Revisiting and revamping some novel results in F-metric spaces

Пересмотр и улучшение некоторых новых результатов в F-метрических пространствах

Ревизија и побољшање неких нових резултата у F-метричким просторима

Zoran D. Mitrovića zoran.mitrovic@etf.unibl.org

University of Banja Luka, Bosnia and Herzegovina

Mudasir Younisb mudasiryouniscuk@gmail.com

University Institute of Technology-RGPV, India

Miloje D. Rajovićc rajovic.m@mfkv.kg.ac.rs

University of Kragujevac, Serbia

This work is licensed under Creative Commons Attribution 4.0 International.

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. The 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. This is what they have in common with ordinary metric spaces. The 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), (Aydi et al, 2019), (Derouiche & Ramoul, 2020), (Jahangir et al, 2021), (Kirk & Shazad, 2014), (Mitrović et al, 2019), (Salem et al, 2020), (Som et al, 2020), (Vujaković et al, 2020), (Vujaković & Radenović, 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) called the mapping T : X → X, defined on a metric space (X, d), an F−contraction if there exist τ > 0 and F satisfying (F1)-(F3) such that

The authors in (Asif et al, 2019) take = {F : (0, +∞) → (−∞, +∞) : F satisfies ₁) and ₂)} .

In 2019, A. Asif et al, (Asif et al, 2019) formulated and proved the fixed-point and common fixed-point results for single-valued Reich-type and Kannan-type F-contractions in the setting of -metric spaces:

Theorem 1. Suppose (f, α) ∈ ×[0, +∞) and (X, ) is an −complete -metric space. Let S, T : X → X be self-mappings. Suppose there exist F ∈ and τ > 0 such that

(1)

for a, b, c ∈ [0, . + ∞) such that a + b + c < 1 with

for all (x, y) ∈ X × X. Then S and T have at most one common fixed point in X.

Corollary 1. Suppose (f, α) ∈ ×[0, +∞) and (X,) is an −complete -metric space Let S, T : X → X be self-mappings. Suppose that k ∈ [0, 1), there exist F ∈ and τ > 0 such that

(2)

with min {(Sx, Ty), (x, y), (x, Sx), (y, Ty)} > 0, for all (x, y) ∈ X × X. Then 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.

Corollary 2. Suppose (f, α) ∈ ×[0, +∞) and (X, ) is an −complete -metric space Let T : X → X be self-mapping. Suppose that for k ∈ [0, 1), there exist F ∈ and τ > 0 such that

(3)

with min { (Tx, Ty), (x, Tx), (y, Ty)} > 0, for all (x, y) ∈ X × X. Then 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, +∞). Then the mapping T is called a Reich-type F-contraction on B (x₀, r) ⊆ X if there exist F ∈ and τ > 0 such that for all x, y ∈ B (x₀, r)

(4)

Theorem 2. Suppose (f, α) ∈ ×[0, +∞) and (X, is an −complete -metric space. Let T be a Reich-type F-contraction on B (x₀, r) ⊆ X. Suppose that for x₀ ∈ X and r > 0, the following conditions are satisfied:

Then S and T have at most one common fixed point in B (x₀, r).

Taking S = T in Theorem 2, the authors in (Asif et al, 2019; Corollary 3.) obtained the following result for single mapping.

Corollary 3. Suppose (f, α) ∈ ×[0, +∞),(F, τ ) ∈ × (0, +∞),(X, ) 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 x₀ ∈ X and r > 0, the following conditions are satisfied:

Then T has at most one fixed point in B (x₀, r).

Corollary 4. Suppose (f, α) ∈ ×[0, +∞),(F, τ ) ∈ × (0, +∞),(X, ) is an −complete -metric space. Let S, T : X → X be a self-mappings and k ∈ [0, 1). Suppose that for x₀ ∈ X and r > 0, the following conditions are satisfied:

Then S and T have at most one common fixed point in B (x₀, r).

Further in the same paper (Asif et al, 2019; Definitions 6, 8, Theorem 5, Corollary 5.), the authors gave the following:

Definition 4. Let (X, ) be a metric space. Let CB (X) be the family of all non-empty closed and bounded subsets of X. Let H : CB (X)×CB (X) → [0, +∞) be a function defined by

(5)

where D (x, B) = inf {(x, y) : y ∈ B}. Then H defines a metric on CB (X) called the Hausdorff-Pompeiu metric induced by .

Definition 5. Let (X, ) be an -metric space. Suppose F ∈ and H : CB (X) × CB (X) → [0, +∞) be the Hausdorff-Pompeiu metric function defined in Definition 2. A mapping T : X → CB (X) is known as a setvalued Reich-type contraction if there is some τ > 0 such that

(6)

for (x, y) ∈ X × X and a, b, c ∈ [0, +∞) such that a + b + c < 1.

Theorem 3. 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. Suppose (f, α) ∈ ×[0, +∞) and (X, ) is an −complete -metric space. Let T : X → CB (X) be a Reich-type F−contraction such that F is right continuous. Suppose that for k ∈ [0, 1), there exist F ∈ and τ > 0 such that

(7)

with min {H (Tx, Ty), (x, Tx), (y, Ty)} > 0 for all (x, y) ∈ X × X.

Then T has a fixed point in X.

In the sequel, we will use the following two results:

Lemma 1. (Mitrović et al, 2019; 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). Then is a d_b−Cauchy sequence in (X, d_b) (resp.−Cauchy sequence in (X, )).

Lemma 2. Let x₀ be a Picard sequence in -metric space inducing by mapping T : X → X and initial point x₀ ∈ X. If (x_n, x_n+1) < (x_n−1, x_n) for all n ∈ then x_n $\ne$ x_m whenever n $\ne$ m.

Proof. Let x_n = x_m for some n, m ∈ with n < m. Then _xn+1 = Tx_n = Tx_m = x_m+1. Further, we get

which is a contradiction.

Firstly, since F : (0, +∞) → (−∞, +∞) it follows that (1) is possible only if (Sx, Ty) > 0 where x, y ∈ X. Also, the condition (F1) yields that a · (x, y) + b · (x, Sx) + c · (y, Ty) > 0 for all x, y ∈ X for which (Sx, Ty) > 0. This means that at least of a, b, c ∈ [0, +∞) must be distinct of 0. Now we can improve the formulation of Theorem 1 and all its corollaries from (Asif et al, 2019) and give new proofs as the following.

Theorem 4. Suppose (f, α) ∈ × [0, +∞) and (X, ) is an −complete −metric space. Let S, T : X → X be self mappings. Suppose there exist a strictly increasing function F : (0, +∞) → (−∞, +∞) and τ > 0 such that (Sx, Ty) > 0 yields

(9)

for a, b, c ∈ [0, +∞) such that a² + b² + c² > 0 and a + b + c < 1.

Then 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, +∞), a² + b² + c² > 0 and a + b + c < 1. Further, we give the proof in several steps:

The step 1.

The step 2.

The 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 $\ne$ x_m whenever n $\ne$ m. This further means that there is (unique) x* ∈ X such that x_n → x* as n → +$\infty$ .

Firstly, let S be continuous. Then 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* $\ne$ 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*) $\ne$ 0.

If the mapping T is continuous, the proof is similar.

The theorem is completely proved.

Remark 1. Our Theorem 11 generalizes, improves, complements and unifies the corresponding Theorem 3 from (Asif et al, 2019) in several directions. First of all, it is worth to notice that some parts of the proof for Theorem 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 Theorem 3 and in both Corollaries 1 and 2 is superfluous.

The next two corollaries follows from our Theorem 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:

Corollary 7. Suppose (f, α) ∈ × [0, +∞) 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).

Then 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. This will be discussed in another of our papers.

The immediate consequences of our Theorem 11 are the following new contractive conditions that complement the ones given in (Collaco & Silva, 1997), (Rhoades, 1977) for usual metric spaces. For more contractive conditions in the framework of metric spaces see (Ćirić, 2003), (Consentino & Vetro, 2014), (Dey et al, 2019), (Karapınar, et al, 2020), (Piri & Kumam, 2014), (Salem et al, 2020), (Wardowski & Dung, 2014). In the sequel we will obtain several new contractive conditions in the framework of −metric spaces.

Corollary 8. Suppose (f, α) ∈ × [0, +∞) and (X, ) is an −complete −metric space. Let S, T : X → X be self mappings. Suppose there exist τ₁ > 0 such that (Sx, Ty) > 0 yields

(13)

for a, b, c ∈ [0, +∞) such that a² + b² + c² > 0 and a + b + c < 1. Then 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 τ₂ > 0 such that (Sx, Ty) > 0 yields

(14)

for a ∈ [0, 1) .

Then S and T have at most one common fixed point in X, if one of the mappings S or T is continuous.

Corollary 10. Suppose (f, α) ∈ × [0, +∞) and (X, ) is an −complete −metric space. Let S, T : X → X be self mappings. Suppose there exist τ₃ > 0 such that (Sx, Ty) > 0 yields

(15)

for b, c ∈ [0, +∞) such that b² + c² > 0 and b + c < 1. Then 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 τ₄ > 0 such that (Sx, Ty) > 0 yields

(16)

for b, c ∈ [0, +∞) such that b² + c² > 0 and b + c < 1.

Then 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 τ₅ > 0 such that (Sx, Ty) > 0 yields

(17)

for b, c ∈ [0, +∞) such that b² + c² > 0 and b + c < 1.

Then S and T have at most one common fixed point in X, if at least one of the mappings S or T is continuous.

Corollary 13. Suppose (f, α) ∈ × [0, +∞) and (X, ) is an −complete −metric space. Let S, T : X → X be self mappings. Suppose there exist τ₆ > 0 such that (Sx, Ty) > 0 yields

(18)

for b, c ∈ [0, +∞) such that b² + c² > 0 and b + c < 1.

Then 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 Theorem 11 and their corollaries.

Example 1. Finally, we give the following simple example that support our Theorem 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 Theorem 11 are satisfied. Indeed, putting in equation (9) b = c = 0, we get for x $\ne$ y :

i.e., e^{−|T x−T y|} > a · e^−|x−y|. Taking x = 2n + 1, y = 2m + 1, n $\ne$ m we further obtain e^−|2n−2m| > a · e^−|2n−2m|. Since n $\ne$ 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 Theorem 3 from (Asif et al, 2019) is not applicable here. This shows that our results are proper generalizations of the ones from (Asif et al, 2019).

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.

FIELD: Mathematics

ARTICLE TYPE: Original scientific paper

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