Best Proximity Points for Generalized 𝜶-𝝍-Proximal Contractive Type Mappings

Abstract

We introduce a new class of non-self-contractive mappings. For such mappings, we study the existence and uniqueness of best proximity points. Several applications and interesting consequences of our obtained results are derived.

1. Introduction and Preliminaries

Let 𝐴 and 𝐵 be two nonempty subsets of a metric space (𝑋,𝑑). An element 𝑥∈𝐴 is said to be a fixed point of a given map 𝑇∶𝐴→𝐵 if 𝑇𝑥=𝑥. Clearly, 𝑇(𝐴)∩𝐴≠∅ is a necessary (but not sufficient) condition for the existence of a fixed point of 𝑇. If 𝑇(𝐴)∩𝐴=∅, then 𝑑(𝑥,𝑇𝑥)>0 for all 𝑥∈𝐴 that is, the set of fixed points of 𝑇 is empty. In a such situation, one often attempts to find an element 𝑥 which is in some sense closest to 𝑇𝑥. Best proximity point analysis has been developed in this direction.

An element 𝑎∈𝐴 is called a best proximity point of 𝑇 if 𝑑(𝑎,𝑇𝑎)=𝑑(𝐴,𝐵), (1) where 𝑑(𝐴,𝐵)=inf𝑑𝑥,𝑦∶𝑥∈𝐴,𝑦∈𝐵. (2) Because of the fact that 𝑑(𝑥,𝑇𝑥)≥𝑑(𝐴,𝐵) for all 𝑥∈𝐴, the global minimum of the mapping 𝑥↦𝑑(𝑥,𝑇𝑥) is attained at a best proximity point. Clearly, if the underlying mapping is a self-mapping, then it can be observed that a best proximity point is essentially a fixed point. The goal of best proximity point theory is to furnish sufficient conditions that assure the existence of such points. For more details on this approach, we refer the reader to [1–12] and references therein.

Recently, Samet et al. [13] introduced a new class of contractive mappings called 𝛼-𝜓-contractive type mappings. Let (𝑋,𝑑) be a metric space.

Definition 1. A self-mapping 𝑇∶𝑋→𝑋 is said to be an 𝛼-𝜓-contraction, where 𝛼∶𝑋×𝑋→[0,∞) and 𝜓 is a (c)-comparison function, if 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋. (3)

Definition 2. A self-mapping 𝑇∶𝑋→𝑋 is said to be 𝛼-admissible, where 𝛼∶𝑋×𝑋→[0,∞), if 𝑥,𝑦∈𝑋,𝛼𝑥,𝑦≥1⟹𝛼𝑇𝑥,𝑇𝑦≥1. (4)

The main results obtained in [13] are the following fixed point theorems.

Theorem 3. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be an 𝛼-𝜓-contractive mapping satisfying the following conditions: (i)𝑇is 𝛼-admissible; (ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1; (iii)𝑇 is continuous. Then, 𝑇 has a fixed point; that is, there exists 𝑥∈𝑋 such that 𝑇𝑥=𝑥.

Theorem 4. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be an 𝛼-𝜓-contractive mapping satisfying the following conditions: (i)𝑇 is 𝛼-admissible; (ii)there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1; (iii)if {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝑋 as 𝑛→∞, then 𝛼(𝑥_𝑛,𝑥)≥1 for all 𝑛. Then, 𝑇 has a fixed point.

Theorem 5. In addition to the hypotheses of Theorem 3 (resp., Theorem 4), suppose that for all (𝑥,𝑦)∈𝑋×𝑋, there exists 𝑧∈𝑋 such that 𝛼(𝑥,𝑧)≥1 and 𝛼(𝑦,𝑧)≥1. Then we have a unique fixed point.

It was shown in [13, 14] that various types of contractive mappings belong to the class of 𝛼-𝜓-contractive type mappings (classical contractive mappings, contractive mappings on ordered metric spaces, cyclic contractive mappings, etc.). For other works in this direction, we refer the reader to [15, 16].

In a very recent paper, Jleli and Samet [17] established some best proximity point results for 𝛼-𝜓-contractive type mappings. Before presenting the main results obtained in [17], we need to fix some notations and recall some definitions.

Let 𝐴 and 𝐵, two nonempty subsets of a metric space (𝑋,𝑑). We will use the following notations: 𝑑(){𝑑()},𝐴𝐴,𝐵∶=inf𝑎,𝑏∶𝑎∈𝐴,𝑏∈𝐵₀∶={𝑎∈𝐴∶𝑑(𝑎,𝑏)=𝑑(𝐴,𝐵)forsome},𝐵𝑏∈𝐵₀∶={𝑏∈𝐵∶𝑑(𝑎,𝑏)=𝑑(𝐴,𝐵)forsome𝑎∈𝐴}. (5)

Definition 6. An element 𝑥^∗∈𝐴 is said to be a best proximity point of the non-self-mapping 𝑇∶𝐴→𝐵 if it satisfies the condition that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (6)

The following concept was introduced in [11].

Definition 7. Let (𝐴,𝐵) be a pair of nonempty subsets of a metric space (𝑋,𝑑) with 𝐴₀≠∅. Then, the pair (𝐴,𝐵) is said to have the 𝑃-property if and only if 𝑑𝑥₁,𝑦₁()𝑑𝑥=𝑑𝐴,𝐵₂,𝑦₂()𝑥=𝑑𝐴,𝐵⟹𝑑₁,𝑥₂=𝑑𝑦₁,𝑦₂, (7) where 𝑥₁,𝑥₂∈𝐴 and 𝑦₁,𝑦₂∈𝐵.

The following concepts were introduced in [17].

Definition 8. Let 𝑇∶𝐴→𝐵 and 𝛼∶𝐴×𝐴→[0,∞). We say that 𝑇 is 𝛼-proximal admissible if 𝛼𝑥₁,𝑥₂𝑑𝑢≥1₁,𝑇𝑥₁()𝑑𝑢=𝑑𝐴,𝐵₂,𝑇𝑥₂()𝑢=𝑑𝐴,𝐵⟹𝛼₁,𝑢₂≥1, (8) for all 𝑥₁,𝑥₂,𝑢₁,𝑢₂∈𝐴.

Clearly, if 𝐴=𝐵, 𝑇 is 𝛼-proximal admissible implies that 𝑇 is 𝛼-admissible.

Definition 9. A non-self-mapping 𝑇∶𝐴→𝐵 is said to be an 𝛼-𝜓-proximal contraction, where 𝛼∶𝐴×𝐴→[0,∞) and 𝜓 is a (c)-comparison function, if 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦,∀𝑥,𝑦∈𝐴. (9)

The main results obtained in [17] are the following.

Theorem 10. Let 𝐴 and 𝐵 be nonempty closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonempty. Let 𝛼∶𝐴×𝐴→[0,∞) and 𝜓 be a (c)-comparison function. Suppose that 𝑇∶𝐴→𝐵 is a non-self-mapping satisfying the following conditions: (i)𝑇(𝐴₀)⊆𝐵₀, and (𝐴,𝐵) satisfies the 𝑃-property; (ii)𝑇 is 𝛼-proximal admissible; (iii)there exist elements 𝑥₀ and 𝑥₁ in 𝐴₀ such that 𝑑𝑥₁,𝑇𝑥₀=𝑑(𝐴,𝐵),𝛼𝑥₀,𝑥₁≥1; (10)(iv)𝑇 is a continuous 𝛼-𝜓-proximal contraction. Then, there exists an element 𝑥^∗∈𝐴₀ such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (11)

Theorem 11. Let 𝐴 and 𝐵 be nonempty closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonempty. Let 𝛼∶𝐴×𝐴→[0,∞) and 𝜓 be a (c)-comparison function. Suppose that 𝑇∶𝐴→𝐵 is a non-self-mapping satisfying the following conditions: (i)𝑇(𝐴₀)⊆𝐵₀, and (𝐴,𝐵) satisfies the 𝑃-property; (ii)𝑇 is 𝛼-proximal admissible; (iii) there exist elements 𝑥₀ and 𝑥₁ in 𝐴₀ such that 𝑑𝑥₁,𝑇𝑥₀=𝑑(𝐴,𝐵),𝛼𝑥₀,𝑥₁≥1; (12)(iv)𝑇 is an 𝛼-𝜓-proximal contraction; (v)if {𝑥_𝑛} is a sequence in 𝐴 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝐴 as 𝑛→∞, then there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝛼(𝑥_𝑛(𝑘),𝑥)≥1 for all 𝑘. Then, there exists an element 𝑥^∗∈𝐴₀ such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (13)

Theorem 12. In addition to the hypotheses of Theorem 10 (resp., Theorem 11), suppose that for all (𝑥,𝑦)∈𝛼⁻¹([0,1[), there exists 𝑧∈𝐴₀ such that 𝛼(𝑥,𝑧)≥1 and 𝛼(𝑦,𝑧)≥1. Then, 𝑇 has a unique best proximity point.

In this paper, we extend and generalize the above results by introducing a new family of non-self-contractive mappings that will be called the class of generalized 𝛼-𝜓-proximal contractive type mappings. For such mappings, we discuss the existence and uniqueness of best proximity points. Various applications and interesting consequences are derived from our main results.

2. Main Results

All the notations presented in the previous section will be used through this paper.

We denote by Ψ the set of nondecreasing functions 𝜓∶[0,∞)→[0,∞) such that ^∞_𝑛=1𝜓^𝑛(𝑡)<∞,∀𝑡>0, (14) where 𝜓^𝑛 is the 𝑛th iterate of 𝜓. These functions are known in the literature as (c)-comparison functions. It is easily proved that if 𝜓 is a (c)-comparison function, then 𝜓(𝑡)<𝑡 for all 𝑡>0.

We introduce the following concept.

Definition 13. A non-self-mapping 𝑇∶𝐴→𝐵 is said to be a generalized 𝛼-𝜓-proximal contraction, where 𝛼∶𝐴×𝐴→[0,∞) and 𝜓∈Ψ, if 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑀𝑥,𝑦,∀𝑥,𝑦∈𝐴, (15) where 𝑀𝑑,1𝑥,𝑦=max𝑥,𝑦2𝑑()(),1𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑦−𝑑𝐴,𝐵2𝑑𝑦,𝑇𝑥+𝑑𝑥,𝑇𝑦−𝑑(𝐴,𝐵). (16)

Our first main result is the following best proximity point theorem.

Theorem 14. Let 𝐴 and 𝐵 be nonempty closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonempty. Let 𝛼∶𝐴×𝐴→[0,∞) and 𝜓∈Ψ. Suppose that 𝑇∶𝐴→𝐵 is a non-self-mapping satisfying the following conditions: (i)𝑇(𝐴₀)⊆𝐵₀, and (𝐴,𝐵) satisfies the 𝑃-property; (ii)𝑇 is 𝛼-proximal admissible; (iii)there exist elements 𝑥₀ and 𝑥₁ in 𝐴₀ such that 𝑑𝑥₁,𝑇𝑥₀=𝑑(𝐴,𝐵),𝛼𝑥₀,𝑥₁≥1; (17)(iv)𝑇 is a continuous generalized 𝛼-𝜓-proximal contraction. Then, there exists an element 𝑥^∗∈𝐴₀ such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (18)

Proof. From condition (iii), there exist elements 𝑥₀ and 𝑥₁ in 𝐴₀ such that 𝑑𝑥₁,𝑇𝑥₀=𝑑(𝐴,𝐵),𝛼𝑥₀,𝑥₁≥1. (19) Since 𝑇(𝐴₀)⊆𝐵₀, there exists 𝑥₂∈𝐴₀ such that 𝑑𝑥₂,𝑇𝑥₁=𝑑(𝐴,𝐵). (20) Now, we have 𝛼𝑥₀,𝑥₁𝑑𝑥≥1,₁,𝑇𝑥₀(),𝑑𝑥=𝑑𝐴,𝐵₂,𝑇𝑥₁=𝑑(𝐴,𝐵). (21) Since 𝑇 is 𝛼-proximal admissible, this implies that 𝛼(𝑥₁,𝑥₂)≥1. Thus, we have 𝑑𝑥₂,𝑇𝑥₁=𝑑(𝐴,𝐵),𝛼𝑥₁,𝑥₂≥1. (22) Again, since 𝑇(𝐴₀)⊆𝐵₀, there exists 𝑥₃∈𝐴₀ such that 𝑑𝑥₃,𝑇𝑥₂=𝑑(𝐴,𝐵). (23) Now, we have 𝛼𝑥₁,𝑥₂𝑑𝑥≥1,₂,𝑇𝑥₁(),𝑑𝑥=𝑑𝐴,𝐵₃,𝑇𝑥₂=𝑑(𝐴,𝐵). (24) Since 𝑇 is 𝛼-proximal admissible, this implies that 𝛼(𝑥₂,𝑥₃)≥1. Thus, we have 𝑑𝑥₃,𝑇𝑥₂=𝑑(𝐴,𝐵),𝛼𝑥₂,𝑥₃≥1. (25) Continuing this process, by induction, we can construct a sequence {𝑥_𝑛}⊂𝐴₀ such that 𝑑𝑥_𝑛+1,𝑇𝑥_𝑛(),𝛼𝑥=𝑑𝐴,𝐵_𝑛,𝑥_𝑛+1≥1,∀𝑛∈ℕ∪{0}. (26) Since (𝐴,𝐵) satisfies the 𝑃-property, we conclude from (26) that 𝑑𝑥_𝑛,𝑥_𝑛+1=𝑑𝑇𝑥_𝑛−1,𝑇𝑥_𝑛,∀𝑛∈ℕ. (27) From condition (iv), that is, 𝑇 is a generalized 𝛼-𝜓-proximal contraction, for all 𝑛∈ℕ, we have 𝛼𝑥_𝑛−1,𝑥_𝑛𝑑𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝜓𝑀𝑥_𝑛−1,𝑥_𝑛. (28) On the other hand, using (26) and (27), we have 𝑀𝑥_𝑛−1,𝑥_𝑛=max𝑑𝑥_𝑛−1,𝑥_𝑛,12𝑑𝑥_𝑛−1,𝑇𝑥_𝑛−1+𝑑𝑥_𝑛,𝑇𝑥_𝑛(),1−𝑑𝐴,𝐵2𝑑𝑥_𝑛,𝑇𝑥_𝑛−1+𝑑𝑥_𝑛−1,𝑇𝑥_𝑛(𝑑𝑥−𝑑𝐴,𝐵)≤max_𝑛−1,𝑥_𝑛,12𝑑𝑥_𝑛−1,𝑥_𝑛+𝑑𝑥_𝑛,𝑇𝑥_𝑛−1+𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑇𝑥_𝑛(),1−𝑑𝐴,𝐵2𝑑(𝐴,𝐵)+𝑑𝑥_𝑛−1,𝑥_𝑛+𝑑𝑥_𝑛,𝑇𝑥_𝑛−1+𝑑𝑇𝑥_𝑛−1,𝑇𝑥_𝑛(𝑑𝑥−𝑑𝐴,𝐵)=max_𝑛−1,𝑥_𝑛,12𝑑𝑥_𝑛−1,𝑥_𝑛()𝑥+𝑑𝐴,𝐵+𝑑_𝑛,𝑥_𝑛+1((),1+𝑑𝐴,𝐵)−𝑑𝐴,𝐵2𝑑(𝐴,𝐵)+𝑑𝑥_𝑛−1,𝑥_𝑛+𝑑(𝐴,𝐵)+𝑑𝑥_𝑛,𝑥_𝑛+1(𝑑𝑥−𝑑𝐴,𝐵)=max_𝑛−1,𝑥_𝑛,12𝑑𝑥_𝑛−1,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛+1,12𝑑𝑥_𝑛−1,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛+1=max𝑑𝑥_𝑛−1,𝑥_𝑛,12𝑑𝑥_𝑛−1,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛+1≤max𝑑𝑥_𝑛−1,𝑥_𝑛,𝑑𝑥_𝑛,𝑥_𝑛+1. (29) Thus, we proved that 𝑀𝑥_𝑛−1,𝑥_𝑛≤max𝑑𝑥_𝑛−1,𝑥_𝑛,𝑑𝑥_𝑛,𝑥_𝑛+1,∀𝑛∈ℕ. (30) Using the above inequality, (26), (27), and (28), and taking in consideration that 𝜓 is a nondecreasing function, we get that 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜓max𝑑𝑥_𝑛,𝑥_𝑛−1,𝑑𝑥_𝑛,𝑥_𝑛+1,∀𝑛∈ℕ. (31) If for some 𝑁∈ℕ∪{0}, we have 𝑥_𝑁+1=𝑥_𝑁, from (26), we get that 𝑑(𝑥_𝑁,𝑇𝑥_𝑁)=𝑑(𝐴,𝐵); that is, 𝑥_𝑁 is a best proximity point. So, we can suppose that 𝑑𝑥_𝑛+1,𝑥_𝑛>0,∀𝑛∈ℕ∪{0}. (32) Suppose that max{𝑑(𝑥_𝑛,𝑥_𝑛−1),𝑑(𝑥_𝑛,𝑥_𝑛+1)}=𝑑(𝑥_𝑛,𝑥_𝑛+1). Using (32) and since 𝜓(𝑡)<𝑡 for all 𝑡>0, we have 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜓𝑑𝑥_𝑛,𝑥_𝑛+1<𝑑𝑥_𝑛,𝑥_𝑛+1, (33) which is a contradiction. Thus, we have max𝑑𝑥_𝑛,𝑥_𝑛−1,𝑑𝑥_𝑛,𝑥_𝑛+1=𝑑𝑥_𝑛,𝑥_𝑛−1,∀𝑛∈ℕ. (34) Now, from (31), we get that 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜓𝑑𝑥_𝑛,𝑥_𝑛−1,∀𝑛∈ℕ. (35) Using the monotony of 𝜓, by induction, it follows from (35) that 𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜓^𝑛𝑑𝑥₁,𝑥₀,∀𝑛∈ℕ∪{0}. (36) Now, we shall prove that {𝑥_𝑛} is a Cauchy sequence in the metric space (𝑋,𝑑). Let 𝜀>0 be fixed. Since ∑^∞_𝑛=1𝜓^𝑛(𝑑(𝑥₁,𝑥₀))<∞, there exists some positive integer ℎ=ℎ(𝜀) such that _𝑘≥ℎ𝜓^𝑘𝑑𝑥₁,𝑥₀<𝜀. (37) Let 𝑚>𝑛>ℎ, using the triangular inequality, (36) and (37), we obtain 𝑑𝑥_𝑛,𝑥_𝑚≤^𝑚−1_𝑘=𝑛𝑑𝑥_𝑘,𝑥_𝑘+1≤^𝑚−1_𝑘=𝑛𝜓^𝑘𝑑𝑥₁,𝑥₀≤_𝑘≥ℎ𝜓^𝑘𝑑𝑥₁,𝑥₀<𝜀. (38) Thus, {𝑥_𝑛} is a Cauchy sequence in the metric space (𝑋,𝑑). Since (𝑋,𝑑) is complete and 𝐴 is closed, there exists some 𝑥^∗∈𝐴 such that 𝑥_𝑛→𝑥^∗ as 𝑛→∞. On the other hand, 𝑇 is a continuous mapping. Then, we have 𝑇𝑥_𝑛→𝑇𝑥^∗ as 𝑛→∞. The continuity of the metric function 𝑑 implies that 𝑑(𝐴,𝐵)=𝑑(𝑥_𝑛+1,𝑇𝑥_𝑛)→𝑑(𝑥^∗,𝑇𝑥^∗) as 𝑛→∞. Therefore, 𝑑(𝑥^∗,𝑇𝑥^∗)=𝑑(𝐴,𝐵). This completes the proof of the theorem.

In the next result, we remove the continuity hypothesis, assuming the following condition in 𝐴:(H) If {𝑥_𝑛} is a sequence in 𝐴 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛 and 𝑥_𝑛→𝑥∈𝐴 as 𝑛→∞, then there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝛼(𝑥_𝑛(𝑘),𝑥)≥1 for all 𝑘.

Theorem 15. Let 𝐴 and 𝐵 be nonempty closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonempty. Let 𝛼∶𝐴×𝐴→[0,∞) and 𝜓∈Ψ. Suppose that 𝑇∶𝐴→𝐵 is a non-self-mapping satisfying the following conditions: (i)𝑇(𝐴₀)⊆𝐵₀, and (𝐴,𝐵) satisfies the 𝑃-property; (ii)𝑇 is 𝛼-proximal admissible; (iii)there exist elements 𝑥₀ and 𝑥₁ in 𝐴₀ such that 𝑑𝑥₁,𝑇𝑥₀=𝑑(𝐴,𝐵),𝛼𝑥₀,𝑥₁≥1; (39)(iv) (H) holds, and 𝑇 is a generalized 𝛼-𝜓-proximal contraction. Then, there exists an element 𝑥^∗∈𝐴₀ such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (40)

Proof. Following the proof of Theorem 14, there exists a Cauchy sequence {𝑥_𝑛}⊂𝐴 such that (26) holds, and 𝑥_𝑛→𝑥^∗∈𝐴 as 𝑛→∞. From the condition (H), there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝛼(𝑥_𝑛(𝑘),𝑥^∗)≥1 for all 𝑘. Since 𝑇 is a generalized 𝛼-𝜓-proximal contraction, we get that 𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑥^∗≤𝛼𝑥_𝑛(𝑘),𝑥^∗𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑥^∗≤𝜓𝑀𝑥_𝑛(𝑘),𝑥^∗,∀𝑘, (41) where 𝑀𝑥_𝑛(𝑘),𝑥^∗=max𝑑𝑥_𝑛(𝑘),𝑥^∗,𝑑𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘)+𝑑𝑥^∗,𝑇𝑥^∗2(),𝑑𝑥−𝑑𝐴,𝐵_𝑛(𝑘),𝑇𝑥^∗+𝑑𝑥^∗,𝑇𝑥_𝑛(𝑘)2−𝑑(𝐴,𝐵). (42) On the other hand, from (26), for all 𝑘, we have 𝑑𝑥^∗,𝑇𝑥^∗≤𝑑𝑥^∗,𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘)+1,𝑇𝑥_𝑛(𝑘)+𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑥^∗=𝑑𝑥^∗,𝑥_𝑛(𝑘)+1+𝑑(𝐴,𝐵)+𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑥^∗. (43) Thus, we have 𝑑𝑇𝑥_𝑛(𝑘),𝑇𝑥^∗≥𝑑𝑥^∗,𝑇𝑥^∗−𝑑𝑥^∗,𝑥_𝑛(𝑘)+1−𝑑(𝐴,𝐵),∀𝑘. (44) Combining (41) with (44), we get that 𝑑𝑥^∗,𝑇𝑥^∗−𝑑𝑥^∗,𝑥_𝑛(𝑘)+1()𝑀𝑥−𝑑𝐴,𝐵≤𝜓_𝑛(𝑘),𝑥^∗,∀𝑘. (45) From (26), for all 𝑘, we have 𝑀𝑥_𝑛(𝑘),𝑥^∗=max𝑑𝑥_𝑛(𝑘),𝑥^∗,𝑑𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘)+𝑑𝑥^∗,𝑇𝑥^∗2(),𝑑𝑥−𝑑𝐴,𝐵_𝑛(𝑘),𝑇𝑥^∗+𝑑𝑥^∗,𝑇𝑥_𝑛(𝑘)2(𝑑𝑥−𝑑𝐴,𝐵)≤max_𝑛(𝑘),𝑥^∗,𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘)+1,𝑇𝑥_𝑛(𝑘)+𝑑𝑥^∗,𝑇𝑥^∗2(),𝑑𝑥−𝑑𝐴,𝐵_𝑛(𝑘),𝑥^∗+𝑑𝑥^∗,𝑇𝑥^∗+𝑑𝑥^∗,𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘)+1,𝑇𝑥_𝑛(𝑘)×(2)⁻¹(𝑑𝑥−𝑑𝐴,𝐵)=max_𝑛(𝑘),𝑥^∗,𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1+𝑑(𝐴,𝐵)+𝑑𝑥^∗,𝑇𝑥^∗2(),𝑑𝑥−𝑑𝐴,𝐵_𝑛(𝑘),𝑥^∗+𝑑𝑥^∗,𝑇𝑥^∗+𝑑𝑥^∗,𝑥_𝑛(𝑘)+1+𝑑(𝐴,𝐵)2(𝑥−𝑑𝐴,𝐵)∶=𝜁_𝑛(𝑘),𝑥^∗. (46) Since 𝜓 is a nondecreasing function, we get from (45) that 𝑑𝑥^∗,𝑇𝑥^∗−𝑑𝑥^∗,𝑥_𝑛(𝑘)+1()𝜁𝑥−𝑑𝐴,𝐵≤𝜓_𝑛(𝑘),𝑥^∗,∀𝑘. (47) Suppose that 𝑑(𝑥^∗,𝑇𝑥^∗)−𝑑(𝐴,𝐵)>0. In this case, we have lim_𝑘→∞𝜁𝑥_𝑛(𝑘),𝑥^∗𝑑𝑥=max0,^∗,𝑇𝑥^∗−𝑑(𝐴,𝐵)2,𝑑𝑥^∗,𝑇𝑥^∗−𝑑(𝐴,𝐵)2, (48) that is, lim_𝑘→∞𝜁𝑥_𝑛(𝑘),𝑥^∗=𝑑𝑥^∗,𝑇𝑥^∗−𝑑(𝐴,𝐵)2. (49) Since 𝑑𝑥^∗,𝑇𝑥^∗−𝑑(𝐴,𝐵)2>0, (50) for 𝑘 large enough, we have 𝜁(𝑥_𝑛(𝑘),𝑥^∗)>0. On the other hand, we have 𝜓(𝑡)<𝑡 for all 𝑡>0. Then, from (47), we get that 𝑑𝑥^∗,𝑇𝑥^∗−𝑑𝑥^∗,𝑥_𝑛(𝑘)+1()𝑥−𝑑𝐴,𝐵<𝜁_𝑛(𝑘),𝑥^∗,for𝑘largeenough. (51) Using (49) and letting 𝑘→∞ in the above inequality, we obtain that 𝑑𝑥^∗,𝑇𝑥^∗()≤𝑑𝑥−𝑑𝐴,𝐵^∗,𝑇𝑥^∗−𝑑(𝐴,𝐵)2, (52) which is a contradiction. Thus, we deduce that 𝑥^∗ is a best proximity point of 𝑇; that is, 𝑑(𝑥^∗,𝑇𝑥^∗)=𝑑(𝐴,𝐵).

The next result gives us a sufficient condition that assures the uniqueness of the best proximity point. We need the following definition.

Definition 16. Let 𝑇∶𝐴→𝐵 be a non-self-mapping and 𝛼∶𝐴×𝐴→[0,∞). We say that 𝑇 is (𝛼,𝑑) regular if for all (𝑥,𝑦)∈𝛼⁻¹([0,1[), there exists 𝑧∈𝐴₀ such that 𝛼(𝑥,𝑧)≥1,𝛼𝑦,𝑧≥1. (53)

Theorem 17. In addition to the hypotheses of Theorem 14 (resp., Theorem 15), suppose that 𝑇 is (𝛼,𝑑) regular. Then, 𝑇 has a unique best proximity point.

Proof. From the proof of Theorem 14 (resp., Theorem 15), we know that the set of best proximity points of 𝑇 is nonempty (𝑥^∗∈𝐴₀ is a best proximity point). Suppose that 𝑦^∗∈𝐴₀ is another best proximity point of 𝑇, that is, 𝑑𝑇𝑥^∗,𝑥^∗=𝑑𝑇𝑦^∗,𝑦^∗=𝑑(𝐴,𝐵). (54) Using the 𝑃-property and (54), we get that 𝑑𝑇𝑥^∗,𝑇𝑦^∗=𝑑𝑥^∗,𝑦^∗. (55) We distinguish two cases.
Case  1. If 𝛼(𝑥^∗,𝑦^∗)≥1.
Since 𝑇 is a generalized 𝛼-𝜓-proximal contraction, using (55), we obtain that 𝑑𝑥^∗,𝑦^∗=𝑑𝑇𝑥^∗,𝑇𝑦^∗≤𝛼𝑥^∗,𝑦^∗𝑑𝑇𝑥^∗,𝑇𝑦^∗≤𝜓𝑀𝑥^∗,𝑦^∗, (56) where from (54) and (55), we have 𝑀𝑥^∗,𝑦^∗=max𝑑𝑥^∗,𝑦^∗,12𝑑𝑥^∗,𝑇𝑥^∗+𝑑𝑦^∗,𝑇𝑦^∗(),1−𝑑𝐴,𝐵2𝑑𝑦^∗,𝑇𝑥^∗+𝑑𝑥^∗,𝑇𝑦^∗(𝑑𝑥−𝑑𝐴,𝐵)=max^∗,𝑦^∗1,0,2𝑑𝑦^∗,𝑇𝑥^∗+𝑑𝑥^∗,𝑇𝑦^∗(𝑑𝑥−𝑑𝐴,𝐵)≤max^∗,𝑦^∗,12𝑑𝑦^∗,𝑇𝑦^∗+𝑑𝑇𝑦^∗,𝑇𝑥^∗+𝑑𝑥^∗,𝑇𝑥^∗+𝑑𝑇𝑥^∗,𝑇𝑦^∗(𝑑𝑥−𝑑𝐴,𝐵)=max^∗,𝑦^∗,𝑑𝑥^∗,𝑦^∗=𝑑𝑥^∗,𝑦^∗. (57) This equality with (56) imply that 𝑑𝑥^∗,𝑦^∗≤𝜓𝑑𝑥^∗,𝑦^∗. (58) Since 𝜓(𝑡)<𝑡 for all 𝑡>0, the above inequality holds only if 𝑑(𝑥^∗,𝑦^∗)=0, that is, 𝑥^∗=𝑦^∗.
Case  2. If 𝛼(𝑥^∗,𝑦^∗)<1.
By hypothesis, there exists 𝑧₀∈𝐴₀ such that 𝛼(𝑥^∗,𝑧₀)≥1 and 𝛼(𝑦^∗,𝑧₀)≥1. Since 𝑇(𝐴₀)⊆𝐵₀, there exists 𝑧₁∈𝐴₀ such that 𝑑𝑧₁,𝑇𝑧₀=𝑑(𝐴,𝐵). (59) Now, we have 𝛼𝑥^∗,𝑧₀𝑑𝑥≥1,_∗,𝑇𝑥^∗(),𝑑𝑧=𝑑𝐴,𝐵₁,𝑇𝑧₀=𝑑(𝐴,𝐵). (60) Since 𝑇 is 𝛼-proximal admissible, we get that 𝛼(𝑥^∗,𝑧₁)≥1. Thus, we have 𝑑𝑧₁,𝑇𝑧₀=𝑑(𝐴,𝐵),𝛼𝑥^∗,𝑧₁≥1. (61) Continuing this process, by induction, we can construct a sequence {𝑧_𝑛} in 𝐴₀ such that 𝑑𝑧_𝑛+1,𝑇𝑧_𝑛=𝑑(𝐴,𝐵),𝛼𝑥^∗,𝑧_𝑛≥1,∀𝑛∈ℕ∪{0}. (62) Using the 𝑃-property and (62), we get that 𝑑𝑧_𝑛+1,𝑥^∗=𝑑𝑇𝑧_𝑛,𝑇𝑥^∗,∀𝑛∈ℕ∪{0}. (63) Since 𝑇 is a generalized 𝛼-𝜓-proximal contraction, we have 𝛼𝑧_𝑛+1,𝑥^∗𝑑𝑇𝑧_𝑛,𝑇𝑥^∗≤𝜓𝑀𝑧_𝑛,𝑥^∗,∀𝑛∈ℕ∪{0}. (64) Combining the above inequality with (63), we get that 𝛼𝑧_𝑛+1,𝑥^∗𝑑𝑧_𝑛+1,𝑥^∗≤𝜓𝑀𝑧_𝑛,𝑥^∗,∀𝑛∈ℕ∪{0}. (65) This implies from (62) that 𝑑𝑧_𝑛+1,𝑥^∗≤𝜓𝑀𝑧_𝑛,𝑥^∗,∀𝑛∈ℕ∪{0}. (66) On the other hand, from (63), for all 𝑛∈ℕ∪{0}, we have 𝑀𝑧_𝑛,𝑥^∗=max𝑑𝑧_𝑛,𝑥^∗,12𝑑𝑧_𝑛,𝑇𝑧_𝑛+𝑑𝑥^∗,𝑇𝑥^∗(),1−𝑑𝐴,𝐵2𝑑𝑥^∗,𝑇𝑧_𝑛+𝑑𝑧_𝑛,𝑇𝑥^∗(𝑑𝑧−𝑑𝐴,𝐵)=max_𝑛,𝑥^∗,12𝑑𝑧_𝑛,𝑥^∗+𝑑𝑥^∗,𝑇𝑥^∗+𝑑𝑇𝑥^∗,𝑇𝑧_𝑛((),1+𝑑𝐴,𝐵)−𝑑𝐴,𝐵2𝑑𝑥^∗,𝑇𝑥^∗+𝑑𝑇𝑥^∗,𝑇𝑧_𝑛+𝑑𝑧_𝑛,𝑥^∗+𝑑𝑥^∗,𝑇𝑥^∗(𝑑𝑧−𝑑𝐴,𝐵)=max_𝑛,𝑥^∗,12𝑑𝑧_𝑛,𝑥^∗+𝑑𝑧_𝑛+1,𝑥^∗≤max𝑑𝑧_𝑛,𝑥^∗,𝑑𝑧_𝑛+1,𝑥^∗. (67) Combining the above inequality with (71), we get that 𝑑𝑧_𝑛+1,𝑥^∗≤𝜓max𝑑𝑧_𝑛,𝑥^∗,𝑑𝑧_𝑛+1,𝑥^∗,∀𝑛∈ℕ∪{0}. (68) Suppose that for some 𝑁, we have 𝑧_𝑁=𝑥^∗. From (63), we get that 𝑧_𝑛=𝑥^∗ for all 𝑛≥𝑁. This implies that 𝑧_𝑛→𝑥^∗ as 𝑛→∞. Now, suppose that 𝑑(𝑧_𝑛,𝑥^∗)>0 for all 𝑛∈ℕ∪{0}. Since 𝜓(𝑡)<𝑡 for all 𝑡>0, the inequality (68) holds only if max𝑑𝑧_𝑛,𝑥^∗,𝑑𝑧_𝑛+1,𝑥^∗=𝑑𝑧_𝑛,𝑥^∗,∀𝑛∈ℕ∪{0}. (69) Now, we have 𝑑𝑧_𝑛+1,𝑥^∗≤𝜓𝑑𝑧_𝑛,𝑥^∗,∀𝑛∈ℕ∪{0}. (70) By induction, we then derive 𝑑𝑧_𝑛,𝑥^∗≤𝜓^𝑛𝑑𝑧₀,𝑥^∗,∀𝑛∈ℕ∪{0}. (71) Letting 𝑛→∞ in (71), we obtain that 𝑧_𝑛→𝑥^∗ as 𝑛→∞. So, in all cases, we have 𝑧_𝑛→𝑥^∗ as 𝑛→∞. Similarly, we can prove that 𝑧_𝑛→𝑦^∗ as 𝑛→∞. By uniqueness of the limit, we obtain that 𝑥^∗=𝑦^∗.

3. Applications

3.1. Standard Best Proximity Point Results

We have the following best proximity point result.

Corollary 18. Let 𝐴 and 𝐵 be nonempty closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonempty. Let 𝜓∈Ψ and suppose that 𝑇∶𝐴→𝐵 is a non-self-mapping satisfying the following conditions: (i)𝑇(𝐴₀)⊆𝐵₀, and (𝐴,𝐵) satisfies the 𝑃-property; (ii)𝑑(𝑇𝑥,𝑇𝑦)≤𝜓(𝑀(𝑥,𝑦)), for all 𝑥,𝑦∈𝐴. Then, there exists a unique element 𝑥^∗∈𝐴₀ such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (72)

Proof. Consider the mapping 𝛼∶𝐴×𝐴→[0,∞) defined by: 𝛼𝑥,𝑦=1,∀𝑥,𝑦∈𝐴. (73) From the definition of 𝛼, clearly 𝑇 is 𝛼-proximal admissible and also it is an 𝛼-𝜓-proximal contraction. On the other hand, for any 𝑥∈𝐴₀, since 𝑇(𝐴₀)⊆𝐵₀, there exists 𝑦∈𝐴₀ such that 𝑑(𝑇𝑥,𝑦)=𝑑(𝐴,𝐵). Moreover, from condition (ii), 𝑇 is a continuous mapping. Now, all the hypotheses of Theorem 14 are satisfied and the existence of the best proximity point follows from Theorem 14. The uniqueness is an immediate consequence of the definition of 𝛼 and Theorem 17.

Taking in Corollary 18  𝜓(𝑡)=𝑘𝑡, where 𝑘∈(0,1), we obtain the following best proximity point result.

Corollary 19. Let 𝐴 and 𝐵 be nonempty closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonempty. Suppose that 𝑇∶𝐴→𝐵 is a non-self-mapping satisfying the following conditions: (i)𝑇(𝐴₀)⊆𝐵₀, and (𝐴,𝐵) satisfies the 𝑃-property; (ii)there exists 𝑘∈(0,1) such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝑘𝑀(𝑥,𝑦), for all 𝑥,𝑦∈𝐴. Then, there exists a unique element 𝑥^∗∈𝐴₀ such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (74)

3.2. Best Proximity Points on a Metric Space Endowed with an Arbitrary Binary Relation

Before presenting our results, we need a few preliminaries.

Let (𝑋,𝑑) be a metric space and ℛ be a binary relation over 𝑋. Denote 𝒮=ℛ∪ℛ⁻¹; (75) this is the symmetric relation attached to 𝑅. Clearly, 𝑥,𝑦∈𝑋,𝑥𝒮𝑦⟺𝑥ℛ𝑦or𝑦ℛ𝑥. (76)

Definition 20. We say that 𝑇∶𝐴→𝐵 is a proximal comparative mapping if 𝑥₁𝒮𝑥₂𝑑𝑢₁,𝑇𝑥₁()𝑑𝑢=𝑑𝐴,𝐵₂,𝑇𝑥₂=𝑑(𝐴,𝐵)⟹𝑢₁𝒮𝑢₂, (77) for all 𝑥₁,𝑥₂,𝑢₁,𝑢₂∈𝐴.

We have the following best proximity point result.

Corollary 21. Let 𝐴 and 𝐵 be nonempty closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonempty. Let ℛ be a binary relation over 𝑋. Suppose that 𝑇∶𝐴→𝐵 is a continuous non-self-mapping satisfying the following conditions: (i)𝑇(𝐴₀)⊆𝐵₀, and (𝐴,𝐵) satisfies the 𝑃-property; (ii)𝑇 is a proximal comparative mapping; (iii)there exist elements 𝑥₀ and 𝑥₁ in 𝐴₀ such that 𝑑𝑥₁,𝑇𝑥₀=𝑑(𝐴,𝐵),𝑥₀𝒮𝑥₁; (78)(iv)there exists 𝜓∈Ψ such that 𝑥,𝑦∈𝐴,𝑥𝒮𝑦⟹𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑀𝑥,𝑦. (79)Then, there exists an element 𝑥^∗∈𝐴₀ such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (80)

Proof. Define the mapping 𝛼∶𝐴×𝐴→[0,∞) by: 𝛼=1𝑥,𝑦if0𝑥𝒮𝑦,otherwise. (81) Suppose that 𝛼𝑥₁,𝑥₂𝑑𝑢≥1,₁,𝑇𝑥₁(),𝑑𝑢=𝑑𝐴,𝐵₂,𝑇𝑥₂=𝑑(𝐴,𝐵), (82) for some 𝑥₁,𝑥₂,𝑢₁,𝑢₂∈𝐴. By the definition of 𝛼, we get that 𝑥₁𝒮𝑥₂,𝑑𝑢₁,𝑇𝑥₁(),𝑑𝑢=𝑑𝐴,𝐵₂,𝑇𝑥₂=𝑑(𝐴,𝐵). (83) Condition (ii) implies that 𝑢₁𝒮𝑢₂, which gives us from the definition of 𝛼 that 𝛼(𝑢₁,𝑢₂)≥1. Thus, we proved that 𝑇 is 𝛼-proximal admissible. Condition (iii) implies that 𝑑𝑥₁,𝑇𝑥₀=𝑑(𝐴,𝐵),𝛼𝑥₀,𝑥₁≥1. (84) Finally, condition (iv) implies that 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑀𝑥,𝑦,∀𝑥,𝑦∈𝐴, (85) that is, 𝑇 is a generalized 𝛼-𝜓-proximal contraction. Now, all the hypotheses of Theorem 14 are satisfied, and the desired result follows immediately from this theorem.

In order to remove the continuity assumption, we need the following condition: (ℋ) if the sequence {𝑥_𝑛} in 𝑋 and the point 𝑥∈𝑋 are such that 𝑥_𝑛𝒮𝑥_𝑛+1 for all 𝑛 and lim_𝑛→∞𝑑(𝑥_𝑛,𝑥)=0, then there exists a subsequence {𝑥_𝑛(𝑘)} of {𝑥_𝑛} such that 𝑥_𝑛(𝑘)𝒮𝑥 for all 𝑘.

Corollary 22. Let 𝐴 and 𝐵 be nonempty closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonempty. Let ℛ be a binary relation over 𝑋. Suppose that 𝑇∶𝐴→𝐵 is a non-self-mapping satisfying the following conditions: (i)𝑇(𝐴₀)⊆𝐵₀, and (𝐴,𝐵) satisfies the 𝑃-property; (ii)𝑇 is a proximal comparative mapping; (iii)there exist elements 𝑥₀ and 𝑥₁ in 𝐴₀ such that 𝑑𝑥₁,𝑇𝑥₀=𝑑(𝐴,𝐵),𝑥₀𝒮𝑥₁; (86)(iv)there exists 𝜓∈Ψ such that 𝑥,𝑦∈𝐴,𝑥𝒮𝑦⟹𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑀𝑥,𝑦, (87)(v)(ℋ) holds. Then, there exists an element 𝑥^∗∈𝐴₀ such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (88)

Proof. The result follows from Theorem 15 by considering the mapping 𝛼 given by (81), and by observing that, condition (ℋ) implies condition (H).

Corollary 23. In addition to the hypotheses of Corollary 21 (resp., Corollary 22), suppose that the following condition holds: for all (𝑥,𝑦)∈𝐴×𝐴 with (𝑥,𝑦)∉𝒮, there exists 𝑧∈𝐴₀ such that 𝑥𝒮𝑧 and 𝑦𝒮𝑧. Then, 𝑇 has a unique best proximity point.

Proof. The result follows from Theorem 17 by considering the mapping 𝛼 given by (81).

3.3. Related Fixed Point Theorems

3.3.1. Fixed Points for Generalized 𝛼-𝜓 Contractive Type Mappings

The concept of generalized 𝛼-𝜓 contractive type mappings was introduced recently in [14].

Definition 24. Let 𝐴 be a nonempty subset of a metric space (𝑋,𝑑) and 𝑇∶𝐴→𝐴 be a self-mapping. We say that 𝑇 is a generalized 𝛼-𝜓 contractive mapping if there exist two functions 𝛼∶𝐴×𝐴→[0,∞) and 𝜓∈Ψ such that for all 𝑥,𝑦∈𝐴, we have 𝛼𝑑𝑑,𝑑()𝑥,𝑦𝑇𝑥,𝑇𝑦≤𝜓max𝑥,𝑦𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑦2,𝑑𝑦,𝑇𝑥+𝑑𝑥,𝑇𝑦2. (89)

Taking 𝐴=𝐵 in Theorems 14–17, we obtain the following fixed point results established in [14].

Corollary 25. Let 𝐴 be a nonempty closed subset of a complete metric space (𝑋,𝑑). Let 𝑇∶𝐴→𝐴 be a generalized 𝛼-𝜓 contractive mapping satisfying the following conditions: (i)𝑇 is 𝛼-admissible; (ii)there exists 𝑥₀∈𝐴 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1; (iii)𝑇 is continuous. Then, 𝑇 has a fixed point.

Corollary 26. Let 𝐴 be a nonempty closed subset of a complete metric space (𝑋,𝑑). Let 𝑇∶𝐴→𝐴 be a generalized 𝛼-𝜓 contractive mapping satisfying the following conditions: (i)𝑇 is 𝛼-admissible; (ii)there exists 𝑥₀∈𝐴 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1; (iii)condition (H) holds. Then, 𝑇 has a fixed point.

Corollary 27. In addition to the hypotheses of Corollary 25 (resp., Corollary 26), suppose that for all (𝑥,𝑦)∈𝛼⁻¹([0,1[), there exists 𝑧∈𝐴 such that 𝛼(𝑥,𝑧)≥1,𝛼𝑦,𝑧≥1. (90) Then, 𝑇 has a unique fixed point.

3.3.2. Fixed Points on a Metric Space Endowed with an Arbitrary Binary Relation

We recall the following concept introduced in [18].

Let 𝐴 be a nonempty closed subset of a complete metric space (𝑋,𝑑). Suppose that 𝑋 is endowed with an arbitrary binary relation ℛ. We denote by 𝒮 the symmetric relation attached to ℛ. Let 𝑇∶𝐴→𝐴 be a given mapping.

Definition 28. We say that 𝑇∶𝐴→𝐴 is a comparative mapping if 𝑇  maps comparable elements into comparable elements, that is, 𝑥,𝑦∈𝐴,𝑥𝒮𝑦⟹𝑇𝑥𝒮𝑇𝑦. (91)

We have the following fixed point theorem.

Corollary 29. Assume that 𝑇∶𝐴→𝐴 is a continuous comparative map, and 𝑑,𝑑()𝑥,𝑦∈𝐴,𝑥𝒮𝑦⟹𝜓max𝑥,𝑦𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑦2,𝑑𝑦,𝑇𝑥+𝑑𝑥,𝑇𝑦2, (92) where 𝜓∈Ψ. Suppose also that there exists 𝑥₀∈𝑋 such that 𝑥₀𝒮𝑇𝑥₀. Then, 𝑇 has a fixed point.

Proof. It follows from Corollary 21 by taking 𝐴=𝐵 and remarking that if 𝐴=𝐵, a comparative map is a proximal comparative map.

Remark that a self-mapping 𝑇∶𝐴→𝐴 satisfying the property (23) is not necessarily continuous (see Example  2.2 in [18]).

Similarly, Taking 𝐴=𝐵 in Corollary 22, we obtain the following fixed point result.

Corollary 30. Assume that 𝑇∶𝐴→𝐴 is a comparative map satisfying (23) for some 𝜓∈Ψ. Suppose also that there exists 𝑥₀∈𝑋 such that 𝑥₀𝒮𝑇𝑥₀. If (ℋ) holds, then 𝑇 has a fixed point.

The uniqueness of the fixed point follows from Corollary 23 by taking 𝐴=𝐵.

Corollary 31. In addition to the hypotheses of Corollary 29 (resp., Corollary 30), suppose that the following condition holds: for all (𝑥,𝑦)∈𝐴×𝐴 with (𝑥,𝑦)∉𝒮, there exists 𝑧∈𝐴 such that 𝑥𝒮𝑧 and 𝑦𝒮𝑧. Then, 𝑇 has a unique fixed point.

Acknowledgment

This work is supported by the Research Center, College of Science, King Saud University.