On Best Proximity Points under the 𝑃-Property on Partially Ordered Metric Spaces

Abstract

Very recently, Abkar and Gabeleh (2013) observed that some best proximity point results under the 𝑃-property can be obtained from the same results in fixed-point theory. In this paper, motivated by this mentioned work, we show that the most best proximity point results on a metric space endowed with a partial order (under the 𝑃-property) can be deduced from existing fixed-point theorems in the literature. We present various model examples to illustrate this point of view.

1. Introduction

Let 𝐴 and 𝐵 be two nonempty subsets of a metric space (𝑋,𝑑). Through this paper, 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}. (1)

Definition 1. An element 𝑥^∗∈𝐴 is said to be a best proximity point of the nonself-mapping 𝑇∶𝐴→𝐵 if and only if it satisfies the condition that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (2)

The notion of 𝑃-property was introduced in [1] as follows.

Definition 2. 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, (3) where 𝑥₁,𝑥₂∈𝐴 and 𝑦₁,𝑦₂∈𝐵.

Various best proximity point results for different classes of nonself-mappings under the 𝑃-property were established recently (see, e.g., [1–6] and references therein). Very recently, Abkar and Gabeleh [7] observed that some best proximity point results under the 𝑃-property can be obtained from the same results in fixed-point theory. The main purpose of this paper is to show that in many cases best proximity point results (under the 𝑃-property) on partially ordered metric spaces can also be deduced from the corresponding fixed-point theorems. We present various model examples to illustrate this point of view.

2. Preliminaries

As model examples, we will consider some known fixed-point theorems in the framework of partially ordered metric spaces.

At first, we need to introduce two classes of real functions. Denote by Ψ the class of all functions 𝜓∶[0,+∞[→[0,+∞[ satisfying the following conditions:  (Ψ₁)𝜓 is continuous and nondecreasing, (Ψ₂)𝜓⁻¹({0})={0}.

We denote by Φ the set of functions 𝜑∶[0,+∞[→[0,+∞[ satisfying the following conditions:  (Φ₁)𝜑 is nondecreasing,  (Φ₂)lim_𝑛→+∞𝜑^𝑛(𝑡)=0 for each 𝑡>0.

Through this paper, ℕ denotes the set of all natural numbers, and ℕ^∗=ℕ⧵{0}.

The following concepts will be useful later.

Definition 3. Let (𝑋,⪯) be a partially ordered set and 𝑇∶𝑋→𝑋 be a giving mapping. We say that 𝑇 is nondecreasing (with respect to ⪯) if and only if 𝑥,𝑦∈𝑋,𝑥⪯𝑦⟹𝑇𝑥⪯𝑇𝑦. (4)

Definition 4. Let (𝑋,⪯) be a partially ordered set and {𝑥_𝑛} be a sequence in 𝑋. We say that {𝑥_𝑛} is nondecreasing (with respect to ⪯) if and only if 𝑥_𝑛⪯𝑥_𝑛+1 for all 𝑛∈ℕ.

Definition 5. Let (𝑋,⪯) be a partially ordered set and 𝑑 be a metric on 𝑋. We say that (𝑋,⪯,𝑑) is regular if and only if the following condition holds: 𝑥_𝑛⊂𝑋isnondecreasinglim_𝑛→+∞𝑑𝑥_𝑛,𝑥=0,𝑥∈𝑋⟹𝑥_𝑛⪯𝑥,∀𝑛∈ℕ. (5)

Definition 6. Let (𝑋,⪯) be a partially ordered set. We say that (𝑋,⪯) is directed if and only if ∀𝑥,𝑦∈𝑋×𝑋,∃𝑧∈𝑋∣(𝑥⪯𝑧)∧𝑦⪯𝑧. (6)

In [8], Harjani and Sadarangani proved the following fixed point result.

Theorem 7. Let (𝑋,⪯) be a partially ordered set and suppose that there exists a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Let 𝑇∶𝑋→𝑋 be a giving mapping. Suppose that the following conditions hold: (i)𝑇 is continuous or (𝑋,⪯,𝑑) is regular,(ii)𝑇 is nondecreasing,(iii)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑇𝑥₀,(iv)for all 𝑥,𝑦∈𝑋 such that 𝑥⪯𝑦, we have 𝜓𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦−𝜑𝑑𝑥,𝑦, (7) where 𝜓,𝜙∈Ψ.
Then 𝑇 has a fixed point. Moreover, the sequence {𝑇^𝑛𝑥₀} converges to this fixed point. If, in addition, (𝑋,⪯) is directed, we have uniqueness of the fixed point.

In [9], Agarwal et al. established the following fixed point result.

Theorem 8. Let (𝑋,⪯) be a partially ordered set and suppose that there exists a metric 𝑑 in X such that (𝑋,𝑑) is a complete metric space. Let 𝑇∶𝑋→𝑋 be a giving mapping. Suppose that the following conditions hold: (i)𝑇 is continuous or (𝑋,⪯,𝑑) is regular,(ii)𝑇 is nondecreasing,(iii)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑇𝑥₀,(iv)for all 𝑥,𝑦∈𝑋 such that 𝑥⪯𝑦, we have 𝑑𝑑(),𝑑𝑇𝑥,𝑇𝑦≤𝜑max𝑥,𝑦,𝑑𝑥,𝑇𝑥,𝑑𝑦,𝑇𝑦𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥2, (8) where 𝜑∈Φ.
Then 𝑇 has a fixed point. Moreover, the sequence {𝑇^𝑛𝑥₀} converges to this fixed point.

The above result was subsequently extended by Turinici in [10].

Through this paper, (𝐴,𝐵) is a pair of nonempty subsets of a metric space (𝑋,𝑑), where 𝑋 is endowed with a partial order ⪯. We denote 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}. (9)

Definition 9. An element 𝑥^∗∈𝐴 is said to be a best proximity point of the nonself-mapping 𝑇∶𝐴→𝐵 if and only if 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (10)

The notion of 𝑃-property was introduced in [1] as follows.

Definition 10. 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, (11) where 𝑥₁,𝑥₂∈𝐴 and 𝑦₁,𝑦₂∈𝐵.

In [11], the authors introduced the concept of proximally increasing mappings.

Definition 11. A nonself-mapping 𝑇∶𝑀⊆𝐴→𝐵 is said to be proximally increasing if and only if 𝑦₁⪯𝑦₂𝑑𝑥₁,𝑇𝑦₁()𝑑𝑥=𝑑𝐴,𝐵₂,𝑇𝑦₂=𝑑(𝐴,𝐵)⟹𝑥₁⪯𝑥₂, (12) where 𝑥₁,𝑥₂∈𝐴,𝑦₁,𝑦₂∈𝑀.

Definition 12. Let (𝑀,𝑁) be a pair of nonempty subsets of 𝑋. A mapping 𝐼∶𝑀→𝑁 is said to be an isometry if and only if 𝑑𝐼𝑥,𝐼𝑦=𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑀. (13)

The following lemmas will be useful later.

Lemma 13. Let (𝑀,𝑁) be a pair of nonempty subsets of 𝑋. Let 𝐼∶𝑀→𝑁 be a bijective mapping. Then 𝐼 is an isometry if and only if 𝐼⁻¹ is an isometry.

Proof. Suppose that 𝐼 is an isometry. Let 𝑧,𝑡∈𝑁,𝑥=𝐼⁻¹𝑧 and 𝑦=𝐼⁻¹𝑡. Since 𝐼 is an isometry, we have 𝑑𝐼𝑥,𝐼𝑦=𝑑𝑥,𝑦, (14) that is, 𝑑(𝑧,𝑡)=𝑑𝐼⁻¹𝑧,𝐼⁻¹𝑡. (15) Then 𝐼⁻¹ is an isometry.

Lemma 14. Consider two nonself-mappings 𝑇∶𝐴₀→𝐵₀ and 𝐼∶𝐴₀→𝐵₀. Suppose that the following conditions hold: (i)𝑇 is proximally increasing,(ii)𝐼∶𝐴₀→𝐵₀ is bijective,(iii)𝑑(𝑥,𝐼𝑥)=𝑑(𝐴,𝐵) for all 𝑥∈𝐴₀.
Then the mapping 𝐼⁻¹𝑇∶𝐴₀→𝐴₀ is nondecreasing (with respect to ⪯).

Proof. Let 𝑥,𝑦∈𝐴₀ such that 𝑥⪯𝑦. We have 𝑑𝐼𝑥⪯𝑦,₋₁𝑇𝑥,𝑇𝑥=𝑑𝐼⁻¹𝑇𝑥,𝐼𝐼⁻¹(𝑇𝑥=𝑑𝐴,𝐵)(from(iii,𝑑𝐼))₋₁𝑇𝑦,𝑇𝑦=𝑑𝐼⁻¹𝑇𝑦,𝐼𝐼⁻¹𝑇𝑦=𝑑(𝐴,𝐵). (16) Since 𝑇 is proximally increasing, we get that 𝐼⁻¹𝑇𝑥⪯𝐼⁻¹𝑇𝑦.

Lemma 15. Consider two nonself-mappings 𝑇∶𝐴₀→𝐵₀ and 𝐼∶𝐴₀→𝐵₀. Suppose that the following conditions hold: (i)𝐼∶𝐴₀→𝐵₀ is bijective,(ii)𝑑(𝑥,𝐼𝑥)=𝑑(𝐴,𝐵) for all 𝑥∈𝐴₀,(iii)there exist 𝑥₀,𝑥₁∈𝐴₀ such that 𝑑(𝑥₁,𝑇𝑥₀)=𝑑(𝐴,𝐵) and 𝑥₀⪯𝑥₁,(iv)the pair (𝐴,𝐵) satisfies the 𝑃-property.
Then 𝑥₀⪯𝐼⁻¹𝑇𝑥₀.

Proof. Using (ii) with 𝑥=𝑥₁, we have 𝑑(𝑥₁,𝐼𝑥₁)=𝑑(𝐴,𝐵). From condition (iii), we have 𝑑(𝑥₁,𝑇𝑥₀)=𝑑(𝐴,𝐵). It follows from the 𝑃 property (condition (iv)) that 𝑑(𝐼𝑥₁,𝑇𝑥₀)=0, which implies that 𝐼𝑥₁=𝑇𝑥₀. From (i) and (iii), we obtain that 𝑥₀⪯𝑥₁=𝐼⁻¹𝑇𝑥₀.

Lemma 16. Let (𝐴,𝐵) be a pair of nonempty closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonempty and (𝐴,𝐵) has the 𝑃 property. Then (𝐴₀,𝐵₀) is a closed pair of subsets of 𝑋.

The proof of Lemma 16 can be found in [5].

Lemma 17. Let (𝐴,𝐵) be a pair of nonempty closed subsets of a metric space (𝑋,𝑑) such that 𝐴₀ is nonempty. Assume that the pair (𝐴,𝐵) has the 𝑃 property. Then there exists a bijective isometry 𝐼∶𝐴₀→𝐵₀ such that 𝑑(𝑥,𝐼𝑥)=𝑑(𝐴,𝐵),∀𝑥∈𝐴₀. (17)

The proof of Lemma 17 can be found in [7].

3. Discussions and Results

In [11], the authors established the following best proximity point result.

Theorem 18. Let 𝑋 be a nonempty set such that (𝑋,⪯) is a partially ordered set and (𝑋,𝑑) is a complete metric space. Let (𝐴,𝐵) be a pair of nonempty closed subsets of the metric space (𝑋,𝑑) such that 𝐴₀≠∅. Let 𝑇∶𝐴→𝐵 be a nonself-mapping. Suppose that the following conditions hold: (i)𝑇(𝐴₀)⊆𝐵₀ and (𝐴,𝐵) satisfy the 𝑃 property,(ii)𝑇 is a continuous on 𝐴,(iii)𝑇 is proximally increasing,(iv)there exist 𝑥₀,𝑥₁∈𝐴₀ such that 𝑑𝑥₁,𝑇𝑥₀=𝑑(𝐴,𝐵),𝑥₀⪯𝑥₁, (18)(v)for all 𝑥,𝑦∈𝐴 such that 𝑥⪯𝑦, we have 𝜓𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦−𝜑𝑑𝑥,𝑦, (19) where 𝜓,𝜑∈Ψ.
Then there exists an element 𝑥^∗∈𝐴 such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (20) Further, the sequence {𝑥_𝑛}, defined by 𝑑𝑥_𝑛+1,𝑇𝑥_𝑛=𝑑(𝐴,𝐵),∀𝑛∈ℕ, (21) converges to the element 𝑥^∗.

We shall prove that the best proximity point result given by Theorem 18 is a consequence of the fixed point result given by Theorem 7.

3.1. Proof of Theorem 18

Denote by 𝑇 the restriction of the mapping 𝑇 to the subset 𝐴₀ of 𝐴. Since 𝑇(𝐴₀)⊆𝐵₀, we have 𝑇∶𝐴₀→𝐵₀. From Lemma 17, there exists a bijective isometry 𝐼∶𝐴₀→𝐵₀ such that 𝑑(𝑥,𝐼𝑥)=𝑑(𝐴,𝐵),∀𝑥∈𝐴₀. (22) Consider the self-mapping 𝑆∶=𝐼⁻¹𝑇∶𝐴₀→𝐴₀. We shall prove that the mapping 𝑆 satisfies all the conditions of Theorem 7.

Claim 1. (𝐴₀,𝑑) is complete.
From Lemma 16, 𝐴₀ is a closed subset of the complete metric space (𝑋,𝑑). Then (𝐴₀,𝑑) is complete.

Claim 2. 𝑆 is continuous and nondecreasing mapping.
From Lemma 13, 𝐼⁻¹ is an isometry, so 𝐼⁻¹ is continuous on 𝐵₀. Since 𝑇 is continuous on 𝐴,𝑇 is continuous on 𝐴₀. Thus, the self-mapping 𝑆=𝐼⁻¹𝑇 is continuous on 𝐴₀. On the other hand, since 𝑇 is proximally increasing, then 𝑇 is also a proximally increasing mapping. Now, our claim follows immediately from Lemma 14.

Claim 3. There exists 𝑎∈𝐴₀ such that 𝑎⪯𝑆𝑎.
We have 𝑑𝑥₁,𝑇𝑥₀(),𝑑𝑥=𝑑𝐴,𝐵₁,𝐼𝑥₁=𝑑(𝐴,𝐵). (23) Using the 𝑃 property, we obtain that 𝑑(𝑇𝑥₀,𝐼𝑥₁)=0, which implies that 𝑇𝑥₀=𝑇𝑥₀=𝐼𝑥₁, that is, 𝑥₁=𝐼⁻¹𝑇𝑥₀=𝑆𝑥₀. Since 𝑥₀⪯𝑥₁, our claim holds with 𝑎=𝑥₀.

Claim 4. The mapping 𝑆 satisfies condition (iv) of Theorem 7.
Let 𝑥,𝑦∈𝐴₀ such that 𝑥⪯𝑦. Since 𝐼⁻¹ is an isometry, it follows from condition (v) that 𝜓𝑑𝑆𝑥,𝑆𝑦=𝜓𝑑𝐼⁻¹𝑇𝑥,𝐼⁻¹𝑑𝑑𝑑.𝑇𝑦=𝜓𝑇𝑥,𝑇𝑦≤𝜓𝑥,𝑦−𝜑𝑥,𝑦 (24) This proves our claim.

Now, the mapping 𝑆 satisfies all the conditions of Theorem 7. We deduce that 𝑆 has a fixed point 𝑥^∗∈𝐴₀, and the sequence {𝑧_𝑛}⊂𝐴₀, defined by 𝑧₀=𝑥₀,𝑧_𝑛+1=𝑆𝑧_𝑛,∀𝑛∈ℕ, (25) converges to 𝑥^∗.

We claim that 𝑥^∗ is a best proximity point of 𝑇. Indeed, we have 0=𝑑𝑥^∗,𝑆𝑥^∗=𝑑𝑥^∗,𝐼⁻¹𝑇𝑥^∗, (26) which implies that 𝑥^∗=𝐼⁻¹𝑇𝑥^∗, that is, 𝐼𝑥^∗=𝑇𝑥^∗=𝑇𝑥^∗. From 𝑑(𝑥^∗,𝐼𝑥^∗)=𝑑(𝐴,𝐵), we obtain that 𝑑(𝑥^∗,𝑇𝑥^∗)=𝑑(𝐴,𝐵). Then 𝑥^∗ is a best proximity point of 𝑇.

Let {𝑥_𝑛}⊂𝐴₀ be the sequence defined by 𝑑𝑥_𝑛+1,𝑇𝑥_𝑛=𝑑(𝐴,𝐵),∀𝑛∈ℕ. (27) Since 𝑑𝑥_𝑛+1,𝐼𝑥_𝑛+1=𝑑(𝐴,𝐵),∀𝑛∈ℕ, (28) from the 𝑃 property, we get that 𝐼𝑥_𝑛+1=𝑇𝑥_𝑛,∀𝑛∈ℕ, (29) which implies that 𝑥_𝑛+1=𝑆𝑥_𝑛,∀𝑛∈ℕ. (30) It follows from (25) and (30) that {𝑥_𝑛}={𝑧_𝑛}→𝑥^∗ as 𝑛→+∞. This makes end to the proof of Theorem 18.

Remark 19. In Theorem 18, the authors considered only the continuous case. However, from Theorem 7, we can also remove the continuity assumption of 𝑇 and replace it by the regularity assumption of (𝐴₀,⪯,𝑑). Moreover, by assuming that (𝐴₀,⪯) is directed, we obtain uniqueness of the best proximity point.

3.2. Additional Result

Using the same techniques, we can obtain a best proximity version of Theorem 8.

Theorem 20. Let 𝑋 be a nonempty set such that (𝑋,⪯) is a partially ordered set and (𝑋,𝑑) is a complete metric space. Let (𝐴,𝐵) be a pair of nonempty closed subsets of the metric space (𝑋,𝑑) such that 𝐴₀≠∅. Let 𝑇∶𝐴→𝐵 be a nonself-mapping. Suppose that the following conditions hold: (i)𝑇(𝐴₀)⊆𝐵₀ and (𝐴,𝐵) satisfy the 𝑃 property, (ii)𝑇 is continuous on 𝐴₀ or (𝐴₀,⪯,𝑑) is regular,(iii)𝑇 is proximally increasing,(iv)there exist 𝑥₀,𝑥₁∈𝐴₀ such that 𝑑𝑥₁,𝑇𝑥₀=𝑑(𝐴,𝐵),𝑥₀⪯𝑥₁, (31)(v)for all 𝑥,𝑦∈𝐴 such that 𝑥⪯𝑦, we have 𝑑𝑇𝑥,𝑇𝑦≤𝜑𝑀_𝑇𝐴,𝐵,𝑥,𝑦, (32) where 𝜑∈Φ and 𝑀_𝑇𝑑()()(),𝑑𝐴,𝐵,𝑥,𝑦∶=max𝑥,𝑦,𝑑𝑥,𝑇𝑥−𝑑𝐴,𝐵,𝑑𝑦,𝑇𝑦−𝑑𝐴,𝐵𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥2−𝑑(𝐴,𝐵). (33)
Then there exists an element 𝑥^∗∈𝐴 such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (34) Further, the sequence {𝑥_𝑛}, defined by 𝑑𝑥_𝑛+1,𝑇𝑥_𝑛=𝑑(𝐴,𝐵),∀𝑛∈ℕ, (35) converges to the element 𝑥^∗.

Proof. We continue to use the same notations as in the proof of Theorem 18. Following the proof of Theorem 18, we have only to check that the self-mapping 𝑆=𝐼⁻¹𝑇∶𝐴₀→𝐴₀ satisfies condition (iv) of Theorem 8. Indeed, let 𝑥,𝑦∈𝐴₀ such that 𝑥⪯𝑦, we have 𝑑𝑆𝑥,𝑆𝑦=𝑑𝐼⁻¹𝑇𝑥,𝐼⁻¹𝑀𝑇𝑦=𝑑𝑇𝑥,𝑇𝑦≤𝜑_𝑇𝐴,𝐵,𝑥,𝑦. (36) On the other hand, one has the evaluation 𝑀_𝑇𝑑()()(),𝑑𝐴,𝐵,𝑥,𝑦=max𝑥,𝑦,𝑑𝑥,𝑇𝑥−𝑑𝐴,𝐵,𝑑𝑦,𝑇𝑦−𝑑𝐴,𝐵𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥2(𝑑,−𝑑𝐴,𝐵)=max𝑥,𝑦,𝛼,𝛽,𝛾 (37) where we denoted (),(),𝑑𝛼∶=𝑑𝑥,𝑇𝑥−𝑑𝐴,𝐵𝛽∶=𝑑𝑦,𝑇𝑦−𝑑𝐴,𝐵𝛾∶=𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥2−𝑑(𝐴,𝐵). (38)
Estimation of 𝛼.
We have ()()()()𝐼𝛼=𝑑𝑥,𝑇𝑥−𝑑𝐴,𝐵≤𝑑𝑥,𝑆𝑥+𝑑𝑆𝑥,𝑇𝑥−𝑑𝐴,𝐵=𝑑𝑥,𝑆𝑥+𝑑⁻¹()()𝐼𝑇𝑥,𝑇𝑥−𝑑𝐴,𝐵=𝑑𝑥,𝑆𝑥+𝑑⁻¹𝑇𝑥,𝐼𝐼⁻¹()()()(𝑇𝑥−𝑑𝐴,𝐵=𝑑𝑥,𝑆𝑥+𝑑𝐴,𝐵−𝑑𝐴,𝐵)(from(().22))=𝑑𝑥,𝑆𝑥 (39)
Estimation of 𝛽.
We have ()()𝐼𝛽=𝑑𝑦,𝑇𝑦−𝑑𝐴,𝐵≤𝑑𝑦,𝑆𝑦+𝑑𝑆𝑦,𝑇𝑦−𝑑𝐴,𝐵=𝑑𝑦,𝑆𝑦+𝑑⁻¹()𝐼𝑇𝑦,𝑇𝑦−𝑑𝐴,𝐵=𝑑𝑦,𝑆𝑦+𝑑⁻¹𝑇𝑦,𝐼𝐼⁻¹().𝑇𝑦−𝑑𝐴,𝐵=𝑑𝑦,𝑆𝑦 (40)
Estimation of 𝛾.
We have 𝑑𝛾=𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥2()≤𝑑−𝑑𝐴,𝐵𝑥,𝑆𝑦+𝑑𝑆𝑦,𝑇𝑦+𝑑𝑦,𝑆𝑥+𝑑𝑆𝑥,𝑇𝑥2()=𝑑𝐼−𝑑𝐴,𝐵𝑥,𝑆𝑦+𝑑⁻¹𝑇𝑦,𝑇𝑦+𝑑𝑦,𝑆𝑥+𝑑𝐼⁻¹𝑇𝑥,𝑇𝑥2()=𝑑()()−𝑑𝐴,𝐵𝑥,𝑆𝑦+𝑑𝐴,𝐵+𝑑𝑦,𝑆𝑥+𝑑𝐴,𝐵2()=𝑑−𝑑𝐴,𝐵𝑥,𝑆𝑦+𝑑𝑦,𝑆𝑥2. (41)
Now, we deduce that 𝑀_𝑇𝑑(),𝑑𝐴,𝐵,𝑥,𝑦≤max𝑥,𝑦,𝑑𝑥,𝑆𝑥,𝑑𝑦,𝑆𝑦𝑥,𝑆𝑦+𝑑𝑦,𝑆𝑥2. (42) Since 𝜑 is nondecreasing, we get that 𝜑𝑀_𝑇𝑑(),𝑑𝐴,𝐵,𝑥,𝑦≤𝜑max𝑥,𝑦,𝑑𝑥,𝑆𝑥,𝑑𝑦,𝑆𝑦𝑥,𝑆𝑦+𝑑𝑦,𝑆𝑥2. (43) Finally, it follows from (36) and (43) that 𝑑𝑑(),𝑑𝑆𝑥,𝑆𝑦≤𝜑max𝑥,𝑦,𝑑𝑥,𝑆𝑥,𝑑𝑦,𝑆𝑦𝑥,𝑆𝑦+𝑑𝑦,𝑆𝑥2. (44) Thus, we proved that the mapping 𝑆 satisfies condition (iv) of Theorem 8. Applying Theorem 8, we obtain the existence of a fixed point of 𝑆. The rest of the proof is exactly like that of Theorem 18, so we omit it.

Acknowledgment

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no RGP-VPP-237.