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International Journal of Analysis
Volume 2013 (2013), Article ID 736451, 6 pages
http://dx.doi.org/10.1155/2013/736451
Research Article

Related Fixed Point of Set-Valued Mappings in Three Menger Spaces

1Centre for Applicable Mathematics and Statistics, University of Central Punjab, Lahore 54770, Pakistan
2R. H. Government Postgraduate College, Kashipur, Udham Singh Nagar, Uttarakhand 244 713, India

Received 22 November 2012; Revised 20 February 2013; Accepted 3 March 2013

Academic Editor: Alexandre Timonov

Copyright © 2013 Ismat Beg and Sunny Chauhan. 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 a related fixed point theorem for set-valued mappings in three Menger spaces. Some examples are also furnished to support the results. Our main result generalizes and extends several known results in the literature.

1. Introduction

Fisher [1] initiated the study of conditions for the existence of a relation connecting the fixed points of two mappings in two different metric spaces (also see [2]). Afterward, Fisher and Tűrkoğlu [3] proved a related fixed point theorem for set-valued mappings in two metric spaces. In 2003, Chourasia and Fisher [4] established a related fixed point theorem for two pairs of set valued mappings in two metric spaces. Several mathematicians have extensively developed metrical related fixed point theorems (see, e.g., [58]). Later on, Pant [9] firstly studied related fixed point theorems for single valued mappings in two complete Menger spaces and generalized the results of Fisher [1, 2]. Pant and Kumar [10] extended the results of Namdeo et al. [7] to two pairs of single-valued mappings.

In this paper, we prove a related fixed point theorem for set-valued mappings in three complete Menger spaces. Our result generalizes the result of Jain et al. [6] and extends the result of Fisher and Tűrkoğlu [3].

2. Preliminaries

Definition 1 (see [11]). A mapping is called a -norm if(i), ,(ii),(iii), whenever and ,(iv), for all .

Examples of -norms are , , and .

Definition 2 (see [11]). A mapping is called distribution function if it is nondecreasing, left continuous with and .

Let be the set of all distribution functions whereas stands for the specific distribution function (also known as Heaviside function) defined by

Definition 3 (see [11]). Let be a nonempty set. An ordered pair is called a PM-space where is a mapping from satisfying the following conditions:(i) if and only if ,(ii),(iii) and ; then for all and .

Every metric space can be realized as a PM-space by considering defined by for all . So PM-spaces offer a wider framework (than that of the metric spaces) and are general enough to cover even wider statistical situations.

Definition 4 (see [12]). A Menger space is a triplet where is a PM-space and is a -norm satisfying the following condition: for all and .

Definition 5 (see [11]). Let be a Menger space and a continuous -norm. A sequence in is said to be(i)convergent to a point in if and only if for every and , there exists a positive integer such that for all .(ii)Cauchy if for every and , there exists a positive integer such that for all .A Menger space in which every Cauchy sequence is convergent is said to be complete.

Lemma 6 (see [13]). Let be a Menger space. If there exists a constant such that for all with fixed , then .

We need the following definition due to Chang et al. [14] for subsequent use in next section.

Definition 7 (see [14]). Let be a Menger space and a nonempty subset of . Then is said to be probabilistically bounded if .

If is itself probabilistically bounded, then is said to be a probabilistically bounded space.

Let be a family of nonempty closed and bounded subsets of a Menger space . For all and for every , we define If the set consists of a single point , we write . Also if the set consists of a single point , we write . It follows immediately from the definition that(i),(ii) if and only if , for all .

3. Main Result

Now we prove a related fixed point theorem for set-valued mappings in three complete Menger spaces.

Theorem 8. Let , , and be three complete Menger spaces, where is a continuous -norm. If is a continuous mapping of into , is a continuous mapping of into , and is a mapping of into satisfying the inequalities for all , , , and , Then has a unique fixed point in , has a unique fixed point in , and has a unique fixed point in . Further , , and .

Proof. Let be an arbitrary point in . We define the sequences in , in , and in inductively as follows. We choose a point in , then a point in , and then a point in . In general, having chosen in , in , and in , choose a point in , then a point in , and then a point in for . Applying inequality (5), we get Using inequality (6), we get on using inequality (8). Further, on using inequality (7), we have on using inequalities (8) and (9). It follows easily by induction on using inequalities (8), (9), and (10) that for all . Since , , and as , it follows that , , and are Cauchy sequences in complete Menger space and so one has a limit in . Similarly, the sequences and are also Cauchy sequences with limits and in the complete Menger spaces and , respectively. Then we have Using the continuity of and , we have and then we see that Now we show that is a fixed point of . Applying inequality (5), we have Since and are continuous, it follows on letting and using (12) that Owing to Lemma 6, and so is a fixed point of . We now have on using (15) and then on using (14). Hence and are fixed points of and , respectively. Further, we see that To prove the uniqueness of , suppose that has another fixed point . Then, using inequality (5), we have Next, using inequality (6), we have and using inequality (7), we obtain It now follows easily from inequalities (22) and (23) that and then Using inequalities (21), (24), and (25), we obtain for all . Since as , it follows that and so is a singleton. It then follows from inequality (25) that is a singleton and from inequality (24) that is a singleton. Using inequality (5) again, we have Next using inequality (6), we have and using inequality (7), we have It follows from inequalities (27), (28), and (29) that , which shows the uniqueness of . The uniqueness of and can be verified similarly.

Example 9. Consider , , and and define for all . Then , , and are three complete Menger spaces with continuous -norm for all . Define the set-valued mappings , and by for all , for all and We can easily verify the inequalities with . Thus all the conditions of Theorem 8 are satisfied and has a unique fixed point in , has a unique fixed point in , and has a unique fixed point in . Hence, , , and , and so , , and .

Corollary 10. Let , and be three complete Menger spaces, where is a continuous -norm. If is a continuous mapping of into , is a continuous mapping of into and is a mapping of into satisfying the inequalities for all , , , , and . Then has a unique fixed point in , has a unique fixed point in and has a unique fixed point in . Further, , and .

Example 11. Let , , and , and be defined as Example 9. Then , , and be three complete Menger spaces with continuous -norm for all . Define the mappings , and by for all , for all and Also the inequalities (32) can be verified with . Thus all the conditions of Corollary 10 are satisfied and has a unique fixed point 1 in , has a unique fixed point 2 in , and has a unique fixed point 4 in . Further, , , and ; that is, , , and .

The following corollaries are the natural results due to Theorem 8 and Corollary 10.

Corollary 12. Let and be two complete Menger spaces, where is a continuous -norm. If is a continuous mapping of into , is a mapping of into satisfying the inequalities
for all , , , and . Then has a unique fixed point in and has a unique fixed point in .

Corollary 13. Let and be two complete Menger spaces, where is a continuous -norm. If is a continuous mapping of into , is a mapping of into satisfying the inequalities for all , , , and . Then has a unique fixed point in and has a unique fixed point in .

Remark 14. Theorem 8 generalizes the result of Jain et al. [6] and Fisher [2]. Corollaries 1013 extend the results of Fisher and Tűrkoğlu [3, Theorem 1, Corollary 1] to PM-spaces.

Acknowledgment

The authors are thankful to anonymous referees for their valuable suggestions and comments to improve the paper.

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