#### Abstract

In this paper, the notion of set-valued -contractions is introduced, and a new fixed point theorem for such contractions is established. An example to illustrate main theorem is given.

#### 1. 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 , where is a nonempty set, is said to be Branciari distance on if and only if it satisfies the following conditions:

For all and for all distinct points , each of them different from and is as follows:

(d1) if and only if

(d2)

(d3)

The pair is called a Branciari distance space, whenever is a Branciari distance on .

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 does not have a topology which is compatible with (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 -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: (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]).

Sarma et al. [15] and Samet [14] (see also [3, 4, 8, 18]) show that Branciari distances have the following topological disadvantages.

(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.

Let be a function.

Consider the following conditions:

(1) is nondecreasing, i.e., , whenever

(2) For any sequence of points in

(3) There exist and such that

(4) is continuous

(5) is strictly increasing

(6) for all with

(7) For any finite sequence

Jleli and Samet [19] introduced the notion of -contractions and generalized the Banach contraction principle in the setting of Branciari distance spaces, where is a function such that (1), (2), and (3) are satisfied.

Since then, Jleli et al. [7] obtained a generalization of result of [19] with conditions (1), (2), (3), and (4). Arshad et al. [20] extended the result of [19] by using the notion of -orbital admissible mapping with conditions (1), (2), and (3). Also, Ahmad et al. [21] extended the result of Jleli and Samet [19] to metric spaces by using conditions (1), (2), and (4). Durmaz and Altun [22] obtained a generalization of the result of Klim and Wardowski [23] by defining the concept of set-valued -contractions with control function satisfying (1), (2), (3), and (6). Abdeljawad et al. [17] introduced the concept of extended Branciari -distance, and they extended the result of Jleli and Samet [19] to extended Branciari -distance spaces with control function satisfying (2) and (3). Cho [24] introduced the notion of generalized set-valued weak -contractions in metric spaces and obtained fixed point results for such contractions with control function satisfying (1), (2), (4), and (7).

Recently, Cho [25] introduced the concept of -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 -contractions.

Very recently, Saleh et al. [26] extended the result of Cho [25] by introducing the concepts of generalized -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 -contractions and set-valued -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 -contractions by using the concept of simulation functions and unified the some existing metric fixed point results. The authors of [29–32] 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 -metric spaces by using the notion of -simulation functions and -simulation functions, respectively.

Let be a function. Consider the following conditions:

(1)

(2)

() , where is continuous and strictly increasing function with

(3) For any sequence

(4) For any sequence with

(5) If for any sequence where , then we have

(6) If for any sequence where , then we have

Let be a function.

Then, we say that
(1) is called a simulation function in the sense of Khojasteh et al. [28] if and only if (), (), and () hold(2) is called a simulation function in the sense of Argoubi et al. [30] if and only if () and () hold(3) is called a simulation function in the sense of Roldán-López-de-Hierro et al. [32] if and only if (), (), and () hold(4) is called a simulation function in the sense of Isik et al. [31] if and only if () and () hold(5) is called a -simulation function [29] if and only if () and () hold(6) is called a -simulation function [33] if and only if () and () hold(7) is calld a *-*simulation function [34] if and only if () and () hold

Denote (resp., , ) by the family of all simulation functions in the sense of Khojasteh et al. (resp., all simulation functions in the sense of Argoubi et al., all -simulation functions).

Note that every simulation function in the sense of Argoubi et al. is a -simulation function. In fact, let . If we take , then .

Proposition 1. *The following are satisfied.
*(1)* and ([29])*(2)* and ([30])*

Denote (resp., ) by the class of all simulations functions in the sense of Roldán-López-de-Hierro et al. (resp., in the sense of Isik et al.). Also, we denote by and the set of all -simulation functions and -simulation functions, respectively.

*Example 1 (see [28, 32, 35, 36]). *Let be a function defined as follows:
(1), where (2), where is a function such that and (3), where is a function such that (4), where is a function such that or is continuous such that if and only if (5) where are continuous functions such that if and only if , and is increasing(6), where is upper semicontinuous with and if and only if (7), where is a function such that for each exists and

Then, .

*Example 2. *Let be a function defined by
Then, () is satisfied.

Let be sequnces of points in such that
Then, we have
Thus, , but , because Hence,

*Example 3 (see [31]). *Let be a function defined as
where .

Then, , but . Hence, Also, we know that and .

Note that . The following examples show that .

*Example 4. *Let be a function defined as
where . Then, () and () are satisfied.

We show that () is not satisfied.

To show this, let be two sequences such that

We may assume that .

Then

Hence, () is not satisfied. Thus,

We now show that () is satisfied.

Let be two sequences such that

Then

Hence, () holds. Thus, Therefore,

Proposition 2. *The following inclusion relations are satisfied.
*(1)*(2)**(3)*

Proposition 3. *If is decreasing in the first coordinate, then *

*Proof. *Let . Then, (2) and (6) hold.

Assume that is decreasing in the first coordinate.

We show that (5) holds.

Let be two sequences such that
where .

From (19), we infer that
which implies
It follows from (19) that
Thus, we obtain
By applying (21) and (23) to (6), we have
Since is decreasing in the first coordinate,
Hence, it follows from (19) and (25) that (5) is satisfied. Thus, .

Note that simulation functions given in Example 1 are -simulation and decreasing functions in the first coordinate. Hence, (see also [33]).

For more details and examples of simulation functions, we refer to [28, 33, 34, 36–40], and for -simulation functions, we refer to [29, 41, 42].

Now, we recall the concept of -simulation function and give the definition of -simulation function.

Let be a function. Consider the following conditions:

(1) is a decreasing function on the first coordinate

(2)

(3)

(4)

(5) For any sequence with

Let be a function.

Then, we say that
(1) is *-*simulation function [25] if and only if it satisfies conditions (), (), and ()(2) is*-*simulation function if and only if it satisfies conditions (), (), (), (), and ()

Denote by the family of all -simulation functions, and by the class of all -simulation functions.

Note that , and .

*Example 5 (see [25]). *Let be functions defined as follows, respectively:
(1) where (2) where is nondecreasing and lower semicontinuous such that , where .

Then, .

*Example 6. *Let be a function defined as follows:
(1) where are continuous functions such that if and only if , and is an increasing function(2), where is upper semicontinuous with and if and only if (3), where is a function such that for each exists and , and Then, , and .

Note that if is satisfied condition
then .

We recall the following definitions which are in [1].

Let be a Branciari distance space, be a sequence, and .

Then, we say that (1) is convergent to (denoted by ) if and only if (2) is Cauchy if and only if (3) is complete if and only if every Cauchy sequence in is convergent to some point in

Let be a Branciari distance space.

We denote by the class of nonempty closed subsets of . Let be the Hausdorff distance on , i.e., for all , where is the distance from the point to the subset .

For , let

Then, we have for all

Lemma 4 (see [43]). *Let be a Branciari distance space, be a Cauchy sequence, and . If there exists a positive integer such that
*(1)*(2)**(3)**(4)**then .*

Lemma 5. *If is a Branciari distance space, then , where is the class of nonempty compact subsets of .*

*Proof. *Let , and let be a sequence such that
It follows from compactness of that there exists a convergent subsequence of .

Let
Since
from Lemma 4, Hence, .

Lemma 6. *Let be a Branciari distance space, and let .**If and , then there exists such that .*

*Proof. *Let .

It follows from definition of infimum that there exists such that . Hence, .

#### 2. Fixed Point Theorems

We denote by the class of all functions such that conditions (2), (4), and (5) hold.

Let be a Branciari distance space.

A set-valued mapping , where is the family of all nonempty subsets of , is called set-valued -contraction with respect to if and only if for all with , and for all , there exists with such that where .

Now, we prove our main result.

Theorem 7. *Let be a complete Branciari distance space, and let be a set-valued -contraction with respect to .*

Then, has a fixed point.

*Proof. *Let be a point, and let be such that .

From (33), there exists with such that
which implies
and so
Again, from (33), there exists with such that
which implies
and so
Inductively, we can find a sequence such that, Since is a decreasing sequence, there exists such that
We now show that .

Assume that .

Then, it follows from that
Let and

Then, and
It follows from (5) that
which is a contradiction.

Thus, we have
and so
We now show that is a Cauchy sequence.

On the contrary, assume that is not a Cauchy sequence.

Then, there exists an for which we can find subsequences and of such that is the smallest index for which It follows from (47) and condition (d3) that
Letting in above inequality, we have
From (33), there exists with such that
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,

It follows from (49), (56), (2), and (4) that
It follows from (5) that
which is a contradiction.

Thus, is a Cauchy sequence.

Since is complete, there exists a point such that
It follows from (33) that there exists with such that
which implies
and hence
Thus, we have
Since
Because and , .

We give an example to illustrate Theorem 7.

*Example 1. *Let and define as follows:

Then, is a complete Branciari distance space but not a metric space (see [9]).

Define a map by and a function by

We now show that is a set valued -contraction with respect to , where .

We consider the following cases.

*Case 1. * and .

For , there exists with such that
and for , there exists with such that

*Case 2. * and .

For , there exists with such that
and for , there exists with such that

*Case 3. * and .

For , there exists with such that
and for , there exists with such that

*Case 4. * and .

For , there exists with such that
and for , there exists with such that

*Case 5. * and .

For , there exists with such that
and for , there exists with such that

*Case 6. * and .