Abstract
Two fixed point theorems for multivalued contractive mappings involving -distance and inequalities of integral type are proved. Our results generalize a few results in the literature. Two illustrative examples are given.
1. Introduction and Preliminaries
In 2002, Branciari [1] generalized the famous Banach contraction principle and proved the following fixed point theorem for the contractive mapping of the integral type.
Theorem 1 (see [1]). Let be a mapping from a complete metric space into itself satisfyingwhere is a constant and is Lebesgue integrable, summable in each compact subset of and for each . Then, has a unique fixed point and for each .
Using Hausdorff metric, Nadler [2] introduced the concept of multivalued contraction and proved a multivalued version of the well-known Banach contraction principle.
Theorem 2 (see [2]). Let be a complete metric space and let be a mapping from into , where is the family of all nonempty closed and bounded subsets of . Assume that there exists such thatThen, has a fixed point.
In the past decades, various fixed point theorems concerning multivalued contractive mappings have been proved. Especially, Feng and Liu [3] generalized Theorem 2 and proved a few fixed point theorems for multivalued contractive mappings without Hausdorff metric.
Theorem 3 (see [3]). Let be a complete metric space and be a multivalued mapping from into , where is the family of all nonempty closed subsets of . Assume that There exist constants with such that, for any , there is satisfying is lower semicontinuous, where .
Then, has a fixed point in .
In 1996, Kada et al. [4] introduced the concept of -distance in a metric space and proved several fixed point theorems for single-valued contractive mappings under -distance. Some other fixed point results concerning -distance can be found in [5–9]. In 2007, Guran [5] deduced the following fixed point theorem, which is a generalization of Theorem 3.
Theorem 4 (see [5]). Let be a complete metric space, be a multivalued mapping, be a -distance on , and . Assume that There exists , with , such that, for any , there is satisfying is lower semicontinuous, where .
Then, has a fixed point in .
Motivated by the results in [1, 3–5], we prove two fixed point results for multivalued contractive mappings of integral type with respect to -distance in complete metric spaces. The results presented in this paper improve Theorems 2–4. Two examples with uncountably many points are included.
Throughout this paper, we denote by the set of positive integers, , , , and
Definition 1 (see [4]). Let be a metric space. A function is called a -distance in if it satisfies the following: For each , a mapping is lower semicontinuous For any , there exists such that and imply
Example 1. Let be a normed linear space with norm , be a positive constant, and be defined byThen, is a -distance in .
Proof. Let . It is clear that holds andwhich implies . For each , put . If and , it follows thatwhich yields . That is, is a -distance in .
Example 2. Let be endowed with the Euclidean metric and be defined bywhere is a constant in . Then, is a -distance in .
Proof. Let . It is clear that holds andwhich implies . For each , put . If and , it is easy to see thatwhich yields . That is, is a -distance in .
Let be a metric space. For any , , , and , putA sequence in is called an orbit of if for all .
Definition 2. Let be a metric space and be a multivalued mapping. A function is said to be Lower semicontinuous in if for each and with T-orbitally lower semicontinuous at if for each orbit of with T-orbitally lower semicontinuous in if it is -orbitally lower semicontinuous at each Obviously, if is lower semicontinuous in , then is -orbitally lower semicontinuous in .
The following lemmas play important roles in this paper.
Lemma 1 (see [10]). Let and be a nonnegative sequence. Then, if and only if .
Lemma 2 (see [4]). Let be a metric space with metric and let be a -distance in . Let and be sequences in , let and be sequences in converging to 0, and let , then the following hold: If and for any , then . In particular, if and , then . If and for any , then converges to . If for any with , then is a Cauchy sequence. If for any , then is a Cauchy sequence.
Lemma 3 (see [11]). Let be a metric space, be a -distance on , and . Suppose that there exists such that . Then, if and only if .
2. Fixed Point Theorems
In this section, we establish fixed point theorems for multivalued contractive mappings (13) and (21), respectively.
Theorem 5. Let be a complete metric space, be a -distance in , and satisfy that for each there is withwhere and are constants in with and .
Then, For each , there exists an orbit of such that for some . if is -orbitally lower semicontinuous at . Moreover, is a fixed point of if .
Proof. Now, we show . Let be an arbitrary point in and . It follows from (13) that there exists such thatContinuing this process, we choose easily a sequence in satisfyingNext, we claim that is a Cauchy sequence. It follows from (15) and thatLetting in (16) and using Lemma 1, we infer thatMaking use of (16), and , we conclude thatwhich together with Lemmas 1 and 2 yields that is a Cauchy sequence. Completeness of implies that there exists some such thatFinally, we show . Since is -orbitally lower semicontinuous at , it follows from (17) thatwhich means that . Thus, Lemma 3 and yield that . This completes the proof.
Theorem 6. Let be a complete metric space, be a -distance in , and satisfy that for each , and there exists withwhere is a constant in and . Then, and hold.
Proof. Now, we show . Let be an arbitrary point in . It follows from (21) that there exists such thatContinuing this process, we construct a sequence in satisfyingNext, we claim that is a Cauchy sequence. It follows from (23) and thatLetting in (24) and using Lemma 1, we infer thatwhich together with (23) implies thatCombining (25) and (26), we conclude thatThe rest of the proof is similar to that of Theorem 5 and is omitted. This completes the proof.
Remark 1. Theorem 5 generalizes Theorems 2–4. The below example demonstrates that Theorem 5 generalizes indeed Theorems 2 and 3.
Example 3. Let be endowed with the Euclidean metric , , , , and be defined byIt is clear that , , andis continuous in . It follows that is lower semicontinuous in . In order to verify (13), for each , we have to consider the following cases.
Case 1. . Clearly, there exists satisfying
Case 2. . It is clear that there exists such that
Case 3. . It follows that there exists such thatHence, (13) holds. That is, the conditions of Theorem 5 are satisfied. It follows from Theorem 5 that has a fixed point .
However, we cannot invoke Theorem 3 to show that the mapping has a fixed point in . Suppose that the conditions of Theorem 3 are satisfied. Take . For , we consider two possible cases as follows: Case 1: . It follows thatwhich implies that , which is impossible because . Case 2: . It is easy to see thatwhich implies that , which is a contradiction because .
It follows from Remark 1 in [3] that Theorem 3 extends Theorem 2. Thus, Theorem 2 is not applicable in proving the existence of fixed points for the multivalued contractive mapping in .
Remark 2. The following example shows that Theorem 6 is different from Theorem 2.
Example 4. Let be endowed with the Euclidean metric , , , and be defined byPut . Clearly, , , is a complete metric space, andis continuous in .
For each , there exists withthat is, (21) holds. It follows from Theorem 6 that has a fixed point .
However, Theorem 2 is useless in proving the existence of fixed points for the mapping in . Suppose that (2) holds. Take and . It follows thatwhich is impossible because .
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 41701616).