# Relational Quasicontractions and Related Fixed Point Theorems

Authors: Nidal H. E. Eljaneid; Faizan A. Khan; Hamid I. A. Mohammed; Aftab Alam
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, some results on fixed points under quasicontractions in the framework of metric space endowed with binary relation are proved. Our newly proved results improve and extend several noted fixed point theorems of the existing literature besides their relation-theoretic analogues. We conclude this article by constructing an example to affirm the efficacy of our results.


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## Body

## 1. Introduction

Given a metric spaceM,ϱ, a mapping ℱ:M⟶M is said to be contraction if it satisfies(1)ϱℱr,ℱt≤kϱr,t,for some0≤k<1.Banach contraction principle [1] plays a key role in the area of metrical fixed point theory. This core result guarantees of the existence and uniqueness of fixed point under the hypotheses that the ambient space remains a complete metric space, whereas the underlying mapping should be a contraction mapping. Many authors extended this classical result employing relatively more general contractive conditions. One of the interesting extensions of contraction mapping was given by Ćirić [2], often referred as quasicontraction. We say that a self-mapping ℱ defined on a metric space M,ϱ is a quasicontraction if for some 0≤q<1 and for all r,t∈M, we have(2)ϱℱr,ℱt≤q⋅maxϱr,t,ϱx,ℱr,ϱt,ℱt,ϱr,ℱt,ϱt,ℱr.Indeed quasicontraction mappings subsume several noted generalized contractions due to Kannan [3], Chatterjea [4], Reich [5], Hardy and Rogers [6], Bianchini [7], Rhoades [8], and Zamfirescu [9]. For further details regarding quasicontractions, we refer some recent works due to Aydi et al. [10], Karapınar et al. [11], Bachar and Khamsi [12], Alfuraidan [13], Fallahi and Aghanians [14], Darko et al. [15], and Karapınar et al. [16].A variant of Banach contraction principle under binary relation is proved by Alam and Imdad [17], wherein the ambient metric space and other involved notions are compatible with an amorphous relation. Under universal relation, the result reduces to classical Banach contraction principle, while under partially ordered relation, the result is transformed into classical order-theoretic metrical fixed point theorems of Ran and Reurings [18] and Nieto and Rodríguez-López [19]. In recent years, various metrical fixed theorems are proved under different types of contractive conditions employing certain binary relations (e.g., [20–34]).The main intent of this article is to prove a fixed point theorem in the setting of metric space endowed with a binary relation under relation-preserving quasicontraction condition. We also deduce some consequences from our newly proved results. An example is also furnished to demonstrate our main results.


## 2. Preliminaries

This section deals with certain relevant notions and basic results which are utilized in our subsequent discussions. For any setM≠∅, a subset ℜ of M2 is termed as a binary relation on M. We sometimes write rℜt instead of r,t∈ℜ, e.g., in case of the relations of “less than equals to” ℜ≔≤ and “greater than equals to” ℜ≔≥ on ℝ, the set of real numbers is expressed, respectively, as r≤t and r≥t.Definition 1.
(see [35]). If ℜ is any binary relation on a set M, then the set(3)ℜ−1:=r,t∈M2:t,r∈ℜremains again a relation on M, which is termed as an inverse relation of ℜ. Also, the set(4)ℜs≔ℜ∪ℜ−1forms again a relation on M, which is called the symmetric-closure of ℜ. Clearly, ℜs remains the least symmetric relation on M among those binary relations which contain ℜ.Definition 2.
(see [17]). Any two elements r and t of a set M are called ℜ-comparative, whereas ℜ remains a relation on M if either r,t∈ℜ or t,r∈ℜ. Usually r,t∈ℜ means that “r and t are ℜ-comparative.”Proposition 1 (see [17]).
Ifℜ remains a relation on M, then(5)r,t∈ℜs⇔r,t∈ℜ.Definition 3.
(see [35]). If ℜ remains a relation on a set M, then ℜ is said to be complete if each pair of elements of M is ℜ-comparative, i.e.,(6)r,t∈ℜ,∀r,t∈M.Definition 4.
(see [35]). If E⊆M and ℜ stands for a relation on M, then the set(7)ℜ|E≔ℜ∩E2is termed as the restriction of ℜ to E. It is clear that ℜ|E remains a relation on E induced by ℜ.Definition 5.
(see [17]). By an ℜ-preserving sequence, whereas ℜ remains a relation on a set M, we meant that the sequence rn⊂M, which satisfy(8)rn,rn+1∈ℜ,∀n∈ℕ0.Definition 6.
(see [17]). Given any set M and a mapping ℱ from M into itself, a relation ℜ on M is termed as ℱ-closed if for any r,t∈M,(9)r,t∈ℜ⇒ℱr,ℱt∈ℜ.Proposition 2 (see [20]).
Given any setM, suppose that ℜ remains a relation on M while ℱ remains a function from M into itself. If ℜ is ℱ-closed, then ℜs must be ℱ-closed.Proposition 3 (see [21]).
Given any setM, suppose that ℜ remains a relation on M while ℱ remains a function from M into itself. If ℜ is ℱ-closed, then ℜ must be ℱn-closed (where n∈ℕ0).Definition 7.
(see [36, 37]). Given any set M, suppose that ℱ remains a function from M into itself and α:M×M⟶0,∞ is a mapping. We say that ℱ is α-admissible if for all r,t∈M,(10)αr,t≥1⇒αℱr,ℱt≥1.
If we define the functionαr,t=1,ifr,t∈ℜ0,ifr,t∉ℜ, then the α-admissibility of ℱ is equivalent to ℱ-closedness of ℜ.Definition 8.
(see [20]). A metric space M,ϱ is termed as ℜ-complete (whereas ℜ remains a relation on M) if each ℜ-preserving Cauchy sequence in M converges.
Obviously, given any metric space, completeness impliesℜ-completeness whatever the relation ℜ. In particular, if ℜ remains the universal relation, then the concepts of ℜ-completeness and usual completeness are equivalent.Definition 9.
(see [20]). A mapping ℱ:M⟶M, whereas M,ϱ remains a metric space while ℜ remains a relation on M, is termed ℜ-continuous at a point r∈M if for any ℜ-preserving sequence rn such that rn⟶ϱr, we have ℱrn⟶ϱℱr. Moreover, ℱ is termed as ℜ-continuous if it is ℜ-continuous at every point of M.
Obviously, for any mapping on a metric space endowed with the relationℜ, continuity implies ℜ-continuity. In particular, if ℜ remains the universal relation, then the concepts of ℜ-continuity and usual continuity are equivalent.Definition 10.
(see [17]). A relation ℜ on a metric space M,ϱ is termed as ϱ-self-closed if each ℜ-preserving sequence rn with rn⟶ϱr has a subsequence rnk whose terms are ℜ-comparative with r, i.e.,(11)rnk,r∈ℜ,∀k∈ℕ0.
In the sequel, for a metric spaceM,ϱ, a relation ℜ and a function ℱ from M into itself, the following notations will be adopted:(i)
Fℱ:= the collection of the fixed points of ℱ(ii)
Mℱ,ℜ:=r∈M:r,ℱr∈ℜ(iii)
δr≔supϱℱnr,ℱmr:n,m∈ℕ


## 3. Main Results

In the following lines, we prove a fixed point theorem for quasicontractions employing a binary relation.Theorem 1.
LetM,ϱ be a metric space while ℜ is a relation on M and ℱ is a function from M into itself. Also, suppose that(1)
aM,ϱ is ℜ-complete.(2)
bℜ is ℱ-closed.(3)
cℱ is ℜ-continuous or ℜ is ϱ-self-closed.(4)
dMℱ,ℜ is nonempty.(5)
e for some q∈0,1/2 and for all r,t∈M with r,t∈ℜ, the following assumption is satisfied:(12)ϱℱr,ℱt≤q⋅maxϱr,t,ϱr,ℱr,ϱt,ℱt,ϱr,ℱt,ϱt,ℱr.Then,ℱ admits a fixed point. Further, the completeness of ℜFℱ implies the uniqueness of fixed point.Proof.
In view of assumptiond, we can choose r0∈Mℱ,ℜ. Define a sequence rn of Picard iteration based at the initial point r0 so that(13)rn=ℱnr0,∀n∈ℕ0.
Asr0,ℱr0∈ℜ, using ℱ-closedness of ℜ and Proposition 3, we get(14)ℱnr0,ℱn+1r0∈ℜ,so that(15)rn,rn+1∈ℜ,∀n∈ℕ0.
It follows that the sequencern is ℜ-preserving.
Denoteϱn≔ϱrn+1,rn. Applying the contractive condition e and using (13) and (15), we obtain for all n∈ℕ0 that(16)ϱn=ϱrn,rn+1=ϱℱrn−1,ℱrn≤q⋅maxϱrn−1,rn,ϱrn−1,ℱrn−1,ϱrn,ℱrn,ϱrn−1,ℱrn,ϱrn,ℱrn−1=q⋅maxϱn−1,ϱn,ϱrn−1,rn+1,0,which gives rise to(17)ϱn≤q⋅ϱn−1+ϱn,so that(18)ϱn≤q1−q⋅ϱn−1=p⋅ϱn−1,∀n∈ℕ.
Asq∈0,1/2, we have p=q/1−q∈0,1. By induction, equation (18) reduces to(19)ϱn≤pϱn−1≤p2ϱn−2≤⋯≤pnϱ0,so that(20)ϱn≤pnϱ0,∀n∈ℕ.
Forn<m, using triangular inequality and (20), we obtain(21)ϱrn,rm≤ϱrn,rn+1+ϱrn+1,rn+2+⋯+ϱrm−1,rm≤pn+pn+1+⋯+pm−1ϱ0=pn1+p+p2+⋯+pn−m+1ϱ0<pn1−pϱ0⟶0,asm,n⟶∞,yielding thereby rn as Cauchy. Hence, rn is an ℜ-preserving Cauchy sequence. By ℜ-completeness of M,ϱ, one can find r∈M satisfying(22)rn⟶ϱr.
Now, we use assumptionc to show that r is a fixed point of ℱ. Suppose that ℱ is ℜ-continuous. Since rn is ℜ-preserving with rn⟶ϱr, therefore ℜ-continuity of ℱ asserts that rn+1=ℱrn⟶ϱℱr. The uniqueness of limit ensures that ℱr=r so that r is a fixed point of ℱ. Next, we assume that ℜ is ϱ-self-closed, then since rn remains an ℜ-preserving sequence converging to r, therefore one can a subsequence rnk of rn satisfying rnk,r∈ℜ for all k∈ℕ0. Obviously, we have(23)rnk⟶ϱr.
On usingrnk,r∈ℜ, symmetry of ϱ and assumption e, we have(24)ϱrnk+1,ℱr=ϱℱrnk,ℱr≤q⋅maxϱrnk,r,ϱrnk,ℱrnk,ϱr,ℱr,ϱrnk,ℱr,ϱr,ℱrnk.
Making use of the triangular inequalityϱrnk,ℱr≤ϱrnk,r+ϱr,ℱr, equation (24) reduces to(25)ϱrnk+1,ℱr≤q⋅ϱrnk,r+ϱrnk,rnk+1+ϱr,ℱr+ϱr,rnk+1.
Using (25) and triangular inequality, we get(26)ϱr,ℱr≤ϱr,rnk+1+ϱrnk+1,ℱr≤1+qϱr,rnk+1+q⋅ϱrnk,r+ϱrnk,rnk+1+ϱr,ℱr,thereby yielding(27)ϱr,ℱr≤1+q1−qϱr,rnk+1+q1−qϱrnk,r+ϱnk⟶0,ask⟶0,(owing to (20) and (23)) so that ℱr=r. Thus, r is a fixed point.
Finally, to prove the uniqueness of fixed point, we taker,t∈Fℱ, then we have(28)ℱr=r,ℱt=t.
AsℜFℱ is complete, we have r,t∈ℜ. Applying contractive condition e on these points, we get(29)ϱr,t=ϱℱr,ℱt≤q⋅maxϱr,t,ϱr,r,ϱt,t,ϱr,t,ϱt,r,so that(30)ϱr,t≤q⋅ϱr,t,which gives rise to(31)r=t.
It follows thatℱ has a unique fixed point. This completes the proof.
Now, we present the following results which ensure that Theorem1 is true for q∈0,1 under some restricted hypotheses.Theorem 2.
LetM,ϱ be a metric space while ℜ is a relation on M and ℱ is a function from M into itself. Also, suppose that(1)
aM,ϱ is ℜ-complete.(2)
bℜ is ℱ-closed and transitive.(3)
cℱ is ℜ-continuous or ℜ is ϱ-self-closed.(4)
d there exists r0∈Mℱ,ℜ such that δr0<∞.(5)
e for some q∈0,1 and for all r,t∈M with r,t∈ℜ, the following assumption is satisfied:(32)ϱℱr,ℱt≤q⋅maxϱr,t,ϱr,ℱr,ϱt,ℱt,ϱr,ℱt,ϱt,ℱr.Then,ℱ admits a fixed point. Further, the completeness of ℜFℱ implies the uniqueness of fixed point.Proof.
Following the lines of the proof of Theorem1, one can show that the sequence rn defined by rn=ℱnr0 is ℜ-preserving. Using (15) and transitivity of ℜ, we obtain(33)rn,rm∈ℜ,∀n,m∈ℕ0withn<m.
Using (33) and assumption e, we get(34)ϱrn,rm≤q⋅maxϱrn−1,rm−1,ϱrn−1,rn,ϱrm−1,rm,ϱrn−1,rm,ϱrn,rm−1,which implies that(35)δrn≤qδrn−1,∀n∈ℕ.
Hence, we have(36)δrn≤qnδr0,∀n∈ℕ,which yields that(37)ϱrn,rn+m≤δrn≤qnδr0,∀n,m∈ℕ.
Asδr0<∞, the sequence rn is Cauchy. Rest of the proof can be completed by proceeding the lines of the proof of Theorem 1.


## 4. Applications

As applications, we deduce the following consequences of Theorem1, which are indeed relation-theoretic versions of some well-known theorems existing in the literature.Corollary 1 (Hardy-Rogers type).
The conclusion of Theorem1 holds under the hypotheses (a)–(d) along with the following:(1)
e1 for some α,η,δ,λ,μ≥0satisfyingα+η+δ+λ+μ<1 and for all r,t∈M with r,t∈ℜ, the following assumption is satisfied:(38)ϱℱr,ℱt≤αϱr,t+ηϱr,ℱr+δϱt,ℱt+λϱt,ℱr+μϱr,ℱt.Proof.
Taker,t∈M with r,t∈ℜ and write q=α+η+δ+λ+μ, and then, we have(39)ϱℱr,ℱt≤αϱr,t+ηϱr,ℱr+δϱt,ℱt+λϱt,ℱr+μϱr,ℱt=q⋅maxϱr,t,ϱr,ℱr,ϱt,ℱt,ϱr,ℱt,ϱt,ℱr.
Thus, the result follows from Theorem1.Corollary 2 (Reich type).
The conclusion of Theorem1 holds under the hypotheses (a)–(d) along with the following:(1)
e2 for some α,η,δ≥0satisfyingα+η+δ<1 and for all r,t∈M with r,t∈ℜ, the following assumption is satisfied:(40)ϱℱr,ℱt≤αϱr,t+ηϱr,ℱr+δϱt,ℱt.Proof.
Choosingλ=μ=0 in Corollary 1, the assumption e1 reduces to e2. Consequently, the result follows from Corollary 1.Corollary 3 (Kannan type).
The conclusion of Theorem1 holds under the hypotheses (a)–(d) along with the following:(1)
e3 for some η,δ≥0satisfyingη+δ<1 and for all r,t∈M with r,t∈ℜ, the following assumption is satisfied:(41)ϱℱr,ℱt≤ηϱr,ℱr+δϱt,ℱt.Proof.
Choosingα=λ=μ=0 in Corollary 1, the assumption e1 reduces to e3. Consequently, the result follows from Corollary 1.Corollary 4 (Chatterjea type).
The conclusion of Theorem1 holds under the hypotheses (a)–(d) along with the following:(1)
e4 for some λ,μ≥0satisfyingλ+μ<1 and for all r,t∈M with r,t∈ℜ, the following assumption is satisfied:(42)ϱℱr,ℱt≤λϱt,ℱr+μϱr,ℱt.Proof.
Choosingα=η=δ=0 in Corollary 1, the assumption e1 reduces to e4. Consequently, the result follows from Corollary 1.Corollary 5 (Bianchini type).
The conclusion of Theorem1 holds under the hypotheses (a)–(d) along with the following:(1)
e5 for some q∈0,1 and for all r,t∈M with r,t∈ℜ, the following assumption is satisfied:(43)ϱℱr,ℱt≤q⋅maxϱr,ℱr,ϱt,ℱt,∀r,t∈Mwithr,t∈ℜ.Proof.
Taker,t∈M with r,t∈ℜ and q∈0,1, then we have(44)ϱℱr,ℱt≤q⋅maxϱr,ℱr,ϱt,ℱt=q⋅maxϱr,t,ϱr,ℱr,ϱt,ℱt,ϱr,ℱt,ϱt,ℱr.
Thus, the result follows from Theorem1.Corollary 6 (Zamfirescu type).
The conclusion of Theorem1 holds under the hypotheses (a)–(d) along with the following:(1)
e6 for some 0≤a<1,0≤b<1/2,0≤c<1/2 and for all r,t∈M with r,t∈ℜ, at least one of the following is true:(i)
ϱℱr,ℱt≤aϱr,t(ii)
ϱℱr,ℱt≤bϱr,ℱr+ϱt,ℱt(iii)
ϱℱr,ℱt≤cϱt,ℱr+ϱr,ℱt


## 5. An Example

In this section, we furnish the following example which demonstrates the importance of Theorem1.Example 1.
ConsiderM≔M1∪M2 with the standard (usual) metric ϱ, where(45)M1=pq:p=0,1,3,9,…;q=1,4,7,…,3k−2,…,M2=pq:p=1,3,9,27,…;q=2,5,8,…,3k−1,….
OnM, define a relation ℜ as follows:(46)ℜ=r,t∈M×M:r−t≥0.
Define the mappingℱ:M⟶M by(47)ℱr=3r5,ifr∈M1,r8,ifr∈M2.
Taker,t∈M with r,t∈ℜ. Then, ℜ is ℱ-closed. Also, by routine calculation, it can easily verify that ℱ satisfies the contractive condition e with q=2/5. Consequently, ℱ has a unique fixed point. Indeed, here ℜ is complete and so ℜFℱ. Indeed, (unique) fixed point of ℱ is r=0.


## 6. Conclusion

In this paper, we have proved the fixed point theorems in the setting of relational metric space under relation-preserving quasicontractions. For possible works, one can prove the variants of the results in the context of relational quasimetric space, relational symmetric space, relational partial metric space, relationalG-metric space, relational generalized metric space, relational JS-metric space, relational cone metric space, relational complex-valued metric space, relational rectangular metric space, relational b-metric space.



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