#### Abstract

In this paper, we obtain new results which have not been encountered before in the literature, in multivalued quasimetric spaces, inspired by Proinov type contractions. We use admissible function as proving theorems. We also give an example that supports our theorems.

#### 1. Introduction and Preliminaries

Fixed point theory has become an important research topic after the famous mathematician Banach’s definition of the metric fixed point [1]. Many theoretical and applied studies have been done on fixed point theory. In the 21st century, the fixed point is still a popular and dynamic research topic. The concept of metric space, which forms the basis of the fixed point theory, is generalized by many researchers and new spaces (-metric, quasimetric, partial metric, fuzzy metric, etc.) are introduced. One of the important generalizations is quasimetric space proved in 1931 as follows.

Definition 1 (see [24]). Let . A function is a quasimetric on if it satisfies the following: for all in this case, the pair is a quasimetric space.

Let be a quasimetric on , and the set . Thus, the family forms a base for a topology on . Moreover, if is a subset of , we denote by the closure of with respect to topology; we say that the subset is -closed if it is closed with respect to .

A sequence in a quasimetric space converges to , (in ) if and only if as . Moreover, we say that the sequence is (1)left-Cauchy if for every there exists such that , whenever (2)right-Cauchy if for every there exists such that , whenever

Thereupon, a quasimetric space is called to be left (resp., right) complete if every left (resp., right) Cauchy sequence converges (to respect ) (see, e.g., [5, 6, 40, 41]).

Nadler [7] is the first who introduced the framework for multivalued contraction mappings. The author proved the important theorem generalized Banach principle using the Hausdorff metric for multivalued mappings. After the proof of Nadler theorem, the theory of multivalued contraction mappings attracted great attention and is used in various branches of mathematics. Multivalent mappings in different spaces are introduced. One of them is multivalued mapping introduced in quasimetric-spaces by Shoaib [8] (see also [9, 10]).

Let be a quasimetric space. We shall denote by the set of all nonempty subsets of , by the set of all nonempty closed bounded subsets of , and let be the set of all compact subsets of .

Definition 2. Let and be a multivalued map. A point is said to be a fixed point of if .
The set of the fixed point of a mapping is denoted by

Lemma 3 is an important condition in the following main results.

Lemma 3 (see [8]). Let and be nonempty closed bounded subsets of a quasimetric space and let . Then, for all , there exists such that .

Nadler [7] stated that if in the metric spaces it is also provided for With similarly thinking, the following lemma can be written.

Lemma 4. Let and be nonempty compact subsets of a quasimetric space , and let Then, for all , there exists such that .

Many researchers have stated different studies on well-known quasimetric spaces, see e.g., [1113]. In recent years, Alqahtani et al. [14] introduced a new generalization in quasimetric spaces and defined -symmetric quasimetric spaces. This definition is as follows.

Definition 5 (see [14]). Assume that is a quasimetric space. If there exists a positive real number such that for all , then, the pair is called a -symmetric quasimetric space.

To simplify the notations, in the following, we will mark by a -symmetric quasimetric space.

It is clear that if , thus becomes a metric space.

Definition 6 (see [8]). Let and . A function , defined by where and ), satisfies all the axioms of quasimetric and is known as the Hausdorff quasimetric induced by the quasimetric .

Example 7. Let be a metric space and a function , where

Then, is a -symmetric quasimetric space, but it is not a metric space.

In the following, we shall collect some main properties of a -symmetric quasimetric space.

Lemma 8 (see [15]). Let , be a sequence in and . Then, (i) is right-Cauchy is left-Cauchy is Cauchy(ii)if is a sequence in and then

Recall the notion of -admissibility introduced in [16, 17].

Definition 9. A map is defined -admissible if for every , we have where is an offered function.

Some authors [1821] introduced by slightly modifying this definition.

Definition 10. Let and . A multivalued mapping is called to be strictly -triangular-admissible on if the following conditions are satisfied:
(wt) for each , and implies
(wa) for each , implies
where .

Definition 11. Let be a -symmetric quasimetric space, and let . The space is said to be strictly -regular if for any sequence such that for all and as , we have for all .

In recent years, researchers working on the fixed point theory seem to focus on introducing new contractions in known spaces. These new contractions are also accepted by many researchers and there are important studies, for example, -contraction ([2226]), -contraction [27], and interpolation contraction [28].

In 2020, Proinov [29] introduced new and interesting contractions in metric spaces. Proinov proved that several fixed point results (Wardowski [22]; Jleli and Samet [27]) observed in recent years are the result of Skof’s fixed point theorem [30], and he introduced a very general fixed point theorem containing the main result of Skof.

Theorem 12 (see [29]). Let be a complete metric space and a map which satisfies the contractive type condition: where are two functions such that (i) for all (ii) is nondecreasing(iii) for each Hence, has a unique fixed point and for all , as .

There are several studies using Proinov’s contractions; some interesting ones are as follows: Alqahtani et al. [31] proposed the Proinov type mappings by involving certain rational expression in dislocated -metrics. Alqahtani et al. [32] introduced the common fixed point of Proinov type contraction via simulation function. Roldán López de Hierro et al. [33] examined multiparametric contractions in -metric spaces, inspired by Proinov results. Alghamdi et al. [34], on the other hand, introduced a new type of contraction using admissible mappings, inspired by Proinov and -contraction.

Besides these, Karapnar et al. [35] combined contractions of Proinov [29] and Górnicki [36] in complete metric spaces and proved new fixed point theorems using admissible functions. Later, Ahmed and Fulga [37] generalized the Górnicki-Proinov type contraction to quasimetric spaces. Erdal et al. [38] published the notion of -interpolative contraction using a combine of interpolative contractions, Proinov type contractions, and ample spectrum contraction. Roldán López de Hierro et al. [39] proposed a new class of contractions in non-Archimedean fuzzy metric spaces, based on the Proinov fixed point results.

#### 2. Main Results

Let us now give an important lemma that we will use in our main results.

Lemma 13 (see [37]). Let be a sequence on such that If the sequence is not left-Cauchy sequence thus there exists an and two subsequences , of such that

Proof. Supposing that the sequence is not left-Cauchy, we can find and the sequences of positive integers , , with for every , such that

On the other hand, since , we can find such that for every , where . Moreover, since the space is supposed to be symmetric, for every . Therefore, for every . Consequently, we have for every . Now, let be the smallest positive integer, greater than , such that

Thus, we have either

In the case of the first inequality holds, and letting , we get . Similarly, in case of the second inequality holds, we can consider so, we also obtain . Now, by the triangle inequality, and taking into account the above considerations, we have and as , we get

We will give multivalued -contractive mappings.

Definition 14. Let , be a -symmetric quasimetric space, a mapping and be a multivalued operator. We say that is a multivalued -contractive mapping if there exist two functions such that for every with and

Theorem 15. Let be a complete -symmetric quasimetric space, and be a multivalued -contractive mapping. Assume that following conditions are satisfied:
() is strictly -admissible and there exist and such that
() if is a sequence in such that for all and as , we have
() is nondecreasing and for all
() for all
Therefore, has a fixed point in .

Proof. Let be an arbitrary point in and such that and . Let now be such that and . Continuing in this way, we can build the sequence of points in , such that , with and , for Moreover, by condition , we have that there exist and such that . Supposing that , if , we get that is a fixed point of . Then, let . As is a strictly -admissible map, we have that . Thus, there exists such that which implies . By continuing this process, we can construct a sequence in such that where for every (as otherwise, if , thus is a fixed point of ) and . Therefore, . From Lemma 3 with , we obtain for each . Keeping in mind and (20) and we get

By hypothesis , we have

Thus, since is a nondecreasing map, for each . So, the sequence is positively decreasing. Then, there exists such that

Assuming that on account of (23), we get a contradiction to supposition as follows:

Therefore, , as a result, .

We prove that the sequence is left-Cauchy. Let us suppose by contradiction that the sequence is not left-Cauchy. Thus, by using Lemma 13, there exist and two subsequences , (,) of such that (7) is fulfilled. From (7), we conclude that and since the mapping is strictly triungular admissible, for every . Substituting and in (7), we obtain for each , then, for any , so that is Because of , we obtain . Therefore, we can write which contradicts the supposition ; then, is left-Cauchy sequence in , so that it is Cauchy sequence using Lemma 8. Therefore, the sequence is Cauchy in the complete -symmetric quasimetric space and so converges to limit . Now, we consider the following cases.

Case 1. If for some , so by triangle inequality of -symmetric quasimetric space and thus, letting , we conclude that that is,
As is closed, we obtain

Case 2. On the contrary, if for every from , we have for all . We claim that Supposing, on the contrary, , there exists such that . Therefore, we obtain

Taking into account the condition , we get . Passing to limit as , we obtain . Therefore,

as is closed,

Example 16. Let be endowed with the 2-symmetric quasimetric , where and a mapping , defined as

We choose two functions with is nondecreasing, and for all where and Let also

We check that is a multivalued -contractive mapping of (20). Actually, if taking into account the way the function is defined, we have consider the case .

Let then, , . We get

So, we obtain

Therefore, (20) fulfilled. Further, all other cases are satisfying, from . Consequently, by Theorem 15, map has a fixed point, this being .

Definition 17. Let and be a multivalued operator. is said to be a multivalued iri type -contractive mapping if there exist two functions such that for every with and where

Theorem 18. Let be a complete -symmetric quasimetric space, and be a multivalued iri type -contractive mapping. Assume that following conditions are satisfied:
() is strictly -triangular-admissible and there exists and such that
() if is a sequence in such that for all and as , we have
() is nondecreasing, and for all
() for all
Therefore, has a fixed point in .

Proof. By condition , and following the lines of the proof of the previous theorem, we have that , for every . Moreover, and from Lemma 3 with , we obtain for each . Keeping in mind and (36), we get As is closed for every , we get that such that , for every .

If so , from assumption , this is a contradiction. Hence, we obtain , and

Similarly, again using , we get

But, the function is nondecreasing map, so that we get for all Therefore, the sequence is positively decreasing, and then, there exists such that . If , from (42), we obtain which contradictions . Therefore, and, as a result,

We claim that is Cauchy sequence. Let us assume by contradiction that the sequence is not left-Cauchy. Then, by Lemma 13, we can find and two subsequences , (with ) of such that (7) holds. Thereupon, we have that for all . Letting and in (9), we get for every where

Keeping in mind the way the sequence was define, let and . Thus,

Letting in the above inequality, and taking into account (44), respectively (7), we get

Moreover, since the function is nondecreasing, taking the limit superior when in (45) we get which contradicts the supposition ; then, is left Cauchy sequence in , so that it is Cauchy sequence using Lemma 8. Therefore, the sequence is Cauchy in the complete -symmetric quasimetric space and so converges to a point . Now, we consider following cases:

Case 1. If for some , so by triangle inequality of -symmetric quasimetric space and thus, letting , we conclude that that is,

Case 2. If we suppose the contrary, that is, for any , from we know that for all . We assert that Suppose, on the contrary, . Thus, there exists such that for every Using (36), we obtain where

Taking into account the condition , we get . Passing to limit as , we obtain a contradiction, then As is compact,

Corollary 19. Let be a -symmetric quasimetric space and be a multivalued mapping satisfying the condition: for every , where the functions and The map admits a fixed point in provided that following conditions hold:
() is nondecreasing, and for all
() for all

Letting , in Corollary 19, we obtain the following result.

Corollary 20. Let be a -symmetric quasimetric space and be a multivalued mapping satisfying the condition: for every , where the functions and The map admits a fixed point in provided that following conditions hold:
() is nondecreasing, and for all ;
() for all .

Corollary 21. Let be a -symmetric quasimetric space and be a multivalued mapping satisfying the condition: for every and where the functions and

The map admits a fixed point in provided that following conditions:

() is nondecreasing, and for all

() for all

Taking , in Corollary 21, we get the following result.

Corollary 22. Let be a -symmetric quasimetric space and be a multivalued mapping satisfying the condition: for every and where the functions and

The map admits a fixed point in provided that following conditions hold:

() is nondecreasing, and for all

() for all

#### 3. Conclusion

In this paper, we expand the very interesting results of Proinov [29] in several ways: First, we involve a more general form of the function by considering multivalued mapping. Secondly, we refine the structure of the considered abstract space with -symmetric quasimetric space. Indeed, quasimetric space is one of the novel extensions of metric space. Besides, -symmetric quasimetric space is more reasonable to work since almost all quasimetric space form -symmetric quasimetric spaces. There are still rooms for the fixed point results in the context of -symmetric quasimetric spaces.

#### Data Availability

No data are used.

#### Disclosure

The authors declare that the study was realized in collaboration with equal responsibility.

#### Conflicts of Interest

The authors declare that they have no competing interests.

#### Authors’ Contributions

All authors read and approved the final manuscript.