COMMON FIXED POINT RESULTS FOR BOYD-WONG AND MEIR-KEELER CONTRACTION IN F-METRIC SPACES (

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COMMON FIXED POINT RESULTS FOR BOYD-WONG AND MEIR-KEELER CONTRACTION INF-METRIC SPACES

(Hasil Titik Tetap Sepunya untuk Pemetaan Pengecutan Boyd-Wong dan Meir-Keeler dalam RuangF-Metrik)

IFFAH NURLATHIFAH FIKRI & ZABIDIN SALLEH^∗

ABSTRACT

A new notion of metric space generalization has been defined by Jleli and Samet, namelyF- metric space, in 2018. The objective of this study is to prove the existence and uniqueness of a common fixed point in the context ofF-metric space. We construct theorems of a common fixed point for commuting mapping pairs with Boyd-Wong and Meir-Keeler contraction in this space.

Moreover, we extend the results of Park and Bae (1981) and Beraet al.(2022) to common fixed point theorems andF-metric space, respectively. The Boyd-Wong contraction is attractive to discuss since we cannot apply the metrizability result on theF-metric space to prove the theorem. The Meir-Keeler contraction is also interesting since it is a generalization of the Boyd-Wong contraction. Lastly, we provide an example of each case to support the findings of our study.

Keywords: F-metric space; commuting mappings; common fixed point; Boyd-Wong contraction; Meir-Keeler contraction

ABSTRAK

Tanggapan baru bagi pengitlakan ruang metrik telah ditakrifkan oleh Jleli dan Samet, iaitu ru- angF-metrik, pada tahun 2018. Objektif kajian ini adalah untuk membuktikan kewujudan dan keunikan titik tetap sepunya dalam konteks ruangF-metrik. Kami membina teorem titik tetap sepunya untuk pasangan pemetaan kalis tukar tertib dengan pemetaan pengecutan Boyd- Wong dan Meir-Keeler dalam ruang ini. Selain itu, kami melanjutkan keputusan Park dan Bae (1981) dan Beraet al.(2022) kepada teorem titik tetap sepunya dan ruangF-metrik, masing- masing. Pemetaan pengecutan Boyd-Wong menarik untuk dibincangkan kerana kita tidak boleh menggunakan hasil kebolehmetrikan pada ruangF-metrik untuk membuktikan teorem berke- naan. Pemetaan pengecutan Meir-Keeler juga menarik kerana ia merupakan pengitlakan bagi pemetaan pengecutan Boyd-Wong. Akhir sekali, kami menyediakan contoh bagi setiap kes untuk menyokong dapatan kajian.

Kata kunci:ruangF-metrik; pemetaan kalis tukar tertib; titik tetap sepunya; Pemetaan pengecutan Boyd-Wong; Pemetaan pengecutan Meir-Keeler

1. Introduction

After the Banach contraction principle (BCP) was proved by Banach and Cacciopoli, many authors introduced a new type of contraction. Some of them are Boyd and Wong (1969) who introduced nonlinear contraction, namely Boyd-Wong contraction, and Meir and Keeler (1969) who discuss Meir-Keeler contraction. Thus, Gopal et al.(2017) introduced the Meir-Keeler contraction as a generalization of the Boyd-Wong contraction. Moreover, the Boyd-Wong and Meir-Keeler contraction class contains the class of Banach contraction (Park & Bae 1981).

On the other hand, Jungck (1976) described the relation between commuting mappings and the common fixed point for contraction. The authors (Karapinar & Agarwal 2022; Gopal &

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Bisht 2017) have explained coincidence and common fixed points for several contractive type pairs by changing the contraction condition in Jungck (1976). The existence of a coincidence and common fixed point for contractive mapping pairs is interesting, because it may fail to have even though we have two continuous mappings, that commute, on compact convex sets (Gopal

& Bisht 2017).

Furthermore, Park (1978) proved common fixed point theorems for contractive mappings in complete metric space. The author applied the equivalent functional condition to replace the Cauchy condition of a contractive iteration in a complete metric space by Geraghty (1973).

Thus, Park and Bae (1981) had an outstanding result in common fixed point theory. They proved the existence of a unique common fixed point that generalized a theorem by Meir and Keeler (1969) in the context of the usual metric space. In addition, Park and Bae (1981) presented the extent of the fixed point theorem for the other contraction, including Boyd-Wong contraction (Boyd & Wong 1969).

Recently, new notions of metric space generalization have been discovered by several authors. One of them is Jleli and Samet (2018) notion which changes the "triangle inequality"

in ordinary metric spaces with another form, namelyF-metric spaces. Since then, many authors have investigatedF-metric space further (Mitrovi´cet al.2019; Alnaseret al.2019; Bera et al.2019; 2022; Asifet al. 2019; Somet al.2020; Jahangiret al.2021; Binbasioglu 2021;

Altun & Erduran 2022). Jahangiret al. (2021) explained the relations of usual metric space withF-metric space. Moreover, the last four years have seen a growing trend towards fixed point (Beraet al.2019; 2022; Jahangir et al.2021) and common fixed point theory (Mitrovi´c et al.2019; Asifet al.2019; Manav & Turkoglu 2019; Binbasioglu 2021) inF-metric spaces.

Furthermore, Som et al. (2020) proved that Boyd-Wong contractive cannot be obtained with metrizability results. In addition, Beraet al.(2022) presented the existence and uniqueness of a fixed point for mapping that satisfies Boyd-Wong type.

In this paper, we extend Beraet al. (2022) result by using the Boyd-Wong type notion to ensure the existence and uniqueness of a common fixed point inF-metric space. Previously, Som et al.(2020) proved that we could obtain a lot of concepts from usual metric space to F-metric space by using the metrizability result. Despite the Boyd-Wong type contraction can not be obtained from the metrizability result, the research on that type of contraction is still interesting. Consequently, the discussion about the common fixed point theorem for Meir- Keeler contraction in the context ofF-metric space is attractive as well. Furthermore, several examples are provided for the two theorems.

2. Preliminaries

GivenS and T are two self-mappings on a nonempty set X. If η, ζ ∈ X exists, such that T ζ =Sζandη=T η=Sηthenζandηis a coincidence point and a common fixed point ofT andS, respectively. LetCP(S, T)andF(S, T)represent the set that contains all coincidence points and common fixed points ofS andT, respectively. And the mappingsT commute with Sif only ifT Sη=ST ηfor everyη ∈X. In addition, letC_S denote the class of self-mapping T such thatT X ⊂SXandT S=ST.

In 2018, Jleli and Samet Jleli and Samet (2018) defined a certain class F that contains functionf : (0,+∞)→Rsatisfies two conditions below:

(F₁) f(a)≤f(b), if0< a < b;

(F₂) we have lim

bn→0f(bn) =−∞, for every sequence(bn)⊂(0,+∞).

Definition 2.1(Jleli & Samet 2018). GivenX ̸=∅anddF :X² →[0,∞)is a mapping, then there exists(g, α)∈ F ×[0,∞)such that for every(η, θ)∈X²satisfies

(df1) dF(η, θ) = 0if only ifη=θ, (df2) dF(η, θ) =dF(θ, η),

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(df₃) ifη ̸=θ, then for anyN₀ ∈N, N₀ ≥2such that for any{v_j}^N_j=1⁰ ⊂Xwith(v₁, v_N₀) = (η, θ), we have

g(dF(η, θ))≤g N0−1

X

j=1

dF(vj, vj+1)

+α.

Then, a mappingdF and the pair(X, dF) are called anF-metric on X andF-metric space, respectively.

Moreover, some concepts onF-metric space presented in Jleli and Samet (2018) as follows Definition 2.2(Jleli & Samet 2018). Given sequence{η_n}is inF-metric space(X, dF).

(i) {η_n}isF-convergent toη^∗ ∈X, if lim

n→∞dF(ηn, η^∗) = 0.

(ii) {η_n}isF-Cauchy, if lim

n→∞dF(η_n, η_m) = 0.

The pair (X, dF) is F-complete F-metric space, if each F-Cauchy sequence in X is F- convergent to a point inX.

Proposition 2.3(Jleli & Samet 2018). Given(X, dF)is anF-metric space andC ̸=∅which is a subset ofX. Then, the equivalent holds for the following statements.

(a) For any sequence{η_n} ⊂C, there exists{η_n_k} ⊂ {η_n}andη∈Csuch that

k→+∞lim dF(ηnk, η) = 0.

(b) CisF-compact.

Furthermore, given(X, dF)is anF-metric space with(g, α). Jleli and Samet (2018) defined d^∗ :X² →[0,∞)by

d^∗(η, θ) = inf

N0−1

X

i=1

dF(vi, vi+1)|N₀ ∈N, N0≥2,{v_i}^N_i=1⁰ ⊆Xwith(vi, vN0) = (η, θ)

, (1) for any(η, θ) ∈ X² and proved that(X, d^∗)is a metric space. Then, according to Somet al.

(2020), (X, dF) is metrizable as regards metricd^∗. Based on these results, we establish the following proposition.

Proposition 2.4. Given(X, dF)is anF-metric space with(g, α)and metricd^∗defined by (1).

GivenAand{η_n}are subsets ofXwhich are a non-empty subset and a sequence, respectively.

(i) {η_n} is a Cauchy sequence as regards metric d^∗ if and only if {η_n} is an F-Cauchy sequence,

(ii) {η_n}is convergent toη∈Xas regards metricd^∗if and only if{η_n}isF-convergent to η∈X,

(iii) Xis complete as regards metricd^∗if and only ifXisF-complete, (iv) Ais compact as regards metricd^∗if and only ifAisF-compact.

Lastly, we define some contractions for two self-mappings that are commonly used.

Definition 2.5. Given(X, dF)is anF-metric space andSis a continuous self-mappings ofX.

AndT is a self-mapping ofX. Then, for everyη, θ∈X,T is said to be (a) S-contraction, ifdF(T η, T θ)≤λdF(Sη, Sθ)for someλ∈(0,1),

(b) S-contractive (Park 1978), ifdF(T η, T θ)< dF(Sη, Sθ)withT η ̸=T θ,and (c) S-nonexpansive, ifdF(T η, T θ)≤dF(Sη, Sθ).

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3. Main Result

3.1. Common fixed point of Boyd-Wong type contraction inF-metric space

Firstly, we define Boyd-Wong contraction for two self-mapping, namelyS andT, inF-metric space withT calledϕ-S-contraction.

Definition 3.1. Given S is a continuous self-mapping of an F-metric space (X, dF) with (g, α) ∈ F × [0,∞) and ϕ is a self-mapping of [0,∞) which nondecreasing upper semi- continuous from right such thatϕ(t) < tfor every t > 0 and for any sequencet_n → t ≥ 0 implieslim sup

n→∞

ϕ(t_n)≤ϕ(t). AnyT ∈C_Sis called anϕ-S-contraction if for anyη, θ∈X

dF(T η, T θ)≤ϕ(dF(Sη, Sθ)). (2)

Secondly, we establish a common fixed point theorem of a self-mappings pair that satisfies ϕ-S-contraction inF-metric space.

Theorem 3.2. Given (X, dF) is an F-completeF-metric space with (g, α) ∈ F ×[0,∞).

SupposeSis a continuous self-mapping ofXandT ∈C_Sis aϕ-S-contraction. Ifϕsatisfies

g(s)> g(ϕ(s)) +α, (3)

for everys∈(0,∞), thenSandT have a unique common fixed point onX.

Proof.Supposeη0is an arbitrary point inX. SinceT(X) ⊆S(X), there exists someη1 ∈X such thatSη1=T η0. In general, we have a sequence{Sη_n}withSηn=T ηn−1for anyn∈N.

By (2), for everyn∈N, we obtain

dF(Sηn+1, Sηn+2) =dF(T ηn, T ηn+1)

≤ϕ(dF(Sη_n, Sη_n+1))

< dF(Sηn, Sηn+1).

As a consequence, we have the sequence{d_F(Sη_n, Sη_n+1)}which is strictly decreasing and bounded below. Hence, lim

n→∞dF(Sη_n, Sη_n+1)exists, namelyp. It is clear thatp≥0. Ifp >0, by the property ofϕ, we obtain

p= lim

n→∞dF(Sηn, Sηn+1)≤lim sup

n→∞ ϕ(dF(Sηn−1, Sηn))≤ϕ(p)< p, which is a contradiction. As a result, we have

0 =p= lim

n→∞dF(Sηn, Sηn+1). (4)

Suppose that{Sη_n}is not anF-Cauchy sequence. Hence, there existsϵ >0and for allk∈N we defined subsequence {Sη_m_k} and{Sη_n_k} of {Sη_n} with mk ≥ nk ≥ k andnk is the smallest number not surpassmksuch that

dF(Sηmk, Sηnk)≥ϵ. (5)

Despite the selection ofn_k, we have

dF(Sηmk, Sηnk−1)< ϵ. (6)

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Using (F₁), (df₃), (5) and (6), we obtain

f(ϵ)≤g(dF(Sη_m_k, Sη_m_k₊₁) +dF(Sη_m_k₊₁, Sη_n_k)) +α

≤g(dF(Sηmk, Sηmk+1) +ϕ(dF(Sηmk, Sηnk−1))) +α

≤g(dF(Sη_m_k, Sη_m_k₊₁) +ϕ(ϵ)) +α.

By lettingk→ ∞, we have g(ϵ)≤g(ϕ(ϵ)) +α,

which contradicts (3). Therefore, the sequence{Sη_n}is anF-Cauchy sequence. Due toXis F-complete, there isη∈Xsuch that

n→∞lim dF(Sηn, η) = 0. (7)

Suppose thatT η̸=Sη. Thus, by (df3), we obtain

g(dF(T η, Sη))≤g(dF(T η, T Sη_n) +dF(T Sη_n, Sη)) +α, (8) for every n ∈ N. If there are Sη1 andSη2 such that dF(Sη1, η) = 0 = dF(Sη2, η), then Sη1 =η=Sη2which is a contradiction. Consequently, we select a sequence{Sη_n_q} ⊆ {Sη_n} such thatdF(Sηnq, η)̸= 0for everyq∈N.By the given condition, (8) and(F₁), we obtain

g(dF(T η, Sη))≤g(dF(T η, T Sηnq) +dF(ST ηnq, Sη)) +α

≤g(ϕ(dF(SSηnq, Sη)) +dF(SSηnq+1, Sη)) +α

≤g(dF(SSηnq, Sη) +dF(SSηnq+1, Sη)) +α.

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By the condition(F₂)and (7), we obtain

g(dF(SSη_n_q, Sη) +dF(SSη_n_q+1, Sη)) +α→ −∞,

asq → ∞, which contradictsdF(T η, Sη) > 0. Thus, we haveSη = T η = z, forz ∈ X.

Moreover, suppose thatSz̸=z. SinceT andSare commuting, then

dF(z, Sz) =dF(T η, ST η) =dF(T η, T Sη)≤ϕ(dF(Sη, SSη))< dF(z, Sz),

which is a contradiction. As a consequence, we havez =Sz =ST η =T Sη = T z.In other words,zis a common fixed point ofT andS. Now, we prove the uniqueness. Suppose these mappings have the other common fixed point, namelyθwithθ̸=z. Thus, we obtain

dF(θ, z) =dF(T θ, T z)≤ϕ(dF(Sθ, Sz))< dF(θ, z), which is a contradiction.□

Now, the example is given to support our result.

Example 3.3. GivenX=RanddF :X² →Ris defined by dF(η, θ) =

(η−θ)², η, θ∈[0,3];

|η−θ|, otherwise.

for any(η, θ)∈X. It is clear thatdF satisfies (df₁) and (df₂). Since 2 =dF(2,4)> dF

2,5

2

+dF

5 2,4

= 7 4,

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then dF does not satisfy triangle inequality. However, dF satisfies (df₃) with g(s) = ln(s), for any s > 0, andα = ln 3. Thus,(X, dF) is an F-metric space. Next, we define metric d^∗ as in Equation (1). obviously, X is complete as regards metric d^∗ since X = R. Using Proposition 2.4, we obtain thatXisF-complete.

GivenSandT is a self-mapping onXdefined by Sη=

_η+4

2 , ifη≥4;

4, ifη <4,

andT η= 4for eachη∈X.We also define a nondecreasing upper semi-continuous from right mappingϕby

ϕ(s) = 1

4s, for anys∈[0,∞),

which satisfies for any sequencesn→s≥0implieslim sup_n→∞ϕ(sn)≤ϕ(s),ϕ(s)< sand g(s) > g(ϕ(s)) +αfor everys >0. Let arbitrary pointsη, θ ∈X. Ifη, θ ∈[4,∞), then we obtain

dF(T η, T θ) = 0≤ϕ(dF(Sη, Sθ)) = 1

8|η−θ|.

Otherwise, we havedF(T η, T θ) = 0≤ϕ(dF(Sη, Sθ)). Hence,T andSsatisfy condition (2) for everyη, θ∈R. Hence, the conditions of Theorem 3.2 are satisfied. Therefore, theSandT have a unique common fixed point, i.e.,z= 4 =S(4) =T(4).

Furthermore, we give the consequences of Theorem 3.2 as follows.

Remark 3.4. By using the same way in Proposition 2.7 in Somet al.(2020), ifT is aϕ-S- contraction (2) in the context ofF-metric space, then we haveT isS-nonexpansive contraction as regards metricd^∗. Accordingly,T need not be aϕ-S-contraction as regards metricd^∗ifT is aϕ-S-contraction inF-metric space. We conclude that Theorem 3.2 cannot be proved by the metrizability result.

Remark 3.5. By replacing continuous self-mappingSwith identity mapping1X, we will get a fixed point theorem for Boyd-Wong contraction in Beraet al.(2022). As a result, we consider that Theorem 3.2 is an extension of Theorem 2.1 (Beraet al.2022).

3.2. Common fixed point of Meir-Keeler type contraction inF-metric space

The following is the Meir-Keeler contraction definition for self-mappingsSandTwithT ∈CS. Definition 3.6. Given S is a continuous self-mapping of an F-metric space (X, dF) with (g, α) ∈ F ×[0,∞)andT ∈CS. A self-mappingT ofXis called an(ϵ, δ)-S-contraction if for anyϵ >0,there existsδ >0such that for allη, θ∈X,

(M1) ϵ≤dF(Sη, Sθ)< ϵ+δimpliesdF(T η, T θ)< ϵ, (M2) T η=T θwheneverSη =Sθ.

Clearly,(ϵ, δ)-S-contraction impliesS-contractive since from (M1),

dF(T η, T θ)< dF(Sη, Sθ), (10)

wheneverSη̸=Sθfor allη, θ∈X.

Theorem 3.7. Given (X, dF) is an F-compact F-metric space with(g, α) ∈ F ×[0,∞).

Suppose thatSis a continuous self-mapping onXandT ∈CSis an(ϵ, δ)-S-contraction, then SandT have a unique common fixed point inX.

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Proof.Supposeη₀ ∈X is an arbitrary point. SinceT(X)⊆S(X), we can construct iteration sequence {Sη_n} with Sη_n = T ηn−1 for everyn ∈ N.If we havedF(Sη_k, Sη_k+1) = 0 for somek ∈ N, thenSη_k =Sη_k+1 = T η_k. In other words,η_k is a coincidence point ofS and T. Suppose thatdF(Sηn, Sηn+1) ̸= 0for alln ∈Nandinf{d_F(Sηn, Sηn+1)} =r for some r >0. By (10), we have

dF(Sη_n+1, Sη_n+2) =dF(T η_n, T η_n+1)< dF(Sη_n, Sη_n+1),

for everyn∈N. Hence,{d_F(Sη_n, Sη_n+1)}is a decreasing sequence. Then, we have

n→∞lim dF(Sηn, Sηn+1) =r.

By (M1), there existsδ >0such that

r≤dF(Sη, Sθ)< r+δ implies dF(T η, T θ)< r, (11) for everyη, θ ∈ X. Since limn→∞dF(Sη_n, Sη_n+1) = r, there existsN₀ ∈ Nsuch that for every natural numberm≥N₀implies

r≤dF(Sηm, Sηm+1)< r+δ. (12)

By (11), we have dF(Sη_m+1, Sη_m+2) = dF(T η_m, T η_m+1) < r, for all m ≥ N₀, which contradicts (12). Then, we havelimn→∞dF(Sη_n, Sη_n+1) = 0.Since(X, dF)is anF-compact F-metric space and{Sη_n} ⊂X, there exists a subsequence{Sη_n_k}of{Sη_n}such that

k→+∞lim dF(Sηnk, η) = 0,

for some η ∈ X. Since S is continuous and T commutes with S, we have {SSη_n_k} = {ST η_n_k₋₁} = {T Sη_n_k₋₁} converges to T η. If we can find somem_k such that SSη_m_k = SSη_m_k₊₁=SSη_m_k₊₂ =. . ., then{SSη_n_k}converges toSSη_m_k andSSη_m_k =SSη_m_k₊₁= ST η_m_k =T Sη_m_k.We obtain a coincidence pointSη_m_k withSSη_m_k =Sη.

SupposedF(SSηnk, SSηnk+1)>0for alln_k. Then, for anyϵ >0there existsN0∈Nsuch that for anyn_k≥N0,dF(SSηnk, Sη)< ₃^ϵ and we can findm_k≥N0such thatSSηmk ̸=Sη.

Assume thatdF(Sη, T η)>0, by (df3) and (10), then we obtain g(dF(Sη, T η))≤g(dF(Sη, ST ηnk) +dF(ST ηnk, T η)) +α

=g(dF(Sη, SSη_n_k₊₁) +dF(T Sη_n_k, T η)) +α

< g(dF(Sη, SSη_n_k₊₁) +dF(SSη_n_k, Sη)) +α.

And by (F₂), then we have

g(dF(Sη, T η))≤g(dF(Sη, SSηnk+1) +dF(SSηnk, Sη)) +α→ −∞,

ask→ ∞which contradictsdF(Sη, T η)>0. Hence, we haveSη =T η. LetSη =T η =ζ and supposedF(Sζ, ζ)>0.By (10), we have

dF(Sζ, ζ) =dF(ST η, T η) =dF(T Sη, T η)< dF(SSη, Sη) =dF(Sζ, ζ),

which is a contradiction. Thus, we obtain Sζ = ζ and T ζ = T Sη = ST η = Sζ = ζ.

Therefore,ζis a common fixed point ofSandT.

Suppose another common fixed point exists that is different from ζ, namely ζ^′. Then by (10), we obtain

dF(ζ, ζ^′) =dF(Sζ, Sζ^′)< dF(T ζ, T ζ^′) =dF(ζ, ζ^′), which contradictsζ ̸=ζ^′. Therefore,ζ is unique.□

In addition, the following example is given to support Theorem 3.7.

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Example 3.8. GivenX= [0,10]anddF :X² →Ris defined by dF(η, θ) =

(η−θ)², η, θ∈[0,2];

|η−θ|, otherwise.

for any(η, θ)∈X². Clearly,dF satisfies (df1) and (df2). However,dF does not satisfy triangle inequality since

3 =dF(0,3)> dF

0,1

2

+dF

1 2,3

= 11 4 .

Moreover, dF satisfies (df3) with g(s) = ln(s) for every s ∈ (0,∞) andα = ln(2).Thus, (X, dF) is an F-metric space. Next, defined d^∗ is metric as in Equation (1). Since X is a finite set, it is clear that(X, d^∗)is a compact metric space. By Proposition 2.4, (X, dF)is an F-compactF-metric space. Let

Sη=

η, η∈[0,8];

8, η∈(8,10]

andT η = 5 for every η ∈ X. Obviously, S is continuous mapping andT ∈ CS. Suppose ϵ > 0is an arbitrary point, there existsδ > 0such that if we haveϵ ≤ dF(Sη, Sθ) < ϵ+δ impliesdF(T η, T θ) = 0< ϵfor allη, θ ∈ X. Thus,T is an(ϵ, δ)-S-contraction. Moreover, since the conditions of Theorem 3.7 hold, T and S have a unique common fixed point, i.e., η^∗ = 5 =S(5) =T(5).

In the following, we establish a proposition regarding the metrizability result.

Proposition 3.9. Given(X, dF)is anF-metric space with(g, α)∈ F ×[0,∞),d^∗ is a metric as in (1) andS, T are a self-mapping onXwithT ∈C_S. IfT is an(ϵ, δ)-S-contraction in the setting ofF-metric space, thenT is an(ϵ, δ)-S-contraction as regards metricd^∗.

Proof. Supposeη, θ ∈ X andϵ > 0are arbitrary points and {s_n}^q_n=1 ⊆ X with(s₁, s_q) = (η, θ) and q ≥ 2. Suppose{T(s_n)}^q_n=1 and{S(s_n)}^q_n=1 are the finite sequences inX. By (M1), there existsδ >0such that

ϵ≤dF(Sη, Sθ)< ϵ+δ implies dF(T η, T θ)< ϵ, (13) wheneverSη̸=Sθ. By (1), we have

d^∗(S(η), S(θ)) = inf{Pq−1

j=1dF(S(s_j), S(s_j+1)) :q∈N, q≥2,(s₁, s_q) = (η, θ)};

d^∗(T(η), T(θ)) = inf{Pq−1

j=1dF(T(s_j), T(s_j+1)) :q ∈N, q≥2,(s₁, s_q) = (η, θ)}. (14) From (13) and (14), we obtain

inf ^q−1

X

j=1

ϵ:q∈N, q≥2

≤d^∗(S(η), S(θ))<inf ^q−1

X

j=1

ϵ+δ :q∈N, q≥2

⇐⇒ ϵ≤d^∗(S(η), S(θ))< ϵ+δ, and

d^∗(T(η), T(θ))<inf ^q−1

X

j=1

ϵ:q∈N, q≥2

⇐⇒ d^∗(T(η), T(θ))< ϵ.

Therefore,T is an(ϵ, δ)-S-contraction as regards metricd^∗.□

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Corollary 3.10. We conclude from Proposition 2.4 and Proposition 3.9 that Meir-Keeler contraction inF-metric space is a direct result of Meir-Keeler contraction for usual metric space.

For further research, we also establish a remark.

Remark 3.11. In Park and Bae (1981), the authors have proved a common fixed point theorem for Meir-Keeler contraction in complete metric space. In other words, they proved it in F-complete F-metric space with f(s) = ln(s) for all s ∈ (0,∞) and α = 0. The triangle inequality property in metric has a significant role in the proof. SinceF-metric may not have triangle inequality property, the common fixed point theorem for Meir-Keeler contraction fails to prove in an F-completeF-metric space. An open question is whether the necessary conditions are sufficient for the theorem to be proven inF-completeF-metric space.

Acknowledgement

The authors delightedly acknowledge the Ministry of Higher Education Malaysia and Universiti Malaysia Terengganu for partially supporting this research under the Fundamental Research Grant Scheme (FRGS) Project Code FRGS/1/2021/STG06/UMT/02/1 and Vote no. 59659.

References

Alnaser L.A., Lateef D., Fouad H.A. & Ahmad J. 2019. Relation theoretic contraction results inF-metric spaces.J.

Nonlinear Sci. Appl.12(5): 337–344.

Altun I. & Erduran A. 2022. Two fixed point results onF-metric spaces.Topology Algebra Appl.10(1): 61–67.

Asif A., Nazam M., Arshad M. & Kim S.O. 2019.F-metric,F-contraction and common fixed-point theorems with applications.Mathematics7(7): 586.

Bera A., Dey L., Som S., Garai H. & Sintunavarat W. 2022. Boyd-Wong contractions inF-metric spaces and applications.Applied General Topology23(1): 157–167.

Bera A., Garai H., Damjanovic B. & Chanda A. 2019. Some interesting result onF-metric spaces.Filomat33(10):

3257–3268.

Binbasioglu D. 2021. Some result of common fixed point for compatible mappingsF-metric spaces.Journal of New Theory37: 1–7.

Boyd D.W. & Wong J.S.W. 1969. On nonlinear contraction.Proc. Amer. Math. Soc.20(2), pp. 458–464.

Cho Y.J., Murthy P. & Jungck G. 1993. A common fixed point theorem of Meir and Keeler type.Internat. J. Math.

Math. Sci.16(4): 669–674.

Geraghty M.A. 1973. On contractive mappings.Proc. Amer. Math. Soc.40(2), pp. 604–608.

Gopal D. & Bisht R.K. 2017. Metrical common fixed points and commuting type mappings. In Gopal D., Kumam, P.&Abbas M. (eds.).Background and Recent Development of Metric Fixed Point Theory: 29-67. Boca Raton, FL: CRC Press.

Gopal D., Patel D.K. & Shukla S. 2017. Banach fixed point theorem and its generalizations. In Gopal D., Kumam, P.

&Abbas M. (eds.).Background and Recent Development of Metric Fixed Point Theory: 1-28. Boca Raton, FL: CRC Press.

Jachymski J.R. 1997. Equivalence of some contractivity properties over metrical structures.Proc. Amer. Math. Soc.

125(8), pp. 2327–2335.

Jahangir F., Haghmaram P. & Nourouzi K. 2021. A note onF-metric spaces.Journal of Fixed Point Theory and Applications23(1): 2.

Jleli M. & Samet B. 2018. On a new generalization of metric spaces.Journal of Fixed Point Theory and Applications 20(3): 128.

Jungck G. 1976. Commuting mappings and fixed points.Amer. Math. Mon.83(4): 261–263.

Karapınar E. & Agarwal R.P. 2022.Fixed Point Theory in Generalized Metric Spaces. Springer Nature.

Manav N. & Turkoglu D. 2019. Common fixed point results on modularF-metric spaces.AIP Conf. Proc.2183.

Matkowski J. 1994. Nonlinear contraction in metrically convex space.Publ. Math. Debrecen45(12): 103–114.

Meir A. & Keeler E. 1969. A theorem on contraction mappings.J. Math. Anal. Appl.28: 326–329.

(10)

Mitrovi´c Z.D., Aydi H., Hussain N. & Mukheimer A. 2019. Reich, Jungck, and Berinde common fixed point results onF-metric spaces and an application.Mathematics7(5): 387.

Park S. 1978. Fixed points ofF-contractive maps.The Rocky Mountain Journal of Mathematics8(4): 743–750.

Park S. & Bae J.S. 1981. Extensions of a fixed point theorem of Meir and Keeler.Ark. Mat.19(1-2): 223–228.

Som S., Bera A. & Dey L.K. 2020. Some remarks on the metrizability ofF-metric spaces.J. Fixed Point Theory Appl.22(1): 17.

Special Interest Group on Modelling and Data Analytics (SIGMDA), Faculty of Ocean Engineering Technology and Informatics,

Universiti Malaysia Terengganu,

21300 Kuala Nerus, Terengganu, Malaysia

E-mail: [email protected], [email protected]^∗

Received: 17 August 2023 Accepted: 15 September 2023

∗Corresponding author