Journal of Operators
Volume 2015 (2015), Article ID 195731, 9 pages
http://dx.doi.org/10.1155/2015/195731

Research Article

Common Fixed Point Theorems of Greguš Type -Weak Contraction for -Weakly Commuting Mappings in 2-Metric Spaces

Penumarthy Parvateesam Murthy and Uma Devi Patel

Department of Pure and Applied Mathematics, Guru Ghasidas Vishwavidyalaya, Bilaspur, Chhattisgarh 495009, India

Received 14 July 2015; Accepted 21 September 2015

Academic Editor: Ram U. Verma

Copyright © 2015 Penumarthy Parvateesam Murthy and Uma Devi Patel. 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.

The main purpose of this paper is to establish a common fixed point theorem for set valued mappings in 2-metric spaces by generalizing a theorem of Abd EL-Monsef et al. (2009) and Murthy and Tas (2009) by using -weak contraction in view of Greguš type condition for set valued mappings using -weakly commuting maps.

1. Introduction and Preliminaries

The weak contraction condition in Hilbert Space was introduced by Alber and Guerre-Delabriere [1]. Later, Rhoades [2] has noticed that the results of Alber and Guerre-Delabriere [1] in Hilbert Spaces are also true in a complete metric space.

Rhoades [2] established a fixed point theorem in a complete metric space by using the following contraction condition.

Let which satisfies the following condition:

where and is a continuous ad nondecreasing function such that if and only if .

Remark 1. In this above result, if , where , then we obtain condition (1) of Banach.

In the recent years, Dutta and Choudhury [3], Zhang and Song [4], and Đorić [5] have given the results in -weak contractive mapping.

The concept of 2-metric space is a natural generalization of the metric space. The concept of 2-metric spaces has been investigated initially in a series of papers (see Gahler [6–8]) and has been developed extensively by Gahler and many others. Gahler defined a 2-metric space as follows.

Definition 2 (see [6]). A 2-metric space on a set with at least three points is nonnegative real-valued mapping satisfying the following conditions:

(1)For two distinct points , there exists a point such that .(2) if at least two of , , and are equal.(3).(4) for all , , , and in .

The function is called a 2-metric for the space and the pair is then called a 2-metric space.

Geometrically, the value of a 2-metric represents the area of a triangle with vertices , , and .

After this, a number of fixed point theorems have been proved for 2-metric spaces by introducing compatible mappings, commuting and weakly commuting mappings. There were some generalizations of metric such as a 2-metric, a -metric, a -metric, a cone metric, and a complex-valued metric. Note that a 2-metric is not a continuous function of its variables, whereas an ordinary metric is. This led Dhage to introduce the notion of a -metric in [9]. But, in 2003, Mustafa and Sims [10] demonstrated that most of the claims concerning the fundamental topological properties of -metric spaces are incorrect. After that, in 2006, Mustafa and Sims [11] introduced the notion of -metric spaces. Only a 2-metric space has not been known to be topologically equivalent to an ordinary metric. Then, there was no easy relationship between results obtained in 2-metric spaces and metric spaces. In particular, the fixed point theorems on 2-metric spaces and metric spaces may be unrelated easily. For more fixed point theorems on 2-metric spaces, the researchers may refer to [12–15].

Throughout this paper, is for a 2-metric space and is the class of all nonempty bounded subsets of .

Definition 3 (see [15]). A sequence in is said to be convergent to a point , denoted by , if for all . The point is called the limit of the sequence in .

Definition 4 (see [15]). A sequence in is said to be Cauchy sequence if , for all .

Definition 5 (see [15]). The space is said to be complete if every Cauchy sequence in converges to a point of .

Let , , and be nonempty sets in . Let and be the functions defined by

If is a singleton set, then In case and are also singleton sets, then

for every , , and From the definition of , we can say that

Also,

for all . Let us note that if at least two of , , and are equal singleton sets.

Definition 6 (see [15]). A sequence of subset of a 2-metric space is said to be convergent to a subset of if,

(1)given , there is a sequence in such that for and ;(2)given , there exists a positive integer such that for , where is the union of all open spheres with centers in and radius

Definition 7 (see [16]). Let and . Then, the pair is said to be weakly commuting if and

for every , and .

Definition 8 (see [16]). Let and . Then, the pair is said to be -weakly commuting if

for every , and and .

Remark 9 (see [16]). If is a single valued function, then Definitions 7 and 8 reduce to the following:

respectively.

Common fixed points of Greguš type [17] have been proved by Diviccaro et al. [18], Fisher and Sessa [19], Mukherjee and Verma [20], Murthy et al. [21], and Singh et al. [14] under weaker conditions. Later, Murthy and Tas [16] generalized and extended the results of Singh et al. [14] and proved a theorem for set valued mapping in 2-metric space.

In this paper, we generalize the results of Abd EL-Monsef et al. [15] and Murthy and Tas [16] by using -weak contraction with Greguš type condition in 2-metric spaces for set valued mapping for -weakly commuting maps.

2. Main Results

Let and be mapping of 2-metric space into itself and are two set valued mappings satisfying the following condition:

for every , , and ,

where

, , , and

(1) is continuous monotone nondecreasing function with if and only if ;(2) is lower semicontinuous, monotone decreasing function with if and only if and for all

Let be an arbitrary point in . Since , then a point such that . Now again, since , for the point we can find a point such that and so on. Inductively, we can construct a sequence in such that

Now we need to prove the following lemma for our main theorem.

Lemma 10. Let be 2-metric space. Let and be self-maps of and satisfying conditions (10) and (11). Then, for every , one has

Proof. Since

we have

Since is nondecreasing function, we can write

Hence, we can write

a contradiction as for each .
Consider

We have

Since is nondecreasing function,

From the above conditions, we have

a contradiction. Hence, we have

Now we are ready to prove a common fixed point theorem by using the concept of -weakly commuting maps theorem as follows.

Theorem 11. Let and be mapping of 2-metric space into itself and are two set valued mappings satisfying conditions (10), (11), and (12) and the following:

(a) or is a complete subspace of .(b)The pair and are R-weakly commuting.

Then , and have unique common fixed point in .

Proof. Let be an arbitrary point in . Since , then there exists a point such that . Now again, since , for the point we can find a point such that and so on. Inductively, we can construct a sequence in such that

Firstly, we have to prove that

For this, assume for by using (11):

where

By (26), we get

If we take , in the above equation by using the property of and function, we can write

The above implies that

a contradiction. So we obtain

By using (28) and (31) and employing the properties of and function, we may write

Again, (31) implies that

Therefore, is a monotone decreasing sequence of nonnegative real number. There exists a nonnegative real number such that

Letting limit in (33), we get

a contradiction with the property of and function. This implies that . Thus, we have

Now, repeating the above process by putting and , we obtain

Hence, for all , we can write

Next, we will show that is a Cauchy sequence. If, otherwise, there exists and sequence of natural numbers and such that, for every natural number ,

corresponding to , we can choose to be the smallest integer such that (41) is satisfied. Then, we have

Putting and in (11), we get

where

Letting limit in the above,

Now

We have to show that

Now, using properties of 2-metric space, we get

Letting , we get

By using properties of 2-metric space, we can write

Letting limit , we have

Again using properties of 2-metric space, we can write

Letting limit , we have

Using (46), (49), (51), and (53), we have

Since is nondecreasing function, .
Therefore,

a contradiction with function; hence, is a Cauchy sequence.
Assume is a complete subspace . Since the sequence is Cauchy, then its subsequence is Cauchy and converges to a point in . Since is complete subspace of , for some ,

According to the construction of sequence, we can have

Letting limit ,

The above implies that

Therefore, we get

Similarly,

Letting limit ,

Therefore, we get

Now we will show that is a coincidence point of and . For and putting and in (11) and (12), we have

where

Letting limit in the above equation, we have

Letting limit in (64), we get

This implies that

Since is monotone nondecreasing function, we can write

a contradiction. Hence, is the coincidence point of and ; that is, Since , for some , we have . If , then, from (11) and (12), putting and , we get

where

Since , , and , we have

Since is monotone nondecreasing function, we can write

a contradiction. Hence, .
Since are -weakly commuting maps, then

which implies that

Again, using (11), putting and , we have

Since is nondecreasing function, we can write

This implies that

a contradiction. Hence, we get and is a common fixed point of and .
Similarly, we can show that is common fixed point of and by assuming is a pair of -weakly commuting maps. Hence, .
For the uniqueness of common fixed point , let be another fixed point of , , , and . By using (11), we have

Since is nondecreasing function, we can write

This implies that

a contradiction. Hence, we have ; that is, that is unique common fixed point of , and in .

As an immediate consequence of the above theorem, we have the following corollaries.

Corollary 12. Let and be mapping of 2-metric space into itself and are two set valued mappings satisfying conditions (a) and (b) of Theorem 11 and the following conditions:

for every , , and ,

where

, , , and is lower semicontinuous, monotone decreasing function with if and only if and for all
Then , and have unique common fixed point in .

Proof. The proof follows from Theorem 11 by taking .

Corollary 13. Let and be mapping of 2-metric space into itself and are two set valued mappings satisfying conditions (a) and (b) of Theorem 11 and the following conditions:

for every , , and ,

where

, , , and Then, , and have unique common fixed point in .

Proof. The proof follows from Corollary 12 by taking , where .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are thankful to the anonymous referee for the comments on this paper which enabled them to present the paper in this form. The first author (P. P. Murthy) is thankful to University Grants Commission, New Delhi, India, for providing financial assistance through Major Research Project: File no. 42-32/2013 (SR).

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