#### Abstract

We introduce *β*-generalized weak contractive multifunctions and give some
results about endpoints of the multifunctions. Also, we give some results about role of a
point in the existence of endpoints.

#### 1. Introduction

Let be a metric space, the collection of all nonempty bounded and closed subsets of , and the Hausdorff metric with respect to ; that is, for all , where . Let be a multifunction. An element is said to be a fixed point of whenever . Also, an element is said to be an endpoint of whenever [1]. We say that has the approximate endpoint property whenever [1]. Let be a mapping. We say that has the approximate endpoint property whenever [1]. Also, the function is called upper semicontinuous whenever for all sequences with [2]. In 2010, Amini-Harandi defined the concept of approximate endpoint property for multifunctions and proved the following result (see [1]).

Theorem 1. *Let be an upper semicontinuous function such that and for all , a complete metric space, and a multifunction satisfing for all . Then has a unique endpoint if and only if has the approximate endpoint property.*

Then Moradi and Khojasteh introduced the concept of generalized weak contractive multifunctions and improved Theorem 1 by providing the following result [3].

Theorem 2. *Let be an upper semicontinuous function such that and for all , a complete metric space, and a generalized weak contractive multifunction; that is, satisfies for all , where . Then has a unique endpoint if and only if has the approximate endpoint property.*

In this paper, we introduce -generalized weak contractive multifunctions, and by adding some conditions to assumptions of the results, we give some results about endpoints of -generalized weak contractive multifunctions. In 2012, the technique of --contractive mappings was introduced by Samet et al. [4]. Later, some authors used it for some subjects in fixed point theory (see for example [5–8]) or generalized it by using the method of --contractive multifunctions (see e.g., [9–12]).

Let be a metric space and a mapping. A multifunction is called -generalized weak contraction whenever there exists a nondecreasing, upper, semicontinuous function such that for all and for all . We say that is -admissible whenever implies that for all , and , where and are subsets of . We say that has the property whenever for each convergent sequence in with and for all , we have . One can find idea of the property for mappings in [13]. We say that has the property whenever for each sequence in with for all , there exists a natural number such that for all . Finally, we say that has the property whenever for each , there exists such that implies that for every there exists such that . A multifunction is called lower semicontinuous at whenever for each sequence in with and every , there exists a sequence in with for all such that [14].

#### 2. Main Results

Now, we are ready to state and prove our main results.

Theorem 3. *Let be a complete metric space, a mapping, and a -admissible, -generalized weak contractive multifunction which has the properties , , and . Suppose that there exist a subset of and such that . Then has an endpoint if and only if has the approximate endpoint property.*

* Proof. *It is clear that if has an endpoint, then has the approximate endpoint property. Conversely, suppose that has the approximate endpoint property. Choose and such that . Since has the approximate endpoint property, for each , there exists such that . Now by using the condition (), choose such that . Also, choose such that , and by continuing this process, we find a sequence in such that and
for all . Since and is -admissible, . By using induction, it is easy to see that for all . Thus, we obtain
for all . If , then
If , then
If , then
and so . Thus, . If
then (other case implies that ). Thus,
for all . We claim that . If , then because is nondecreasing. On the other hand, since for all , we have which is a contradiction. Hence, . Let for all . If there exists a natural number such that , then it is easy to see that for all , and so . Now suppose that for all . In this case, we have for all . Hence, is a decreasing sequence, and so there exists such that . If , then for all , and so for all . Since is upper and semicontinuous, we obtain which is a contradiction. Thus, . Now, we prove that is a Cauchy sequence. If is not a Cauchy sequence, then there exist and natural numbers such that and for all . Also, we choose as small as possible such that
Thus, for all . Hence, . Since has the property , we obtain
for all . Since , . In fact,
and so . Since is upper semicontinuous, by using we obtain
which is a contradiction, and so is a Cauchy sequence. Choose such that . Now, note that
for all , and so
for all , and so . Since has the property , we obtain
for all . If , then we have
for all . This implies that , and so
If or , then it is easy to see that . Thus, is an endpoint of .

Next example shows that a -generalized weak contractive multifunction is not necessarily a generalized weak contractive multifunction.

*Example 4. *Let . Define by for all . Suppose that is an arbitrary upper semicontinuous function such that for all . If and , then and . Hence,
Thus, is not a generalized weak contractive multifunction. Now, suppose that for all and define by for all subsets and of . Then, we have
for all . Thus, is a -generalized weak contractive multifunction.

Next example shows that there are multifunctions which satisfy the conditions of Theorem 3, while they are not generalized weak contractive multifunctions.

*Example 5. *Let and let . Define by
If and , then
where is an arbitrary upper semicontinuous function such that for all . Thus, is not a generalized weak contractive multifunction. Now, we show that satisfies all conditions of Theorem 3. For this aim, define and whenever and are subsets of and otherwise. First suppose that or that . If , then and . But, , and so . If or , then , or and so . Hence, . Now, suppose that . In this case, we have , , and . Thus, , and so
Therefore, is a -generalized weak contractive multifunction. Now, we show that is -admissible. If , then , and so and for all and . Thus, for all and . Now, suppose and . Then, and . Hence, . Now, we show that satisfies the condition . First note that, for each , there exists such that . Now, we show that for each there exists such that . If , then , , and
Since for we have , . Thus,
If , then and . Hence,
It is easy to check that satisfies the conditions and . Note that, is the endpoint of .

Now, we add an assumption to obtain uniqueness of endpoint. In this way, we introduce a new notion. Let be a set and a map. We say that the set has the property whenever for all subsets and of with or .

Corollary 6. *Let be a complete metric space, a mapping, and a -admissible, -generalized weak contractive multifunction which has the properties , , and . Suppose that there exist a subset of and such that . If has the approximate endpoint property and has the property , then has a unique endpoint.*

*Proof. *By using Theorem 3, has a endpoint. If has two distinct endpoints and , then because has the property . Hence,
which is a contradiction. Thus, has a unique endpoint.

In Example 5, has a unique endpoint, while does has not the property . Also, has the property , while is not lower semicontinuous. To see this, consider the sequence defined by for and put and . Then and . Let be an arbitrary sequence in such that for all . Then, and for all . But, for sufficiently large and for all since , . This implies that is not lower semicontinuous.

Corollary 7. *Let be a complete metric space, a mapping, and a -admissible multifunction which has the properties , , and . Suppose that has the property , and there exist a subset of , and such that and for all . Then has a unique endpoint if and only if has the approximate endpoint property.*

* Proof. *It is sufficient that we define for all . Then, Theorem 3 and Corollary 6 guarantee the result.

It has been proved that lower semicontinuity of the multifunction and the property are independent conditions [9]. We can replace lower semicontinuity of the multifunction instead of the property to obtain the next result. Its proof is similar to the proof of Theorem 3.

Theorem 8. *Let be a complete metric space, a mapping, and a lower semicontinuous, -admissible, -generalized weak contractive multifunction which has the properties and . Suppose that there exist a subset of and such that . Then has the approximate endpoint property if and only if has an endpoint.*

Corollary 9. *Let be a complete metric space, a mapping, and a lower semicontinuous, -admissible, -generalized weak contractive multifunction which has the properties and . Suppose that there exist a subset of and such that . If has the approximate endpoint property and has the property , then has a unique endpoint.*

Corollary 10. *Let be a complete metric space, a mapping, and a -admissible multifunction which has the properties , , and . Suppose that has the property , and there exist a subset of , and such that and for all . If has the approximate endpoint property, then .*

* Proof. *If we put , then, by using Theorem 2.10 in [9], has a fixed point. Since has the approximate endpoint property, by using Corollary 7, has a unique endpoint such . Let . If , then . If , then because has the property . Also, we have
But, . Thus, , and so .

Next corollary shows us the role of a point in the existence of endpoints.

Corollary 11. *Let be a complete metric space, a fixed element, and a multifunction such that has the property and for all subsets and of with and all and . Assume that for all with , where is a nondecreasing upper semicontinuous function such that for all . Suppose that there exist a subset of and such that . Assume that for each convergent sequence in with and , for all , one has . Also, for each sequence in with for all , there exists a natural number such that for all . Then has an endpoint if and only if has the approximate endpoint property.*

* Proof. *It is sufficient we define by whenever and otherwise, and then we use Theorem 3.

Corollary 12. *Let be a complete metric space, a fixed element and a lower semicontinuous multifunction such that has the property and for all subsets and of with and all and . Assume that
**
for all with , where is a nondecreasing upper semicontinuous function such that for all . Suppose that there exist a subset of and such that . Assume that for each convergent sequence in with and for all , we have . Then has an endpoint if and only if has the approximate endpoint property.*

* Proof. *It is sufficient to define by whenever and otherwise, and then we use Theorem 8.

Let be an ordered metric space. Define the order on arbitrary subsets and of by if and only if for each there exists such that . It is easy to check that is a partially ordered set.

Theorem 13. *Let be a complete ordered metric space and a closed and bounded valued multifunction on such that has the property and for all subsets and of with and all and . Assume that for all with , where is a nondecreasing upper semicontinuous function such that for all . Suppose that there exist a subset of and such that . Assume that for each convergent sequence in with and , for all , one has . Also, for each sequence in with for all , there exists a natural number such that for all . Then has an endpoint if and only if has the approximate endpoint property.*

* Proof. *Define whenever and otherwise, and then we use Theorem 3.

Corollary 14. *Let be a complete ordered metric space and a closed and bounded valued multifunction on such that has the property and for all subsets and of with , all , and . Assume that for all with , where is a nondecreasing upper semicontinuous function such that for all . Suppose that there exist a subset of and such that . Assume that for each convergent sequence in with and , for all , one has . Also, for each sequence in with for all , there exists a natural number such that for all . If has the approximate endpoint property and for all subsets and of with or , then has a unique endpoint.*

* Proof. *Define whenever and otherwise, and then we use Corollary 6.

Let be a metric space and a multifunction. We say that is an -multifunction whenever for each there exists such that . It is obvious that each -multifunction is an multifunction which has the property . Thus, one can conclude similar results to above ones for -multifunctions. Here, we provide some ones. Although by considering -multifunction we restrict ourselves, we obtain strange results with respect to above ones. One can prove the following by reading exactly the proofs of similar above results.

Theorem 15. *Let be a complete metric space, a mapping, and a -admissible, -generalized weak contractive -multifunction which has the properties and . Suppose that there exist a subset of and such that . Then has an endpoint, and so has the approximate endpoint property.*

Theorem 16. *Let be a complete metric space, a mapping, and a lower semicontinuous, -admissible, and -generalized weak contractive -multifunction which has the property . Suppose that there exist a subset of and such that . Then has an endpoint, and so has the approximate endpoint property.*

The next result is a consequence of Theorem 15.

Corollary 17. *Let be a complete metric space, a fixed element, and an -multifunction such that for all subsets and of with , all , and . Assume that for all with , where is a nondecreasing upper semicontinuous function such that for all . Suppose that there exist a subset of and such that . Assume that for each convergent sequence in with and for all one has . Also, for each sequence in with for all , there exists a natural number such that for all . Then has an endpoint, and so has the approximate endpoint property.*

The next result is a consequence of Theorem 16.

Corollary 18. *Let be a complete ordered metric space and a closed and bounded valued lower semicontinuous -multifunction on such that for all subsets and of with , all , and . Assume that for all with , where is a nondecreasing upper semicontinuous function such that for all . Suppose that there exist a subset of and such that . Assume that for each sequence in with for all , there exists a natural number such that for all . Then has an endpoint, and so has the approximate endpoint property.*

#### Acknowledgments

This work was completed while the second author (Dr. Gopal) was visiting the Azarbaijan University of Shahid Madani, Azarshahr, Tabriz, Iran, during the summer of 2012. He thanks Professor Sh. Rezapour and the University for their hospitality and support. The second author gratefully acknowledges the support from the CSIR, govternment of India, Grant no.-25(0215)/13/EMR-II.