Best proximity points in ℱ-metric spaces with applications

Durdana Lateef

doi:10.1515/dema-2022-0191

Artikel Open Access

Best proximity points in ℱ-metric spaces with applications

Durdana Lateef

Veröffentlicht/Copyright: 12. April 2023

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Aus der Zeitschrift Demonstratio Mathematica Band 56 Heft 1

Abstract

The aim of this article is to introduce α - ψ -proximal contraction in the setting of ℱ-metric space and prove the existence of best proximity points for these contractions. As applications of our main results, we obtain coupled best proximity points on ℱ-metric space equipped with an arbitrary binary relation.

Keywords: best proximity point; α-ψ-proximal contraction; non-self mappings; coupled best proximity point

MSC 2010: 46S40; 47H10; 54H25

1 Introduction

The Banach contraction principle [1] is the first result which was introduced by Stefan Banach in 1922 in which notions of fixed point and metric space play an important role. Let Θ , Φ be non-empty subsets of metric space ( ℳ , σ ) . A point ω ∈ ℳ is called a fixed point of ℒ : Θ → Φ if ℒ ω = ω . Because of significance and simplicity of the concept of the fixed point, this conception has been improved and lengthened in many distinct ways. In 2010, Basha [2] gave the conception of best proximity point and extended the famous Banach’s contraction principle. For more particular on this perspective, we refer the readers to [3–15]. On the other hand, the well-known extensions of the notion of metric spaces have been done by Bakhtin [16] which was conventionally given by Czerwik [17] in 1993 by generalizing the Banach contraction principle. Jleli and Samet [18] introduced a novel metric space known as ℱ -metric space to extend the classical metric space and b -metric space. Later on, Al-Mezel et al. [19] introduced the notion of generalized ( α β - ψ )-contractions in ℱ -metric spaces with the help of α -admissibility of the mapping and obtained fixed-point results in this generalized space.

In this article, we introduce α - ψ -proximal contraction in the background of ℱ -metric space and prove the existence of best proximity points for these contractions.

2 Preliminaries

We state this section with definition of b -metric space in this manner.

Definition 2.1

(See [17]). Let ℳ ≠ ∅ and s ≥ 1 . A function σ b : ℳ × ℳ → [ 0 , ∞ ) is called a b -metric if the following assertions hold:

σ b ( ω , ϖ ) = 0 if and only if ω = ϖ ;
σ b ( ω , ϖ ) = σ b ( ϖ , ω ) ,
σ b ( ω , ν ) ≤ s [ σ b ( ω , ϖ ) + σ b ( ϖ , ν ) ] ,
∀ ω , ϖ , ν ∈ ℳ .

The pair ( ℳ , σ b ) is called a b -metric space.

Jleli and Samet [18] established a fascinating extension of a metric space as follows.

Let ℱ be the set of functions f : ( 0 , + ∞ ) → R satisfying the following conditions:

0 < s < ι ⇒ f ( s ) ≤ f ( ι ) ,
for all { ι n } ⊆ R + , lim n → ∞ ı n = 0 ⇔ lim n → ∞ f ( ı n ) = − ∞ .

Definition 2.2

[18] Let ℳ ≠ ∅ and let σ ℱ : ℳ × ℳ → [ 0 , + ∞ ) . Assume that ∃ ( f , α ) ∈ ℱ × [ 0 , + ∞ ) such that

( ω , ϖ ) ∈ ℳ × ℳ , σ ℱ ( ω , ϖ ) = 0 if and only if ω = ϖ .
σ ℱ ( ω , ϖ ) = σ ℱ ( ϖ , ω ) , ∀ ( ω , ϖ ) ∈ ℳ × ℳ .
For every ( ω , ϖ ) ∈ ℳ × ℳ , for every N ∈ N , N ≥ 2 , and for every ( u i ) i = 1 N ⊂ ℳ , with ( u 1 , u N ) = ( ω , ϖ ) , we have
σ ℱ ( ω , ϖ ) > 0 ⇒ f ( σ ℱ ( ω , ϖ ) ) ≤ f ∑ i = 1 N − 1 σ ℱ ( u i , u i + 1 ) + α .
Then ( ℳ , σ ℱ ) is called an ℱ -metric space.

Example 2.1

(See [18]) The function σ ℱ : R × R → [ 0 , + ∞ )

σ ℱ ( ω , ϖ ) = ( ω − ϖ ) 2 if ( ω , ϖ ) ∈ [ 0 , 3 ] × [ 0 , 3 ] ∣ ω − ϖ ∣ if ( ω , ϖ ) ∉ [ 0 , 3 ] × [ 0 , 3 ] ,

with f ( ι ) = ln ( ι ) and α = ln ( 3 ) , is an ℱ -metric.

Remark 2.1

It is clear from the definition that any metric space is an ℱ -metric space, but the inverse is not true in general.

Definition 2.3

(See [18]) Let ( ℳ , σ ℱ ) be an ℱ -metric space.

Let { κ n } ⊆ ℳ . The sequence { ω n } is said to be ℱ -convergent to ω ∈ ℳ if { ω n } is convergent to ω regarding ℱ -metric σ ℱ .
The sequence { ω n } is called ℱ -Cauchy, iff
lim n , m → ∞ σ ℱ ( ω n , ω m ) = 0 .
If every ℱ -Cauchy sequence in ℳ is ℱ -converges to a point in ℳ , then ( ℳ , σ ℱ ) is ℱ -complete.

Theorem 2.1

[18] Let ( ℳ , d ℱ ) be an ℱ -metric space and ℒ : ℳ → ℳ and assume that the conditions given below are satisfied:

( ℳ , σ ℱ ) is ℱ -complete,
∃ k ∈ ( 0 , 1 ) , such that
σ ℱ ( ℒ ( ω ) , ℒ ( ϖ ) ) ≤ k σ ℱ ( ω , ϖ ) .
Then ℒ has a unique fixed point ω ∗ ∈ ℳ . Moreover, for any ω 0 ∈ ℳ , the sequence { ω n } ⊂ ℳ defined by
ω n + 1 = ℒ ( ω n ) , n ∈ N ,
is ℱ -convergent to ω ∗ .

For more characteristics in these ways, we mention the researchers to [18–32].

Motivated with Basha [2], we define the notion of best proximity point in the context of ℱ -metric space in this way.

Definition 2.4

Let ( ℳ , σ ℱ ) is ℱ -metric space and Θ , Φ ∈ N ( ℳ ) . An element ω ∗ ∈ Θ is professed to be a best proximity point of ℒ : Θ → Φ if this assertion hold

σ ℱ ( ω ∗ , ℒ ω ∗ ) ≤ σ ℱ ( Θ , Φ ) .

Consistent with Eldred and Veeramani [3], we give the ℱ -distance between the pair ( Θ , Φ ) of two nonempty sets which satisfy the property P .

Definition 2.5

Let ( ℳ , σ ℱ ) is ℱ -metric space and Θ , Φ ∈ N ( ℳ ) , then σ ℱ ( Θ , Φ ) is ℱ -distance between two nonempty sets Θ and Φ . Now define Θ 0 and Φ 0 by

Θ 0 = { ω ∈ Θ : ∃ u ∈ Φ such that σ ℱ ( ω , u ) = σ ℱ ( Θ , Φ ) } Φ 0 = { u ∈ Φ : ∃ ω ∈ Θ such that σ ℱ ( ω , u ) = σ ℱ ( Θ , Φ ) } .

Then ( Θ , Φ ) is called to have the property P if Θ 0 ≠ ∅ and

ω , ϖ ∈ Θ 0 , u , v ∈ Φ 0 , σ ℱ ( ω , u ) = σ ℱ ( ϖ , v ) = σ ℱ ( Θ , Φ ) ⇒ σ ℱ ( ω , ϖ ) = σ ℱ ( u , v ) .

Definition 2.6

Let ( ℳ , σ ℱ ) is ℱ -metric space and Θ , Φ ∈ N ( ℳ ) . A mapping ℒ : Θ → Φ is called α -proximal admissible ( α -prox admis) if ∃ a function α : Θ × Θ → [ 0 , ∞ ) such that

α ( ω , ϖ ) ≥ 1 σ ℱ ( u , ℒ ω ) = σ ℱ ( Θ , Φ ) σ ℱ ( v , ℒ ϖ ) = σ ℱ ( Θ , Φ ) ⇒ α ( u , v ) ≥ 1 ,

where ω , ϖ , u , v ∈ Θ .

3 Best proximity point results in ℱ -metric spaces

We represent by Ψ the collection of non-decreasing functions ψ : [ 0 , ∞ ) → [ 0 , ∞ ) such that ∑ n = 1 ∞ ψ n ( ı ) < ∞ for each ı > 0 . If ψ ∈ Ψ , then ψ ( ı ) < ı for all ı > 0 . We also denote N ( ℳ ) and C l ( ℳ ) as set of non-empty subsets of ℳ and closed subsets of ℳ , respectively.

Definition 3.1

Let ( ℳ , σ ℱ ) is ℱ -metric space and Θ , Φ ∈ N ( ℳ ) . A mapping ℒ : Θ → Φ is said to be an α - ψ -proximal contraction if there exists ψ ∈ Ψ and α : Θ × Θ → [ 0 , ∞ ) such that

(1) α ( ω , ϖ ) σ ℱ ( ℒ ω , ℒ ϖ ) ≤ ψ ( σ ℱ ( ω , ϖ ) )

∀ ω , ϖ ∈ Θ .

Theorem 3.1

Let ( ℳ , σ ℱ ) is complete ℱ -metric space and Θ , Φ ∈ C l ( ℳ ) such that Θ 0 ≠ ∅ . Let α : Θ × Θ → [ 0 , ∞ ) and ψ ∈ Ψ . Assume that ℒ : Θ → Φ be an α - ψ -proximal contraction and α -prox admis satisfying these assertions:

ℒ ( Θ 0 ) ⊆ Φ 0 and ( Θ , Φ ) fulfils the property P;
∃ ω 0 , ω 1 ∈ Θ 0 such that
σ ℱ ( ω 1 , ℒ ω 0 ) = σ ℱ ( Θ , Φ ) , and α ( ω 0 , ω 1 ) ≥ 1 .
ℒ is continuous.

Then ∃ ω ∗ ∈ Θ such that σ ℱ ( ω ∗ , ℒ ω ∗ ) ≤ σ ℱ ( Θ , Φ ) .

Proof

By the hypothesis (ii), ∃ ω 0 , ω 1 ∈ Θ 0 such that

(2) σ ℱ ( ω 1 , ℒ ω 0 ) = σ ℱ ( Θ , Φ ) , α ( ω 0 , ω 1 ) ≥ 1 .

Since ℒ ( Θ 0 ) ⊆ Φ 0 , ∃ ω 2 ∈ Θ 0 such that

σ ℱ ( ω 2 , ℒ ω 1 ) = σ ℱ ( Θ , Φ ) .

Now, we have α ( ω 0 , ω 1 ) ≥ 1 , σ ℱ ( ω 1 , ℒ ω 0 ) = σ ℱ ( Θ , Φ ) , and σ ℱ ( ω 2 , ℒ ω 1 ) = σ ℱ ( Θ , Φ ) . As the mapping ℒ is α -prox admis, we obtain α ( ω 1 , ω 2 ) ≥ 1 . Hence,

(3) σ ℱ ( ω 2 , ℒ ω 1 ) = σ ℱ ( Θ , Φ ) , α ( ω 1 , ω 2 ) ≥ 1 .

Again since ℒ ( Θ 0 ) ⊆ Φ 0 , ∃ ω 3 ∈ Θ 0 such that

σ ℱ ( ω 3 , ℒ ω 2 ) = σ ℱ ( Θ , Φ ) .

Now, we have α ( ω 1 , ω 2 ) ≥ 1 , σ ℱ ( ω 2 , ℒ ω 1 ) = σ ℱ ( Θ , Φ ) and σ ℱ ( ω 3 , ℒ ω 2 ) = σ ℱ ( Θ , Φ ) . As the mapping ℒ is α -prox admis, we obtain α ( ω 2 , ω 3 ) ≥ 1 . Hence,

(4) σ ℱ ( ω 3 , ℒ ω 2 ) = σ ℱ ( Θ , Φ ) , α ( ω 2 , ω 3 ) ≥ 1 .

By pursuing in this way, by induction, we can generate { ω n } ⊂ Θ 0 such that

(5) σ ℱ ( ω n + 1 , ℒ ω n ) = σ ℱ ( Θ , Φ ) , α ( ω n , ω n + 1 ) ≥ 1 ,

∀ n ∈ N ∪ { 0 } . Assume that ω k = ω k + 1 for some k . From (5), we have

σ ℱ ( ω k , ℒ ω k ) = σ ℱ ( ω k + 1 , ℒ ω k ) = σ ℱ ( Θ , Φ ) ,

i.e., ω k is a best proximity point of ℒ . Hence, we assume that σ ℱ ( ω n − 1 , ω n ) > 0 for all n ∈ N ∪ { 0 } . As ( Θ , Φ ) satisfies the property P , we summarize from (5) that

σ ℱ ( ω n , ω n + 1 ) = σ ℱ ( ℒ ω n − 1 , ℒ ω n ) ,

∀ n ∈ N ∪ { 0 } . So by (1), we have

(6) σ ℱ ( ω n , ω n + 1 ) ≤ α ( ω n , ω n + 1 ) σ ℱ ( ω n , ω n + 1 ) = α ( ω n , ω n + 1 ) σ ℱ ( ℒ ω n − 1 , ℒ ω n ) ≤ ψ ( σ ℱ ( ω n − 1 , ω n ) )

∀ n ≥ 0 . By using the monotonicity of ψ and (6), we obtain

σ ℱ ( ω n , ω n + 1 ) ≤ ψ n ( σ ℱ ( ω 0 , ω 1 ) )

∀ n ∈ N ∪ { 0 } . Let ε > 0 be fixed and ( f , α ) ∈ ℱ × [ 0 , + ∞ ) be such that (D 3 ) is satisfied. By ( ℱ 2 ), ∃ δ > 0 such that

(7) 0 < ı < δ ⇒ f ( ı ) < f ( ε ) − α .

Let n ( ε ) ∈ N such that ∑ n ≥ n ( δ ) ψ n ( σ ℱ ( ω 0 , ω 1 ) ) < δ . Hence, by (6), (7), and ( ℱ 1 ), we have

(8) f ∑ i = n m − 1 ψ i ( σ ℱ ( ω 0 , ω 1 ) ) ≤ f ∑ n ≥ n ( δ ) ψ n ( σ ℱ ( ω 0 , ω 1 ) ) < f ( ε ) − α

for m > n > n ( ε ) . By using ( D 3 ) and (8), we obtain σ ℱ ( ω n , ω m ) > 0 , m > n > n ( ε ) implies

f ( σ ℱ ( ω m , ω n ) ) ≤ f ∑ i = n m − 1 σ ℱ ( ω i , ω i + 1 ) + α ≤ f ∑ i = n m − 1 ψ i ( σ ℱ ( ω 0 , ω 1 ) ) + α < f ( ε ) ,

which implies by ( ℱ 1 ) that σ ℱ ( ω m , ω n ) < ε , m > n > n ( ε ) . This proves that { ω n } is ℱ -Cauchy. Since ( ℳ , σ ℱ ) is ℱ -complete and Θ is closed, ∃ ω ∗ ∈ Θ such that { ω n } is ℱ -convergent to ω ∗ , i.e.,

(9) lim n → ∞ σ ℱ ( ω n , ω ∗ ) = 0 ,

i.e., ℒ ω n → ℒ ω ∗ as n → ∞ . By using the continuity of σ ℱ , we obtain

σ ℱ ( Θ , Φ ) = σ ℱ ( ω n + 1 , ℒ ω n ) → σ ℱ ( ω ∗ , ℒ ω ∗ )

as n → ∞ . Therefore, σ ℱ ( ω ∗ , ℒ ω ∗ ) = σ ℱ ( Θ , Φ ) .□

If { ω n } ⊆ Θ such that α ( ω n , ω n + 1 ) ≥ 1 , for all n and ω n → ω ∈ Θ as n → ∞ , then ∃ { ω n ( k ) } of { ω n } such that α ( ω n ( k ) , ω ) ≥ 1 , for all k .

Theorem 3.2

Let ( ℳ , σ ℱ ) be a complete ℱ -metric space and Θ , Φ ∈ C l ( ℳ ) such that Θ 0 ≠ ∅ . Let α : Θ × Θ → [ 0 , ∞ ) and ψ ∈ Ψ . Assume that ℒ : Θ → Φ be an α - ψ -proximal contraction and α -proximal admissible satisfies these assertions:

ℒ ( Θ 0 ) ⊆ Φ 0 and ( Θ , Φ ) satisfies the property P;
∃ ω 0 , ω 1 ∈ Θ 0 such that
σ ℱ ( ω 1 , ℒ ω 0 ) = σ ℱ ( Θ , Φ ) , and α ( ω 0 , ω 1 ) ≥ 1 .
( J ) holds.

Then ∃ ω ∗ ∈ ℳ such that σ ℱ ( ω ∗ , ℒ ω ∗ ) ≤ σ ℱ ( Θ , Φ ) .

Proof

Backing the result of Theorem 3.1, ∃ { ω n } ⊂ Θ such that (1) holds and { ω n } is ℱ -convergent to ω ∗ , i.e.,

lim n → ∞ σ ℱ ( ω n , ω ∗ ) = 0 .

By the property ( κ ), ∃ { ω n ( k ) } of { ω n } such that α ( ω n ( k ) , ω ∗ ) ≥ 1 , for all k . We declare that ℒ ω n ( k ) → ℒ ω ∗ as k → ∞ . So by (1), we obtain

σ ℱ ( ℒ ω n ( k ) , ℒ ω ∗ ) ≤ α ( ω n ( k ) , ω ∗ ) σ ℱ ( ℒ ω n ( k ) , ℒ ω ∗ ) ≤ ψ ( σ ℱ ( ω n ( k ) , ω ∗ ) )

∀ k . By taking k → ∞ and using the continuity of σ ℱ , we have

σ F ( Θ , Φ ) = σ ℱ ( ω n ( k ) + 1 , ℒ ω n ( k ) ) → σ ℱ ( ω ∗ , ℒ ω ∗ )

as n → ∞ . Therefore,

σ ℱ ( ω ∗ , ℒ ω ∗ ) = σ F ( Θ , Φ )

thus proved.□

Definition 3.2

Let ℒ : Θ → Φ and α : Θ × Θ → [ 0 , ∞ ) . The mapping ℒ is said to be ( α , σ F ) -regular if for all ( ω , ϖ ) ∈ α − 1 [ 0 , 1 ) , ∃ ϱ ∈ Θ 0 such that

α ( ω , ϖ ) ≥ 1 and α ( ϖ , ϱ ) ≥ 1 .

Theorem 3.3

Besides to the supposition of Theorem 3.1 (respectively Theorem 3.2), assume that ℒ is ( α , σ ) -regular. Then, ∃ ω ∗ ∈ Θ such that σ ℱ ( ω ∗ , ℒ ω ∗ ) ≤ σ ℱ ( Θ , Φ ) , which is unique.

Proof

It is clear from the Theorem 3.1 that the set of best proximity points of ℒ is non-empty. Assume that ∃ ϖ ∗ ∈ Θ 0 of ℒ , i.e.,

(10) σ ℱ ( ℒ ω ∗ , ω ∗ ) = σ ℱ ( ℒ ϖ ∗ , ϖ ∗ ) = σ ℱ ( Θ , Φ ) .

By using the property P and (10), we obtain that

(11) σ ℱ ( ℒ ω ∗ , ℒ ϖ ∗ ) = σ ℱ ( ω ∗ , ϖ ∗ ) .

We discuss two cases.

Case 1. If α ( ω ∗ , ϖ ∗ ) ≥ 1 , by using (10), we obtain that

σ ℱ ( ω ∗ , ϖ ∗ ) = σ ℱ ( ℒ ω ∗ , ℒ ϖ ∗ ) ≤ α ( ω ∗ , ϖ ∗ ) σ ℱ ( ℒ ω ∗ , ℒ ϖ ∗ ) ≤ ψ ( σ ℱ ( ω ∗ , ϖ ∗ ) ) .

Since ψ ( ι ) < ι , for all ι > 0 , the aforementioned inequality satisfies only if σ ℱ ( ω ∗ , ϖ ∗ ) = 0 , i.e., ω ∗ = ϖ ∗ .

Case 2. If α ( ω ∗ , ϖ ∗ ) < 1 .

By supposition, ∃ ϱ 0 ∈ Θ 0 such that α ( ω ∗ , ϱ 0 ) ≥ 1 and α ( ϖ ∗ , ϱ 0 ) ≥ 1 . Since ℒ ( Θ 0 ) ⊆ Φ 0 , there exists ϱ 1 ∈ Θ 0 such that

σ ℱ ( ϱ 1 , ℒ ϱ 0 ) = σ ℱ ( Θ , Φ ) .

Now, we have

α ( ω ∗ , ϱ 0 ) ≥ 1

σ ℱ ( ω ∗ , ℒ ω ∗ ) = σ ℱ ( Θ , Φ ) ,

σ ℱ ( ϱ 1 , ℒ ϱ 0 ) = σ ℱ ( Θ , Φ ) .

As ℒ is α -prox admis, so we have α ( ω ∗ , ϱ 1 ) ≥ 1 . Hence,

σ ℱ ( ϱ 1 , ℒ ϱ 0 ) = σ ℱ ( Θ , Φ ) and α ( ω ∗ , ϱ 1 ) ≥ 1 .

By pursuing in this way, we can generate a sequence { ϱ n } in Θ 0 such that

(12) σ ℱ ( ϱ n + 1 , ℒ ϱ n ) = σ ℱ ( Θ , Φ ) and α ( ω ∗ , ϱ n ) ≥ 1

∀ n ≥ 0 . By property P and (12), we obtain that

(13) σ ℱ ( ϱ n + 1 , ω ∗ ) = σ ℱ ( ℒ ϱ n , ℒ ω ∗ )

∀ n ∈ N ∪ { 0 } . Since ℒ is an α – ψ -proximal contraction, we have

σ ℱ ( ϱ n + 1 , ω ∗ ) = σ ℱ ( ℒ ϱ n , ℒ ω ∗ ) ≤ α ( ϱ n , ω ∗ ) σ ℱ ( ℒ ϱ n , ℒ ω ∗ ) ≤ ψ ( σ ℱ ( ϱ n , ω ∗ ) )

∀ n ≥ 0 . By induction, we can obtain

(14) σ ℱ ( ϱ n , ω ∗ ) ≤ ψ n ( σ ℱ ( ϱ 0 , ω ∗ ) )

∀ n ≥ 0 . Assume that ϱ 0 = ω ∗ . Then by (13), we obtain

σ ℱ ( ϱ 1 , ω ∗ ) = σ ℱ ( ℒ ϱ 0 , ℒ ω ∗ ) = σ ℱ ( ℒ ω ∗ , ℒ ω ∗ ) = 0 ,

that is, ϱ 1 = ω ∗ . By pursuing in this way and inductively, we have ϱ n = ω ∗ , for all n ≥ 0 . Assume σ ℱ ( ϱ 0 , ω ∗ ) > 0 . By taking limit as n → ∞ in ( 14), we establish that ϱ n → ω ∗ whenever n → ∞ . Thus, in all the discussed cases, we obtain ϱ n → ω ∗ as n → ∞ . Likewise, we can show that ϱ n → ϖ ∗ as n → ∞ . By uniqueness of the limit, we obtain that ω ∗ = ϖ ∗ .□

4 Applications

Theorem 4.1

Let ( ℳ , σ ℱ ) is complete ℱ -metric space, Θ , Φ ∈ C l ( ℳ ) such that Θ 0 ≠ ∅ . Let ψ ∈ Ψ . Suppose that ℒ : Θ → Φ satisfying

ℒ ( Θ 0 ) ⊆ Φ 0 and ( Θ , Φ ) satisfying the property P;
σ ℱ ( ℒ ω , ℒ ϖ ) ≤ ψ ( σ ℱ ( ω , ϖ ) ) , for all ω , ϖ ∈ Θ .

Then ∃ ω ∗ ∈ ℳ such that σ ℱ ( ω ∗ , ℒ ω ∗ ) ≤ σ ℱ ( Θ , Φ ) .

Proof

Define α : Θ × Θ → [ 0 , ∞ ) by

α ( ω , ϖ ) = 1

∀ ω , ϖ ∈ Θ . Evidently ℒ is α -prox admis by the definition of α , and also it is an α – ψ -proximal contraction. Otherwise, for any ω ∈ Θ 0 , since ℒ ( Θ 0 ) ⊆ Φ 0 , ∃ ϖ ∈ Θ 0 such that σ ( ℒ ω , ϖ ) = σ ( Θ , Φ ) . Furthermore, from the hypothesis (ii), we obtain

σ ℱ ( ℒ ω , ℒ ϖ ) ≤ ψ ( σ ℱ ( ω , ϖ ) ) < σ ℱ ( ω , ϖ ) .

From the aforementioned inequality, we have ℒ is a continuous. Thus, all the assumptions of Theorem 3.1 are fulfilled and ∃ is the best proximity point of ℒ directly from Theorem 3.1. Furthermore, from the definition of α and from Theorem 3, we obtain that this best proximity point is unique.□

If we take ψ ( ı ) = k ı , where 0 < k < 1 in Theorem 4.1, we establish this result.

Theorem 4.2

Let ( ℳ , σ ℱ ) is complete ℱ -metric space, Θ , Φ ∈ C l ( ℳ ) such that Θ 0 ≠ ∅ . Let ψ ∈ Ψ . Assume that ℒ : Θ → Φ satisfying these assertions:

ℒ ( Θ 0 ) ⊆ Φ 0 and ( Θ , Φ ) satisfies the property P;
∃ k ∈ ( 0 , 1 ) such that σ ℱ ( ℒ ω , ℒ ϖ ) ≤ k σ ℱ ( ω , ϖ ) , for all ω , ϖ ∈ Θ .

Then ∃ ω ∗ ∈ ℳ such that σ ℱ ( ω ∗ , ℒ ω ∗ ) ≤ σ ℱ ( Θ , Φ ) .

4.1 Results on ℱ -metric space endowed with binary relation

Let ( ℳ , σ ℱ ) be an ℱ -metric space and R be any binary relation on ℳ and let

S = R ∪ R − 1 .

Evidently,

ω , ϖ ∈ ℳ , ω S ϖ if and only if ω R ϖ or ϖ R ω .

Definition 4.1

A mapping ℒ : Θ → Φ is called a proximal comparative mapping if

ω 1 S ω 2 σ F ( u 1 , ℒ u 1 ) = σ F ( Θ , Φ ) σ F ( u 2 , ℒ u 2 ) = σ F ( Θ , Φ ) ⇒ u 1 S u 2

∀ ω 1 , ω 2 , u 1 , u 2 ∈ Θ .

Theorem 4.3

Let ( ℳ , σ ℱ ) is complete ℱ - metric space, Θ , Φ ∈ C l ( ℳ ) such that Θ 0 ≠ ∅ . Let R be any binary relation on ℳ . Assume that ℒ : Θ → Φ is continuous satisfying

ℒ ( Θ 0 ) ⊆ Φ 0 and ( Θ , Φ ) satisfies the P -Property;
ℒ is a proximal comparative mapping,
∃ ω 0 , ω 1 ∈ Θ 0 such that
σ ℱ ( ω 1 , ℒ ω 0 ) = σ ℱ ( Θ , Φ ) , and ω 0 S ω 1 ,
∃ ψ ∈ Ψ such that
(15) ω , ϖ ∈ Θ , ω S ϖ implies σ ℱ ( ℒ ω , ℒ ϖ ) ≤ ψ ( σ ℱ ( ω , ϖ ) ) ,

Then ∃ ω ∗ ∈ ℳ such that σ ℱ ( ω ∗ , ℒ ω ∗ ) ≤ σ ℱ ( Θ , Φ ) .

Proof

Define α : Θ × Θ → [ 0 , ∞ ) by:

α ( ω , ϖ ) = 1 if ω S ϖ 0 otherwise.

Suppose that

α ( ω 1 , ω 2 ) ≥ 1 σ ℱ ( u 1 , ℒ ω 1 ) = σ ℱ ( Θ , Φ ) σ ℱ ( u 2 , ℒ ω 2 ) = σ ℱ ( Θ , Φ )

for some ω 1 , ω 2 , u 1 , u 2 ∈ Θ . By the definition of α , we obtain that

ω 1 S ω 2 , σ ℱ ( u 1 , ℒ ω 1 ) = σ ℱ ( Θ , Φ ) σ ℱ ( u 2 , ℒ ω 2 ) = σ ℱ ( Θ , Φ ) .

Then by supposition (ii), we obtain that u 1 S u 2 . Now by the definition of α , we have α ( u 1 , u 2 ) ≥ 1 . Thus, we established that ℒ is α -prox admis. Supposition (iii) yields

σ ℱ ( ω 1 , ℒ ω 0 ) = σ ℱ ( Θ , Φ )

and α ( ω 0 , ω 1 ) ≥ 1 . Finally, condition (iv) implies that

α ( ω , ϖ ) σ ℱ ( ℒ ω , ℒ ϖ ) ≤ ψ ( σ ℱ ( ω , ϖ ) )

i.e., ℒ is an α – ψ -proximal contraction. Thus, all the assumption of Theorem 3.1 hold, and the required result comes directly from this result.□

If we want to omit the continuity of ℒ , then we use this assumption:

if { ω n } in ℳ and ω ∈ ℳ are such that ω n S ω n + 1 , for all n ≥ 0 and lim n → ∞ σ ℱ ( ω n , ω ) = 0 , then ∃ { ω n ( k ) } of { ω n } such that ω n ( k ) S ω for all k .

Theorem 4.4

Let Θ , Φ ∈ C l ( ℳ ) , where ( ℳ , σ ℱ ) is complete ℱ -metric space such that Θ 0 ≠ ∅ . Let ℛ be a binary relation over ℳ . Suppose that ℒ : Θ → Φ satisfying

ℒ ( Θ 0 ) ⊆ Φ 0 and ( Θ , Φ ) fulfils the property P;
ℒ is a proximal comparative mapping,
∃ ω 0 , ω 1 ∈ Θ 0 such that
σ ℱ ( ω 1 , ℒ ω 0 ) = σ ℱ ( Θ , Φ ) , and ω 0 S ω 1 ,
∃ ψ ∈ Ψ such that
ω , ϖ ∈ Θ , ω S ϖ implies σ ℱ ( ℒ ω , ℒ ϖ ) ≤ ψ ( σ ℱ ( ω , ϖ ) ) ,
( ℋ ) holds.

Then ∃ ω ∗ ∈ ℳ such that σ ℱ ( ω ∗ , ℒ ω ∗ ) ≤ σ ℱ ( Θ , Φ ) .

Proof

If we consider α : Θ × Θ → [ 0 , ∞ ) given by

α ( ω , ϖ ) = 1 if ω S ϖ 0 otherwise.

and by observing that assertion ( H ) yields condition (J), then from Theorem 3.2 we obtain the conclusion.□

Theorem 4.5

Addition to the assumptions of Theorem 4.3 (respectively Theorem 4.4),assume that these conditions fulfils: for all ( ω , ϖ ) ∈ Θ × Θ with ( ω , ϖ ) ∉ S , ∃ ϱ ∈ Θ 0 such that ω S ϱ and ϖ S ϱ . Then there exists ω ∗ ∈ ℳ such that σ ℱ ( ω ∗ , ℒ ω ∗ ) ≤ σ ℱ ( Θ , Φ ) which is unique.

4.2 Coupled best proximity points results

Definition 4.2

A point ( ω ∗ , ϖ ∗ ) ∈ Θ × Θ is professed to be a coupled best proximity point of J if

σ ℱ ( ω ∗ , J ( ω ∗ , ϖ ∗ ) ) = σ ℱ ( ϖ ∗ , J ( ϖ ∗ , ω ∗ ) ) = σ ℱ ( Θ , Φ ) .

We establish the following notions.

M = ℳ × ℳ , ℵ 1 = Θ × Θ , ℵ 2 = Φ × Φ .

Define ℒ : ℵ 1 → ℵ 2 by

ℒ ( ω , ϖ ) = ( J ( ω , ϖ ) , J ( ϖ , ω ) ) ,

∀ ( ω , ϖ ) ∈ ℵ 1 . We supply the product set M with σ ℱ ∕ given by:

σ ℱ ∕ ( ( ω , ϖ ) , ( u , v ) ) = σ ℱ ( ω , u ) + σ ℱ ( ϖ , v ) 2 .

Evidently, if ( ℳ , σ ℱ ) is ℱ -complete, then ( M , σ ℱ ∕ ) is F -complete.

Definition 4.3

A mapping J : Θ × Θ → Φ is professed to be bi-proximal comparative (bi-prox comp) mapping if

ω 1 S ω 2 , ϖ 1 S ϖ 2 σ F ( u 1 , J ( ω 1 , ϖ 1 ) ) = σ F ( Θ , Φ ) σ F ( u 2 , J ( ω 2 , ϖ 2 ) ) = σ F ( Θ , Φ ) ⇒ u 1 S u 2

∀ ω 1 , ω 2 , ϖ 1 , ϖ 2 , u 1 , u 2 ∈ Θ .

Theorem 4.6

Let Θ , Φ ∈ C l ( ℳ ) , where ( ℳ , σ ℱ ) is complete ℱ -metric space such that Θ 0 ≠ ∅ and R be a binary relation on ℳ . Assume that the mapping J : Θ × Θ → Φ is continuous satisfying

J ( Θ 0 × Θ 0 ) ⊆ Φ 0 and ( Θ , Φ ) fulfils the property P,
J is a bi-prox comp mapping,
∃ ω 0 , ϖ 0 , ω 1 , ϖ 1 ∈ Θ 0 such that
σ ℱ ( ω 1 , J ( ω 0 , ϖ 0 ) ) = σ ℱ ( ϖ 1 , J ( ϖ 0 , ω 0 ) ) = σ ℱ ( Θ , Φ ) , and ω 0 S ω 1 , ϖ 0 S ϖ 1 ,
∃ a ψ ∈ Ψ in such that
(16) ω , ϖ , u , v ∈ Θ , ω S u , ϖ S v ⇒ σ ℱ ( J ( ω , ϖ ) , J ( u , v ) ) ≤ ψ σ ℱ ( ω , u ) + σ ℱ ( ϖ , v ) 2 .

Then J possess a coupled best proximity point.

Proof

Define a binary relation R 2 on ℳ by ( ω , ϖ ), ( u , v ) ∈ M , ( ω , ϖ ) R 2 ( u , v ) if and only if ω S u , ϖ S v . If we represent by S 2 the symmetric relation devoted to R 2 , evidently, we obtain S 2 = R 2 . We assert that ℒ : Θ × Θ → Φ has a best proximity point ( ω ∗ , ϖ ∗ ) ∈ Θ 0 × Θ 0 , such that

σ ℱ ∕ ( ( ω ∗ , ϖ ∗ ) , ℒ ( ω ∗ , ϖ ∗ ) ) = σ ℱ ∕ ( ℵ 1 , ℵ 2 ) .

Represent by:

A 0 = { ( ℓ 1 , ℓ 2 ) ∈ ℵ 1 , σ ℱ ∕ ( ( ℓ 1 , ℓ 2 ) , ( ℏ 1 , ℏ 2 ) ) = σ ℱ ∕ ( ℵ 1 , ℵ 2 ) for some ( ℏ 1 , ℏ 2 ) ∈ ℬ 0 } ℬ 0 = { ( ℏ 1 , ℏ 2 ) ∈ ℵ 2 , σ ℱ ∕ ( ( ℓ 1 , ℓ 2 ) , ( ℏ 1 , ℏ 2 ) ) = σ ℱ ∕ ( ℵ 1 , ℵ 2 ) for some ( ℓ 1 , ℓ 2 ) ∈ A 0 } .

We can observe that

σ ℱ ∕ ( ℵ 1 , ℵ 2 ) = σ ℱ ( Θ , Φ ) .

In fact, we have

σ ℱ ∕ ( ℵ 1 , ℵ 2 ) = inf { σ ℱ ∕ ( ( ℓ 1 , ℓ 2 ) , ( ℏ 1 , ℏ 2 ) ) : ( ℓ 1 , ℓ 2 ) ∈ ℵ 1 , ( ℏ 1 , ℏ 2 ) ∈ ℵ 2 } = 1 2 inf { σ ℱ ( ( ℓ 1 , ℏ 1 ) + ( ℓ 2 , ℏ 2 ) ) : ( ℓ 1 , ℏ 1 ) ∈ ℵ 1 × ℵ 2 , ( ℓ 2 , ℏ 2 ) ∈ ℵ 1 × ℵ 2 } = 1 2 ( inf { σ ℱ ( ( ℓ 1 , ℏ 1 ) : ( ℓ 1 , ℏ 1 ) ) ∈ Θ × Φ } + inf { σ ℱ ( ( ℓ 2 , ℏ 2 ) ) : ( ℓ 2 , ℏ 2 ) ∈ Θ × Φ } ) = 1 2 ( σ ℱ ( Θ , Φ ) + σ ℱ ( Θ , Φ ) ) = σ ℱ ( Θ , Φ ) .

Now, let ( ℓ 1 , ℓ 2 ) ∈ A 0 . Then there exists ( ℏ 1 , ℏ 2 ) ∈ ℵ 2 such that

σ ℱ ∕ ( ( ℓ 1 , ℓ 2 ) , ( ℏ 1 , ℏ 2 ) ) = σ ℱ ∕ ( ℵ 1 , ℵ 2 ) ,

that is,

σ ℱ ( ( ℓ 1 , ℏ 1 ) + ( ℓ 2 , ℏ 2 ) ) = 2 σ ℱ ( Θ , Φ ) .

Thus, we have

σ ℱ ( ( ℓ 1 , ℏ 1 ) + ( ℓ 2 , ℏ 2 ) ) = 2 σ ℱ ( Θ , Φ ) σ ℱ ( ℓ 1 , ℏ 1 ) ≥ σ ℱ ( Θ , Φ ) σ ℱ ( ℓ 2 , ℏ 2 ) ≥ σ ℱ ( Θ , Φ ) ,

which implies that

σ ℱ ( ℓ 1 , ℏ 1 ) = σ ℱ ( ℓ 2 , ℏ 2 ) = σ ℱ ( Θ , Φ ) .

This implies that ( ℓ 1 , ℓ 2 ) ∈ Θ 0 × Θ 0 . Similarly, if ( ℓ 1 , ℓ 2 ) ∈ Θ 0 × Θ 0 , we have ( ℓ 1 , ℓ 2 ) ∈ A 0 . Thus, we proved that Θ 0 × Θ 0 = A 0 . Likewise, we can prove that Φ 0 × Φ 0 = ℬ 0 . Since Θ 0 ≠ ∅ , then A 0 ≠ ∅ . Otherwise, from (i), we have

ℒ ( ℵ 1 ) = { ( J ( ω , ϖ ) , J ( ϖ , ω ) ) : ( ω , ϖ ) ∈ Θ 0 × Θ 0 } ⊂ J ( Θ 0 × Θ 0 ) × J ( Θ 0 × Θ 0 ) ⊆ ℬ 0 .

Suppose now that for some ( ℓ 1 , ℓ 2 ) , ( ω 1 , ω 2 ) ∈ ℵ 1 , ( ℏ 1 , ℏ 2 ) , ( ϖ 1 , ϖ 2 ) ∈ ℵ 2 , we have

σ ℱ ∕ ( ( ℓ 1 , ℓ 2 ) , ( ℏ 1 , ℏ 2 ) ) = σ ℱ ∕ ( ℵ 1 , ℵ 2 ) σ ℱ ∕ ( ( ω 1 , ω 2 ) , ( ϖ 1 , ϖ 2 ) ) = σ ℱ ∕ ( ℵ 1 , ℵ 2 ) .

This implies that

σ ℱ ( ( ℓ 1 , ℏ 1 ) ) = σ ℱ ( ( ℓ 2 , ℏ 2 ) ) = σ ℱ ( Θ , Φ ) σ ℱ ( ( ω 1 , ϖ 1 ) ) = σ ℱ ( ( ω 2 , ϖ 2 ) ) = σ ℱ ( Θ , Φ ) .

Since ( Θ , Φ ) fulfils the property P , we obtain that

σ ℱ ( ( ℓ 1 , ω 1 ) ) = σ ℱ ( ( ϖ 1 , ℏ 1 ) )

and

σ ℱ ( ( ℓ 2 , ω 2 ) ) = σ ℱ ( ( ϖ 2 , ℏ 2 ) ) ,

which implies that

σ ℱ ∕ ( ( ℓ 1 , ℓ 2 ) , ( ω 1 , ω 2 ) ) = σ ℱ ∕ ( ( ℏ 1 , ℏ 2 ) , ( ϖ 1 , ϖ 2 ) ) .

Thus, we showed that ( ℵ 1 , ℵ 2 ) fulfils the property P . Assume that for some

( ℓ 1 , ℓ 2 ) , ( ω 1 , ω 2 ) , ( u 1 , u 2 ) , ( v 1 , v 2 ) ∈ ℵ 1 ,

we have

( ℓ 1 , ℓ 2 ) S 2 ( ω 1 , ω 2 )

σ ℱ ∕ ( ( u 1 , u 2 ) , ℒ ( ℓ 1 , ℓ 2 ) ) = σ ℱ ∕ ( ℵ 1 , ℵ 2 )

σ ℱ ∕ ( ( v 1 , v 2 ) , ℒ ( ω 1 , ω 2 ) ) = σ ℱ ∕ ( ℵ 1 , ℵ 2 ) .

This implies that

ℓ 1 S ω 1 , ℓ 2 S ω 2

σ ℱ ( u 1 , J ( ℓ 1 , ℓ 2 ) ) = σ ℱ ( Θ , Φ )

σ ℱ ( v 1 , J ( ω 1 , ω 2 ) ) = σ ℱ ( Θ , Φ ) ,

and

ℓ 2 S ω 2 , ℓ 1 S ω 1

σ ℱ ( u 2 , J ( ℓ 2 , ℓ 1 ) ) = σ ℱ ( Θ , Φ )

σ ℱ ( v 2 , J ( ω 2 , ω 1 ) ) = σ ℱ ( Θ , Φ ) .

Since J is a bi-prox comp, so we have

u 1 S v 1 , u 2 S v 2 ,

that is,

( u 1 , u 2 ) S 2 ( v 1 , v 2 ) .

which shows ℒ is a prox comp. Now, from condition (iii), we obtain

σ ℱ ( ω 1 , J ( ω 0 , ϖ 0 ) ) + σ ℱ ( ϖ 1 , J ( ϖ 0 , ω 0 ) ) = 2 σ ℱ ( Θ , Φ )

and

( ω 0 , ϖ 0 ) S 2 ( ω 1 , ϖ 1 ) ,

which implies that

σ ℱ ∕ ( ( ω 1 , ϖ 1 ) , ℒ ( ω 0 , ϖ 0 ) ) = σ ℱ ∕ ( ℵ 1 , ℵ 2 )

and

( ω 0 , ϖ 0 ) S 2 ( ω 1 , ϖ 1 ) .

Moreover, if ( ω , ϖ ) , ( u , v ) ∈ ℵ 1 are such that

( ω 0 , ϖ 0 ) S 2 ( u , v ) ,

i.e., ω S u and ϖ S v , from condition (iv), we have

(17) σ ℱ ( J ( ω , ϖ ) , J ( u , v ) ) ≤ ψ σ ℱ ( ω , u ) + σ ℱ ( ϖ , v ) 2

and

(18) σ ℱ ( J ( ϖ , ω ) , J ( v , u ) ) ≤ ψ σ ℱ ( ω , u ) + σ ℱ ( ϖ , v ) 2 .

Adding (17) to (18), we obtain that

σ ℱ ∕ ( ( J ( ω , ϖ ) , J ( ϖ , ω ) ) , ( J ( u , v ) , J ( v , u ) ) ) ≤ ψ ( σ ℱ ∕ ( ( ω , ϖ ) , ( u , v ) ) ) ,

i.e.,

σ ℱ ∕ ( ℒ ( ω , ϖ ) , ℒ ( u , v ) ) ≤ ψ ( σ ℱ ∕ ( ( ω , ϖ ) , ( u , v ) ) ) .

Now, all the assumptions of Theorem 4.3 hold, and thus, we conclude then that J possess a best proximity point ( ω ∗ , ϖ ∗ ) ∈ A 0 , that is, ( ω ∗ , ϖ ∗ ) ∈ Θ 0 × Θ 0 , which satisfies

σ ℱ ∕ ( ( ω ∗ , ϖ ∗ ) , ℒ ( ω ∗ , ϖ ∗ ) ) = σ ℱ ∕ ( ℵ 1 , ℵ 2 ) .

As we already proved that

σ ℱ ∕ ( ℵ 1 , ℵ 2 ) = σ ℱ ( Θ , Φ ) ,

so the aforementioned equality implies immediately that

□ σ ℱ ( ω , J ( ω ∗ , ϖ ∗ ) ) = σ ℱ ∕ ( ϖ ∗ , J ( ϖ ∗ , ω ∗ ) ) = σ ℱ ( Θ , Φ ) .

Theorem 4.7

Let ( ℳ , σ ℱ ) is complete ℱ - metric space, Θ , Φ ∈ C l ( ℳ ) such that Θ 0 ≠ ∅ . Let R is binary relation over ℳ . Assume that the mapping J : Θ × Θ → Φ is continuous satisfying

J ( Θ 0 × Θ 0 ) ⊆ Φ 0 and ( Θ , Φ ) fulfils the property P ;
J is a bi-proximal comparative mapping,
there exist ω 0 , ϖ 0 , ω 1 , ϖ 1 ∈ Θ 0 such that
σ ℱ ( ω 1 , J ( ω 0 , ϖ 0 ) ) = σ ℱ ( ϖ 1 , J ( ϖ 0 , ω 0 ) ) = σ ℱ ( Θ , Φ ) , and ω 0 S ω 1 , ϖ 0 S ϖ 1 ,
∃ a ψ ∈ Ψ in such that
ω , ϖ , u , v ∈ Θ , ω S u , ϖ S v ⇒ σ ℱ ( J ( ω , ϖ ) , J ( u , v ) ) ≤ ψ σ ℱ ( ω , u ) + σ ℱ ( ϖ , v ) 2 ,
(H) holds.

Then J has a coupled best proximity point.

Theorem 4.8

Besides the assumptions of Theorem 4.6 (respectively Theorem 4.7), assume that this condition holds: ∀ ( ω , ϖ ) ∈ Θ × Θ , ∃ ϱ ∈ Θ 0 such that ω S ϱ and ϖ S ϱ . Then ℒ has a unique best proximity point ( ω ∗ , ϖ ∗ ) ∈ Θ × Θ . Moreover, we have ω ∗ = ϖ ∗ .

Proof

Suppose ( ω , ϖ ), ( u , v ) ∈ Θ × Θ . By the assumptions, there exists z 1 ∈ Θ 0 such that ωSϱ 1 and u S ϱ 1 . Likewise, there exists ϱ 2 ∈ Θ 0 such that ϖ S ϱ 2 and v S ϱ 2 . Hence, we obtain ( ω , ϖ ) S 2 ( ϱ 1 , ϱ 2 ) and ( u , v ) S 2 ( ϱ 1 , ϱ 2 ) , where ( ϱ 1 , ϱ 2 ) ∈ Θ 0 × Θ 0 . Now, by Theorem 4.5, we establish that ℒ has a unique best proximity point, i.e., a unique coupled best proximity point of ℒ .□

By setting Θ = Φ in Theorem 3.1, we establish this result.

Theorem 4.9

Let ( ℳ , σ ℱ ) is complete ℱ -metric space and Θ ∈ C l ( ℳ ) . Assume that ℒ : Θ → Θ satisfies the condition:

σ ℱ ( ℒ ω , ℒ ϖ ) ≤ ψ ( σ ℱ ( ω , ϖ ) )

∀ ω , ϖ ∈ Θ , where ψ ∈ Ψ . Then there exists ω ∗ ∈ Θ such that ℒ ω ∗ = ω ∗ , which is unique.

By setting ψ ( ı ) = k ı for some k ∈ ( 0 , 1 ) and ı > 0 in Theorem 4.9, we obtain the main result of Jleli and Samet [18].

Theorem 4.10

Let ( ℳ , σ ℱ ) is complete ℱ -metric space and Θ ∈ C l ( ℳ ) . Assume that ℒ : Θ → Θ satisfies the condition:

σ ℱ ( ℒ ω , ℒ ϖ ) ≤ k σ ℱ ( ω , ϖ ) ,

∀ ω , ϖ ∈ Θ , where k ∈ ( 0 , 1 ) . Then ∃ ω ∗ ∈ Θ such that ℒ ω ∗ = ω ∗ , which is unique.

Author contributions: The author has accepted responsibility for the entire content of this article and approved its submission.
Conflict of interest: The author states no conflict of interest.

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Received: 2021-12-06

Revised: 2022-10-24

Accepted: 2022-11-30

Published Online: 2023-04-12

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best proximity point; α-ψ-proximal contraction; non-self mappings; coupled best proximity point

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