Generalized Proximal 𝜓-Contraction Mappings and Best Proximity Points

Abstract

We generalized the notion of proximal contractions of the first and the second kinds and established the best proximity point theorems for these classes. Our results improve and extend recent result of Sadiq Basha (2011) and some authors.

1. Introduction

The significance of fixed point theory stems from the fact that it furnishes a unified treatment and is a vital tool for solving equations of form 𝑇𝑥=𝑥 where 𝑇 is a self-mapping defined on a subset of a metric space, a normed linear space, topological vector space or some suitable space. Some applications of fixed point theory can be found in [1–12]. However, almost all such results dilate upon the existence of a fixed point for self-mappings. Nevertheless, if 𝑇 is a non-self-mapping, then it is probable that the equation 𝑇𝑥=𝑥 has no solution, in which case best approximation theorems explore the existence of an approximate solution whereas best proximity point theorems analyze the existence of an approximate solution that is optimal. A classical best approximation theorem was introduced by Fan [13]; that is, if 𝐴 is a nonempty compact convex subset of a Hausdorff locally convex topological vector space 𝐵 and 𝑇∶𝐴→𝐵 is a continuous mapping, then there exists an element 𝑥∈𝐴 such that 𝑑(𝑥,𝑇𝑥)=𝑑(𝑇𝑥,𝐴). Afterward, several authors, including Prolla [14], Reich [15], Sehgal, and Singh [16, 17], have derived extensions of Fan’s theorem in many directions. Other works of the existence of a best proximity point for contractions can be seen in [18–21]. In 2005, Eldred et al. [22] have obtained best proximity point theorems for relatively nonexpansive mappings. Best proximity point theorems for several types of contractions have been established in [23–36].

Recently, Sadiq Basha in [37] gave necessary and sufficient to claimed that the existence of best proximity point for proximal contraction of first kind and the second kind which are non-self mapping analogues of contraction self-mappings and also established some best proximity and convergence theorem as follow.

Theorem 1.1 (see [37, Theorem 3.1]). Let (𝑋,𝑑) be a complete metric space and let 𝐴 and 𝐵 be nonempty, closed subsets of 𝑋. Further, suppose that 𝐴₀ and 𝐵₀ are nonempty. Let 𝑆∶𝐴→𝐵, 𝑇∶𝐵→𝐴 and 𝑔∶𝐴∪𝐵→𝐴∪𝐵 satisfy the following conditions.(a)𝑆 and 𝑇 are proximal contractions of first kind.(b)𝑔 is an isometry.(c) The pair (𝑆,𝑇) is a proximal cyclic contraction.(d)𝑆(𝐴₀)⊆𝐵₀,𝑇(𝐵₀)⊆𝐴₀.(e)𝐴₀⊆𝑔(𝐴₀) and 𝐵₀⊆𝑔(𝐵₀).Then, there exists a unique point 𝑥∈𝐴 and there exists a unique point 𝑦∈𝐵 such that 𝑑𝑔𝑥,𝑆𝑥=𝑑𝑔𝑦,𝑇𝑦=𝑑𝑥,𝑦=𝑑(𝐴,𝐵). (1.1) Moreover, for any fixed 𝑥₀∈𝐴₀, the sequence {𝑥_𝑛}, defined by 𝑑𝑔𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑(𝐴,𝐵), (1.2) converges to the element 𝑥. For any fixed 𝑦₀∈𝐵₀, the sequence {𝑦_𝑛}, defined by 𝑑𝑔𝑦_𝑛+1,𝑇𝑦_𝑛=𝑑(𝐴,𝐵), (1.3) converges to the element 𝑦.
On the other hand, a sequence {𝑢_𝑛} in 𝐴 converges to 𝑥 if there is a sequence of positive numbers {𝜖_𝑛} such that lim_𝑛→∞𝜖_𝑛=0,𝑑𝑢_𝑛+1,𝑧_𝑛+1≤𝜖_𝑛, (1.4) where 𝑧_𝑛+1∈𝐴 satisfies the condition that 𝑑(𝑧_𝑛+1,𝑆𝑢_𝑛)=𝑑(𝐴,𝐵).

Theorem 1.2 (see [37, Theorem 3.4]). Let (𝑋,𝑑) be a complete metric space and let 𝐴 and 𝐵 be nonempty, closed subsets of 𝑋. Further, suppose that 𝐴₀ and 𝐵₀ are nonempty. Let 𝑆∶𝐴→𝐵 and 𝑔∶𝐴→𝐴 satisfy the following conditions.(a)𝑆 is proximal contractions of first and second kinds.(b)𝑔 is an isometry.(c)𝑆 preserves isometric distance with respect to 𝑔.(d)𝑆(𝐴₀)⊆𝐵₀.(e)𝐴₀⊆𝑔(𝐴₀).Then, there exists a unique point 𝑥∈𝐴 such that 𝑑𝑔𝑥,𝑆𝑥=𝑑(𝐴,𝐵). (1.5) Moreover, for any fixed 𝑥₀∈𝐴₀, the sequence {𝑥_𝑛}, defined by 𝑑𝑔𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑(𝐴,𝐵), (1.6) converges to the element 𝑥.
On the other hand, a sequence {𝑢_𝑛} in 𝐴 converges to 𝑥 if there is a sequence of positive numbers {𝜖_𝑛} such that lim_𝑛→∞𝜖_𝑛=0,𝑑𝑢_𝑛+1,𝑧_𝑛+1≤𝜖_𝑛, (1.7) where z_𝑛+1∈𝐴 satisfies the condition that 𝑑(𝑧_𝑛+1,𝑆𝑢_𝑛)=𝑑(𝐴,𝐵).

The aim of this paper is to introduce the new classes of proximal contractions which are more general than class of proximal contraction of first and second kinds, by giving the necessary condition to have best proximity points and we also give some illustrative examples of our main results. The results of this paper are extension and generalizations of main result of Sadiq Basha in [37] and some results in the literature.

2. Preliminaries

Given nonvoid subsets 𝐴 and 𝐵 of a metric space (𝑋,𝑑), we recall the following notations and notions that will be used in what follows: 𝑑()𝑑,𝐴𝐴,𝐵∶=inf𝑥,𝑦∶𝑥∈𝐴,𝑦∈𝐵₀∶=𝑥∈𝐴∶𝑑𝑥,𝑦=𝑑(𝐴,𝐵)forsome,𝐵𝑦∈𝐵₀∶=𝑦∈𝐵∶𝑑𝑥,𝑦=𝑑(𝐴,𝐵)forsome𝑥∈𝐴. (2.1)

If 𝐴∩𝐵≠∅, then 𝐴₀ and 𝐵₀ are nonempty. Further, it is interesting to notice that 𝐴₀ and 𝐵₀ are contained in the boundaries of 𝐴 and 𝐵, respectively, provided 𝐴 and 𝐵 are closed subsets of a normed linear space such that 𝑑(𝐴,𝐵)>0 (see [31]).

Definition 2.1 ([37, Definition 2.2]). A mapping 𝑆∶𝐴→𝐵 is said to be a proximal contraction of the first kind if there exists 𝛼∈[0,1) such that 𝑑(𝑢,𝑆𝑥)=𝑑𝑣,𝑆𝑦=𝑑(𝐴,𝐵)⇒𝑑(𝑢,𝑣)≤𝛼𝑑𝑥,𝑦 (2.2) for all 𝑢,𝑣,𝑥,𝑦∈𝐴.

It is easy to see that a self-mapping that is a proximal contraction of the first kind is precisely a contraction. However, a non-self-proximal contraction is not necessarily a contraction.

Definition 2.2 (see [37, Definition 2.3]). A mapping 𝑆∶𝐴→𝐵 is said to be a proximal contraction of the second kind if there exists 𝛼∈[0,1) such that 𝑑(𝑢,𝑆𝑥)=𝑑𝑣,𝑆𝑦=𝑑(𝐴,𝐵)⇒𝑑(𝑆𝑢,𝑆𝑣)≤𝛼𝑑𝑆𝑥,𝑆𝑦 (2.3) for all 𝑢,𝑣,𝑥,𝑦∈𝐴.

Definition 2.3. Let 𝑆∶𝐴→𝐵 and 𝑇∶𝐵→𝐴. The pair (𝑆,𝑇) is said to be a proximal cyclic contraction pair if there exists a nonnegative number 𝛼<1 such that 𝑑(𝑎,𝑆𝑥)=𝑑𝑏,𝑇𝑦=𝑑(𝐴,𝐵)⇒𝑑(𝑎,𝑏)≤𝛼𝑑𝑥,𝑦+(1−𝛼)𝑑(𝐴,𝐵) (2.4) for all 𝑎,𝑥∈𝐴 and 𝑏,𝑦∈𝐵.

Definition 2.4. Leting 𝑆∶𝐴→𝐵 and an isometry 𝑔∶𝐴→𝐴, the mapping 𝑆 is said to preserve isometric distance with respect to 𝑔 if 𝑑𝑆𝑔𝑥,𝑆𝑔𝑦=𝑑𝑆𝑥,𝑆𝑦 (2.5) for all 𝑥,𝑦∈𝐴.

Definition 2.5. A point 𝑥∈𝐴 is said to be a best proximity point of the mapping 𝑆∶𝐴→𝐵 if it satisfies the condition that 𝑑(𝑥,𝑆𝑥)=𝑑(𝐴,𝐵). (2.6)

It can be observed that a best proximity reduces to a fixed point if the underlying mapping is a self-mapping.

Definition 2.6. 𝐴 is said to be approximatively compact with respect to 𝐵 if every sequence {𝑥_𝑛} in 𝐴 satisfies the condition that 𝑑(𝑦,𝑥_𝑛)→𝑑(𝑦,𝐴) for some 𝑦∈𝐵 has a convergent subsequence.

We observe that every set is approximatively compact with respect to itself and that every compact set is approximatively compact. Moreover, 𝐴₀ and 𝐵₀ are nonempty set if 𝐴 is compact and 𝐵 is approximatively compact with respect to 𝐴.

3. Main Results

Definition 3.1. A mapping 𝑆∶𝐴→𝐵 is said to be a generalized proximal 𝜓-contraction of the first kind, if for all 𝑢,𝑣,𝑥,𝑦∈𝐴 satisfies 𝑑(𝑢,𝑆𝑥)=𝑑𝑣,𝑆𝑦=𝑑(𝐴,𝐵)⇒𝑑(𝑢,𝑣)≤𝜓𝑑𝑥,𝑦, (3.1) where 𝜓∶[0,∞)→[0,∞) is an upper semicontinuous function from the right such that 𝜓(𝑡)<𝑡 for all 𝑡>0.

Definition 3.2. A mapping 𝑆∶𝐴→𝐵 is said to be a generalized proximal 𝜓-contraction of the second kind, if for all 𝑢,𝑣,𝑥,𝑦∈𝐴 satisfies 𝑑(𝑢,𝑆𝑥)=𝑑𝑣,𝑆𝑦=𝑑(𝐴,𝐵)⇒𝑑(𝑆𝑢,𝑆𝑣)≤𝜓𝑑𝑆𝑥,𝑆𝑦, (3.2) where 𝜓∶[0,∞)→[0,∞) is a upper semicontinuous from the right such that 𝜓(𝑡)<𝑡 for all 𝑡>0.

It is easy to see that if we take 𝜓(𝑡)=𝛼𝑡, where 𝛼∈[0,1), then a generalized proximal 𝜓-contraction of the first kind and generalized proximal 𝜓-contraction of the second kind reduce to a proximal contraction of the first kind Definition 2.1 and a proximal contraction of the second kind Definition 2.2, respectively. Moreover, it is easy to see that a self-mapping generalized proximal 𝜓-contraction of the first kind and the second kind reduces to the condition of Boy and Wong' s fixed point theorem [3].

Next, we extend the result of Sadiq Basha [37] and the Banach's contraction principle to the case of non-self-mappings which satisfy generalized proximal 𝜓-contraction condition.

Theorem 3.3. Let (𝑋,𝑑) be a complete metric space and let 𝐴 and 𝐵 be nonempty, closed subsets of 𝑋 such that 𝐴₀ and 𝐵₀ are nonempty. Let 𝑆∶𝐴→𝐵, 𝑇∶𝐵→𝐴, and 𝑔∶𝐴∪𝐵→𝐴∪𝐵 satisfy the following conditions:(a)𝑆 and 𝑇 are generalized proximal 𝜓-contraction of the first kind;(b)𝑔 is an isometry;(c) The pair (𝑆,𝑇) is a proximal cyclic contraction;(d)𝑆(𝐴₀)⊆𝐵₀,𝑇(𝐵₀)⊆𝐴₀;(e)𝐴₀⊆𝑔(𝐴₀) and 𝐵₀⊆𝑔(𝐵₀).Then, there exists a unique point 𝑥∈𝐴 and there exists a unique point 𝑦∈𝐵 such that 𝑑𝑔𝑥,𝑆𝑥=𝑑𝑔𝑦,𝑇𝑦=𝑑𝑥,𝑦=𝑑(𝐴,𝐵). (3.3) Moreover, for any fixed 𝑥₀∈𝐴₀, the sequence {𝑥_𝑛}, defined by 𝑑𝑔𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑(𝐴,𝐵), (3.4) converges to the element 𝑥. For any fixed 𝑦₀∈𝐵₀, the sequence {𝑦_𝑛}, defined by 𝑑𝑔𝑦_𝑛+1,𝑇𝑦_𝑛=𝑑(𝐴,𝐵), (3.5) converges to the element 𝑦.
On the other hand, a sequence {𝑢_𝑛} in 𝐴 converges to 𝑥 if there is a sequence of positive numbers {𝜖_𝑛} such that lim_𝑛→∞𝜖_𝑛=0,𝑑𝑢_𝑛+1,𝑧_𝑛+1≤𝜖_𝑛, (3.6) where 𝑧_𝑛+1∈𝐴 satisfies the condition that 𝑑(g𝑧_𝑛+1,𝑆𝑢_𝑛)=𝑑(𝐴,𝐵).

Proof. Let 𝑥₀ be a fixed element in 𝐴₀. In view of the fact that 𝑆(𝐴₀)⊆𝐵₀ and 𝐴₀⊆𝑔(𝐴₀), it is ascertained that there exists an element 𝑥₁∈𝐴₀ such that 𝑑𝑔𝑥₁,𝑆𝑥₀=𝑑(𝐴,𝐵). (3.7) Again, since 𝑆(𝐴₀)⊆𝐵₀ and 𝐴₀⊆𝑔(𝐴₀), there exists an element 𝑥₂∈𝐴₀ such that 𝑑𝑔𝑥₂,𝑆𝑥₁=𝑑(𝐴,𝐵). (3.8) By similar fashion, we can find 𝑥_𝑛 in 𝐴₀. Having chosen 𝑥_𝑛, one can determine an element 𝑥_𝑛+1∈𝐴₀ such that 𝑑𝑔𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑(𝐴,𝐵). (3.9) Because of the facts that 𝑆(𝐴₀)⊆𝐵₀ and 𝐴₀⊆𝑔(𝐴₀), by a generalized proximal 𝜓-contraction of the first kind of 𝑆, 𝑔 is an isometry and property of 𝜓, for each 𝑛∈ℕ, we have 𝑑𝑥_𝑛+1,𝑥_𝑛=𝑑𝑔𝑥_𝑛+1,𝑔𝑥_𝑛≤𝜓𝑑𝑥_𝑛,𝑥_𝑛−1≤𝑑𝑥_𝑛,𝑥_𝑛−1. (3.10) This means that the sequence {𝑑(𝑥_𝑛+1,𝑥_𝑛)} is nonincreasing and bounded. Hence there exists 𝑟≥0 such that lim_𝑛→∞𝑑𝑥_𝑛+1,𝑥_𝑛=𝑟. (3.11) If 𝑟>0, then 𝑟=lim_𝑛→∞𝑑𝑥_𝑛+1,𝑥_𝑛≤lim_𝑛→∞𝜓𝑑𝑥_𝑛,𝑥_𝑛−1=𝜓(𝑟)<𝑟, (3.12) which is a contradiction unless 𝑟=0. Therefore, 𝛼_𝑛∶=lim_𝑛→∞𝑑𝑥_𝑛+1,𝑥_𝑛=0. (3.13) We claim that {𝑥_𝑛} is a Cauchy sequence. Suppose that {𝑥_𝑛} is not a Cauchy sequence. Then there exists 𝜀>0 and subsequence {𝑥_𝑚𝑘},{𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑛_𝑘>𝑚_𝑘≥𝑘 with 𝑟_𝑘∶=𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘≥𝜀,𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘−1<𝜀 (3.14) for 𝑘∈{1,2,3,…}. Thus 𝜀≤𝑟_𝑘≤𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘−1+𝑑𝑥_𝑛𝑘−1,𝑥_𝑛𝑘<𝜀+𝛼_𝑛𝑘−1. (3.15) It follows from (3.13) that lim_𝑘→∞𝑟_𝑘=𝜀. (3.16) On the other hand, by constructing the sequence {𝑥_𝑛}, we have 𝑑𝑔𝑥_𝑚𝑘+1,𝑆𝑥_𝑚𝑘=𝑑(𝐴,𝐵),𝑑𝑔𝑥_𝑛𝑘+1,𝑆𝑥_𝑛k=𝑑(𝐴,𝐵). (3.17) Sine 𝑆 is a generalized proximal 𝜓-contraction of the first kind and 𝑔 is an isometry, we have 𝑑𝑥_𝑚𝑘+1,𝑥_𝑛𝑘+1=𝑑𝑔𝑥_𝑚𝑘+1,𝑔𝑥_𝑛𝑘+1≤𝜓𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘. (3.18) Notice also that 𝜀≤𝑟_𝑘≤𝑑𝑥_𝑚𝑘,𝑥_𝑚𝑘+1+𝑑𝑥_𝑛𝑘+1,𝑥_𝑛𝑘+𝑑𝑥_𝑚𝑘+1,𝑥_𝑛𝑘+1=𝛼_𝑚𝑘+𝛼_𝑛𝑘+𝑑𝑥_𝑚𝑘+1,𝑥_𝑛𝑘+1≤𝛼_𝑚𝑘+𝛼_𝑛𝑘+𝜓𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘. (3.19) Taking 𝑘→∞ in above inequality, by (3.13), (3.16), and property of 𝜓, we get 𝜀≤𝜓(𝜀). Therefore, 𝜀=0, which is a contradiction. So we obtain the claim and hence converge to some element 𝑥∈𝐴. Similarly, in view of the fact that 𝑇(𝐵₀)⊆𝐴₀ and 𝐴₀⊆𝑔(𝐴₀), we can conclude that there is a sequence {𝑦_𝑛} such that 𝑑(𝑔𝑦_𝑛+1,𝑆𝑦_𝑛)=𝑑(𝐴,𝐵) and converge to some element 𝑦∈𝐵. Since the pair (𝑆,𝑇) is a proximal cyclic contraction and 𝑔 is an isometry, we have 𝑑𝑥_𝑛+1,𝑦_𝑛+1=𝑑𝑔𝑥_𝑛+1,𝑔𝑦_𝑛+1≤𝛼𝑑𝑥_𝑛,𝑦_𝑛+(1−𝛼)𝑑(𝐴,𝐵). (3.20) We take limit in (3.20) as 𝑛→∞; it follows that 𝑑𝑥,𝑦=𝑑(𝐴,𝐵), (3.21) so, we concluded that 𝑥∈𝐴₀ and 𝑦∈𝐵₀. Since 𝑆(𝐴₀)⊆𝐵₀ and 𝑇(𝐵₀)⊆𝐴₀, there is 𝑢∈𝐴 and 𝑣∈𝐵 such that 𝑑(𝑢,𝑆𝑥)=𝑑(𝐴,𝐵)𝑑𝑣,𝑇𝑦=𝑑(𝐴,𝐵). (3.22)(3.23) From (3.9), (3.22), and the notion of generalized proximal 𝜓-contraction of first kind of 𝑆, we get 𝑑𝑢,𝑔𝑥_𝑛+1≤𝜓𝑑𝑥,𝑥_𝑛. (3.24) Letting 𝑛→∞, we get 𝑑(𝑢,𝑔𝑥)≤𝜓(0)=0 and thus 𝑢=𝑔𝑥. Therefore 𝑑𝑔𝑥,𝑆𝑥=𝑑(𝐴,𝐵). (3.25) Similarly, we can show that 𝑣=𝑔𝑦 and then 𝑑𝑔𝑦,𝑇𝑦=𝑑(A,𝐵). (3.26) From (3.21), (3.25), and (3.26), we get 𝑑𝑥,𝑦=𝑑𝑔𝑥,𝑆𝑥=𝑑𝑔𝑦,𝑇𝑦=𝑑(𝐴,𝐵). (3.27)
Next, to prove the uniqueness, let us suppose that there exist 𝑥^∗∈𝐴 and 𝑦^∗∈𝐵 with 𝑥≠𝑥^∗,𝑦≠𝑦^∗ such that 𝑑𝑔𝑥^∗,𝑆𝑥^∗(),𝑑=𝑑𝐴,𝐵𝑔𝑦^∗,𝑇𝑦^∗=𝑑(𝐴,𝐵). (3.28) Since 𝑔 is an isometry, 𝑆 and 𝑇 are generalized proximal 𝜓-contractions of the first kind and the property of 𝜓; it follows that 𝑑𝑥,𝑥^∗=𝑑𝑔𝑥,𝑔𝑥^∗≤𝜓𝑑𝑥,𝑥^∗<𝑑𝑥,𝑥^∗,𝑑𝑦,𝑦^∗=𝑑𝑔𝑦,𝑔𝑦^∗≤𝜓𝑑𝑦,𝑦^∗<𝑑𝑦,𝑦^∗, (3.29) which is a contradiction, so we have 𝑥=𝑥^∗ and 𝑦=𝑦^∗. On the other hand, let {𝑢_𝑛} be a sequence in 𝐴 and let {𝜖_𝑛} be a sequence of positive real numbers such that lim_𝑛→∞𝜖_𝑛=0,𝑑𝑢_𝑛+1,𝑧_𝑛+1≤𝜖_𝑛, (3.30) where 𝑧_𝑛+1∈𝐴 satisfies the condition that 𝑑(𝑔𝑧_𝑛+1,𝑆𝑢_𝑛)=𝑑(𝐴,𝐵). Since 𝑆 is a generalized proximal 𝜓-contraction of first kind and 𝑔 is an isometry, we have 𝑑𝑥_𝑛+1,𝑧_𝑛+1≤𝜓𝑑𝑥_𝑛,𝑢_𝑛. (3.31) Given 𝜖>0, we choose a positive integer 𝑁 such that 𝜖_𝑛≤𝜖 for all 𝑛≥𝑁; we obtain that 𝑑𝑥_𝑛+1,𝑢_𝑛+1≤𝑑𝑥_𝑛+1,𝑧_𝑛+1+𝑑𝑧_𝑛+1,𝑢_𝑛+1≤𝜓𝑑𝑥_𝑛,𝑢_𝑛+𝜖_𝑛. (3.32) Therefore, we get 𝑑𝑢_𝑛+1,𝑥≤𝑑𝑢_𝑛+1,𝑥_𝑛+1+𝑑𝑥_𝑛+1𝑑𝑥,𝑥≤𝜓_𝑛,𝑢_𝑛+𝜖_𝑛+𝑑𝑥_𝑛+1,𝑥. (3.33) We claim that 𝑑(𝑢_𝑛,𝑥)→0 as 𝑛→∞; supposing the contrary, by inequality (3.33) and property of 𝜓, we get lim_𝑛→∞𝑑𝑢_𝑛+1,𝑥≤lim_𝑛→∞𝑑𝑢_𝑛+1,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥≤lim_𝑛→∞𝜓𝑑𝑥_𝑛,𝑢_𝑛+𝜖_𝑛+𝑑𝑥_𝑛+1,𝑥=𝜓lim_𝑛→∞𝑑𝑥_𝑛,𝑢_𝑛<lim_𝑛→∞𝑑𝑥_𝑛,𝑢_𝑛≤lim_𝑛→∞𝑑𝑥_𝑛,𝑥+𝑑𝑥,𝑢_𝑛=lim_𝑛→∞𝑑𝑥,𝑢_𝑛, (3.34) which is a contradiction, so we have {𝑢_𝑛} is convergent and it converges to 𝑥. This completes the proof of the theorem.

If 𝑔 is assumed to be the identity mapping, then by Theorem 3.3, we obtain the following corollary.

Corollary 3.4. Let (𝑋,𝑑) be a complete metric space and let 𝐴 and 𝐵 be nonempty, closed subsets of 𝑋. Further, suppose that 𝐴₀ and 𝐵₀ are nonempty. Let 𝑆∶𝐴→𝐵, 𝑇∶𝐵→𝐴 and g∶𝐴∪𝐵→𝐴∪𝐵 satisfy the following conditions:(a)𝑆 and 𝑇 are generalized proximal 𝜓-contraction of the first kind;(b)𝑆(𝐴₀)⊆𝐵₀,𝑇(𝐵₀)⊆𝐴₀;(c) the pair (𝑆,𝑇) is a proximal cyclic contraction.Then, there exists a unique point 𝑥∈𝐴 and there exists a unique point 𝑦∈𝐵 such that 𝑑𝑔𝑥,𝑆𝑥=𝑑𝑔𝑦,𝑇𝑦=𝑑𝑥,𝑦=𝑑(𝐴,𝐵). (3.35)

If we take 𝜓(𝑡)=𝛼𝑡, where 0≤𝛼<1, we obtain following corollary.

Corollary 3.5 (see [37, Theorem 3.1]). Let (𝑋,𝑑) be a complete metric space and 𝐴 and 𝐵 be non-empty, closed subsets of 𝑋. Further, suppose that 𝐴₀ and 𝐵₀ are non-empty. Let 𝑆∶𝐴→𝐵, 𝑇∶𝐵→𝐴 and 𝑔∶𝐴∪𝐵→𝐴∪𝐵 satisfy the following conditions:(a)𝑆 and 𝑇 are proximal contractions of first kind;(b)𝑔 is an isometry;(c) the pair (𝑆,𝑇) is a proximal cyclic contraction;(d)𝑆(𝐴₀)⊆𝐵₀,𝑇(𝐵₀)⊆𝐴₀;(e)𝐴₀⊆𝑔(𝐴₀) and 𝐵₀⊆𝑔(𝐵₀).Then, there exists a unique point 𝑥∈𝐴 and there exists a unique point 𝑦∈𝐵 such that 𝑑𝑔𝑥,𝑆𝑥=𝑑𝑔𝑦,𝑇𝑦=𝑑𝑥,𝑦=𝑑(𝐴,𝐵). (3.36) Moreover, for any fixed 𝑥₀∈𝐴₀, the sequence {𝑥_𝑛}, defined by 𝑑𝑔𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑(𝐴,𝐵), (3.37) converges to the element 𝑥. For any fixed 𝑦₀∈𝐵₀, the sequence {𝑦_𝑛}, defined by 𝑑𝑔𝑦_𝑛+1,𝑇𝑦_𝑛=𝑑(𝐴,𝐵), (3.38) converges to the element 𝑦.

If 𝑔 is assumed to be the identity mapping in Corollary 3.5, we obtain the following corollary.

Corollary 3.6. Let (𝑋,𝑑) be a complete metric space and let 𝐴 and 𝐵 be nonempty, closed subsets of 𝑋. Further, suppose that 𝐴₀ and 𝐵₀ are nonempty. Let 𝑆∶𝐴→𝐵, 𝑇∶𝐵→𝐴, and 𝑔∶𝐴∪𝐵→𝐴∪𝐵 satisfy the following conditions:(a)𝑆 and 𝑇 are proximal contractions of first kind;(b)𝑆(𝐴₀)⊆𝐵₀,𝑇(𝐵₀)⊆𝐴₀;(c) the pair (𝑆,𝑇) is a proximal cyclic contraction.Then, there exists a unique point 𝑥∈𝐴 and there exists a unique point 𝑦∈𝐵 such that 𝑑𝑔𝑥,𝑆𝑥=𝑑𝑔𝑦,𝑇𝑦=𝑑𝑥,𝑦=𝑑(𝐴,𝐵). (3.39)

For a self-mapping, Theorem 3.3 includes the Boy and Wong' s fixed point theorem [3] as follows.

Corollary 3.7. Let (𝑋,𝑑) be a complete metric space and let 𝑇∶𝑋→𝑋 be a mapping that satisfies 𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑑(𝑥,𝑦)) for all 𝑥,𝑦∈𝑋, where 𝜓∶[0,∞)→[0,∞) is an upper semicontinuous function from the right such that 𝜓(𝑡)<𝑡 for all 𝑡>0. Then 𝑇 has a unique fixed point 𝑣∈𝑋. Moreover, for each 𝑥∈𝑋,{𝑇^𝑛𝑥} converges to 𝑣.

Next, we give an example to show that Definition 3.1 is different form Definition 2.1; moreover we give an example which supports Theorem 3.3.

Example 3.8. Consider the complete metric space ℝ² with metric defined by 𝑑𝑥₁,𝑦₁,𝑥₂,𝑦₂=𝑥₁−𝑥₂+𝑦₁−𝑦₂, (3.40) for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈ℝ². Let 𝐴=0,𝑦∶0≤𝑦≤1,𝐵=1,𝑦∶0≤𝑦≤1. (3.41) Then 𝑑(𝐴,𝐵)=1. Define the mappings 𝑆∶𝐴→𝐵 as follows: 𝑆=𝑦0,𝑦1,𝑦−²2. (3.42) First, we show that 𝑆 is generalized proximal 𝜓-contraction of the first kind with the function 𝜓∶[0,∞)→[0,∞) defined by 𝜓(𝑡)=𝑡𝑡−²2,0≤𝑡≤1,𝑡−1,𝑡>1. (3.43) Let (0,𝑥₁),(0,𝑥₂),(0,𝑎₁) and (0,𝑎₂) be elements in 𝐴 satisfying 𝑑0,𝑥₁,𝑆0,𝑎₁=𝑑(𝐴,𝐵)=1,𝑑0,𝑥₂,𝑆0,𝑎₂=𝑑(𝐴,𝐵)=1. (3.44) It follows that 𝑥_𝑖=𝑎_𝑖−𝑎²_𝑖2for𝑖=1,2. (3.45) Without loss of generality, we may assume that 𝑎₁−𝑎₂>0, so we have 𝑑0,𝑥₁,0,𝑥₂=𝑑0,𝑎₁−𝑎²₁2,0,𝑎₂−𝑎²₂2=𝑎₁−𝑎²₁2−𝑎₂−𝑎²₂2=𝑎₁−𝑎₂−𝑎²₁2−𝑎²₂2≤𝑎₁−𝑎₂−12𝑎₁−𝑎₂²=𝜓𝑑0,𝑎₁,0,𝑎₂. (3.46) Thus 𝑆 is a generalized proximal 𝜓-contraction of the first kind.
Next, we prove that 𝑆 is not a proximal contraction. Suppose 𝑆 is proximal contraction then for each (0,𝑥),(0,𝑦),(0,𝑎),(0,𝑏)∈𝐴 satisfying 𝑑((0,𝑥),𝑆(0,𝑎))=𝑑(𝐴,𝐵)=1,𝑑0,𝑦,𝑆(0,𝑏)=𝑑(𝐴,𝐵)=1, (3.47) there exists 𝑘∈[0,1) such that 𝑑(0,𝑥),0,𝑦≤𝑘𝑑((0,𝑎),(0,𝑏)). (3.48) From (3.47), we get 𝑎𝑥=𝑎−²2𝑏,𝑦=𝑏−²2, (3.49) and thus 𝑎𝑎−²2−𝑏𝑏−²2(),),(||.=𝑑0,𝑥0,𝑦≤𝑘𝑑((0,𝑎0,𝑏))=𝑘𝑎−𝑏 (3.50) Letting 𝑏=0 with 𝑎≠0, we get 1=lim_𝑎→0+𝑎1−2≤𝑘<1, (3.51) which is a contradiction. Therefore 𝑆 is not a proximal contraction and Definition 3.1 is different form Definition 2.1.

Example 3.9. Consider the complete metric space ℝ² with Euclidean metric. Let ,.𝐴=0,𝑦∶𝑦∈ℝ𝐵=1,𝑦∶𝑦∈ℝ (3.52) Define two mappings 𝑆∶𝐴→𝐵, 𝑇∶𝐵→𝐴 and 𝑔∶𝐴∪𝐵→𝐴∪𝐵 as follows: 𝑆=𝑦0,𝑦1,4=𝑦,𝑇1,𝑦0,4,𝑔𝑥,𝑦=𝑥,−𝑦. (3.53) Then it is easy to see that 𝑑(𝐴,𝐵)=1, 𝐴₀=𝐴, 𝐵₀=𝐵 and the mapping 𝑔 is an isometry.
Next, we claim that 𝑆 and 𝑇 are generalized proximal 𝜓-contractions of the first kind. Consider a function 𝜓∶[0,∞)→[0,∞) defined by 𝜓(𝑡)=𝑡/2 for all 𝑡≥0. If (0,𝑦₁),(0,𝑦₂)∈𝐴 such that 𝑑𝑎,𝑆0,𝑦₁=𝑑(𝐴,𝐵)=1,𝑑𝑏,𝑆0,𝑦₂=𝑑(𝐴,𝐵)=1 (3.54) for all 𝑎,𝑏∈𝐴, then we have 𝑦𝑎=0,¹4𝑦,𝑏=0,²4. (3.55) Because, 𝑑()𝑦𝑎,𝑏=𝑑0,¹4,𝑦0,²4=𝑦₁4−𝑦₂4=14𝑦₁−𝑦₂≤12𝑦₁−𝑦₂=12𝑑0,𝑦₁,0,𝑦₂=𝜓𝑑0,𝑦₁,0,𝑦₂. (3.56) Hence 𝑆 is a generalized proximal 𝜓-contraction of the first kind. If (1,𝑦₁),(1,𝑦₂)∈𝐵 such that 𝑑𝑎,𝑇1,𝑦₁=𝑑(𝐴,𝐵)=1,𝑑𝑏,𝑇1,𝑦₂=𝑑(𝐴,𝐵)=1 (3.57) for all 𝑎,𝑏∈𝐵, then we get 𝑦𝑎=1,¹4𝑦,𝑏=1,²4. (3.58) In the same way, we can see that 𝑇 is a generalized proximal 𝜓-contraction of the first kind. Moreover, the pair (𝑆,𝑇) forms a proximal cyclic contraction and other hypotheses of Theorem 3.3 are also satisfied. Further, it is easy to see that the unique element (0,0)∈𝐴 and (1,0)∈𝐵 such that 𝑑𝑔(0,0),𝑆(0,0)=𝑑𝑔(1,0),𝑇(1,0)=𝑑((0,0),(1,0))=𝑑(𝐴,𝐵). (3.59)
Next, we establish a best proximity point theorem for non-self-mappings which are generalized proximal 𝜓-contractions of the first kind and the second kind.

Theorem 3.10. Let (𝑋,𝑑) be a complete metric space and let 𝐴 and 𝐵 be non-empty, closed subsets of 𝑋. Further, suppose that 𝐴₀ and 𝐵₀ are non-empty. Let 𝑆∶𝐴→𝐵 and 𝑔∶𝐴→𝐴 satisfy the following conditions:(a)𝑆 is a generalized proximal 𝜓-contraction of first and second kinds;(b)𝑔 is an isometry;(c)𝑆 preserves isometric distance with respect to 𝑔;(d)𝑆(𝐴₀)⊆𝐵₀;(e)𝐴₀⊆𝑔(𝐴₀).Then, there exists a unique point 𝑥∈𝐴 such that 𝑑𝑔𝑥,𝑆𝑥=𝑑(𝐴,𝐵). (3.60) Moreover, for any fixed 𝑥₀∈𝐴₀, the sequence {𝑥_𝑛}, defined by 𝑑𝑔𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑(𝐴,𝐵), (3.61) converges to the element 𝑥.
On the other hand, a sequence {𝑢_𝑛} in 𝐴 converges to 𝑥 if there is a sequence of positive numbers {𝜖_𝑛} such that lim_𝑛→∞𝜖_𝑛=0,𝑑𝑢_𝑛+1,𝑧_𝑛+1≤𝜖_𝑛, (3.62) where 𝑧_𝑛+1∈𝐴 satisfies the condition that 𝑑(g𝑧_𝑛+1,𝑆𝑢_𝑛)=𝑑(𝐴,𝐵).

Proof. Since 𝑆(𝐴₀)⊆𝐵₀ and 𝐴₀⊆𝑔(𝐴₀), similarly in the proof of Theorem 3.3, we can construct the sequence {𝑥_𝑛} of element in 𝐴₀ such that 𝑑𝑔𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑(𝐴,𝐵) (3.63) for nonnegative number 𝑛. It follows from 𝑔 that is an isometry and the virtue of a generalized proximal 𝜓-contraction of the first kind of 𝑆; we see that 𝑑𝑥_𝑛,x_𝑛+1=𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1≤𝜓𝑑𝑥_𝑛,𝑥_𝑛−1 (3.64) for all 𝑛∈ℕ. Similarly to the proof of Theorem 3.3, we can conclude that the sequence {𝑥_𝑛} is a Cauchy sequence and converges to some 𝑥∈𝐴. Since 𝑆 is a generalized proximal 𝜓-contraction of the second kind and preserves isometric distance with respect to 𝑔 that 𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛+1=𝑑𝑆𝑔𝑥_𝑛,𝑆𝑔𝑥_𝑛+1≤𝜓𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛≤𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛, (3.65) this means that the sequence {𝑑(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛)} is nonincreasing and bounded below. Hence, there exists 𝑟≥0 such that lim_𝑛→∞𝑑𝑆𝑥_𝑛+1,𝑆𝑥_𝑛=𝑟. (3.66) If 𝑟>0, then 𝑟=lim_𝑛→∞𝑑𝑆𝑥_𝑛+1,𝑆𝑥_𝑛≤lim_𝑛→∞𝜓𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛=𝜓(𝑟)<𝑟, (3.67) which is a contradiction, unless 𝑟=0. Therefore 𝛽_𝑛∶=lim_𝑛→∞𝑑𝑆𝑥_𝑛+1,𝑆𝑥_𝑛=0. (3.68) We claim that {𝑆𝑥_𝑛} is a Cauchy sequence. Suppose that {𝑆𝑥_𝑛} is not a Cauchy sequence. Then there exists 𝜀>0 and subsequence {𝑆𝑥_𝑚𝑘},{𝑆𝑥_𝑛𝑘} of {𝑆𝑥_𝑛} such that 𝑛_𝑘>𝑚_𝑘≥𝑘 with 𝑟_𝑘∶=𝑑𝑆𝑥_𝑚𝑘,𝑆𝑥_𝑛k≥𝜀,𝑑𝑆𝑥_𝑚𝑘,𝑆𝑥_𝑛𝑘−1<𝜀 (3.69) for 𝑘∈{1,2,3,…}. Thus 𝜀≤𝑟_𝑘≤𝑑𝑆𝑥_𝑚𝑘,𝑆𝑥_𝑛𝑘−1+𝑑𝑆𝑥_𝑛𝑘−1,𝑆𝑥_𝑛𝑘<𝜀+𝛽_𝑛𝑘−1, (3.70) it follows from (3.68) that lim_𝑘→∞𝑟_𝑘=𝜀. (3.71) Notice also that 𝜀≤𝑟_𝑘≤𝑑𝑆𝑥_𝑚k,𝑆𝑥_𝑚𝑘+1+𝑑𝑆𝑥_𝑛𝑘+1,𝑆𝑥_𝑛𝑘+𝑑𝑆𝑥_𝑚𝑘+1,𝑆𝑥_𝑛𝑘+1=𝛽_𝑚𝑘+𝛽_𝑛𝑘+𝑑𝑆𝑥_𝑚𝑘+1,𝑆𝑥_𝑛𝑘+1≤𝛽_𝑚𝑘+𝛽_𝑛𝑘+𝜓𝑑𝑆𝑥_𝑚𝑘,𝑆𝑥_𝑛𝑘. (3.72) Taking 𝑘→∞ in previous inequality, by (3.68), (3.71), and property of 𝜓, we get 𝜀≤𝜓(𝜀). Hence, 𝜀=0, which is a contradiction. So we obtain the claim and then it converges to some 𝑦∈𝐵. Therefore, we can conclude that 𝑑𝑔𝑥,𝑦=lim_𝑛→∞𝑑𝑔𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑(𝐴,𝐵). (3.73) That is 𝑔𝑥∈𝐴₀. Since 𝐴₀⊆𝑔(𝐴₀), we have 𝑔𝑥=𝑔𝑧 for some 𝑧∈𝐴₀ and then 𝑑(𝑔𝑥,𝑔𝑧)=0. By the fact that 𝑔 is an isometry, we have 𝑑(𝑥,𝑧)=𝑑(𝑔𝑥,𝑔𝑧)=0. Hence 𝑥=𝑧 and so 𝑥 becomes to a point in 𝐴₀. As 𝑆(𝐴₀)⊆𝐵₀ that 𝑑(𝑢,𝑆𝑥)=𝑑(𝐴,𝐵) (3.74) for some 𝑢∈𝐴. It follows from (3.63) and (3.74) that 𝑆 is a generalized proximal 𝜓-contraction of the first kind that 𝑑𝑢,𝑔𝑥_𝑛+1≤𝜓𝑑𝑥,𝑥_𝑛 (3.75) for all 𝑛∈ℕ. Taking limit as 𝑛→∞, we get the sequence {𝑔𝑥_𝑛} converging to a point 𝑢. By the fact that 𝑔 is continuous, we have 𝑔𝑥_𝑛→𝑔𝑥as𝑛→∞. (3.76) By the uniqueness of limit of the sequence, we conclude that 𝑢=𝑔𝑥. Therefore, it results that 𝑑(𝑔𝑥,𝑆𝑥)=𝑑(𝑢,𝑆𝑥)=𝑑(𝐴,𝐵). The uniqueness and the remaining part of the proof follow as in Theorem 3.3. This completes the proof of the theorem.

If 𝑔 is assumed to be the identity mapping, then by Theorem 3.10, we obtain the following corollary.

Corollary 3.11. Let (𝑋,𝑑) be a complete metric space and let 𝐴 and 𝐵 be nonempty, closed subsets of 𝑋. Further, suppose that 𝐴₀ and 𝐵₀ are nonempty. Let 𝑆∶𝐴→𝐵 satisfy the following conditions:(a)𝑆 is a generalized proximal 𝜓-contraction of first and second kinds;(b)𝑆(𝐴₀)⊆𝐵₀.Then, there exists a unique point 𝑥∈𝐴 such that 𝑑(𝑥,𝑆𝑥)=𝑑(𝐴,𝐵). (3.77) Moreover, for any fixed 𝑥₀∈𝐴₀, the sequence {𝑥_𝑛}, defined by 𝑑𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑(𝐴,𝐵), (3.78) converges to the best proximity point 𝑥 of 𝑆.

If we take 𝜓(𝑡)=𝛼𝑡, where 0≤𝛼<1 in Theorem 3.10, we obtain following corollary.

Corollary 3.12 (see [37, Theorem 3.4]). Let (𝑋,𝑑) be a complete metric space and let 𝐴 and 𝐵 be non-empty, closed subsets of 𝑋. Further, suppose that 𝐴₀ and 𝐵₀ are non-empty. Let 𝑆∶𝐴→𝐵 and 𝑔∶𝐴→𝐴 satisfy the following conditions:(a)𝑆 is a proximal contraction of first and second kinds;(b)𝑔 is an isometry;(c)𝑆 preserves isometric distance with respect to 𝑔;(d)𝑆(𝐴₀)⊆𝐵₀;(e)𝐴₀⊆𝑔(𝐴₀).Then, there exists a unique point 𝑥∈𝐴 such that 𝑑𝑔𝑥,𝑆𝑥=𝑑(𝐴,𝐵). (3.79) Moreover, for any fixed 𝑥₀∈𝐴₀, the sequence {𝑥_𝑛}, defined by 𝑑𝑔𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑(𝐴,𝐵), (3.80) converges to the element 𝑥.

If 𝑔 is assumed to be the identity mapping in Corollary 3.12, we obtain the following corollary.

Corollary 3.13. Let (𝑋,𝑑) be a complete metric space and let 𝐴 and 𝐵 be non-empty, closed subsets of 𝑋. Further, suppose that 𝐴₀ and 𝐵₀ are non-empty. Let 𝑆∶𝐴→𝐵 satisfy the following conditions:(a)𝑆 is a proximal contraction of first and second kinds;(b)𝑆(𝐴₀)⊆𝐵₀.Then, there exists a unique point 𝑥∈𝐴 such that 𝑑(𝑥,𝑆𝑥)=𝑑(𝐴,𝐵). (3.81) Moreover, for any fixed 𝑥₀∈𝐴₀, the sequence {𝑥_𝑛}, defined by 𝑑𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑(𝐴,𝐵), (3.82) converges to the best proximity point 𝑥 of 𝑆.

Acknowledgments

W. Sanhan would like to thank Faculty of Liberal Arts and Science, Kasetsart University, Kamphaeng-Saen Campus and Centre of Excellence in Mathematics, CHE, Sriayudthaya Rd., Bangkok, Thailand. C. Mongkolkeha was supported from the Thailand Research Fund through the the Royal Golden Jubilee Ph.D. Program (Grant no. PHD/0029/2553). P. Kumam was supported by the Commission on Higher Education, the Thailand Research Fund, and the King Mongkuts University of Technology Thonburi (Grant no. MRG5580213).