# Some Common Fixed Point Theorems for Weakly Compatible Mappings in Metric Spaces

- M. A. Ahmed
^{1}Email author

**2009**:804734

https://doi.org/10.1155/2009/804734

© M. A. Ahmed. 2009

**Received: **23 October 2008

**Accepted: **18 January 2009

**Published: **3 February 2009

## Abstract

We establish a common fixed point theorem for weakly compatible mappings generalizing a result of Khan and Kubiaczyk (1988). Also, an example is given to support our generalization. We also prove common fixed point theorems for weakly compatible mappings in metric and compact metric spaces.

## 1. Introduction

In the last years, fixed point theorems have been applied to show the existence and uniqueness of the solutions of differential equations, integral equations and many other branches mathematics (see, e.g., [1–3]). Some common fixed point theorems for weakly commuting, compatible, -compatible and weakly compatible mappings under different contractive conditions in metric spaces have appeared in [4–15]. Throughout this paper, is a metric space.

where , for some and , for some .

If for some , we denote , and for , and , respectively. Also, if , then one can deduce that .

We need the following definitions and lemmas.

Definition 1.1 (see [16]).

A sequence
of nonempty subsets of
is said to be *convergent* to
if:

(i)each point in is the limit of a convergent sequence , where is in for ( := the set of all positive integers),

(ii)for arbitrary , there exists an integer such that for , where denotes the set of all points in for which there exists a point in , depending on such that .

is then said to be the *limit* of the sequence
.

Definition 1.2 (see [9]).

A set-valued function
is said to be *continuous* if for any sequence
in
with
, it yields
.

Lemma 1.3 (see [16]).

If and are sequences in converging to and in , respectively, then the sequence converges to .

Lemma 1.4 (see [16]).

Let be a sequence in and let be a point in such that . Then the sequence converges to the set in .

Lemma 1.5 (see [9]).

Lemma 1.6 (see [17]).

Let be a right continuous function such that for every . Then, for every , where denotes the -times repeated composition of with itself.

Definition 1.7 (see [15]).

The mappings
and
are *weakly commuting on*
if
and
for all
.

Definition 1.8 (see [13]).

The mappings
and
are said to be
-*compatible* if
whenever
is a sequence in
such that
,
and
for some
.

Definition 1.9 (see [13]).

The mappings
and
are *weakly compatible* if they commute at coincidence points, that is, for each point
such that
, then
(note that the equation
implies that
is a singleton).

If is a single-valued mapping, then Definition 1.7 (resp., Definitions 1.8 and 1.9) reduces to the concept of weak commutativity (resp., compatibility and weak compatibility) for single-valued mappings due to Sessa [18] (resp., Jungck [11, 12]).

but the converse of these implications may not be true (see, [13, 15]).

Throughtout this paper, we assume that is the set of all functions satisfying the following conditions:

(i) is upper semi-continuous continuous at a point from the right, and non-decreasing in each coodinate variable,

Theorem 1.10 (see [19]).

where satisfies (i) and for each , and , with . Then and have a unique common fixed point such that .

In the present paper, we are concerned with the following:

(1)replacing the commutativity of the mappings in Theorem 1.10 by the weak compatibility of a pair of mappings to obtain a common fixed point theorem metric spaces without the continuity assumption of the mappings,

(2)giving an example to support our generalization of Theorem 1.10,

(3)establishing another common fixed point theorem for two families of set-valued mappings and two single-valued mappings,

(4)proving a common fixed point theorem for weakly compatible mappings under a strict contractive condition on compact metric spaces.

## 2. Main Results

In this section, we establish a common fixed point theorem in metric spaces generalizing Theorems 1.10. Also, an example is introduced to support our generalization. We prove a common fixed point theorem for two families of set-valued mappings and two single-valued mappings. Finally, we establish a common fixed point theorem under a strict contractive condition on compact metric spaces.

First we state and prove the following.

Theorem 2.1.

Let be two sefmaps of a metric space and let be two set-valued mappings with

then there is a point such that .

Proof.

then . Therefore, is a Cauchy sequence.

as . So, is a Cauchy sequence. Hence, for some . But by (2.3), so that . Consequently, . Moreover, we have, for , that . Therefore, . So, we have by Lemma 1.4 that . In like manner it follows that and .

This contradiction demands that . Similarly, if the pair is weakly compatible, one can deduce that . Therefore, we get that .

The proof, assuming the completeness of , is similar to the above.

Now, we give an example to show the greater generality of Theorem 2.1 over Theorem 1.10.

Example 2.2.

for all
. It is clear that
is a complete metric space. Since
is a closed subset of
, so
is complete. We note that
is a
-compatible pair and therefore a weakly compatible pair. Also,
and
, that is,
and
are weakly compatible. On the other hand, if
, so that
even though
, that is,
is not a
-compatible pair. We know that
is the unique common fixed point of
and
. Hence the hypotheses of Theorem 2.1 are satisfied. Theorem 1.10 is not applicable because
for all
, and the maps *I*, *J* and *G* are not continuous at
.

In Theorem 2.1, if the mappings and are replaced by and , where is an index set, we obtain the following.

Theorem 2.3.

then there is a point such that for each .

Proof.

Inspired by the work of Chang [9], we state the following theorem on compact metric spaces.

Theorem 2.4.

holds whenever the right-hand side of (2.20) is positive, then there is a unique point in such that .

## Declarations

### Acknowledgment

The author wishes to thank the refrees for their comments which improved the original manuscript.

## Authors’ Affiliations

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