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Journal of Applied Mathematics
Volume 2012, Article ID 643729, 14 pages
http://dx.doi.org/10.1155/2012/643729
Research Article

Best Proximity Point Theorems for Some New Cyclic Mappings

1Department of Applied Mathematics, National Hsinchu University of Education, Taiwan
2Department of Applied Mathematics, Chung Yuan Christian University, Taiwan

Received 26 February 2012; Accepted 16 June 2012

Academic Editor: Pablo González-Vera

Copyright © 2012 Chi-Ming Chen and Chao-Hung Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By using the stronger Meir-Keeler mapping, we introduce the concepts of the sMK-G-cyclic mappings, sMK-K-cyclic mappings, and sMK-C-cyclic mappings, and then we prove some best proximity point theorems for these various types of contractions. Our results generalize or improve many recent best proximity point theorems in the literature (e.g., Elderd and Veeramani, 2006; Sadiq Basha et al., 2011).

1. Introduction and Preliminaries

Let 𝐴 and 𝐵 be nonempty subsets of a metric space (𝑋,𝑑). Consider a mapping 𝑇𝐴𝐵𝐴𝐵, 𝑇 is called a cyclic map if 𝑇(𝐴)𝐵 and 𝑇(𝐵)𝐴. 𝑥𝐴 is called a best proximity point of 𝑇 in 𝐴 if 𝑑(𝑥,𝑇𝑥)=𝑑(𝐴,𝐵) is satisfied, where 𝑑(𝐴,𝐵)=inf{𝑑(𝑥,𝑦)𝑥𝐴,𝑦𝐵}. 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 𝐵 be nonempty closed convex subsets of a uniformly convex Banach space. Suppose 𝑓𝐴𝐵𝐴𝐵 is a cyclic contraction, that is, 𝑓(𝐴)𝐵 and 𝑓(𝐵)𝐴, and there exists 𝑘(0,1) such that 𝑑(𝑓𝑥,𝑓𝑦)𝑘𝑑(𝑥,𝑦)+(1𝑘)𝑑(𝐴,𝐵)forevery𝑥𝐴,𝑦𝐵.(1.1) Then there exists a unique best proximity point in 𝐴. Further, for each 𝑥𝐴, {𝑓2𝑛𝑥} converges to the best proximity point.

Later, best proximity point theorems for various types of contractions have been obtained in [37]. Particularly, in [8], the authors prove some best proximity point theorems for 𝐾-cyclic mappings and 𝐶-cyclic mappings in the frameworks of metric spaces and uniformly convex Banach spaces, thereby furnishing an optimal approximate solution to the equations of the form 𝑇𝑥=𝑥, where 𝑇 is a non-self-𝐾-cyclic mapping or a non-self-𝐶-cyclic mapping.

Definition 1.2 (see [8]). A pair of mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 is said to form a 𝐾-cyclic mapping between 𝐴 and 𝐵 if there exists a nonnegative real number 𝑘<1/2 such that []𝑑(𝑇𝑥,𝑆𝑦)𝑘𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑆𝑦)+(12𝑘)𝑑(𝐴,𝐵),(1.2) for 𝑥𝐴 and 𝑦𝐵.

Definition 1.3 1.3 (see [8]). A pair of mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 is said to form a 𝐶-cyclic mapping between 𝐴 and 𝐵 if there exists a nonnegative real number 𝑘<1/2 such that []𝑑(𝑇𝑥,𝑆𝑦)𝑘𝑑(𝑥,𝑆𝑦)+𝑑(𝑦,𝑇𝑥)+(12𝑘)𝑑(𝐴,𝐵),(1.3) for 𝑥𝐴 and 𝑦𝐵.

In this paper, we also recall the notion of Meir-Keeler mapping (see [9]). A function 𝜙[0,)[0,) is said to be a Meir-Keeler mapping if, for each 𝜂>0, there exists 𝛿>0 such that, for 𝑡[0,) with 𝜂𝑡<𝜂+𝛿, we have 𝜙(𝑡)<𝜂. Generalization of the above function has been a heavily investigated branch of research. In this study, we introduce the below notion of the stronger Meir-Keeler function 𝜓[0,)[0,1/2).

Definition 1.4. We call 𝜓[0,)[0,1/2) a stronger Meir-Keeler mapping if the mapping 𝜓 satisfies the following condition: 𝜂>0𝛿>0𝛾𝜂10,2[𝑡0,)𝜂𝑡<𝛿+𝜂𝜓(𝑡)<𝛾𝜂.(1.4)

The following provides two example of a stronger Meir-Keeler mapping.

Example 1.5. Let 𝜓[0,)[0,1/2) be defined by 𝜓(𝑡)=0,if𝑡1,𝑡121,if1<𝑡<2,3,if𝑡2.(1.5) Then 𝜓 is a stronger Meir-Keeler mapping which is not a Meir-Keeler function.

Example 1.6. Let 𝜓[0,)[0,1/2) be defined by 𝑡𝜓(𝑡)=.3𝑡+1(1.6) Then 𝜓 is a stronger Meir-Keeler mapping.

In this paper, by using the stronger Meir-Keeler mapping, we introduce the concepts of the sMK-𝐺-cyclic mappings, sMK-𝐾-cyclic mappings and sMK-𝐶-cyclic mappings, and then we prove some best proximity point theorems for these various types of contractions. Our results generalize or improve many recent best proximity point theorems in the literature (e.g., [2, 8]).

2. sMK-G-Cyclic Mappings

In this section, we prove the best proximity point theorems for the sMK-𝐺-cyclic non-self mappings.

Definition 2.1. Let (𝑋,𝑑) be a metric space, and let 𝐴 and 𝐵 be nonempty subsets of 𝑋. A pair of mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 is said to form an sMK-𝐺-cyclic mapping between 𝐴 and 𝐵 if there is a stronger Meir-Keeler function 𝜓+[0,1/2) in 𝑋 such that for 𝑥𝐴 and 𝑦𝐵, [],𝑑(𝑇𝑥,𝑆𝑦)𝑑(𝐴,𝐵)𝜓(𝑑(𝑥,𝑦))𝐺(𝑥,𝑦)2𝑑(𝐴,𝐵)(2.1) where 𝐺(𝑥,𝑦)=max{𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑆𝑦),𝑑(𝑥,𝑆𝑦),𝑑(𝑦,𝑇𝑥)}.

Lemma 2.2. Let 𝐴 and 𝐵 be nonempty subsets of a metric space (𝑋,𝑑). Suppose that the mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐺-cyclic mapping between 𝐴 and 𝐵. Then there exists a sequence {𝑥𝑛} in 𝑋 such that lim𝑛𝑑𝑥𝑛,𝑥𝑛+1=𝑑(𝐴,𝐵).(2.2)

Proof. Let 𝑥0𝐴 be given and let 𝑥2𝑛+1=𝑇𝑥2𝑛 and 𝑥2𝑛+2=𝑆𝑥2𝑛+1 for each 𝑛{0}. Taking into account (2.1) and the definition of the stronger Meir-Keeler function 𝜓+[0,1/2), we have that for each 𝑛{0}𝑑𝑥2𝑛+1,𝑥2𝑛+2𝑑(𝐴,𝐵)=𝑑𝑇𝑥2𝑛,𝑆𝑥2𝑛+1𝑑𝑥𝑑(𝐴,𝐵)𝜓2𝑛,𝑥2𝑛+1𝐺𝑥2𝑛,𝑥2𝑛+1,2𝑑(𝐴,𝐵)(2.3) where 𝐺𝑥2𝑛,𝑥2𝑛+1𝑑𝑥=max2𝑛,𝑥2𝑛+1𝑥,𝑑2𝑛,𝑇𝑥2𝑛𝑥,𝑑2𝑛+1,𝑆𝑥2𝑛+1𝑥,𝑑2𝑛,𝑆𝑥2𝑛+1,𝑑𝑥2𝑛+1,𝑇𝑥2𝑛𝑑𝑥=max2𝑛,𝑥2𝑛+1𝑥,𝑑2𝑛,𝑥2𝑛+1𝑥,𝑑2𝑛+1,𝑥2𝑛+2𝑥,𝑑2𝑛,𝑥2𝑛+2𝑥,𝑑2𝑛+1,𝑥2𝑛+1𝑑𝑥max2𝑛,𝑥2𝑛+1𝑥,𝑑2𝑛,𝑥2𝑛+1𝑥,𝑑2𝑛+1,𝑥2𝑛+2𝑥,𝑑2𝑛,𝑥2𝑛+1𝑥+𝑑2𝑛+1,𝑥2𝑛+2𝑑𝑥,02max2𝑛,𝑥2𝑛+1𝑥,𝑑2𝑛+1,𝑥2𝑛+2.(2.4) Taking into account (2.3) and (2.4), we have that for each 𝑛{0}𝑑𝑥2𝑛+1,𝑥2𝑛+2𝑑𝑥𝑑(𝐴,𝐵)𝜓2𝑛,𝑥2𝑛+1𝑑𝑥2max2𝑛,𝑥2𝑛+1𝑥,𝑑2𝑛+1,𝑥2𝑛+2𝑑𝑥𝑑(𝐴,𝐵)<max2𝑛,𝑥2𝑛+1𝑥,𝑑2𝑛+1,𝑥2𝑛+2𝑑(𝐴,𝐵),(2.5) and so we conclude that 𝑑𝑥2𝑛+1,𝑥2𝑛+2𝑥𝑑(𝐴,𝐵)<𝑑2𝑛,𝑥2𝑛+1𝑑(𝐴,𝐵),(2.6) and, for each 𝑛, 𝑑𝑥2𝑛,𝑥2𝑛+1𝑑(𝐴,𝐵)=𝑑𝑆𝑥2𝑛1,𝑇𝑥2𝑛𝑑(𝐴,𝐵)=𝑑𝑇𝑥2𝑛,𝑆𝑥2𝑛1𝑑𝑥𝑑(𝐴,𝐵)𝜓2𝑛,𝑥2𝑛1𝐺𝑥2𝑛,𝑥2𝑛1,2𝑑(𝐴,𝐵)(2.7) where 𝐺𝑥2𝑛,𝑥2𝑛1𝑑𝑥=max2𝑛,𝑥2𝑛1𝑥,𝑑2𝑛,𝑇𝑥2𝑛𝑥,𝑑2𝑛1,𝑆𝑥2𝑛1𝑥,𝑑2𝑛,𝑆𝑥2𝑛1𝑥,𝑑2𝑛1,𝑇𝑥2𝑛𝑑𝑥max2𝑛,𝑥2𝑛1𝑥,𝑑2𝑛,𝑥2𝑛+1𝑥,𝑑2𝑛1,𝑥2𝑛𝑥,𝑑2𝑛,𝑥2𝑛𝑥,𝑑2𝑛1,𝑥2𝑛+1𝑑𝑥max2𝑛,𝑥2𝑛1𝑥,𝑑2𝑛,𝑥2𝑛+1𝑥,𝑑2𝑛1,𝑥2𝑛𝑥,0,𝑑2𝑛1,𝑥2𝑛𝑥+𝑑2𝑛,𝑥2𝑛+1𝑑𝑥2max2𝑛1,𝑥2𝑛𝑥,𝑑2𝑛,𝑥2𝑛+1.(2.8) Taking into account (2.7) and (2.8), we have that for each 𝑛𝑑𝑥2𝑛,𝑥2𝑛+1𝑑𝑥𝑑(𝐴,𝐵)𝜓2𝑛1,𝑥2𝑛𝑑𝑥2max2𝑛,𝑥2𝑛1𝑥,𝑑2𝑛,𝑥2𝑛+1𝑑𝑥𝑑(𝐴,𝐵)<max2𝑛1,𝑥2𝑛𝑥,𝑑2𝑛,𝑥2𝑛+1𝑑(𝐴,𝐵),(2.9) and so we conclude that 𝑑𝑥2𝑛+1,𝑥2𝑛+2𝑥𝑑(𝐴,𝐵)<𝑑2𝑛,𝑥2𝑛+1𝑑(𝐴,𝐵).(2.10) Generally, by (2.6) and (2.10), we have that for each 𝑛𝑑𝑥𝑛+1,𝑥𝑛+2𝑥<𝑑𝑛,𝑥𝑛+1,𝑑𝑥𝑛+1,𝑥𝑛+2𝑑𝑥𝑑(𝐴,𝐵)𝜓𝑛,𝑥𝑛+1𝑑𝑥2𝑛,𝑥𝑛+1.𝑑(𝐴,𝐵)(2.11) Thus the sequence {𝑑(𝑥𝑛,𝑥𝑛+1)}𝑛{0} is decreasing and bounded below and hence it is convergent. Let lim𝑛𝑑(𝑥𝑛,𝑥𝑛+1)=𝜂0. Then there exists 𝑛0 and 𝛿>0 such that for all 𝑛 with 𝑛𝑛0𝑥𝜂𝑑𝑛,𝑥𝑛+1<𝜂+𝛿.(2.12) Taking into account (2.12) and the definition of stronger Meir-Keeler function 𝜓, corresponding to 𝜂 use, there exists 𝛾𝜂[0,1/2) such that 𝜓𝑑𝑥𝑛,𝑥𝑛+1<𝛾𝜂𝑛𝑛0.(2.13) Thus, we can deduce that for each 𝑛 with 𝑛𝑛0+1𝑑𝑥𝑛,𝑥𝑛+1𝑑𝑥𝑑(𝐴,𝐵)𝜓𝑛1,𝑥𝑛𝑑𝑥2𝑛1,𝑥𝑛𝑑(𝐴,𝐵)<𝛾𝜂𝑑𝑥2𝑛1,𝑥𝑛,𝑑(𝐴,𝐵)(2.14) and so 𝑑𝑥𝑛,𝑥𝑛+1𝑑(𝐴,𝐵)<𝛾𝜂𝑑𝑥2𝑛1,𝑥𝑛<𝑑(𝐴,𝐵)2𝛾𝜂2𝑑𝑥𝑛2,𝑥𝑛1<𝑑(𝐴,𝐵)<2𝛾𝜂𝑛𝑛0𝑑𝑥𝑛0,𝑥𝑛0+1.𝑑(𝐴,𝐵)(2.15) Since 𝛾𝜂[0,1/2), we get lim𝑛𝑑𝑥𝑛,𝑥𝑛+1𝑑(𝐴,𝐵)=0,(2.16) that is, lim𝑛𝑑(𝑥𝑛,𝑥𝑛+1)=𝑑(𝐴,𝐵).

Lemma 2.3. Let 𝐴 and 𝐵 be nonempty closed subsets of a metric space (𝑋,𝑑). Suppose that the mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐺-cyclic mapping between 𝐴 and 𝐵. For a fixed point 𝑥0𝐴, let 𝑥2𝑛+1=𝑇𝑥2𝑛 and 𝑥2𝑛+2=𝑆𝑥2𝑛+1. Then the sequence {𝑥𝑛} is bounded.

Proof. It follows from Lemma 2.2 that {𝑑(𝑥2𝑛1,𝑥2𝑛)} is convergent and hence it is bounded. Since 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐺-cyclic mapping between 𝐴 and 𝐵, there is a stronger Meir-Keeler function 𝜓+[0,1/2) in 𝑋 such that 𝑑𝑥2𝑛,𝑇𝑥0=𝑑𝑆𝑥2𝑛1,𝑇𝑥0=𝑑𝑇𝑥0,𝑆𝑥2𝑛1𝑑𝑥𝜓0,𝑥2𝑛1𝐺𝑑𝑥0,𝑥2𝑛12𝑑(𝐴,𝐵)+𝑑(𝐴,𝐵),(2.17) where 𝐺𝑑𝑥0,𝑥2𝑛1𝑑𝑥=max0,𝑥2𝑛1𝑥,𝑑0,𝑇𝑥0𝑥,𝑑2𝑛1,𝑆𝑥2𝑛1𝑥,𝑑0,𝑆𝑥2𝑛1𝑥,𝑑2𝑛1,𝑇𝑥0𝑑𝑥=max0,𝑥2𝑛1𝑥,𝑑0,𝑇𝑥0𝑥,𝑑2𝑛1,𝑥2𝑛𝑥,𝑑0,𝑥2𝑛𝑥,𝑑2𝑛1,𝑇𝑥0𝑑𝑥max0,𝑇𝑥0+𝑑𝑇𝑥0,𝑥2𝑛𝑥+𝑑2𝑛,𝑥2𝑛1𝑥,𝑑0,𝑇𝑥0𝑥,𝑑2𝑛1,𝑥2𝑛,𝑑𝑥0,𝑇𝑥0+𝑑𝑇𝑥0,𝑥2𝑛𝑥,𝑑2𝑛1,𝑥2𝑛𝑥+𝑑2𝑛,𝑇𝑥0𝑥=𝑑0,𝑇𝑥0+𝑑𝑇𝑥0,𝑥2𝑛𝑥+𝑑2𝑛,𝑥2𝑛1.(2.18) Taking into account (2.17) and (2.18), we get 𝑑𝑥2𝑛,𝑇𝑥0𝜓𝑑𝑥0,𝑥2𝑛1𝑑𝑥1𝜓0,𝑥2𝑛1𝑑𝑥0,𝑇𝑥0𝑥+𝑑2𝑛,𝑥2𝑛1+𝑑𝑥12𝜓0,𝑥2𝑛1𝑑𝑥1𝜓0,𝑥2𝑛1𝑑(𝐴,𝐵)(2.19) Therefore, the sequence {𝑥2𝑛} is bounded. Similarly, it can be shown that {𝑥2𝑛+1} is also bounded. So we complete the proof.

Theorem 2.4. Let 𝐴 and 𝐵 be nonempty closed subsets of a metric space. Let the mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐺-cyclic mapping between 𝐴 and 𝐵. For a fixed point 𝑥0𝐴, let 𝑥2𝑛+1=𝑇𝑥2𝑛 and 𝑥2𝑛+2=𝑆𝑥2𝑛+1. Suppose that the sequence {𝑥2𝑛} has a subsequence converging to some element 𝑥 in 𝐴. Then, 𝑥 is a best proximity point of 𝑇.

Proof. Suppose that a subsequence {𝑥2𝑛𝑘} converges to 𝑥 in 𝐴. It follows from Lemma 2.2 that 𝑑(𝑥2𝑛𝑘1,𝑥2𝑛𝑘) converges to 𝑑(𝐴,𝐵). Since 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐺-cyclic mapping between 𝐴 and 𝐵 and taking into account (2.13), we have that for each 2𝑛𝑘 with 2𝑛𝑘𝑛0+1𝑑𝑥2𝑛𝑘,𝑇𝑥=𝑑𝑇𝑥,𝑥2𝑛𝑘𝑑𝜓𝑥,𝑥2𝑛𝑘1𝐺𝑥,𝑥2𝑛𝑘12𝑑(𝐴,𝐵)+𝑑(𝐴,𝐵)<𝛾𝜂𝐺𝑥,𝑥2𝑛𝑘12𝑑(𝐴,𝐵)+𝑑(𝐴,𝐵),(2.20) where 𝐺𝑥,𝑥2𝑛𝑘1𝑑=max𝑥,𝑥2𝑛𝑘1𝑥,𝑑(𝑥,𝑇𝑥),𝑑2𝑛𝑘1,𝑆𝑥2𝑛𝑘1,𝑑𝑥,𝑆𝑥2𝑛𝑘1𝑥,𝑑2𝑛𝑘1𝑑,𝑇𝑥=max𝑥,𝑥2𝑛𝑘1𝑥,𝑑(𝑥,𝑇𝑥),𝑑2𝑛𝑘1,𝑥2𝑛𝑘,𝑑𝑥,𝑥2𝑛𝑘𝑥,𝑑2𝑛𝑘1𝑑,𝑇𝑥=max𝑥,𝑥2𝑛𝑘𝑥+𝑑2𝑛𝑘,𝑥2𝑛𝑘1𝑥,𝑑(𝑥,𝑇𝑥),𝑑2𝑛𝑘1,𝑥2𝑛𝑘,𝑑𝑥,𝑥2𝑛𝑘𝑥,𝑑2𝑛𝑘𝑥,𝑇𝑥+𝑑2𝑛𝑘1,𝑥2𝑛𝑘𝑑𝑥,𝑥2𝑛𝑘𝑥+𝑑2𝑛𝑘,𝑥2𝑛𝑘1𝑥+𝑑2𝑛𝑘.,𝑇𝑥(2.21) Following from (2.20) and (2.21), we obtain that 𝑑𝑥2𝑛𝑘,𝑇𝑥𝛾𝜂𝑑𝑥,𝑥2𝑛𝑘𝑥+𝑑2𝑛𝑘,𝑥2𝑛𝑘1𝑥+𝑑2𝑛𝑘,𝑇𝑥2𝑑(𝐴,𝐵)+𝑑(𝐴,𝐵),(2.22) that is, we have that 𝑑𝑥(𝐴,𝐵)𝑑2𝑛𝑘𝛾,𝑇𝑥𝜂1𝛾𝜂𝑑𝑥,𝑥2𝑛𝑘𝑥+𝑑2𝑛𝑘,𝑥2𝑛𝑘1+𝛾1𝜂1𝛾𝜂𝑑(𝐴,𝐵),(2.23) letting 𝑘. Then we conclude that 𝛾𝑑(𝐴,𝐵)𝑑(𝑥,𝑇𝑥)𝜂1𝛾𝜂[]+𝛾𝑑(𝐴,𝐵)+01𝜂1𝛾𝜂𝑑(𝐴,𝐵).(2.24) Therefore, 𝑑(𝑥,𝑇𝑥)=𝑑(𝐴,𝐵), that is, 𝑥 is a best proximity point of 𝑇.

3. sMK-K-Cyclic Mappings

In this section, we prove the best proximity point theorems for the sMK-𝐾-cyclic non-self mappings.

Definition 3.1. Let (𝑋,𝑑) be a metric space, and let 𝐴 and 𝐵 be nonempty subsets of 𝑋. A pair of mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 is said to form an sMK-𝐾-cyclic mapping between 𝐴 and 𝐵 if there is a stronger Meir-Keeler function 𝜓+[0,1/2) in 𝑋 such that, for 𝑥𝐴 and 𝑦𝐵, [],𝑑(𝑇𝑥,𝑆𝑦)𝑑(𝐴,𝐵)𝜓(𝑑(𝑥,𝑦))𝐾(𝑥,𝑦)2𝑑(𝐴,𝐵)(3.1) where 𝐾(𝑥,𝑦)=𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑆𝑦).

Lemma 3.2. Let 𝐴 and 𝐵 be nonempty subsets of a metric space (𝑋,𝑑). Suppose that the mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐾-cyclic mapping between 𝐴 and 𝐵. Then there exists a sequence {𝑥𝑛} in 𝑋 such that lim𝑛𝑑𝑥𝑛,𝑥𝑛+1=𝑑(𝐴,𝐵).(3.2)

Proof. Let 𝑥0𝐴 be given and let 𝑥2𝑛+1=𝑇𝑥2𝑛 and 𝑥2𝑛+2=𝑆𝑥2𝑛+1 for each 𝑛{0}. Taking into account (3.1) and the definition of the stronger Meir-Keeler function 𝜓+[0,1/2), we have that 𝑛{0}𝑑𝑥2𝑛+1,𝑥2𝑛+2𝑑(𝐴,𝐵)=𝑑𝑇𝑥2𝑛,𝑆𝑥2𝑛+1𝑑𝑥𝑑(𝐴,𝐵)𝜓2𝑛,𝑥2𝑛+1𝐾𝑥2𝑛,𝑥2𝑛+1,2𝑑(𝐴,𝐵)(3.3) where 𝐾𝑥2𝑛,𝑥2𝑛+1𝑥=𝑑2𝑛,𝑇𝑥2𝑛𝑥+𝑑2𝑛+1,𝑆𝑥2𝑛+1𝑥=𝑑2𝑛,𝑥2𝑛+1𝑥+𝑑2𝑛+1,𝑥2𝑛+2.(3.4) Taking into account (3.3) and (3.4), we have that 𝑑𝑥2𝑛+1,𝑥2𝑛+2𝑥<𝑑2𝑛,𝑥2𝑛+1.(3.5) Similarly, we can conclude that 𝑑𝑥2𝑛,𝑥2𝑛+1𝑥<𝑑2𝑛1,𝑥2𝑛.(3.6) Generally, by (3.5) and (3.6), we have that for each 𝑛{0}𝑑𝑥𝑛+1,𝑥𝑛+2𝑥<𝑑𝑛,𝑥𝑛+1.(3.7) Thus the sequence {𝑑(𝑥𝑛,𝑥𝑛+1)}𝑛{0} is decreasing and bounded below and hence it is convergent. Let lim𝑛𝑑(𝑥𝑛,𝑥𝑛+1)𝑑(𝐴,𝐵)=𝜂0. Then there exists 𝑛0 and 𝛿>0 such that for all 𝑛 with 𝑛𝑛0𝑥𝜂𝑑𝑛,𝑥𝑛+1<𝜂+𝛿.(3.8) Taking into account (3.8) and the definition of stronger Meir-Keeler function 𝜓, corresponding to 𝜂 use, there exists 𝛾𝜂[0,1/2) such that 𝜓𝑑𝑥𝑛,𝑥𝑛+1<𝛾𝜂𝑛𝑛0.(3.9) Thus, we can deduce that for each 𝑛 with 𝑛𝑛0+1𝑑𝑥𝑛,𝑥𝑛+1𝑑𝑥𝑑(𝐴,𝐵)𝜓𝑛1,𝑥𝑛𝐾𝑥𝑛1,𝑥𝑛2𝑑(𝐴,𝐵)<𝛾𝜂𝑑𝑥𝑛1,𝑇𝑥𝑛1𝑥+𝑑𝑛,𝑆𝑥𝑛2𝑑(𝐴,𝐵)=𝛾𝜂𝑑𝑥𝑛1,𝑥𝑛𝑥+𝑑𝑛,𝑥𝑛+1,2𝑑(𝐴,𝐵)(3.10) that is, 𝑑𝑥𝑛,𝑥𝑛+1𝛾𝑑(𝐴,𝐵)<𝜂1𝛾𝜂𝑑𝑥𝑛1,𝑥𝑛,𝑑(𝐴,𝐵)(3.11) since 𝛾𝜂[0,1/2). Therefore we get that for each 𝑛 with 𝑛𝑛0+1𝑑𝑥𝑛,𝑥𝑛+1𝛾𝑑(𝐴,𝐵)<𝜂1𝛾𝜂𝑑𝑥𝑛1,𝑥𝑛<𝛾𝑑(𝐴,𝐵)𝜂1𝛾𝜂2𝑑𝑥𝑛2,𝑥𝑛1<𝛾𝑑(𝐴,𝐵)<𝜂1𝛾𝜂𝑛𝑛0𝑑𝑥𝑛0,𝑥𝑛0+1.𝑑(𝐴,𝐵)(3.12) Since 𝛾𝜂[0,1/2), we get lim𝑛𝑑𝑥𝑛,𝑥𝑛+1𝑑(𝐴,𝐵)=0,(3.13) that is, lim𝑛𝑑(𝑥𝑛,𝑥𝑛+1)=𝑑(𝐴,𝐵).

Lemma 3.3. Let 𝐴 and 𝐵 be nonempty closed subsets of a metric space (𝑋,𝑑). Suppose that the mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐾-cyclic mapping between 𝐴 and 𝐵. For a fixed point 𝑥0𝐴, let 𝑥2𝑛+1=𝑇𝑥2𝑛 and 𝑥2𝑛+2=𝑆𝑥2𝑛+1. Then the sequence {𝑥𝑛} is bounded.

Proof. It follows from Lemma 3.2 that {𝑑(𝑥2𝑛1,𝑥2𝑛)} is convergent and hence it is bounded. Since 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐾-cyclic mapping between 𝐴 and 𝐵, there is a stronger Meir-Keeler function 𝜓+[0,1/2) in 𝑋 such that, for 𝑥0𝐴 and 𝑥2𝑛1𝐵, 𝑑𝑥2𝑛,𝑇𝑥0𝑑(𝐴,𝐵)=𝑑𝑆𝑥2𝑛1,𝑇𝑥0𝑑(𝐴,𝐵)=𝑑𝑇𝑥0,𝑆𝑥2𝑛1𝑑𝑥𝑑(𝐴,𝐵)𝜓0,𝑥2𝑛1𝐾𝑥0,𝑥2𝑛1,2𝑑(𝐴,𝐵)(3.14) where 𝐾(𝑥0,𝑥2𝑛1)=𝑑(𝑥0,𝑇𝑥0)+𝑑(𝑥2𝑛1,𝑆𝑥2𝑛1). So we get that 𝑑𝑥2𝑛,𝑇𝑥0𝑑𝑥𝜓0,𝑥2𝑛1𝑑𝑥0,𝑇𝑥0𝑥+𝑑2𝑛1,𝑥2𝑛+𝑑𝑥12𝜓0,𝑥2𝑛1𝑑(𝐴,𝐵).(3.15) Therefore, the sequence {𝑥2𝑛} is bounded. Similarly, it can be shown that {𝑥2𝑛+1} is also bounded. So we complete the proof.

Theorem 3.4. Let 𝐴 and 𝐵 be nonempty closed subsets of a metric space. Let the mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐾-cyclic mapping between 𝐴 and 𝐵. For a fixed point 𝑥0𝐴, let 𝑥2𝑛+1=𝑇𝑥2𝑛 and 𝑥2𝑛+2=𝑆𝑥2𝑛+1. Suppose that the sequence {𝑥2𝑛} has a subsequence converging to some element 𝑥 in 𝐴. Then, 𝑥 is a best proximity point of 𝑇.

Proof. Suppose that a subsequence {𝑥2𝑛𝑘} converges to 𝑥 in 𝐴. It follows from Lemma 2.2 that 𝑑(𝑥2𝑛𝑘1,𝑥2𝑛𝑘) converges to 𝑑(𝐴,𝐵). Since 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐾-cyclic mapping between 𝐴 and 𝐵 and taking into account (3.9), we have that for each 2𝑛𝑘 with 2𝑛𝑘𝑛0+1𝑑𝑥2𝑛𝑘,𝑇𝑥=𝑑𝑇𝑥,𝑥2𝑛𝑘𝑑𝜓𝑥,𝑥2𝑛𝑘1𝐾𝑥,𝑥2𝑛𝑘12𝑑(𝐴,𝐵)+𝑑(𝐴,𝐵)<𝛾𝜂𝐾𝑥,𝑥2𝑛𝑘12𝑑(𝐴,𝐵)+𝑑(𝐴,𝐵),(3.16) where 𝐾𝑥,𝑥2𝑛𝑘1𝑥=𝑑(𝑥,𝑇𝑥)+𝑑2𝑛𝑘1,𝑆𝑥2𝑛𝑘1𝑥=𝑑(𝑥,𝑇𝑥)+𝑑2𝑛𝑘1,𝑥2𝑛𝑘.(3.17) Following from (3.16) and (3.17), we obtain that for each 2𝑛𝑘 with 2𝑛𝑘𝑛0+1𝑑𝑥(𝐴,𝐵)𝑑2𝑛𝑘,𝑇𝑥𝛾𝜂𝑑𝑥(𝑥,𝑇𝑥)+𝑑2𝑛𝑘,𝑥2𝑛𝑘1+12𝛾𝜂𝑑(𝐴,𝐵),(3.18) Letting 𝑘. Then we conclude that 𝑑(𝑥,𝑇𝑥)=𝑑(𝐴,𝐵), that is, 𝑥 is a best proximity point of 𝑇.

4. sMK-C-Cyclic Mappings

In this section, we prove the best proximity point theorems for the sMK-𝐶-cyclic non-self mappings.

Definition 4.1. Let (𝑋,𝑑) be a metric space, and let 𝐴 and 𝐵 be nonempty subsets of 𝑋. A pair of mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 is said to form an sMK-𝐶-cyclic mapping between 𝐴 and 𝐵 if there is a stronger Meir-Keeler function 𝜓+[0,1/2) in 𝑋 such that, for 𝑥𝐴 and 𝑦𝐵, [],𝑑(𝑇𝑥,𝑆𝑦)𝑑(𝐴,𝐵)𝜓(𝑑(𝑥,𝑦))𝐶(𝑥,𝑦)2𝑑(𝐴,𝐵)(4.1) where 𝐶(𝑥,𝑦)=𝑑(𝑥,𝑆𝑦)+𝑑(𝑦,𝑇𝑥).

Lemma 4.2. Let 𝐴 and 𝐵 be nonempty subsets of a metric space (𝑋,𝑑). Suppose that the mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐶-cyclic mapping between 𝐴 and 𝐵. Then there exists a sequence {𝑥𝑛} in 𝑋 such that lim𝑛𝑑𝑥𝑛,𝑥𝑛+1=𝑑(𝐴,𝐵).(4.2)

Proof. Let 𝑥0𝐴 be given and let 𝑥2𝑛+1=𝑇𝑥2𝑛 and 𝑥2𝑛+2=𝑆𝑥2𝑛+1 for each 𝑛{0}. Taking into account (4.1) and the definition of the stronger Meir-Keeler function 𝜓+[0,1/2), we have that 𝑛{0}𝑑𝑥2𝑛+1,𝑥2𝑛+2𝑑(𝐴,𝐵)=𝑑𝑇𝑥2𝑛,𝑆𝑥2𝑛+1𝑑𝑥𝑑(𝐴,𝐵)𝜓2𝑛,𝑥2𝑛+1𝐶𝑥2𝑛,𝑥2𝑛+1,2𝑑(𝐴,𝐵)(4.3) where 𝐶𝑥2𝑛,𝑥2𝑛+1𝑥=𝑑2𝑛,𝑆𝑥2𝑛+1𝑥+𝑑2𝑛+1,𝑇𝑥2𝑛𝑥=𝑑2𝑛,𝑥2𝑛+2𝑥+𝑑2𝑛+1,𝑥2𝑛+1𝑥𝑑2𝑛,𝑥2𝑛+1𝑥+𝑑2𝑛+1,𝑥2𝑛+2.(4.4) Taking into account (4.3) and (4.4), we conclude that 𝑑𝑥2𝑛+1,𝑥2𝑛+2𝑥<𝑑2𝑛,𝑥2𝑛+1.(4.5) Similarly, we can conclude that 𝑑𝑥2𝑛,𝑥2𝑛+1𝑥<𝑑2𝑛1,𝑥2𝑛.(4.6) Generally, by (4.5) and (4.6), we have that for each 𝑛{0}𝑑𝑥𝑛+1,𝑥𝑛+2𝑥<𝑑𝑛,𝑥𝑛+1.(4.7) Thus the sequence {𝑑(𝑥𝑛,𝑥𝑛+1)}𝑛{0} is decreasing and bounded below and hence it is convergent. Let lim𝑛𝑑(𝑥𝑛,𝑥𝑛+1)=𝜂0. Then there exists 𝑛0 and 𝛿>0 such that for all 𝑛 with 𝑛𝑛0𝑥𝜂𝑑𝑛,𝑥𝑛+1<𝜂+𝛿.(4.8) Taking into account (4.5) and the definition of stronger Meir-Keeler function 𝜓, corresponding to 𝜂 use, there exists 𝛾𝜂[0,1/2) such that 𝜓𝑑𝑥𝑛,𝑥𝑛+1<𝛾𝜂𝑛𝑛0.(4.9) Thus, we can deduce that for each 𝑛 with 𝑛𝑛0+1𝑑𝑥𝑛,𝑥𝑛+1𝑑𝑥𝑑(𝐴,𝐵)𝜓𝑛1,𝑥𝑛𝐶𝑥𝑛1,𝑥𝑛2𝑑(𝐴,𝐵)<𝛾𝜂𝑑𝑥𝑛1,𝑆𝑥𝑛𝑥+𝑑𝑛,𝑇𝑥𝑛12𝑑(𝐴,𝐵)=𝛾𝜂𝑑𝑥𝑛1,𝑥𝑛+1𝑥+𝑑𝑛,𝑥𝑛2𝑑(𝐴,𝐵)𝛾𝜂𝑑𝑥𝑛1,𝑥𝑛𝑥+𝑑𝑛,𝑥𝑛+1,+02𝑑(𝐴,𝐵)(4.10) that is, 𝑑𝑥𝑛,𝑥𝑛+1𝛾𝑑(𝐴,𝐵)<𝜂1𝛾𝜂𝑑𝑥𝑛1,𝑥𝑛,𝑑(𝐴,𝐵)(4.11) since 𝛾𝜂[0,1). Therefore we get that for each 𝑛 with 𝑛𝑛0+1𝑑𝑥𝑛,𝑥𝑛+1𝛾𝑑(𝐴,𝐵)<𝜂1𝛾𝜂𝑑𝑥𝑛1,𝑥𝑛<𝛾𝑑(𝐴,𝐵)𝜂1𝛾𝜂2𝑑𝑥𝑛2,𝑥𝑛1<𝛾𝑑(𝐴,𝐵)<𝜂1𝛾𝜂𝑛𝑛0𝑑𝑥𝑛0,𝑥𝑛0+1.𝑑(𝐴,𝐵)(4.12) Since 𝛾𝜂[0,1/2), we obtain that lim𝑛𝑑(𝑥𝑛,𝑥𝑛+1)=𝑑(𝐴,𝐵).

Lemma 4.3. Let 𝐴 and 𝐵 be nonempty closed subsets of a metric space (𝑋,𝑑). Suppose that the mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐶-cyclic mapping between 𝐴 and 𝐵. For a fixed point 𝑥0𝐴, let 𝑥2𝑛+1=𝑇𝑥2𝑛 and 𝑥2𝑛+2=𝑆𝑥2𝑛+1. Then the sequence {𝑥𝑛} is bounded.

Proof. It follows from Lemma 4.2 that {𝑑(𝑥2𝑛1,𝑥2𝑛)} is convergent and hence it is bounded. Since 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐶-cyclic mapping between 𝐴 and 𝐵, there is a stronger Meir-Keeler function 𝜓+[0,1/2) in 𝑋 such that for 𝑥0𝐴 and 𝑥2𝑛1𝐵, 𝑑𝑥2𝑛,𝑇𝑥0𝑑(𝐴,𝐵)=𝑑𝑆𝑥2𝑛1,𝑇𝑥0𝑑(𝐴,𝐵)=𝑑𝑇𝑥0,𝑆𝑥2𝑛1𝑑𝑥𝑑(𝐴,𝐵)𝜓0,𝑥2𝑛1𝐶𝑥0,𝑥2𝑛1,2𝑑(𝐴,𝐵)(4.13) where 𝐶𝑥0,𝑥2𝑛1𝑥=𝑑0,𝑆𝑥2𝑛1𝑥+𝑑2𝑛1,𝑇𝑥0𝑥=𝑑0,𝑥2𝑛𝑥+𝑑2𝑛1,𝑇𝑥0.(4.14) So we get that 𝑑𝑥2𝑛,𝑇𝑥0𝑑𝑥𝜓0,𝑥2𝑛1𝑑𝑥0,𝑥2𝑛𝑥+𝑑2𝑛1,𝑇𝑥0+𝑑𝑥12𝜓0,𝑥2𝑛1𝑑𝑥𝑑(𝐴,𝐵)𝜓0,𝑥2𝑛1𝑑𝑥2𝑛1,𝑥2𝑛𝑥+2𝑑2𝑛,𝑇𝑥0𝑥+𝑑0,𝑇𝑥0+𝑑𝑥12𝜓0,𝑥2𝑛1𝑑(𝐴,𝐵).(4.15) Therefore, the sequence {𝑥2𝑛} is bounded. Similarly, it can be shown that {𝑥2𝑛+1} is also bounded. So we complete the proof.

Theorem 4.4. Let 𝐴 and 𝐵 be nonempty closed subsets of a metric space. Let the mappings 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐶-cyclic mapping between 𝐴 and 𝐵. For a fixed point 𝑥0𝐴, let 𝑥2𝑛+1=𝑇𝑥2𝑛 and 𝑥2𝑛+2=𝑆𝑥2𝑛+1. Suppose that the sequence {𝑥2𝑛} has a subsequence converging to some element 𝑥 in 𝐴. Then, 𝑥 is a best proximity point of 𝑇.

Proof. Suppose that a subsequence {𝑥2𝑛𝑘} converges to 𝑥 in 𝐴. It follows from Lemma 2.2 that 𝑑(𝑥2𝑛𝑘1,𝑥2𝑛𝑘) converges to 𝑑(𝐴,𝐵). Since 𝑇𝐴𝐵 and 𝑆𝐵𝐴 form an sMK-𝐶-cyclic mapping between 𝐴 and 𝐵 and taking into account (4.9), we have that, for each 2𝑛𝑘 with 2𝑛𝑘𝑛0+1, 𝑑𝑥2𝑛𝑘,𝑇𝑥=𝑑𝑇𝑥,𝑥2𝑛𝑘𝑑𝜓𝑥,𝑥2𝑛𝑘1𝐶𝑥,𝑥2𝑛𝑘12𝑑(𝐴,𝐵)+𝑑(𝐴,𝐵)<𝛾𝜂𝐶𝑥,𝑥2𝑛𝑘12𝑑(𝐴,𝐵)+𝑑(𝐴,𝐵),(4.16) where 𝐶𝑥,𝑥2𝑛𝑘1=𝑑𝑥,𝑆𝑥2𝑛𝑘1𝑥+𝑑2𝑛𝑘1,𝑇𝑥=𝑑𝑥,𝑥2𝑛𝑘𝑥+𝑑2𝑛𝑘1.,𝑇𝑥(4.17) Following from (4.16) and (4.17), we obtain that 𝑑𝑥2𝑛𝑘,𝑇𝑥𝛾𝜂𝑑𝑥,𝑥2𝑛𝑘𝑥+𝑑2𝑛𝑘,𝑥2𝑛𝑘1𝑥+𝑑2𝑛𝑘,𝑇𝑥2𝑑(𝐴,𝐵)+𝑑(𝐴,𝐵),(4.18) that is, we have that 𝑥𝑑(𝐴,𝐵)𝑑2𝑛𝑘𝛾,𝑇𝑥𝜂1𝛾𝜂𝑑𝑥,𝑥2𝑛𝑘𝑥+𝑑2𝑛𝑘,𝑥2𝑛𝑘1+𝛾1𝜂1𝛾𝜂𝑑(𝐴,𝐵).(4.19) Letting 𝑘. Then we conclude that 𝛾𝑑(𝐴,𝐵)𝑑(𝑥,𝑇𝑥)𝜂1𝛾𝜂[]+𝛾𝑑(𝐴,𝐵)+01𝜂1𝛾𝜂𝑑(𝐴,𝐵).(4.20) Therefore, 𝑑(𝑥,𝑇𝑥)=𝑑(𝐴,𝐵), that is, 𝑥 is a best proximity point of 𝑇.

Acknowledgment

The authors would like to thank the referee(s) for many useful comments and suggestions for the improvement of the paper.

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