixed point.

Theorem 2.2Letbe a partially ordered set andGbe aG-metric onXsuch thatis a completeG-metric space. Suppose that there exist, andsuch that

for allwithand. Suppose also thatFis continuous and has the mixedg-monotone property, andgis continuous and commutes withF. If there existsuch thatand, thenFandghave a coupled coincidence point, that is, there existssuch thatand.

Proof Given satisfying and , we shall construct iterative sequences and in the following way: Since , we can choose such that and . Analogously, we choose such that and due to the same reasoning. Since F has the mixed g-monotone property, we conclude that and . By repeating this process, we derive the iterative sequence

and

If for some we have , then and , that is, F and g have a coincidence point. So, we assume that for all . Thus, we have either or . We set

for all . Due to the property (G2), we have for all . By using inequality (2.3), we obtain

Taking (2.4) into account, (2.5) becomes

Since for all , it follows that is monotone decreasing. Therefore, there is some such that .

Now, we assert that . Suppose, on the contrary, that . Letting in (2.6) and using the properties of the map ϕ, we get

which is contradiction. Thus . Hence

Next, we prove that and are Cauchy sequences in the G-metric space . Suppose, on the contrary, that at least one of and is not a Cauchy sequence in . Then there exist and sequences of natural numbers and such that for every natural number k, and

Now, corresponding to , we choose to be the smallest for which (2.8) holds. Hence

Using the rectangle inequality (property (G5)), we get

Letting in the above inequality and using (2.7) yields

Again, by the rectangle inequality, we have

Using the fact that for any , we obtain from properties (G2)-(G4)

Next, using inequality (2.3), we have

Now, using (2.7),(2.10), the properties of the function ϕ, and letting in (2.11), we get

which is a contradiction. Thus, we have proven that and are Cauchy sequences in the G-metric space . Now, since is complete, there are such that and are respectively G-convergent to x and y. That is from Proposition 1.4, we have

Using the continuity of g, we get from Proposition 1.7

Since and , employing the commutativity of F and g yields

Now, we shall show that and .

The mapping F is continuous, and since the sequences and are respectively G-convergent to x and y, using Definition 1.9, the sequence is G-convergent to . Therefore, from (2.13), is G-convergent to . By uniqueness of the limit and using (2.12), we have . Similarly, we can show that . Hence, is a coupled coincidence point of F and g. This completes the proof. □

The following example illustrates that Theorem 2.2 is an extension of Theorem 1.16.

Example 2.3 Let us reconsider Example 2.1. Define a map by

and by for all . Then . We observe that

and

Then, the statement (2.3) of Theorem 2.2 is satisfied for and is the desired coupled coincidence point.

In the next theorem, we omit the continuity hypothesis of F.

Theorem 2.4Letbe a partially ordered set andGbe aG-metric onXsuch thatisg-ordered complete. Suppose that there existand mappingsandsuch that

for allwithand. Suppose also thatis complete, Fhas the mixedg-monotone property and. If there existsuch thatand, thenFandghave a coupled coincidence point.

Proof Proceeding exactly as in Theorem 2.2, we have that and are Cauchy sequences in the complete G-metric space . Then, there exist such that and . Since is non-decreasing and is non-increasing, using the regularity of , we have and for all . If and for some , then and , which implies that and , that is, is a coupled coincidence point of F and g. Then, we suppose that for all . Using the rectangle inequality, (2.15) and property for all , we get

Letting in the above inequality, we obtain

which implies that and . Thus we proved that is a coupled coincidence point of F and g. □

Corollary 2.5Letbe a partially ordered set andGbe aG-metric onXsuch thatis a completeG-metric space. Suppose that there exist, andsuch that

for allwithand. Suppose also thatFis continuous, has the mixedg-monotone property, andgis continuous and commutes withF. If there existsuch thatand, thenFandghave a coupled coincidence point.

Proof Taking with in Theorem 2.4, we obtain Corollary 2.5. □

Corollary 2.6Letbe a partially ordered set andGbe aG-metric onXsuch thatisg-ordered complete. Suppose that there exist, andsuch that

for allwithand. Suppose also thatis complete, Fhas the mixedg-monotone property, . If there existsuch thatand, thenFandghave a coupled coincidence point.

Proof Taking with in Theorem 2.4, we obtain Corollary 2.6 □

Remark 2.7 Taking (the identity mapping) in Corollary 2.5, we obtain [[2], Theorem 3.1]. Taking in Corollary 2.6, we obtain [[2], Theorem 3.2].

Now we shall prove the existence and uniqueness theorem of a coupled common fixed point. If is a partially ordered set, we endow the product set with the partial order ∇ defined by

Theorem 2.8In addition to the hypothesis of Theorem 2.2, suppose that for all, , there existssuch thatis comparable withand. Suppose also thatϕis a non-decreasing function. ThenFandghave a unique coupled common fixed point, that is, there exists a uniquesuch that

Proof From Theorem 2.2, the set of coupled coincidences is non-empty. We shall show that if and are coupled coincidence points, that is, if , , and , then

By assumption, there exists such that is comparable with and . Without loss of generality, we can assume that

and

Put , and choose such that and . Then, similarly as in the proof of Theorem 2.2, we can inductively define sequences and in X by and .

Further, set , , , and, in the same way, define the sequences , , and . Since

then and . Using that F is a mixed g-monotone mapping, one can show easily that and for all . Thus from (2.15), we get

Without loss of generality, we can suppose that for all . Since ϕ is non-decreasing, from the previous inequality, we get

for each . Letting in the above inequality and using Lemma 1.15, we obtain

Analogously, we derive that

Hence, from (2.17), (2.18) and the uniqueness of the limit, we get and . Hence the equalities in (2.16) are satisfied. Since and , by commutativity of F and g, we have

Denote and , then by (2.19), we get

Thus, is a coincidence point. Then, from (2.16) with and , we have and , that is,

From (2.20), (2.21), we get

Then, is a coupled common fixed point of F and g.

To prove the uniqueness, assume that is another coupled common fixed point. Then by (2.16), we have and . □

Theorem 2.9Letbe a partially ordered set andGbe aG-metric onXsuch thatis a completeG-metric space andis regular. Suppose that there existandhaving the mixed monotone property such that

for allwithand. If there existsuch thatand, thenFhas a coupled fixed point. Furthermore, if, then, that is, .

Proof Following the proof of Theorem 2.4 with , we have only to show that . Since , we get .

Thus, we have . Suppose that . Using inequality (2.15), we have

a contradiction. Thus, and . □

The authors declare that they have no competing interests.

All authors have contributed equally. All authors read and approved the final manuscript.

1. Aydi, H, Damjanović, B, Samet, B, Shatanawi, W: Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces. Math. Comput. Model.. 54, 2443–2450 (2011). Publisher Full Text

2. Choudhury, BS, Maity, P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model.. 54(1-2), 73–79 (2011). Publisher Full Text

3. Mustafa, Z, Sims, B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal.. 7(2), 289–297 (2006)

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

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

6. Berinde, V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal.. 74, 7347–7355 (2011). Publisher Full Text

7. Berinde, V: Coupled coincidence point theorems for mixed monotone nonlinear operators. Comput. Math. Appl. doi:10.1016/j.camwa.2012.02.012 (2012)

8. Abbas, M, Sintunavarat, W, Kumam, P: Coupled fixed point of generalized contractive mappings on partially ordered G-metric spaces. Fixed Point Theory Appl.. 2012, Article ID 31 (2012)

9. Aydi, H, Postolache, M, Shatanawi, W: Coupled fixed point results for -weakly contractive mappings in ordered G-metric spaces. Comput. Math. Appl.. 63(1), 298–309 (2012). Publisher Full Text

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

11. Aydi, H, Karapinar, E, Shatanawi, W: Tripled common fixed point results for generalized contractions in ordered generalized metric spaces. Fixed Point Theory Appl.. 2012, Article ID 101 (2012)

12. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math.. 3, 133–181 (1922)

13. Ding, H-S, Li, L: Coupled fixed point theorems in partially ordered cone metric spaces. Filomat. 25(2), 137–149 (2011). Publisher Full Text

14. 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

15. Karapınar, E, Luong, NV, Thuan, NX, Hai, TT: Coupled coincidence points for mixed monotone operators in partially ordered metric spaces. Arabian Journal of Mathematics. 1, 329–339 (2012). Publisher Full Text

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

17. Karapınar, E: Couple fixed point on cone metric spaces. Gazi University Journal of Science. 24, 51–58 (2011)

18. Mustafa, Z, Aydi, H, Karapınar, E: On common fixed points in image-metric spaces using (E.A) property. Comput. Math. Appl. doi:10.1016/j.camwa.2012.03.051 (2012)

19. Mustafa, Z, Obiedat, H, Awawdeh, F: Some fixed point theorem for mapping on complete G-metric spaces. Fixed Point Theory Appl.. 2008, Article ID 189870 (2008)

20. Mustafa, Z, Khandaqji, M, Shatanawi, W: Fixed point results on complete G-metric spaces. Studia Sci. Math. Hung.. 48, 304–319 (2011)

21. Mustafa, Z, Sims, B: Fixed point theorems for contractive mappings in complete G-metric spaces. Fixed Point Theory Appl.. 2009, Article ID 917175 (2009)

22. Mustafa, Z, Shatanawi, W, Bataineh, M: Existence of fixed point results in G-metric spaces. Int. J. Math. Math. Sci.. 2009, Article ID 283028 (2009)

23. Luong, NV, Thuan, NX: Coupled fixed point theorems in partially ordered G-metric spaces. Math. Comput. Model.. 55, 1601–1609 (2012). Publisher Full Text

24. Shatanawi, W: Fixed point theory for contractive mappings satisfying Φ-maps in G-metric spaces. Fixed Point Theory Appl.. 2010, Article ID 181650 (2010)

25. Shatanawi, W: Some fixed point theorems in ordered G-metric spaces and applications. Abstr. Appl. Anal.. 2011, Article ID 126205 (2011)

26. Shatanawi, W: Coupled fixed point theorems in generalized metric spaces. Hacet. J. Math. Stat.. 40(3), 441–447 (2011)

27. Shatanawi, W, Abbas, M, Nazir, T: Common coupled coincidence and coupled fixed point results in two generalized metric spaces. Fixed Point Theory Appl.. 2011, Article ID 80 (2011)

28. Tahat, N, Aydi, H, Karapınar, E, Shatanawi, W: Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces. Fixed Point Theory Appl.. 2012, 48 (2012). BioMed Central Full Text

29. Fréchet, M: Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo. 22, 1–74 doi:10.1007/BF03018603 (1906)

    doi:10.1007/BF03018603

    Publisher Full Text

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

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

32. Matthews, SG: Partial metric topology. Research Report 212. Dept. of Computer Science, University of Warwick (1992)

33. Guo, D, Lakshmikantham, V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal., Theory Methods Appl.. 11, 623–632 (1987). Publisher Full Text

34. Fenwick, DH, Batycky, RP: Using metric space methods to analyse reservoir uncertainty. Banff, Alberta, Canada. (2011)