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The Scientific World Journal
Volume 2013 (2013), Article ID 810732, 7 pages
http://dx.doi.org/10.1155/2013/810732
Research Article

g-Weak Contraction in Ordered Cone Rectangular Metric Spaces

1Department of Mathematics, Govt. S.G.S.P.G. College Ganj Basoda, Vidisha 464221, India
2Department of Mathematics, Choithram College of Professional Studies, Dhar Road, Indore 453001, India
3Department of Applied Mathematics, Shri Vaishnav Institute of Technology & Science, Gram Baroli Sanwer Road, Indore 453331, India

Received 13 April 2013; Accepted 12 May 2013

Academic Editors: T. Ozawa and S. A. Tersian

Copyright © 2013 S. K. Malhotra et al. 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 prove some common fixed-point theorems for the ordered g-weak contractions in cone rectangular metric spaces without assuming the normality of cone. Our results generalize some recent results from cone metric and cone rectangular metric spaces into ordered cone rectangular metric spaces. Examples are provided which illustrate the results.

1. Introduction and Preliminaries

There are a number of generalizations of metric spaces. One such generalization is obtained by replacing the real valued metric function with a vector valued metric function. In the mid-20th century (see [1]), the notions of K-metric and K-normed spaces were introduced, in such spaces an ordered Banach space instead of the real numbers was used as a codomain for metric function. Indeed, this idea of replacement of real numbers by an ordered “set” can be seen in [2, 3] (see also references therein). Huang and Zhang [4] reintroduced such spaces under the name of cone metric spaces, defining convergent and Cauchy sequence in terms of interior points of underlying cone. They proved the basic version of the fixed-point theorem with the assumption that the cone is normal. Subsequently several authors (see, e.g., [514]) generalized the results of Huang and Zhang. In [13], Rezapour and Hamlbarani removed the normality of cone and proved the results of Huang and Zhang in nonnormal cone metric spaces.

In [15], Branciari introduced a class of generalized metric spaces with replacing triangular inequality by similar ones which involve four or more points instead of three and improved Banach contraction principle. Azam and Arshad [16] proved fixed-point result for Kannan-type contraction in rectangular metric spaces. After the work of Huang and Zhang [4], Azam et al. [17] introduced the notion of cone rectangular metric spaces and proved fixed-point result for Banach-type contraction in cone rectangular space. Samet and Vetro [18] obtained the fixed-point results in c-chainable cone rectangular metric spaces.

Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton’s approximation method [19] and in optimization theory [20]. The existence of fixed point in partially ordered sets was investigated by Ran and Reurings [21] and then by Nieto and Rodríguez-López [22]. Fixed-point results in ordered cone metric spaces were obtained by several authors (see, e.g., [11, 2325]). Very recently, Malhotra et al. [26] proved the fixed-point results in ordered cone rectangular metric spaces for Reich-type contractions.

The notion of -weak contraction is introduced by Vetro (see [14]) in cone metric spaces. In this paper, we prove some common fixed point theorems for -weak contractions in ordered cone rectangular metric spaces. Our results generalize and extend the results of Huang and Zhang [4], Azam et al. [17], Azam and Arshad [16], Malhotra et al. [26], and the result of Vetro [14] on ordered cone rectangular metric spaces.

We need the following definitions and results, consistent with [4, 20].

Definition 1 (see [4]). Let be a real Banach space and a subset of . The set is called a cone if(i)is closed, nonempty, and ; here is the zero vector of ; (ii), , ;(iii) and .
Given a cone , we define a partial ordering “” with respect to by if and only if . We write to indicate that but , while if and only if , where denotes the interior of .
Let be a cone in a real Banach space ; then is called normal, if there exists a constant such that for all , The least positive number satisfying the above inequality is called the normal constant of .

Definition 2 (see [4]). Let be a nonempty set and a real Banach space. Suppose that the mapping satisfies(i) for all and if and only if ; (ii) for all ; (iii) for all .
Then, is called a cone metric on , and is called a cone metric space. In the following, we always suppose that is a real Banach space, and is a solid cone in ; that is, and “” is partial ordering with respect to .

The concept of cone metric space is more general than that of a metric space because each metric space is a cone metric space with and .

For examples and basic properties of normal and nonnormal cones and cone metric spaces, we refer to [4, 13].

The following remark will be useful in sequel.

Remark 3 (see [27]). Let be a cone in a real Banach space , and we then have the following (a)If and then .(b)If and then .(c)If for each , then .(d)If and , then there exist such that, for all , we have .(e)If for each and , then .(f)If , where , then .

Definition 4 (see [17]). Let be a nonempty set. Suppose that the mapping satisfies (i) for all and if and only if ; (ii) for all ; (iii) for all and for all distinct points (rectangular property).
Then, is called a cone rectangular metric on , and is called a cone rectangular metric space. Let be a sequence in and . If for every , with there is such that for all , , then is said to be convergent, converges to , and is the limit of . We denote this by or , as . If for every with there is such that for all and we have , then is called a Cauchy sequence in . If every Cauchy sequence is convergent in , then is called a complete cone rectangular metric space. If the underlying cone is normal, then is called normal cone rectangular metric space.

Example 5. Let , , and .
Define as follows: Now is a cone rectangular metric space but is not a cone metric space because it lacks the triangular property: as .

Note that in the above example is a normal cone rectangular metric space. The following is an example of non-normal cone rectangular metric space.

Example 6. Let , with , and for . Then, this cone is not normal (see [13]).
Define as follows: Then is nonnormal cone rectangular metric space but is not a cone metric space because it lacks the triangular property.

Definition 7 (see [5]). Let and be self-mappings of a nonempty set and . The pair is called weakly compatible if for all . If for some in , then is called a coincidence point of and , and is called a point of coincidence of and .

Definition 8. If a nonempty set is equipped with a partial order “” and mapping such that is a cone rectangular metric space, then is called an ordered cone rectangular metric space. Let be two mappings. The mapping is called nondecreasing with respect to “”, if for each , implies . The mapping is called -nondecreasing if for each , implies . A subset of is called well ordered if for all the elements of are comparable; that is, for all either or . is called -well ordered if all the elements of are -comparable; that is, for all either or .

In the trivial case, that is, for (the identity mapping of ), the -well orderedness reduces into well orderedness. But, for nontrivial cases, that is, when the concepts of -well orderedness and well orderedness are independent.

Example 9. Let , let “” be a partial order relation on defined by ,. Let , and be defined by . Then it is clear that is not well ordered but it is -well ordered, while is not -well ordered but it is well ordered.

Let be an ordered cone rectangular metric space two mappings. The mapping is called ordered Reich-type contraction if for all with such that and If (5) is satisfied for all , then is called Reich contraction.

The mapping is called an ordered -weak contraction if for all with , where , , and are nonnegative constants such that . If inequality (6) is satisfied for all , then is called a -weak contraction.

Note that for (the identity mapping of ) the ordered -weak contraction reduces into the ordered Reich contraction.

Now, we can state our main results.

2. Main Results

Theorem 10. Let be an ordered cone rectangular metric space two mappings such that and is complete. Suppose that the following conditions are satisfied: (i) is an ordered -weak contraction, that is, satisfies (6); (ii) is -nondecreasing; (iii)there exists such that ; (iv)if were any nondecreasing sequence in converging to some , then for all and .
Then, and have a coincidence point. Furthermore, if and are weakly compatible then they have a common fixed point. In addition, the set of common fixed points of and is -well ordered if and only if the common fixed point of and is unique.

Proof. Starting with given , we define a sequence as follows: let (which is possible as ). As , we have , and as is -nondecreasing, we obtain . Again, therefore let . Since and is -nondecreasing, we obtain . On repeating this process, we obtain Thus, is a nondecreasing sequence with respect to .
We will show that and have a point of coincidence. If, for any , we have ; therefore, is a point of coincidence of and with coincidence point . Therefore, we assume that for all .
As, for all , it follows from (6) that For simplicity, set for all ; then it follows from above inequality that where (as ). By repeating this process, we obtain If for any and positive integer , then as , it follows from (6) that that is, Repeating this process times, we obtain a contradiction. Therefore, we can assume that for all distinct .
Again, as , we obtain from (6) and (10) that
that is, As , we have where .
For the sequence , we consider in two cases.
If is odd say , then using rectangular inequality and (10), we obtain Therefore, If is even, say , then using rectangular inequality and (10) and (16), we obtain Therefore, As and , we have and . So, it follows from (18), (20), and (a), (d) of Remark 3 that for every with there exists such that and for all . Thus, is a Cauchy sequence in . As is complete, there exist such that We will show that .
Now, From (iv), we have ; that is,; therefore it follows from (6) that that is, In view of (10), (21), and (a), (d) of Remark 3, for every with , there exists such that , for all . Therefore, it follows from above inequality that Therefore, again with same arguments, from (22) and (c) of Remark 3, we obtain that ; that is,. Thus, is a coincidence point and is point of coincidence of and .
Now suppose that and are weakly compatible; then we have . As , therefore using (6) we obtain As , it follows from (f) of Remark 3 and the above inequality that ; that is, . Thus, is a common fixed point of and .
Suppose the set of common fixed points of and , that is, , is -well ordered and . As is -well ordered, let, for example, . Then, it follows from (6) that As , it follows from (f) of Remark 3 that ; that is,. Therefore, the common fixed point of and is unique. For converse, let common fixed point be unique, then will be singleton and therefore -well ordered.

Remark 11. For the existence of common fixed point of and , Vetro [14] used the condition Here, we have used the weak compatibility of mappings and , and it is obvious that the condition used by Vetro implies the weak compatibility of mappings and .

Taking (identity mapping of ), we obtain the main result of [26].

Corollary 12. Let be an ordered complete cone rectangular metric space and a mapping such that the following conditions are satisfied: (i) is an ordered Reich-type contraction, that is, satisfies (5); (ii) is nondecreasing with respect to “”; (iii)there exists such that ; (iv)if is any nondecreasing sequence in converging to some then , for all .
Then, has a fixed point. In addition, the set of fixed points of is well ordered if and only if the fixed point of is unique.

With suitable values of control constants , , and , we obtain the following generalizations of Theorems 2.1 and 2.3 of Abbas and Jungck [5] on ordered cone rectangular metric spaces.

Corollary 13. Let be an ordered cone rectangular metric space two mappings such that and is complete. Suppose that the following conditions are satisfied: (i) for all with , where ; (ii) is -nondecreasing; (iii)there exists such that ; (iv)if were any nondecreasing sequence in converging to some then , for all and .
Then, and have a coincidence point. Furthermore, if and are weakly compatible then they have a common fixed point. In addition, the set of common fixed points of and is -well ordered if and only if the common fixed point of and is unique.

Corollary 14. Let be an ordered cone rectangular metric space and two mappings such that and is complete. Suppose that the following conditions are satisfied: (i) for all with , where ; (ii) is -nondecreasing; (iii)there exists such that ; (iv)if were any nondecreasing sequence in converging to some , then for all and .
Then, and have a coincidence point. Furthermore, if and are weakly compatible then they have a common fixed point. In addition, the set of common fixed points of and is -well ordered if and only if the common fixed point of and is unique.

The following example shows that the results of this paper are a proper generalization of the results of Malhotra et al. [26] and Vetro [14].

Example 15. Let and with , for . Define as follows: Then, is a complete nonnormal cone rectangular metric space but not cone metric space. Define mappings and partial order “” on as follows: Then it is easy to verify that is an ordered -weak contraction in with . Indeed, we have to check the validity of (6) only for . Then, therefore, (6) holds for , . Again, therefore, (6) holds for arbitrary , , and such that .
All other conditions of Theorem 10 are satisfied and is the unique common fixed point of and . Note that is not an ordered Reich-type contraction. Indeed, for point there are no such that condition (5) is satisfied. Therefore, the results of Malhotra et al. [26] are not applicable here.

The following example illustrates the crucial role of weak compatibility of mappings for the existence of common fixed point in Theorem 10.

Example 16. Let be the cone rectangular metric space as in Example 15. Then, is a complete nonnormal cone rectangular metric space but not cone metric space. Define mappings and partial order “” on as follows: Then, it is easy to verify that is an ordered -weak contraction in with . Indeed, we have to check the validity of (6) only for . Then, Therefore, (6) holds for , . Again, Therefore, (6) holds for arbitrary , , and such that .
Similarly, for , and condition (6) holds for arbitrary , , and such that .
All other conditions of Theorem 10 (except and are weakly compatible) are satisfied and is a coincidence point of and . Note that ; that is, is a coincidence point of and but ; therefore, and are not weakly compatible and have no common fixed point.

In the following theorem, the conditions on , “nondecreasing” and “completeness of space,” are replaced by another condition.

Theorem 17. Let be an ordered cone rectangular metric space and is two mappings such that . Suppose that the following conditions are satisfied: (A) is an ordered -weak contraction that satisfies (6); (B)there exists such that and for all .
Then, and have a coincidence point. Furthermore, if and are weakly compatible, then they have a common fixed point. In addition, the set of common fixed points of and is -well ordered if and only if the common fixed point of and is unique.

Proof. Let for all and for some (which is possible, since ); then for all . If , then ; that is, is a coincidence point of and . If , then by assumption (B) , so , and by (A), we obtain a contradiction. Therefore, we must have ; that is,, and so is a coincidence point of and .
The existence, necessary and sufficient condition for uniqueness of common fixed point follows from a similar process as used in Theorem 10.

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