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Journal of Operators
Volume 2013 (2013), Article ID 290525, 8 pages
http://dx.doi.org/10.1155/2013/290525
Research Article

Remarks on Some Recent Coupled Coincidence Point Results in Symmetric -Metric Spaces

Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia

Received 24 March 2013; Accepted 27 April 2013

Academic Editor: Antun Milas

Copyright © 2013 Stojan Radenović. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We use a method of reducing coupled coincidence point results in (ordered) symmetric -metric spaces to the respective results for mappings with one variable, even obtaining (in some cases) more general theorems. Our results generalize, extend, unify, and complement recent coupled coincidence point theorems in this frame, established by Cho et al. (2012), Aydi et al. (2011), and Choudhury and Maity (2011). Also, by using our method several recent tripled coincidence point results in ordered symmetric -metric spaces can be reduced to the coincidence point results with one variable.

1. Introduction and Preliminaries

In 2004, Mustafa and Sims introduced a new notion of generalized metric space called G-metric space, where to every triplet of elements a nonnegative real number is assigned [1]. Fixed point theory, as well as coupled and tripled cases, in such spaces were studied in [26]. In particular, Banach contraction mapping principle was established in these works.

Fixed point theory has also developed rapidly in metric and cone metric spaces endowed with a partial ordering (see [7, 8] and references therein). Fixed point problems have also been considered in partially ordered G-metric spaces [911].

For more details on the following definitions and results concerning G-metric spaces, we refer the reader to [1, 9, 1220].

Definition 1. Let be a nonempty set, and let be a function satisfying the following properties:(a) if ;(b) for all , , and with ;(c) for all , , and , with ;(d) (symmetry in all three variables); and(e) for all , , , and .
Then the function is called a G-metric on and the pair is called a G-metric space.

Definition 2. Let be a G-metric space and let be a sequence of points in .(i)A point is said to be the limit of a sequence if , and one says that the sequence is G-convergent to .(ii)The sequence is said to be a G-Cauchy sequence if, for every , there is a positive integer such that , for all ; that is, , as .(iii) is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in is G-convergent in .

Proposition 3 (see [1]). Let be a G-metric space, and let be a sequence of points in . Then the following are equivalent.(1)The sequence   is G-convergent to .(2)  as  .(3)  as  .(4)  as  .

Definition 4 (see [1, 10]). A G-metric on is said to be symmetric if for all . Every G-metric on defines a metric on by For a symmetric G-metric space, one obtains However, for an arbitrary G-metric on , just the following inequality holds: , for all .

Definition 5. In this work, one will consider the following three classes of mappings [3, 21]: is continuous and nondecreasing and , is lower semicontinuous and , and.
For weak -contractions in the frame of metric spaces see [19, 21].
At first we need the following well-known definitions and results (see, e.g., [9, 15, 22]).

Definition 6. Let be a partially ordered nonempty set, and let be two mappings. The mapping has the mixed -monotone property if for any the following hold: Note that, if , the identity mapping, then is said to have the mixed monotone property.

Remark 7. If is a partially ordered set, then is also partially ordered with

Definition 8. Let and . An element is called a coupled coincidence point of and if while is called a coupled point of coincidence of mappings and . Moreover, is called a coupled common fixed point of and if

Remark 9. Otherwise, is a coupled coincidence point of and if and only if is a coincidence point of the mappings and which are defined by

Definition 10. Mappings are said to be compatible in a G-metric space if whenever is a sequence in such that in .
It is easy to prove that and are compatible in if and only if they are compatible in the associated metric space .

Definition 11. Let be a G-metric space, and let and be two mappings. One says that and are compatible if whenever and are such that The proof of the following lemma is immediate (for (2) see [18]).

Lemma 12. Let be a symmetric G-metric space. Define by Then is a symmetric G-metric space.
Let . Define and extend to by using the symmetry in the variables. Then it is clear that is an asymmetric G-metric space. It is not hard to see that is not a G-metric space.

Remark 13. One can prove that mappings and are compatible in if and only if and are compatible in .

Definition 14. (1) Let be a nonempty set. Then is called an ordered G-metric space if(i) is a G-metric space and(ii) is a partially ordered set.
(2) Let be a partially ordered G-metric space. One says that is regular if the following hypotheses hold:(i)if a nondecreasing sequence is such that as , then for all ,(ii)if a nonincreasing sequence is such that as , then for all .

2. Main Results

Now, we are ready to state and prove our first result.

Theorem 15. Let be a partially ordered symmetric Gmetric space, , and . Assume that there exist and such that for all , , , , , and for which or . Assume that and satisfy the following conditions:; has the mixed -monotone property; and are continuous and compatible and is G-complete, or is regular and one of or is G-complete;there exist such thatThen and have a coupled coincidence point.

Remark 16. (a) Obviously, the condition (1) from [23] is equivalent to the condition (13). Hence, by using a new symmetric G-metric space we have obtained a method of reducing coupled coincidence and coupled fixed point results in (ordered) symmetric G-metric spaces to the respective results for mappings with one variable, even obtaining (in some cases) more general theorems. We note that this method cannot be applied in the case of asymmetric G-metric spaces (see (2) of Lemma 12). For other details of coupled case in ordered metric spaces see also [22].
(b) Also, we note that Theorem 3.1. from [23] holds if and are compatible instead of commuting (see Step 3 in [23]). Indeed, since then because and are compatible.
Therefore, now we have that is, . Similarly, we obtain that .

Assertions similar to the following lemma were used in the frame of metric spaces in the course of proofs of several fixed point results in various papers (see, e.g., [9, 21]). This lemma holds in every G-metric space.

Lemma 17. Let be a G-metric space, and let be a sequence in such that . If is not a G-Cauchy sequence in , then there exist and two sequences and of positive integers such that the following four sequences tend to when :

The following lemma is crucial for the proof of Theorem 15, and it holds in every G-metric space.

Lemma 18. Let be a partially ordered G-metric space, and let and be two self-mappings on . Assume that there exist and such that for all , , , , , , , , and for which or . If the following conditions hold:(i) is a -nondecreasing with respect to and ;(ii)there exists such that ;(iii) and are continuous and compatible, and is G-complete or is regular, and one of or is G-complete.
Then and have a coincidence point in .

Proof. If , then is a coincidence point of and . Therefore, let . Since , we obtain a Jungck sequence for all , where , and by induction we get that . If for some , then is a coincidence point of and . Therefore, suppose that for each . Now, we will prove the following:(1) as ;(2) is a G-Cauchy sequence.
Indeed, by putting , , and in (19) we get and since the function is nondecreasing, it follows that ; that is, there exists . If , we get from the previous relation ; that is, which is a contradiction. Hence, we obtain that .
Further, using Lemma 17 we shall prove that is a G-Cauchy sequence. Suppose this is not the case. Then, by Lemma 17 there exist and two sequences and of positive integers such that the following sequences tend to when : Putting , , and in (19) we have that is, Letting , we get ; that is, . Since , we get , which is a contradiction. We have proved that is a G-Cauchy sequence in .
In case (iii), since is G-complete, there exists such that . Then we have Further, according to Definition 1 (e) and since and are continuous and compatible, we get It follows that is a coincidence point for and .
In case , it follows that , (in both cases when or is G-complete), and then , and by the contractive condition (19) we have By taking limit as in the above inequality, we obtain and hence .

Proof of Theorem 15. Firstly, (13) implies for all , and from for which or .
Further,(1) implies that ;(2) implies that is nondecreasing with respect to and ;(3) implies that and are continuous and compatible and is G-complete,
or implies that is regular and one of or is G-complete;(4) implies that there exists such that or .
All conditions of Lemma 18 for the ordered G-metric space are satisfied. Therefore, the mappings and have a coincidence point in . According to Remark 9 the mappings and have a coupled coincidence point. The proof of Theorem 15 is complete.

Our second main result is the following.

Theorem 19. Let be a partially ordered symmetric G-metric space, , and . Assume that there exists such that for all , , , , , and for which or . Assume that and satisfy the following conditions:(1);(2) has the mixed monotone property;(3) and are continuous and compatible and is G-complete, or is regular and one of or is G-complete;(4)there exist such thatThen and have a coupled coincidence point.

For the proof of Theorem 19 we use the following.

Lemma 20. Let be a partially ordered G-metric space, and let and be two self-mappings on . Assume that there exists such that for all , , , , , , , , and for which or . If the following conditions hold:(i) is nondecreasing with respect to and ;(ii)there exists such that ;(iii) and are continuous and compatible, and is G-complete or is regular and one of or is G-complete,
Then and have a coincidence point in .

Proof. If , then is a coincidence point of and . Therefore, let . Since , we obtain a Jungck sequence for all , where , and by induction we get that . If for some , then is a coincidence point of and . Therefore, suppose that for each . Now, we will prove that as .
Indeed, by putting , , and in (31) we get That is, there exists . If , we get which is a contradiction. Hence, we obtain that .
Further, by using Lemma 17, we shall prove that is a G-Cauchy sequence. Suppose this is not the case. Then, by Lemma 17 there exist and two sequences and of positive integers such that the following sequences tend to when : Putting , , and in (31) we have that is, Letting , we obtain Hence, we get , which is a contradiction. We have proved that is a G-Cauchy sequence in .
Now, in case , since is G-complete, there exists such that . Then we have and because it follows that is a coincidence point for and .
In case  , it follows that , (in both cases when or is G-complete), and then , and by the contractive condition (31) we have By taking the limit as in the above inequality we obtain and hence .

Proof of Theorem 19. The proof is very similar to the proof of Theorem 15. Namely, the contractive condition (29) for the mappings and is equivalent to the following condition: for the mappings and . The proof is further an immediate consequence of Lemma 20.

Remark 21. We have obtained that, in case of symmetric G-metric spaces, Theorem 19 generalizes both results (Theorems 3.1 and 3.2) from [3]. Also, Example 22 shows that this generalization is proper. It is clear that our new method with ordered symmetric G-metric spaces and with mappings and implies that all results from [3] can be reduced to known results with one variable.
The following example supports both of our theorems, the first with and and the second with .

Example 22. Let be endowed with the complete G-metric for all , , and and with the usual order. Consider the mappings and . All the conditions of Theorems 15 and 19 are satisfied. In particular, the mapping has the mixed -monotone property, and we will check that and are compatible.
Let and be two sequences in such that Then and , wherefrom it follows that . Then and similarly Also, and do not commute, and therefore a coupled coincidence point of and cannot be obtained by Theorem 3.1 from [3].
The contractive condition (13) is satisfied with and which follows from for all , , , , , and for which or .
Hence, Similarly, the condition (29) is satisfied with ; that is, for all , , , , , and for which or . There exists a coupled coincidence point of the mappings and .

Acknowledgments

The author is very grateful to the reviewer for the useful remarks and interesting comments. He is thankful to the Ministry of Education, Science and Technological Development of Serbia.

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