#### Abstract

In this paper, we introduce extended -gauge spaces and the extended family of generalized extended pseudo--distances. Moreover, we define the sequential completeness and construct the Caristi-type -contractions in the framework of extended -gauge spaces. Furthermore, we develop periodic and fixed point results in this new setting endowed with a graph. The obtained results of this paper not only generalize but also unify and improve the existing results in the corresponding literature.

#### 1. Introduction and Preliminaries

The famous Caristi fixed point theorem [1] states that a self-mapping on a complete metric space possesses a fixed point in if for all , where is a lower semicontinuous function.

Indeed, Caristi [1] observed these results when he searched for alternative proof of the outstanding fixed point theorem of Banach. It is known also Caristi-Kirk fixed point theorem [2]. In fact, Caristi’s theorem is equivalent to metric completeness [3]. For some other contributions to this topic, we refer to [4–10].

In view of extending the concept of Banach contraction, Banach -contraction was introduced by Jachymaski [11] in complete metric space accompanied with the graph where the set of vertex matches with the metric space (see also [12–22]).

The notion of metric space has been refined and extended in several distinct directions, by many authors [23–25]. Among all, the notion of gauge space was initiated by Dugundji [26] as a generalization of a metric space. In 1973, Reilly [27] studied quasi-gauge spaces and proved that it generalizes topological spaces, quasi uniform spaces, and quasi metric spaces. This notion was extended as -gauge spaces by Ali et al. [28] in 2015. For further facts on gauge spaces, we recommend the reader to [29–36].

In 2013, Wlodarczyk and Plebaniak [37] have given the notion of left (right) -families of generalized pseudo distances in quasi-gauge spaces that generalizes the abovementioned distances and provides powerful and useful tools for finding solutions to various problems of nonlinear analysis.

This paper is aimed at introducing extended -gauge spaces and the extended -family of generalized extended pseudo--distances generated by . Moreover, by using extended -family, we define the extended -sequential completeness and construct the Caristi-type -contractions . Furthermore, we investigate periodic and fixed point results for these mappings in the new setting endowed with a graph, which generalizes and improves the existing results in the literature of fixed point theory.

In what follows, we recollect some essential concepts and basic results which shall be used in the sequel. For a nonempty set , we use the notation to denote the set of all nonempty subsets of the space . If is a multivalued map, then the sets of all fixed points are denoted by Fix , that is, Fix . In addition, the set of all periodic points of is denoted by Per , that is, Per , where = (-times). A dynamic process of the system starting at is a sequence defined by .

One of the most interesting extension of a metric is the notion of -metric [38, 39].

*Definition 1. *Let be a nonempty set and . A map is -metric, if it satisfies the following properties:
(a)(b)(c)for all . Here, the pair is called -metric space.

Indeed, -metric is one of the most interesting and original generalizations of the notion metric. As it is seen obviously, in the case of , the notions -metric and standard metric coincide. On the other hand, despite the standard metric, -metric is not continuous despite metric. Further, an open (closed) ball is not an open (closed) set. For more details on -metric and interesting examples, we refer to, e.g., [40–49].

In 2017, Kamran et al. [50] refined the notion of -metric under the name “extended -metric.”

*Definition 2. *Suppose be a nonvoid set and let . A map is said to be an extended *-*metric, if it satisfies the following properties:
(a)(b)(c), for all For given extended -metric on , a pair is called extended -metric space.

*Definition 3. *Let be a nonvoid set. The map is called to be pseudo metric, if it satisfies the following properties:
(a)(b)(c), for all The pair is said to be pseudo metric space.

In 2015, Ali et al. [28] have defined gauge spaces in the setting of -pseudo metrics called -gauge spaces. In order to introduce extended -gauge spaces, we start here with the introduction of the notion of extended pseudo--metric.

*Definition 4. *Let be a nonempty set and A map is an extended pseudo--metric, if it satisfies the following properties:
(a)(b)(c) for all The pair is called extended pseudo--metric space.

*Example 1. *Let . Define and for all as follows:
for all Then, is an extended pseudo--metric on . Indeed, and . Further, holds.

*Example 2. *Let and with . Define as follows:
for all Further, holds. In conclusion, is an extended pseudo--metric on . Notice that ; thus, is not a pseudo metric on . This example shows that extended pseudo--metric is more general than pseudo metric.

*Definition 5. *Each family of extended pseudo--metrics , , is called an extended -gauge on .

*Definition 6. *Let the family be an extended -gauge on . The topology whose subbase is defined by the family of all balls is called the topology induced by on .

*Definition 7. *Suppose be a topological space and be an extended -gauge on such that . Then, the topological space is called to be an extended -gauge space, which is denoted by .

*Remark 8. *(a) Each gauge space is -gauge space (for =1), and each -gauge space is an extended -gauge space (for , for each , where ). Therefore, we can term extended -gauge spaces as the largest general spaces. (b) We observe that if , for each , where , the above definitions turn down to the corresponding definitions in -gauge spaces, and if for each , the above definitions turn down to the corresponding definitions in gauge spaces.

We now introduce the notion of extended -families of generalized extended pseudo--distances on (extended -families is the generalization of extended quasi--gauges).

*Definition 9. *Let be an extended -gauge space. The family where is said to be the extended -family of generalized extended pseudo*-*-distances on (for short, extended -family on ) if there exists , where such that for each and for all , the following hold:

(J1)

(J2) For each sequences and in fulfilling
the following holds:
We denote .

We mention here some trivial properties of extended -families in the following remark.

*Remark 10. *Let be an extended -gauge space. Then, for each and for all , the following hold:
(a)(b)Let . If and , then is an extended pseudo--metric(c)There exist examples of which show that the maps are not an extended pseudo--metrics (see following Example 3)

*Example 3. *Suppose . Let be the family of pseudo--metric where be defined as in Example 1.

Let the set . Let satisfies , where . Let and for all define as follows:
and . Then, .

We observe that for all ; thus, condition holds. Indeed, condition will not hold in case if there exists some such that , , , and . However, then this implies the existence of with , and on other hand, , which is impossible.

Now suppose that (4) and (5) are satisfied by the sequences and in . Then, (5) implies

Thus, (6) is satisfied by the sequences and . Therefore, is an extended -family on .

Now, using extended -family on , we establish the following concepts of extended -completeness in the extended -gauge space .

*Definition 11. *Let be an extended -gauge space. Let be the extended -family on . A sequence is extended -cauchy sequence in if, for all ,

*Definition 12. *Let be an extended -gauge space. Let be the extended -family on . The sequence is called to be extended -convergent to if , where

*Definition 13. *Let be an extended -gauge space. Let be the extended -family on . If , where Then, the sequence in is extended -convergent in .

*Definition 14. *Let be an extended -gauge space. Let be the extended -family on . The space is called -sequentially complete extended -gauge space, if every extended -Cauchy in is an extended -convergent in .

*Remark 15. *Let be an extended -gauge space.
(a)For each subsequence of , where is an extended -convergent in , we have (b)We observe that if for all , where and , the above definitions of completeness reduce to the corresponding definitions in -gauge spaces (see [28])

*Definition 16. *Let be an extended -gauge space. The map , where is called to be an extended -closed map on if for each sequence in , which is extended -converging in , i.e., and its subsequences and satisfy has the property that there exists such that .

*Definition 17. *Let be an extended -gauge space, and let be the extended -family on . A set is a -closed in if , where , is the -closure in , which indicates the set of all for which there exists a sequence in which -converges to .

Define . Thus, denotes the class of all nonempty -closed subsets of .

*Definition 18. *Let be an extended -gauge space, let be the extended -family on , and let, for each , and for all ,
Define on the distance of Hausdorff type, where as follows:
for each and for all .

In this paper, is a directed set and be an extended -gauge space enriched with the graph where the set of vertices coincides with set and the set of edges contains . Also, is such that no two edges are parallel.

#### 2. Periodic and Fixed Point Theorems

Our main results for multivalued mappings are now given below.

Theorem 19. *Let be an extended -gauge space. Let , where , be the extended -family on such that is extended -sequentially complete. Let be a multivalued edge preserving map and , be a lower semicontinuous function such that for each and where , we have, for all ,
**Assume, moreover, that the following conditions hold:
*(i)*There exist and such that *(ii)*For each and , there exists such that
for all . Then, the following statements hold:*(I)*For any , is extended -convergent sequence in ; thus, *(II)*Furthermore, assume that for some is an extended -closed map on . Then,**(a _{1}) Fix*

*(a*

_{2})*Proof. *(I) We first show that is an extended -Cauchy sequence in .

By assumption (i), there exists and such that . Now using (13), we can write, for each ,

Now by using assumption (ii) and (15), we have for each and such that

As is edge preserving, we can write . Proceeding in the same manner, we have a sequence such that and for each and for all , we have

This implies that the sequence is a nonincreasing sequence; hence, there exits such that as . Now for and each , we have

Letting , we have . This implies that is an extended -cauchy sequence in , i.e., for all and for each ,

Now, since is extended -sequentially complete -gauge space, we have extended -convergent in , i.e., for all , we have, for all and for each ,

Thus, from (19) and (20), fixing , defining and , and applying to these sequences, we get, for all and for each ,

This implies .

(II) To prove , let be arbitrary and fixed. Since and we have thus defining , we can write

Also, its subsequences satisfy, for all , and are extended -convergent to each point . Now, using the fact below,

And the supposition that for some is an extended -closed map on , we have

Thus, holds. The assertion follows from and the fact that . Hence, the theorem is proved.☐

Theorem 20. *Let be an extended -gauge space. Let , where , be the extended -family on such that is extended -sequentially complete. Let be a multivalued edge preserving map and , be a lower semicontinuous function such that for each and where , we have, for each ,
**Assume, moreover, that the following condition holds:
*(i)*There exist and such that **Then, the following statements hold:
*(I)*For any , is extended -convergent sequence in ; thus, *(II)*Furthermore, assume that for some is an extended -closed map on . Then,**(b _{1}) Fix*

*(b*

_{2})*Proof. *(I) We first show that is an extended -Cauchy sequence in . By assumption (i), there exist and such that . Now using (28), we can write, for each As is edge preserving, we can write . Proceeding in the same manner, we have a sequence such that and for each and for all , we have
This implies that the sequence is a nonincreasing sequence; hence, there exits such that as . Now for and each , we have
Letting , we have . This implies that is an extended -cauchy sequence in , i.e., for all and for each ,
Now, since is extended -sequentially complete -gauge space, we have extended -convergent in , i.e., for all , we have, for all and for each ,
Thus, from (32) and (33), fixing , defining and , and applying to these sequences, we get, for all and for each ,
This implies .

(II) To prove , let be arbitrary and fixed. Since and we have
thus defining , we can write
Also, its subsequences
satisfy, for all ,
and are extended -convergent to each point . Now, using the fact below,
And the supposition that for some is an extended -closed map on , we have
Thus, holds. The assertion follows from and the fact that . Hence, the theorem is proved.☐

Theorem 21. *Let be an extended -gauge space. Let , where , be the extended -family on such that is extended -sequentially complete. Let be a multivalued edge preserving map and , be a upper semicontinuous function such that for each and where , we have, for each ,
**Assume, moreover, that the following conditions hold:
*(i)*There exist and such that *(ii)*For each and , there exists such that for each ,**Then, the following statements hold:
*(I)*For any , is extended -convergent sequence in ; thus, *(II)*Furthermore, assume that for some is an extended -closed map on . Then,**(c _{1}) Fix*

*(c*

_{2})*Proof. *(I) We first show that is an extended -Cauchy sequence in . By assumption (i), there exist and such that . Now using (41), we can write, for each ,
Now by using assumption (ii) and (43), we have for each and such that
As is edge preserving, we can write . Proceeding as above, we have a sequence such that , and for each