Fuzzy Fixed Point Results in F-Metric Spaces with Applications

In this paper, some concepts of F-metric spaces are used to study a few fuzzy fixed point theorems. Consequently, corresponding fixed point theorems of multivalued and single-valued mappings are discussed. Moreover, one of our obtained results is applied to establish some conditions for existence of solutions of fuzzy Cauchy problems. It is hoped that the established ideas in this work will awake new research directions in fuzzy fixed point theory and related hybrid models in the framework of F-metric spaces.


Introduction
One of the challenges in mathematical modeling of practical phenomena relates to the indeterminacy induced by our inability to categorize events with adequate precision. It has been understood that classical mathematics cannot cope effectively with imprecisions. As a result, the concept of fuzzy set was initiated by Zadeh [1] in 1965 as one of the uncertainty approaches to construct mathematical models compatible with real world problems in engineering, life science, economics, medicine, language theory, and so on. The basic ideas of fuzzy set have been extended in different directions. In particular, the notion of fixed point results for fuzzy set-valued mappings and fuzzy contractions was initiated by Heilpern [2] who proved a fixed point theorem parallel to the Banach fixed point theorem (see [3]) in the frame of fuzzy set. Thereafter, several authors have studied and applied fuzzy fixed point results in different settings [4,5], see, for example [6][7][8][9][10][11][12][13][14] and the references therein.
Not long ago, Jleli and Samet [9] initiated the concepts of F-metric spaces and obtained a generalization of the Banach fixed point theorem. Meanwhile, researchers have picked keen interests in establishing and improving different results in F-metric spaces, see, for instance, [15][16][17].
The aim of this paper is twofold. First, we study some common fuzzy fixed point results in the setting of F-complete F-metric spaces. Consequently, corresponding fixed point theorems of multivalued and single-valued mappings are derived. Thereafter, one of our obtained results is applied to discuss some solvability conditions of fuzzy initial value problems. As far as we know, in the setting of F-metric spaces and fuzzy mappings, the results presented herein are new and fundamental. On this note, it can be improved upon when discussed in other generalized hybrid models within the scope of fuzzy mathematics.

Preliminaries
In this section, we record some specific definitions and results that will be useful in what follows hereafter.
Definition 4 (see [9]). Let ðX, DÞ be an F -metric space and fx n g n∈ℕ be a sequence in X: In the sequel, we shall adopt the following notations and definitions in the setting of F-metric space. We shall denote an F-metric by D F so that ðX, D F Þ represents an F-metric space. Let Cð2 X Þ be the set of all nonempty compact subsets of X and A, B ∈ Cð2 X Þ. Then Then, the Hausdorff metric H F on Cð2 X Þ induced by the metric D F is defined as Definition 5 (see [1,2]). Let X be an arbitrary nonempty set. Then, a fuzzy set in X is a function with domain X and values in ½0, 1 = I. If B is a fuzzy set in X and x ∈ X, then the function values BðxÞ is called the grade of membership of x ∈ X.
The α -level set of B, denoted by ½B α , is defined as Here, M denotes the closure of the crisp set M. Also, the family of fuzzy sets in a metric space X shall be denoted by I X .
Let the set of all nonempty bounded proximal sets in X be denoted by P r ðXÞ and the set of all nonempty closed and bounded subsets of X be represented by CBðXÞ. Since every compact set is proximal and any proximal set is closed, we have the inclusions: Definition 7 (see [2]). Let X be an arbitrary set and Y a metric space. A mapping B : X ⟶ I X is called a fuzzy mapping. A fuzzy mapping B is a fuzzy subset of X × Y with membership value BðxÞðyÞ : Definition 8 (see [2,7]). Let A and B be fuzzy mappings from The point u is known as a common fuzzy fixed point of A and B if u ∈ ½Au α ∩ ½Bu α : Definition 9 (see [18,19]). A nondecreasing function φ : ½0, ∞Þ ⟶ ½0,∞Þ is said to be a comparison function, if φ n ðtÞ ⟶ 0 as n ⟶ ∞ for every t ∈ ½0,∞Þ, where φ n ðtÞ denotes the nth iterate of φ.
Denote by Ω the set of all comparison functions.
Proof. Let x 0 ∈ X be arbitrary. By hypothesis, there exists α A ðx 0 Þ ∈ ð0, 1 such that ½Ax 0 αAðx 0 Þ ∈ Cð2 X Þ. Since ½Ax 0 αAðx0Þ is a nonempty compact subset of X, there exists Similarly, we can find α B ðx 1 Þ ∈ ð0, 1 such that ½Bx 1 αBðx1Þ ∈ Cð2 X Þ and by compactness of ½Bx 1 αBðx1Þ , we can choose Therefore, using (10) together with (9), we have ð11Þ which is a contradiction. It follows that max fDðx 0 , By continuous repetition of the above steps, we generate a sequence fx n g n∈ℕ of elements of X with such that Consequently, by induction, for all n ∈ ℕ, we have Let η > 0 be a given positive number and ð f , ρÞ ∈ F × ½0,∞Þ such that condition ðD 3 Þ is satisfied. By ðF 2 Þ, there exists λ > 0 such that Let nðηÞ ∈ ℕ such that 0 < ∑ n≥nðηÞ φ n ðD F ðx 0 , x 1 ÞÞ < λ. Hence, by (17) and (F 1 ), we get Now, for D F ðx k , x n Þ > 0, by (D 3 ), (16), and (18), we obtain It follows from (F 1 ) that This shows that fx n gn ∈ ℕ is F-Cauchy. Hence, F-completeness of ðX, D F Þ implies that there exists u ∈ X such that x n ⟶ u as n ⟶ ∞. Now, to prove that u ∈ ½Au αAðuÞ , assume that D F ðu, ½Au αAðuÞ Þ > 0. Then by (D 3 ), Now, we analyze (21) under the following cases: then (21) becomes Since x n ⟶ u as n ⟶ ∞, then by (F 2 ) and the properties of ϕ ∈ Ω, which is a contradiction. then, Hence, by (F 2 ) and the properties of then, By condition (F 1 ), from (28), for ρ = 0, we have As n ⟶ ∞ in (29), we obtain which is a contradiction.
Next, we give an example to support the validity of the hypotheses of Theorem 12.
Theorem 18. Let ðX, D F Þ be an F -complete F -metric space and A, B : X ⟶ I X be fuzzy mappings. Assume that for every x ∈ X, there exist α A ðxÞ, α B ðxÞ ∈ ð0, 1 such that ½Ax αAðxÞ , ½Bx αBðxÞ ∈Cð2 X Þ. Suppose also that the following conditions hold: (i) The function f ∈ F is assumed to be continuous. In addition, suppose φ ∈ Ω satisfies f ðtÞ > f ðφðtÞÞ + ρ for all t ∈ ð0,∞Þ (ii) And for all x,y ∈ X, we have Then, there exists u ∈ X such that u ∈ ½Au α A ðuÞ ∩ ½Bu α B ðuÞ :

Journal of Function Spaces
Proof. Following the proof of Theorem 12, we obtain that fx n g n∈ℕ is an F-Cauchy sequence in the F-complete metric space ðX, D F Þ. Therefore, there exists u ∈ X such that Now, to prove that u ∈ ½Au α A ðuÞ , we argue by contradiction. So assume D F ðu, ½Au αAðuÞ Þ > 0. Then by (D 3 ), and inequation (41), we get Taking the limit in (43) as n → ∞ and using (42) together with the continuity of f and φ, we have which is a contradiction to the condition on f . It follows that D F ðu, ½Au αAðuÞ Þ = 0. On similar steps, we can show that D F ðu, ½Bu αBðuÞ Þ = 0. Consequently, we have

Consequences
In this section, we apply Theorems 12 and 18 to deduce some fixed point results of multivalued and single-valued mappings in the context of F-metric spaces. To this end, recall that a point u ∈ X is called a fixed point of a multivalued (single-valued) mapping T on X, if u ∈ Tu ðu = TuÞ.
Corollary 20. Let ðX, D F Þ be an F -complete F -metric space and S, T : X ⟶ Cð2 X Þ be multivalued mappings. Suppose that the following condition holds: for all x, y ∈ X, where φ ∈ Ω. Then, there exists u ∈ X such that u ∈ Su ∩ Tu.
Proof. Let α A , α B : X ⟶ ð0, 1 be any two arbitrary mappings, and consider two fuzzy set-valued maps A, B : X ⟶ I X defined as follows: Then, for all x ∈ X, we have Similarly, ½Bx α BðxÞ = Tx. Consequently, Theorem 12 can be applied to find u ∈ X such that u ∈ ½Au α A ðuÞ ∩ ½Bu α B ðuÞ = Su ∩ Tu.
Following the proof of Corollary 20, we can also apply Theorem 18 to establish the following result.

Corollary 21.
Let ðX, D F Þ be an F -complete F -metric space and S, T : X ⟶ Cð2 X Þ be multivalued mappings. Suppose that the following conditions hold: (i) The function f ∈ F is assumed to be continuous. In addition, suppose φ ∈ Ω satisfies f ðtÞ > f ðφðtÞÞ + ρ for all t ∈ ð0,∞Þ (ii) And for all x, y ∈ X, we have 6 Journal of Function Spaces Then, there exists u ∈ X such that u ∈ Su ∩ Tu.

Corollary 22.
Let ðX, D F Þ be an F -complete F -metric space and g, h : X ⟶ X be single-valued mappings. Suppose that the following condition holds: for all x, y ∈ X, where φ ∈ Ω. Then, there exists u ∈ X such that u = gðuÞ = hðuÞ.
Following the proof of Corollary 22, one can also employ Theorem 18 to establish the following result.

Corollary 23.
Let ðX, D F Þ be an F -complete F -metric space and g, h : X ⟶ X be single-valued mappings. Suppose that the following conditions hold: (i) The function f ∈ F is assumed to be continuous. In addition, suppose that φ ∈ Ω satisfies f ðtÞ > f ð φðtÞ ðtÞÞ + ρ for all t ∈ ð0,∞Þ and for all x, y ∈ X (ii) And for all x, y ∈ X, we have Then, there exists u ∈ X such that u = gðuÞ = hðuÞ.
In the following, we apply Corollary 22 to deduce the main result of Jlei and Samet [9].
Corollary 24 (see [9]). Let ðX, D F Þ be an F -complete F -metric space and h : X ⟶ X be a single-valued mapping. If for all x, y ∈ X, there exists λ ∈ ð0, 1Þ such that then there exists u ∈ X such that hðuÞ = u.
Remark 25. It is obvious that more consequences of Theorems 12 and 18 can be obtained, but we skip them due to the length of the paper.

Application to Fuzzy Initial Value Problems
Fuzzy differential equations (FDEs) and fuzzy integral equations (FIEs) play significant roles in modeling dynamic systems in which uncertainties or vague notions flourish. These concepts have been established in different theoretical directions, and a large number of applications in practical problems have been studied (see, for example, [20][21][22]). Several techniques for studying FDEs have been presented. The first most popular is using the Hukuhara differentiability (H-differentiability) for fuzzy valued functions (see [20,23,24]). On the other hand, the concept of FIEs was initiated by Kaleva [22] and Seikkala [25]. In the study of existence and uniqueness conditions for solutions of FDEs and FIEs, many authors have applied different fixed point theorems. By using the classical Banach fixed point theorem, Subrahmanyam and Sudarsanam [26] proved an existence and uniqueness result for some Volterra integral equations involving fuzzy set-valued mappings. With the help of Shaulder's fixed point theorem and Arzela-Ascoli's theorem, Allahviranloo et al. [27] studied the existence and uniqueness conditions of solutions of some nonlinear fuzzy Volterra integral equations. In [28], the authors discussed some existence results for a fuzzy initial value problem (FIVP) by employing some contractive-like mapping techniques. Congxin and Shiji [29] studied a Cauchy problem of fuzzy differential equation on the basis of the definition of Hdifferentiability for fuzzy set-valued mappings. They obtained the existence and uniqueness theorem for the Cauchy problem under some generalized Lipschitz condition. Similarly, Villamizar-Roa et al. [30] studied the existence 7 Journal of Function Spaces and uniqueness of solution of FIVP in the setting of generalized Hukuhara derivatives. For some intricacies involved in the theory of fuzzy differential equations, the interested reader may consult [22,24,31].
In this section, using the ideas of fuzzy mappings in an F -complete F-metric space, we provide some conditions for the existence of solutions of a FIVP. In line with the existence methods, our technique is connected with studying the existence of solutions of the equivalent Volterra integral reformulation of the FIVP.
First, in what follows, we recall a few known results that are needed in the sequel. For most of these basic concepts, we follow [30,32]. Let P κ ðℝÞ denote the family of nonempty compact subsets of ℝ. Define addition and multiplication in P κ ðℝÞ as usual, that is, for A, B ∈ P κ ðℝÞ and η ∈ ℝ, we have The Hausdorff metric H in P κ ðℝÞ is defined as It is well known that the couple ðP κ ðℝÞ, HÞ is a complete metric space. Moreover, the metric H satisfies the following properties for all A, B, C, D ∈ P κ ðℝÞ: In general, A + ð−AÞ ≠ f0g, where ð−1ÞA = f−a : a ∈ Ag, and hence P κ ðℝÞ is not a linear space (cf. [30]).
Definition 26. A fuzzy number in ℝ is a function x : ℝ ⟶ ½0, 1 having the following properties: (i) x is normal, that is, there exists t 0 ∈ ℝ such that xðt 0 Þ = 1 (ii) x is fuzzy convex, that is (iii) x is upper semicontinuous, that is, ½x α is closed for all α ∈ ½0, 1 (iv) ½x 0 = ft ∈ R : xðtÞ > 0g is compact Throughout this section, we shall denote the set of all fuzzy numbers in ℝ by I 1 . The set ½x α = ft ∈ ℝ : xðtÞ ≥ αg = ½x l α , x r α denotes the α-level set of x ∈ I 1 . It follows from (i) to (iv) that ½x α ∈ P κ ðℝÞ.
The supremum on I 1 is defined as for every x, y ∈ I 1 , where x r α − x l α = diamð½x α Þ is called the diameter of ½x α .
We shall call Cð½a, b, I 1 Þ the set of all continuous fuzzy functions defined on ½a, b. It is verifiable that Cð½a, b, I 1 Þ is an F-complete F-metric space with respect to the F-metric: The following lemma summarizes some basic properties of the integral of fuzzy functions.
Lemma 27 (see [22]). Let x, y : ½a, b ⟶ I 1 be fuzzy functions and η ∈ ℝ. Then, Definition 28 (see [30]). Let I n denote the set of all fuzzy numbers in ℝ n and x, y, z ∈ I n . An element z is called the Hukuhara difference (or H-difference) of x and y, if it satisfies the equation x = y + z. If the H-difference of x and y exists, it is denoted by x ⊖ H y (or x − y). It is easy to see that x ⊖ H x = f0g, and if x ⊖ H y exists, it is unique.
Definition 29 (see [30]). Let g : ða, bÞ ⟶ I n . The function g is said to be strongly generalized differentiable (or GH-differentiable) at t 0 ∈ ða, bÞ, if there exists an element g G ′ ðt 0 Þ ∈ I n such that there exists the Hukuhara differences: Here, the limit is taken in the metric space ðI n , DÞ, and at the end points of ða, bÞ, only one-sided derivatives are considered.