Common Fixed Points of Generalized Contractive Hybrid Pairs in Symmetric Spaces

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 symmetric on a set is a mapping such that

(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

(1.2)

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 becompatible 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.

Consider with usual metric.

• (a) Define and as: and

  

  (1.3)

then and are weakly compatible.

• (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.

Let be self maps of a symmetric space , and let be maps from into such that the pairs and are If

(2.1)

for each for which where

(2.2)

then , and have a unique common fixed point.

Proof.

By hypothesis there exist points in such that , and . Also, Therefore by (2.2) we have

(2.3)

Now we claim that . For, otherwise, by (2.1),

(2.4)

a contradiction and hence Obviously, Thus (2.2) gives

(2.5)

Next we claim that If not, then (2.1) implies

(2.6)

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.

Let be self maps of a symmetric space and let be maps from into such that the pairs and are If

(2.7)

for each for which where

(2.8)

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.

Let be self maps of a symmetric space and let be maps from into such that the pairs and are . If

(2.9)

for each for which where

(2.10)

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

Proof.

Note that

(2.11)

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

Corollary 2.4.

Let be a self map on a symmetric space and let be a map from into such that and are . If

(2.12)

for each for which where

(2.13)

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.

Let be self maps of a symmetric space and let be maps from into such that the pairs and are If

(2.14)

for each for which ,

(2.15)

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

Proof.

By hypothesis there exist points in such that and Therefore by (2.15) we have

(2.16)

Now we show that . Suppose not. Then condition (2.14) implies that

(2.17)

which is a contradiction and hence Note that, Thus (2.15) gives

(2.18)

Now we claim that If not, then condition (2.14) implies that

(2.19)

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,

) if is such that

(2.20)

then .

Theorem 2.6.

Let be self maps of a symmetric space and let be maps from into such that the pairs and are . If

(2.21)

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

Proof.

By hypothesis there exist points in such that , and Also, First we show that . Suppose not. Then condition (2.21) implies that

(2.22)

which, from implies that this further implies that, a contradiction. Hence the claim follows. Also, Next we claim that If not, then condition (2.21) implies that

(2.23)

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.

Let be self maps of symmetric space and let be maps from into such that the pairs and are If for a control function one has

(2.24)

for each for which right-hand side of (2.24) is not equal to zero where

(2.25)

then , and have a unique common fixed point.

Proof.

By hypothesis there exist points in such that , and Also, using the triangle inequality, we obtain Therefore by (2.25) we have

(2.26)

Now we show that . Suppose not. Then condition (2.24) implies that

(2.27)

which is a contradiction. Therefore which further implies that, Hence the claim follows. Again, Therefore by (2.25) we have

(2.28)

Next we claim that If not, then condition (2.24) implies

(2.29)

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.

Let be self maps of a symmetric space , and let be maps from into such that the pairs and are If

(2.30)

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.

By hypothesis there exist points in such that , and Also, using the triangle inequality, we obtain, Now we claim that . For, otherwise, by (2.30),

(2.31)

which is a contradiction. Therefore Hence the claim follows. Again, Now we claim that If not, then condition (2.30) implies that

(2.32)

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.

Let Define by

(2.33)

Note that is symmetric but not a metric on .

Define by

(2.34)

and as follows:

(2.35)

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.

(b)[19, Theorem ], [20, Theorem ], [21, Theorem ], and [22, Theorem ] are special cases of Corollary 2.2. Moreover, [23, Theorem ] and [24, Theorem ] also become special cases of Corollary 2.2.

• (c) ([25, Theorem ]) is a special case of Theorem 2.1. Theorem 2.1 also generalizes ([26, Theorem ]) and ( [27, Theorems and ]).

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