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

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.

Following [9, 16], we define,

(1.1)

For all , we define

(1.2)

where , for some and , for some .

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

It follows immediately from the definition of that, for every ,

(1.3)

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]).

For any , it yields that .

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]).

It can be seen that

(1.4)

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,

(ii)For each , .

Theorem 1.10 (see [19]).

Let be mappings of a complete metric space into and be a mapping of into itself such that and are continuous, , , , and for all ,

(1.5)

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

(2.1)

Suppose that one of and is complete and the pairs and are weakly compatible. If there exists a function such that for all ,

(2.2)

then there is a point such that .

Proof.

Let be an arbitrary point in . By (2.1), we choose a point in such that and for this point there exists a point in such that . Continuing this manner we can define a sequence as follows:

(2.3)

for . For simplicity, we put for . By (2.2) and (2.3), we have that

(2.4)

If , then

(2.5)

This contradiction demands that

(2.6)

Similarly, one can deduce that

(2.7)

So, for each , we obtain that

(2.8)

where . By (2.8) and Lemma 1.6, we obtain that . Since

(2.9)

then . Therefore, is a Cauchy sequence.

Let be an arbitrary point in for . Then and is a Cauchy sequence. We assume without loss of generality that is complete. Let be the sequence defined by (2.3). But for all . Hence, we find that

(2.10)

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 .

Since, for ,

(2.11)

and as , we get from Lemma 1.3 that

(2.12)

This is absurd. So, . But , so such that . If , , then we have

(2.13)

We must conclude that .

Since and the pair is weakly compatible, so . Using the inequality (2.2), we have

(2.14)

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.

To see that is unique, suppose that . If , then

(2.15)

which is inadmissible. So, .

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

Example 2.2.

Let endowed with the Euclidean metric . Assume that for every . Define and as follows:

(2.16)

We have that and . Moreover, if . If , then and . So, we obtain that

(2.17)

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.

Let be a metric space, and let be selfmaps of , and for , be set-valued mappings with and . Suppose that one of and is complete and for the pairs and are weakly compatible. If there exists a function such that, for all ,

(2.18)

then there is a point such that for each .

Proof.

Using Theorem 2.1, we obtain for any , there is a unique point such that and . For all

(2.19)

This yields that .

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

Theorem 2.4.

Let be a compact metric space, selfmaps of set-valued functions with and Suppose that the pairs , are weakly compatible and the functions , are continuous. If there exists a function , and for all , the following inequality:

(2.20)

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