Abstract

In this paper, we have established some fixed-point results for the class of multivalued -contractions in the setting of extended -metric space. An example is furnished to show the validity of our results. The results we have obtained generalize/extend many recent results by Asl, Bota, Samreen et al., and those contained therein.

1. Introduction and Preliminaries

One of the important and pioneering results is the celebrated Banach contraction in metric fixed-point theory. Generalizations in the existence of solutions of differential, integral, and integrodifferential equations are mostly based on creating outstanding generalizations in the metric fixed-point theory. These generalizations are obtained by enriching metric structure of underlying space and/or generalizing contraction condition. Bakhtin [1] and Czerwik [2] extended first time the idea of metric space by modifying the triangle inequality and called it a -metric space. Kamran et al. [3], in 2017, further generalized the idea of -metric and introduced an extended -metric (Eb-metric) space. They weakened the triangle inequality of metric and established fixed-point results for a class of contractions. Following the idea of Eb-metric space, a number of authors have published several results in this direction (see, e.g., [4, 5]). To have some insight about miscellaneous generalizations of metric, we refer the readers to a recent article [6] and for some work on -metric, see [7–17].

In 1976, Nadler [18] extended first time the idea of Banach contraction principle for multivalued mappings. He used the set of all closed and bounded subsets of a metric space and the Hausdorff metric on it. Some of the important generalization of Nadler’s result can be seen in ([19–21]). Subashi and Gjini [22] further generalized the concept of extended -metric space to multivalued mappings by using extended Hausdorff -metric. Unlike Nadler, they used , the set of all compact subsets of an Eb-metric space .

In this paper, we have discussed the multivalued -contractions on Eb-metric spaces and proved some fixed-point results. The first section of the paper consists of some essential definitions and preliminaries. The second section is dedicated to some fixed-point results for multivalued mappings where extended -comparison function has been used. In the last section, some well-known theorems are mentioned which are direct consequences of our main result.

The core reason behind adding this section is to recollect some essential concepts and results which are valuable throughout this paper.

Definition 1. ([23], Czerwik) For any nonempty set , a -metric on is a function such that the following axioms hold:
: if and only if .
: .
: such that

The pair is then termed as -metric space with coefficient . Evidently, we can see that the collection of -metric spaces is a superclass of the collection of metric spaces.

A comparison function is an increasing function such that for all ([24]).

A nonnegative real-valued function on is called a -comparison function if it is increasing, and for every and , the series converges.

It is evident from the definition that a -comparison function is a comparison function but the converse may not be true in general (see for example [25]).

Let us consider a -metric space and an increasing nonnegative function on . We call a map to be a -comparison function if for all the series converges ([25, 26]).

The function is an example of -comparison function if for a -metric space . Note that for the defined -comparison function becomes equivalent to the definition of a comparison function.

In the following, the authors enriched the notion of -metric space by amending the triangle inequality

Definition 2. ([3], Kamran al.) Consider a map where . An extended -metric (Eb-metric) on is a function which satisfies
EB1: if and only if .
EB2: .
EB3: .

The pair is then termed as an extended -metric (Eb-metric) space.

If for some , then Definition 2 reduces to the definition of -metric space with coefficient .

Definition 3. ([3], Kamran al.) Let us consider an Eb-metric space. A sequence in is said to be (i)convergent which converges to in if and only if as ; we write (ii)a Cauchy sequence if as

We say that an Eb-metric space is complete if every Cauchy sequence in converges in . We note that the extended -metric is not a continuous functional in general and every convergent sequence converges to a single point.

Next, we define the concept of -orbital lower semicontinuity (lsc in short) in the case of Eb-metric space which we will use

Definition 4. [27]. Let , , and the orbit of , A real-valued function on is said to be a -orbitally at if and implies In case if is multivalued, then the orbit of at is given as .

2. Main Results

For some technical reasons, Samreen et al., introduced another class of comparison functions for Eb-metric spaces given as follows

Definition 5. Let be an Eb-metric space. A nonnegative increasing real-valued function on is called an extended -comparison function if there exists a mapping such that for some and the infinite series converges for all and for every . Here, for We say that is an extended -comparison function for at .

Remark 6. It can be easily seen that by taking (a constant), Definition 5 coincides with the definition of a -comparison function for an arbitrary self-map on . Every extended -comparison function is also a comparison function for some ; i.e., if for every , then by setting , we have

Example 7. Let be an Eb-metric space, a self-map on , and exists for . Define as

Then, by using ratio test, one can easily see that the series converges.where is a distance from a point to a set and .

Definition 8. [22] Let be an Eb-metric space and . An extended Pompeiu-Hausdorff metric induced by is a function defined as:

Theorem 9. [22] Let be a complete Eb-metric space. Then, is a complete Eb-metric space with respect to the metric

The following lemma is trivial.

Lemma 10. Let be an Eb-metric space and . Then, for any and for every , there exist such that

Now we are able to state our main result.

Theorem 11. Let be a continuous functional on such that is an Eb-metric space. Let be a closed subset of and be such that . Assume that for all and ;

Moreover, the inequality (1) strictly holds if and only if and is an extended -comparison function for at . Then, there exists in such that , where . Furthermore, is a point fixed under the map if and only if the map is -orbitally at .

Proof. Let and . Then, because if it is equal, then is a fixed point of . By using (1) for , we obtain

Choose such that

Now, and ; then, by Lemma 2, there exists such that

Again, ; otherwise, is fixed under the map . By using (1), we obtain

Choose such that while the second inequality is due to (4). By Lemma 10, for and , such that

Continuing in the same way, we get

If , then by using (6) and the triangle inequality in Eb-metric, we obtain,

But is an extended -comparison function, so the series converges. Let be the sum of the series. By setting , from inequality (7), we obtain which further implies that . Hence, is a Cauchy sequence in . But is a closed subset of complete space so there exists such that

Using the definition of an extended Hausdorff -metric and (1), we have

But as which infers that .

Assume that is -orbitally lsc at . Then,

Hence, . But is closed, so and thus, is fixed under the map . Conversely, if is a point fixed under the map , then .

Remark 12. Note that Theorem 11 extends/generalizes the main result by Samreen et al. (, Theorem 15.9) to the case of multivalued mappings. Moreover, Theorem 11 includes main results such as by Czerwik (Theorem 9 [2]) and Samreen et al. (Theorem 3.10 (6) [28]) as special cases when the self-mapping is taken on a -metric space. It also invokes some of the results by Proinov [29] and Hicks and Rhoades [30] in the case of metric space.

Example 13. Let and be defined as . Then, is an Eb-metric space with . Define by ; then, for each and , we have . For every and , we obtain

If we define by , then fulfilled all the conditions present in our main Theorem 11. So in such that as we can see here that .

3. Consequences

In this section, we will discuss an important consequence of Theorem 11 which involves multivalued contractions on Eb-metric spaces. The obtained result generalizes some results by Asl et al. (Theorem 2.1 [31]) and Bota et al. (Theorem 9 [32]).where is an extended -comparison function for at . Then, in such that . Additionally, is a point in fixed under the map if and only if the map is -orbitally lsc at .for every . Then, as . Additionally, is a point fixed under the map if and only if the map is -orbitally lsc at .for all . Here, denotes the class of all extended -comparison functions.Theorem 4. Let be a continuous functional on such that is a complete Eb-metric space. Suppose is a contractive multivalued operator of type (Eb) satisfies the following: (i) is -admissible(ii)There exist and such that

Corollary 14. (Theorem 3.9) Let be a continuous functional on such that is a complete Eb-metric space. Let be a map such that . Assume that for every

Proof. The assertion simply follows by taking a self-map and then using Theorem 11.

Theorem 15. Let be a continuous functional on such that is a complete Eb-metric space. Let be such that the orbit of , is a subset of . Suppose that exists and is a constant so that for all . Assume that

Proof. Define by . By taking a self-map, Example 7 invokes that is an extended -comparison function for at . Hence, the result follows from Theorem 11.

Remark 16. Note that Theorem 15 generalizes Theorem 9 [30] for multivalued mappings in the case of Eb-metric spaces.

Definition 17. Let be a map such that is an Eb-metric space. A multivalued mapping is said to be a -admissible map if there exists a real-valued mapping on which is nonnegative and implies that for all . Note that is defined by

Definition 18. [32] Let be an Eb-metric space. A multivalued mapping is said to be a -contractive multivalued operator of type (Eb) if there exist two functions and such that [33]

Then, such that as where . Furthermore, the point is fixed under the map if and only if the function is -orbitally lsc at .

Proof. Since is -admissible and for , so . By using infimum property, for and ,

Thus, which further implies that . Again, by using the same property, for and , . Continue the similar process to obtain

The contractive condition (8) thus implies which becomes equivalent to the following condition: for every and . Thus, all the conditions of Theorem 11 are satisfied and so the assertions follow.

Remark 19. 1. Note that Theorem 4.2 in becomes a special case of Theorem 4 for a self-map. Also, for a selfmap and , Theorem 4 reduces to Theorem 2, 1 [33].

Data Availability

No data is used.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

M. S., W. U., and E. K. contributed in writing, reviewing, and editing the manuscript. All authors contributed equally and significantly in writing this article. All authors have read and agreed to the published version of the manuscript.