Fixed Point Theorems for Set-Valued L-Contractions in Branciari Distance Spaces

Branciari [1] introduced a Branciari distance by replacing the triangle inequality in a metric with the rectangular inequality as follows. A map d : X × X⟶ 1⁄20,∞Þ, where X is a nonempty set, is said to be Branciari distance on X if and only if it satisfies the following conditions: For all x, y ∈ X and for all distinct points u, v ∈ X, each of them different from x and y is as follows: (d1) dðx, yÞ = 0 if and only if x = y (d2) dðx, yÞ = dðy, xÞ (d3) dðx, yÞ ≤ dðx, uÞ + dðu, vÞ + dðv, yÞ The pair ðX, dÞ is called a Branciari distance space, whenever d is a Branciari distance on X. In many papers, for example, [2–8], it is called generalized metric space, Branciari metric space, or rectangular metric space. However, these names do not reflect and indicate the meaning well of the notion of Branciari distance spaces because Branciari distance can not reduce to the standard metric. Further it is well known that a Branciari distance space ðX, dÞ does not have a topology which is compatible with d (see [8]). For these reasons, we rename and use it as Branciari distance space. Branciari [1] extended the Banach contraction principle to Branciari distance space. After that, a lot of authors, for example, [2–15] and references therein, obtained fixed point results in such spaces. Jain et al. [16] obtained fixed point results in extended Branciari b-distance spaces [17] by defining the notion of certain contractive conditions, and they gave an application to nonlinear fractional differential equations. Branciari [1] investigated the existence of fixed points with the following two conditions:


Introduction and Preliminaries
Branciari [1] introduced a Branciari distance by replacing the triangle inequality in a metric with the rectangular inequality as follows.
A map d : X × X ⟶ ½0,∞Þ, where X is a nonempty set, is said to be Branciari distance on X if and only if it satisfies the following conditions: For all x, y ∈ X and for all distinct points u, v ∈ X, each of them different from x and y is as follows: (d1) dðx, yÞ = 0 if and only if x = y (d2) dðx, yÞ = dðy, xÞ (d3) dðx, yÞ ≤ dðx, uÞ + dðu, vÞ + dðv, yÞ The pair ðX, dÞ is called a Branciari distance space, whenever d is a Branciari distance on X.
In many papers, for example, [2][3][4][5][6][7][8], it is called generalized metric space, Branciari metric space, or rectangular metric space. However, these names do not reflect and indicate the meaning well of the notion of Branciari distance spaces because Branciari distance can not reduce to the standard metric. Further it is well known that a Branciari distance space ðX, dÞ does not have a topology which is compatible with d (see [8]). For these reasons, we rename and use it as Branciari distance space.
Branciari [1] investigated the existence of fixed points with the following two conditions: (i) The topology of a Branciari distance space is a Hausdorff topological space (ii) Any Branciari distance is continuous in each coordinates However, it is known that the above two conditions are not correct (see [14,15]).
(B1) A Branciari distance does not need to be continuous in each coordinates (B2) A convergent sequence in Branciari distance spaces does not need to be Cauchy (B3) The topology of a Branciari distance space does not need to be a Hausdorff topological space (B4) An open ball does not need to be an open set Note that it follows from (B3) that the uniqueness of limits can not be guaranteed.
In despite of the above toplogical feature, the existence of fixed points can be investigated without additional conditions such as continuity of Branciari distances or/ and Hausdorffness of the topology of Branciari distance spaces. This is why researchers are interested in Branciari distance spaces.
Recently, Cho [25] introduced the concept of L-contractions in Branciari distance spaces and established a fixed point theorem for such contractions. He unified concepts of some contractions which exist in literature including θ-contractions.
Very recently, Saleh et al. [26] extended the result of Cho [25] by introducing the concepts of generalized L-contractions in Branciari distance spaces. Aydi et al. [27] extended the result of Cho [25] to partial metric spaces.
In the paper, we introduce notions of set-valued L -contractions and set-valued L * -contractions in Branciari distance spaces and prove the existence of fixed points for both type of contractions.
Khojasteh et al. [28] introduced the notion of Z-contractions by using the concept of simulation functions and unified the some existing metric fixed point results. The authors of [29][30][31][32] gave generalizations of simulation functions and obtained generalizations of results of [28]. Moreover, Demma et al. [33] and Yamaod and Sintunavarat [34] extended the results of [28] to b-metric spaces by using the notion of b-simulation functions and s-simulation functions, respectively.
We show that (ζ5) is not satisfied.

Abstract and Applied Analysis
To show this, let ft n g, fs n g ⊂ ð0,∞Þ be two sequences such that We may assume that s n < t n , ∀n = 1, 2, 3, ⋯. Then Hence, (ζ5) is not satisfied. Thus, ζ ∉ Z K : We now show that (ζ6) is satisfied. Let fa n g, fb n g ⊂ ð0,∞Þ be two sequences such that a n < b n ∀n = 1, 2, 3, ⋯and lim n⟶∞ a n = lim Hence, (ζ6) holds. Thus, ζ ∈ Z R : Therefore, Z R ≠ Z K : The following inclusion relations are satisfied.
Proof. Let ζ ∈ Z s . Then, (ζ2) and (ζ6) hold. Assume that ζ ∈ Z s is decreasing in the first coordinate. We show that (ζ5) holds. Let ft n g, fs n g ⊂ ð0,∞Þ be two sequences such that where b > 1.
We recall the following definitions which are in [1]. Let ðX, dÞ be a Branciari distance space, fx n g ⊂ X be a sequence, and x ∈ X.
Then, we say that (1) fx n g is convergent to x (denoted by lim n⟶∞ x n = x) if and only if lim n⟶∞ dðx, x n Þ = 0 (2) fx n g is Cauchy if and only if lim n,m⟶∞ dðx n , x m Þ = 0 Lemma 4 (see [43]). Let ðX, dÞ be a Branciari distance space, fx n g ⊂ X be a Cauchy sequence, and x, y ∈ X. If there exists a positive integer N such that Proof. Let A ∈ CðXÞ, and let fx n g ⊂ A be a sequence such that It follows from compactness of A that there exists a convergent subsequence fx nðkÞ g of fx n g. Let Since from Lemma 4, x = a ∈ A: Hence, A ∈ CLðXÞ. Lemma 6. Let ðX, dÞ be a Branciari distance space, and let A, B ∈ CLðXÞ. If a ∈ A and dða, BÞ < c, then there exists b ∈ B such that dða, bÞ < c.
Let ðX, dÞ be a Branciari distance space. A set-valued mapping T : X ⟶ 2 X , where 2 X is the family of all nonempty subsets of X, is called set-valued L -contraction with respect to ξ ∈ L if and only if for all x, y ∈ X with dðx, yÞ > 0, and for all u ∈ Tx, there exists v ∈ Ty with dðu, vÞ > 0 such that where θ ∈ Θ. Now, we prove our main result.

Theorem 7.
Let ðX, dÞ be a complete Branciari distance space, and let T : X ⟶ CLðXÞ be a set-valued L -contraction with respect to ξ ∈ L.
Then, T has a fixed point.
Proof. Let x 0 ∈ X be a point, and let x 1 ∈ Tx 0 be such that d which implies and so Again, from (33), there exists x 3 ∈ Tx 2 with dðx 2 , x 3 Þ > 0 such that which implies and so Inductively, we can find a sequence fx n g ⊂ X such that, ∀n = 1, 2, 3, ⋯, x n−1 ≠ x n , x n ∈ Tx n−1 and d x n , x n+1 ð Þ< d x n−1 , x n ð Þ: ð40Þ Since fdðx n−1 , x n Þg is a decreasing sequence, there exists r ≥ 0 such that We now show that r = 0. Assume that r ≠ 0. Then, it follows from ðθ2Þ that Let t n = θðdðx n , x n+1 ÞÞ and s n = θðdðx n−1 , x n ÞÞ∀n = 1, 2, 3, ⋯: Then, t n ≤ s n ∀n = 1, 2, 3, ⋯ and It follows from (ξ5) that which is a contradiction. Thus, we have and so We now show that fx n g is a Cauchy sequence. On the contrary, assume that fx n g is not a Cauchy sequence.
Then, there exists an ε > 0 for which we can find subsequences fx mðkÞ g and fx nðkÞ g of fx n g such that mðkÞ is the smallest index for which mðkÞ > nðkÞ > k∀k = 1, 2, 3, ⋯ It follows from (47) and condition (d3) that Letting k ⟶ ∞ in above inequality, we have which implies So Taking limit supremum in above inequality and using (49), we have We deduce that Taking limit infimum in above inequality and using (53), we have It follows from (53) and (55) that Let Then, t k ≤ s k ∀k = 1, 2, 3, ⋯: It follows from (49), (56), (θ2), and (θ4) that It follows from (ξ5) that which is a contradiction. Thus, fx n g is a Cauchy sequence.
Since X is complete, there exists a point x * ∈ X such that It follows from (33) that there exists y n ∈ Tx * with dðx n+1 , y n Þ > 0 such that which implies and hence Thus, we have Since Because Tx * ∈ CLðXÞ and fy n g ⊂ Tx * , x * ∈ Tx * .
We give an example to illustrate Theorem 7.

Abstract and Applied Analysis
We consider the following cases. Case 1. x = 1 and y = 2.

Corollary 8.
Let ðX, dÞ be a complete Branciari distance space, and let T : X ⟶ CLðXÞ be a set valued map such that for all x, y ∈ X with dðx, yÞ > 0 and inf z∈Tx dðz, TyÞ > 0 where ξ ∈ L * , θ ∈ Θ and r > 1: Then, T has a fixed point.

Conclusion
One can unify and merge some existing fixed point theorems by using L-simulation functions and L * -simulation functions in Branciari distance spaces. One can obtain some concequence of the main theorem by applying L-simulation functions and L * -simulation functions given in Example 1 and Example 2. Further, one can derive all the results of the paper in the setting of metric spaces.

Suggestion
We suggest that the b-simulation function can be extended in a similar way to the one in which the simulation function is extended to the L-simulation function. The main theorem can be extended and generalized to b-metric space, Branciari b-distance space, and extended Branciari b-distance space using certain extended simulation functions, and the existing fixed point theorem can be interpreted.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.