A Common End Point Theorem for Set-Valued Generalized -Weak Contraction

Abstract

We introduce the class of generalized -weak contractive set-valued mappings on a metric space. We establish that such mappings have a unique common end point under certain weak conditions. The theorem obtained generalizes several recent results on single-valued as well as certain set-valued mappings.

1. Introduction and Preliminaries

Alber and Guerre-Delabriere [1] defined weakly contractive maps on a Hilbert space and established a fixed point theorem for such a map. Afterwards, Rhoades [2], using the notion of weakly contractive maps, obtained a fixed point theorem in a complete metric space. Dutta and Choudhury [3] generalized the weak contractive condition and proved a fixed point theorem for a selfmap, which in turn generalizes theorem 1 in [2] and the corresponding result in [1]. The study of common fixed points of mappings satisfying certain contractive conditions has been at the center of vigorous research activity. Beg and Abbas [4] obtained a common fixed point theorem extending weak contractive condition for two maps. In this direction, Zhang and Song [5] introduced the concept of a generalized -weak contraction condition and obtained a common fixed point for two maps, and Ðorić [6] proved a common fixed point theorem for generalized -weak contractions. On the other hand, there are many theorems in the existing literature which deal with fixed point of multivalued mappings. In some cases, multivalued mapping defined on a nonempty set assumes a compact value for each in . There are the situations when, for each in , is assumed to be closed and bounded subset of . To prove existence of fixed point of such mappings, it is essential for mappings to satisfy certain contractive conditions which involve Hausdorff metric.

The aim of this paper is to obtain the common end point, a special case of fixed point, of two multivlaued mappings without appeal to continuity of any map involved therein. It is also noted that our results do not require any commutativity condition to prove an existence of common end point of two mappings. These results extend, unify, and improve the earlier comparable results of a number of authors.

Let be a metric space, and let be the class of all nonempty bounded subsets of . We define the functions and as follows:

(1.1)

where denotes the set of all positive real numbers. For and , we write and , respectively. Clearly, . We appeal to the fact that if and only if for and

(1.2)

for . A point is called a fixed point of if . If there exists a point such that , then is termed as an end point of the mapping .

2. Main Results

In this section, we established an end point theorem which is a generalization of fixed point theorem for generalized -weak contractions. The idea is in line with Theorem 2.1 in [6] and theorem 1 in [5].

Definition 2.1.

Two set-valued mappings are said to satisfy the property of generalized-weak contraction if the inequality

(2.1)

where

(2.2)

holds for all and for given functions .

Theorem 2.2.

Let be a complete metric space, and let be two set-valued mappings that satisfy the property of generalized -weak contraction, where

(a) is a continuous monotone nondecreasing function with if and only if ,

(b) is a lower semicontinuous function with if and only if

then there exists the unique point such that .

Proof.

We construct the convergent sequence in and prove that the limit point of that sequence is a unique common fixed point for and . For a given and nonnegative integer let

(2.3)

and let

(2.4)

The sequencesandare convergent. Suppose that is an odd number. Substituting and in (2.1) and using properties of functions and , we obtain

(2.5)

which implies that

(2.6)

Now from (2.2) and from triangle inequality for , we have

(2.7)

If , then

(2.8)

From (2.6) and (2.8) it follows that

(2.9)

It furthermore implies that

(2.10)

which is a contradiction. So, we have

(2.11)

Similarly, we can obtain inequalities (2.11) also in the case when is an even number. Therefore, the sequence defined in (2.4) is monotone nonincreasing and bounded. Let when . From (2.11), we have

(2.12)

Letting in inequality

(2.13)

we obtain

(2.14)

which is a contradiction unless . Hence,

(2.15)

From (2.15) and (2.3), it follows that

(2.16)

The sequenceis a Cauchy sequence. First, we prove that for each there exists such that

(2.17)

Suppose opposite that (2.17) does not hold then there exists for which we can find nonnegative integer sequences and , such that is the smallest element of the sequence for which

(2.18)

This means that

(2.19)

From (2.19) and triangle inequality for , we have

(2.20)

Letting and using (2.15), we can conclude that

(2.21)

Moreover, from

(2.22)

using (2.15) and (2.21), we get

(2.23)

and from

(2.24)

using (2.15) and (2.23), we get

(2.25)

Also, from the definition of (2.2) and from (2.15), (2.23), and (2.25), we have

(2.26)

Putting , in (2.1), we have

(2.27)

Letting and using (2.23), (2.26), we get

(2.28)

which is a contradiction with .

Therefore, conclusion (2.17) is true. From the construction of the sequence , it follows that the same conclusion holds for . Thus, for each there exists such that

(2.29)

From (2.4) and (2.29), we conclude that is a Cauchy sequence.

In complete metric space , there exists such that as .

The pointis end point of. As the limit point is independent of the choice of , we also get

(2.30)

From

(2.31)

we have as . Since

(2.32)

letting and using (2.30), we obtain

(2.33)

which implies . Hence, or .

The pointis also end point for. It is easy to see that . Using that is fixed point for , we have

(2.34)

and using an argument similar to the above, we conclude that or .

The pointis a unique end point forand. If there exists another fixed point , then and from

(2.35)

we conclude that .

The proof is completed.

The Theorem 2.2 established that set-valued mappings and under weak condition (2.1) have the unique common end point . Now, we give an example to support our result.

Example 2.3.

Consider as a subspace of real line with usual metric, . Let be defined as

(2.36)

and take as and.

From Tables 1 and 2, it is easy to verify that mappings and satisfy condition (2.1).

Therefore, and satisfy the property of generalized . Note that and have unique common end point. . Also, note that for condition (2.1), which became analog to condition (2.1) in [5], does not hold. For example, while .

Table 1

Table 2

Remark 2.4.

The Theorem 2.2 generalizes recent results on single-valued weak contractions given in [3, 5, 6]. The example above shows that function in (2.1) gives an improvement over condition (2.1) in [5].