( α − ψ )-Contractive Mappings on Generalized Quasimetric Spaces

In the last decade, quasimetric spaces have been one of the interesting topics for the researchers in the field of fixed point theory due to two reasons. The first reason is that the assumptions of quasimetric are weaker than the more generalmetric. Consequently, the obtained fixed point results in this space are more general and hence the corresponding results in metric space are covered. The second reason is the fact that fixed point problems in G-metric space (introduced by Mustafa and Sims [1]) can be reduced to related fixed point problems in the context of quasimetric space (see, e.g., [2, 3]). Very recently, Lin et al. [4] introduced the notion of generalized quasimetric spaces and investigated the existence of a certain operator on such spaces. In this paper [4], the authors assumed that the generalized quasimetric space is Hausdorff to get a fixed point. In this paper, we examine the existence of (α-ψ)-contractive mappings in the context of generalized quasimetric space without the Hausdorffness assumption. Consequently, our results extend, improve, and generalize several results in the literature. In what follows we recall the basic definitions and results on the topics for the sake of completeness. Throughout the paper, the symbols R, N, and N 0 denote the real numbers, the natural numbers, and the positive integers, respectively. Let X be a nonempty set and let d : X × X → [0,∞). Then d is called a distance function if, for every x, y, z ∈ X, it satisfies


Introduction and Preliminaries
In the last decade, quasimetric spaces have been one of the interesting topics for the researchers in the field of fixed point theory due to two reasons.The first reason is that the assumptions of quasimetric are weaker than the more general metric.Consequently, the obtained fixed point results in this space are more general and hence the corresponding results in metric space are covered.The second reason is the fact that fixed point problems in -metric space (introduced by Mustafa and Sims [1]) can be reduced to related fixed point problems in the context of quasimetric space (see, e.g., [2,3]).Very recently, Lin et al. [4] introduced the notion of generalized quasimetric spaces and investigated the existence of a certain operator on such spaces.In this paper [4], the authors assumed that the generalized quasimetric space is Hausdorff to get a fixed point.
In this paper, we examine the existence of (-)-contractive mappings in the context of generalized quasimetric space without the Hausdorffness assumption.Consequently, our results extend, improve, and generalize several results in the literature.
In what follows we recall the basic definitions and results on the topics for the sake of completeness.Throughout the paper, the symbols R, N, and N 0 denote the real numbers, the natural numbers, and the positive integers, respectively.
One of the very natural generalizations of the notion of a metric was introduced by Branciari [5] in 2000 by replacing the triangle inequality assumption of a metric with a weaker condition, quadrilateral inequality.
Then (, ) is called a generalized metric space (or shortly g.m.s).
We present an example to show that not every generalized metric on a set  is a metric on .
Regarding the weakness of the topology of generalized metric space, mentioned above, the authors add some additional conditions to get the analog of existing fixed point results in the literature; see, e.g., [8][9][10][11][12][13][14][15].Very recently, Suzuki [16] underlined the importance of generalized metric space by emphasizing that generalized metric space and metric space have no compatible topology.
The following is the definition of the notion of generalized quasimetric space defined by Lin et al. [4] Definition 4. Let  be a nonempty set and let  :  ×  → [0, ∞) be a mapping such that, for all ,  ∈  and for all distinct points , V ∈  each of them different from  and , one has Then (, ) is called a generalized quasimetric space (or shortly g.q.m.s).
It is evident that any generalized metric space is a generalized quasimetric space, but the converse is not true in general.We give an example to show that not every generalized quasimetric on a set  is a generalized metric on .
Remark 8.A sequence {  } in a g.q.m.s is Cauchy if and only if it is left-Cauchy and right-Cauchy.
Definition 9 (see [4]).Let (, ) be a g.q.m.s.We say that (1) (, ) is left-complete if and only if each left-Cauchy sequence in  is convergent; (2) (, ) is right-complete if and only if each right-Cauchy sequence in  is convergent; (3) (, ) is complete if and only if each Cauchy sequence in  is convergent.
Notice that, in the literature in several reports for fixed point results in generalized metric space, an additional but superfluous condition, "Hausdorffness, " was assumed.Recently, Jleli and Samet [17], Kirk and Shahzad [18], Karapınar [19], Kadeburg, and Radenović [7], and Aydi et al. [20] reported new some fixed point results by removing the assumption of Hausdorffness in the context of generalized metric spaces.The following crucial lemma is inspired from [7,17].
Lemma 10.Let (, ) be a generalized quasimetric space and let {  } be a Cauchy sequence in  such that   ̸ =   whenever  ̸ = .Then the sequence {  } can converge to at most one point.
Proof.Given  > 0, since {  } is a Cauchy sequence, there exists  0 ∈ N such that  (  ,   ) < , ∀,  >  0 . ( We use the method of Reductio ad absurdum.Suppose, on the contrary, that there exist two distinct points  and  in  such that the sequence {  } converges to  and ; that is, lim By assumption for any  ∈ N,   ̸ =   , and since  ̸ = , there exists  1 ∈ N such that   ̸ =  and   ̸ =  for any  >  1 ≥  0 .Due to quadrilateral inequality, we have Letting ,  → ∞, we can obtain that (, ) = 0 by regarding ( 5) and (6).Hence, we get  =  which is a contradiction.

Main Results
In this section, we state and prove the main result of this paper.We start by introducing the following family of functions.
These functions are known in the literature as (c)-comparison functions.It is easily proved that if  is a (c)-comparison function, then () <  for any  > 0. For more details about such function, we refer the reader to [21,22].In this study, we discuss the notion of -admissible mappings; see, e.g., [23][24][25][26][27].The following definition was introduced in [23].
Then  has a periodic point.
Proof.Due to statement (ii) of theorem, there exists  0 ∈  which is an arbitrary point such that ( 0 ,  0 ) ≥ 1 and ( 0 ,  0 ) ≥ 1.We will construct a sequence {  } in  by  +1 =   =  +1  0 for all  ≥ 0. If we have   0 =   0 +1 for some  0 , then  =   0 is a fixed point of .Hence, for the rest of the proof, we presume that Since  is -admissible, we have Utilizing the expression above, we obtain that By repeating the same steps with starting with the assumption ( 1 ,  0 ) = ( 0 ,  0 ) ≥ 1, we conclude that In a similar way, we derive that Recursively, we get that Analogously, we can easily derive that Step 1.We will show that lim  → ∞ (  ,  +1 ) = 0 and lim  → ∞ (  ,  +2 ) = 0. Regarding ( 8) and ( 12), we deduce that for all  ≥ 1.
In the same way {  } is a left-Cauchy sequence in (, ).So it is a Cauchy sequence.Since  is a complete g.q.m.s, there exists  ∈  such that lim Also, we can easily see that   ̸ =   for whenever  ̸ = .Indeed, if   =   , for some ,  ∈ N with  < , then which is a contradiction.Analogously, we derive the same conclusion for the case  > .Therefore, we conclude that the sequence {  } cannot have two limits due to Lemma 10.
Then  has a periodic point.
In what follows we give an example to illustrate Theorem 13.
Example 15.In Example 5 define the mapping  :  →  as First, we can see easily that the classic Branciari contraction [5] cannot be applied in this case since Now we define the mapping  from  ×  → [0, ∞) by for all ,  ∈ .For () = /2, where  ≥ 0, we have Obviously  is -admissible and also for  0 =  we have Finally  is continuous.Therefore  satisfies in Theorem 13 and we can see that  has two fixed points  and 3.
Theorem 16.Adding Property (E) to the hypotheses of Theorem 13 (res.Theorem 14), one obtains existence of a fixed point of .
Proof.Suppose that  is a periodic point of ; that is,    = .
Consequently, we get the following contradiction: (, ) < (, ).Hence, the assumption that  is not a fixed point of  is not true and thus  =  −1  is a fixed point of .
To assure the uniqueness of the fixed point, we will consider the following properties.

Theorem 17. Adding property (U) to the hypotheses of Theorem 16, one obtains uniqueness of the fixed point of 𝑓.
Proof.Suppose that  and V are two distinct fixed points of .By property (), (, V) = (, V) ≥ 1.