Abstract

The purpose of this paper is to present the existence of the best period proximity point for cyclic weaker Meir-Keeler contractions and asymptotic cyclic weaker Meir-Keeler contractions in metric spaces.

1. Introduction and Preliminaries

Throughout this paper, by ℝ+ we denote the set of all nonnegative numbers, while β„• is the set of all natural numbers. Let 𝐴 and 𝐡 be nonempty subsets of a metric space (𝑋,𝑑). Consider a mapping π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡, 𝑓 is called a cyclic map if 𝑓(𝐴)βŠ†π΅ and 𝑓(𝐡)βŠ†π΄. A point π‘₯ in 𝐴 is called a best proximity point of 𝑓 in 𝐴 if 𝑑(π‘₯,𝑓π‘₯)=𝑑(𝐴,𝐡) is satisfied, where 𝑑(𝐴,𝐡)=inf{𝑑(π‘₯,𝑦)∢π‘₯∈𝐴,π‘¦βˆˆπ΅}, and π‘₯∈𝐴 is called a best periodic proximity point of 𝑓 in 𝐴 if 𝑑(π‘₯,𝑓2πœ…+1π‘₯)=𝑑(𝐴,𝐡) is satisfied, for some πœ…βˆˆβ„•βˆͺ{0}. In 2005, Eldred et al. [1] proved the existence of a best proximity point for relatively nonexpansive mappings using the notion of proximal normal structure. In 2006, Eldred and Veeramani [2] proved the following existence theorem.

Theorem 1.1 (see Theorem  3.10 in [2]). Let 𝐴 and 𝐡 be nonempty closed convex subsets of a uniformly convex Banach space. Suppose π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is a cyclic contraction, that is, 𝑓(𝐴)βŠ†π΅ and 𝑓(𝐡)βŠ†π΄, and there exists π‘˜βˆˆ(0,1) such that 𝑑(𝑓π‘₯,𝑓𝑦)β‰€π‘˜π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)𝑑(𝐴,𝐡)foreveryπ‘₯∈𝐴,π‘¦βˆˆπ΅.(1.1) Then there exists a unique best proximity point in 𝐴. Further, for each π‘₯∈𝐴, {𝑓2𝑛π‘₯} converges to the best proximity point.

In this paper, we also recall the notion of Meir-Keeler type mapping. A mapping πœ“βˆΆβ„+→ℝ+ is said to be a Meir-Keeler-type mapping (see [3]) if for each πœ‚>0, there exists 𝛿>0 such that for π‘‘βˆˆβ„+ with πœ‚β‰€π‘‘<πœ‚+𝛿, we have πœ“(𝑑)<πœ‚.

In the recent, Eldred et al. [1] introduced the below notion of cyclic Meir-Keeler contraction.

Definition 1.2 (see [1]). Let (𝑋,𝑑) be a metric space, and let 𝐴 and 𝐡 be nonempty subsets of 𝑋. Then π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is called a cyclic Meir-Keeler contraction if the following are satisfied: (i)𝑓(𝐴)βŠ‚π΅ and 𝑓(𝐡)βŠ‚π΄;(ii) for every πœ€>0, there exists 𝛿>0 such that𝑑(π‘₯,𝑦)<𝑑(𝐴,𝐡)+πœ€+𝛿implies𝑑(𝑓π‘₯,𝑓𝑦)<𝑑(𝐴,𝐡)+πœ€(1.2) for all π‘₯∈𝐴 and π‘¦βˆˆπ΅.
In the recent, Di Bari et al. [4] proved the following best proximity point theorem.

Theorem 1.3 (see [4]). Let 𝑋 be a uniformly convex Banach space, and let 𝐴 and 𝐡 be nonempty subsets of 𝑋. Suppose 𝐴 is closed and convex and π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is a cyclic Meir-Keeler contraction. Then there exists a unique best proximity point in 𝐴. Further, for each π‘₯∈𝐴, {𝑓2𝑛π‘₯} converges to best proximity point.

Later, many authors studied this subject, and many results on best proximity points are proved. (see, e.g., [5–10]). In this study, we will introduce the new concepts of cyclic weaker Meir-Keeler contractions and asymptotic cyclic weaker Meir-Keeler contractions in metric spaces, and the purpose of this paper is to present the existence of the best period proximity point for these contractions.

2. The Best Periodic Proximity Points for Cyclic Weaker Meir-Keeler Contractions

In this section, we first introduce the below notions of the weaker Meir-Keeler-type mapping, πœ‘-mapping, and cyclic weaker Meir-Keeler contraction in metric spaces.

Definition 2.1. Let (𝑋,𝑑) be a metric space, and πœ‘βˆΆβ„+→ℝ+. Then πœ‘ is called a weaker Meir-Keeler-type mapping in 𝑋 if for each πœ‚>0, there exists 𝛿>0 such that for π‘₯,π‘¦βˆˆπ‘‹ with πœ‚β‰€π‘‘(π‘₯,𝑦)<𝛿+πœ‚, there exists 𝑛0βˆˆβ„• such that πœ‘π‘›0(𝑑(π‘₯,𝑦))<πœ‚.

The following provides an example of a weaker Meir-Keeler-type mapping that is not a Meir-Keeler-type mapping in a metric space(𝑋,𝑑).

Example 2.2. Let 𝑋=ℝ2, and we define π‘‘βˆΆπ‘‹Γ—π‘‹β†’β„+ by 𝑑||π‘₯(π‘₯,𝑦)=1βˆ’π‘¦1||+||π‘₯2βˆ’π‘¦2||ξ€·π‘₯βˆ€π‘₯=1,π‘₯2𝑦,𝑦=1,𝑦2ξ€Έβˆˆπ‘‹.(2.1) If πœ‘βˆΆβ„+→ℝ+, ⎧βŽͺ⎨βŽͺβŽ©πœ‘(𝑑)=0if𝑑≀1,2𝑑if1<𝑑<2,1if𝑑β‰₯2,(2.2) where𝑑=𝑑(π‘₯,𝑦), π‘₯,π‘¦βˆˆπ‘‹, then πœ‘ is a weaker Meir-Keeler-type mapping that is not a Meir-Keeler-type mapping in 𝑋.

Definition 2.3. Let (𝑋,𝑑) be a metric space. A mapping πœ‘βˆΆβ„+→ℝ+ is called a πœ‘-mapping in 𝑋 if the mapping πœ‘βˆΆβ„+→ℝ+ satisfies the following conditions:(πœ‘1)πœ‘ is a weaker Meir-Keeler-type mapping in 𝑋;(πœ‘2) for all 𝑑>0, {πœ‘π‘›(𝑑)}π‘›βˆˆβ„• is nonincreasing;(πœ‘3) for all 𝑑>0, πœ‘(𝑑)>0 and πœ‘(0)=0.
The following provides two examples of a πœ‘-mapping.

Example 2.4. Let 𝑋=ℝ2,and we define π‘‘βˆΆπ‘‹Γ—π‘‹β†’β„+ by ||π‘₯𝑑(π‘₯,𝑦)=1βˆ’π‘¦1||+||π‘₯2βˆ’π‘¦2||ξ€·π‘₯βˆ€π‘₯=1,π‘₯2𝑦,𝑦=1,𝑦2ξ€Έβˆˆπ‘‹.(2.3) Let πœ‘βˆΆβ„+→ℝ+ be 1πœ‘(𝑑)=2π‘‘βˆ€π‘‘βˆˆβ„+.(2.4) Then πœ‘βˆΆβ„+→ℝ+ is a πœ‘-mapping in 𝑋.

Example 2.5. Let 𝑋=[0,4], and we define π‘‘βˆΆπ‘‹Γ—π‘‹β†’β„+ by ||||𝑑(π‘₯,𝑦)=π‘₯βˆ’π‘¦βˆ€π‘₯,π‘¦βˆˆπ‘‹.(2.5) If πœ‘βˆΆβ„+→ℝ+, ⎧βŽͺ⎨βŽͺ⎩3πœ‘(𝑑)=4𝑑12𝑑if0≀t≀1,if1<𝑑<2,if2≀𝑑≀4,(2.6) where 𝑑=𝑑(π‘₯,𝑦), π‘₯,π‘¦βˆˆπ‘‹, then πœ‘ is a πœ‘-mapping in 𝑋.

Definition 2.6. Let (𝑋,𝑑) be a metric space, and let 𝐴 and 𝐡 be nonempty subsets of 𝑋. Then π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is called a cyclic weaker Meir-Keeler contraction if the following conditions hold:(1)𝑓(𝐴)βŠ‚π΅ and 𝑓(𝐡)βŠ‚π΄;(2) there is a πœ‘-mapping πœ‘βˆΆβ„+→ℝ+ in 𝑋 such that for all π‘›βˆˆβ„• and π‘₯∈𝐴, π‘¦βˆˆπ΅ with 𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡)>0,𝑑(𝑓𝑛π‘₯,𝑓𝑛𝑦)βˆ’π‘‘(𝐴,𝐡)<πœ‘π‘›π‘‘ξ‚€ξ‚(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡)),π‘₯,π‘¦βˆ’π‘‘(𝐴,𝐡)=0implies𝑑(𝑓𝑛π‘₯,𝑓𝑛𝑦)βˆ’π‘‘(𝐴,𝐡)=0.(2.7)
The following provides an example of a cyclic weaker Meir-Keeler contraction.

Example 2.7. Let 𝐴=[βˆ’2,0] and 𝐡=[0,2] in the metric space (ℝ,𝑑), where 𝑑(π‘₯,𝑦)=|π‘₯βˆ’π‘¦|. Define 𝑓(π‘₯)=βˆ’π‘₯4βˆ€π‘₯∈𝐴βˆͺ𝐡.(2.8) Let πœ‘βˆΆβ„+→ℝ+ be defined by ⎧βŽͺ⎨βŽͺ⎩3πœ‘(𝑑)=4𝑑if0≀t≀1,2𝑑if1<𝑑<2,1if2≀𝑑≀4,(2.9) where 𝑑=𝑑(π‘₯,𝑦), π‘₯∈𝐴, π‘¦βˆˆπ΅. Then all conditions (1) and (2) of Definition 2.6 and therefore 𝑓 are a cyclic weaker Meir-Keeler contraction. Notice that 𝑑(𝐴,𝐡)=0.
Now, we are in this position to state the following results.

Lemma 2.8. Let (𝑋,𝑑) be a metric space, and let 𝐴, 𝐡 be nonempty subsets of 𝑋. Suppose π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is a cyclic weaker Meir-Keeler contraction. Then limπ‘›β†’βˆžπ‘‘(𝑓𝑛π‘₯,𝑓𝑛+1π‘₯)=𝑑(𝐴,𝐡) holds.
Proof. Since π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is a cyclic weaker Meir-Keeler contraction, there is a πœ‘-mapping πœ‘βˆΆβ„+→ℝ+ in 𝑋 such that𝑑(𝑓𝑛π‘₯,𝑓𝑛𝑦)βˆ’π‘‘(𝐴,𝐡)<πœ‘π‘›(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡)),(2.10) for allπ‘›βˆˆβ„•andπ‘₯∈𝐴,π‘¦βˆˆπ΅.
Since {πœ‘π‘›(𝑑(π‘₯,𝑦))}π‘›βˆˆβ„• is nonincreasing, hence we also conclude {πœ‘π‘›(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))}π‘›βˆˆβ„• is nonincreasing, and it must converge to some πœ‚β‰₯0. We claim that πœ‚=0. On the contrary, assume that πœ‚>0. By the definition of the weaker Meir-Keeler-type mapping πœ‘, corresponding to πœ‚ use, there exists 𝛿>0 such that for π‘₯,π‘¦βˆˆπ‘‹ with πœ‚β‰€π‘‘(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡)<𝛿+πœ‚, there exists 𝑛0βˆˆβ„• such that πœ‘π‘›0(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))<πœ‚. Since limπ‘›β†’βˆžπœ‘π‘›(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))=πœ‚, there exists π‘š0βˆˆβ„• such that πœ‚β‰€πœ‘π‘š(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))<𝛿+πœ‚, forall π‘šβ‰₯π‘š0. Thus, we conclude that πœ‘π‘š0+𝑛0(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))<πœ‚. So we get a contradiction. So limπ‘›β†’βˆžπœ‘π‘›(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))=0, and so limπ‘›β†’βˆžπ‘‘(𝑓𝑛π‘₯,𝑓𝑛𝑦)βˆ’π‘‘(𝐴,𝐡)=0, that is, limπ‘›β†’βˆžπ‘‘(𝑓𝑛π‘₯,𝑓𝑛𝑦)=𝑑(𝐴,𝐡). Thus, we also conclude that limπ‘›β†’βˆžπ‘‘(𝑓𝑛π‘₯,𝑓𝑛+1π‘₯)=𝑑(𝐴,𝐡).

Applying above Lemma 2.8, it is easy to conclude the following theorem.

Theorem 2.9. Let (𝑋,𝑑) be a metric space, and let 𝐴,𝐡 be nonempty subsets of 𝑋. Suppose π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is a cyclic weaker Meir-Keeler contraction and if for some π‘₯∈𝐴, the sequence {𝑓2𝑛+1π‘₯} converges to π‘₯∈𝐴, then π‘₯ is a best periodic proximity point of 𝑓 in 𝐴.
Proof. By the definition of the weaker Meir-Keeler-type mapping πœ‘βˆΆβ„+→ℝ+ in 𝑋, there exists 𝑛0βˆˆβ„• such that πœ‘π‘›0(πœ‚)β‰€πœ‚ for each πœ‚>0. Since {𝑓2𝑛+1π‘₯} converges toπ‘₯∈𝐴, corresponding to above 𝑛0 use, we have 𝑑(𝐴,𝐡)≀𝑑π‘₯,𝑓2𝑛0+1π‘₯≀𝑑π‘₯,𝑓2𝑛+1π‘₯𝑓+𝑑2𝑛+1π‘₯,𝑓2𝑛0+1π‘₯ξ€Έξ€·βˆ’π‘‘(𝐴,𝐡)+𝑑(𝐴,𝐡)≀𝑑π‘₯,𝑓2𝑛+1π‘₯ξ€Έ+πœ‘2𝑛0+1𝑑𝑓2(π‘›βˆ’π‘›0)π‘₯,π‘₯ξ€Έξ€Έξ€·βˆ’π‘‘(𝐴,𝐡)+𝑑(𝐴,𝐡)≀𝑑π‘₯,𝑓2𝑛+1π‘₯ξ€Έ+πœ‘2𝑛0𝑑𝑓2(π‘›βˆ’π‘›0)π‘₯,π‘₯ξ€Έξ€Έξ€·βˆ’π‘‘(𝐴,𝐡)+𝑑(𝐴,𝐡)≀𝑑π‘₯,𝑓2𝑛+1π‘₯𝑓+𝑑2(π‘›βˆ’π‘›0)π‘₯,π‘₯ξ€Έξ€·βˆ’π‘‘(𝐴,𝐡)+𝑑(𝐴,𝐡)≀𝑑π‘₯,𝑓2𝑛+1π‘₯𝑓+𝑑2(π‘›βˆ’π‘›0)π‘₯,𝑓2(π‘›βˆ’π‘›0)+1π‘₯𝑓+𝑑2(π‘›βˆ’π‘›0)+1π‘₯,π‘₯ξ€Έ,(2.11) Letting π‘›β†’βˆž. Then 𝑑(𝐴,𝐡)=𝑑(π‘₯,𝑓2𝑛0+1π‘₯). Thus π‘₯ is a best period proximity point of𝑓 in 𝐴.

3. The Best Periodic Proximity Points for Asymptotic Cyclic Weaker Meir-Keeler Contractions

In this section, we introduce the below notions of the asymptotic cyclic weaker Meir-Keeler-type sequence and asymptotic cyclic weaker Meir-Keeler contraction in a metric space (𝑋,𝑑).

Definition 3.1. Let (𝑋,𝑑) be a metric space. A sequence {πœ‘π‘›|πœ‘π‘›βˆΆβ„+→ℝ+}π‘›βˆˆβ„• in 𝑋 is called an asymptotic weaker Meir-Keeler-type sequence if {πœ‘π‘›|πœ‘π‘›βˆΆβ„+→ℝ+}π‘›βˆˆβ„• satisfies the following conditions:(C1) for each πœ‚>0, there exists 𝛿>0 such that for π‘₯,π‘¦βˆˆπ‘‹ with πœ‚β‰€π‘‘(π‘₯,𝑦)<𝛿+πœ‚, there exists 2𝑛0βˆˆβ„• such that πœ‘2𝑛0(𝑑(π‘₯,𝑦))<πœ‚;(C2) for all π‘›βˆˆβ„• and 𝑑>0, {πœ‘π‘›(𝑑)}π‘›βˆˆβ„• is nonincreasing;(C3) for all π‘›βˆˆβ„•, πœ‘π‘›(0)=0 and πœ‘π‘›(𝑑)>0, 𝑑>0.

Example 3.2. Let 𝑋=ℝ2 and we define π‘‘βˆΆπ‘‹Γ—π‘‹β†’β„+ by 𝑑||π‘₯(π‘₯,𝑦)=1βˆ’π‘¦1||+||π‘₯2βˆ’π‘¦2||ξ€·π‘₯βˆ€π‘₯=1,π‘₯2𝑦,𝑦=1,𝑦2ξ€Έβˆˆπ‘‹.(3.1) Let πœ‘π‘›βˆΆβ„+→ℝ+ be πœ‘π‘›1(𝑑)=2π‘›π‘‘βˆ€π‘‘βˆˆβ„+,π‘›βˆˆβ„•,(3.2) where 𝑑=𝑑(π‘₯,𝑦), π‘₯,π‘¦βˆˆπ‘‹. Then {πœ‘π‘›βˆ£πœ‘π‘›βˆΆβ„+→ℝ+}π‘›βˆˆβ„• is an asymptotic weaker Meir-Keeler-type sequence in a metric space (𝑋,𝑑).

Definition 3.3. Let (𝑋,𝑑) be a metric space, and let 𝐴 and 𝐡 be nonempty subsets of 𝑋. Then π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is an asymptotic cyclic weaker Meir-Keeler contraction if the following conditions hold:
(1)𝑓(𝐴)βŠ‚π΅ and 𝑓(𝐡)βŠ‚π΄;(2) there is an asymptotic weaker Meir-Keeler-type sequence {πœ‘π‘›|πœ‘π‘›βˆΆβ„+→ℝ+}π‘›βˆˆβ„• such that for all π‘›βˆˆβ„• and π‘₯∈𝐴, π‘¦βˆˆπ΅ with 𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡)>0,𝑑(𝑓𝑛π‘₯,𝑓𝑛𝑦)βˆ’π‘‘(𝐴,𝐡)<πœ‘π‘›(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡)),𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡)=0implies𝑑(𝑓𝑛π‘₯,𝑓𝑛𝑦)βˆ’π‘‘(𝐴,𝐡)=0.(3.3)
Now, we are in this position to state the following results.

Lemma 3.4. Let (𝑋,𝑑) be a metric space and 𝐴,𝐡 nonempty subsets of 𝑋. Suppose π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is an asymptotic cyclic weaker Meir-Keeler contraction. Then limπ‘›β†’βˆžπ‘‘(𝑓𝑛π‘₯,𝑓𝑛+1π‘₯)=𝑑(𝐴,𝐡) holds.
Proof. Since π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is an asymptotic cyclic weaker Meir-Keeler contraction, there is an asymptotic weaker Meir-Keeler-type sequence {πœ‘π‘›|πœ‘π‘›βˆΆβ„+→ℝ+}π‘›βˆˆβ„• such that𝑑(𝑓𝑛π‘₯,𝑓𝑛𝑦)βˆ’π‘‘(𝐴,𝐡)<πœ‘π‘›(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡)),(3.4) for all π‘›βˆˆβ„• and π‘₯∈𝐴, π‘¦βˆˆπ΅.
Since {πœ‘π‘›(𝑑(π‘₯,𝑦))}π‘›βˆˆβ„• is nonincreasing, hence we also conclude {πœ‘π‘›(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))}π‘›βˆˆβ„• is nonincreasing, and it must converge to some πœ‚β‰₯0. We claim that πœ‚=0. On the contrary, assume that πœ‚>0. By the definition of asymptotic weaker Meir-Keeler-type sequence, corresponding to πœ‚ use, there exists 𝛿>0 such that for π‘₯,π‘¦βˆˆπ‘‹ with πœ‚β‰€π‘‘(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡)<𝛿+πœ‚, there exists 2𝑛0βˆˆβ„• such that πœ‘2𝑛0(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))<πœ‚. Since limπ‘›β†’βˆžπœ‘π‘›(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))=πœ‚, there exists π‘š0βˆˆβ„• such that πœ‚β‰€πœ‘π‘š(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))<𝛿+πœ‚, for all π‘šβ‰₯π‘š0. Thus, we conclude that πœ“π‘š0+2𝑛0(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))<πœ‚. So we get a contradiction. Therefore, limπ‘›β†’βˆžπœ‘π‘›(𝑑(π‘₯,𝑦)βˆ’π‘‘(𝐴,𝐡))=0, and so limπ‘›β†’βˆžπ‘‘(𝑓𝑛π‘₯,𝑓𝑛𝑦)βˆ’π‘‘(𝐴,𝐡)=0, that is, limπ‘›β†’βˆžπ‘‘(𝑓𝑛π‘₯,𝑓𝑛𝑦)=𝑑(𝐴,𝐡). Thus, we also conclude that limπ‘›β†’βˆžπ‘‘(𝑓𝑛π‘₯,𝑓𝑛+1π‘₯)=𝑑(𝐴,𝐡).

Applying above Lemma 3.4, we are easy to conclude the following theorem.

Theorem 3.5. Let (𝑋,𝑑) be a metric space and 𝐴, 𝐡 nonempty subsets of 𝑋. Suppose π‘“βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is an asymptotic cyclic weaker Meir-Keeler contraction, and if for some π‘₯∈𝐴, the sequence {𝑓2𝑛+1π‘₯} converges to π‘₯∈𝐴, then π‘₯ is a best periodic proximity point of 𝑓 in 𝐴.
Proof. By the definition of the asymptotic weaker Meir-Keeler-type sequence {πœ‘π‘›|πœ‘π‘›βˆΆβ„+→ℝ+}π‘›βˆˆβ„•, thus there exists 2𝑛0βˆˆβ„• such that πœ‘2𝑛0(πœ‚)β‰€πœ‚ for each πœ‚>0. Since {𝑓2𝑛+1π‘₯} converges to π‘₯∈𝐴, corresponding to above 2𝑛0 use, we have 𝑑(𝐴,𝐡)≀𝑑π‘₯,𝑓2𝑛0+1π‘₯≀𝑑π‘₯,𝑓2𝑛+1π‘₯𝑓+𝑑2𝑛+1π‘₯,𝑓2𝑛0+1π‘₯ξ€Έξ€·βˆ’π‘‘(𝐴,𝐡)+𝑑(𝐴,𝐡)≀𝑑π‘₯,𝑓2𝑛+1π‘₯ξ€Έ+πœ‘2𝑛0+1𝑑𝑓2(π‘›βˆ’π‘›0)π‘₯,π‘₯ξ€Έξ€Έξ€·βˆ’π‘‘(𝐴,𝐡)+𝑑(𝐴,𝐡)≀𝑑π‘₯,𝑓2𝑛+1π‘₯ξ€Έ+πœ‘2𝑛0𝑑𝑓2(π‘›βˆ’π‘›0)π‘₯,π‘₯ξ€Έξ€Έξ€·βˆ’π‘‘(𝐴,𝐡)+𝑑(𝐴,𝐡)≀𝑑π‘₯,𝑓2𝑛+1π‘₯𝑓+𝑑2(π‘›βˆ’π‘›0)π‘₯,π‘₯ξ€Έξ€·βˆ’π‘‘(𝐴,𝐡)+𝑑(𝐴,𝐡)≀𝑑π‘₯,𝑓2𝑛+1π‘₯𝑓+𝑑2(π‘›βˆ’π‘›0)π‘₯,𝑓2(π‘›βˆ’π‘›0)+1π‘₯𝑓+𝑑2(π‘›βˆ’π‘›0)+1π‘₯,π‘₯ξ€Έ.(3.5) Letting π‘›β†’βˆž. Then 𝑑(𝐴,𝐡)=𝑑(π‘₯,𝑓2𝑛0+1π‘₯). Thus π‘₯ is a best period proximity point of 𝑓 in 𝐴.

Acknowledgments

The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper. This paper was supported by the National Science Council of the Republic of China.