# Existence of Best Proximity Point with an Application to Nonlinear Integral Equations

Authors: Shagun Sharma; Sumit Chandok
Journal: Journal of Mathematics
 (2021)

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## Abstract

Using the idea of modifiedϱ-proximal admissible mappings, we derive some new best proximity point results for ϱ−ϑ-contraction mappings in metric spaces. We also provide some illustrations to back up our work. As a result of our findings, several fixed-point results for such mappings are also found. We obtain the existence of a solution for nonlinear integral equations as an application.


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## Body

## 1. Introduction and Preliminaries

Problems originating in several disciplines of mathematical analysis, such as obtaining the existence of a solution for integral and differential equations, are solved using fixed-point theory. The study of fixed points for nonself-mapping, on the contrary, is also fascinating. A nonself-contractionℑ:ζ⟶£ does not necessarily have a fixed point for two given nonempty closed subsets ζ and £ of a complete metric space ℵ. In this case, it is important to identify a point u′∈ζ such that ℘u′,ℑu′ is minimum. Essentially, if(1)℘u′,ℑu′=distζ,£=inf℘u′,v˜:u′∈ζ,v˜∈£,where ℘u′,ℑu′ is the minimum value distζ,£ and hence u′ is an approximate solution of equation ℑu′=u′ with least possible error, such a solution is known as the best proximity point of mapping ℑ.Various academics have discovered a number of best proximity point theorems and associated fixed-point results in metric or normed linear spaces (see [1–7] and references cited therein). Inspired by the work of Geraghty [8] and Kutbi and Sintunavarat [9], we introduced a new class of nonself-contractive mappings known as ϱ−ϑ-contraction. The goal of this work is to use the modified ϱ-proximal admissible mappings concept to obtain some best proximity point outcomes for ϱ−ϑ-contraction mappings in metric space. On a metric space enriched with an arbitrary binary relation, some optimal proximity point results are proved. For such a class of mappings, we also obtain certain fixed-point findings. As an application of our obtained results, we find the solution of a nonlinear integral equation.The set of natural numbers and the set of real numbers are denoted byℕ and ℝ, respectively, throughout this work.Letζ and £ be two nonempty subsets of ℵ,℘, a metric space. Define(2)℘ζ,£=inf℘u′,v˜:u′∈ζ,v˜∈£,ζ0=u′∈ζ:there exists somev˜∈£such that℘u′,v˜=℘ζ,₤,E0=v˜∈£:there exists someu′∈ζsuch that℘u′,v˜=℘ζ,₤.Definition 1.
Letζ,£ be a pair of nonempty subsets of a metric space ℵ,℘ with ζ0≠∅. Then, the pair ζ,£ is said to have ₤-property (see [10]) if and only if, for any u′1,u′2∈ζ and v˜1,v˜2∈£,(3)℘u′1,v˜1=℘ζ,£,℘u′2,v˜2=℘ζ,£,then℘u′1,u′2=℘v˜1,v˜2.
Many authors have refined the concept ofϱ-admissible mappings developed by Samet [10] for proximal mappings (see [11, 12]).Definition 2.
Letζ and £ be two nonempty subsets of a metric space ℵ,℘. A mapping ℑ:ζ⟶£ is called modified ϱ-proximal admissible if there exists a mapping ϱ:£×£⟶0,∞ such that ϱℑu′0,ℑu′1≥1:(4)℘x′1,ℑu′0=℘ζ,£,℘x′2,ℑu′1=℘ζ,£,thenϱℑx′1,ℑx′2≥1,for all u′0,u′1,x′1,x′2∈ζ.
Letℵ,℘ be a metric space and ® be a binary relation over ℵ. Denote(5)κ=®+®−1is the symmetric, transitive relation attached to ®. Clearly,(6)u′,v˜∈ℵ,u′κv˜⇔u′®v˜orv˜®u′.Definition 3.
Letζ and £ be two nonempty subsets of metric space ℵ,℘. A mapping ℑ:ζ⟶£ is called modified proximal comparative mapping; if ℑu′0κℑu′1,(7)℘x′1,ℑu′0=℘ζ,£,℘x′2,ℑu′1=℘ζ,£,thenℑx′1κℑx′2,for all u′0,u′1,x′1,x′2∈ζ.Definition 4.
Letℵ be a metric space. A mapping ϱ:ℵ×ℵ⟶0,∞ is called transitive [13] if it satisfies the following condition:(8)ϱx′,y′≥1,ϱy′,z′≥1impliesϱx′,z′≥1,for all x′,y′,z′∈ℵ.


## 2. Main Results

To start with, we have the following definition:Definition 5.
Letζ and £ be two nonempty subsets of a metric space ℵ,℘. A mapping ℑ:ζ⟶£ is an ϱ−ϑ contraction mapping if there exist two functions ϱ:£×£⟶0,∞ and ϑ:ℵ⟶0,1 for which ϑℑu′≤ϑu′ and limsupϑu′<1 for all u′∈ℵ such that(9)ϱℑu′1,ℑu′2℘ℑu′1,ℑu′2≤ϑℑu′1℘u′1,u′2,for all u′1,u′2∈ζ.Theorem 1.
Letζ and £ be two nonempty closed subsets of a complete metric space ℵ such that ζ0 is nonempty. Let ϱ:£×£⟶0,∞ be transitive and ℑ:ζ⟶£ be continuous mapping satisfying the following assertions:(i)
ℑ is ϱ−ϑ contraction(ii)
ℑζ0⊆E0 and ζ,£ satisfy the ₤-property(iii)
ℑ is modified ϱ-proximal admissible
Furthermore, suppose that there existsu′0,u′1∈ζ0 and ℑu′0∈E0 such that ℘u′1,ℑu′0=℘ζ,£ and ϱℑu′0,ℑu′1≥1. Then, the mapping ℑ has a best proximity point.Proof.
By assumption, there existsu′0,u′1∈ζ0⊆ζ and ℑu′0∈ℑζ0⊆E0⊆£ such that ℘u′1,ℑu′0=℘ζ,£ and ϱℑu′0,ℑu′1≥1.
Sinceu′1∈ζ0, then ℑu′1∈ℑζ0⊆E0. By definition of E0, there exists u′2∈ζ0 such that ℘u′2,ℑu′1=℘ζ,£. Since ℑ is modified ϱ-proximal admissible, and ϱℑu′0,ℑu′1≥1, we obtain(10)℘u′1,ℑu′0=℘ζ,£,℘u′2,ℑu′1=℘ζ,£,thenϱℑu′1,ℑu′2≥1.
Again,u′2∈ζ0; then, ℑu′2∈ℑζ0⊆E0. By definition of E0, there exists u′3∈ζ0 such that ℘u′3,ℑu′2=℘ζ,£. Since ℑ is modified ϱ-proximal admissible, and ϱℑu′1,ℑu′2≥1, we have(11)℘u′2,ℑu′1=℘ζ,£,℘u′3,ℑu′2=℘ζ,£,thenϱℑu′2,ℑu′3≥1.
Continuing in this fashion, we can construct a sequenceu′n such that(12)℘u′n+1,ℑu′n=℘ζ,£,℘u′n,ℑu′n−1=℘ζ,£,ϱℑu′n,ℑu′n+1≥1.
As a pairζ,£ satisfies ₤-property, we have(13)℘u′n,u′n+1=℘ℑu′n−1,ℑu′n.
Forn∈ℕ, we have(14)℘u′n,u′n+1=℘ℑu′n−1,ℑu′n≤ϱℑu′n−1,ℑu′n℘ℑu′n−1,ℑu′n≤ϑℑu′n−1℘u′n−1,u′n≤ϑu′n−1℘u′n−1,u′n<℘u′n−1,u′n,for all n∈ℕ. Therefore, the sequence ℘u′n−1,u′n is strictly decreasing and(15)limn⟶+∞℘u′n−1,u′n=d′,for some d′≥0.
Next, we claim thatd′=0. Assume, on the contrary, that d′>0. On taking limit as n⟶+∞ in equation (14), we obtain that(16)limn⟶+∞℘u′n,u′n+1℘u′n−1,u′n≤ϑu′n−1,1≤ϑu′n−1,which is a contradiction. Therefore, d′=0.
Next, we show thatu′n is a Cauchy sequence. Using triangle inequality, (12) and (13), we have, for all m,n∈ℕ and m>n,(17)℘u′m,u′n≤℘u′m,u′m+1+℘u′m+1,u′n+1+℘u′n+1,u′n=℘u′m,u′m+1+℘ℑu′m,ℑu′n+℘u′n+1,u′n≤ϱℑu′m,ℑu′n℘u′m,u′m+1+ϱℑu′m,ℑu′n℘ℑu′m,ℑu′n+ϱℑu′m,ℑu′n℘u′n+1,u′n≤ϱℑu′m,ℑu′n℘u′m,u′m+1+℘u′n+1,u′n+ϑℑu′m℘u′m,u′n≤ϱℑu′m,ℑu′n℘u′m,u′m+1+℘u′n+1,u′n+ϑu′m℘u′m,u′n.
It implies that(18)1−ϑu′m℘u′m,u′n≤ϱℑu′m,ℑu′n℘u′m,u′m+1+℘u′n+1,u′n.
On taking limit asm,n⟶+∞ in (18), we obtain that(19)limm,n⟶+∞℘u′m,u′n=0.
Therefore,u′n is a Cauchy sequence in ζ. Since ζ is a closed subset of a complete metric ℵ, we obtain(20)limn⟶+∞u′n=u′,for some u′∈ζ. Since ℑ is continuous,(21)limn⟶+∞ℑu′n=ℑu′,for some ℑu′∈£. By (12), we obtain(22)℘u′,ℑu′=limn⟶+∞℘u′n+1,ℑu′n=℘ζ,£.
Hence,u′ is a best proximity point of ℑ.
The following hypotheses can be used to substituteℑ’s continuity hypothesis in Theorem 1.
(ℜ) If u′n is a sequence in ℵ such that ϱℑu′n,ℑu′n+1≥1, for all n and u′n⟶u′∈ℵ as n⟶∞, then there exists a subsequence u′nk of u′n such that ϱℑu′nk,ℑu′≥1, for all k.Theorem 2.
Letζ and £ be two nonempty closed subsets of a complete metric space ℵ such that ζ0 is nonempty. Let ϱ:£×£⟶0,∞ be transitive and ℑ:ζ⟶£ be a mapping satisfying the following assertions:(i)
ℑ is ϱ−ϑ contraction(ii)
ℑζ0⊆E0 and ζ,£ satisft the ₤-property(iii)
ℑ is modified ϱ-proximal admissible
Furthermore, suppose that (ℜ) holds and there exists u′0,u′1∈ζ0 and ℑu′0∈E0 such that ℘u′1,ℑu′0=℘ζ,£ and ϱℑu′0,ℑu′1≥1. Then, the mapping ℑ has a best proximity point.Proof.
Following the proof of Theorem1, the sequence u′n is Cauchy and converges to some u′∗ in ℵ,℘.
From our assumption and using (12), there exists a subsequence u′nk of u′n such that ϱℑu′nk,ℑu′∗≥1, for all k.
Now, we shall show thatℑ has a best proximity point. Using triangle inequality and (12), we obtain(23)℘u′nk+1,ℑu′∗≤℘u′nk+1,ℑu′nk+℘ℑu′nk,ℑu′∗≤℘ζ,£+ϱℑu′nk,ℑu′∗℘ℑu′nk,ℑu′∗≤℘ζ,£+ϑℑu′nk℘u′nk,u′∗≤℘ζ,£+ϑu′nk℘u′nk,u′∗.
Again, by triangle inequality and equation (23), we have(24)℘ζ,£≤℘u′∗,ℑu′∗≤℘u′∗,u′nk+1+℘u′nk+1,ℑu′∗≤℘u′∗,u′nk+1+℘ζ,£+ϑu′nk℘u′nk,u′∗.
Takingk⟶+∞ in (24), we have(25)℘ζ,£=℘u′∗,ℑu′∗.
This shows thatℑ has a best proximity point.


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

Using modified proximal comparative mapping, we establish some optimal proximity point findings on a metric space equipped with an arbitrary binary relation in this section.Theorem 3.
Letζ and £ be two nonempty closed subsets of a complete metric space ℵ,℘ such that ζ0 is nonempty and ® be a binary relation over ℵ. Suppose that ℑ:ζ⟶£ is a continuous mapping satisfying the following assertions:(i)
There existsϑ:ℵ⟶0,1 such that, for all u′1,u′2∈ζ, u′1κu′2,(26)℘ℑu′1,ℑu′2≤ϑu′1℘u′1,u′2.(ii)
ℑζ0⊆E0 and ζ,£ satisfy the ₤-property.(iii)
ℑ is modified proximal comparative mapping.
If there existsu′3,u′4∈ζ0 and ℑu′3∈E0 such that ℘u′4,ℑu′3=℘ζ,£, u′3κu′4, then the mapping ℑ has a best proximity point.Proof.
Define a mappingϱ:£×£⟶ℝ by(27)ϱℑu′1,ℑu′2=2,ifℑu′1κℑu′2,1,otherwise.
Suppose thatϱℑu′3,ℑu′4≥1, such that(28)℘u′4,ℑu′3=℘ζ,£,℘u′5,ℑu′4=℘ζ,£.
Hold for someu′3,u′4,u′5∈ζ. By definition of ϱ, we obtain(29)ℑu′3κℑu′4,℘u′4,ℑu′3=℘ζ,£,℘u′5,ℑu′4=℘ζ,£.
Condition (iii) implies thatℑu′4κℑu′5, which gives us from the definition of ϱ that ϱℑu′4,ℑu′5≥1. This shows ℑ is a modified ϱ-proximal admissible mapping.
Furthermore, using assumption, we have(30)℘u′4,ℑu′3=℘ζ,£,ϱℑu′3,ℑu′4≥1.
Condition (i) implies that(31)ϱℑu′1,ℑu′2℘ℑu′1,ℑu′2≤ϑu′1℘u′1,u′2,∀u′1,u′2∈ζ.
Thus, all the conditions of Theorem1 are satisfied; then, ℑ has a best proximity point.
We may replace the continuity hypothesis ofℑ in Theorem 3 by the following hypothesis.
(ℜ∗) If u′n is a sequence in ℵ such that ℑu′nκℑu′n+1 for all n and u′n⟶u′∈ℵ as n⟶+∞, then there exists a subsequence u′nk of u′n such that ℑu′nkκℑu′, for all k.Theorem 4.
Letζ and £ be two nonempty closed subsets of a complete metric space ℵ,℘ such that ζ0 is nonempty and ® be a binary relation over ℵ. Suppose that ℑ:ζ⟶£ is a mapping satisfying the following assertions:(i)
There existsϑ:ℵ⟶0,1 such that, for all u′1,u′2∈ζ, u′1κu′2,(32)℘ℑu′1,ℑu′2≤ϑu′1℘u′1,u′2.(ii)
ℑζ0⊆E0 and ζ,£ satisfy the ₤-property.(iii)
ℑ is modified proximal comparative mapping.
Furthermore, we assume thatℜ∗ holds and there exists u′3,u′4∈ζ0 and ℑu′3∈E0 such that ℘u′4,ℑu′3=ℑζ,£, u′3κu′4. Then, the mapping ℑ has a best proximity point.Proof.
By evaluating the mappingϱ defined in Theorem 3 and noting that condition ℜ∗ implies condition ℜ, the result is derived from Theorem 2.


## 4. Consequences and Related Results

In this section, we obtain some results on the existence of fixed points as a result of our findings.If we takeζ=£=ℵ in Theorem 1, we obtain the following result:Corollary 1 (see Theorem 10 in [9]).
Letℵ be a complete metric space. Let ϱ:ℵ×ℵ⟶0,∞ be transitive and ℑ:ℵ⟶ℵ be continuous mapping satisfying the following assertions:(i)
ℑ is ϱ−ϑ contraction(ii)
ℑ is ϱ admissible(iii)
There existsu′0∈ζ such that ϱu′0,ℑu′0≥1
Then, the mappingℑ has a fixed point.
By consideringϱu′1,u′2=1, for all u′1,u′2∈ζ and ϑu′1=k;k∈0,1, for all u′1∈ℵ in Corollary 1, we get the famous Banach contraction theorem.Corollary 2 (see [14]).
Letℵ be a complete metric space and ℑ:ℵ⟶ℵ be a mapping satisfying the following assertion:(33)℘ℑu′1,ℑu′2≤k℘u′1,u′2,for all u′1,u′2∈ℵ. Then, the mapping ℑ has a fixed point.
If we takeζ=£=ℵ in Theorem 2, we obtain the following result.Corollary 3 (see Theorem 12 in [9]).
Letℵ be a complete metric space. Let ϱ:ℵ×ℵ⟶0,∞ be transitive and ℑ:ℵ⟶ℵ be a mapping satisfying the following assertions:(i)
ℑ is ϱ−ϑ contraction(ii)
ℑ is ϱ admissible(iii)
There existsu′0∈ℵ such that ϱu′0,ℑu′0≥1(iv)
(ℜ) holds
Then, the mappingℑ has a fixed point.
If we takeζ=£=ℵ in Theorems 3 and 4, we obtain the following fixed-point results:Corollary 4.
Suppose thatℑ:ℵ⟶ℵ is a continuous mapping on a metric space ℵ,℘ with a binary relation ® over ℵ, satisfying(34)℘ℑu′1,ℑu′2≤ϑu′1℘u′1,u′2,for u′1,u′2∈ℵ, u′1κu′2, where ϑ:ℵ⟶0,1. If there exists u′∗∈ℵ a such that u′∗κℑu′∗, then the mapping ℑ has a fixed point.Corollary 5.
Suppose thatℑ:ℵ⟶ℵ is a self-mapping on a metric space ℵ,℘ with a binary relation ® over ℵ, satisfying(35)℘ℑu′1,ℑu′2≤ϑu′1℘u′1,u′2,for u′1,u′2∈ℵ, u′1κu′2, where ϑ:ℵ⟶0,1. If there exists u′∗∈ℵ such that u′∗κℑu′∗ and ℜ∗ holds, then the mapping ℑ has a fixed point.


## 5. Examples

We give several illustrations that support our findings in this section.Example 1.
Considerℵ=ℝ2 with metric(36)℘v˜,u′∗=v˜1−v˜22+u′1∗−u′2∗2,for all v˜=v˜1,u′1∗,u′∗=v˜2,u′2∗∈ℝ2.
Supposeζ=1/2,u′1:0≤u′1≤1 and £=0,u′1:0≤u′1≤1, such that ℘ζ,£=1/2.
Defineℑ:ζ⟶£ by(37)ℑu′1=0,1,u′1=12,1,0,a′4:0≤a′≤u′1,otherwise,for all u′1∈ζ.
Defineϱ:£×£⟶0,∞ by(38)ϱ0,u′1∗,0,u′2∗=1,ifu′1∗,u′2∗∈0,14,2,otherwise,then ϱ is transitive. If z′1=1/2,u′1 and z′2=1/2,u′2 in ζ, for u′1,u′2∈0,1/2, then(39)ℑz′1=0,a′4:0≤a′≤u′1,ℑz′2=0,a′4:0≤a′≤u′2.
Also,ϱℑz′1,ℑz′2=1 and ℘u′1,ℑz′1=1/2=℘ζ,£ and ℘u′2,ℑz′2=1/2=℘ζ,£ if and only if u′1,u′2∈1/2,x′/4;0≤x′≤1/2. Then, ϱℑu′1,ℑu′2=1. This shows ℑ is a modified ϱ-proximal admissible and ℑ is continuous. Since ζ0=ζ and E0=£, then ℑζ0⊆E0, for each u′1∈ζ0.
Next, we prove thatℑ is ϱ−ϑ contraction. Let ϑt′=1/3, for all t′∈ℵ. Take z′1=1/2,u′1 and z′2=1/2,u′2 in ζ, where 0≤u′1,u′2≤1/2. Consider(40)℘ℑz′1,ℑz′2=0−02+u′14−u′242=u′14−u′242=u′14−u′24=14u′1−u′2≤13u′1−u′2.
It implies that(41)ϱℑz′1,ℑz′2℘ℑz′1,ℑz′2≤ϑz′1℘z′1,z′2,for all z′1,z′2∈ζ. Hence, ℑ is a ϱ−ϑ contraction. All conditions of Theorem 1 are satisfied and ℑ has a best proximity point 1/2,1.Example 2.
Considerℵ=ℝ2 with metric(42)℘v˜,u′∗=v˜1−v˜22+u′1∗−u′2∗2,for all v˜=v˜1,u′1∗,u′∗=v˜2,u′2∗∈ℝ2. Suppose that(43)ζ=u′1,u′2:u′12+u′22=3andu′2≥0,£=u′1,u′2:u′12+u′22=1andu′2≥0.
Then,℘ζ,£=2. Define ℑ:ζ⟶£ by(44)ℑu′1,u′2=u′1,u′23,for all u′1,u′2∈ζ.
Defineϱ:ℵ×ℵ⟶0,∞ by(45)ϱu′1,u′1∗,u′2,u′2∗=2,for all u′1,u′1∗,u′2,u′2∗∈ℵ. Hence, ϱ is transitive. If z′1,z′2∈ζ, then ϱℑz′1,ℑz′2=2>1,℘u′1,ℑz′1=2=℘ζ,£ and ℘u′2,ℑz′2=2=℘ζ,£ if and only if u′1,u′2∈0,3,3,0. Therefore, ϱℑu′1,ℑu′2=2>1. This shows ℑ is a modified ϱ-proximal admissible and ℑ is continuous. Since ζ0=ζ and E0=£, then ℑζ0⊆E0, for each u′1∈ζ0.
Next, we prove thatℑ is ϱ−ϑ contraction.
Letϑt′=1/2. Take z′1=x′1,y′1andz′2=x′2,y′2 in ζ; then,(46)℘ℑz′1,ℑz′2=x′13−x′232+y′13−y′232=132x′1−x′22+132y′1−y′22=13x′1−x′22+y′1−y′22=13℘z′1,z′2.
It implies that(47)ϱℑz′1,ℑz′2℘ℑz′1,ℑz′2≤ϑz′1℘z′1,z′2,for all z′1,z′2∈ℵ. Therefore, ℑ is a ϱ−ϑ contraction. All conditions of Theorem 1 are satisfied and ℑ has a best proximity point 3,0.


## 6. Application to Integral Equations

In this section, we obtain the solution of integral equation as an application of our obtained results.If we takeζ=£=ℵ in Theorem 1, we obtain the solution of nonlinear integral equation.Theorem 5.
Let∁a′,b′ be the set of all continuous functions on closed interval a′,b′, with metric defined by(48)℘u′,v′=supt′∈a′,b′u′t′−v′t′,for all u′,v′∈∁a′,b′. Consider the nonlinear integral equation:(49)u′t′=υt′+∫a′b′ωt′,x′,u′x′dx′,where t′∈a′,b′, υ:a′,b′⟶ℝ, and ω:a′,b′×a′,b′×u′a′,b′⟶ℝ for each u′∈∁a′,b′. Suppose that the following statements hold:(i)
υ is continuous on a′,b′ and ωt′,x′,u′x′ is integrable with respect to x′ on a′,b′(ii)
ℑu′∈∁a′,b′, for all u′∈∁a′,b′, where ℑu′t′=υt′+∫a′b′ωt′,x′,u′x′dx′, for all t′∈a′,b′(iii)
For allu′∈∁a′,b′, u′x′≥0, and ℑu′x′≥0, for all x′∈a′,b′(iv)
For allx′,t′∈a′,b′ and u′,v′∈∁a′,b′, ωt′,x′,u′x′−ωt′,x′,v′x′≤k′/b′−a′maxu′x′−v′x′; k′∈0,1(v)
There existsu′1∈∁a′,b′ such that u′1t′≥0 and ℑu′1t′≥0, for all t′∈a′,b′
Then, nonlinear integral equation (49) has a solution in ∁a′,b′.Proof.
Define a mappingℑ:∁a′,b′⟶∁a′,b′ by(50)ℑu′t′=υt′+∫a′b′ωt′,x′,u′x′dx′,for all u′∈∁a′,b′ and for all t′∈a′,b′. Then, ℑ is a continuous mapping.
Define a mappingϱ:∁a′,b′×∁a′,b′⟶ℝ by(51)ϱu′,v′=2,ifu′x′,v′x′∈0,∞,x′∈a′,b′,1,otherwise.
We shall show thatℑ is a modified ϱ-proximal admissible mapping. Indeed, for u′,v′∈∁a′,b′ such that ϱℑu′,ℑv′≥1, we have u′x′,v′x′≥0, for all x′∈a′,b′. It follows from condition (iii) that ℑu′x′,ℑv′x′≥0.
Therefore,ϱℑu′x′,ℑv′x′>1, and hence, ℑ is a modified ϱ-proximal admissible mapping and ϱ is transitive. Now, we claim that ℑ is a ϱ−ϑ contraction. Let ϑx′=k′, for all x′∈a′,b′. Consider(52)ℑu′x′−ℑv′x′=∫a′b′ωt′,x′,u′x′dx′−∫a′b′ωt′,x′,v′x′dx′≤∫a′b′ωt′,x′,u′x′−ωt′,x′,v′x′dx′≤k′b′−a′maxu′x′−v′x′∫a′b′dx′≤k′b′−a′b′−a′maxu′x′−v′x′≤k′℘u′,v′≤ϑu′℘u′,v′.
It implies that(53)ϱℑu′,ℑv′℘ℑu′,ℑv′≤ϑu′℘u′,v′.
Then,ℑ is a ϱ−ϑ contraction. Let ℑu′n⊆∁a′,b′ such that ϱℑu′n,ℑu′n+1≥1 and limn⟶+∞ℑu′n=ℑu′∈∁a′,b′. Then, u′x′,u′nx′∈0,∞, for all x′∈a′,b′ and n≥0. Therefore, ϱℑu′n,ℑu′>1, for all n≥1. Therefore, we conclude all the hypotheses of Theorem 1 are satisfied. Thus, equation (49) has a solution u′∈∁a′,b′.


## 7. Conclusion

Using the concept of modifiedϱ-proximal admissible in the setting of metric space ℵ,℘, we find some novel best proximity point results for ϱ−ϑ-contraction mappings. Many known results in the literature are also generalized by our findings.



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