Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 846315, 6 pages

http://dx.doi.org/10.1155/2013/846315

## Common Fixed Points for JH-Operators and Occasionally Weakly g-Biased Pairs under Relaxed Condition on Probabilistic Metric Space

^{1}Department of Applied Science (Mathematics), Bipin Tripathi Kumaun Institute of Technology Dwarahat (Almora), Uttarakhand Technical University, Dehradun, Uttarakhand 263653, India^{2}Department of Mathematics and DST-CIMS, Banaras Hindu University, Varanasi 221005, India

Received 14 May 2013; Revised 2 July 2013; Accepted 3 July 2013

Academic Editor: C. Cuevas

Copyright © 2013 Arvind Bhatt and Harish Chandra. 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 obtain some fixed point theorems for JH-operators and occasionally weakly g-biased maps on a set *X* together with the function without using the triangle inequality and without using the symmetric condition. Our results extend the results of Bhatt et al. (2010).

#### 1. Introduction

Fixed point theory in probabilistic metric spaces can be considered as a part of probabilistic analysis, which is a one of the emerging areas of interdisciplinary mathematical research with many diverse applications. The theory of probabilistic metric spaces was introduced by Menger [1] in connection with some measurements in Physics. Over the years, the theory has found several important applications in the investigation of physical quantities in quantum particle physics and string theory as studied by El Naschie [2, 3]. The area of probabilistic metric spaces is also of fundamental importance in probabilistic functional analysis. The first effort in this direction was made by Sehgal [4], who, in his doctoral dissertation, initiated the study of contraction mapping theorems in probabilistic metric spaces. Since then, Sehgal and Bharucha-Reid [5] obtained a generalization of Banach Contraction Principle on a complete Menger space which is an important step in the development of fixed point theorems in Mengar space.

Sessa [6] initiated the tradition of improving commutativity in fixed point theorems by introducing the notion of weakly commuting maps in metric spaces. Jungck [7] soon enlarged this concept to compatible maps. The notion of compatible mappings in a Mengar space has been introduced by Mishra [8]. After this, Jungck [9] gave the concept of weakly compatible maps. Aamri and El Moutawakil [10] introduced the (E.A) property and thus generalized the concept of noncompatible maps. The results obtained in the metric fixed point theory by using the notion of non-compatible maps or the (E.A) property are very interesting. Al-Thagafi and Shahzad [11] defined the concept of occasionally weakly compatible mappings which is more general than the concept of weakly compatible maps. Bhatt et al. [12] have given application of occasionally weakly compatible mappings in dynamical system. Pathak and Hussain [13] defined the concept P-operators. Hussain et al. [14] gave the concept of JH-operators and occasionally weakly g-biased.

In this paper, we obtain some fixed point theorems for JH-operators and occasionally weakly biased pairs under relaxed condition on . Our results extend the results of Bhatt et al. [12].

We begin with the following basic definitions of concepts relating to probabilistic metric spaces for ready reference and also for the sake of completeness.

*Definition 1 (see [15]). *A real valued function on the set of real numbers is called a distribution function if it is nondecreasing, left continuous with and .

The Heaviside function is a distribution function defined by

*Definition 2. * Let be a nonempty set and let denote the set of all distribution functions defined on . is a mapping from into satisfying the following condition:
where defined by for all , and is a function such that if and only if , (symmetric and triangle conditions are not required). A topology on is given by if and only if for each , for some , where .

*Definition 3 (see [16, 17]). *Let be a non-empty set. A point in is called a coincidence point of and if and only if . In this case is called a point of coincidence of and .

Let and denote the sets of coincidence points and points of coincidence, respectively, of the pair . For a space satisfying (2) and , the diameter of is defined by

Here we extend the concept of JH-operators and occasionally weakly g-biased pairs and the space satisfying condition (2).

*Definition 4. * Let be a non-empty set together with the function satisfying condition (2). Two self-maps and of a space are called JH-operators if and only if there is a point in such that

*Definition 5. * Let be a non-empty set together with the function satisfying condition (2). Two self-maps and of a space are called weakly g-biased, if and only whenever .

*Definition 6. * Let be a non-empty set together with the function satisfying condition (2). Two self-maps and of a space are called occasionally weakly g-biased, if and only if there exists some such that and .

*Example 7. * Let and , where
Define by

In this example and . Here = . For . Similarly = . For , . Therefore . Now we can easily show that . Therefore, an occasionally weakly compatible and a nontrivial weakly g-biased pair are occasionally weakly g-biased pairs, but the converse does not hold.

#### 2. Section II

We note that every symmetric (semimetric) space [18] can be realized as a probabilistic semi-metric space by taking defined by for all , in . So probabilistic semi-metric spaces provide a wider framework than that of the symmetric spaces and are better suited in many situations. In this paper we have relaxed the symmetric condition from probabilistic semimetric space. In this section, we prove some fixed point theorems for a pair of JH-operator on space without imposing the restriction of the triangle inequality or symmetry on . In this section, we also prove some fixed point theorems for a pair of Occasionally weakly biased on space without imposing the restriction of the triangle inequality and symmetry only on point of coincidence and image of point of coincidence.

Theorem 8. * Let be a non-empty set together with the function satisfying condition (2). Suppose and are JH-operators on satisfying the following condition:
**
for all with and , where , , and . Then and have a unique common fixed point.*

*Proof. *We claim that and have a unique point of coincidence . If possible, suppose there is another point of coincidence and . Then . So from (7) we get
This is a contradiction, which implies that . Hence we get . Therefore there exists a unique element in such that . Thus implies that , and hence is a unique common fixed point of and .

Let a function be defined by satisfying condition , for all .

Theorem 9. * Let be a non-empty set together with the function satisfying the condition (2). If and are occasionally weakly g-biased on , suppose
**
for some point of coincidence of and
**
for some and . Then and have a unique common fixed point.*

*Proof. * Since and are occasionally weakly biased, there exists some such that and . We claim that is the unique common fixed point of and . For if then from (9) and (10) we get
Because and are occasionally weakly biased, hence,
by using condition (9),
Since satisfying the condition , for all . Therefore, which is a contradiction. Therefore . Hence which further implies that . Thus is a common fixed point of and .

For uniqueness, suppose that such that and and . Then (10) gives
Let . Then we get . Similarly, we get . So, . This is a contradiction. Therefore, . Therefore, the common fixed point of and is unique.

*Example 10. * Letting and defined as
and , where
Define by and if .

One has if and if .

In this example we observe that , where are occasionally weakly g-biased pairs and . Now for , , . Example 10 is the unique common fixed point of and .

Corollary 11. *Let be a non-empty set together with the function satisfying condition (2). If and are occasionally weakly g-biased on , suppose
**
whenever is point of coincidence of and
**
for some and . Then and have a unique common fixed point.*

The proof of the following theorem can be easily obtained by replacing condition (10) by condition (20), the proof of Theorem 9.

Theorem 12. *Let be a non-empty set together with the function satisfying condition (2). If and are occasionally weakly g-biased on , Suppose
**
whenever is point of coincidence of and
**
for some and . Then and have a unique common fixed point.*

#### 3. Section III

In this section, we prove several fixed point theorems for four self-mappings on , where satisfying condition (2). We begin with the following theorem.

Theorem 13. * Let be a non-empty set and satisfying condition (2). Suppose that , , , and are self-mappings of and that the pairs and are each JH-operators on . If
**
whenever and are points of coincidence of and , respectively, and
**
for each for which , then , , , and have a unique common fixed point.*

*Proof. *By hypothesis there exist points such that and . Suppose that for all . Then from (22),
This is a contradiction. Hence for all . This implies that . So . Moreover, if there is another point such that , then, using (22), it follows that or , and is the unique point of coincidence of and .

Thus . This implies that , and hence is a unique common fixed point of and . Similarly is a unique fixed point of and . Suppose . Using (21) and (22) we get
This is a contradiction. Therefore, and is the unique common fixed point of , , , and .

Let the control function be a continuous nondecreasing function such that and . Let a function be defined by satisfying the condition , for all .

Theorem 14. * Let be a non-empty set and satisfying condition (2). Suppose that , , , and are self-mappings of and that the pairs and are each JH-operators on . If
**
whenever and are points of coincidence of and , respectively, and
**
where,
**
for each for which , then , , , and have a unique common fixed point.*

*Proof. *By hypothesis there exist points , in such that and . We claim that . Suppose that . Then from (25) and (26), we get
which is a contradiction. Therefore , which further implies that . Hence the claim follows that is, . Now from the repeated use of condition (26) we can show that , , and and have a unique common fixed point.

Define such that,

Theorem 15. * Let be a non-empty set and satisfying condition (2). Suppose that , , , and are self-mappings of and that the pairs and are each JH-operators on . If
**
whenever and are points of coincidence of and , respectively, and
**
for all , then , , and and have a unique common fixed point.*

*Proof. * It follows from the given assumptions that there exists a point such that , and there exists another point for which . Suppose that . Then, from (30) we have
Since and are points of coincidence of and , respectively, hence, from (30) we get,

Therefore, from we get . This shows that . Suppose that there exists another point such that . Then, using (30) one obtains . Hence is the unique point of coincidence of and . . This implies that , and hence is a unique common fixed point of and . Similarly, there exists a unique point such that . It then follows that , and is a common fixed point of , , , and , and is unique.

#### 4. Application to Dynamic Programming

Throughout in this section, we assume that and are Banach spaces, is a state space, and is a decision space. We denote by the set of all bounded real valued functions defined on .

As suggested by Bellman and Lee [19], the basic form of the functional equations arising in dynamic programming is where and represent the state and decision vectors, respectively; represents the transformation of the process, and represents the optimal return function with initial state (here opt denotes maximum or minimum).

We now study the existence and uniqueness of a common solution of the following functional equations arising in dynamic programming: where and .

As an application of Corollary 11, the existence and uniqueness of a common solution of the functional equations arising in dynamic programming can be established which extends Theorem 18 [12].

*Definition 16. * Let be a non-empty set and a function such that

Corollary 17. * Let be a non-empty set and a function satisfying condition (35). If and are JH-operators self-mappings of and
**
where a nondecreasing function satisfying the condition for each , then and have a unique common fixed point.*

*Proof. *The proof of this corollary can be easily obtained.

We now present main result of this section.

Theorem 18. * Suppose that the following conditions (i), (ii), (iii), and (iv) are satisfied.*(i) * and are bounded. *(ii) * for all and , where is a nondecreasing function satisfying the condition for each and and are defined as follows:
*(iii) *If there is a point for some implies ,**then is the unique common solution of (34).*

*Proof. *For any let
From conditions (i), (ii), (iii), it follows that and are self-mappings of .

Letting , be any two points of , and any positive number then there exist such that
Subtracting (42) from (39) and using (ii), we have
Similarly, from (40) and (41), we get
Hence
Since (42) is true for any and any positive number,
Therefore, from Corollary 17, is the unique common fixed point of and ; that is, is the unique common solution of functional equation (34).

#### Acknowledgment

Authors are grateful to referee for careful reading of this paper and for valuable comments.

#### References

- K. Menger, “Statistical metrices,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 28, no. 12, pp. 535–537, 1942. View at Publisher · View at Google Scholar - M. S. El Naschie, “On the uncertainty of Cantorian geometry and the two-slit experiment,”
*Chaos, Solitons & Fractals*, vol. 9, no. 3, pp. 517–529, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. S. El Naschie, “On the unification of heterotic strings, M theory and $\epsilon $(
^{∞}) theory,”*Chaos, Solitons & Fractals*, vol. 11, no. 14, pp. 2397–2408, 2000. View at Publisher · View at Google Scholar · View at Scopus - V. M. Sehgal,
*Some fixed point theorems in functional analysis and probability [Ph.D. dissertation]*, Wayne State University, Detroit, Mich, USA, 1966. - V. M. Sehgal and A. T. Bharucha-Reid, “Fixed points of contraction mappings on probabilistic metric spaces,”
*Mathematical Systems Theory*, vol. 6, pp. 97–102, 1972. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Sessa, “On a weak commutativity condition of mappings in fixed point considerations,”
*Publications de l'Institut Mathématique*, vol. 32(46), pp. 149–153, 1982. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jungck, “Compatible mappings and common fixed points,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 9, no. 4, pp. 771–779, 1986. View at Publisher · View at Google Scholar · View at MathSciNet - S. N. Mishra, “Common fixed points of compatible mappings in PM-spaces,”
*Mathematica Japonica*, vol. 36, no. 2, pp. 283–289, 1991. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jungck, “Common fixed points for noncontinuous nonself maps on nonmetric spaces,”
*Far East Journal of Mathematical Sciences*, vol. 4, no. 2, pp. 199–215, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Aamri and D. El Moutawakil, “Some new common fixed point theorems under strict contractive conditions,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 181–188, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Al-Thagafi and N. Shahzad, “Generalized $I$-nonexpansive selfmaps and invariant approximations,”
*Acta Mathematica Sinica*, vol. 24, no. 5, pp. 867–876, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - A. Bhatt, H. Chandra, and D. R. Sahu, “Common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 73, no. 1, pp. 176–182, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. K. Pathak and N. Hussain, “Common fixed points for $P$-operator pair with applications,”
*Applied Mathematics and Computation*, vol. 217, no. 7, pp. 3137–3143, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - N. Hussain, M. A. Khamsi, and A. Latif, “Common fixed points for $JH$-operators and occasionally weakly biased pairs under relaxed conditions,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 6, pp. 2133–2140, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - B. Schweizer and A. Skalar,
*Statistical Metric Spaces*, North Holland, Amsterdam, The Netherlands, 1983. - G. Jungck and B. E. Rhoades, “Fixed point theorems for occasionally weakly compatible mappings,”
*Fixed Point Theory*, vol. 7, no. 2, pp. 286–296, 2006. View at Google Scholar - G. Jungck and N. Hussain, “Compatible maps and invariant approximations,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 2, pp. 1003–1012, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. A. Wilson, “On semi-metric spaces,”
*American Journal of Mathematics*, vol. 53, no. 2, pp. 361–373, 1931. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Bellman and E. S. Lee, “Functional equation arising in dynamic programming,”
*Aequationes Mathematicae*, vol. 17, pp. 1–18, 1979. View at Google Scholar