Multivalued Fixed Point Results for Two Families of Mappings in Modular-Like Metric Spaces with Applications

Department of Mathematics, Faculty of Science and Technology, Women University of Azad Jammu and Kashmir, Bagh, Pakistan Department of Mathematics and Statistics, Riphah International University, Islamabad, Pakistan Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, P. O. Box 48940, Leioa, 48940 Bizkaia, Spain Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia China Medical University Hospital, China Medical University, Taichung 40402, Taiwan Department of Mathematics, Daejin University, Kyunggi 11159, Republic of Korea


Introduction and Preliminaries
If the image of a point x under two mappings is x itself, then x is called a common fixed point of those mappings. eory of fixed point has a basic role in analysis (see ). Chistyakov [7] established the concept of modular metric spaces and showed briefly about modular convergence, convex modular, equivalent metrics, abstract convex cone, and metric semigroup. Padcharoen et al. [16] introduced the concept of α-type F-contractions in modular metric spaces and discussed some related results. Further results on these spaces in different directions can be seen in [6,13,14].
In this paper, we establish some common fixed point theorems for two families of set-valued mappings satisfying a generalized contraction on a sequence only in a more generalized setting of modular-like metric spaces. New results can be established in dislocated metric spaces, ordered spaces, partial metric spaces, and metric spaces as a consequence of our findings. To support our results, some applications and examples are discussed. We give the following preliminary concepts which will be used in our results.
Definition 1. (see [16]). Let A ≠ ϕ. A function w: (0, ∞) × A × A ⟶ [0, ∞) is called a modular-like metric on A if for all a, b, c ∈ A, l > 0, and w l (a, b) � w(l, a, b), it satisfies (i) w l (a, b) � w l (b, a) for all l > 0 (ii) w l (a, b) � 0 for all l > 0 and then a � b (iii) w l+u (a, b) ≤ w l (a, c) + w u (c, b) for all l, u > 0 If we replace (ii) by w l (a, b) � 0 for all l > 0 if and only if a � b, then w becomes a modular metric on A. If we replace (ii) by w l (a, b) � 0 for some l > 0 and then a � b, then w becomes a regular modular metric on A. For g ∈ A and ε > 0, We will use m.l.m. space instead of modular-like metric space.
(i) E ⊆ A is known as w-complete if for any sequence (a n ) n∈N in E and for some l > 0, w l (a m , a n ) ⟶ 0 as m, n ⟶ ∞ implies w l (a n , a) ⟶ 0 as n ⟶ ∞ for some a ∈ E (ii) e sequence (a n ) n∈N in A is known as w-Cauchy for some l > 0 if w l (a m , a n ) ⟶ 0 as m, n ⟶ ∞ (iii) e sequence (a n ) n∈N in A is known as w-convergent to a ∈ A for some l > 0 if and only if w l (a n , a) ⟶ 0 as n ⟶ ∞ Definition 3. Let (A, w) be an m.l.m. space and E⊆A. A member p 0 which belongs to E is said to be a best ap- If each g ∈ A has a best approximation in E, then E is known as a proximinal set. P(A) is equal to the family of proximinal sets in A. Let A � R + ∪ 0 { } and w l (g, p) � 1/l(g + p) for all l > 0. Define a set E � [4,6]; then, for each y ∈ A, w l (y, E) � w l (y, [4,6]) � inf u∈ [4,6] w l (y, u) � w l (y, 4). (2) Hence, 4 is the best approximation in E for each y ∈ A. Also, [4,6] is a proximinal set. is known as w l -Hausdorff metric. e pair (P(A), H w l ) is called the w l -Hausdorff metric space. Let A � R + ∪ 0 { } and w l (g, p) � (1/l)(g + p) for all l > 0. If P � [5,6] and O � [9,10], then H w l (P, O) � (13/l).

Definition 5.
Let (X, w) be a modular-like metric space.
en, we will say that w satisfies the Δ M -condition if it is the case that lim n,m⟶∞ w p (x n , x m ) � 0, for p � m − n implies lim n,m⟶∞ w l (x n , x m ) � 0(m, n ∈ N, m > n) for some l > 0.

Main Results
Let (I, w) be an m.l.m. space and c 0 ∈ I; let S σ : σ ∈ Ω and T β : β ∈ Φ be two families of multifunctions from I to P(I). Let c 1 ∈ S a c 0 be an element such that Let In this way, we get a sequence Theorem 1. Let (I, w) be a complete m.l.m. space. Assume that w is regular and satisfies the Δ M -condition. Let c 0 ∈ I, α: I × I ⟶ [0, ∞), T β : β ∈ Φ , and S σ : σ ∈ Ω be the families of α * -dominated set-valued functions on w. Suppose there exist τ > 0 and U ∈ 5 such that is implies then from (10), we have a contradiction. erefore, for all i ∈ N ∪ 0 { }. Hence, from (10), we have Similarly, we have (14) and (15), we have Repeating these steps, we get Similarly, we have Inequalities (17) and (18) can jointly be written as Taking the limit as n ⟶ ∞ in (19), we have Since U ∈ 5, lim n⟶∞ w 1 c n , c n+1 � 0.
Hence, T β S σ (c n ) is a Cauchy sequence in w. Since (I, w) is a regular complete modular-like metric space, there exists c � ∈ I such that T β S σ (c n ) ⟶ c � as n ⟶ ∞, and so, lim n⟶∞ w 1 c n , c � � 0.

Applications in Graph Theory
Jachymski [11] developed a relation between fixed point theory and graph theory by the induction of graphic contractions. Hussain et al. [9] established some results for the new type of contractions endowed with a graph and also showed an application. Further useful results on the graph can be seen in [34,35,40].

Results on Single-Valued Mappings
In this section, some consequences of our results related to single-valued mappings in m.l.m. spaces have been discussed. Let (I, w) be an m.l.m. space, c 0 ∈ I, and S σ , T β : I ⟶ I be two families of mappings. Let c 1 � S σ c 0 , generalized rational-type Wardowski's contraction. Dominated mappings are applied to find out the fixed point results. Applications in the subject of integral equations and graph theory are presented. Moreover, we investigate our results in new generalized modular-like metric spaces. Many consequences of our results in dislocated metric spaces, metric spaces, and partial metric spaces (even with a partial order) can be established easily.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
Each author contributed equally to this paper, read, and approved the final manuscript.