Common Fixed Point of Multivalued Generalized -Weak Contractive Mappings

Abstract

Fixed point and coincidence results are presented for multivalued generalized -weak contractive mappings on complete metric spaces, where is a lower semicontinuous function with and for all . Our results extend previous results by Zhang and Song (2009), as well as by Rhoades (2001), Nadler (1969), and Daffer and Kaneko (1995).

1. Introduction

Let be a metric space. We denote the family of all nonempty closed and bounded subsets of by .

A mapping is said to be -weak contractive if there exists a map with and for all such that

(1.1)

for all .

Also two mappings are called generalized -weak contractions if there exists a map with and for all such that

(1.2)

for all , where

(1.3)

A mapping is said to be a weak contraction if there exists such that

(1.4)

for all , where denotes the Hausdorff metric on induced by , that is,

(1.5)

for all , and where

(1.6)

A mapping is said to be -weak contractive if there exists a map with and for all such that

(1.7)

for all .

The concepts of weak and -weak contractive mappings were defined by Daffer and Kaneko [1] in 1995.

Many authors have studied fixed points for multivalued mappings. Among many others, see, for example, [1–4], and the references therein.

In the following theorem, Nadler [3] extended the Banach Contraction Principle to multivalued mappings.

Theorem 1.1.

Let be a complete metric space. Suppose is a contraction mapping in the sense that for some ,

(1.8)

for all . Then there exists a point such that (i.e., is a fixed point of ).

Daffer and Kaneko [1] proved the existence of a fixed point for a multivalued weak contraction mapping of a complete metric space into .

Theorem 1.2.

Let be a complete metric space and be such that

(1.9)

for some and for all (i.e., weak contraction). If is lower semicontinuous (l.s.c.), then there exists such that .

In Section 3 we extend Nadler and Daffer-Kaneko's theorems to multivalued generalized weak contraction mappings (see Definition 2.1).

Rhoades [5, Theorem ] proved the following fixed point theorem for -weak contractive single valued mappings, giving another generalization of the Banach Contraction Principle.

Theorem 1.3.

Let be a complete metric space, and let be a mapping such that

(1.10)

for every (i.e., -weak contractive), where is a continuous and nondecreasing function with and for all . Then has a unique fixed point.

By choosing , -weak contractions become mappings of Boyd and Wong type [6], and by defining for and , then -weak contractions become mappings of Reich type [7].

Recently Zhang and Song [8] proved the following theorem on the existence of a common fixed point for two single valued generalized -weak contraction mappings.

Theorem 1.4.

Let be a complete metric space, and let be two mappings such that for all

(1.11)

(i.e., generalized -weak contractions), where is an l.s.c. function with and for all . Then there exists a unique point such that .

In Section 4, we extend Theorem 1.3 by assuming to be only l.s.c., and extend Theorem 1.4 to multivalued mappings.

2. Preliminaries

In this paper, denotes a complete metric space and denotes the Hausdorff metric on .

Definition 2.1.

Two mappings are called generalized weak contractions if there exists such that

(2.1)

for all .

Definition 2.2.

Two mappings are called generalized -weak contractive if there exists a map with and for all such that

(2.2)

for all .

In the proof of our main results, we will use the following well-known lemma, and refer to Nadler [3] or Assad and Kirk [9] for its proof.

Lemma 2.3.

If and , then for each , there exists such that

(2.3)

3. Extension of Nadler and Daffer-Kaneko's Theorems

The following theorem extends Nadler and Daffer-Kaneko's Theorems to a coincidence theorem, without assuming to be l.s.c.

Theorem 3.1.

Let be a complete metric space, and let be two multivalued mappings such that for all ,

(3.1)

where (i.e., multivalued generalized weak contractions). Then there exists a point such that and (i.e., and have a common fixed point). Moreover, if either or is single valued, then this common fixed point is unique.

Proof.

Obviously if and only if is a common fixed point of and .

Let be such that . Let and . By Lemma 2.3, there exists such that . Again by using Lemma 2.3, there exists such that . By induction and using Lemma 2.3, we can find in this way a sequence in such that and

(3.2)

and and

(3.3)

It follows that

(3.4)

since if otherwise , then and so . Hence and this is a contradiction.

Similarly,

(3.5)

From (3.4) and (3.5), we conclude that

(3.6)

for all . Since and (3.6) holds, is a Cauchy sequence. Since is complete, there exists such that

We have

(3.7)

Letting in the above inequality, we conclude that . So . Since , we have .

Similarly, . Therefore, and have a common fixed point.

Furthermore, if is single valued, then this common fixed point is unique. In fact, if and are two common fixed points for and , then

(3.8)

Since , , and so .

Remark 3.2.

The last part of the proof of Theorem 3.1 shows that if are multivalued and is a common fixed point, and or is a singleton, then the common fixed point of and is unique.

By taking in Theorem 3.1, we get the following corollary that extends the Daffer and Kaneko theorem (Theorem 1.2).

Corollary 3.3.

Let be a complete metric space and let be such that

(3.9)

for some and for all (i.e., weak contraction). Then there exists such that .

Example 3.4.

Let be endowed with the Euclidean metric. Let be defined by and . Obviously,

(3.10)

So and have a common fixed point (), and since is single valued, this fixed point is unique.

4. Extension of Rhoades and Zhang-Song's Theorems

First we extend Zhang and Song's theorem (Theorem 1.4) to the case where one of the mappings is multivalued.

Theorem 4.1.

Let be a complete metric space and let and be two mappings such that for all ,

(4.1)

(i.e., generalized -weak contractive) where is l.s.c. with and for all . Then there exists a unique point such that .

Proof.

Unicity of the common fixed point follows from (4.1).

Obviously if and only if is a common fixed point of and .

Let and . Let . By Lemma 2.3, there exists such that

(4.2)

We let . Inductively, we let , and by Lemma 2.3, we choose such that

(4.3)

We break the argument into four steps.

Step 1.

Proof.

Using (4.1) and (4.3),

(4.4)

where

(4.5)

So . Hence by (4.4),

(4.6)

Also

(4.7)

where

(4.8)

So . Hence by (4.7),

(4.9)

Therefore, by (4.6) and (4.9), we conclude that

(4.10)

for all .

Therefore, the sequence is monotone nonincreasing and bounded below. So there exists such that

(4.11)

Since is l.s.c.,

(4.12)

By (4.4), we conclude that

(4.13)

and so . Hence .

Step 2.

is a bounded sequence.

Proof.

If were unbounded, then by Step 1, and are unbounded. We choose the sequence such that , is even and minimal in the sense that , and , and similarly is odd and minimal in the sense that , and is even and minimal in the sense that and , and is odd and minimal in the sense that and .

Obviously for every . By Step 1, there exists such that for all we have . So for every , we have and

(4.14)

Hence Also

(4.15)

and this shows that

So if is odd, then

(4.16)

where

(4.17)

and this shows that Since is l.s.c. and (4.16) holds, we have . So and this is a contradiction.

Step 3.

is Cauchy.

Proof.

Let . Since is bounded, for all . Obviously is decreasing. So there exists such that . We need to show that .

For every , there exists such that and

(4.18)

By (4.18), we conclude that

(4.19)

From Step 1 and (4.19), we have

(4.20)

So we may assume that for every , is odd and is even. Hence

(4.21)

where

(4.22)

This inequality shows that Since is l.s.c. and (4.21) holds, we have . Hence and so . Therefore, is a Cauchy sequence.

Step 4.

and have a common fixed point.

Proof.

Since is complete and is Cauchy, there exists such that For every

(4.23)

where

(4.24)

and this shows that

Since is l.s.c. and (4.23) holds, letting in (4.23) we get

(4.25)

So and hence . Since , then .Also

(4.26)

where

(4.27)

So from (4.26), we have

(4.28)

Thus , and hence . Therefore, .

Remark 4.2.

In the proof of Theorem in Zhang and Song [8], the boundedness of the sequence is used, but not proved. Also, for the proof that is a Cauchy sequence, the monotonicity of is used, without being explicitly mentioned.

In our proof of Theorem 4.1, which is different from [8, Theorem ], is not assumed to be nondecreasing.

The following theorem extends Rhoades' theorem by assuming to be only l.s.c..

Theorem 4.3.

Let be a complete metric space, and let be a mapping such that

(4.29)

for every (i.e., -weak contractive), where is an l.s.c. function with and for all . Then has a unique fixed point.

Proof.

The proof is similar to the proof of Theorem 4.1, by taking , and replacing with .

Remark 4.4.

With a similar proof as in Theorem 4.1, in Theorem 4.3 we can replace the inequality (4.29) by the following inequality (4.30) for two single valued mappings .

(4.30)

5. Conclusion and Future Directions

We have extended Nadler and Daffer-Kaneko's theorems to a coincidence theorem without assuming the lower semicontinuity of the mapping .

We have also extended Rhoades' theorem by assuming to be only l.s.c., as well as Zhang and Song's theorem to the case where one of the mappings is multivalued. Future directions to be pursued in the context of this research include the investigation of the case where both mappings in Zhang and Song's theorem are multivalued.