# Common Fixed Points of Weakly Contractive and Strongly Expansive Mappings in Topological Spaces

- M H Shah
^{1}, - N Hussain
^{2}and - A R Khan
^{3}Email author

**2010**:746045

https://doi.org/10.1155/2010/746045

© M. H. Shah et al. 2010

**Received: **17 May 2010

**Accepted: **21 July 2010

**Published: **4 August 2010

## Abstract

## Keywords

## 1. Introduction

It is well known that if is a compact metric space and is a weakly contractive mapping (see Section 2 for the definition), then has a fixed point in (see [1, p. 17]). In late sixties, Furi and Vignoli [2] extended this result to -condensing mappings acting on a bounded complete metric space (see [3] for the definition). A generalized version of Furi-Vignoli's theorem using the notion of weakly -contractive mappings acting on a topological space was proved in [4] (see also [5]).

On the other hand, in [6] while examining KKM maps, the authors introduced a new concept of lower (upper) semicontinuous function (see Definition 2.1, Section 2) which is more general than the classical one. In [7], the authors used this definition of lower semicontinuity to redefine weakly -contractive mappings and strongly -expansive mappings (see Definition 2.6, Section 2 ) to formulate and prove several results for fixed points.

In this article, we have used the notions of weakly -contractive mappings ( where is a topological space) to prove a version of the above-mentioned fixed point theorem [7, Theorem ] for common fixed points (see Theorem 3.1). We also prove a common fixed point theorem under the assumption that certain iteration of the mappings in question is weakly -contractive. As a corollary to this fact, we get an extension (to common fixed points) of [7, Theorem ] for Banach spaces with a quasimodulus endowed with a suitable transitive binary relation. The most interesting result of this section is Theorem 3.8 wherein the strongly -expansive condition on (with some other conditions) implies that and have a unique common fixed point.

In Section 4, we define a new class of noncommuting self-maps and prove some common fixed point results for this new class of mappings.

## 2. Preliminaries

Definition 2.1 (see [6]).

*lower semi-continuous from above*(

*lsca*) at if for any net convergent to with

A function is said to be lsca if it is lsca at every

The following lemmas state some properties of lsca mappings. The first one is an analogue of Weierstrass boundedness theorem and the second one is about the composition of a continuous function and a function lsca.

Lemma 2.3 (see [6]).

Let be a compact topological space and a function lsca. Then there exists such that

Lemma 2.4 (see [7]).

Let be a topological space and a continuous function. If is a function lsca, then the composition function is also lsca.

Proof.

Remark 2.5 (see [6]).

Definition 2.6 (see [7]).

Let be a topological space and be lsca. The mapping is said to be

(i)weakly -contractive if for all such that

(ii)strongly -expansive if for all such that

If is a metric space with metric and , then we call respectively, weakly contractive and strongly expansive.

Let . The set of fixed points of (resp., ) is denoted by (resp., ). A point is a coincidence point (common fixed point) of and if . The set of coincidence points of and is denoted by Maps are called (1) commuting if for all , (2) weakly compatible [8] if they commute at their coincidence points, that is, if whenever , and (3) occasionally weakly compatible [9] if for some

## 3. Common Fixed Point Theorems for Commuting Maps

In this section we extend some results in [7] to the setting of two mappings having a unique common fixed point.

Theorem 3.1.

(i) is continuous and weakly -contractive or

(ii) is continuous and weakly -contractive with ,

then and have a unique common fixed point.

Proof.

Now if or is continuous and since is lsca, then by Remark 2.5, is lsca. So by Lemma 2.3, has a minimum at, say,

*Suppose that*

*is continuous and weakly*

*-contractive*. Then as is continuous. Now observe that if is continuous, and , then We show that Suppose that ; then

- (ii)

which is a contradiction. Hence the claim follows.

which is false. Thus and have a unique common fixed point.

If , then Theorem 3.1(i) reduces to [7, Theorem ].

Corollary 3.2 (see [7, Theorem ]).

holds for every countable set then has a unique fixed point.

Example 3.3.

We verify the hypothesis of Theorem 3.1.

(i)Observe that and are, clearly, continuous by their definition.

So by Theorem 3.1, and have a unique common fixed point.

Corollary 3.4.

(i) is continuous and weakly contractive or

(ii) is continuous and weakly contractive with ,

then and have a unique common fixed point.

Proof.

It is immediate from Theorem 3.1 with .

Corollary 3.5.

(i) is continuous and weakly contractive or

(ii) is continuous and weakly -contractive with

then and have a unique common fixed point.

Proof.

It is immediate from Theorem 3.1.

Theorem 3.6.

then and have a unique common fixed point.

Proof.

Now since is lsca and if or is continuous, then by Remark 2.5 would be lsca and hence by Lemma 2.3, would have a minimum, say, at

*Suppose that*

*is continuous and*

*weakly*

*-contractive*. Then as is continuous. Now observe that is continuous, and implies that is continuous and and so for some . We show that . Suppose that for any , then

which is a contradiction. Hence the claim follows.

a contradiction, hence Thus is a common fixed point of and and hence of and

*-*contractive, then we have

which is false. Thus and have a unique common fixed point which obviously is a unique common fixed point of and .

Part (2). The conclusion follows if we set in part (3).

Part (1). The conclusion follows if we set and in part (3).

A nice consequence of Theorem 3.6 is the following theorem where is taken as a Banach space equipped with a transitive binary relation.

Theorem 3.7.

Let be a Banach space with a transitive binary relation such that for with Suppose, further, that the mappings are such that the following conditions are satisfied:

(iii)A is bounded linear operator and for some and for all such that with

for all with commuting on and if one of the conditions, (1)–(3), of Theorem 3.6 holds, then and have a unique common fixed point.

So is weakly contractive. Since is continuous (as is bounded and contractive) by Theorem 3.6, and have a unique common fixed point.

b Suppose that and is contractive for all with commuting on and being contractive The proof now follows if we mutually interchange in (a) above

Theorem 3.8.

and , commute on If is a homeomorphism and strongly -expansive, then and have a unique common fixed point.

Proof.

and for every Since is continuous and weakly -contractive, by Theorem 3.1, the mappings and have a unique common fixed point, say, . Since implies that so is a unique common fixed point of and

The following example illustrates Theorem 3.8.

Example 3.9.

is strongly
-expansive. Also
is lower semi-continuous and hence *lsca*. Thus all the conditions of Theorem 3.8 are satisfied and
and
have a unique common fixed point.

## 4. Occasionally Banach Operator Pair and Weak F-Contractions

In this section, we define a new class of noncommuting self-maps and prove some common fixed point results for this new class of maps.

The pair is called a Banach operator pair [10] if the set is -invariant, namely, . Obviously, commuting pair is a Banach operator pair but converse is not true, in general; see [10–13]. If is a Banach operator pair, then need not be a Banach operator pair.

Definition 4.1.

Clearly, Banach operator pair (BOP) is occasionally Banach operator pair (OBOP) but not conversely, in general.

Example 4.2.

Let with usual norm. Define by and , for and . and . Obviously is OBOP but not BOP as . Further, is not weakly compatible and hence not commuting.

Example 4.3.

imply that is OBOP. Further, note that and . Hence is not occasionally weakly compatible pair.

Definition 4.4.

Here we extend this concept to the space satisfying condition (4.5).

Definition 4.5.

*occasionally Banach operator pair*on iff there is a point in such that and

Theorem 4.6.

If is continuous and weakly -contractive, satisfies condition (4.5), and the pair is occasionally Banach operator pair, then and have a unique common fixed point.

Proof.

Therefore, . That is, is unique common fixed point of and .

Corollary 4.7.

If is continuous and weakly contractive and the pair is occasionally Banach operator pair, then and have a unique common fixed point.

Proof.

It is immediate from Theorem 4.6 with .

Corollary 4.8.

If is continuous and weakly contractive and the pair is occasionally Banach operator pair, then and have a unique common fixed point.

Proof.

It is immediate from Theorem 4.6.

Theorem 4.6 holds for a Banach operator pair without condition (4.5) as follows.

Theorem 4.9.

If is continuous and weakly -contractive and the pair is a Banach operator pair, then and have a unique common fixed point.

Proof.

By Corollary 3.2, is a singleton. Let . As is a Banach operator pair, by definition . Thus and hence . That is, is unique common fixed point of and .

Corollary 4.10.

If is continuous and weakly contractive and the pair is a Banach operator pair, then and have a unique common fixed point.

## Declarations

## Authors’ Affiliations

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