Open Access

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

Journal of Inequalities and Applications20102010:746045

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

Received: 17 May 2010

Accepted: 21 July 2010

Published: 4 August 2010

Abstract

Using the notion of weakly -contractive mappings, we prove several new common fixed point theorems for commuting as well as noncommuting mappings on a topological space X. By analogy, we obtain a common fixed point theorem of mappings which are strongly -expansive on X.

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]).

Let be a topological space. A function is said to be lower semi-continuous from above (lsca) at if for any net convergent to with
(2.1)
we have
(2.2)

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

Example 2.2.
  1. (i)
    Let . Define by
    (2.3)
     
Let be a sequence of nonnegative terms such that converges to . Then
(2.4)
Similarly, if is a sequence in of negative terms such that converges to , then
(2.5)
Thus, is lsca at
  1. (ii)
    Every lower semi-continuous function is lsca but not conversely. One can check that the function with defined below is lsca at but is not lower semi-continuous at :
    (2.6)
     

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.

Fix and consider a net in convergent to such that
(2.7)
Set and Then since is continuous, and lsca implies that
(2.8)

with for Thus and is lsca.

Remark 2.5 (see [6]).

Let be topological space. Let be a continuous function and lsca. Then defined by is also lsca. For this, let be a net in convergent to Since is continuous, Suppose that
(2.9)
Then since is lsca, we have
(2.10)

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.

Let be a topological space, , and self-mappings such that for every countable set
(3.1)

and , commute on If

(i) is continuous and weakly -contractive or

(ii) is continuous and weakly -contractive with ,

then and have a unique common fixed point.

Proof.

Let and define the sequence by setting for Let Then
(3.2)
so by hypothesis is compact. Define by
(3.3)

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,

(i   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
(3.4)
a contradiction to the minimality of at Having one can see that Indeed, if then we have
(3.5)
a contradiction.
  1. (ii)
      Suppose that   is continuous and weakly -contractive with . Then as is continuous. Put ; then is continuous, and implies that . We claim that for otherwise we will have
    (3.6)
     

which is a contradiction. Hence the claim follows.

Now suppose that then we have
(3.7)

a contradiction, hence

In both cases, uniqueness follows from the contractive conditions: suppose there exists such that Then we have
(3.8)

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 ]).

Let be a topological space, , and continuous and weakly -contractive. If the implication
(3.9)

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

Example 3.3.

Let be the Banach space of all null real sequences. Define
(3.10)
Let and a sequence such that
(3.11)
with as Define the mappings by
(3.12)
where and are such that for ,
(3.13)
(3.14)
and for
(3.15)

We verify the hypothesis of Theorem 3.1.

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

(ii)For we have

(3.16)
Since the sequences and are null sequences, there exists such that
(3.17)
Hence
(3.18)
This implies that and are weakly contractive. Thus and are continuous and weakly contractive. Next suppose that for any countable set we have
(3.19)
then by the definition of , we can consider Hence closure of being closed subset of a compact set is compact. Also
(3.20)

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

Corollary 3.4.

Let ( be a metric space, , and self-mappings such that for every countable set
(3.21)

and , commute on If

(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.

Let be a compact metric space, , and self-mappings such that for every countable set
(3.22)

and , commute on If

(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.

Let be a topological space, , and self-mappings such that for every countable set
  1. (1)

    is relatively compact;

     
  2. (2)

    is relatively compact for some ;

     
  3. (3)

    is relatively compact for some .

     
And , commute on Further, if
(3.23)

then and have a unique common fixed point.

Proof.

Part (3): we proceed as in Theorem 3.1. Let for some and define the sequence by setting for Let Then
(3.24)
so by hypothesis (3), is compact. Define by
(3.25)

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

(i)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
(3.26)
a contradiction to the minimality of at Therefore, for some . One can check that . Suppose that , then we have
(3.27)
a contradiction. Thus is a common fixed point of and and hence of and
  1. (ii)
    Suppose that   is continuous and   weakly -contractive with . Then as is continuous. Put Then continuous and imply that . We claim that for otherwise we will have
    (3.28)
     

which is a contradiction. Hence the claim follows.

Now suppose that then we have
(3.29)

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

Now we establish the uniqueness of Suppose there exists such that for some . Now if is continuous and is weakly - contractive, then we have
(3.30)
and if is continuous and is weakly - contractive, then we have
(3.31)

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:

(i) and for all

(ii) then

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

If either
(3.32)

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.

Proof.
  1. (a)
    Suppose that for all with commuting on and is contractive Then we have
    (3.33)
     
Next
(3.34)
Therefore, after -steps, , we get
(3.35)
Hence,
(3.36)

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.

Let be a topological space, with closed and Let be mappings such that for every countable set
(3.37)

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

Proof.

Suppose that is a homeomorphism and strongly -expansive. Let with Then there exists such that and or and Since is strongly -expansive, we have
(3.38)
or
(3.39)
So is a weakly -contractive mapping. Choose any countable subset of and set Suppose that
(3.40)
Then for some and we get
(3.41)
So by hypothesis is compact and since is a homeomorphism, is compact. Since for every and , we have
(3.42)
for every Thus
(3.43)

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.

Let with the River metric defined by
(3.44)
where , , and denotes the Euclidean metric on Then is a topological space with a topology induced by the metric . Consider the sets defined by
(3.45)
Let the mappings be defined by and for Then is clearly a homeomorphism and for an arbitrary countable subset of and ,
(3.46)
If and only if Indeed, if such that , then
(3.47)
Further, for every Set where is the Radial metric defined by
(3.48)
and ; Now for since
(3.49)

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 [1013]. If is a Banach operator pair, then need not be a Banach operator pair.

Definition 4.1.

The pair is called occasionally Banach operator pair if
(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.

Let with usual norm and . Define by
(4.2)
(4.3)
Here and implies that is not Banach operator pair. Similarly, is not Banach operator pair. Further,
(4.4)

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

Definition 4.4.

Let be a nonempty set and be a mapping such that
(4.5)
For a space satisfying (4.5) and the diameter of is defined by
(4.6)

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

Definition 4.5.

Let be a space satisfying (4.5). The pair is called occasionally Banach operator pair on iff there is a point in such that and
(4.7)

Theorem 4.6.

Let be a topological space, , and self-mappings such that for every countable set
(4.8)

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.

By Corollary 3.2, is a singleton. Let . Then, by our hypothesis,
(4.9)

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

Corollary 4.7.

Let ( be a metric space, , and self-mappings such that for every countable set
(4.10)

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.

Let be a compact metric space, , and self-mappings such that for every countable set
(4.11)

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.

Let be a topological space, , and self-mappings such that for every countable set
(4.12)

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.

Let ( be a metric space, , and self-mappings such that for every countable set
(4.13)

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

Declarations

Acknowledgments

N. Hussain thanks the Deanship of Scientific Research, King Abdulaziz University for the support of the Research Project no. (3-74/430). A. R. Khan is grateful to the King Fahd University of Petroleum Minerals and SABIC for the support of the Research Project no. SB

Authors’ Affiliations

(1)
Department of Mathematical Sciences, LUMS, DHA Lahore, Pakistan
(2)
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
(3)
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

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Copyright

© M. H. Shah et al. 2010

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