Fixed Points and Stability for Integral-Type Multivalued Contractive Mappings

Abstract

The existence and iterative approximations of fixed points concerning two classes of integral-type multivalued contractive mappings in complete metric spaces are proved, and the stability of fixed point sets relative to these multivalued contractive mappings is established. The results obtained in this article generalize and improve some known results in the literature. An illustrative example is given.

1. Introduction

The famous Banach fixed point theorem has both various extensions and valuable applications in a mass of differential equations, difference equations, functional equations, matrix equations, and integral equations ([1–26]). In 2002, Branciari [3] obtained an interesting integral-type fixed point theorem for the contractive mapping of integral type, which is an integral version of the Banach contraction mapping.

Theorem 1 (see [3]). Let be a mapping from a complete metric space into itself satisfying where is a constant and is Lebesgue integrable, summable on each compact subset of and for each . Then, has a unique fixed point such that for each .

Later, the researchers [1, 2, 6, 7, 9, 10, 12, 14, 17–21, 26] generalized Theorem 1 from different directions and got a lot of fixed point results for various contractive mappings of integral type.

In 1969, Nadler [15] gave a multivalued analog of the Banach fixed point theorem by using the Hausdorff metric and introducing the multivalued contraction mapping, that is, he presented a nice fixed point theorem for the multivalued contraction mapping.

Theorem 2 (see [10]). Let be a complete metric space and be a multivalued contraction mapping, that is, there exists a constant satisfying Then, has a fixed point in .

Czerwik [5] and Gordji et al. [8] extended Theorem 2 and proved fixed point theorems for some multivalued contractive mappings, which include (2) as special cases. The researchers [4, 11, 13, 22–24] gained fixed point theorems for several multivalued contractive mappings and studied also the stability of fixed point sets with respect to the multivalued contractive mappings. Lim [11] established the stability of fixed point sets associated with the multivalued contraction mappings in Theorem 2. Choudhury et al. [24] proved that a uniformly convergent sequence of multivalued contractions has stable fixed point sets.

By combining the ideas of Nadler, Branciari, and Lim, in this article, we study the existence and iterative approximations of fixed points concerning two classes of integral-type multivalued contractive mappings in complete metric spaces and present stability of fixed point sets relative to a sequence of integral-type multivalued contractive mappings. Our results generalize and unify a few results in [5, 8, 11, 15]. An example is also presented to illustrate the efficiency of our results.

2. Preliminaries

Throughout this paper, denotes the set of all positive integers, , and

Assume that is a metric space, stands for the family of all nonempty closed subsets of , and denotes the family of all nonempty closed bounded subsets of . For and , , define

A sequence in is called an orbit of at if for each .

Lemma 3 (see [12]). Let and be a nonnegative sequence and . Then

Lemma 4 (see [12]). Let and be a nonnegative sequence. Then if and only if

It follows from [13] the following.

Lemma 5. Assume that is a metric space and . Then, for any and , there exists such that

Lemma 6 (see [13]). Assume that is a metric space. Then

Lemma 7. Assume that is a metric space, , and converges to . Then

Proof. It follows from Lemma 6 that that is This completes the proof.☐

Lemma 8. Let and . Then

Proof. (see (12)). Let . It follows that and there exist a sequence in satisfying Thus, (14) and mean that which yields that On account of (15) and Lemma 3, we infer that Clearly, (12) follows from (17) and (18). The proof of (13) is similar to that of (12) and is omitted. This completes the proof.☐

3. Fixed Point Theorems and an Example

Now, we investigate the existence and iterative approximations of fixed points for the integral-type multivalued contractive mappings (19) and (42), respectively.

Theorem 9. Assume that is a complete metric space and satisfies that where is a constant in and . Then, for each , there exists an orbit of at such that it converges to some fixed point of and

Proof. For any in and in , Lemma 5 guarantees that Note that which together with (19), (21), and yields. and Lemma 5 reveals that Notice that which together with (19), (25), and infers and Making use of (24) and (28), we deduce Continuing the process, we obtain an order of at satisfying Thus, (30), , and mean Lemma 4 gives By (31) and , we obtain It is clear that (33), , , and Lemma 8 guarantee that is Lemma 4 ensures Hence, is a Cauchy sequence.
Completeness of means that there exists a point in with Letting in (33) and using (37) and Lemma 3, we arrive at that is, (20) holds.
Next, we prove that . From (32), (37), and Lemmas 6 and 7, we get immediately that is Taking into account (19), (37), (40), and Lemmas 3 and 7, we gain which yields because . That is, . This completes the proof.☐

Note that . It follows from Theorem 9 the following.

Theorem 10. Assume that is a complete metric space and satisfies where is a constant in and . Then, for each , there exists an orbit of at such that it converges to some fixed point of and (20) holds.

Remark 11. In case and , where is a constant in , then Theorem 12 reduces to a result, which generalizes Theorems 1 and 2 in [5], Theorem 2.1 in [8], and Theorem 5 in [15]. The example below shows that Theorem 10 extends properly Theorem 5 in [15].

Example 1. Let be endowed with the Euclidean metric . Let , , and be defined by

It is clear that . Let with . In order to verify (19), we consider below two possible cases:

Case 1. . Clearly,

Case 2. and . It follows that

That is, (19) holds. Thus, Theorem 10 ensures that has a fixed point . Observe that which means that Theorem 5 in [15] cannot be used to show the existence of fixed points of .

4. On Stability of Fixed Point Sets

Now, we discuss the stability of fixed point sets for the integral-type multivalued contractive mappings (19) and (42), respectively. Put .

Theorem 12. Assume that is a complete metric space and satisfy where is a constant in and . Then, .

Proof. Without loss of generality, we assume that . Note that Theorem 9 yields for . Put in . Lemma 5 guarantees which together with (47) and (49) and implies and Lemma 5 ensures with Note that which together with (47), (53), and gives and From (52) and (56), we deduce that Continuing the process, we obtain an order of at satisfying (48) and In light of (58), , and , we have Combining (60) and Lemma 4, we get By (59) and , we infer It follows from (62), , , and Lemma 8 that that is which together with Lemma 4 means that is, is a Cauchy sequence.
Completeness of guarantees Letting in (62) and making use of (66) and Lemma 3, we deduce Observe that (61), (66), and Lemma 7 ensure that is By virtue of (47), (66), (69), , and Lemmas 3 and 7, we have which means because , that is, .
Taking advantage of (48), (59), (67), , and Lemma 4, we get It follows from Lemma 8 that and Reversing the roles of and , we also conclude Using (73), (74), Lemma 8, and , we obtain This completes the proof.☐

Theorem 13. Assume that is a complete metric space and satisfy where is a constant in and . Assume that Then, .

Proof. Let . It follows from Theorem 12.34 in [27] and that there exists with where is the Lebesgue measure of . (77) guarantees that there exists satisfying By means of (78), (79), , and Theorem 12, we conclude which implies that Thus, follows from (81) and Lemma 4. This completes the proof.☐

Theorems 12 and 13 infer immediately the following.

Theorem 14. Assume that is a complete metric space and satisfy that where is a constant in and . Then, .

Theorem 15. Assume that is a complete metric space and satisfy (77) and where is a constant in and . Then, .

Remark 16. Theorems 14 and 15 extend, respectively, Lemma 1 and Theorem 1 in [11].

5. Conclusion

In this paper, we introduce two classes of integral-type multivalued contractive mappings, which include some known multivalued contractive mappings as special cases, and prove the existence, iterative approximations, and stability of fixed points for these integral-type multivalued contractive mappings under certain conditions. Our results extend several known results in the literature.

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 competing interests.

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) The authors thank the referees for useful comments and suggestions.