A PROPOSAL FOR REVISITING SOME FIXED POINT RESULTS IN DISLOCATED QUASI-METRIC SPACES

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we prove some new fixed point theorems in a dislocated quasi-metric spaces for a self mapping, which unify and generalize some existing relevant fixed point theorems. Moreover, many examples are provided to illustrate our improvements.


INTRODUCTION
In 1886, Pointcaré presented one of the most dynamic research subjects in nonlinear analysis, which is the notion of the fixed point. In 1922, Banach introduced a powerful tool in nonlinear analysis, which is the Banach contraction principle [1]. Since then, this contraction principle has been generalized in several directions and in different spaces see e.g., [2][3][4][5] end the references therein. In 2000, Hitzler et al introduced the concepts of dislocated metric spaces and established a fixed point theorem, which generalized the Banach contraction principle in such spaces [6]. Afterward, various generalizations of those spaces are introduced and many fixed point results were established see e.g., [7][8][9][10]12] and references therein.
Here, we recall some relevant definitions which will be needed in our subsequent discussion. Definition 1.1. Let X be a non empty set and d : X × X → R + be a function such that (1) d(x, y) = d(y, x) = 0 implies x = y, Then, d is called dislocated quasi-metric (or simply dq-metric) on X.
Here, x is called dq-limit of sequence {x n } and we write x n → x as n → ∞. Definition 1.4. Let (X, d 1 ) and (Y, d 2 ) be two dq-metric spaces, the function f : X → Y is said to be continuous if for each sequence {x n } ⊂ X which dq-converges to x in X, the sequence Recently, Anu [13] proved the following interesting generalization of Banach contraction principle for a continuous self mapping in dislocated quasi-metric space.
for all x, y ∈ X with d(x, y) = 0, and where α, β , γ, δ , µ ∈ R + verifying Then, the self-mapping T has a unique fixed point.
The purpose of this paper is to extend some results concerning generalized contractions of

MAIN RESULTS
Here, we provide two new contraction conditions of fixed point theorems in dq-metric spaces.
for all x, y ∈ X with d(x, y) = 0 and where λ ∈ [0, 1 2 ). Then, T has a unique fixed point.
Proof. Let T be a self mapping of X such that the condition (2.1) holds. We consider Next, we will distinguish the following two cases : Ty) d(x,y)+d(x,T x) and in this case, we consider η ∈ X. First, if T η = η, the mapping T has a fixed point. Next, we assume that T η = η. Thus, by taking x = η and y = T η in inequality (2.1), we obtain Therefore, for all n ∈ N * , we have where the quantity M(x n−1 , x n ) is given by (2.5) In addition, we use the triangular inequality to get that, for all n ∈ N * , we have Combining inequalities (2.5) and (2.6), we deduce Next, it follows from (2.4) and (2.7) that for n ∈ N * , we have which implies that for n ∈ N * , we find Then, since 2λ ∈ [0, 1) and Then, {x n } is a Cauchy sequence in a complete space X and there exists u ∈ X such that Let us now consider n ∈ N * , we have In addition, we have Keeping in mind the following inequality the inequality (2.15) leads to Passing to limit in (2.17) as n → ∞ and using (2.14), we get Thus, since λ , 2λ 2 ∈ [0, 1 2 ), it follows from (2.18) and (2.14) that Hence, Tu = u, and then T has at least one fixed point in X, which finishes the existence part.
For the uniqueness, let u, v ∈ X two fixed points of T such that u = v. From (2.1), we have (2.20) Since λ ∈ [0, 1 2 ), the above inequality implies that d(u, u) = 0, and similarly, we have On the other hand, we use (2.1) to conclude that (2.22) Lastly, since 2λ ∈ [0, 1), the inequalities (2.21) and (2.22 which is a contradiction. Then we have one and only one fixed point. Example 2.2. Consider the set X = 0, 1 9 , 100 endowed with the dq-metric d given by d(x, y) = x + 2 y, ∀ x, y ∈ X.
We construct a mapping T : X → X by T 0 = 0, T 100 = 1 9 and T 1 9 = 0. For λ = 1 3 , it is clear that all the assumptions of Theorem 2.1 hold, and then, 0 is the unique fixed point of T .
We note here that from Theorem 2.1, we can deduce immediately Theorem 1.1. Now, we give a new result similar to Theorem 1.1, in which we omit the continuity assumption of T . Theorem 2.3. Let (X, d) be a complete dq-metric space and T : X → X a self-mapping. If for all x, y ∈ X with d(x, y) = 0 and where α, β , γ, δ , µ ∈ R + such that Then, the self-mapping T has a unique fixed point.
, the inequality (2.25) can be written as follows.
Hence, T satisfies the conditions of Theorem 2.1 and then T has a unique fixed point in X.
We now state another result, which generalized Theorem 2.1.
Theorem 2.4. Let (X, d) be a complete dq-metric space and T : X → X a self mapping. If for all x, y ∈ X with d(x, y) = 0, λ ∈ [0, 1 a ) and a ≥ 2. Then, T has a unique fixed point in X.
Proof. It can be obtained in a similar way to that used in the proof of Theorem 2.1.
Finally, we give the following example illustrating the main result Theorem 2.4.
For λ = 1 3 , a = 3, all assumptions of Theorem 2.4 hold. Then, 0 is the unique fixed point of T .

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.