Some periodic and fixed point theorems on quasi-b-gauge spaces

*Correspondence: hassen.aydi@isima.rnu.tn; baak@hanyang.ac.kr 2Institut Supérieur d’Informatique et des Techniques de Communication, Université de Sousse, H. Sousse, 4000, Tunisia 5Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea Full list of author information is available at the end of the article Abstract The notions of a quasi-b-gauge space (U,Qs; ) and a left (right) Js; -family of generalized quasi-pseudo-b-distances generated by (U,Qs; ) are introduced. Moreover, by using this left (right) Js; -family, we define the left (right) Js; -sequential completeness, and we initiate the Nadler type contractions for set-valued mappings T : U → ClJs; (U) and the Banach type contractions for single-valued mappings T : U → U, which are not necessarily continuous. Furthermore, we develop novel periodic and fixed point results for these mappings in the new setting, which generalize and improve the existing fixed point results in the literature. Examples validating our obtained results are also given.


Introduction
For U = ∅, let 2 U indicate the set of all nonempty subsets of the space U. If T : U → 2 U is a set-valued mapping, then Fix(T) = {u ∈ U : u ∈ T(u)} denotes the set of all fixed points of T, and Per(T) = {u ∈ U : u ∈ T [k] (u) for some k in N} denotes the set of all periodic points of T, where T [k] = T • T • T • · · · • T (k-times). A dynamic process of the system (U, T) starting at z 0 ∈ U, is a sequence {z m : m ∈ {0} ∪ N} described by z m ∈ T(z m-1 ) for all m ∈ N.
Recall that for a map T : U → U, a sequence (z m : m ∈ {0} ∪ N) beginning at z 0 ∈ U such that z m = T [m] (z 0 ) for all m ∈ {0} ∪ N is called a Picard iteration.
In 1966, Dugundji [1] initiated the idea of gauge spaces, which generalizes metric spaces (or more generally, pseudo-metric spaces). Gauge spaces have the characteristic that even the distance between two distinct points of the space may be zero. This simple characterization has been the center of interest for many researchers worldwide. In 1973, Reilly [2] initiated quasi-gauge spaces. According to his definition, quasi-gauge spaces generalize quasi-pseudo metric space by replacing a single quasi-pseudo metric space with the family of such spaces on the set. In this way, he was also able to show that quasi-gauge spaces generalize gauge spaces. In 2015, Ali et al. [3] introduced the concept of b-gauge spaces and obtained some fixed point results. For further facts on gauge spaces, we recommend the readers to Agarwal et. al. [4], Frigon [5], Chis and Precup [6], Chifu and Petrusel [7], Lazara and Petrusel [8], Cherichi et al. [9,10] and Jleli et al. [11].
For a long time, it was unknown how to define the distances, which generalize metrics, or pseudo metrics. These distances provide important and powerful tools in finding the solutions to various crucial problems in fixed point theory. In this direction, the works done by Kada et al. [12], Suzuki [13], and Lin and Du [14] in metric spaces are appreciable.
Wlodarczyk and Plebaniak [15] introduced the notion of left (right) J -families of generalized quasi-pseudo distances in quasi-gauge spaces, which generalized the abovementioned distances and provided useful and important aid for solving numerous problems of nonlinear analysis. To know how these families work and help in proving different results of fixed point theory, see [16][17][18][19][20][21].
The famous fixed point results due to Banach [22] called the Banach contraction principle has attained its fame in the case of single-valued mappings and attracted various authors for many years. The principle assures the uniqueness and existence of a fixed point of certain self-maps on a complete metric space and gives a powerful tool to estimate the fixed point. Nadler [23] extended the Banach contraction principle to the case of multivalued mappings using the idea of a Hausdorff metric. Their analogs in more general spaces are important, fascinating and challenging for most researchers. For further results on the subject, see [24][25][26].
This paper aims to introduce the notions of a quasi-b-gauge space (U, Q s; ) and left (right) J s; -families of generalized quasi-pseudo-b-distances generated by (U, Q s; ). Moreover, by using these left (right) J s; -families, we introduce the concept of left (right) J s; -sequential completeness. We also investigate the Nadler type contractions for setvalued maps T : U → Cl J s; (U) and Banach type contractions for single-valued maps T : U → U (that are not necessarily continuous). Furthermore, we present new periodic and fixed point results for such mappings in the new setting, which generalize and complement the existing fixed point results in he literature. Some examples are also provided in support of the main results.

Preliminaries
The following concepts are useful in the entire paper. The famous Banach contraction principle [22] states that: Let (U, q) be a complete metric space. Suppose that T : U → U is a contraction mapping, i.e., there exists μ ∈ [0, 1) such that for all e, f ∈ U. Then (i) T has a unique fixed point g in U; (ii) for each g 0 ∈ U, the sequence {g m = T [m] (g 0 ) : m ∈ N} converges to such a fixed point. Recall that the Hausdorff metric H q on the class of all nonempty closed and bounded subsets CB(U) in the metric space (U, q) is described for all A, B ∈ CB(U) in the following way The main result of Nadler [23] for set-valued mappings is: Let (U, q) be a complete metric space and let T : U → CB(U) satisfy a (H q , μ)contraction, i.e., there exists μ ∈ [0, 1) such that for all e, f ∈ U. Then there is g ∈ U such that g ∈ T(g) (that is, Fix(T) = ∅).
On the other hand, Bakhtin [27] introduced the notion of a b-metric space in 1989, and Czerwik [28] presented it formally in 1993 in order to generalize the Banach contraction principle.

Definition 2.1 ([27]) Let U be a nonempty set and
for all e, f ∈ U as: Then q is a b-metric on U, where s = 2.
We observe that q is not a metric on U, since the triangular inequality does not hold. Also, we note from the definition of a b-metric space that when s = 1, both the concepts of a metric space and a b-metric space coincide. Thus, the class of b-metric spaces is bigger than the class of metric spaces.
is called a quasi-pseudo metric if it fulfils the following for all e, f , g ∈ U: (a) q(e, e) = 0; (b) q(e, g) ≤ q(e, f ) + q(f , g). The pair (U, q) is called a quasi-pseudo metric space.
Then q is a quasi-pseudo metric on U and (U, q) is a quasi-pseudo metric space.
Since the symmetric property does not hold (i.e., q(e, f ) = d(f , e)), (U, q) is not a pseudo metric space, and hence it is not a metric space.

Main results
In 2015, Ali et al. [3] has defined gauge spaces in the local of b s -pseudo metrics, called b-gauge spaces. In order to introduce quasi-b-gauge spaces, we start the introduction of the notion of a quasi-pseudo-b metric. (a) q(e, e) = 0; for all e, f ∈ U.
Then q is a quasi-pseudo-b-metric on U with s = p ≥ 1. Since symmetry property does not hold, q is not a pseudo-b-metric, and hence it is not a b-metric.
Then q is a quasi-pseudo-b-metric on U. Indeed, q(e, e) = 0 for all e ∈ U. Further, q(e, g) ≤ 2{q(e, f ) + q(f , g)} holds for all e, f , g ∈ U and for s = 2. Also, (U, q) is a Hausdorff quasipseudo-b-metric space. Definition 3.7 Suppose (U, T ) is a topological space and Q s; is a quasi-b-gauge on U such that T = T (Q s; ). Then the topological space (U, Q s; ) is called to be a quasi-b-gauge space. We note that (U, Q s; ) is the Hausdorff if Q s; is separating.
Remark 3.8 (a) Each topological space and quasi-uniform space is a quasi-gauge space [2]. Also, each quasi-gauge space is a quasi-b-gauge space (for s = 1). Therefore, in the asymmetric structure, we can term a quasi-b-gauge space as the largest general space. (b) We observe that if s = 1, the above definitions turn down to agree with the definitions in quasi-gauge spaces.
We now establish the notion of left (right) J s; -families of generalized quasi-pseudo-bdistances on U [left (right) J s; -families are the generalizations of quasi-b-gauges].
if the following statements hold for all β ∈ and for all x, y, z ∈ U: the following hold: Now, we mention some trivial properties of left (right) J s; -families.
, which shows that the maps J β , β ∈ are not quasi-pseudo-b metric (see Example 3.12 below). (d) We note that if s = 1, the above definition reduces to the corresponding definition in quasi-gauge spaces.
for all e, f ∈ U.
Proof By incorporating the definition of the left (right) J s; -family of generalized quasipseudo-b-distances on U in the proof of Proposition 3.11 of [15], the proof of our result can easily be obtained.
Example 3.12 Let (U, Q s; ) be a quasi-b-gauge space, where U contains at least two distinct points and Q s; = {q β : β ∈ } is the family of quasi-pseudo-b metrics q β : Let the set F ⊂ U contain at least two distinct, arbitrary and fixed points, and let for all e, f ∈ U as: (3.10) Thus, the condition (J 1 ) holds. Indeed, the condition (J 1 ) does not hold only if there are some β ∈ and e, f ,  (3.6). Then in particular (3.6) yields that there exists m 1 = m 1 (β) ∈ N such that for all m ≥ m 1 , for all β ∈ , and for all 0 < < d β , we have (3.11) By (3.11) and (3.10), denoting m = min{m 1 (β) : β ∈ }, we have for all m ≥ m Let there exist m ∈ N such that for all m ≥ m , for all β ∈ , and for all 0 < < d β , we have    (v m :m∈N) .
(b) We observe that if s = 1 for all β ∈ , the above definitions turn down to agree with the definitions in quasi-gauge spaces. Remark 3. 19 We note that if s = 1, the above definition reduces to the corresponding definition in quasi-gauge spaces.
In a quasi-b-gauge space, we describe the left (right) Hausdorff type quasi-b-distances and Nadler type left (right) contractions in the following way. Definition 3.20 Let (U, Q s; ) be a quasi-b-gauge space, and let J s; = {J β : β ∈ } be a left (right) J s; -family on U. Let ζ ∈ {1, 2, 3} and suppose that for all β ∈ , for all u ∈ U, and for all V ∈ 2 U , (3.14) if for all β ∈ and for all x, y ∈ U: , μ)-contraction on U), but the converse is not generally true.
We now extend the above theorems to the Banach type single-valued left (right)contractions.
, μ)-contraction on U) if for all β ∈ and for all x, y ∈ U: As a result of Definition 3.23 and Theorem 3.22, we now have the following theorem. (3.33) The set-valued map T is defined by [5,6] for u = 3. In fact, we observe We can also write (3.36) in the form that there exists m 0 ∈ N such that for all > 0 and for all n > m ≥ m 0 , we have J(z m , z n ) < and so, in particular in view of (3.38), (3.32), and (3.33), this implies that there exists m 1 ≥ m 0 such that for all 0 < and for all n > m ≥ m 1 , we have J z m , z n = q z m , z n = 0 < .
Hence, for each x, y ∈ U such that Tx = Ty and m ∈ N, we obtain q m (Tx, Ty) ≤ μq m (x, y) where μ < 1. For Tx = Ty, we have q m (Tx, Ty) = 0, so (4.6) holds. Hence, by Theorem (3.24), the operator T has a fixed point, that is, the integral equation (4.3) has at least one solution.