Revisiting and revamping some novel results in F-metric spaces

: Introduction/purpose: is article establishes several new contractive conditions in the context of so-called -metric spaces. e main purpose was to generalize, extend, improve, complement, unify and enrich the already published results in the existing literature. We used only the property (F1) of Wardowski as well as one well–known lemma for the proof that Picard sequence is an -Cauchy in the framework of -metric space. Methods: Fixed point metric theory methods were used. Results: New results are enunciated concerning the F-contraction of two mappings S and T in the context of −complete metric spaces. Conclusions: e obtained results represent sharp and signiﬁcant improvements of some recently published ones. At the end of the paper, an example is given, claiming that the results presented in this paper are proper generalizations of recent developments.


Introduction and preliminaries
It is exactly one hundred years since S. Banach (Banach, 1922) proved the famous principle of contraction in his doctoral dissertation. Since then, many researchers have been trying to generalize that significant result in many directions. In one direction, new classes of metric spaces were created and the renowned results were extended to these spaces. Among them, b-metric and -metric spaces stand out. e former ones were introduced by Bakhtin (Bakhtin, 1989) and Czerwik (Czerwik, 1993) and the latter were recently introduced by Jleli and Samet (Jleli & Samet, 2018). Not that these two cases of spaces are intangible. Namely, there is a b-metric space that is not F-metric, and vie versa, there is an -metric that is not b-metric. Note that convergence, Cauchyness and completeness of both types of spaces are defined for ordinary metric spaces. Also, it is worth mentioning that b-metric and -metric do not have to be continuous functions with two variables as is the case with ordinary metric. In both types of spaces, a convergent sequence is a Cauchy and it has a unique limit. is is what they have in common with ordinary metric spaces. e continuity of mapping in both classes of spaces is sequential, i.e., the same as in ordinary metric spaces. Let us now list the definitions of each of the mentioned types of spaces. For more new details on -metric spaces and new developments in the metric fixed point theory, one can see some noteworthy papers (Asif et al, 2019), , (Derouiche & Ramoul, 2020), (Jahangir et al, 2021), (Kirk & Shazad, 2014), , (Salem et al, 2020), (Som et al, 2020), , , (Younis et al, 2019a), (Younis et al, 2019b). Definition 1. (Bakhtin, 1989;Czerwik, 1993) Let X be a nonempty set and s ≥ 1 be a given real number. A function d b : X × X → [0, +∞) is said to be a b-metric with the coefficient s if for all x, y, z # X the following conditions are satisfied: Let be the set of functions f : (0, +∞) → (−∞, +∞) satisfying the following conditions: 1 ) f is non-decreasing, 2 ) For every sequence {t n } # (0, +∞), we have Definition 2. (Jleli & Samet, 2018) Let X be a (nonempty) set. A function : X × X → [0, +∞) is called a -metric on X if there exists (f, α) # × [0, +∞) such that for all x, y # X the following conditions hold: In this case, the pair (X, ) is called a −metric space. Wardowski (Wardowski, 2012) considered a nonlinear function F : (0, +∞) → (−∞, +∞) with the following characteristics: Wardowski (Wardowski, 2012)  (2) with min { (Sx, Ty), (x, y), (x, Sx), (y, Ty)} > 0, for all (x, y) # X × X. en S and T have at most one common fixed point in X.
By replacing S with T, the authors obtained the following result for single mapping.

en T have at most one fixed point in X.
Definition 3.

Let (X, be an −complete -metric space and S, T : X → X be self-mappings. Suppose that a + b + c < 1 for a, b, c # [0, +∞). en the mapping T is called a Reich-type F-contraction on B (x 0 , r) # X if there exist F # and τ > 0 such that for all x, y # B (x 0 , r)
is an −complete -metric space. Let T be a Reichtype F-contraction on B (x 0 , r) # X. Suppose that for x 0 # X and r > 0, the following conditions are satisfied: en S and T have at most one common fixed point in B (x 0 , r). Taking S = T in eorem 2, the authors in (Asif et al, 2019; Corollary 3.) obtained the following result for single mapping.
is an −complete -metric space and T : X → X is a self-mapping. Suppose that a + b + c < 1 for a, b, c # [0, +∞). Suppose that for x 0 # X and r > 0, the following conditions are satisfied: en T has at most one fixed point in B (x 0 , r). (X, ) is an −complete -metric space. Let S, T : X → X be a self-mappings and k # [0, 1). Suppose that for x 0 # X and r > 0, the following conditions are satisfied: en S and T have at most one common fixed point in B (x 0 , r). Further in the same paper (Asif et al, 2019;Definitions 6, 8, eorem 5, Corollary 5.), the authors gave the following:

Let (X, ) be an −complete -metric space and (f, α) # ×[0, +∞). If the mapping T : X → CB (X) is a set-valued Reich-type F-contraction such that F is right continuous, then T has a fixed point in X.
Corollary 5 In the sequel, we will use the following two results: Lemma 1. ; Lemma 1.) Let (X, d b ) (resp. (X, ) be a b-metric (resp. −metric) space and the sequence in it such that (8) for all n # ,where λ # [0, 1). en is a d b −Cauchy sequence in (X, d b ) (resp. −Cauchy sequence in (X, )).
Lemma 2. Let x 0 be a Picard sequence in -metric space inducing by mapping T : X → X and initial point x 0 # X. If (x n , x n+1 ) < (x n−1 , x n ) for all n # then x n x m whenever n m. Proof. Let x n = x m for some n, m ∈ with n < m. en xn+1 = Tx n = Tx m = x m+1 . Further, we get which is a contradiction.

en S and T have at most one common fixed point in X, if at least one of the mappings S or T is continuous.
Proof. Already, we first eliminate the function F. Indeed, from (9) if (Sx, Ty) > 0 follows (10) where a, b, c ∈ [0, +∞), a 2 + b 2 + c 2 > 0 and a + b + c < 1. Further, we give the proof in several steps: e step 1. e step 2. e step 3.
According to Lemmas 9 and 10, we have that the sequence {x n } is a −Cauchy in an −complete -metric space (X, ) and x n x m whenever n m. is further means that there is (unique) x* ∈ X such that x n → x* as n → + .
Firstly, let S be continuous. en x 2n+1 = Sx 2n → Sx* = x* since in each −metric space the subsequence of each convergent sequence converges to the unique limit. Now, we will prove that also Tx* = x*. Indeed, if Tx* x* then by using (10) with x = y = x* we get Finally, we obtain that (1 − c) · (x*, Tx*) < 0 which is a contradiction because we suppose (x*, Tx*) 0. If the mapping T is continuous, the proof is similar. e theorem is completely proved. Remark 1. Our eorem 11 generalizes, improves, complements and unifies the corresponding eorem 3 from (Asif et al, 2019) in several directions. First of all, it is worth to notice that some parts of the proof for eorem 3 are doubtful. Namely, the authors in their proof use that -metric is a continuous function with two variables ( (x n , y n ) → (x, y) if (x n , x) → 0 and (y n , y) → 0), which is not case. Also, it is clear that the function F in their eorem 3 and in both Corollaries 1 and 2 is superfluous. e next two corollaries follows from our eorem 11. Corollary 6. Suppose (f, α) # × [0, +∞) and (X, ) is an −complete −metric space. Let S, T : X → X be self mappings. Suppose there exist strictly increasing function F : (0, +∞) → (−∞, +∞) and τ > 0 such that (Sx, Ty) > 0 yields (11) for k # [0, 1). By replacing S with T, we get the following result for single mapping: +∞) and (X, ) is an −complete −metric space. Let T : X → X be self mapping. Suppose there exist strictly increasing function F : (0, +∞) → (−∞, +∞) and τ > 0 such that (Tx, Ty) > 0 yields (12) for k # [0, 1). en T has at most one fixed point in X, if it is continuous. Remark 2. Now we give the following Important Notice: It is useful to note that the other results from (Asif et al, 2019) can be repaired and supplemented in the same or similar way. It should also be said that the results on Hausdorff-Pompeiu metric given in (Asif et al, 2019) are dubious. is will be discussed in another of our papers.
Corollary 8. Suppose (f, α) # × [0, +∞) and (X, ) is an −complete −metric space. Let S, T : X → X be self mappings. Suppose there exist τ 1 > 0 such that (Sx, Ty) > 0 yields (13) for a, b, c # [0, +∞) such that a 2 + b 2 + c 2 > 0 and a + b + c < 1. en S and T have at most one common fixed point in X, if one of the mappings S or T is continuous.
Corollary 9. Suppose (f, α) # × [0, +∞) and (X, ) is an −complete −metric space. Let S, T : X → X be self mappings. Suppose there exist τ 2 > 0 such that (Sx, Ty) > 0 yields (14) for a # [0, 1) . en S and T have at most one common fixed point in X, if one of the mappings S or T is continuous. +∞) and (X, ) is an −complete −metric space. Let S, T : X → X be self mappings. Suppose there exist τ 3 > 0 such that (Sx, Ty) > 0 yields (15) for b, c # [0, +∞) such that b 2 + c 2 > 0 and b + c < 1. en S and T have at most one common fixed point in X, if one of the mappings S or T is continuous.
Corollary 11. Suppose (f, α) # × [0, +∞) and (X, ) is an −complete −metric space. Let S, T : X → X be self mappings. Suppose there exist τ 4 > 0 such that (Sx, Ty) > 0 yields (16) for b, c # [0, +∞) such that b 2 + c 2 > 0 and b + c < 1. en S and T have at most one common fixed point in X, if one of the mappings S or T is continuous. Corollary 12. Suppose (f, α) # × [0, +∞) and (X, ) is an −complete −metric space. Let S, T : X → X be self mappings. Suppose there exist τ 5 > 0 such that (Sx, Ty) > 0 yields (17) for b, c # [0, +∞) such that b 2 + c 2 > 0 and b + c < 1. en S and T have at most one common fixed point in X, if at least one of the mappings S or T is continuous. +∞) and (X, ) is an −complete −metric space. Let S, T : X → X be self mappings. Suppose there exist τ 6 > 0 such that (Sx, Ty) > 0 yields (18) for b, c # [0, +∞) such that b 2 + c 2 > 0 and b + c < 1. en S and T have at most one common fixed point in X, if one of the mappings S or T is continuous. Proof. As each of the functions is strictly increasing on (0, +∞), the proof immediately follows by our eorem 11 and their corollaries.
Example 1. Finally, we give the following simple example that support our eorem 11 with S = T. Suppose that X = {2n + 1 : n ∈ } . Define the −metric given by the following It is clear that is a −metric and F is strictly increasing on (0, +∞). All the conditions of eorem 11 are satisfied. Indeed, putting in equation (9) b = c = 0, we get for x y : i.e., e −|T x−T y| > a · e −|x−y| . Taking x = 2n + 1, y = 2m + 1, n m we further obtain e −|2n−2m| > a · e −|2n−2m| . Since n m this means that there exists a ∈ [0, 1) such that (9) holds true, i.e., T has a unique fixed point in X = {2n + 1 : n ∈ }, which is x = 3. Note that lim r→+0 F (r) = −1, then eorem 3 from (Asif et al, 2019) is not applicable here. is shows that our results are proper generalizations of the ones from (Asif et al, 2019).

Conclusion
In this article, we obtained several new contractive conditions in the framework of -metric spaces. Our results improve, extend, complement, generalize, and unify various recent developments in the context of metric spaces. An example shows that the main results of (Asif et al, 2019) are not applicable in our case. We think that this is a useful contribution in the framework of .-contraction introduced by D. Wardowski.