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 haveSince is nondecreasing function, we can writeHence, we can write a contradiction as for each .
Consider We haveSince is nondecreasing function, From the above conditions, we havea 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.