# Common Fixed Points of Generalized Contractive Hybrid Pairs in Symmetric Spaces

- Mujahid Abbas
^{1}and - Abdul Rahim Khan
^{2}Email author

**2009**:869407

**DOI: **10.1155/2009/869407

© M. Abbas and A. R. Khan. 2009

**Received: **16 April 2009

**Accepted: **10 November 2009

**Published: **3 December 2009

## Abstract

Several fixed point theorems for hybrid pairs of single-valued and multivalued occasionally weakly compatible maps satisfying generalized contractive conditions are established in a symmetric space.

## 1. Introduction and Preliminaries

In 1968, Kannan [1] proved a fixed point theorem for a map satisfying a contractive condition that did not require continuity at each point. This paper was a genesis for a multitude of fixed point papers over the next two decades. Sessa [2] coined the term weakly commuting maps. Jungck [3] generalized the notion of weak commutativity by introducing compatible maps and then weakly compatible maps [4]. Al-Thagafi and Shahzad [5] gave a definition which is proper generalization of nontrivial weakly compatible maps which have coincidence points. Jungck and Rhoades [6] studied fixed point results for occasionally weakly compatible (owc) maps. Recently, Zhang [7] obtained common fixed point theorems for some new generalized contractive type mappings. Abbas and Rhoades [8] obtained common fixed point theorems for hybrid pairs of single-valued and multivalued owc maps defined on a symmetric space (see also [9]). For other related fixed point results in symmetric spaces and their applications, we refer to [10–15]. The aim of this paper is to obtain fixed point theorems involving hybrid pairs of single-valued and multivalued owc maps satisfying a generalized contractive condition in the frame work of a symmetric space.

Definition 1.1.

A set
together with a symmetric
is called a *symmetric space*.

We will use the following notations, throughout this paper, where is a symmetric space, and , and is the class of all nonempty bounded subsets of The diameter of is denoted and defined by

Clearly, For and we write and respectively. We appeal to the fact that if and only if for

Recall that is called a coincidence point (resp., common fixed point) of and if (resp., ).

Definition 1.2.

Maps
and
are said to be*compatible* if
for each
and
whenever
is a sequence in
such that
(
) and
for some
[21].

Definition 1.3.

Maps
and
are said to be *weakly compatible* if
whenever

Definition 1.4.

Maps
and
are said to be *owc* if and only if there exists some point
in
such that
and

Example 1.5.

- (a)Define and as: and(1.3)

- (b)Define by(1.4)

It can be easily verified that is coincidence point of and but and are not weakly compatible there, as . Hence and are not compatible. However, the pair is occasionally weakly compatible, since the pair is weakly compatible at

Assume that satisfies the following.

(i) and for each .

(ii) is nondecreasing on

Define, satisfies above

Let satisfy the following.

(iii) for each .

(iv) is nondecreasing on

Define, satisfies above

For some examples of mappings which satisfy we refer to [7].

## 2. Common Fixed Point Theorems

In the sequel we shall consider, which is defined on where stands for a real number to the left of and assume that the mapping satisfies above.

Theorem 2.1.

then , and have a unique common fixed point.

Proof.

which is a contradiction and the claim follows. Similarly, we obtain Thus , and have a common fixed point. Uniqueness follows from (2.1).

Corollary 2.2.

and , then have a unique common fixed point.

Proof.

Since (2.7) is a special case of (2.1), the result follows from Theorem 2.1.

Corollary 2.3.

where and Then , and have a unique common fixed point.

Proof.

So, (2.9) is a special case of (2.1) and hence the result follows from Theorem 2.1.

Corollary 2.4.

Then and have a unique common fixed point.

Proof.

Condition (2.12) is a special case of condition (2.1) with and Therefore the result follows from Theorem 2.1.

Theorem 2.5.

where , and , then and have a unique common fixed point.

Proof.

which is a contradiction, and hence the claim follows. Similarly, we obtain Thus , and have a common fixed point. Uniqueness follows easily from (2.14).

Define such that

) is nondecreasing in the 4th and 5th variables,

then .

Theorem 2.6.

for all for which where then , and have a unique common fixed point.

Proof.

which, from and implies that this further implies that Hence the claim follows. Similarly, it can be shown that which proves that is a common fixed point of , and . Uniqueness follows from (2.21) and ( ).

A control function is a continuous monotonically increasing function that satisfies and, if and only if

Let be such that for each

Theorem 2.7.

then , and have a unique common fixed point.

Proof.

which is a contradiction. Therefore which further implies that Hence the claim follows. Similarly, it can be shown that which proves the result.

Set is continuous and nondecreasing mapping with if and only if

The following theorem generalizes [16, Theorem ].

Theorem 2.8.

for all , for which right-hand side of (2.30) is not equal to zero where then , and have a unique common fixed point.

Proof.

which is a contradiction, and hence the claim follows. Similarly, it can be shown that which, proves that is a common fixed point of , and . Uniqueness follows easily from (2.30).

Example 2.9.

Note that is symmetric but not a metric on .

Clearly, but and but they show that is not weakly compatible. On the other hand, gives that Hence is occasionally weakly compatible. Note that , , , and they imply that is not weakly compatible Now gives that . Hence is occasionally weakly compatible. As and so is the unique common fixed point of , and

Remark 2.10 s.

Weakly compatible maps are occasionally weakly compatible but converse is not true in general. The class of symmetric spaces is more general than that of metric spaces. Therefore the following results can be viewed as special cases of our results:

(a)([17, Theorem ] and [18, Theorem ]) are special cases of Theorem 2.7.

(d)[28, Theorem ] becomes special case of Corollary 2.4.

## Declarations

### Acknowledgments

The authors are thankful to the referees for their critical remarks to improve this paper. The second author gratefully acknowledges the support provided by King Fahad University of Petroleum and Minerals during this research.

## Authors’ Affiliations

## References

- Kannan R:
**Some results on fixed points.***Bulletin of the Calcutta Mathematical Society*1968,**60:**71–76.MathSciNetMATHGoogle Scholar - Sessa S:
**On a weak commutativity condition of mappings in fixed point considerations.***Publications de l'Institut Mathématique. Nouvelle Série*1982,**32(46):**149–153.MathSciNetMATHGoogle Scholar - Jungck G:
**Compatible mappings and common fixed points.***International Journal of Mathematics and Mathematical Sciences*1986,**9**(4):771–779. 10.1155/S0161171286000935MathSciNetView ArticleMATHGoogle Scholar - Jungck G:
**Common fixed points for noncontinuous nonself maps on nonmetric spaces.***Far East Journal of Mathematical Sciences*1996,**4**(2):199–215.MathSciNetMATHGoogle Scholar - Al-Thagafi MA, Shahzad N:
**Generalized -nonexpansive selfmaps and invariant approximations.***Acta Mathematica Sinica*2008,**24**(5):867–876. 10.1007/s10114-007-5598-xMathSciNetView ArticleMATHGoogle Scholar - Jungck G, Rhoades BE:
**Fixed point theorems for occasionally weakly compatible mappings.***Fixed Point Theory*2006,**7**(2):287–296.MathSciNetMATHGoogle Scholar - Zhang X:
**Common fixed point theorems for some new generalized contractive type mappings.***Journal of Mathematical Analysis and Applications*2007,**333**(2):780–786. 10.1016/j.jmaa.2006.11.028MathSciNetView ArticleMATHGoogle Scholar - Abbas M, Rhoades BE:
**Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings defined on symmetric spaces.***Panamerican Mathematical Journal*2008,**18**(1):55–62.MathSciNetMATHGoogle Scholar - Abbas M, Rhoades BE:
**Common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive condition.***Mathematical Communications*2008,**13**(2):295–301.MathSciNetMATHGoogle Scholar - Aliouche A:
**A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type.***Journal of Mathematical Analysis and Applications*2006,**322**(2):796–802. 10.1016/j.jmaa.2005.09.068MathSciNetView ArticleMATHGoogle Scholar - Chandra H, Bhatt A:
**Some fixed point theorems for set valued maps in symmetric spaces.***International Journal of Mathematical Analysis*2009,**3**(17):839–846.MathSciNetMATHGoogle Scholar - Cho S-H, Lee G-Y, Bae J-S:
**On coincidence and fixed-point theorems in symmetric spaces.***Fixed Point Theory and Applications*2008,**2008:**-9.Google Scholar - Hicks TL, Rhoades BE:
**Fixed point theory in symmetric spaces with applications to probabilistic spaces.***Nonlinear Analysis: Theory, Methods & Applications*1999,**36**(3):331–344. 10.1016/S0362-546X(98)00002-9MathSciNetView ArticleMATHGoogle Scholar - Imdad M, Ali J:
**Common fixed point theorems in symmetric spaces employing a new implicit function and common property (E.A).***Bulletin of the Belgian Mathematical Society. Simon Stevin*2009,**16:**421–433.MathSciNetMATHGoogle Scholar - Pathak HK, Tiwari R, Khan MS:
**A common fixed point theorem satisfying integral type implicit relations.***Applied Mathematics E-Notes*2007,**7:**222–228.MathSciNetMATHGoogle Scholar - Beg I, Abbas M:
**Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition.***Fixed Point Theory and Applications*2006,**2006:**-7.Google Scholar - Chang TH:
**Common fixed point theorems for multivalued mappings.***Mathematica Japonica*1995,**41**(2):311–320.MathSciNetMATHGoogle Scholar - Shrivastava PK, Bawa NPS, Nigam SK:
**Fixed point theorems for hybrid contractions.***Varāhmihir Journal of Mathematical Sciences*2002,**2**(2):275–281.MathSciNetMATHGoogle Scholar - Azam A, Beg I:
**Coincidence points of compatible multivalued mappings.***Demonstratio Mathematica*1996,**29**(1):17–22.MathSciNetMATHGoogle Scholar - Kamran T:
**Common coincidence points of R -weakly commuting maps.***International Journal of Mathematics and Mathematical Sciences*2001,**26**(3):179–182. 10.1155/S0161171201005245MathSciNetView ArticleMATHGoogle Scholar - Jungck G, Rhoades BE:
**Fixed points for set valued functions without continuity.***Indian Journal of Pure and Applied Mathematics*1998,**29**(3):227–238.MathSciNetMATHGoogle Scholar - Hadžić O:
**Common fixed point theorems for single-valued and multivalued mappings.***Review of Research. Faculty of Science. Mathematics Series*1988,**18**(2):145–151.MathSciNetMATHGoogle Scholar - Kaneko H, Sessa S:
**Fixed point theorems for compatible multi-valued and single-valued mappings.***International Journal of Mathematics and Mathematical Sciences*1989,**12**(2):257–262. 10.1155/S0161171289000293MathSciNetView ArticleMATHGoogle Scholar - Kaneko H:
**A common fixed point of weakly commuting multi-valued mappings.***Mathematica Japonica*1988,**33**(5):741–744.MathSciNetMATHGoogle Scholar - Fisher B:
**Common fixed points for set-valued mappings.***Indian Journal of Mathematics*1983,**25**(3):265–270.MathSciNetMATHGoogle Scholar - Sessa S, Fisher B:
**On common fixed points of weakly commuting mappings and set-valued mappings.***International Journal of Mathematics and Mathematical Sciences*1986,**9**(2):323–329. 10.1155/S0161171286000406MathSciNetView ArticleMATHGoogle Scholar - Fisher B:
**Common fixed point theorem for commutative mappings and set valued mappings.***Journal of University of Kuwait*1984,**11:**15–21.MATHGoogle Scholar - Dhage BC:
**Common fixed point theorems for coincidentally commuting pairs of nonself mappings in metrically convex metric spaces.***Analele Ştiinţifice ale Universităţii Al. I. Cuza din Iaşi. Serie Nouă. Matematică*2003,**49**(1):45–60.MathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.