F -Metric, F-Contraction and Common Fixed-Point Theorems with Applications

In this paper, we noticed that the existence of fixed points of F-contractions, in F -metric space, can be ensured without the third condition (F3) imposed on the Wardowski function F : (0, ∞) → R. We obtain fixed points as well as common fixed-point results for Reich-type F-contractions for both single and set-valued mappings in F -metric spaces. To show the usability of our results, we present two examples. Also, an application to functional equations is presented. The application shows the role of fixed-point theorems in dynamic programming, which is widely used in computer programming and optimization. Our results extend and generalize the previous results in the existing literature.


Introduction
In recent years, many authors have presented interesting generalizations of metric spaces (see for example [1][2][3][4][5][6][7][8][9][10][11]). Among them, Jleli et al. [12] presented the idea of F -metric space and, while comparing it with metric spaces, they proved that every metric space is an F -metric space but the converse is not true, confirming that F -metric space is more general than the metric space. With the help of concrete examples, they obtained a similar result for s-relaxed metric space. They discussed a relation between b-metric and F -metric spaces, defined a natural topology on these spaces and proved that after imposing a sufficient condition, the closed ball is closed with respect to the given topology. Finally, they established a fixed-point theorem of the Banach contraction in the frame of F -metric spaces.
In this article, we relax the restrictions on Wardowski's mapping [13] by eliminating the third condition and prove common fixed-point results of Reich-type F-contractions for both single and set-valued mappings in F -metric spaces.
This article is organized into seven sections. This first section contains a short history of the literature providing a motivation for this article. Section 2 contains some basic definitions which help readers to understand our results. In Section 3, we established theorems of fixed points and common fixed points of single-valued F-contractions in F -metric spaces. An example is provided to explain our results. Section 4 deals with fixed-point theorems of F-contractions with respect to closed balls in F -metric spaces along with an example. In Section 5, the common fixed-point theorems of multi-valued modified F-contractions are proved in F -metric spaces. Section 6 is concerned with an application of the mentioned results to the functional equations in dynamic programming. Section 7 consists of conclusions.

Basic Relevant Notions
Wardowski [13] considered a nonlinear function F : (0, ∞) → R with the following characteristics: for any sequence {t n } ⊂ (0, ∞), we have Wardowski [13] called the mapping T : X → X, defined on a metric space X = (X, d), an F-contraction if there exist τ > 0 and F satisfying (F1)-(F3) such that In what follows, we let Definition 1. [12] Suppose A is a non-empty set and (g, α) ∈ B × [0, ∞). Let the function d : Then d is known as an F -metric on A and the pair (A, d) is called an F -metric space. Example 1. [12] Let A = N (the set of natural numbers) and d : for all (a, b) ∈ A × A. Then d is an F -metric on A.
Example 2. [12] Let A = N and d : for all (a, b) ∈ A × A. Then d is an F -metric on A.

Definition 2.
[12] Suppose {a n } is a sequence in A. Then {a n } is an F -Cauchy sequence if lim n,m→∞ d(a n , a m ) = 0.
Definition 4. [12] Let (A, d) be an F -metric space and B be a non-empty subset of A. Then the following statements are equivalent: For any sequence {a n } ⊂ B, we have lim n→∞ d(a n , a) = 0, a ∈ A =⇒ a ∈ B.
Then g has a unique fixed point a * ∈ A. Moreover, for any a 0 ∈ A, the sequence defined by a n+1 = g(a n ), n ∈ N is F -convergent to a * .

Theorem 2.
[11] Suppose A is a complete metric space with metric d, and let g : A → A be a function such that d(g(a), g(b)) ≤ αd(a, b) + βd(a, g(a)) + γd(b, g(b)) for all a, b ∈ A, where α, β, γ are non-negative and satisfy α + β + γ < 1. Then g has a unique fixed point.

Fixed Points of Reich-Type F-Contractions in F -Metric Spaces
In this section, we construct fixed-point and common fixed-point results for single-valued Reich-type and Kannan-type F-contractions in the setting of F -metric spaces.
By (4) and (5), we write By (d3) and the above inequality, we obtain To prove that z * is the fixed point of S, assume d(Sz * , z * ) > 0. Then By (F1) and letting j → ∞, we have which is a contradiction. Hence d(Sz * , z * ) = 0, i.e., Sz * = z * . Following the same steps, we get Now we show the uniqueness. Assume that z * * is also a common fixed point of S and T and z * = z * * . Then It can be easily verified that d is an F -metric and F satisfies (F1)-(F2). Fix b = c = 0 and (x, y) ∈ X × X. Suppose m = n, then For τ ∈ (0, ln e) = (0, 1), the inequality (1) holds true. Moreover, it is clear that X 1 is the only common fixed point of S and T.
Taking a = 0 in Theorem 3, we get the following result of Kannan-type F-contractions. Ty) with min{d(Sx, Ty), d(x, Sx), d(y, Ty)} > 0, for all (x, y) ∈ X × X. Then S and T have at most one common fixed point in X.
Replacing S with T, we get the following result of single mappings. Ty) with min{d(Tx, Ty), d(x, Tx), d(y, Ty)} > 0, for all (x, y) ∈ X × X. Then T has at most one fixed point in X.

Fixed Points of Reich-Type F-Contractions on F -Closed Balls
This portion of the paper deals with the fixed-point theorems of Reich-type F-contractions that hold true only on the closed balls rather than on the whole space X.
Definition 5. Let (X, d) be an F -complete F -metric space and S, T : X → X be self-mappings. Suppose that a + b + c < 1 for a, b, c ∈ [0, ∞). Then the mapping T is called a Reich-type F-contraction on B(x 0 , r) ⊆ X if there exist F ∈ B and τ > 0 such that τ + F(d(Sx, Ty)) ≤ F ad(x, y) + bd(x, Sx) + cd(y, Ty) , ∀x, y ∈ B(x 0 , r).
Suppose that for x 0 ∈ X and r > 0, the following conditions are satisfied: Then S and T have at most one common fixed point in B(x 0 , r).
We need to show that (x n ) is in B(x 0 , r) for all n ∈ N. We show it by mathematical induction. By (b), we write d(x 0 , x 1 ) < r.
Therefore, (x n ) ⊂ B(x 0 , r) for all n ∈ N. Now by (6), we have Following the same steps of proof of Theorem 3 and using (a), we obtain that the sequence (x n ) is F -convergent to some z * in B(x 0 , r). Furthermore, z * can be proved as common fixed point of S and T in the same way as in Theorem 3.
Taking S = T in Theorem 4, we get the following result of single mappings.
is an F -complete F -metric space and T : X → X is a self-mapping. Suppose that a + b + c < 1 for a, b, c ∈ [0, ∞). Suppose that for x 0 ∈ X and r > 0, the following conditions are satisfied: τ + F(d(Tx, Ty)) ≤ F ad(x, y) + bd(x, Tx) + cd(y, Ty) , for all x, y ∈ B(x 0 , r), Then T has at most one fixed point in B(x 0 , r).
and define d by
Hence, condition (b) holds only for B(x 0 , r) and not on X × X. Moreover, 0 ∈ B(x 0 , r) is the fixed point of T.
Let S, T : X → X be self-mappings and κ ∈ [0, 1). Suppose that for x 0 ∈ X and r > 0, the following conditions are satisfied: Ty)) , for all x, y ∈ B(x 0 , r), Then S and T have at most one common fixed point in B(x 0 , r).

Fixed Points of Set-Valued Reich-Type F-Contractions in F -Metric Spaces
This section is concerned with the fixed points of set-valued Reich-type F-contractions in F -metric spaces. Definition 7. Let (X, d) be an F -metric space [23,24]. Suppose F ∈ B and H : CB(X) × CB(X) → [0, ∞) be the Hausdorff metric function defined in Definition 6. A mapping T : X → CB(X) is known as a set-valued Reich-type F-contraction if there is some τ > 0 such that 2τ + F H(Tx, Ty) ≤ F ad(x, y) + bd(x, Tx) + cd(y, Ty) (A) for (x, y) ∈ X × X and a, b, c ∈ [0, ∞) such that a + b + c < 1.

Theorem 5.
Let (X, d) be an F -complete F -metric space and ( f , α) ∈ B × [0, ∞). If the mapping T : X → CB(X) is a set-valued Reich-type F-contraction such that F is right continuous, then T has a fixed point in X.
Proof. Suppose for a fixed natural number n 0 , x n 0 ∈ X. If x n 0 ∈ Tx n 0 , then x n 0 is the fixed point of T. Let x 0 be an initial guess in X. Choose a point x 1 in X such that x 1 ∈ Tx 0 and x 1 / ∈ Tx 1 , continuing in this manner, we can define a sequence (x n ) such that x n+1 ∈ Tx n and x n+1 / ∈ Tx n+1 for all n ≥ 0. Assume that for x n in X, x n / ∈ Tx n for all n ≥ 0, then Tx n−1 = Tx n . Since F is right continuous, there exists some real value h > 1 such that F hH(Tx n−1 , Tx n ) ≤ F H(Tx n−1 , Tx n ) + τ.

Since
D(x n , Tx n ) ≤ H(Tx n−1 , Tx n ) < hH(Tx n−1 , Tx n ), we have by (F1) Since F is right continuous, we can write by Remark 1, We also have by (12), inf y∈Tx n F d(x n , y) ≤ F H(Tx n−1 , Tx n ) + τ.
Using (F1) and letting n → ∞, we get which is a contradiction, hence d(z * , Tz * ) = 0, i.e., z * is the fixed point of T.
be a Reich-type F-contraction such that F is right continuous. Suppose that for k ∈ [0, 1), there exist F ∈ B and with min{d(Tx, Ty), d(x, Tx), d(y, Ty)} > 0, for all (x, y) ∈ X × X. Then T has a fixed point in X.

Applications
In this part, we use the obtained results, to show the existence of a unique common solution of functional equations in dynamic programming.
The problems of dynamic programming comprise two main parts. One is the state space, which is a collection of parameters representing different states, including transitional states, initial states, and action states. The other is the decision space that is the series of decisions taken to solve the problems. This setting formulates the problems of computer programming and mathematical optimization. In particular, the problems of dynamic programming are transformed into the problems of functional equations: g 1 a, b, p(η(a, b)) for a ∈ A, where U and V are Banach spaces such that A ⊆ U and B ⊆ V and Assume that the decision space and state space are A and B respectively. Our aim is to show that the Equations (16) and (17) Clearly, if the functions G, g 1 and g 2 are bounded then S and T are well-defined. (C3) For τ : R + → R + , (a, b) ∈ A × B, h, k ∈ W(A) and t ∈ A, we have where Tk) for α, β, γ ∈ [0, ∞) such that α + β + γ < 1, where min{d(Sh, Tk), M(h, k)} > 0. Now we prove the following theorem. Proof. By [6], we know that (W(A), d) is an F -complete F -metric space, where d is given by (18). By (C1), S and T are self-mappings on W(A). Suppose λ is an arbitrary positive number and h 1 , h 2 ∈ W(A). Take a ∈ A and b 1 , b 2 ∈ B such that Sh j < G(a, b j ) + g 1 a, b j , h j (η(a, b j )) + λ Th j < G(a, b j ) + g 2 a, b j , h j (η(a, b j )) + λ and Sh 1 ≥ G(a, b 2 ) + g 1 a, b 2 , h 1 (η(a, b 2 )) (24) Th 2 ≥ G(a, b 1 ) + g 2 a, b 1 , h 2 (η(a, b 1 )) .
Combining the above two inequalities, we get This implies that the mapping F : R + → R defined by F(a) = ln a is an element of B, and τ + F(d(Sh 1 , Th 2 )) ≤ F(M(h 1 , h 2 )).
Now it is clear that all the conditions of Theorem 3 are fulfilled, so by applying Theorem 3, S and T have at most one common and bounded solution of the Equations (16) and (17).

Conclusions
Our results extend and generalize the existence of fixed points of Reich-type F-contractions in F -metric spaces and F -closed balls with relaxed conditions. We get fixed-point results of set-valued Reich-type F-contractions. Finally, we applied the results to obtain a fixed-point theorem in dynamic programming. Acknowledgments: The authors are very grateful to the anonymous reviewers for their valuable suggestions which helped improving this paper.

Conflicts of Interest:
The authors declare no conflict of interest.