Abstract and Applied Analysis

Abstract and Applied Analysis / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 2438193 | https://doi.org/10.1155/2020/2438193

Seong-Hoon Cho, "Fixed Point Theorems for Generalized Set-Valued Weak -Contractions in Complete Metric Spaces", Abstract and Applied Analysis, vol. 2020, Article ID 2438193, 8 pages, 2020. https://doi.org/10.1155/2020/2438193

Fixed Point Theorems for Generalized Set-Valued Weak -Contractions in Complete Metric Spaces

Academic Editor: Jozef Banas
Received01 Apr 2020
Revised28 Jun 2020
Accepted29 Jul 2020
Published01 Sep 2020

Abstract

In this paper, the notion of generalized set-valued weak -contractions is introduced and a new fixed point theorem for such contractions is established in the setting metric spaces. The main result is a generalization of fixed point theorems in the literature. An example and an application to generalized differential equation are given to support the validity of the main theorem.

1. Introduction and Preliminaries

Jleli and Samet [1] introduced the notion of -contractions in Branciari metric spaces [2] and obtained a generalization of the Banach contraction principle, where is a function satisfying the following conditions:

(1) is nondecreasing

(2)For any sequence ,

(3)There exist and such that

Consider the following conditions:

(4) is continuous on

(5) is strictly increasing

(6)For each finite sequence ,

(7)For any with ,

Denote by the class of all functions satisfying conditions

Let be the class of all functions satisfying conditionsand the family of all functions satisfying conditionsand let be the family of all functions satisfying conditions

Indeed, is also an extension of the family in [3], which satisfies conditions

Jleli et al. [3] obtained a generalization of the result of [1] with a function , and Ahmad et al. [4] gave an extention of the result of Jleli and Samet [1] to metric spaces with a simple condition () instead of (3).

Since then, Al-Sulami et al. [5] introduced the concept of -contractions by using a binary relation and obtained a generalization of the result of [1]. Also, Ahmad et al. [6] gave -fuzzy fixed point results by introducing the concept of generalized -contraction fuzzy mappings in metric spaces.

Let be a metric space and be the family of nonempty closed and bounded subsets of , and let be the Pompeiu-Hausdorff distance on , i.e., for all ,where is the distance from the point to the subset .

For , let

Then, we have for all

Recall that the metric space is complete if and only if is complete.

Nadler [7] extended the Banach contraction principle to set-valued mappings by using Pompeiu-Hausdorff distance.

Since then, many researchers (for example, [818] and reference therein) have been studying the fixed point results of set-valued maps in not only metric spaces but also various abstract spaces.

In particular, Hançer et al. [19] extended the result of [1] to set-valued mappings defined on metric spaces. They proved the following theorem.

Theorem 1 [19]. Letbe a complete metric space and letbe a set-valued map. Assume that there existssuch that for allwith,where .
Then, has a fixed point.

Recently, Durmaz [20] generalized the result of Hançer et al. [19] as follows.

Theorem 2 [20]. Letbe a complete metric space and letbe a set-valued map. Assume that there existssuch that for allwith,where and .
Then, has a fixed point.

On the other hand, Khojasteh et al. [21] gave an extension of Banach contraction principle to -metric spaces by introducing the concept of -metrics, where is a continuous mapping with respect to each variable.

The difference between the concept of -contraction in the sense of Khojasteh et al. and the concept of -contraction in the sense of Jleli and Samet is that the concept of -contraction in the sense of Khojasteh is defined in -metric space and the concept of -contraction in the sense of Jleli and Samet is defined in metric space.

In the paper, we introduce the concept of generalized set-valued weak -contraction maps, and we establish a new fixed point theorem for such set-valued contraction maps in the setting of metric spaces.

Lemma 3. Letbe a metric space, and letand. If, then there existssuch that

Proof. Let .
Since , there exists such that , by the definition of infimum. Hence, .

2. Fixed Point Theorems

A function , where is metric space, is called lower semicontinuous function if for each and each sequence with , we have

Let is a strictly increasing and lower semicontinuous function such that

For any sequence

Let be a metric space.

A set-valued mapping is called set-valued weak-contraction if there exists a function such that for all with and for any , there exists with such that

A set-valued mapping is called generalized set-valued weak-contraction if there exists a function such that for all with and for any , there exists with such thatwhere

Remark 4. For, . In fact, if, then. Becauseis strictly increasing,. It follows from (1) that.

Now, we prove our main result.

Theorem 5. Letbe a complete metric space, and letbe a generalized set-valued weak-contraction. If, thenhas a fixed point.

Proof. Let be a point, and let be such that .
Then, it follows from (17) that there exists with such thatwhere .
We deduce that .
If , then it follows from (18) and Remark 4 thatwhich is a contradiction.
Thus, we have , and so,Since , it follows from Remark 4 that .
Hence,Since is nondecreasing,Again, there exists with such thatwhere .
We obtain that .
Assume that .
Then,which is a contradiction.
Thus, we have , and so,which impliesSince is nondecreasing,Inductively, we have a sequence such that, for all Since is a decreasing sequence, there exists such thatWe now show that .
Assume that .
By letting in (28), and using lower semi-continuity of and continuity of , we haveSince , it follows from Remark 4 that .
Hence, we havewhich is a contradiction.
Thus, we haveand so,It follows from (28) thatHence,which impliesGiven , there exists a positive integer such that for all ,Hence, we have that for all ,which impliesHence, is a Cauchy sequence.
Since is complete, there exists a point such thatSince and , it follows from (17) that there exists such thatwhere
We infer thatand so,Let .
Then,Hence,Letting in (41), and using continuity of and lower semicontinuity of , we haveAssume that .
Then, , and so, by Remark 4. Thus, we havewhich is a contradiction.
Hence, we have

Corollary 6. Letbe a complete metric space, and letbe a set-valued map such thatIf either or , then has a fixed point.

Proof. Suppose that .
Since is strictly increasing, there exists the inverse function such thatLet .
Then, it follows from the condition thatBy Lemma 3, there exists such thatHence,By Theorem 5, has a fixed point.
Assume that .
Let .
It follows from (7) thatBecause is nondecreasing,and so,LetBy the definition of infimum, there exists such thatThus,By Theorem 5, has a fixed point.

Corollary 7. Letbe a complete metric space, and letbe a set-valued map such thatIf either or , then has a fixed point.

Corollary 8. Letbe a complete metric space, and letbe a set-valued map satisfying the following condition: there exists such that for all with and for any , there exists with such thatwhere
Then, has a fixed point.

Proof. We define a function byThen, is nondecreasing and lower semicontinuous such that (1) and (2) hold. It follows from (60) that for all with and for any , there exists with withBy Theorem 5, has a fixed point.

Corollary 9. Let be a complete metric space, and letbe a set-valued map such thatwhere .
If either or , then has a fixed point.

Corollary 10. Letbe a complete metric space, and letbe a set-valued map such thatwhere
If either or , then has a fixed point.

Corollary 11. Letbe a complete metric space, and letbe a set-valued weak-contraction. If, thenhas a fixed point.

Corollary 12. Letbe a complete metric space, and letbe a set-valued map such thatIf either or , then has a fixed point.

Corollary 13. Letbe a complete metric space, and letbe a set-valued map such thatIf either or , then has a fixed point.

Corollary 14. Letbe a complete metric space, and letbe a set-valued map satisfying the following condition: there exist such that for all with and for any , there exists with such thatwhere
Then, has a fixed point.

Corollary 15. Letbe a complete metric space, and letbe a set-valued map such that for allwithand for any, there existssuch thatwhere is nondecreasing and lower semicontinuous such that . Then, has a fixed point.

Proof. Let
From (68), we have that for all with and for any , there exists such thatLet , where is nondecreasing and lower semicontinuous such that .
Then, is nondecreasing and lower semicontinuous, andIt follows from (69) that for all with and for any , there exists such thatBy Theorem 5, has a fixed point.

Remark 16. Corollary 15is an extention of Theorem 1 of [22] to set-valued maps and is a modification of Theorem 18of [23].

By taking , where in Theorem 5, we obtain the following result.

Corollary 17. Letbe a complete metric space, and letbe a set-valued map. If for allwithand for any, there existswithsuch thatwhere , then has a fixed point.

We give an example to illustrate Theorem 5. This example shows that the main theorem is a genuine generalization of Nadler’s contraction principle, Theorem 1 and Theorem 2.

Let , and let for all .

Define a map byand a function by

Let .

Then, is strictly increasing and lower semicontinuous satisfying conditions (1) and (2).

We now show that is generalized set-valued weak -contraction.

Let .

We have that

Then, for , there exists such that

If and , then we have

For , there exists such that

Hence, is a generalized set-valued weak -contraction.

Thus, all hypotheses of Theorem 5 are satisfied, and has a fixed point .

Note that Nadler’s contraction principle is not satisfied. In fact, if , thenwhich implies

Hence,and hence,which is a contradiction. Thus, Nadler’s contraction principle does not hold.

Also, note that Theorem 1 and Theorem 2 cannot be used here. In fact, ifwhere , thenand so, . Hence, condition (12) is not satisfied. Since , condition (11) does not hold. Thus, Theorem 1 and Theorem 2 are not applicable here.

3. Application to Initial Value Problem

Let be an origin-centered closed ball, and let the family of nonempty closed convex subsets of with Hausdorff metric generated by the norm of .

Let be a continuous map.

Let be the space of all continuous functions from into , and letfor all .

Then, is a complete metric space.

Note that every continuous function from a compact space into a metric space is bounded.

Consider the following initial value problem:

The existence of solutions for generalized differential equation (86) under certain conditions on is proved by Casting [24], Filippove [25], and Herrmes [26].

We give an application of our fixed point result to the problem on the existence of solutions for generalized differential equation (86).

Theorem 18. Suppose that for eachandthere exists withand such thatThen, the initial value problem (86) has a solution.

Proof. Define a mapping byLet and be given.
By assumption, there exists with such thatHence,