# Recent Fixed-Point Results forθ−Contraction Mappings in Rectangular M−Metric Spaces with Supportive Application

Authors: Mustafa Mudhesh; Hasanen A. Hammad; Habes Alsamir; Muhammad Arshad; Eskandar Ameer
Journal: Journal of Mathematics
 (2021)

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## Abstract

The goal of this manuscript is to present a new fixed-point theorem onθ−contraction mappings in the setting of rectangular M-metric spaces (RMMSs). Also, a nontrivial example to illustrate our main result has been given. Moreover, some related sequences with θ−contraction mappings have been discussed. Ultimately, our theoretical result has been implicated to study the existence and uniqueness of the solution to a nonlinear integral equation (NIE).


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## Body

## 1. Introduction

Fixed-point theory is one of the branches of functional analysis known as “Banach contraction principle (BCP)” [1] which plays a main role in several branches of mathematics and applied sciences. Its core subject is concerned with the stipulations for the existence of one or more fixed points of a mapping T from a topological space ℵ into itself, whereas we can obtain a fixed point such that T1=1 and 1∈ℵ. Later, this principle has been extended by using different forms of contractive conditions in various spaces (for example, see [2–7]).The idea of a generalized metric space (GMS) (rectangular metric space or Branciari metric space) was introduced by Branciari [8] in 2000, where the triangle inequality is replaced by the inequality d1,J≤d1,ℓ+dℓ,ℏ+dℏ,J for all pairwise distinct points ı,J,ℓ,ℏ∈ℵ. Under this space, many fixed-point results were proposed and nice applications were discussed (see [9–13]).Recently, GMSs extended to recent versions such asM−metric spaces (MMSs), rectangular metric spaces (RMSs), extended rectangular b-metric spaces (ERbMSs), partial rectangular b-metric spaces (PRbMSs), and partial rectangular metric spaces (PRMSs) (see [8, 14–17]).In 2018, a new version of a metric space was introduced by Özg ür et al. [18], called a rectangular M−metric space (RMMS) which is also extended (MMS). Shukla [17] defined a PRMS which is the generalized RMS. In such spaces, the authors [17, 18] discussed and presented some fixed-point theorems for self-mappings. They presented some topological concepts about open balls and convergence and defined the notions of a circle and a fixed circle using these concepts with an application to fixed-circle problems. For more details of generalizing fixed-point results, see [8, 14, 17, 19–27].Continuing in the same direction, in this manuscript, a new fixed-point theorem forθ−contraction mapping in RMMSs is proved and some related sequences are introduced. To support the theoretical result, a nontrivial example is obtained and the existence of a solution to an NIE has been found.


## 2. Preliminaries

In this part, we consider for all1,J∈ℵ, mı,J=minm1,1,mJ,J, Mı,J=maxm1,1,mJ,J, mrı,J=minmr1,1,mrJ,J, and Mrı,J=maxmr1,1,mrJ,J.Now, we give some previous results in the gradation of spaces.Definition 1.
(see [17], PRMS). Let ℵ be a nonempty set. A mapping p:ℵ×ℵ⟶ℝ+ is said to be a PRM on ℵ if for any 1,J∈ℵ and all distinct points ℓ,ℏ∈ℵ/1,J, the following stipulations hold:(i)
RP11=J⇔p1,J=p1,1=pJ,J(ii)
RP2p1,1≤p1,J(iii)
RP3p1,J=pJ,1(iv)
RP4p1,J≤pı,ℓ+pℓ,ℏ+pℏ,J−pℓ,ℓ−pℏ,ℏ
The pairℵ,p is called a PRMS.The above space was extended to an MMS by Asadi et al. [14] as follows.Definition 2.
(MMS). Assume thatℵ is a nonempty set. If the function m:ℵ×ℵ⟶ℝ+ satisfies the following hypotheses, for all ı,J,ℓ∈ℵ, then the pair ℵ,m is called an MMS:(i)
MM1m1,1=mJ,J=m1,J⇔1=J(ii)
MM2mı,J≤m1,J(iii)
MM3m1,J=mJ,1(iv)
MM4m1,J−mı,J≤m1,ℓ−mı,ℓ+mℓ,J−mℓ,JÖzgür et al. [18] generalized an MMS to a RMMS as follows.Definition 3.
(RMMS). Letℵ be a nonempty set. If the function mr:ℵ×ℵ⟶ℝ+ verifies the following assumptions, for all 1,J∈ℵ, then the pair ℵ,mr is called a RMMS:(i)
RM1mr1,J=mrı,J=Mrı,J⇔1=J(ii)
RM2mrı,J≤mr1,J(iii)
RM3mr1,J=mrJ,1(iv)
RM4mr1,J−mrı,J≤mr1,ℓ−mrı,ℓ+mrℓ,ℏ−mrℓ,ℏ+mrℏ,J−mrℏ,J, for all ℓ,ℏ∈ℵ/1,JIt is clear that every MMS is a RMMS.The results given follow from Özgür et al. [18].Remark 1.
Letℵ,mr be a RMMS. Clearly, for every 1,J∈ℵ, we have(1)
0≤Mrı,J+mrı,J=mr1,1+mrJ,J(2)
0≤Mrı,J−mrı,J=mr1,1−mrJ,JAlso, it can be easily verified under special cases that(3)
Mrı,J−mrı,J≤Mrı,ℓ−mrı,ℓ+Mrℓ,ℏ−mrℓ,ℏ+Mrℏ,J−mrℏ,JExample 1.
Assume thatℂ is the set of all complex numbers, and consider the set ℵφ=z∈ℂ:argz=φ∪0 for a fixed φ∈0,2π and define the function mr:ℵ×ℵ⟶0,∞ by mrı,J=ı+J/2 for all ı,J∈ℵφ. Then, ℵφ,mr is a RMMS.Proposition 1.
(see [18]). Suppose that ℵ,d is a RMS and a function ξ:0,∞⟶α,∞ is a one-to-one and nondecreasing function with ξ0=α so that ξı+J+ℏ=ξı+ξJ+ξℏ−2α, for all ı,J,ℏ∈ℵ. Then, the function mr:ℵ×ℵ⟶0,∞ which is defined by mrı,J=ξdı,J, for all ı,J∈ℵ with the set ℵ, is a RMMS.Example 2.
Letℵ,d be a RMS and a function ξ:0,∞⟶α,∞ be defined as ξt=mt+n, with ξ0=α for all t∈0,∞, and mrı,J=mdı,J+n is a RMMS with the set ℵ.Definition 4.
Assume thatℵ,mr is a RMMS; a sequence ın∈ℵ is said to be(1)
Converging to a point1 iff(1)limn⟶∞mr1n,1−mrın,ı,ı=0.(2)
mr-Cauchy sequence iff(2)limn,m⟶∞mr1n,1m−mrın,ım,limn,m⟶∞Mrın,ım−mrın,ım,exist and is finite.(3)
A RMMS calledmr−complete if every mr−Cauchy sequence ın converges to a point ı so that(3)limn⟶∞mrın,ı−mrın,ı=0,limn⟶∞Mrın,ı−mrın,ı=0.Lemma 1.
(see [18]). Assume that ın⟶ı and Jn⟶J as n⟶∞ in a RMMS ℵ,mr. Then,(4)limn⟶∞mrın,Jn−mrın,Jn=mrı,J−mrı,J.Lemma 2.
(see [18]). Assume that ın⟶ı as n⟶∞ in a RMMS ℵ,mr. Then,(5)limn⟶∞mrın,J−mrın.J=mrı,J−mrı,Jfor allJ∈ℵ.Lemma 3.
(see [18]). Suppose that ın⟶ı and Jn⟶J as n⟶∞ in a RMMS ℵ,mr. Then, mrı,J=mrı,J. Furthermore, if mrı,ı=mrJ,J, then ı=J.Lemma 4.
(see [18]). Let ın be a sequence in a RMMS ℵ,mr. If there is r∈0,1 so that mrın+1,ın≤rmrın,ın−1 for all n∈ℕ, then(i)
limn⟶∞mrın,ın−1=0(ii)
limn⟶∞mrın,ın=0(iii)
limn,m⟶∞mrın,ım=0(iv)
ın is an mr-Cauchy sequenceLemma 5.
(see [18]). Let ℵ,mr be a RMMS and T be a self-mapping on ℵ. If there is k∈0,1 so that mrTı,TJ≤kmrı,J for all ı,J∈ℵ, then Tın⟶Tu as n⟶∞, where ınn≥0 is a sequence defined by ın+1=Tın so that ın⟶u as n⟶∞.Theorem 1.
(see [18]). Suppose that T is a self-mapping on a RMMS ℵ,mr. If there is k∈0,1 so that(6)mrTı,TJ≤kmrı,Jfor allı,J∈ℵ,then T has a unique fixed point u in ℵ, where mru,u=0.Recently, Jleli and Samet [10] established the fixed-point theorem as follows.Theorem 2.
(see [10]). Let ℵ,d be a complete GMS and T:ℵ⟶ℵ be a given map. Suppose that there is θ∈Θ and k∈0,1 such that(7)ı,J∈ℵ,dTl,TJ≠0⇒θdTı,TJ≤θdı,Jk,then T has a unique fixed point.Theorem2 was extended by Jleli et al. [11] as follows.Theorem 3.
(see [11]). Let ℵ,d be a complete GMS and T:ℵ⟶ℵ be a given map. Suppose that there is θ∈Θ and k∈0,1 so that(8)ı,J∈ℵ,dTı,TJ≠0⇒θdTı,TJ≤θMı,Jk,where(9)Mı,J=maxdı,J,dı,Tı,dJ,TJ,then T has a unique fixed point.In Theorems2 and 3, the authors considered Θ is the set of all functions θ:0,∞⟶1,∞ so that the following conditions hold:(i)
Θ1θ is nondecreasing(ii)
Θ2 for each sequence tn⊂0,∞, limn⟶∞θtn=1 iff limn⟶∞tn=0+(iii)
Θ3 there is r∈0,1 and L∈0,∞ so that limt⟶0+θt−1/tr=L(iv)
Θ4θ is continuous


## 3. Main Result

We introduce in this part some new fixed-point results viaθ−contraction mapping in RMMSs which generalizes the results of Jleli et al. [11].Theorem 4.
Letℵ,mr be a complete RMMS and T:ℵ⟶ℵ be a given mapping. Suppose that there is θ∈Θ and k∈0,1 so that(10)ı,J∈ℵ,mrTı,TJ≠0⇒θmrTı,TJ≤θNı,Jk,where(11)Nı,J=maxmrı,J,mrı,Tı,mrJ,TJ,and θ is as defined in the above section. Then, T has a unique fixed point.Proof.
Letı0∈ℵ be an arbitrary value and define a sequence ınn∈ℕ by letting ı1=Tı0, ı2=Tı1,ı3=Tı2,…,ın+1=Tın=Tn+1ı0. If for some q∈ℕ, we have Tqı=Tq+1ı, then Tqı is a FP of T . So we suppose that mrTnı,Tn+1ı>0∀n∈ℕ. Now, from (10), we have(12)θmrTnı,Tn+1ı≤θNTn−1ı,Tnık,(13)NTn−1ı,Tnı=maxmrTn−1ı,Tnı,mrTn−1ı,TTn−1ımrTnı,TTnı,(14)NTn−1ı,Tnı=maxmrTn−1ı,Tnı,mrTn−1ı,TnımrTnı,Tn+1ı,=maxmrTn−1ı,Tnı,mrTnı,Tn+1ı.
IfNTn−11,Tn1=mrTn1,Tn+1ı, then from (12), one can write(15)θmrTnı,Tn+1ı≤θmrTnı,Tn+1ık.
Takingln in both sides, we obtain(16)lnθmrTn1,Tn+11≤klnθmrTn1,Tn+11,which is a contradiction because k∈0,1. Thus, we have(17)NTn−1ı,Tnı=mrTn−1ı,Tnı.
It follows from (12) that(18)θmrTnı,Tn+1ı≤θmrTn−1ı,Tnık≤θmrTn−2ı,Tn−1ık2⋮≤θmrı,Tıkn.This leads to(19)1≤θmrTnı,Tn+1ı≤θmrı,Tıkn,∀n∈ℕ.
Asn⟶∞ in (19), we obtain that(20)θmrTnı,Tn+1ı⟶1,asn⟶∞,which implies from Θ2 that limn⟶∞mrTnı,Tn+1ı=0. Now, from condition Θ3, there are r∈0,1 and L∈0,∞ so that(21)limn⟶∞θmrTnı,Tn+1ı−1mrTnı,Tn+1ır=L.
Suppose thatL<∞. In this case, let B=L/2>0. From the definition of the limit, there is n0∈ℕ such that(22)θmrTnı,Tn+1ı−1mrTnı,Tn+1ır−L≤B,∀n≥n0.
This implies that(23)θmrTnı,Tn+1ı−1mrTnı,Tn+1ır≥L−B=B,∀n≥n0.
Thus,(24)nmrTnı,Tn+1ır≤AnθmrTnı,Tn+1ı−1,∀n≥n0,where A=1/B.
Now, considerL=∞ and B>0 as an arbitrary positive number. From the definition of the limit, there is n0∈ℕ so that(25)θmrTnı,Tn+1ı−1mrTnı,Tn+1ır≥B,∀n≥n0.
This leads to(26)nmrTnı,Tn+1ır≤AnθmrTnı,Tn+1ı−1,∀n≥n0,where A=1/B.
Thus, in both cases, there existA>0 and n0∈ℕ so that(27)nmrTnı,Tn+1ır≤AnθmrTnı,Tn+1ı−1,∀n≥n0.
Using (14), we obtain(28)nmrTnı,Tn+1ır≤Anθmrı,Tıkn−1,∀n≥n0.
Taking limit asn⟶∞ in the above inequality, one can obtain(29)limn⟶∞nmrTnı,Tn+1ır=0.
Thus, there isn1∈ℕ so that(30)mrTnı,Tn+1ı≤1n1/r,∀n≥n1.
Now, we will discuss the following two steps.
St1. When Tnı=Tmı for some integers n≠m with m>n, we have Tm−nTnı=Tnı. Choose J=Tnı and p=m−n. Then, TpJ=J and J is a periodic point of T. By (10) and (14), we obtain(31)θmrJ,TJ=θmrTpJ,Tp+1J≤θmrJ,TJkp,
Asp⟶∞, we have mrJ,TJ=0. On the other hand, we can write(32)θmrJ,J=θmrTpJ,TpJ≤θmrTp−1J,Tp−1Jk⋮≤θmrTJ,TJkp−1≤θmrJ,Jkp,which leads to(33)lnθmrJ,J≤kp−1lnθmrTJ,TJ≤kplnθmrJ,J,or equivalently(34)θmrJ,J=θmrTJ,TJ.This yields mrJ,J=mrTJ,TJ=0. Hence, TJ=J, that is, J is a fixed point of T.
St2. When Tnı≠Tmı for every n,m∈ℕ with n≠m, we shall prove that T has a periodic point. Suppose on the contrary, utilizing (10) and (11), we obtain(35)θmrTnı,Tn+2ı≤θNTn−1ı,Tn+1ık,where(36)NTn−1ı,Tn+1ı=maxmrTn−1ı,Tn+1ı,mrTn−1ı,TnımrTn+1ı,Tn+2ı.
Sinceθ is continuous and nondecreasing, using (35) and (36), one can write(37)θmrTnı,Tn+2ı≤maxθmrTn−1ı,Tn+1ıθmrTn−1ı,TnıθmrTn+1ı,Tn+2ık.
LetI be the set of n∈ℕ such that(38)un=maxθmrTn−1ı,Tn+1ı,θmrTn−1ı,Tnı,θmrTn+1ı,Tn+2ı=θmrTn−1ı,Tn+1ı.
IfI<∞, there exists N∈ℕ such that for every n≥N,(39)maxθmrTn−1ı,Tn+1ı,θmrTn−1ı,Tnı,θmrTn+1ı,Tn+2ı=maxθmrTn−1ı,Tnı,θmrTn+1ı,Tn+2ı.
In this case, we obtain from (37) that(40)1≤θmrTnı,Tn+2ı≤maxθmrTn−1ı,Tnı,θmrTn+1ı,Tn+2ık,for all n≥N. Letting n⟶∞ in the above inequality and using (20), we obtain(41)θmrTnı,Tn+2ı⟶1,asn⟶∞.
IfI=∞, we can find a subsequence of un, which we denote also by un, such that(42)un=θmrTn−1ı,Tn+1ı,fornlarge enough.
In this case, we obtain from (37) that(43)1≤θmrTnı,Tn+2ı≤θmrTn−1ı,Tn+1ık≤θmrTn−2ı,Tnık2≤…≤θmrı,T2ıkn,for n large enough. Letting n⟶∞ in the above inequality, we have(44)θmrTnı,Tn+2ı⟶1,asn⟶∞.
Then, in both cases, (44) holds. Using (44) and property Θ2, we obtain(45)limn⟶∞mrTnı,Tn+2ı=0.
Similarly, from conditionΘ3, there exists n2∈ℕ such that(46)mrTnı,Tn+2ı≤1n1/r,for alln≥n2.
LetN′=maxn0,n1. If m>2 is odd, then we consider m=2q+1, where q≥1, Using (30) for all n≥N´, we can obtain(47)mrTnı,Tn+mı−mrTnıTn+mı≤mrTnı,Tn+1ı−mrTnıTn+1ı+mrTn+1ı,Tn+2ı−mrTn+1ıTn+2ı+⋯+mrTn+2qı,Tn+2q+1ı−mrTn+2qıTn+2q+1ı≤mrTnı,Tn+1ı+mrTn+1ı,Tn+2ı+⋯+mrTn+2qı,Tn+2q+1ı≤1n1/r+1n+11/r+⋯+1n+2q1/r≤∑i=n∞1i1/r.
Now, ifm>2 is even, then writing m=2q, where q≥2 and using (30) and (46), for all n≥N´, we obtain(48)mrTnx,Tn+mx−mrTnxTn+mx≤mrTnx,Tn+2x−mrTnxTn+2x+mrTn+2x,Tn+3x−mrTn+2xTn+3x+⋯+mrTn+2q−1x,Tn+2qx−mrTn+2q−1xTn+2qx,≤mrTnx,Tn+2x+mrTn+2x,Tn+3x+⋯+mrTn+2q−1x,Tn+2qx≤1n1/r+1n+21/r+⋯+1n+2q−11/r≤∑i=n∞1i1/r.
Combining (47) and (48), we have(49)mrTnı,Tn+mı−mrTnıTn+mı≤∑i=n∞1i1/r,for alln≥N′,m∈ℕ.Since the series ∑i=n∞1/i1/r is convergent, we find that mrTnı,Tn+mı−mrTnıTn+mı is also convergent as n,m⟶∞, which implies that(50)mrTnı,Tn+mı−mrTnıTn+mı≤mrTnı,Tn+mı.
Thus, the sequenceTnı is mr−Cauchy in the mr−complete RMMS ℵ,mr. Hence, there is some u∈ℵ so that Tnı⟶u as n⟶∞. Without restriction of the generality, we can suppose that Tnı≠u for all n.
Assume thatmru,Tu>0. Using (10), we obtain(51)θmrTn+11,Tu≤θNTnı,uk,for alln∈ℕ,where(52)NTnı,u=maxmrTnı,u,mrTnı,Tn+1ı,mru,Tu.
Taking limit asn⟶∞ in the above inequality and using Θ4 and Lemmas 1–3, we have(53)θmru,Tu≤θmru,Tuk<θmru,Tu,which is a contradiction; therefore, Tu=u, which is also a contradiction with the supposition that T does not have a periodic point. Thus, T has a periodic point, say u, of period q. Assume that the set of all fixed points of T is empty. Then, we have(54)q>1andmru,Tu>0.
Using (10), we obtain that(55)θmru,Tu=θmrTqu,Tq+1u≤θmru,Tukq<θmru,Tu,which is a contradiction. Thus, the set of fixed points of T is nonempty, that is, T has more than one fixed point.
Now, assume thatu,v∈ℵ are two fixed points of T so that mru,v=mrTu,Tv>0. Utilizing (10), we obtain(56)θmru,v=θmrTu,Tv≤θmru,vk<θmru,v,which is a contradiction. Then, T has a unique fixed point. This completes the proof.Theorem4 is still valid if we replace condition (10) with one of the following stipulations: for all ı,J,u∈ℵ with mru,u=0,(i)
There is0<k<1 so that(57)mrT1,TJ≤kmaxmrı,J,mrı,Tı,mrJ,TJ.(ii)
There is0≤k<1/2 so that(58)mrTı,TJ≤kmrı,Tı+mrJ,TJ.(iii)
There is0<k<3−1/2 so that(59)mrTı,TJ≤kmrı,TJ+mrJ,Tı.(iv)
There areθ∈Θ and k∈0,1 so that(60)mrTı,TJ≠0⟹θmrTı,TJ≤θmrı,Jk.We note thatΘ contains a large class of functions. For instance,(61)θt=2−2πarctan1tα,0<α<1,t>0.So we conclude the following corollary from Theorem4.Corollary 1.
Letℵ,mr be a complete RMMS and T:ℵ⟶ℵ be a self-mapping. If there exists θ∈Θ and α,k∈0,1 such that for all ı,J∈ℵ and Tı≠TJ,(62)2−2πarctan1mrTı,TJα≤2−2πarctan1Nı,Jαk,where(63)Nı,J=maxmrı,J,mrı,TJ,mrJ,TJ.
Then,T has a unique fixed point.The example below justifies all requirements of Theorem4.Example 3.
Letℵφ=0∪z∈ℂ:argz=φ for a fixed φ, where 0≤φ<2π. Define the function mr:ℵ×ℵ⟶0,∞ by mrı,J=ı+J/2 for all ı,J∈ℵφ. Then, ℵφ,mr is a RMMS. Take φ=0,π/2 and the RMMS becomes ℵ0,π/2,mr such that ℵ0,π/2=0,∞∪Ai=z⌣=ni:1≤n<∞ and mrz⌣1,z⌣2=z⌣1+z⌣1/2 for all z⌣1,z⌣2∈ℵ0,π/2. Let θ:0,∞⟶1,∞ be defined by θt=et; it is clear that θ∈Θ. Define T:ℵ0,π/2⟶ℵ0,π/2 by(64)Tz⌣=z⌣2+n,z⌣∈Ai,1≤n<∞,z⌣2,z⌣∈0,1,z⌣2n,z⌣∈1,∞,2≤n<∞.
Now, forz⌣∈Ai, 1≤n<∞, we have(65)θmrTz⌣1,Tz⌣2=emrTz⌣1,Tz⌣2=θz⌣1/2+n+z⌣2/2+n2=θ12+nz⌣1+z⌣22=θ12+nmrz⌣1,z⌣2=e1/2+nmrz⌣1,z⌣2=emrz⌣1,z⌣21/2+n≤eNz⌣1,z⌣21/2+n=θNz⌣1,z⌣2k.
Then,T satisfies the contraction with 0<k=1/2+n<1.On the other hand, forz⌣∈0,1, we have(66)θmrTz⌣1,Tz⌣2=θz⌣1/2+z⌣2/22=θ12z⌣1+z⌣22=θ12mrz⌣1,z⌣2=e1/2mrz⌣1,z⌣2=emrz⌣1,z⌣21/2≤eNz⌣1,z⌣21/2=θNz⌣1,z⌣2k.Similarly, forz⌣∈1,∞, 2≤n<∞, we obtain(67)θmrTz⌣1,Tz⌣2=θz⌣1/2n+z⌣2/2n2=θ12nz⌣1+z⌣22=θ12nmrz⌣1,z⌣2=e1/2nmrz⌣1,z⌣2=emrz⌣1,z⌣21/2n≤eNz⌣1,z⌣21/2n=θNz⌣1,z⌣2k,where k=1/2n∈0,1. Hence, the conditions of Theorem 4 are fulfilled and z⌣=0 is a unique fixed point of T.


## 4. Supportive Application

In this section, we apply the theoretical results of Theorem4 to discuss the existence of solution to a NIE:(68)ıt=∫0t£t,rϖr,ırdr,t∈0,D,D>0,where pounds:0,D×0,D⟶ℝ and ϖ:0,D×ℝ⟶ℝ are continuous functions.Define the operatorT:ℵ⟶ℵ by(69)Tıt=∫0t£t,rϖr,ırdr,where ℵ=C0,D is the space of all continuous functions defined from 0,D to ℝ equipped with(70)mrı,J=ı+J2=maxt∈0,Dıt+Jt2,∀ı,J∈ℵ.Obviously,ℵ,mr is a complete RMMS.We will consider Problem (68) under the following assumptions:(i)
The functionpounds:0,D×0,D⟶ℝ is continuous and maxt,r∈0,Dpoundst,r≤1(ii)
The functionϖ:0,D×ℝ⟶ℝ is continuous,(71)ϖr,ır+ϖr,Jr<λ2e−λPı,J,∀ı,J∈ℵ,r∈0,D,whereλ≥1 and Pı,J=maxı+J/2,ı+Tı/2,J+TJ/2(iii)
For allı,J∈ℵ, ıλ=maxt∈0,Dıte−λt and(72)mrλı,J=ıλ+Jλλ=maxt∈0,Dıt+Jtλe−λt,so that the pairℵ,mrλ is a complete RMMS with λ=2(iv)
For allı,J∈ℵ, define the function θ∈Θ by θt=et, where t>0Now, we can state and prove the important theorem in this part as follows.Theorem 5.
Via hypotheses (i)–(iv), Problem (68) has a unique solution in ℵ.Proof.
Define the operatorT:ℵ⟶ℵ as in (69). If T has a unique fixed point, then the NIE (68) has a unique solution. From Hypotheses (i)–(iii), we can write(73)Tır+TJr=∫0t£t,rϖr,ırdr+∫0t£t,rϖr,Jrdr≤∫0t£t,rϖr,ırdr+∫0t£t,rϖr,Jrdr≤∫0t£t,rϖr,ır+ϖr,Jrdr≤∫0t|ϖr,ır|+ϖr,Jrdr<∫0tλ2e−λPır,Jrdr=λ2e−λ∫0terλmaxır+Jrλe−rλ,ır+Tırλe−rλ,Jr+TJrλe−rλdr≤λ2e−λ∫0terλmaxmrλı,J,mrλı,Tı,mrλJ,TJdr=λ2e−λNı,J∫0terλdr=λ2e−λNı,J1λeλt−1=λe−λNı,Jeλt−λe−λNı,J≤λe−λNı,Jeλt.
Consequently, we have(74)Tır+TJrλe−λt≤e−λNı,J.This implies that(75)mrλTı,TJ=maxt∈0,DTır+TJrλe−λt≤e−λNı,J.
By Hypothesis (iv), we obtain(76)emrλTı,TJ≤ee−λNı,J=eNı,Je−λ/2,yielding(77)θmrλTı,TJ≤θNı,Jk,∀ı,J∈ℵ,where k=e−λ/2∈0,1 with λ=2. Thus, all conditions of Theorem 4 are fulfilled. So T has a unique fixed point which is a unique solution to NIE (68).


## 5. Conclusions

In this paper, we established fixed points forθ−contraction self-mapping in RMMS. Our new results are extensions of recent fixed-point theorems of Jalili et al. [11]. Also, we gave an example to clarify the obtained results. Finally, we applied our main result to study the existence and uniqueness of a solution for a NIE. The new concepts lead to further investigations and applications.



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