New Fixed Point Results in F-Quasi-Metric Spaces and an Application

The main goal of the present paper is to obtain several fixed point theorems in the framework ofF-quasi-metric spaces, which is an extension of F-metric spaces. Also, a Hausdorff δ-distance in these spaces is introduced, and a coincidence point theorem regarding this distance is proved. We also present some examples for the validity of the given results and consider an application to the Volterra-type integral equation.


Introduction and Preliminaries
In the last century, nonlinear functional analysis has experienced many advances. One of these improvements is the introduction of various spaces and is the proof of fixed point results in these spaces along with its applications in engineering science. One of these spaces is function weighted metric space introduced by Jleli and Samet [1]. This is a generalization of metric spaces.
In the sequel, we apply F for the set of all nondecreasing functions that are logarithmic-like.
Definition 3 [5]. Consider a mapping δ q : X × X ⟶ ½0, +∞Þ satisfies the properties ðη 1 Þ and ðη 3 Þ from the definition of a F-metric. Then, δ q is named a F-quasi-metric on X and ðX, δ q Þ is called a F-quasi-metric space.
Definition 5 [5]. Consider a F-quasi-metric space ðX, δ q Þ with a sequence fx n g therein. Then, fx n g is said to be a right-convergent sequence (left-convergent sequence) to x ∈ X if lim n⟶∞ δ q ðx, x n Þ = 0 ( lim n⟶∞ δ q ðx n , xÞ = 0). Further, fx n g is said to be a biconvergent sequence (in summary, convergent sequence) when it is both right-convergent and left-convergent.
Definition 7 [5]. Consider a F-quasi-metric space ðX, δ q Þ with a sequence fx n g therein. Then, fx n g is a right-Cauchy sequence (a left-Cauchy sequence) if lim n,m⟶∞ δ q ðx n , x m Þ = 0 ( lim n,m⟶∞ δ q ðx m , x n Þ = 0). With this interpretation, fx n g is bi-Cauchy sequence (in summary, Cauchy sequence) if it is both left-Cauchy sequence and right-Cauchy sequence. Now, a F-quasi-metric space ðX, δ q Þ is named rightcomplete (left-complete) if every right-Cauchy sequence (left-Cauchy sequence) in X is a right-convergent sequence (left-convergent sequence) in X. Further, ðX, δ q Þ is bicomplete (in short, complete) if it is both left-complete and right-complete.
On the other hand, Bhaskar and Lakshmikantham [6] defined the notion of coupled fixed point and presented several coupled fixed point propositions for a mixed monotone mapping in partially ordered matric spaces. Also, they studied the existence and uniqueness of a solution to a periodic boundary value problem. For more details on coupled, tripled, and n-tupled fixed point assertions, one can see [7] and references therein.
Definition 9 [8,9]. Let F : X × X ⟶ X and g : X ⟶ X be two optional mappings. An element ðu, vÞ ∈ X × X is said to be a coupled coincidence point of F and g if Fðu, vÞ = gu and Fðv, uÞ = gv. Further, an element ðu, vÞ ∈ X × X is named a common fixed point of F and g if Fðu, vÞ = gu = u and Fðv, uÞ = gv = v.
Note that if g is the identity mapping, then ðx, yÞ is called a coupled fixed point of F [6].
Definition 10 [9]. Let F : X × X ⟶ X and g : X ⟶ X be two optional mappings. Then F and g is said to be commutative if Fðgu, gvÞ = gðFðu, vÞÞ for every u, v ∈ X.
In this paper, we introduce several common fixed point and common coupled fixed point theorems in such spaces and prove them. In Section 2, we prove a common fixed point theorem and a common coupled fixed point result in this space. In Section 3, we obtain a coincidence point result for single-valued and multivalued mappings regarding a Hausdorff δ-distance. Ultimately, as an application of these results, the existence of solution of the Volterra-type integral equation is investigated in Section 4.

F-Quasi-Metric Space and Fixed Point Theory
Theorem 11. Let ðX, δ q Þ be a bicomplete F-quasi-metric space. Also, let g, T : X ⟶ X be two arbitrary mappings so that T and g are commutative, TðXÞ ⊂ gðXÞ, gðXÞ is closed, and for every x, y ∈ X, where k ∈ ð0, 1Þ. Then T and g contain a unique common fixed point in X.
Proof. Due to TðXÞ ⊂ gðXÞ, we select a point x 1 ∈ X such that Tx 0 = gx 1 for a given x 0 ∈ X. By continuing this process, we can construct a sequence y n in X by y n = Tx n = gx n+1 for n = 0, 1, ⋯. First, note that T and g possess a unique coincidence point. On the contrary, assume that u 1 , v 1 ∈ X are two different coincidence points of T and g. Then, δ q ðu 2 , v 2 Þ > 0 with gu 1 = Tu 1 = u 2 and gv 1 = Tv 1 = v 2 . Now, by (2), we get which is a contradiction. Assume ðf , BÞ ∈ F × ½0, +∞Þ so that (η 3 ) is complied. For an arbitrary ε > 0 and because of (η 3 ), there exists γ > 0 such that Now, let fy n g be a sequence in X. Without loss of totality, suppose that δ q ðTx 0 , Tx 1 Þ > 0. Otherwise, x 1 is a coincidence point of T and g. Now, using (2), we obtain 2 Advances in Mathematical Physics which implies by induction that for every n in ℕ. Hence, for every m and n in ℕ so that m > n, we get Since there is some N ∈ ℕ so that 0 < ðk n /1 − kÞδ q ðTx 0 , Tx 1 Þ < γ for every n ≥ N. Hence, from (4) and (η 1 ), we observe that for m > n ≥ N. Employing (η 3 ) together with (9), we obtain It follows that δ q ðTx n , Tx m Þ < ε. Therefore, fy n g = fTx n g is right-Cauchy. Similarly, by changing the order of the pairs ðx i+1 , x i Þ in the above process, we conclude that fy n g is also a left-Cauchy sequence. Hence, it is a Cauchy sequence. Now, since ðX, δ q Þ is a bicomplete space, there exists z ∈ X such that fy n g is convergent to z. Since fTx n g = fgx n+1 g ⊂ gðXÞ and gðXÞ is closed, we have lim n⟶∞ δðgx n , gzÞ = 0. As a next step, we show that z is a coincidence point of T and g. On the contrary, consider δ q ðTz, gzÞ > 0. Then we have As n ⟶ ∞ in the inequality above, we obtain which is a contradiction. Hence, δ q ðTz, gzÞ = 0; that is, z is a unique coincidence point of T and g. Therefore, g and T contian a unique point of coincidence w = gz = Tz. By commutativity of the mapping T and g, we have gw = gðgzÞ = gTðzÞ = TgðzÞ = Tw. Hence, gw is another point of coincidence of g and T. Now, by the uniqueness of the point of coincidence of g and T, we have w = gw = Tw; that is, g and T contain a unique common fixed point. This completes the proof.
In the sequel, denote for simplicity X × ⋯× X by X n , where X is a nonempty set and n ∈ ℕ.
Lemma 12. Consider a F-quasi-metric space ðX, δ q Þ. Then, the following assertions hold: (2) The mapping f : X n ⟶ X and g : X ⟶ X contain an n-tuple common fixed point iff the mapping F : X n ⟶ X n and G : X n ⟶ X n defined by possess a common fixed point in X n .
Proof. Clearly, Δ q satisfies in (η 1 ). We show that Δ q satisfies in (η 3 ). For every ðx i,j Þ ⊂ X for 1 ≤ i ≤ N and 1 ≤ j ≤ n, consider Then, we have where f j ∈ F and B j ∈ ½0, +∞Þ. Therefore, we obtain The proofs of (2) and (3) are straightforward and left to the reader.
Remember that Lemma 12 is a two-way relationship. Consequently, we can establish n-tuple fixed point propositions 3 Advances in Mathematical Physics from fixed point assertions and conversely. Now, set n = 2 in Lemma 12. Then, we have the following theorem.
Theorem 13. Let ðX, δ q Þ be a bicomplete F-quasi-metric space. Also, let g : X ⟶ X and T : X 2 ⟶ X be two arbitrary mappings so that T and g are commutative, TðX 2 Þ ⊂ gðXÞ, gðXÞ is closed, and for all ðx, yÞ and ðx * , y * Þ in X 2 , where k ∈ ð0, 1Þ. Then, T and g contain a unique common coupled fixed point in X 2 .
Proof. Let us define Δ q : Further, we consider F : X 2 ⟶ X 2 by Fðx, yÞ = ðTðx, yÞ, Tðy, xÞÞ and G : X 2 ⟶ X 2 by Gðx, yÞ = ðgx, gyÞ for all x, y ∈ X. Using Lemma 12, ðX 2 , Δ q Þ is a bicomplete F-quasimetric space. Also, ðx, yÞ ∈ X 2 is a common coupled fixed point of T and g iff it is a common fixed point of F and G.
On the other hand, from (18), we have either or Now, by Theorem 11, F and G have a common fixed point and by Lemma 12, T and g have a common coupled fixed point.
Clearly T and g are commutative. Also, we have Therefore, by letting k = 1/2, all of the hypotheses of Theorem 13 hold. Thus, T and g possess a common coupled fixed point in X 2 .

Fixed Point Theorem and Hausdorff δ q -Distance
Let us start with the following definition: Consider a F-quasi-metric space ðX, δ q Þ, and denote the family of all nonempty bounded closed subsets of X by CBðXÞ. Then, Hð·, · Þ is said to be a Hausdorff δ q -distance on CBðXÞ, if where δ q ðx, BÞ = inf fδ q ðx, yÞ, y ∈ Bg.
Definition 15 [10]. Let X be a nonempty set, g : X ⟶ X be a single-valued mapping, and T : X ⟶ CBðXÞ be a multivalued mapping. Also, let w = gx ∈ Tx for some x ∈ X. Then w is said to be a point of coincidence of g and T, and x is said to be a coincidence point of g and T.
Theorem 16. Let ðX, δ q Þ be a bicomplete F-quasi-metric space. Also, let g : X ⟶ X be a single-valued mapping and T : X ⟶ CBðXÞbe a multivalued mapping so that TðXÞ ⊂ g ðXÞ, gðXÞ is closed, and g is continuous. Assume that there exists k ∈ ð0, 1Þ such that for all x, y ∈ X. Then T and g have a coincidence point in X.
Proof. Due to TðXÞ ⊂ gðXÞ, we select a point x 1 ∈ X such that gx 1 ∈ Tx 0 for a given x 0 ∈ X. By continuing this procedure, we can construct a sequence x n in X such that gx n+1 ∈ Tx n for n = 0, 1, ⋯. Suppose that ðf , BÞ ∈ F × ½0, +∞Þ so that (η 3 ) holds. For an arbitrary ε > 0 and due to (η 3 ), there exists γ > 0 such that 4 Advances in Mathematical Physics Consider the sequence fgx n g ⊂ X. Now, without loss of generality, suppose that H δ q ðTx 0 , Tx 1 Þ > 0. Otherwise, x 1 is a coincidence point of T and g. Now, from (24), we have Hence, we have δ q ðgx n , gx n+1 Þ ≤ k n δ q ðgx 0 , x 1 Þ for all n ∈ ℕ. Now, let m and n be two natural numbers with m > n. Then, we have On the other hand, since lim for n ≥ N. Hence, by (25) and (η 1 ), we have for all m > n ≥ N. Employing (η 3 ) together with (29), we obtain Now, by (η 1 ), we have δ q ðgx n , gx m Þ < ε. This proves that fgx n g is right-Cauchy. Similarly, by changing the order of the pairs ðx i+1 , x i Þ in the above process, we conclude that fgx n g is also a left-Cauchy sequence. Therefore, it is a Cauchy sequence. Note that ðX, δ q Þ is bicomplete and gðXÞ is closed. Thus, there exists x ∈ X such that lim n⟶∞ gx n = gx. Now, we shall show that gx ∈ Tx. For this purpose, using (24), we have Thus, Hence, gx ∈ Tx. Consequently, T and g possess a point of coincidence.
Now, we consider that the function f ðtÞ = ln t for every t ∈ I, B = 0, and k = e −∥x∥ B . Therefore, all assertions of Theorem 11 hold. As a result, Theorem 11 confirms the existence of fixed point of T so that this fixed point is the answer of the integral equation.

Data Availability
No data were used to support this study.