huggingchat/papers-content / 2602 /2602.00645.md

|

40.5 kB

Title: Best Proximity Point Results for Perimetric Contractions

URL Source: https://arxiv.org/html/2602.00645

Markdown Content: Abstract 1Introduction 2Perimetric proximal contractions References Best Proximity Point Results for Perimetric Contractions Hiranmoy Garai1, Evgeniy Petrov2,∗, Pratikshan Mondal3, Lakshmi Kanta Dey4 1 Hiranmoy Garai, Department of Mathematics, Government General Degree College Ranibandh, Bankura, India. hiran.garai24@gmail.com 2 Evgeniy Petrov, Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Batiuka str. 19, 84116 Slovyansk, Ukraine. eugeniy.petrov@gmail.com 4 Pratikshan Mondal, Department of Mathematics, A.B.N. Seal College, Cooch Behar, West Bengal, India real.analysis77@gmail.com 3 Lakshmi Kanta Dey, Department of Mathematics, National Institute of Technology Durgapur, India. lakshmikdey@yahoo.co.in Abstract.

This paper has two aims, first one is to introduce special kind of proximal contractions guaranteeing a finite number of best proximity points, and second one is to derive best proximity point results for perimetric contractions. To meet these two aims, we introduce two new proximal contractions: perimetric proximal contractions of the first and the second kind, and derive best proximity point results for these mappings. We establish that for these particular mappings, best proximity points are not necessarily unique; however, we provide an upper bound, proving that at most two such points can exist. To establish the validity of our results, we provide illustrative examples demonstrating that these newly defined mappings can possess unique or exactly two best proximity points.

Key words and phrases: Best proximity point; perimetric proximal contractions; approximative compactness. 2020 Mathematics Subject Classification: 47 H 10 , 54 H 25 . ∗ Corresponding author

1.Introduction

Fixed point theory is an invaluable tool to find solutions of equations of the form 𝑇 𝑥

𝑥 ( 𝑇 being a self-map on a non empty set 𝐴 , say). However, if 𝑇 isn’t a self-map, specifically if the domain and co-domain of 𝑇 are disjoint, fixed point theory isn’t applicable for solving such equations. In fact, exact solutions to these equations are unavailable, so we aim for an optimal solution instead. If 𝐴 and 𝐵 are disjoint non-empty subsets of a metric space ( 𝑋 , 𝑑 ) , and 𝑇 : 𝐴 → 𝐵 is a mapping, then finding an optimal solution to 𝑇 𝑥

𝑥 means finding 𝑥 ∗ ∈ 𝐴 such that 𝑑 ( 𝑥 ∗ , 𝑇 𝑥 ∗ )

𝑑 ( 𝐴 , 𝐵 ) , where 𝑑 ( 𝐴 , 𝐵 )

inf { 𝑑 ( 𝑎 , 𝑏 ) : 𝑎 ∈ 𝐴 , 𝑏 ∈ 𝐵 } . Such a point 𝑥 ∗ ∈ 𝐴 , if it exists, satisfying 𝑑 ( 𝑥 ∗ , 𝑇 𝑥 ∗ )

𝑑 ( 𝐴 , 𝐵 ) , is referred to as a best proximity point of 𝑇 . The concept of best proximity points was initially considered by Fan [9] and Reich [18, 19], and subsequently explored extensively by mathematicians in various directions throughout the last few years, see [10, 11, 5, 4, 14, 13]. Among these various directions, most involve different types of contraction conditions, some of which have been used previously in fixed point theory. The various contraction conditions specifically developed for best proximity point results include many interesting contractions. Notably, the proximal contractions introduced by Sadiq Basha [3] are among the most significant ones.

Definition 1.1.

([3, p. 1774, Definition 2.2]) Let 𝐴 and 𝐵 be disjoint non-empty subsets of a metric space ( 𝑋 , 𝑑 ) . A mapping 𝑇 : 𝐴 → 𝐵 is said to be a proximal contraction of the first kind if there exists a constant 𝛼 ∈ [ 0 , 1 ) such that for all 𝑢 1 , 𝑢 2 , 𝑥 1 , 𝑥 2 ∈ 𝐴 , the following hold:

𝑑 ( 𝑢 1 , 𝑇 𝑥 1 )

𝑑 ( 𝐴 , 𝐵 )

𝑑 ( 𝑢 2 , 𝑇 𝑥 2 )

𝑑 ( 𝐴 , 𝐵 ) } ⟹ 𝑑 ( 𝑢 1 , 𝑢 2 ) ≤ 𝛼 𝑑 ( 𝑥 1 , 𝑥 2 ) .

Definition 1.2.

([3, p. 1774, Definition 2.4]) Let 𝐴 and 𝐵 be disjoint non-empty subsets of a metric space ( 𝑋 , 𝑑 ) . A mapping 𝑇 : 𝐴 → 𝐵 is said to be a proximal contraction of the second kind if there exists a constant 𝛼 ∈ [ 0 , 1 ) such that for all 𝑢 1 , 𝑢 2 , 𝑥 1 , 𝑥 2 ∈ 𝐴 , the following hold:

𝑑 ( 𝑢 1 , 𝑇 𝑥 1 )

𝑑 ( 𝐴 , 𝐵 )

𝑑 ( 𝑢 2 , 𝑇 𝑥 2 )

𝑑 ( 𝐴 , 𝐵 ) } ⟹ 𝑑 ( 𝑇 𝑢 1 , 𝑇 𝑢 2 ) ≤ 𝛼 𝑑 ( 𝑇 𝑥 1 , 𝑇 𝑥 2 ) .

The key best proximity point findings concerning these proximal contractions as follows, with definitions of 𝐴 0 and 𝐵 0 recalled beforehand:

For a metric space ( 𝑋 , 𝑑 ) and two non-empty subsets 𝐴 , 𝐵 of 𝑋 , we use the notations 𝐴 0 and 𝐵 0 as follows:

𝐴 0

{ 𝑥 ∈ 𝐴 : 𝑑 ( 𝑥 , 𝑦 )

𝑑 ( 𝐴 , 𝐵 ) for some 𝑦 ∈ 𝐵 }

𝐵 0

{ 𝑦 ∈ 𝐵 : 𝑑 ( 𝑥 , 𝑦 )

𝑑 ( 𝐴 , 𝐵 ) for some 𝑥 ∈ 𝐴 } .

Theorem 1.3.

([3, p. 1778, Corollary 3.4]) If ( 𝑋 , 𝑑 ) is complete; 𝐴 , 𝐵 are closed, 𝐴 0 , 𝐵 0 are non-empty; 𝑇 ( 𝐴 0 ) ⊆ 𝐵 0 and 𝑇 is a continuous proximal contraction of the first kind, then 𝑇 has a unique best proximity point in 𝐴 .

Theorem 1.4.

([3, p. 1777, Corollary 3.2]) If ( 𝑋 , 𝑑 ) is complete; 𝐴 , 𝐵 are closed, 𝐴 0 , 𝐵 0 are non-empty; 𝐴 is approximatively compact with respect to 𝐵 ; 𝑇 ( 𝐴 0 ) ⊆ 𝐵 0 and 𝑇 is a continuous proximal contraction of the second kind, then 𝑇 has a unique best proximity point in 𝐴 .

Following Basha’s pioneering work, proximal contractions have been explored in various ways (see [5, 8, 12]). It’s noteworthy that most best proximity point results conclude with the existence of either a unique or infinitely many best proximity points.

On the other hand, a recent notable development in fixed point theory is the concept of mappings that contract perimeters of triangles, introduced by Petrov [15] in 2023 .

Definition 1.5.

([15, p. 2, Definition 2.1]) Let ( 𝑋 , 𝑑 ) be a metric space with at least three points. Then 𝑇 : 𝑋 → 𝑋 is said to be a mapping contracting perimeters of triangles on 𝑋 if there exists 𝛼 ∈ [ 0 , 1 ) such that the inequality

𝑑 ( 𝑇 𝑥 , 𝑇 𝑦 ) + 𝑑 ( 𝑇 𝑦 , 𝑇 𝑧 ) + 𝑑 ( 𝑇 𝑧 , 𝑇 𝑥 ) ≤ 𝛼 ( 𝑑 ( 𝑥 , 𝑦 ) + 𝑑 ( 𝑦 , 𝑧 ) + 𝑑 ( 𝑧 , 𝑥 ) )

(1.1)

holds for all three pairwise distinct points 𝑥 , 𝑦 , 𝑧 ∈ 𝑋 .

This pioneering approach is complemented by a foundational fixed point existence result, which goes as follows:

Theorem 1.6.

([15, p. 3, Theorem 2.4]) Let ( 𝑋 , 𝑑 ) , | 𝑋 | ≥ 3 , be a complete metric space and let 𝑇 : 𝑋 → 𝑋 be a mapping contracting perimeters of triangle on 𝑋 . Then 𝑇 has a fixed point in 𝑋 if and only if 𝑇 does not possess any periodic points of prime period 2 . The number of fixed points is at most two.

This area has been further explored by mathematicians in various ways (see [7, 17, 16, 6, 1, 2]), though each effort is limited to fixed points only. A striking aspect of these new mappings is that they often yield a finite number of fixed points, unlike most existing results which focus on unique or infinite fixed points.

These facts raise two questions. First, why are perimeter-contracting mappings restricted to fixed points, and not extended to best proximity points? The second question is: can we have best proximity point results with a finite number of best proximity points? With these questions in mind, we explore the perimetric counterparts of Sadiq Basha’s proximal contractions, seeking answers to both. More precisely, we propose two kinds of perimetric proximal contractions: perimetric proximal contraction of the first kind and perimetric proximal contraction of the second kind. After this, we obtain two results guaranteeing a finite number of best proximity points, which aren’t necessarily unique- in fact, we show the number of best proximity points is at most two. These results are accompanied by examples to illustrate their validity.

2.Perimetric proximal contractions

Here, we define perimetric proximal contractions of the first and the second kind formally.

Definition 2.1.

Let 𝐴 , 𝐵 be two non-empty subsets of a metric space ( 𝑋 , 𝑑 ) , and let 𝑇 : 𝐴 → 𝐵 be a mapping. Then 𝑇 is called a perimetric proximal contraction of the first kind if there exists 𝛼 ∈ [ 0 , 1 ) such that for all 𝑢 1 , 𝑢 2 , 𝑢 3 , 𝑥 1 , 𝑥 2 , 𝑥 3 ∈ 𝐴 with 𝑥 1 , 𝑥 2 , 𝑥 3 are pairwise distinct, the following hold:

𝑑 ( 𝑢 1 , 𝑇 𝑥 1 )

In the following example we show that a perimetric proximal contraction of the first kind is not obligatory proximal contraction of the first kind.

Example 2.3.

Consider a metric space ( ℝ , 𝑑 ) , where 𝑑 is the Euclidean distance on ℝ . Let 𝐴

{ − 3 , 0 , 3 , 4 } and 𝐵

[ − 2 , − 1 ] ∪ [ 1 , 2 ] . Let also

𝑇 ( − 3 )

− 2 , 𝑇 ( 0 )

3 2 , 𝑇 ( 3 )

2 , 𝑇 ( 4 )

The definition requires the inequality to hold for 𝑥 1 , 𝑥 2 , 𝑥 3 that are pairwise distinct. Since there are exactly three valid 𝑥 values in our pairs { − 3 , 3 , 4 } , there is only one possible set of points { 𝑢 1 , 𝑢 2 , 𝑢 3 } (up to permutation) that satisfies the necessary condition. Specifically:

{ 𝑢 1 , 𝑢 2 , 𝑢 3 }

{ − 3 , 3 , 0 } and { 𝑥 1 , 𝑥 2 , 𝑥 3 }

{ − 3 , 3 , 4 } .

Further,

𝑃 ( 𝑢 1 , 𝑢 2 , 𝑢 3 )

𝑑 ( 𝑢 1 , 𝑢 2 ) + 𝑑 ( 𝑢 2 , 𝑢 3 ) + 𝑑 ( 𝑢 3 , 𝑢 1 )

𝑑 ( − 3 , 3 ) + 𝑑 ( 3 , 0 ) + 𝑑 ( 0 , − 3 )

6 + 3 + 3

12 ,

and

𝑃 ( 𝑥 1 , 𝑥 2 , 𝑥 3 )

𝑑 ( 𝑥 1 , 𝑥 2 ) + 𝑑 ( 𝑥 2 , 𝑥 3 ) + 𝑑 ( 𝑥 3 , 𝑥 1 )

𝑑 ( − 3 , 3 ) + 𝑑 ( 3 , 4 ) + 𝑑 ( 4 , − 3 )

6 + 1 + 7

14 .

Clearly, the inequality in Definition 2.1 holds for all 𝛼 ∈ [ 6 7 , 1 ) . Thus, 𝑇 is a perimetric proximal contraction of the first kind. Also, it is easy to see that 𝐴 0

{ − 3 , 0 , 3 } , 𝐵 0

{ − 2 , − 1 , 1 , 2 } , 𝑇 ( 𝐴 0 )

{ − 2 , 1.5 , 2 } and 𝑇 ( 𝐴 0 ) ⊈ 𝐵 0 . We remark that 𝑇 has two best proximity points − 3 and 3 .

Since 𝑑 ( − 3 , 𝑇 ( − 3 ) )

1 and 𝑑 ( 3 , 𝑇 ( 3 ) )

1 but 𝑑 ( − 3 , 3 )

𝑑 ( − 3 , 3 ) , by Definition 1.1 we have that 𝑇 is not a proximal contraction of the first kind.

Moving on, we present our main results on best proximity points. First, we formulate a special condition for a mapping 𝑇 , which will be crucial for our findings. We designate this special condition as Condition Λ .

Condition Λ : Let 𝐴 , 𝐵 be two non-empty subsets of a metric space ( 𝑋 , 𝑑 ) and let 𝑇 : 𝐴 → 𝐵 be a mapping. Then, 𝑇 is said to satisfy Condition Λ if there does not exist any pair of points ( 𝑥 , 𝑦 ) where 𝑥 ≠ 𝑦 , of 𝐴 such that 𝑑 ( 𝑥 , 𝑇 𝑦 )

𝑑 ( 𝑦 , 𝑇 𝑥 )

𝑑 ( 𝐴 , 𝐵 ) .

We now present our first result.

Theorem 2.4.

Suppose that ( 𝑋 , 𝑑 ) is a complete metric space and 𝐴 , 𝐵 are closed subsets of 𝑋 . Let | 𝐴 0 | ≥ 3 . Let 𝑇 : 𝐴 → 𝐵 be continuous perimetric proximal contraction of the first kind such that 𝑇 ( 𝐴 0 ) ⊆ 𝐵 0 . Then if 𝑇 satisfies Condition Λ , 𝑇 possesses a best proximity point. Moreover, the best proximity points of 𝑇 are at most two in number.

Proof.

Since | 𝐴 0 | ≥ 3 , we can choose an element 𝑢 0 ∈ 𝐴 0 . Then, 𝑇 𝑢 0 ∈ 𝑇 ( 𝐴 0 ) ⊆ 𝐵 0 , so there exists 𝑢 1 ∈ 𝐴 0 such that 𝑑 ( 𝑢 1 , 𝑇 𝑢 0 )

𝑑 ( 𝐴 , 𝐵 ) . Again, 𝑇 𝑢 1 ∈ 𝑇 ( 𝐴 0 ) ⊆ 𝐵 0 , so there exists 𝑢 2 ∈ 𝐴 0 such that 𝑑 ( 𝑢 2 , 𝑇 𝑢 1 )

𝑑 ( 𝐴 , 𝐵 ) . Continuing in this way, we obtain a sequence { 𝑢 𝑛 } having the properties that 𝑢 𝑛 ∈ 𝐴 0 and 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝐴 , 𝐵 ) for all 𝑛 ∈ ℕ .

If 𝑢 𝑛

𝑢 𝑛 + 1 for some 𝑛 ∈ ℕ , it follows that 𝑢 𝑛 is a best proximity point of 𝑇 , as 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝑢 𝑛 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝐴 , 𝐵 ) .

Next, we assume that 𝑢 𝑛 ≠ 𝑢 𝑛 + 1 for all 𝑛 ∈ ℕ . Under this assumption, we claim that 𝑢 𝑛 , 𝑢 𝑛 + 1 , 𝑢 𝑛 + 2 are pairwise distinct for all 𝑛 ∈ ℕ . If not, then we must have 𝑢 𝑛

𝑢 𝑛 + 2 for some 𝑛 ∈ ℕ . Then 𝑑 ( 𝑢 𝑛 , 𝑇 𝑢 𝑛 + 1 )

𝑑 ( 𝑢 𝑛 + 2 , 𝑇 𝑢 𝑛 + 1 )

𝑑 ( 𝐴 , 𝐵 ) and also ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝐴 , 𝐵 ) . Hence, 𝑇 does not satisfy Condition Λ , contradicting the assumption made earlier. Thus, any three consecutive terms of { 𝑢 𝑛 } are pairwise distinct.

Now, since 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝐴 , 𝐵 )

𝑑 ( 𝑢 𝑛 + 2 , 𝑇 𝑢 𝑛 + 1 )

𝑑 ( 𝑢 𝑛 + 3 , 𝑇 𝑢 𝑛 + 2 ) , we have

𝑑 ( 𝑢 𝑛 + 1 , 𝑢 𝑛 + 2 ) + 𝑑 ( 𝑢 𝑛 + 2 , 𝑢 𝑛 + 3 ) +

𝑑 ( 𝑢 𝑛 + 3 , 𝑢 𝑛 + 1 ) ≤

𝛼 { 𝑑 ( 𝑢 𝑛 , 𝑢 𝑛 + 1 ) + 𝑑 ( 𝑢 𝑛 + 1 , 𝑢 𝑛 + 2 ) + 𝑑 ( 𝑢 𝑛 + 2 , 𝑢 𝑛 ) } .

(2.1)

Let 𝑡 𝑛

𝑑 ( 𝑢 𝑛 , 𝑢 𝑛 + 1 ) + 𝑑 ( 𝑢 𝑛 + 1 , 𝑢 𝑛 + 2 ) + 𝑑 ( 𝑢 𝑛 + 2 , 𝑢 𝑛 ) . Then from (2.1), we get

𝑡 𝑛 + 1 ≤ 𝛼 𝑡 𝑛 for all 𝑛 ∈ ℕ .

(2.2)

Further, from (2.2), we can write

𝑡 1

𝑡 2

⋯

𝑡 𝑛

⋯ .

(2.3)

If all terms of { 𝑢 𝑛 } are not distinct, then there is a minimal natural number 𝑗 ( ≥ 3 ) such that 𝑢 𝑗

𝑢 𝑖 for some 𝑖 ∈ ℕ with 0 ≤ 𝑖 ≤ 𝑗 − 2 . This leads to 𝑢 𝑗 + 1

𝑢 𝑖 + 1 , 𝑢 𝑗 + 2

𝑢 𝑖 + 2 , ⋯ , and so, 𝑡 𝑗

𝑡 𝑖 , contradicting (2.3). Hence, all terms of { 𝑢 𝑛 } are distinct. From (2.2), it follows that

𝑡 𝑛 + 1 ≤ 𝛼 𝑡 𝑛 ≤ 𝛼 2 𝑡 𝑛 − 1 ≤ ⋯ ≤ 𝛼 𝑛 𝑡 1 ,

which yields 𝑑 ( 𝑢 𝑛 + 1 , 𝑢 𝑛 + 2 ) ≤ 𝑡 𝑛 + 1 ≤ 𝛼 𝑛 𝑡 1 .

With 0 ≤ 𝛼 < 1 , { 𝑢 𝑛 } is a Cauchy sequence in 𝐴 , a closed subset of the complete metric space ( 𝑋 , 𝑑 ) , so there exists 𝑢 ∈ 𝐴 such that 𝑢 𝑛 → 𝑢 as 𝑛 → ∞ . Given that 𝑇 is continuous, we have 𝑇 𝑢 𝑛 → 𝑇 𝑢 , and consequently, 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 ) → 𝑑 ( 𝑢 , 𝑇 𝑢 ) as 𝑛 → ∞ . Thus, we have 𝑑 ( 𝑢 , 𝑇 𝑢 )

𝑑 ( 𝐴 , 𝐵 ) , implying that 𝑢 is a best proximity point of 𝑇 .

For the last part of the proof, we assume that 𝑇 has three distinct (which are pairwise distinct as well) best proximity points 𝑥 , 𝑦 , 𝑧 ∈ 𝐴 . Then, we have 𝑑 ( 𝑥 , 𝑇 𝑥 )

𝑑 ( 𝐴 , 𝐵 )

𝑑 ( 𝑦 , 𝑇 𝑦 )

𝑑 ( 𝑧 , 𝑇 𝑧 ) and so,

𝑑 ( 𝑥 , 𝑦 ) + 𝑑 ( 𝑦 , 𝑧 ) + 𝑑 ( 𝑧 , 𝑥 )

≤ 𝛼 { 𝑑 ( 𝑥 , 𝑦 ) + 𝑑 ( 𝑦 , 𝑧 ) + 𝑑 ( 𝑧 , 𝑥 ) }

< 𝑑 ( 𝑥 , 𝑦 ) + 𝑑 ( 𝑦 , 𝑧 ) + 𝑑 ( 𝑧 , 𝑥 ) ,

leading to a contradiction, and thus the proof is concluded. ∎

From Theorem 2.4, we have the following corollary:

Corollary 2.5.

The sufficiency holds in Theorem 1.6.

Proof.

Indeed, let us set 𝐴

𝐵

𝑋 in Theorem 2.4. Then it follows from Definition 2.2 that 𝑑 ( 𝐴 , 𝐵 )

0 , 𝑢 1

𝑇 𝑥 1 , 𝑢 2

𝑇 𝑥 2 , 𝑢 3

𝑇 𝑥 3 and, hence, 𝑇 is a mapping contracting the perimeters of triangles. The completeness of the space 𝑋 ensures that every Cauchy sequence converges to a limit that belongs to the set 𝑋 . In the same way, the closedness of 𝐴 and 𝐵 provided similar property for these subsets. Clearly, 𝐴 0

𝐵 0

𝑋 and the inclusion 𝑇 ( 𝐴 0 ) ⊆ 𝐵 0 holds. One can easily see that in the case 𝐴

𝐵

𝑋 , Condition Λ is equivalent to the condition that there is no periodic point of prime period 2 and the existence of a best proximity point is equivalent to the existence of a fixed point for the mapping 𝑇 . It remains only to note that any mapping contracting the perimeters of triangles is continuous ([15, p. 2, Proposition 2.3]). ∎

Our next result is for perimetric proximal contraction of the second kind. Before this, we recall the definition of approximatively compact set.

Definition 2.6.

([3, p. 1774, Definition 2.1]) Let 𝐴 , 𝐵 be two non-empty subsets of a metric space ( 𝑋 , 𝑑 ) . Then 𝐴 is said to be approximatively compact with respect to 𝐵 if every sequence { 𝑢 𝑛 } of 𝐴 satisfying the condition 𝑑 ( 𝑢 , 𝑢 𝑛 ) → 𝑑 ( 𝑢 , 𝐴 ) for some 𝑢 ∈ 𝐵 , has a convergent subsequence.

Theorem 2.7.

The first conclusion of Theorem 2.4 remains valid if 𝑇 is a perimetric proximal contraction of the second kind instead of the first kind with other conditions of Theorem 2.4, and 𝐴 is approximatively compact with respect to 𝐵 . Further, 𝑇 has at most two best proximity points when it is injective.

Proof.

Following the proof of Theorem 2.4, we construct a sequence { 𝑢 𝑛 } in 𝐴 such that 𝑢 𝑛 ∈ 𝐴 0 , 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝐴 , 𝐵 ) for all 𝑛 ∈ ℕ . If 𝑢 𝑛

𝑢 𝑛 + 1 for some 𝑛 , then 𝑢 𝑛 is a best proximity point of 𝑇 . So, we assume that 𝑢 𝑛 ≠ 𝑢 𝑛 + 1 for all 𝑛 ∈ ℕ , and we can then show, as before, that any three consecutive terms of { 𝑢 𝑛 } are pairwise distinct. If 𝑇 𝑢 𝑛

𝑇 𝑢 𝑛 + 1 for some 𝑛 , then we have 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 + 1 )

𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝐴 , 𝐵 ) , showing that 𝑢 𝑛 + 1 is a best proximity point of 𝑇 . We therefore assume that 𝑇 𝑢 𝑛 ≠ 𝑇 𝑢 𝑛 + 1 for every 𝑛 . Again, if 𝑇 𝑢 𝑛

𝑇 𝑢 𝑛 + 2 for some 𝑛 , then we have 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 + 2 )

𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝐴 , 𝐵 ) and also, we have 𝑑 ( 𝑢 𝑛 + 2 , 𝑇 𝑢 𝑛 + 1 )

𝑑 ( 𝐴 , 𝐵 ) , showing that 𝑇 does not satisfy Condition Λ which leads to a contradiction. Hence, any three consecutive terms of { 𝑇 𝑢 𝑛 } are pairwise distinct.

Following the lines of proof of Theorem 2.4, we can show that { 𝑇 𝑢 𝑛 } is a Cauchy sequence in the closed subset 𝐵 of 𝑋 , and hence there exists 𝑣 ∈ 𝐵 such that 𝑇 𝑢 𝑛 → 𝑣 as 𝑛 → ∞ . Then, we have

𝑑 ( 𝑣 , 𝐴 ) ≤ 𝑑 ( 𝑣 , 𝑢 𝑛 + 1 )
≤ 𝑑 ( 𝑣 , 𝑇 𝑢 𝑛 ) + 𝑑 ( 𝑇 𝑢 𝑛 , 𝑢 𝑛 + 1 )

𝑑 ( 𝑣 , 𝑇 𝑢 𝑛 ) + 𝑑 ( 𝐴 , 𝐵 )

≤ 𝑑 ( 𝑣 , 𝑇 𝑢 𝑛 ) + 𝑑 ( 𝑣 , 𝐴 ) .

Using the fact that lim 𝑛 → ∞ 𝑑 ( 𝑣 , 𝑇 𝑢 𝑛 )

0 , we obtain lim 𝑛 → ∞ 𝑑 ( 𝑣 , 𝑢 𝑛 + 1 )

𝑑 ( 𝑣 , 𝐴 ) . As 𝐴 is approximatively compact with respect to 𝐵 , { 𝑢 𝑛 } has a convergent subsequence { 𝑢 𝑛 ( 𝑘 ) } converging to 𝑢 ∈ 𝐴 . Then

𝑑 ( 𝑢 , 𝑣 )

lim 𝑘 → ∞ 𝑑 ( 𝑢 𝑛 ( 𝑘 ) , 𝑇 𝑢 𝑛 ( 𝑘 ) − 1 )

𝑑 ( 𝐴 , 𝐵 ) ,

(2.4)

and so 𝑢 ∈ 𝐴 0 . Further, since 𝑇 is continuous, we have lim 𝑘 → ∞ 𝑇 𝑢 𝑛 ( 𝑘 )

𝑇 𝑢 , and so 𝑇 𝑢

𝑣 . Thus from (2.4), we get 𝑑 ( 𝑢 , 𝑇 𝑢 )

𝑑 ( 𝐴 , 𝐵 ) , showing that 𝑢 is a best proximity point of 𝑇 .

Suppose that 𝑇 is injective. We show that 𝑇 has at most two best proximity points. Assume, if possible, that 𝑇 has three distinct best proximity points 𝑥 , 𝑦 , 𝑧 ∈ 𝐴 . By injectivity of 𝑇 , the points 𝑇 𝑥 , 𝑇 𝑦 and 𝑇 𝑧 are pairwise distinct. Then from Definition 2.2, we get

𝑑 ( 𝑇 𝑥 , 𝑇 𝑦 ) + 𝑑 ( 𝑇 𝑦 , 𝑇 𝑧 ) + 𝑑 ( 𝑇 𝑧 , 𝑇 𝑥 )

≤ 𝛼 { 𝑑 ( 𝑇 𝑥 , 𝑇 𝑦 ) + 𝑑 ( 𝑇 𝑦 , 𝑇 𝑧 ) + 𝑑 ( 𝑇 𝑧 , 𝑇 𝑥 ) }

< 𝑑 ( 𝑇 𝑥 , 𝑇 𝑦 ) + 𝑑 ( 𝑇 𝑦 , 𝑇 𝑧 ) + 𝑑 ( 𝑇 𝑧 , 𝑇 𝑥 ) ,

resulting in a contradiction. Hence, 𝑇 has at most two best proximity points. ∎

Corollary 2.8.

Let ( 𝑋 , 𝑑 ) , | 𝑋 | ⩾ 3 , be a complete metric space and let 𝑇 : 𝑋 → 𝑋 be a continuous mapping such that the inequality

𝑑 ( 𝑇 2 𝑥 , 𝑇 2 𝑦 ) + 𝑑 ( 𝑇 2 𝑦 , 𝑇 2 𝑧 ) + 𝑑 ( 𝑇 2 𝑧 , 𝑇 2 𝑥 ) ⩽ 𝛼 { 𝑑 ( 𝑇 𝑥 , 𝑇 𝑦 ) + 𝑑 ( 𝑇 𝑦 , 𝑇 𝑧 ) + 𝑑 ( 𝑇 𝑧 , 𝑇 𝑥 ) }

holds for all 𝑥 , 𝑦 , 𝑧 ∈ 𝑋 with 𝑇 𝑥 , 𝑇 𝑦 , 𝑇 𝑧 being pairwise distinct and some 0 ⩽ 𝛼 < 1 . Suppose also that 𝑇 does not possess any periodic points of prime period 2 . Then 𝑇 has a fixed point in 𝑋 . The number of fixed points is at most two.

Proof.

The proof is similar to the proof of Corollary 2.5. ∎

Example 2.9.

Consider a metric space ( ℝ 2 , 𝑑 ) , where 𝑑 is the Euclidean metric on ℝ 2 . Let 𝐴

{ 𝑝 , 𝑞 , 𝑟 , 𝑠 } , 𝐵

{ 𝑎 , 𝑏 , 𝑐 } , where 𝑎

( 0 , 0 ) , 𝑏

( 0 , 1 ) , 𝑐

( 0 , 2 ) , 𝑝

( 1 , 2 ) , 𝑞

( 1 , 1 ) , 𝑟

( 1 , 0 ) , 𝑠

( 2 , 0 ) , and let

𝑇 ( 𝑝 )

𝑐 , 𝑇 ( 𝑞 )

𝑏 , 𝑇 ( 𝑟 )

𝑏 , 𝑇 ( 𝑠 )

𝑎 .

It is clear that 𝑑 ( 𝐴 , 𝐵 )

1 𝑑 ( 𝑎 , 𝑟 )

𝑑 ( 𝑏 , 𝑞 )

𝑑 ( 𝑐 , 𝑝 ) , 𝐴 0

{ 𝑝 , 𝑞 , 𝑟 } , 𝐵 0

{ 𝑎 , 𝑏 , 𝑐 } .

{ 𝑝 , 𝑞 , 𝑟 } with { 𝑥 1 , 𝑥 2 , 𝑥 3 }

{ 𝑝 , 𝑞 , 𝑠 } ,

{ 𝑢 1 , 𝑢 2 , 𝑢 3 }

{ 𝑞 , 𝑞 , 𝑟 } with { 𝑥 1 , 𝑥 2 , 𝑥 3 }

{ 𝑞 , 𝑟 , 𝑠 } ,

{ 𝑢 1 , 𝑢 2 , 𝑢 3 }

{ 𝑝 , 𝑞 , 𝑞 } with { 𝑥 1 , 𝑥 2 , 𝑥 3 }

{ 𝑝 , 𝑞 , 𝑟 } ,

and

{ 𝑢 1 , 𝑢 2 , 𝑢 3 }

{ 𝑝 , 𝑞 , 𝑟 } with { 𝑥 1 , 𝑥 2 , 𝑥 3 }

{ 𝑝 , 𝑟 , 𝑠 } .

For all 𝑥 , 𝑦 , 𝑧 ∈ 𝑋 set 𝑃 ( 𝑥 , 𝑦 , 𝑧 )

Clearly, in each case the inequality in Definition 2.1 holds with some 𝛼 < 1 .

Thus, 𝑇 is a perimetric proximal contraction of the first kind. Additionally, from the form of 𝑇 , it is evident that there are no distinct points 𝑥 , 𝑦 ∈ 𝐴 for which the equalities 𝑑 ( 𝑥 , 𝑇 𝑦 )

𝑑 ( 𝐴 , 𝐵 )

𝑑 ( 𝑦 , 𝑇 𝑥 ) hold, which ensures that 𝑇 satisfies Condition Λ . Also, it is easy to see that 𝑇 ( 𝐴 0 )

{ 𝑏 , 𝑐 } ⊆ 𝐵 0 . Accordingly, all the assumptions of Theorem 2.4 hold, so by virtue of Theorem 2.4, 𝑇 has best proximity points, limited to a maximum of two. We remark that 𝑇 has two best proximity points 𝑝 and 𝑞 .

Example 2.10.

Consider the complete metric space ( ℝ , 𝑑 ) , where 𝑑 is the Euclidean metric on ℝ . Let us choose 𝐴

{ 𝑥 ∈ ℕ : 𝑥

7 + 4 𝑛 , 𝑛

0 , 1 , 2 , 3 , ⋯ } and 𝐵

{ 𝑥 ∈ ℕ : 𝑥

2 𝑛 , 𝑛

1 , 2 , 3 , ⋯ } . Then 𝑑 ( 𝐴 , 𝐵 )

1 and 𝐴 0

𝐴 and 𝐵 0

{ 6 , 8 , 10 , ⋯ } . Let us define 𝑇 : 𝐴 → 𝐵 by

𝑇 𝑥

{ 6 , if 𝑥

7

12 , if 𝑥

11

2 𝑛 + 2 , if 𝑥

7 + 4 𝑛 , 𝑛 ≥ 2 .

Let 𝑢 1 , 𝑢 2 , 𝑢 3 , 𝑥 1 , 𝑥 2 , 𝑥 3 ∈ 𝐴 where 𝑥 1 , 𝑥 2 , 𝑥 3 are pairwise distinct with 𝑑 ( 𝑢 1 , 𝑇 𝑥 1 )

𝑑 ( 𝐴 , 𝐵 )

𝑑 ( 𝑢 2 , 𝑇 𝑥 2 )

𝑑 ( 𝑢 3 , 𝑇 𝑥 3 ) . Then, we have the following possibilities:

Case I: 𝑢 1

𝑥 1

7 ; 𝑢 2

𝑥 2

11 ; and 𝑥 3 ≠ 7 , 11 .

Then, 𝑥 3

7 + 4 𝑥 3 ′ where 𝑥 3 ′ ∈ ℕ with 𝑥 3 ′ ≥ 2 . Then, the relation 𝑑 ( 𝑢 3 , 𝑇 𝑥 3 )

𝑑 ( 𝐴 , 𝐵 ) yields either 𝑢 3

2 𝑥 3 ′ + 1 or 𝑢 3

2 𝑥 3 ′ + 3 . Then,

𝑑 ( 𝑢 1 , 𝑢 2 ) + 𝑑 ( 𝑢 2 , 𝑢 3 ) + 𝑑 ( 𝑢 3 , 𝑢 1 )

4 + | 11 − 𝑢 3 | + | 𝑢 3 − 7 |

4 + | 𝑢 3 − 11 | + | 𝑢 3 − 7 |

≤ 4 + max ⁡ { | 2 𝑢 3 − 18 | , 4 }

{ 4 + max ⁡ { | 4 𝑥 3 ′ − 16 | , 4 } if 𝑢 3

2 𝑥 3 ′ + 1

4 + max ⁡ { | 4 𝑥 3 ′ − 12 | , 4 } if 𝑢 3

2 𝑥 3 ′ + 3 ,

and

𝑑 ( 𝑥 1 , 𝑥 2 ) + 𝑑 ( 𝑥 2 , 𝑥 3 ) + 𝑑 ( 𝑥 3 , 𝑥 1 )

4 + | 11 − 𝑥 3 | + | 𝑥 3 − 7 |

4 + | 4 − 4 𝑥 3 ′ | + 4 𝑥 3 ′

8 𝑥 3 ′ .

Hence, the inequality

𝑑 ( 𝑢 1 , 𝑢 2 ) + 𝑑 ( 𝑢 2 , 𝑢 3 ) + 𝑑 ( 𝑢 3 , 𝑢 1 ) ≤ 𝛼 ( 𝑑 ( 𝑥 1 , 𝑥 2 ) + 𝑑 ( 𝑥 2 , 𝑥 3 ) + 𝑑 ( 𝑥 3 , 𝑥 1 ) )

(2.5)

holds for 𝛼

2 3 .

Case II: 𝑢 1

𝑥 1

7 ; and 𝑥 2 , 𝑥 3 ≠ 7 , 11 .

Then, 𝑥 2

7 + 4 𝑥 2 ′ , 𝑥 3

7 + 4 𝑥 3 ′ where 𝑥 2 ′ , 𝑥 3 ′ ∈ ℕ with 𝑥 2 ′ , 𝑥 3 ′ ≥ 2 . Then the relations 𝑑 ( 𝑢 2 , 𝑇 𝑥 2 )

𝑑 ( 𝐴 , 𝐵 ) and 𝑑 ( 𝑢 3 , 𝑇 𝑥 3 )

𝑑 ( 𝐴 , 𝐵 ) yield either 𝑢 2

2 𝑥 2 ′ + 1 or 𝑢 2

2 𝑥 2 ′ + 3 ; and either 𝑢 3

2 𝑥 3 ′ + 1 or 𝑢 3

2 𝑥 3 ′ + 3 . In any possibilities, we always have 𝑢 2 , 𝑢 3 ≥ 7 . Without loss of generality, assume that 𝑥 3

𝑥 2 , then 𝑢 3 ≥ 𝑢 2 ≥ 𝑢 1 and 𝑥 3 ′

𝑥 2 ′ ≥ 2 . Hence,

𝑑 ( 𝑢 1 , 𝑢 2 ) + 𝑑 ( 𝑢 2 , 𝑢 3 ) + 𝑑 ( 𝑢 3 , 𝑢 1 ) 𝑑 ( 𝑥 1 , 𝑥 2 ) + 𝑑 ( 𝑥 2 , 𝑥 3 ) + 𝑑 ( 𝑥 3 , 𝑥 1 )

2 ( 𝑢 3 − 7 ) 2 ( 𝑥 3 − 7 )

𝑢 3 − 7 𝑥 3 − 7

𝑢 3 − 7 4 𝑥 3 ′

{ 2 𝑥 3 ′ − 6 4 𝑥 3 ′ if 𝑢 3

2 𝑥 3 ′ + 1

2 𝑥 3 ′ − 4 4 𝑥 3 ′ if 𝑢 3

2 𝑥 3 ′ + 3 ,

≤ 1 2 − 1 𝑥 3 ′ .

Hence, inequality (2.5) holds for 𝛼

2 3 .

Case III: 𝑢 1

𝑥 1

11 ; and 𝑥 2 , 𝑥 3 ≠ 7 , 11 .

Then similar to Case II, we have 𝑥 2

7 + 4 𝑥 2 ′ , 𝑥 3

7 + 4 𝑥 3 ′ where 𝑥 2 ′ , 𝑥 3 ′ ∈ ℕ with 𝑥 2 ′ , 𝑥 3 ′ ≥ 2 , and these together with the relations 𝑑 ( 𝑢 2 , 𝑇 𝑥 2 )

𝑑 ( 𝐴 , 𝐵 ) and 𝑑 ( 𝑢 3 , 𝑇 𝑥 3 )

𝑑 ( 𝐴 , 𝐵 ) imply either 𝑢 2

2 𝑥 2 ′ + 1 or 𝑢 2

2 𝑥 2 ′ + 3 ; and either 𝑢 3

2 𝑥 3 ′ + 1 or 𝑢 3

2 𝑥 3 ′ + 3 . Without loss of generality, assume that 𝑥 3

𝑥 2 , then 𝑢 3 ≥ 𝑢 2 and 𝑥 3 ′

𝑥 2 ′ ≥ 2 . Hence,

𝑑 ( 𝑢 1 , 𝑢 2 ) + 𝑑 ( 𝑢 2 , 𝑢 3 ) + 𝑑 ( 𝑢 3 , 𝑢 1 )

| 𝑢 2 − 11 | + | 𝑢 2 − 𝑢 3 | + | 11 − 𝑢 3 |

| 𝑢 2 − 𝑢 3 | + | 11 − 𝑢 2 | + | 11 − 𝑢 3 |

≤ 𝑢 3 − 𝑢 2 + max ⁡ { | 𝑢 2 + 𝑢 3 − 22 | , 𝑢 3 − 𝑢 2 }

and, similarly,

𝑑 ( 𝑥 1 , 𝑥 2 ) + 𝑑 ( 𝑥 2 , 𝑥 3 ) + 𝑑 ( 𝑥 3 , 𝑥 1 )

2 | 𝑥 3 − 𝑥 1 |

4 𝑥 3 ′ − 4 .

Analogously we can show inequality (2.5) for 𝛼

2 3 .

Case IV: 𝑢 1 , 𝑢 2 , 𝑢 3 ≠ 7 , 11 and 𝑥 1 , 𝑥 2 , 𝑥 3 ≠ 7 , 11 .

Here as in the above cases, we can show that inequality (2.5) holds for 𝛼

2 3 .

Thus, 𝑇 is a perimetric proximal contraction of the first kind. The equalities 𝑑 ( 𝑥 , 𝑇 𝑦 )

1 𝑑 ( 𝑦 , 𝑇 𝑥 ) for distinct 𝑥 , 𝑦 ∈ 𝐴 would require 𝑥 and 𝑦 to be negative integers or fractions - an impossibility here. Hence, by contraposition, there are no distinct points 𝑥 , 𝑦 ∈ 𝐴 with 𝑑 ( 𝑥 , 𝑇 𝑦 )

𝑑 ( 𝐴 , 𝐵 )

𝑑 ( 𝑦 , 𝑇 𝑥 ) , i.e., 𝑇 satisfies Condition Λ . Further, it is easy to verify that 𝑇 ( 𝐴 0 ) ⊆ 𝐵 0 .

Consequently, Theorem 2.4 applies, ensuring 𝑇 has best proximity points and their number does not exceed two. Indeed 7 and 11 are the best proximity points of 𝑇 .

Example 2.11.

Consider the complete metric space ( ℝ 2 , 𝑑 ) , equipped with the Euclidean metric 𝑑 , defined by 𝑑 ( 𝑥 , 𝑦 )

{ | 𝑥 1 − 𝑥 2 | 2 + | 𝑦 1 − 𝑦 2 | 2 } 1 2 , 𝑥

( 𝑥 1 , 𝑦 1 ) , 𝑦

( 𝑥 2 , 𝑦 2 ) ∈ ℝ 2 . We choose 𝐴

{ ( 𝑥 , 1 ) ∈ ℝ 2 : − 1 ≤ 𝑥 ≤ 1 } and 𝐵

{ ( 𝑥 , 0 ) ∈ ℝ 2 : − 1 ≤ 𝑥 ≤ 1 } . Then 𝑑 ( 𝐴 , 𝐵 )

1 , 𝐴 0

𝐴 and 𝐵 0

𝐵 . We define a mapping 𝑇 : 𝐴 → 𝐵 by 𝑇 ( 𝑥 , 𝑦 )

( 𝑥 10 , 0 ) for all ( 𝑥 , 𝑦 ) ∈ 𝐴 .

Then, for 𝑢

( 𝑢 1 , 1 ) , 𝑥

( 𝑥 1 , 1 ) ∈ 𝐴 , the condition 𝑑 ( 𝑢 , 𝑇 𝑥 )

𝑑 ( 𝐴 , 𝐵 )

1 yields 𝑢 1

𝑥 1 10 . With this in hand, a simple argument establishes that 𝑇 is a perimetric proximal contraction of the second kind, with 𝛼

2 3 . Further, taking 𝑥

( 𝑥 1 , 1 ) , 𝑦

( 𝑥 2 , 1 ) ∈ 𝐴 with 𝑥 ≠ 𝑦 , the equalities 𝑑 ( 𝑥 , 𝑇 𝑦 )

𝑑 ( 𝐴 , 𝐵 ) and 𝑑 ( 𝑦 , 𝑇 𝑥 )

𝑑 ( 𝐴 , 𝐵 ) imply 𝑥 1

𝑥 2 10 and 𝑥 2

𝑥 1 10 , which is untenable here. Thus, there do not exist distinct points 𝑥 , 𝑦 ∈ 𝐴 satisfying 𝑑 ( 𝑥 , 𝑇 𝑦 )

𝑑 ( 𝐴 , 𝐵 )

𝑑 ( 𝑦 , 𝑇 𝑥 ) , ensuring that 𝑇 satisfies Condition Λ . Moreover, it is evident that 𝐴 is approximately compact with respect to 𝐵 . In addition, one readily observes that 𝑇 ( 𝐴 0 ) ⊆ 𝐵 0 , and 𝑇 is injective. Invoking Theorem 2.7, we conclude that 𝑇 admits a best proximity point, with a count of at most two; in this case, 𝑇 has exactly one best proximity point, viz., ( 0 , 1 ) .

References [1] A. Banerjee, P. Mondal, and L.K. Dey.Perimetric contraction on quadrilaterals.Science & Technology Asia, 30(4):55–67, 2025. [2] A. Banerjee, P. Mondal, and L.K. Dey.Perimetric contractions of Kannan and Chatterjea type in quadrilaterals.To appear in Mat. Vesn., 2025. [3] S. Sadiq Basha.Best proximity point theorems.J. Approx. Theory, 163:1772–1781, 2011. [4] S. Sadiq Basha.Best proximity points: optimal solutions.J. Optim. Theory Appl., 151(1):210–216, 2011. [5] S. Sadiq Basha.Best proximity point theorems for some special proximal contractions.Numer. Funct. Anal. Optim., 40(10):1182–1193, 2019. [6] C. Bey, E. Petrov, and R. Salimov.On three-point generalizations of Banach and Edelstein fixed point theorems.Filomat, 39(1):185–195, 2025. [7] R.K. Bisht.Perimetric contraction principle on quadrilaterals in semimetric spaces with triangle functions.Acta Math. Hungar., 175(2):1–14, 2025. [8] J. Chen, S. Xiao, H. Wang, and S. Deng.Best proximity point for the proximal nonexpansive mapping on the starshaped sets.Fixed Point Theory Appl., 2015: 19, 2015. [9] K. Fan.Extension of two fixed point theorem of F.E. Browder.Math Z., 112:234–240, 1969. [10] J. Anuradha and P. Veeramani.Proximal pointwise contraction.Topology Appl., 156(18):2942–2948, 2009. [11] A. Kostić, V. Rakǒcević, and S. Radenović.Best proximity points involving simulation functions with 𝑤 0 -distance.Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 113(2):715–727, 2019. [12] P. Mondal, H. Garai, and L.K. Dey.On proximal contractions via implicit relations and best proximity points.Miskolc Math. Notes, 22(2):783–798, 2021. [13] S. Mondal and L.K. Dey.Some common best proximity point theorems in a complete metric space.Afr. Mat., 28:85–97, 2017. [14] E. Karapınar.Best proximity points of cyclic mappings.Appl. Math. Lett., 25(11):1761–1766, 2012. [15] E. Petrov.Fixed point theorem for mappings contracting perimeters of triangles.J. Fixed Point Theory Appl., 25(3), 2023.Article Number: 74. [16] E. Petrov and R.K. Bisht.Fixed point theorem for generalized Kannan type mappings.Rend.Circ. Mat. Palermo, II. Ser,, 73(2024):2895–2912, 2024. [17] E. Petrov, R. Salimov, and R.K. Bisht.On generalizations of some fixed point theorems in semimetric spaces with triangle functions.Front. Appl. Math. Stat., 10, 2024.Article: 1392560. [18] S. Reich.Fixed points in locally convex spaces.Maths Z., 125:17–31, 1972. [19] S. Reich.Approximate selections, best approximations, fixed points, and invariant sets.J. Math. Anal. Appl., 62:104–113, 1978. Generated on Sat Jan 31 09:03:16 2026 by LaTeXML Report Issue Report Issue for Selection

Fixed point theory is an invaluable tool to find solutions of equations of the form 𝑇 ​ 𝑥

𝑥 means finding 𝑥 ∗ ∈ 𝐴 such that 𝑑 ​ ( 𝑥 ∗ , 𝑇 ​ 𝑥 ∗ )

𝑑 ​ ( 𝐴 , 𝐵 ) , where 𝑑 ​ ( 𝐴 , 𝐵 )

inf { 𝑑 ​ ( 𝑎 , 𝑏 ) : 𝑎 ∈ 𝐴 , 𝑏 ∈ 𝐵 } . Such a point 𝑥 ∗ ∈ 𝐴 , if it exists, satisfying 𝑑 ​ ( 𝑥 ∗ , 𝑇 ​ 𝑥 ∗ )

𝑑 ​ ( 𝑢 1 , 𝑇 ​ 𝑥 1 )

𝑑 ​ ( 𝑢 2 , 𝑇 ​ 𝑥 2 )

𝑑 ​ ( 𝑢 1 , 𝑇 ​ 𝑥 1 )

𝑑 ​ ( 𝑢 2 , 𝑇 ​ 𝑥 2 )

𝐴 0

{ 𝑥 ∈ 𝐴 : 𝑑 ​ ( 𝑥 , 𝑦 )

𝑑 ​ ( 𝐴 , 𝐵 ) ​ for some ​ 𝑦 ∈ 𝐵 } 𝐵 0

{ 𝑦 ∈ 𝐵 : 𝑑 ​ ( 𝑥 , 𝑦 )

𝑑 ​ ( 𝑢 1 , 𝑇 ​ 𝑥 1 )

𝑑 ​ ( 𝑢 2 , 𝑇 ​ 𝑥 2 )

𝑑 ​ ( 𝑢 3 , 𝑇 ​ 𝑥 3 )

𝑑 ​ ( 𝑢 1 , 𝑇 ​ 𝑥 1 )

𝑑 ​ ( 𝑢 2 , 𝑇 ​ 𝑥 2 )

𝑑 ​ ( 𝑢 3 , 𝑇 ​ 𝑥 3 )

𝑑 ​ ( 𝐴 , 𝐵 ) , 𝑑 ​ ( 𝑢 2 , 𝑇 ​ 𝑥 2 )

𝑑 ​ ( 𝐴 , 𝐵 ) , 𝑑 ​ ( 𝑢 3 , 𝑇 ​ 𝑥 3 )

Consider a metric space ( ℝ , 𝑑 ) , where 𝑑 is the Euclidean distance on ℝ . Let 𝐴

{ − 3 , 0 , 3 , 4 } and 𝐵

𝑇 ​ ( − 3 )

− 2 , 𝑇 ​ ( 0 )

3 2 , 𝑇 ​ ( 3 )

2 , 𝑇 ​ ( 4 )

Let us show that 𝑇 is a perimetric proximal contraction of the first kind. It is easy to see that 𝑑 ​ ( 𝐴 , 𝐵 )

1 . Next, we identify all pairs ( 𝑢 , 𝑥 ) ∈ 𝐴 × 𝐴 that satisfy the condition 𝑑 ​ ( 𝑢 , 𝑇 ​ 𝑥 )

𝑑 ​ ( 𝐴 , 𝐵 )

If 𝑥

− 3 , 𝑇 ​ 𝑥

− 2 . We need 𝑑 ​ ( 𝑢 , − 2 )

1 . The only solution in 𝐴 is 𝑢

If 𝑥

0 , 𝑇 ​ 𝑥

1.5 . We need 𝑑 ​ ( 𝑢 , 1.5 )

If 𝑥

3 , 𝑇 ​ 𝑥

2 . We need 𝑑 ​ ( 𝑢 , 2 )

1 . The only solution in 𝐴 is 𝑢

If 𝑥

4 , 𝑇 ​ 𝑥

1 . We need 𝑑 ​ ( 𝑢 , 1 )

1 . The only solution in 𝐴 is 𝑢

{ 𝑢 1 , 𝑢 2 , 𝑢 3 }

{ − 3 , 3 , 0 } and { 𝑥 1 , 𝑥 2 , 𝑥 3 }

𝑃 ​ ( 𝑢 1 , 𝑢 2 , 𝑢 3 )

𝑑 ​ ( 𝑢 1 , 𝑢 2 ) + 𝑑 ​ ( 𝑢 2 , 𝑢 3 ) + 𝑑 ​ ( 𝑢 3 , 𝑢 1 )

𝑑 ​ ( − 3 , 3 ) + 𝑑 ​ ( 3 , 0 ) + 𝑑 ​ ( 0 , − 3 )

6 + 3 + 3

𝑃 ​ ( 𝑥 1 , 𝑥 2 , 𝑥 3 )

𝑑 ​ ( 𝑥 1 , 𝑥 2 ) + 𝑑 ​ ( 𝑥 2 , 𝑥 3 ) + 𝑑 ​ ( 𝑥 3 , 𝑥 1 )

𝑑 ​ ( − 3 , 3 ) + 𝑑 ​ ( 3 , 4 ) + 𝑑 ​ ( 4 , − 3 )

6 + 1 + 7

Clearly, the inequality in Definition 2.1 holds for all 𝛼 ∈ [ 6 7 , 1 ) . Thus, 𝑇 is a perimetric proximal contraction of the first kind. Also, it is easy to see that 𝐴 0

{ − 3 , 0 , 3 } , 𝐵 0

{ − 2 , − 1 , 1 , 2 } , 𝑇 ​ ( 𝐴 0 )

Since 𝑑 ​ ( − 3 , 𝑇 ​ ( − 3 ) )

1 and 𝑑 ​ ( 3 , 𝑇 ​ ( 3 ) )

1 but 𝑑 ​ ( − 3 , 3 )

𝑑 ​ ( 𝑦 , 𝑇 ​ 𝑥 )

Since | 𝐴 0 | ≥ 3 , we can choose an element 𝑢 0 ∈ 𝐴 0 . Then, 𝑇 ​ 𝑢 0 ∈ 𝑇 ​ ( 𝐴 0 ) ⊆ 𝐵 0 , so there exists 𝑢 1 ∈ 𝐴 0 such that 𝑑 ​ ( 𝑢 1 , 𝑇 ​ 𝑢 0 )

𝑑 ​ ( 𝐴 , 𝐵 ) . Again, 𝑇 ​ 𝑢 1 ∈ 𝑇 ​ ( 𝐴 0 ) ⊆ 𝐵 0 , so there exists 𝑢 2 ∈ 𝐴 0 such that 𝑑 ​ ( 𝑢 2 , 𝑇 ​ 𝑢 1 )

𝑑 ​ ( 𝐴 , 𝐵 ) . Continuing in this way, we obtain a sequence { 𝑢 𝑛 } having the properties that 𝑢 𝑛 ∈ 𝐴 0 and 𝑑 ​ ( 𝑢 𝑛 + 1 , 𝑇 ​ 𝑢 𝑛 )

If 𝑢 𝑛

𝑢 𝑛 + 1 for some 𝑛 ∈ ℕ , it follows that 𝑢 𝑛 is a best proximity point of 𝑇 , as 𝑑 ​ ( 𝑢 𝑛 + 1 , 𝑇 ​ 𝑢 𝑛 )

𝑑 ​ ( 𝑢 𝑛 , 𝑇 ​ 𝑢 𝑛 )

Next, we assume that 𝑢 𝑛 ≠ 𝑢 𝑛 + 1 for all 𝑛 ∈ ℕ . Under this assumption, we claim that 𝑢 𝑛 , 𝑢 𝑛 + 1 , 𝑢 𝑛 + 2 are pairwise distinct for all 𝑛 ∈ ℕ . If not, then we must have 𝑢 𝑛

𝑢 𝑛 + 2 for some 𝑛 ∈ ℕ . Then 𝑑 ​ ( 𝑢 𝑛 , 𝑇 ​ 𝑢 𝑛 + 1 )

𝑑 ​ ( 𝑢 𝑛 + 2 , 𝑇 ​ 𝑢 𝑛 + 1 )

𝑑 ​ ( 𝐴 , 𝐵 ) and also ( 𝑢 𝑛 + 1 , 𝑇 ​ 𝑢 𝑛 )

Now, since 𝑑 ​ ( 𝑢 𝑛 + 1 , 𝑇 ​ 𝑢 𝑛 )

𝑑 ​ ( 𝐴 , 𝐵 )

𝑑 ​ ( 𝑢 𝑛 + 2 , 𝑇 ​ 𝑢 𝑛 + 1 )

Let 𝑡 𝑛

If all terms of { 𝑢 𝑛 } are not distinct, then there is a minimal natural number 𝑗 ( ≥ 3 ) such that 𝑢 𝑗

𝑢 𝑖 for some 𝑖 ∈ ℕ with 0 ≤ 𝑖 ≤ 𝑗 − 2 . This leads to 𝑢 𝑗 + 1

𝑢 𝑖 + 1 , 𝑢 𝑗 + 2

𝑢 𝑖 + 2 , ⋯ , and so, 𝑡 𝑗

For the last part of the proof, we assume that 𝑇 has three distinct (which are pairwise distinct as well) best proximity points 𝑥 , 𝑦 , 𝑧 ∈ 𝐴 . Then, we have 𝑑 ​ ( 𝑥 , 𝑇 ​ 𝑥 )

Fixed point theory is an invaluable tool to find solutions of equations of the form 𝑇 𝑥

𝑥 means finding 𝑥 ∗ ∈ 𝐴 such that 𝑑 ( 𝑥 ∗ , 𝑇 𝑥 ∗ )

𝑑 ( 𝐴 , 𝐵 ) , where 𝑑 ( 𝐴 , 𝐵 )

inf { 𝑑 ( 𝑎 , 𝑏 ) : 𝑎 ∈ 𝐴 , 𝑏 ∈ 𝐵 } . Such a point 𝑥 ∗ ∈ 𝐴 , if it exists, satisfying 𝑑 ( 𝑥 ∗ , 𝑇 𝑥 ∗ )

𝑑 ( 𝑢 1 , 𝑇 𝑥 1 )

𝑑 ( 𝑢 2 , 𝑇 𝑥 2 )

𝑑 ( 𝑢 1 , 𝑇 𝑥 1 )

𝑑 ( 𝑢 2 , 𝑇 𝑥 2 )

{ 𝑥 ∈ 𝐴 : 𝑑 ( 𝑥 , 𝑦 )

𝑑 ( 𝐴 , 𝐵 ) for some 𝑦 ∈ 𝐵 }

𝐵 0

{ 𝑦 ∈ 𝐵 : 𝑑 ( 𝑥 , 𝑦 )

𝑑 ( 𝑢 1 , 𝑇 𝑥 1 )

𝑑 ( 𝑢 2 , 𝑇 𝑥 2 )

𝑑 ( 𝑢 3 , 𝑇 𝑥 3 )

𝑑 ( 𝑢 1 , 𝑇 𝑥 1 )

𝑑 ( 𝑢 2 , 𝑇 𝑥 2 )

𝑑 ( 𝑢 3 , 𝑇 𝑥 3 )

𝑑 ( 𝐴 , 𝐵 ) , 𝑑 ( 𝑢 2 , 𝑇 𝑥 2 )

𝑑 ( 𝐴 , 𝐵 ) , 𝑑 ( 𝑢 3 , 𝑇 𝑥 3 )

𝑇 ( − 3 )

− 2 , 𝑇 ( 0 )

3 2 , 𝑇 ( 3 )

2 , 𝑇 ( 4 )

Let us show that 𝑇 is a perimetric proximal contraction of the first kind. It is easy to see that 𝑑 ( 𝐴 , 𝐵 )

1 . Next, we identify all pairs ( 𝑢 , 𝑥 ) ∈ 𝐴 × 𝐴 that satisfy the condition 𝑑 ( 𝑢 , 𝑇 𝑥 )

𝑑 ( 𝐴 , 𝐵 )

− 3 , 𝑇 𝑥

− 2 . We need 𝑑 ( 𝑢 , − 2 )

0 , 𝑇 𝑥

1.5 . We need 𝑑 ( 𝑢 , 1.5 )

3 , 𝑇 𝑥

2 . We need 𝑑 ( 𝑢 , 2 )

4 , 𝑇 𝑥

1 . We need 𝑑 ( 𝑢 , 1 )

𝑃 ( 𝑢 1 , 𝑢 2 , 𝑢 3 )

𝑑 ( 𝑢 1 , 𝑢 2 ) + 𝑑 ( 𝑢 2 , 𝑢 3 ) + 𝑑 ( 𝑢 3 , 𝑢 1 )

𝑑 ( − 3 , 3 ) + 𝑑 ( 3 , 0 ) + 𝑑 ( 0 , − 3 )

𝑃 ( 𝑥 1 , 𝑥 2 , 𝑥 3 )

𝑑 ( 𝑥 1 , 𝑥 2 ) + 𝑑 ( 𝑥 2 , 𝑥 3 ) + 𝑑 ( 𝑥 3 , 𝑥 1 )

𝑑 ( − 3 , 3 ) + 𝑑 ( 3 , 4 ) + 𝑑 ( 4 , − 3 )

{ − 2 , − 1 , 1 , 2 } , 𝑇 ( 𝐴 0 )

Since 𝑑 ( − 3 , 𝑇 ( − 3 ) )

1 and 𝑑 ( 3 , 𝑇 ( 3 ) )

1 but 𝑑 ( − 3 , 3 )

𝑑 ( 𝑦 , 𝑇 𝑥 )

Since | 𝐴 0 | ≥ 3 , we can choose an element 𝑢 0 ∈ 𝐴 0 . Then, 𝑇 𝑢 0 ∈ 𝑇 ( 𝐴 0 ) ⊆ 𝐵 0 , so there exists 𝑢 1 ∈ 𝐴 0 such that 𝑑 ( 𝑢 1 , 𝑇 𝑢 0 )

𝑑 ( 𝐴 , 𝐵 ) . Again, 𝑇 𝑢 1 ∈ 𝑇 ( 𝐴 0 ) ⊆ 𝐵 0 , so there exists 𝑢 2 ∈ 𝐴 0 such that 𝑑 ( 𝑢 2 , 𝑇 𝑢 1 )

𝑑 ( 𝐴 , 𝐵 ) . Continuing in this way, we obtain a sequence { 𝑢 𝑛 } having the properties that 𝑢 𝑛 ∈ 𝐴 0 and 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑢 𝑛 + 1 for some 𝑛 ∈ ℕ , it follows that 𝑢 𝑛 is a best proximity point of 𝑇 , as 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝑢 𝑛 , 𝑇 𝑢 𝑛 )

𝑢 𝑛 + 2 for some 𝑛 ∈ ℕ . Then 𝑑 ( 𝑢 𝑛 , 𝑇 𝑢 𝑛 + 1 )

𝑑 ( 𝑢 𝑛 + 2 , 𝑇 𝑢 𝑛 + 1 )

𝑑 ( 𝐴 , 𝐵 ) and also ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

Now, since 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝐴 , 𝐵 )

𝑑 ( 𝑢 𝑛 + 2 , 𝑇 𝑢 𝑛 + 1 )

For the last part of the proof, we assume that 𝑇 has three distinct (which are pairwise distinct as well) best proximity points 𝑥 , 𝑦 , 𝑧 ∈ 𝐴 . Then, we have 𝑑 ( 𝑥 , 𝑇 𝑥 )

𝑑 ( 𝐴 , 𝐵 )

𝑑 ( 𝑦 , 𝑇 𝑦 )

𝑋 in Theorem 2.4. Then it follows from Definition 2.2 that 𝑑 ( 𝐴 , 𝐵 )

𝑇 𝑥 1 , 𝑢 2

𝑇 𝑥 2 , 𝑢 3

𝑋 and the inclusion 𝑇 ( 𝐴 0 ) ⊆ 𝐵 0 holds. One can easily see that in the case 𝐴

Following the proof of Theorem 2.4, we construct a sequence { 𝑢 𝑛 } in 𝐴 such that 𝑢 𝑛 ∈ 𝐴 0 , 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝐴 , 𝐵 ) for all 𝑛 ∈ ℕ . If 𝑢 𝑛

𝑢 𝑛 + 1 for some 𝑛 , then 𝑢 𝑛 is a best proximity point of 𝑇 . So, we assume that 𝑢 𝑛 ≠ 𝑢 𝑛 + 1 for all 𝑛 ∈ ℕ , and we can then show, as before, that any three consecutive terms of { 𝑢 𝑛 } are pairwise distinct. If 𝑇 𝑢 𝑛

𝑇 𝑢 𝑛 + 1 for some 𝑛 , then we have 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 + 1 )

𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝐴 , 𝐵 ) , showing that 𝑢 𝑛 + 1 is a best proximity point of 𝑇 . We therefore assume that 𝑇 𝑢 𝑛 ≠ 𝑇 𝑢 𝑛 + 1 for every 𝑛 . Again, if 𝑇 𝑢 𝑛

𝑇 𝑢 𝑛 + 2 for some 𝑛 , then we have 𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 + 2 )

𝑑 ( 𝑢 𝑛 + 1 , 𝑇 𝑢 𝑛 )

𝑑 ( 𝐴 , 𝐵 ) and also, we have 𝑑 ( 𝑢 𝑛 + 2 , 𝑇 𝑢 𝑛 + 1 )

𝑑 ( 𝑣 , 𝐴 ) ≤ 𝑑 ( 𝑣 , 𝑢 𝑛 + 1 )
≤ 𝑑 ( 𝑣 , 𝑇 𝑢 𝑛 ) + 𝑑 ( 𝑇 𝑢 𝑛 , 𝑢 𝑛 + 1 )

Using the fact that lim 𝑛 → ∞ 𝑑 ( 𝑣 , 𝑇 𝑢 𝑛 )

0 , we obtain lim 𝑛 → ∞ 𝑑 ( 𝑣 , 𝑢 𝑛 + 1 )

𝑑 ( 𝑢 , 𝑣 )

lim 𝑘 → ∞ 𝑑 ( 𝑢 𝑛 ( 𝑘 ) , 𝑇 𝑢 𝑛 ( 𝑘 ) − 1 )

and so 𝑢 ∈ 𝐴 0 . Further, since 𝑇 is continuous, we have lim 𝑘 → ∞ 𝑇 𝑢 𝑛 ( 𝑘 )

𝑇 𝑢 , and so 𝑇 𝑢

𝑣 . Thus from (2.4), we get 𝑑 ( 𝑢 , 𝑇 𝑢 )