Remarks on F-weak contractions and discontinuity at the xed point

The aim of this paper is to generalize celebrated results due to Wardowski[8, 9] and also to provide yet new solutions to the once open problem[6, p.242] on the existence of a contractive mapping which possesses a xed point but is not continuous at the xed point via generalized F ∗∗-weak contractions. Finally, an example is given to illustrate our results.


Introduction and Preliminary Notes
The rst contractive denition is that of Banach [1], which states that if G satises, d(Gx, Gy) ≤ σd(x, y) for each x, y ∈ X, where σ ∈ [0, 1), then G has a unique xed point in X. This hypothesis has numerous applications but endures from one disadvantage, the denition requires G be continuous throughout X. Kannan [3] gave an example of a contractive concept in 1968, which did not require that G be continuous. There after several authors made several extensions of this result. Rhoades [5] compared 250 contractive denitions, and he found that while most of the contractive denitions do not require the mapping to be continuous over the whole domain, they all require continuous mapping over the entire domain, they all require continuous mapping at the xed point. Rhoades [6] [9] and Alfaqih [7]. Motivated by the studies above, we are investigating new contractive conditions to get another solution to the open question. We recall the following denitions, which are necessary in the next section, before stating our main results. Denition 1.1. [8] Let be the family of all functions F : (0, inf ty) → R satisfying the following conditions: Example 1.1. [8] The following functions belong to F: Denition 1.2. [8] Let (X, d) be a metric space. A self-mapping G on X is said to be an F -contraction if there exist σ > 0 and F ∈ such that (for all x, y ∈ X) Wardowski [8] proved the nding as follows: Theorem 1.1. Every F -contraction mapping G dened on a complete metric space (X, d) has a unique xed point. Moreover, for any x ∈ X, the sequence {G n x} converges to the xed point of G.
In 2014, Wardowski and Dung [9] utilized the same lesson of auxiliary functions to present the idea of F -weak contractions as follows: Denition 1.3. [9] Let (X, d) be a metric space. A self-mapping G on X is said to be an F -weak contraction if there exist σ > 0 and F ∈ such that (for all x, y ∈ X) Wardowski and Dung [9] demonstrated the following hypothesis: Alfaqih, et al. [7] interpret that, theorem(1.2) will survive without F 1 and F 3 assumptions besides eliminating one way inference of F 2 assumption and present the following Auxiliary class functions. Let be the set of all functions F : (0, ∞) → R satisfying the following condition: F * 2 : for every sequence β n ⊆ (0, ∞), lim n→∞ F (β n ) = −∞ ⇒ lim n→∞ β n = 0.
Next, Alfaqih, et al. [7] introduce the notion of F * -weak contraction mappings, which is as follows Alfaqih, et al. [7] proved the subsequent theorem: Now, we give the following lemma which will be used eectively in proving this study's key theorem.
The aim of this article is to present the idea of F * * -weak mapping and obtain new solutions to the open question on the existence of F * * -weak contractive conditions that are strong enough to generate a xed point but which do not require the mapping to be continuous at the point.

Main Results
First, we present the idea of F * * -weak mapping of the contraction, which operates as follows. Denition 2.1. Let (X, d) be a metric space. A map G : X → X is said to be an F * * -weak contraction on (X, d) if there exist F ∈ and σ > 0 such that, for all x, y ∈ X satisfying d(Gx, Gy) > 0, the following holds: σ + F (d(Gx, Gy)) ≤ F (max d(x, y), d(x, Gx), d(y, Gy), d(x, Gy), d(y, Gx)) Remark 2.1. Every F -contraction is an F * * -weak contraction but inverse implication does not hold. Now, we are able to construct this section's main theorem. Theorem 2.1. Let (X, d) be a complete metric space and G : X → X is a F * * -weak contraction. If F is continuous and σ > 0 then, (a) G has unique xed point w ∈ X (b) lim n→∞ G n (x) = w for all x ∈ X Additionally, G is continuous at w if and only if lim n→∞ (max d(x, y), d(x, Gx), d(y, Gy), d(x, Gy), d(y, Gx)) = 0 Proof. Let x 0 ∈ X be an arbitrary and xed. We dene a sequence {x n } ⊆ X by x n+1 = Gx n . If there exist n 0 ∈ N ∪ {0} such that x n 0 +1 = x n 0 then Gx n 0 = x n 0 . This proves that x n 0 is a xed point of G. Now we assume that x n+1 = x n for all n ∈ N ∪ {0}. It follows from denition(2.1) that for each n ∈ N, For all n ∈ N. Taking the limit as n → ∞ in eq.(2) we get Now, we prove {x n } is a Cauchy sequence. Contradictory, suppose that {x n } isn't a Cauchy sequence.
Assuming k → ∞ and using eq.(4) and eq.(5), we get As F is continuous so by using eq.(4), eq.(5) and eq.(6), we obtain F ( ) ≤ F ( ) − , which is a contradiction. Hence {x n } is a cauchy sequence. Since X is complete, there exist w ∈ X such that lim n→∞ x n = w. Now we prove w is an unique xed point of G. Let Q = {n ∈ N ∪ {0} : x n = Gw}. Here we consider two cases: (a)Q is innite: In the event that Q is innite at that point there exists a subsequence {x n(k) } ⊆ {x n } which converges to Gw. Since limit is unique we get Gw = w.
(b)Q is nite: If Q is nite then d(x n , Gw) > 0 for innitesimal n ∈ N ∪ {0}. Therefore there exist a subsequence {x n(k) } ⊆ {x n } in such a manner that d(x n(k) , Gw) > 0 for every k ∈ N ∪ {0}. Now by utilizing eq.(1), we If d(Gw, w) > 0, then by letting k → ∞ in eq. (7), we obtain F (d(w, Gw)) ≤ F (d(w, Gw)) − σ. Which is a contradiction, therefore d(Gw, w) = 0. Thus we get Gw = w. That means w is a xed point of G. To prove the uniqueness, let u be some other xed point of G. Then denition(2.1) gives F (d(w, u)) ≤ F (d(w, u)) − σ a contradiction. Hence G has unique point.
and G is continuous at its xed point nonetheless it's discontinuous at its domain.

Conict of Interests
The author declare that there is no conict of interests regarding the publication of this paper.