# Near-Coincidence Point Results in Norm Interval Spaces via Simulation Functions

Authors: Misbah Ullah; Muhammad Sarwar; Hassen Aydi; Yaé Ulrich Gaba
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Recently, Wu in 2018 established interesting results in the framework of interval spaces. He initiated the idea of near-fixed points and proved some related basic results in metric interval, norm interval, and hyperspaces. In 2015, Khojasteh et al. gave the concept of simulation functions and studied some fixed-point results in metric spaces. Motivated by this work, we give some near-coincidence point results in norm interval spaces using the concept given by Khojasteh et al. Examples are also provided for the validation of the results.


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## Body

## 1. Introduction

Many researchers are still showing high interest in the field of metric fixed-point theory. They are working in different directions and generalizing the remarkable results in this area [1–4]. The first one who took interest in this area was Poincare. Later, Brouwer established a (topological) fixed-point theorem. The metric fixed-point theory attracts researchers due to its applications in both applied and pure mathematics. There are many applications of metric fixed-point theory in the existence of solutions for nonlinear systems. In 1922, Banach [5] established a remarkable result, known as the Banach contraction principle (BCP).This BCP was modified and generalized in different forms and structures. Among them, there are dislocated quasi metric spaces [6], cone metric spaces [7], generalized metric spaces [8], controlled metric spaces [9], orthogonal partial b-metric spaces [10], etc.Khojasteh et al. [11] modified the contractive condition by introducing the concept of a simulation function S:0,∞×0,∞⟶ℝ. Later, Roldan Lopez de Hierro et al. [12, 13] extended the stated concept and investigated some coincidence point results in metric spaces. With the help of a simulation-type function, Argoubi et al. [14] gave interesting results in partial ordered metric spaces. Alharbi et al. [15] made a generalization by combining the concept of simulation and admissible functions in the related literature. Alsubaie et al. [16] proved some common fixed-point results for two mappings in the setting of metric spaces by using the concept of a simulation function. Alqahtani et al. [17] proved fixed-point results by introducing the concept of a bilateral contraction which is a combination of Ćirić- and Caristi-type contractions. In [18], the authors studied the existence and uniqueness of a common fixed point in the setting of b-metric spaces, by using the concept of extended Z-contractions associated with an ψ−simulation function. In [19], the authors established results on the existence of best proximity points of certain mappings using simulation functions in complete metric spaces. Later, Karapinar [20] presented some fixed-point results by defining a new contractive condition via admissible mappings imbedded in a simulation function.Recently, Wu [21] initiated the concept of interval spaces. These spaces contain all closed bounded intervals over the set ℝ. Over the interval spaces, he defined a metric, as well as a norm using the equivalence relation Ω=. These spaces are called metric intervals and norm interval spaces, respectively. He studied near-fixed-point results in metric intervals, as well as in norm interval spaces. After this, he gave the concept of a hyperspace, which is a space containing all possible subsets of a vector space. He defined null sets, as well as the equivalence relation Ω= in a hyperspace and defined a norm over this type of spaces. He also presented near-fixed-point results in hyperspaces. For more details, see [22–24].Inspired by the work done in [11, 21, 22], we established some near-coincidence point results in metric interval and hyperspaces [25] via a simulation function. We also presented some near-coincidence point results in norm interval spaces via a simulation function. For validation of results and definitions, some examples are provided.


## 2. Preliminaries

In this section, some basic definitions and results are stated related to the existing literature.


### 2.1. Interval Spaces

LetI be the set containing all close bounded intervals of the form σ,υ, where σ,υ∈ℝ and σ≤υ. Also, σ∈ℝ is considered as an element σ,σ∈I [21].The binary operation of addition and scaler multiplication is stated as follows:(1)σ,υ⊕σ′,υ′=σ+σ′,υ+υ′,kσ,υ=kσ,kυ,k≥0,kυ,kσ,k<0.Due to the inverse property, the above space does not fulfill the condition of a conventional vector space. Forσ,υ∈I, the subtraction(2)σ,υ⊖σ,υ=σ,υ⊕−υ,−σ=σ−υ,υ−σdoes not give the zero element 0,0. So the inverse of σ,υ does not exist. For the above deficiency, the null set was defined by Wu [21] as follows.


### 2.2. Null Set

The null set contains all the elements of the type−σ,σ, and so it is defined as follows:(3)Ω=σ,υ⊖σ,υ;σ,υis an element ofI,or(4)Ω=−α,α;α≥0.


### 2.3. Binary RelationΩ=

We writeσ,υΩ=σ′,υ′ iff there exist ω1,ω2∈Ω such that(5)σ,υ⊕ω1=σ′,υ′⊕ω2.Clearly, we can haveσ,υ=σ′,υ′⇒σ,υΩ=σ′,υ′ by taking ω1=ω2=0,0. However, the converse is not true in general.Proposition 1.
(see [21]). Ω= is an equivalence relation.
According to the equivalence relationΩ=, the equivalence class of almost identical intervals is defined as σ,υ=p,q∈I:σ,υΩ=p,q for any σ,υ∈I.


### 2.4. Norm Interval Space

The pairI,⋅ fulfilling the following axioms is called a norm interval space [21]:(i)
σ,υ=0 implies σ,υ∈Ω(ii)
ασ,υ=ασ,υ(iii)
σ,υ⊕σ′,υ′≤σ,υ+σ′,υ′ for all σ,υ,σ′,υ′∈I, where I contains all close bounded intervals over ℝ with the null set Ω and ⋅ is a real-valued mapping on IWe say that the null condition is satisfied by⋅ if the condition (iii) is replaced by(6)σ,υ=0if and only ifσ,υ∈Ω.⋅ is said to satisfy the null equalities if for all ω1,ω2∈Ω and σ,υ,σ′,υ′∈I, the following equalities hold:(1)
σ,υ⊕ω1⊖σ′,υ′⊕ω2=σ,υ⊖σ′,υ′(2)
σ,υ⊕ω1⊖σ′,υ′=σ,υ⊖σ′,υ′(3)
σ,υ⊖σ′,υ′⊕ω2=σ,υ⊖σ′,υ′Definition 1.
IfI,⋅ is a norm interval space, then(i)
The mapping⋅ is said to satisfy the null super-inequality if(7)σ,υ⊕ω≥σ,υ,for anyσ,υ∈Iandω∈Ω.(ii)
The mapping⋅ is said to satisfy the null sub-inequality if(8)σ,υ⊕ω≤σ,υ,for anyσ,υ∈Iandω∈Ω.(iii)
The mapping⋅ is said to satisfy the null equality if(9)σ,υ⊕ω=σ,υ,for anyσ,υ∈Iandω∈Ω.Example 1.
Let⋅ be a nonnegative real-valued function defined on I by(10)σ,υ=σ+υ.
Then,I,⋅ forms a norm interval space such that ⋅ satisfies the null equality.Proposition 2.
(see [21]). Let I,⋅ be a norm interval space such that ⋅ satisfies the null super-inequality. Then, for any σ,υ,σ′,υ′,σ1,υ1,σ2,υ2,…,σm,υm, we have(11)σ,υ⊖σ′,υ′≤σ,υ⊖σ1,υ1+σ1,υ1⊖σ2,υ2+⋯+σm,υm⊖υ,υ′.Proposition 3.
LetI,⋅ be a norm interval space; then, the following hold:(i)
If⋅ satisfies the null equality, then for all σ,υ,σ′,υ′∈I,(12)σ,υΩ=σ′,υ′impliesσ,υ=σ′,υ′.(ii)
For anyσ,υ,σ′,υ′∈I,(13)σ,υ⊖σ′,υ′=0,impliesσ,υΩ=σ′,υ′.(iii)
If⋅ satisfies the null super-inequality and null condition, then for any σ,υ,σ′,υ′∈I,(14)σ,υΩ=σ′,υ′,impliesσ,υ⊖σ′,υ′=0.
For proof of the above propositions, see [21].Definition 2.
LetI,⋅ be a norm interval space. A sequence σn,υnn=1+∞ is said to converge to a limit σ,υ if and only if(15)limn⟶∞σn,υn⊖σ,υ=0.Proposition 4.
Consider a norm interval spaceI,⋅ with the null set Ω.(i)
If the null super-inequality holds for⋅, then the convergence of the sequence σn,υnn=1+∞ to σ,υ and σ′,υ′ simultaneously implies σ,υ=σ′,υ′(ii)
If the null equality holds for⋅ and the sequence σn,υnn=1+∞ converges to σ,υ, then for any σ′,υ′∈σ,υ, the given sequence will also converge to σ′,υ′Definition 3.
Consider the norm interval spaceI,⋅ with the null set Ω, where ⋅ satisfies the null equality. If σ,υ∈I is the limit of the sequence σn,υnn=1+∞, then σ,υ is called the class limit. We can also write(16)limn⟶∞σn,υn=σ,υ.Proposition 5.
(see [21]). In the norm interval space I,⋅ if the null super-inequality holds for ⋅, the class limit is unique.Definition 4.
A sequenceσn,υnn=1+∞ in a norm interval space I,⋅ is called a Cauchy sequence if and only if for any ɛ≥0, there exists K∈ℕ such that(17)σn,υn⊖σm,υm<ɛ,for m,n>K with m≠n. If every Cauchy sequence is convergent in I, then I is complete.Definition 5.
A complete norm interval spaceI,⋅ is called a Banach interval space.Example 2.
Let⋅ be a nonnegative real-valued function defined on I by(18)σ,υ=σ+υ.
Then,I,⋅ forms a Banach interval space such that ⋅ satisfies the null equality.Definition 6.
LetF be a self-mapping on I. Then, the point σo,υo∈I is called a near-fixed point of F if and only if Fσo,υoΩ=σo,υo.Definition 7.
(see [11, 12]). A function S:0,∞×0,∞⟶ℝ is called a simulation function if the following conditions hold:S1. S0,0=0S2. Sα,β<β−α for all α,β>0S3. If αn,βn are two sequences in 0,∞ such that limn⟶∞αn=limn⟶∞βn>0 and αn<βn for all n∈ℕ, then(19)limn⟶∞supSαn,βn<0.
ByS2, we must have(20)Sα,α<0.
The following are some interesting examples of simulation functions:(i)
Sα,β=χβ−Υα for all α,β∈0,∞ where χ and γ are continuous on 0,∞ such that γα=χα if and only if α=0 and γα<α≤χα for all α>0. If we take γβ=λβ and χα=α, then Sα,β=λβ−α.(ii)
Sα,β=β−χβ−α for all α,β∈0,∞ where χ is continuous on 0,∞ such that χα=0 if and only if α=0 (see Example 2.2 in [11]).(iii)
Sα,β=βχβ−α for all α,β∈0,∞ where χ is a mapping such that limα⟶r+χt<1 for all r>0 [12].(iv)
Sα,β=ηβ−α for all α,β∈0,∞, where η is an upper semicontinuous function so that ηα<α for all α>0 and η0=0 [12].


## 2.1. Interval Spaces

LetI be the set containing all close bounded intervals of the form σ,υ, where σ,υ∈ℝ and σ≤υ. Also, σ∈ℝ is considered as an element σ,σ∈I [21].The binary operation of addition and scaler multiplication is stated as follows:(1)σ,υ⊕σ′,υ′=σ+σ′,υ+υ′,kσ,υ=kσ,kυ,k≥0,kυ,kσ,k<0.Due to the inverse property, the above space does not fulfill the condition of a conventional vector space. Forσ,υ∈I, the subtraction(2)σ,υ⊖σ,υ=σ,υ⊕−υ,−σ=σ−υ,υ−σdoes not give the zero element 0,0. So the inverse of σ,υ does not exist. For the above deficiency, the null set was defined by Wu [21] as follows.


## 2.2. Null Set

The null set contains all the elements of the type−σ,σ, and so it is defined as follows:(3)Ω=σ,υ⊖σ,υ;σ,υis an element ofI,or(4)Ω=−α,α;α≥0.


## 2.3. Binary RelationΩ=

We writeσ,υΩ=σ′,υ′ iff there exist ω1,ω2∈Ω such that(5)σ,υ⊕ω1=σ′,υ′⊕ω2.Clearly, we can haveσ,υ=σ′,υ′⇒σ,υΩ=σ′,υ′ by taking ω1=ω2=0,0. However, the converse is not true in general.Proposition 1.
(see [21]). Ω= is an equivalence relation.
According to the equivalence relationΩ=, the equivalence class of almost identical intervals is defined as σ,υ=p,q∈I:σ,υΩ=p,q for any σ,υ∈I.


## 2.4. Norm Interval Space

The pairI,⋅ fulfilling the following axioms is called a norm interval space [21]:(i)
σ,υ=0 implies σ,υ∈Ω(ii)
ασ,υ=ασ,υ(iii)
σ,υ⊕σ′,υ′≤σ,υ+σ′,υ′ for all σ,υ,σ′,υ′∈I, where I contains all close bounded intervals over ℝ with the null set Ω and ⋅ is a real-valued mapping on IWe say that the null condition is satisfied by⋅ if the condition (iii) is replaced by(6)σ,υ=0if and only ifσ,υ∈Ω.⋅ is said to satisfy the null equalities if for all ω1,ω2∈Ω and σ,υ,σ′,υ′∈I, the following equalities hold:(1)
σ,υ⊕ω1⊖σ′,υ′⊕ω2=σ,υ⊖σ′,υ′(2)
σ,υ⊕ω1⊖σ′,υ′=σ,υ⊖σ′,υ′(3)
σ,υ⊖σ′,υ′⊕ω2=σ,υ⊖σ′,υ′Definition 1.
IfI,⋅ is a norm interval space, then(i)
The mapping⋅ is said to satisfy the null super-inequality if(7)σ,υ⊕ω≥σ,υ,for anyσ,υ∈Iandω∈Ω.(ii)
The mapping⋅ is said to satisfy the null sub-inequality if(8)σ,υ⊕ω≤σ,υ,for anyσ,υ∈Iandω∈Ω.(iii)
The mapping⋅ is said to satisfy the null equality if(9)σ,υ⊕ω=σ,υ,for anyσ,υ∈Iandω∈Ω.Example 1.
Let⋅ be a nonnegative real-valued function defined on I by(10)σ,υ=σ+υ.
Then,I,⋅ forms a norm interval space such that ⋅ satisfies the null equality.Proposition 2.
(see [21]). Let I,⋅ be a norm interval space such that ⋅ satisfies the null super-inequality. Then, for any σ,υ,σ′,υ′,σ1,υ1,σ2,υ2,…,σm,υm, we have(11)σ,υ⊖σ′,υ′≤σ,υ⊖σ1,υ1+σ1,υ1⊖σ2,υ2+⋯+σm,υm⊖υ,υ′.Proposition 3.
LetI,⋅ be a norm interval space; then, the following hold:(i)
If⋅ satisfies the null equality, then for all σ,υ,σ′,υ′∈I,(12)σ,υΩ=σ′,υ′impliesσ,υ=σ′,υ′.(ii)
For anyσ,υ,σ′,υ′∈I,(13)σ,υ⊖σ′,υ′=0,impliesσ,υΩ=σ′,υ′.(iii)
If⋅ satisfies the null super-inequality and null condition, then for any σ,υ,σ′,υ′∈I,(14)σ,υΩ=σ′,υ′,impliesσ,υ⊖σ′,υ′=0.
For proof of the above propositions, see [21].Definition 2.
LetI,⋅ be a norm interval space. A sequence σn,υnn=1+∞ is said to converge to a limit σ,υ if and only if(15)limn⟶∞σn,υn⊖σ,υ=0.Proposition 4.
Consider a norm interval spaceI,⋅ with the null set Ω.(i)
If the null super-inequality holds for⋅, then the convergence of the sequence σn,υnn=1+∞ to σ,υ and σ′,υ′ simultaneously implies σ,υ=σ′,υ′(ii)
If the null equality holds for⋅ and the sequence σn,υnn=1+∞ converges to σ,υ, then for any σ′,υ′∈σ,υ, the given sequence will also converge to σ′,υ′Definition 3.
Consider the norm interval spaceI,⋅ with the null set Ω, where ⋅ satisfies the null equality. If σ,υ∈I is the limit of the sequence σn,υnn=1+∞, then σ,υ is called the class limit. We can also write(16)limn⟶∞σn,υn=σ,υ.Proposition 5.
(see [21]). In the norm interval space I,⋅ if the null super-inequality holds for ⋅, the class limit is unique.Definition 4.
A sequenceσn,υnn=1+∞ in a norm interval space I,⋅ is called a Cauchy sequence if and only if for any ɛ≥0, there exists K∈ℕ such that(17)σn,υn⊖σm,υm<ɛ,for m,n>K with m≠n. If every Cauchy sequence is convergent in I, then I is complete.Definition 5.
A complete norm interval spaceI,⋅ is called a Banach interval space.Example 2.
Let⋅ be a nonnegative real-valued function defined on I by(18)σ,υ=σ+υ.
Then,I,⋅ forms a Banach interval space such that ⋅ satisfies the null equality.Definition 6.
LetF be a self-mapping on I. Then, the point σo,υo∈I is called a near-fixed point of F if and only if Fσo,υoΩ=σo,υo.Definition 7.
(see [11, 12]). A function S:0,∞×0,∞⟶ℝ is called a simulation function if the following conditions hold:S1. S0,0=0S2. Sα,β<β−α for all α,β>0S3. If αn,βn are two sequences in 0,∞ such that limn⟶∞αn=limn⟶∞βn>0 and αn<βn for all n∈ℕ, then(19)limn⟶∞supSαn,βn<0.
ByS2, we must have(20)Sα,α<0.
The following are some interesting examples of simulation functions:(i)
Sα,β=χβ−Υα for all α,β∈0,∞ where χ and γ are continuous on 0,∞ such that γα=χα if and only if α=0 and γα<α≤χα for all α>0. If we take γβ=λβ and χα=α, then Sα,β=λβ−α.(ii)
Sα,β=β−χβ−α for all α,β∈0,∞ where χ is continuous on 0,∞ such that χα=0 if and only if α=0 (see Example 2.2 in [11]).(iii)
Sα,β=βχβ−α for all α,β∈0,∞ where χ is a mapping such that limα⟶r+χt<1 for all r>0 [12].(iv)
Sα,β=ηβ−α for all α,β∈0,∞, where η is an upper semicontinuous function so that ηα<α for all α>0 and η0=0 [12].


## 3. Results and Discussion

Proposition 6.
In an interval spaceI with the null set Ω, σ,υΩ=σ′,υ′ iff σ′−σ=υ−υ′.Proof.
Let us suppose thatσ,υΩ=σ′,υ′; then, by definition, there exist −k,k and −h,h in Ω such that(21)σ,υ⊕−k,k=σ′,υ′⊕−h,h,that is,(22)σ−k,υ+k=σ′−h,υ′+h.
This implies that(23)σ−k=σ′−h,υ+k=υ′+h,σ−σ′=k−h,υ−υ′=−k+h,σ−σ′=k−h,υ−υ′=−k−h.
Putting the values ofk−h from the 1st equality in the second equality, we have(24)υ−υ′=−σ−σ′,impliesσ′−σ=υ−υ′.
Conversely, now, let us suppose thatσ′−σ=υ−υ′; then, we have to show that(25)σ,υΩ=σ′,υ′.
Hence,(26)σ′−σ=υ−υ′=k,σ′−σ=k,υ−υ′=k,σ=σ′−k,υ=υ′+k,which implies(27)σ,υ=σ′−k,υ′+k,σ,υ=σ′,υ′⊕−k,k,and so from the last equality, we have(28)σ,υΩ=σ′,υ′.Example 3.
Taking the intervals3,7 and 4,6, we have 4−3=7−6=1 and hence by the above function, we have 3,7Ω=4,6. For verification, take ω1=0.0 and ω2=−1,1. Then,(29)3,7⊕0,0=4,6⊕−1,1.Definition 8.
For a pointσ,υ∈I, if Fσ,υΩ=gσ,υ, then the point σ,υ is called a near-coincidence point of F and g.Example 4.
Taking the functionFx,y=x2−1,2y2+1 and gx,y=x2,2y2, then we can verify that 3,5 is the near-coincidence point for the functions defined above.Definition 9.
IfF and g are two self-mappings over I,⋅ such that(30)limn⟶∞Fgσn,υn⊖gFσn,υn=0,then the mappings are called compatible.Definition 10.
IfFgσ,υΩ=gFσ,υ for all σ,υ∈I,d, then F and g are called commuting mappings.Definition 11.
F is a Z⋅,g-contraction in I,⋅ corresponding to a simulation function S∈Z if(31)SFσ,υ⊖Fσ′,υ′,gσ,υ⊖gσ′,υ′≥0,for all σ,υ,σ′,υ′∈I such that gσ,υΩ≠gσ′,υ′.Example 5.
Define the mappingsF and g as Fx,y=x2−1,2y2+1andgx,y=x2,2y2; then, F satisfies the criteria of Z⋅,g-contraction in I,⋅ according to the simulation function Ss,t=λt−s, where λ≥1.Definition 12.
For a sequenceσn,υn in the Banach interval space I,⋅, if(32)gσn+1,υn+1Ω=Fσn,υn,for alln≥0,then the sequence is known as a Picard F,g sequence at the point σo,υo.Theorem 1.
LetFσ,υ=f1σ,f2υ and Gσ,υ=g1σ,g2υ be two self-mappings over the interval space I, where f1σ≤f2υ and g1σ≤g2υ for all σ≤υ. If σ is a coincidence point for f1 and g1 and υ is a coincidence point for f2 and g2, then σ,υ is a near-coincidence point for F and G.Proof.
Asσ and υ are coincidence points for f1, g1 and f2, g2, respectively, we have(33)f1σ=g1σ,f2υ=g2υ.
This implies that(34)f1σ,f2υ=g1σ,g2υ,f1σ,f2υΩ=g1σ,g2υ,Fσ,υΩ=Gσ,υ.
Hence, a near-coincidence point for the mappingsF and G is σ,υ over the interval space I. The converse of the above statement is not true in general, because f1σ,f2υΩ=g1σ,g2υ does not imply f1σ=g1σandf2υ=g2υ.Example 6.
Taking the functionFx,y=x2,y+1 and Gx,y=x,y2+1, then clearly −1 is a coincidence point for x2 and x and 1 is a coincidence point for y+1 and y2+1. So, −1,1 is a near-coincidence point for F and G since F−1,1=1,2=1,2=G1,2. For justifying the converse of the above statement, we can verify that −1/2,1/2 is a near-coincidence point for F and G, but −1/2 and 1/2 are not coincidence points for x2 and x and y+1 and y2+1, respectively. As F−1/2,1/2=1/4,3/2 and G−1/2,1/2=1/2,5/4, to prove that −1/2,1/2 is a near-coincidence point; we have to show 1/4,3/2Ω=1/2,5/4. Taking ω1=0,0 and ω2=−1/4,1/4, we have(35)14,32⊕0,0=12,54⊕−14,14.Lemma 1.
LetFσ,υ=f1σ,f2υ and Gσ,υ=g1σ,g2υ be two self-mappings over the interval space I, where f1σ≤f2υ and g1σ≤g2υ for all σ≤υ. If g1σ−f1σ=f2υ−g2υ for some σ,υ∈I, then σ,υ is a near-coincidence point for F and G.Proof.
Using Proposition6, g1σ−f1σ=f2υ−g2υ implies that f1σ,f2υΩ=g1σ,g2υ, i.e.,(36)Fσ,υΩ=Gσ,υ.
Hence, it is proved thatσ,υ is a near-coincidence point for F and G.Lemma 2.
Consider a Banach interval spaceI with a Z⋅,g-contraction F. If σ,ν and σ′,ν′ both are the near-coincidence points for F and g, then(37)Fσ,νΩ=gσ,νΩ=gσ′,ν′Ω=Fσ′,ν′.
Furthermore, the equivalence class of a near-coincidence point is unique ifF or g is injective.Proof.
Letσ,ν and σ′,ν′ be two near-coincidence points of F and g. Then, we have(38)Fσ,νΩ=gσ,ν,Fσ′,ν′Ω=gσ′,ν′.
In the above requirement, the two equalities are clear. We only need to show thatgσ,νΩ=gσ′,ν′. On the contrary, let us suppose that gσ,νΩ=gσ′,ν′; so we have(39)gσ,ν⊖gσ′,ν′≥0.
As the mappingF is a Z⋅,g-contraction, by definition, we have(40)0≤SFσ,ν⊖Fσ′,ν′,gσ,ν⊖gσ′,ν′=Sgσ,ν⊖gσ′,ν′,gσ,ν⊖gσ′,ν′.
The last inequality is a contradiction to (20) in the definition of the simulation function, i.e., Sr,r<0, where r>0. So our supposition is wrong and we accept that gσ,νΩ=gσ′,ν′.
Hence, we prove that(41)Fσ,νΩ=gσ,νΩ=gσ′,ν′Ω=Fσ′,ν′.
Furthermore, letF be injective; then, the equivalence class of a near-coincidence point is unique. By the above work, we have(42)Fσ,νΩ=gσ,νΩ=gσ′,ν′Ω=Fσ′,ν′.
It implies that(43)Fσ,νΩ=Fσ′,ν′.
AsF is injective, σ,νΩ=σ′,ν′. It further implies that σ,ν=σ′,ν′.Theorem 2.
Consider az⋅,g-contraction F in the Banach interval space I,⋅ where ⋅ satisfies the null equality and F and g are continuous and compatible mappings. Assume that the space is satisfying the CLRF,g property. Then, a near-coincidence point exists for F and g.Proof.
As the spaceI,⋅ satisfies the CLRF,g property, i.e., there exists a Picard sequence σn,νn, such that(44)gσn+1,νn+1Ω=Fσn,νn,for alln≥0.
There are two possibilities: either the sequenceσn,νn contains a near-coincidence point, or it converges to the near-coincidence point. We will take the case that the sequence does not contain a near-coincidence point. Hence,(45)gσn,νnΩ=Fσn,νnΩ=gσn+1,νn+1,for alln≥0.
The result will be proved in the following steps.
First of all, we will show that(46)limn⟶∞gσn,νn⊖gσn+1,νn+1=0.
AsF is a z⋅,g-contraction, by CLRF,g property and condition (ii) of a simulation function, we have(47)0≤SFσn,νn⊖Fσn+1,νn+1,gσn,νn⊖gσn+1,νn+1=Sgσn+1,νn+1⊖gσn+2,νn+2,gσn,νn⊖gσn+1,νn+1<gσn,νn⊖gσn+1,νn+1−gσn+1,νn+1⊖gσn+2,νn+2.
This implies that(48)0<gσn+1,νn+1⊖gσn+2,νn+2<gσn,νn⊖gσn+1,νn+1.
The sequencegσn,νn⊖gσn+1,νn+1 is nonnegative and decreasing, so it converges to a limit, say Ł, i.e.,(49)limn⟶∞gσn,νn⊖gσn+1,νn+1=Ł.
We have to show thatŁ=0. On the contrary, let us suppose that Ł>0. Consider the sequences with the same limit rn=gσn+1,νn+1⊖gσn+2,νn+2 and sn=gσn,νn⊖gσn+1,νn+1 such that rn<sn for all n∈ℕ.
Now, by condition (iii) of the simulation function, we have(50)0>limsupn⟶∞Srn,sn=limsupn⟶∞Sgσn+1,νn+1⊖gσn+2,νn+2,gσn,νn⊖gσn+1,νn+1.
It is a contradiction because(51)Sgσn+1,νn+1⊖gσn+2,νn+2,gσn,νn⊖gσn+1,νn+1>0.
Thus,Ł=0. That is,(52)limn⟶∞gσn,νn⊖gσn+1,νn+1=0.
Next, we will show that the sequencegσn,νn is a Cauchy sequence. Let us suppose, on the contrary, that gσn,νn is not Cauchy. So there will exist ɛo>0 such that for all N∈ℕ, there exist positive integers m,n such that(53)gσn,νn⊖gσm,νm>ɛo.
We can construct two partial subsequencesgσnk,νnk and gσmk,ν__k such that no≤nk≤mk and(54)gσnk,νnk⊖gσmk,νmk>ɛo,for allk∈ℕ.
Letmk be the smallest positive integer in nk,nk+1,nk+2,…. Then,(55)gσmk−1,νmk−1⊖gσnk,νnk≤ɛo,for allk∈ℕ.
Also,mk>nk from (54), so mk≥nk+1 for all k∈ℕ. But mk=nk+1 is not possible taking into account (52) and (54) simultaneously. So, we have mk≥nk+2 for any k∈ℕ. It follows that nk+1<mk<mk+1 for all k∈ℕ. From (54) and (55), we have(56)ɛo<gσmk,νmk⊖gσnk,νnk≤gσmk,νmk⊖gσmk−1,νmk−1+gσmk−1,νmk−1⊖gσnk,νnk≤gσmk,νmk⊖gσmk−1,νmk−1+ɛo,for allk∈ℕ.
Therefore,(57)limk⟶∞gσmk,νmk⊖gσnk,νnk=ɛo.
Also,(58)limk⟶∞gσmk+1,νmk+1⊖gσnk+1,νnk+1=ɛo.
AsF is a Zd,g-contraction associated with S,(59)0≤SFσmk,νmk⊖Fσnk,νnk,gσmk,νmk⊖gσnk,νnk=Sgσmk+1,νmk+1⊖gσnk+1,νnk+1,gσmk,νmk⊖gσnk,νnk<gσmk,νmk⊖gσnk,νnk−gσmk+1,νmk+1⊖gσnk+1,νnk+1.
Thus,(60)0<gσmk+1,νmk+1⊖gσnk+1,νnk+1<gσmk,νmk⊖gσnk,νnk.
Let(61)rn=gσmk+1,νmk+1⊖gσnk+1,νnk+1,sn=gσmk,νmk⊖gσnk,νnk.
Clearly,rn,sn>0, limn⟶∞rn=limn⟶∞sn=ɛo, and rn<sn.
So byS3,(62)0≤limsupk⟶∞Sgσmk+1,νmk+1⊖gσnk+1,νnk+1,gσmk,νmk⊖gσnk,νnk<0,which is a contradiction. Thus, gσn,νn is a Cauchy sequence in I,d.
That is,gσn,νn is a Cauchy sequence. Now, as the space is complete, the sequence gσn,νn will converge to a limit σ,ν. Since the mappings F and g are continuous, one writes(63)gσn,νn⟶σ,ν,impliesggσn,νn⟶gσ,ν,gσn,νn⟶σ,ν,impliesFgσn,νn⟶Fσ,ν.
The compatibility of the mappings yields that(64)limn⟶∞Fgσn,νn⊖gFσn,νn=0.
Consider(65)Fσ,ν⊖gσ,ν=limn⟶∞Fgσn,νn⊖ggσn+1,νn+1=limn⟶∞Fgσn,νn⊖gFσn,νn,Fσ,ν⊖gσ,ν=0.
From the above function, we haveFσ,νΩ=gσ,ν; i.e., σ,ν is a near-coincidence point of F and g.Example 7.
Consider the two continuous self-mappingsF and g in the Banach interval space I,⋅ defined by(66)Fσ,υ=2σ−4,2υ+4.gσ,υ=σ−2,υ+2.
The functionF is a z⋅,g-contraction according to the simulation function Ss,t=λt−s, where λ≥2. Also, the functions F and g are compatible. The sequence −1/n,1/n is a Picard sequence, i.e.,(67)gσn+1,υn+1Ω=Fσn,υn,for alln≥2,g−1n+1,1n+1Ω=F−1n,1n,−1n+1−2,1n+1+2Ω=−2n−4,2n+4.
We can easily show thatgσn+1,υn+1Ω=Fσn,υn,for alln≥0, by taking ω1=−2n2+3n+2/nn+1,2n2+3n+2/nn+1 and ω2=0,0. Then,(68)−1n+1−2,1n+1+2⊕−2n2+3n+2nn+1,2n2+3n+2nn+1=−2n−4,2n+4⊕0,0.
If we replace the compatibility of mappings by commuting mappings, then the following corollary can be stated.Corollary 1.
Consider the continuous and commuting mappingsF and g in the Banach interval space I,⋅ such that the criteria of Z-contraction is satisfied by F. Assume that CLRF,g property holds in I; then, a near-coincidence point exists for F and g.Corollary 2.
Consider a Banach interval spaceI,⋅ with two self-mappings F and g. Then, a near-coincidence point exists for F and g if(69)Fσ,υ⊖Fσ′,υ′≤λgσ,υ⊖gσ′,υ′,for all σ,υ,σ′,υ′∈I, where gσ,υΩ≠gσ′,υ′ and λ∈0,1.Proof.
Taking the simulation functionSσ,υ=λυ−σ for all σ,υ∈0,∞ and λ∈0,1, according to the above condition, we have(70)Fσ,υ⊖Fσ′,υ′≤λgσ,υ⊖gσ′,υ′,for allσ,υ,σ′,υ′∈I.
It implies that(71)0≤λgσ,υ⊖gσ′,υ′−Fσ,υ⊖Fσ′,υ′≤SFσ,υ⊖Fσ′,υ′,gσ,υ⊖gσ′,υ′.
The last inequality allows to say thatF is a Z⋅−contraction, and hence, by Theorem 2, there will be a near-coincidence point for F and g.Corollary 3.
Consider a Banach interval spaceI,⋅ with self-mappings F and g such that(72)Fσ,υ⊖Fσ′,υ′≤gσ,υ⊖gσ′,υ′−Φgσ,υ⊖gσ′,υ′∀σ,υ,σ′,υ′∈I,where Φ is a lower semicontinuous function defined on 0,∞ so that Φ−10=0; then, F and g have a near-coincidence point in I.Proof.
It suffices to take the simulation functionSσ,υ=υ−Φυ−σ for all σ,υ∈0,∞. Then, we can easily prove that F is a z−contraction. So by Theorem 2, there exists a near-coincidence point for F and g.



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