Some variants of Wardowski fixed point theorem

The purpose of this paper is to consider some F-contraction mappings in a dualistic partial metric space and to provide sufficient related conditions for the existence of a fixed point. The obtained results are extensions of several ones existing in the literature. Moreover, we present examples and an application to support our results.


Introduction
The study of metric fixed point theory was initiated by Banach (1922). It has been enriched by introducing several structures, generalizing underlying metric space, and the contractive condition. The Banach contraction principle has been proved in several different abstract metric spaces (see [1][2][3][4][5][6][7][8][9][10]). Before the publication of the paper of Mustafa et al. [5], the metric functions were defined on X 2 , Mustafa et al. [5] introduced a metric function defined on X 3 , and it is known as a G-metric. Mustafa et al. [5] proved the Banach contraction principle (BCP) in G-metric spaces and gave several examples to establish its superiority over the BCP in metric spaces. Sedghi et al. [11] introduced another abstract metric defined on X 3 , called an S-metric, and proved the analogue of the BCP.
Working on network topologies in computer science, Matthews [12] observed that the self-distance of a point need not be zero. This led him to introduce a partial metric and hence obtain an analogue of the BCP. Later, many researchers worked on partial metric spaces and produced several papers on partial metric topologies and existence of fixed points of self-mappings (see [13][14][15][16][17]).
Neill [18] observed during his study on dual topologies that a partial metric may have negative value, that is, the range set of a partial metric can be considered as a real line. It is known as a dualistic partial metric. Oltra et al. [19] presented an analogue of the Banach fixed point result in complete dualistic partial metric spaces using some convergence properties of sequences. Subsequently, Nazam et al. [20][21][22] and Pitea [23] published some papers addressing the existence of fixed points of self-mappings defined on dualistic partial metric spaces.
In this paper, we investigate fixed points of F-contractions under the rules practiced in dualistic partial metric spaces. We also discuss how this differs from the classical Fcontraction introduced by Wardowski [24].

Preliminaries
Let T : → be a self-mapping on a nonempty set . The sequence { n } in with an initial point 0 given as n = T( n-1 ) for all n ∈ N is said to be a Picard iterative sequence. The partial metric space was defined by Matthews [12].
The notion of a partial metric was extended by Neill [18] in the following. Definition 2 ([18]) A dualistic partial metric on a nonempty set is a function V : × → R fulfilling the following properties for all , , ν ∈ : Then ( , V) is said to be a dualistic partial metric space.
Remark 1 Each partial metric is a dualistic partial metric. The converse does not hold in general. For this, consider Then V m is a dualistic partial metric (see [21]). Here V m is not a partial metric since for any < 0, < 0 it implies V m ( , ) < 0.
Note that in a dualistic partial metric, V( , ) = 0 does not necessarily imply = . Indeed, V m (-1, 0) = 0. This case creates a problem in obtaining a fixed point of a selfmapping in a dualistic partial metric space. To resolve this problem, in this paper we em-ploy our proposed F-contraction (defined in the next section) along with axioms (V 2 ) and (V 1 ) to get a fixed point.
We give now some examples of dualistic partial metrics. The first is related to the classic one.
Example 1 If ( , d) is a metric space and c ∈ R is an arbitrary constant, then is a dualistic partial metric on .
Next, we use a partial metric to define a dualistic partial metric space. verifies axioms (V 1 ) -(V 4 ). So it is dualistic partial metric on . We emphasize that V( , ) may have negative values.
Then ( , V) is a dualistic partial metric space.
Neill [18] ensured that each dualistic partial metric V on generates a T 0 topology τ [V] on having as a base the family of V-balls {B V ( , ω) : ∈ , ω > 0}, where If ( , V) is a dualistic partial metric space, then the function is a quasi-metric on such that τ (V) = τ (d s V ) and d s V ( , ) = max{d * ( , ), d * ( , )} is a metric on . It is said to be an induced metric on ( , V).
The following describes the convergence criteria in a dualistic partial metric space established by Oltra et al. [19].
V( n , m ) exists and is finite.
Remark 2 For a sequence { n } ⊂ , the convergence with respect to an induced metric space ( , d s V ) may not lead to the convergence with respect to the dualistic partial metric space ( , V). Clearly, This implies that n → -1 with respect to ( , d s V ). On the other hand, consider That is, It is a contradiction. Thus, n -1 with respect to ( , V).
With regard to the connections between convergence and Cauchyness in dualistic partial metric spaces, we recall the next lemma. (1) F is strictly increasing;

Main results
(2) For any sequence of positive terms {a n }, lim n→∞ a n = 0 ⇐⇒ lim n→∞ F(a n ) = -∞; Example 4 The following functions F : (0, ∞) → R given for all ∈ (0, ∞) as: In [24], Wardowski introduced a remarkable contraction known as F-contraction and presented a method to obtain fixed points of such contractions in complete metric spaces. The following theorem is the variant of Wardowski's fixed point theorem [24] in the complete dualistic partial metric spaces. Theorem 1 is a useful generalization of the mentioned theorem.
Theorem 1 Let (M, V) be a complete dualistic partial metric space, F ∈ F , and T : M → M be a continuous mapping for which there exists τ > 0 such that, for all j, k from M, the following implication holds: Then T possesses a unique fixed point.
Proof Let j 0 be an element from M and j n+1 = Tj n for all n ∈ N. If there is n o ∈ N for which j n o +1 = j n o , then j n o is a fixed point of T. Therefore, we may assume that j n = j n+1 for any n ∈ N. Given that V(j n , j n+1 ) = 0 for any n ∈ N. By the use of the contractive condition, we obtain, for n ≥ 1, which, by taking advantage of the monotonicity of F, {|V(j n , j n+1 )|} is decreasing. Let r ≥ 0 be its limit. By contractive condition (3.1), we have This in turns implies that Letting n → ∞ and using (F 2 ), we get This yields that lim n→∞ V(j n+1 , j n ) = 0 = r. (3.4) Focusing now on the self-distance, we have, for n ∈ N, and so {|V(j n , j n )|} is a sequence decreasing to some l ≥ 0. Arguing as above, we get that l = 0. By the third property of functions in F , there is θ ∈ (0, 1) such that lim n→∞ |V n | θ × F(|V n |) = 0. Consider V n = V(j n , j n+1 ) for n ∈ N. By applying repeatedly inequality (3.2), it follows that Letting n → ∞ and taking advantage of the properties of the function F, we get that Denote by V n = V(j n , j n ) for n ∈ N. Similarly, there is N 2 ∈ N such that, for any n ≥ N 2 , Having in mind inequalities (3.5) and (3.6), we have, for m > n ≥ max{N 1 , N 2 }, Taking the limit to ∞, it follows that {d * (j n , j m )} → 0. Applying an analogous procedure, we get that {d * (j m , j n )} → 0, hence d s V (j n , j m ) → 0, so {j n } is a Cauchy sequence in the complete metric space (M, d s V ). Let j be its limit. Obviously, Since T is a continuous mapping, {Tj n } converges to Tj. This implies that V(Tj n , Tj n ) → V(Tj, Tj). Since V(Tj n , Tj n ) → V(j, j) = 0, it follows that V(Tj, Tj) = 0.

In view that
by considering n → ∞ and using the continuity of T, we obtain V(j, Tj) = 0.
Thus V(Tj, Tj) = V(j, j) = V(j, Tj), so Tj = j. Let us focus now on the uniqueness of the fixed point of T. Assume that j and k are two distinct fixed points of T. If V(j, k) = 0, then the following relations hold true: which is a contradiction. Therefore, V(j, k) = 0. Similarly, it can be proved that V(j, j) = 0 and V(k, k) = 0. It follows that j = k, and so the fixed point is unique. We continue with an example with respect to this theorem. Example 5 does not only explain Theorem 1, but also shows that the Wardowski fixed point theorem [24] is not applicable for = .
Example 5 Let A = (-∞, 0]. Define the mapping Then (A, V) is a complete dualistic partial metric space. Let τ > 0 and define the mapping: if ∈ Q(a set of rational numbers), if ∈ Q (a set of irrational numbers). The following theorem is for Kannan type F-contraction in the dualistic partial metric spaces. It is a useful generalization of [40,Theorem 2]. Theorem 2 Let ( , V) be a complete dualistic partial metric space, F ∈ F , and T : → be a continuous mapping for which there exists τ > 0 such that, for all j and k in , the following implication holds: Then T admits a unique fixed point.
Proof For j 0 from , set j n+1 = Tj n for n ∈ N. If there is s ∈ N for which j s+1 = j s , then j s is a fixed point of T. We presume that j n = j n+1 for any n ∈ N. Given V(j n , j n+1 ) = 0 for any n ∈ N. By the use of the contractive condition, we obtain, for n ≥ 1, from which, by taking advantage of the monotony of F, we have that {|V(j n , j n+1 )|} is a decreasing sequence. Let r ≥ 0 be its limit. By applying inequality (3.8) and keeping in mind also that {|V(j n , j n+1 )|} is a decreasing sequence, we get, for n ≥ 1, (3.9) It follows that lim n→∞ F(|V(j n , j n+1 )|) = -∞. Having in mind F ∈ F , we obtain that lim n→∞ |V(j n , j n+1 )| = 0.
Let us now analyze the self-distance. Presume that there exists n 0 ∈ N such that V(j n , j n ) = 0 for any n ≥ n 0 , where n 0 ∈ N. The contractive condition imposes From (3.9), it follows that Letting n → ∞ and using the properties of the function F, we get that lim n→∞ V(j n , j n ) = 0.
As in the proof of Theorem 1, we obtain that {j n } is a Cauchy sequence and so it is convergent to some j. From the preceding lines, V(j, j) = 0. Since T is continuous, {Tj n } converges to Tj, and therefore V(Tj n , Tj n ) → V(Tj, Tj). On the other hand, V(Tj n , Tj n ) → V(j, j) = 0. We have obtained that V(Tj, Tj) = 0.
In view that and also that by considering n → ∞ and using the continuity of T, we obtain V(j, Tj) = 0. We conclude that Tj = j, and so T has a fixed point. Assume now that the set {n ∈ N, V(j n , j n ) = 0} is not bounded. In this case, there exists a subsequence of {j n }, {j n k } for which V(j n k , j n k ) = 0, k ∈ N. We repeat the same lines as in the proof of Theorem 1 for this subsequence {j n k }, and we obtain that {j n k } is convergent to j ∈ with V(j, j) = 0. The equality j = Tj follows as in the case when {n ∈ N, V(j n , j n ) = 0} is bounded.
Summing up, we have proved that T has a fixed point. Let us focus now on the uniqueness of the fixed point of T. Assume that j and k are two distinct fixed points of T. Now suppose that V(j, j) = 0. Then the following holds: which is impossible. Hence, V(j, j) = 0. Similarly, it can be proved that V(k, k) = 0.
If V(j, k) = 0, then the following holds: and it is not well defined. Therefore, V(j, k) = 0. It follows that j = k, and so the fixed point is unique.
To support our results, we give the next example.
Example 6 Let = (-∞, 0] and define the function V : × → R by According to [21], the pair ( , V) is a complete dualistic partial metric space. Consider the mapping T : → given as We claim that T fulfills the conditions of Theorem 2 for τ = ln Case IV. j = k ∈ (-4, 0]. We have so, this case is also true.
In this case, one writes The inequality in Theorem 2 is fulfilled. By applying this result, we get that T has a unique fixed point, which is j = -1 2 . On the other hand, it can be easily verified that the inequality in Theorem 1 is not satisfied in the case j = k = 0.

Application to the boundary value problem
In this section, we apply Theorem 1 to obtain the existence of the solution to the following boundary value problem. For more details on the applications of BCP, readers are referred to [41,42]. (ĺ) -ǔ(ĺ) + L, L ∈ R.
The pair (C, V) is a complete dualistic partial metric space. The associated operator T : C → C to the boundary value problem (4.1) has the following form: for allĺ ∈ [0, 1]. Note that problem (4.1) has a solution iff the operator T has a fixed point.   Taking ln on both sides, we have ln V T( ), T(ǔ) ≤ ln e -τ V( ,ǔ) .
Define F( ) = ln( ) for all ∈ C, and simplifying the last inequality, we have Hence, the application of Theorem 1 ensures that T has at least one fixed point q(ĺ) ∈ C, that is, T(q(ĺ) = q(ĺ) which is a solution of (4.2).

Conclusion
This paper has extended the classical F-contraction from the case d(j, k) > 0 to V(j, k) = 0. There are self-mappings defined on dualistic partial metric spaces which have no fixed points, but satisfy contractive conditions given in [24,33,35,37]. However, such mappings do not satisfy (3.1) and (3.7). Thus, contractive conditions (3.1) and (3.7) are refinements of the ordinary (Banach type and Kannan type) F-contractions. This paper has also presented a new method to show the existence of fixed points of F-contractions in dualistic partial metric spaces.