# On Nonunique Fixed Point Theorems via Interpolative Chatterjea Type Suzuki Contraction in Quasi-Partial b-Metric Space

Authors: Pragati Gautam; Shiv Raj Singh; Santosh Kumar; Swapnil Verma
Journal: Journal of Mathematics
 (2022)

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## Abstract

In the present research paper, Chatterjea type contraction is defined and discussed in the framework of quasi-partial b-metric space. Further, some common fixed point results are proved using the notion of interpolation. The results are extended to fixed point theorems for modified Chatterjea type Suzuki contraction using w-admissible maps. The results proved are new and unique supported by application which will enrich the existing literature.


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## Body

## 1. Introduction and Preliminaries

In 1922, Banach [1] gave one of the important result called Banach contraction principle and discussed its importance in metric fixed point theory, i.e., let H be a self map on a nonempty set G and G,d be a complete metric space. If there exists a constant κ∈0,1 such that(1)dHσ,Hυ≤κdσ,υ,for allσ,υ∈G,then it possesses a unique fixed point in G. Afterwards, many generalizations of Banach contraction principle came in the literature (see [2–4]) In 1968, Kannan [5] removed the continuity property from this principle and defined a new variant.Theorem 1 (see [5]).
LetG,d be a complete metric space and a self map T:G⟶G be a Kannan contraction mapping, i.e.,(2)dTσ,Tυ≤ϱdσ,Tσ+dυ,Tυ,for all σ,υ∈G, where ϱ∈0,1/2. Then, T admits a unique fixed point in G.In 1972, Chatterjea [6] generalized the contraction mapping on a complete metric space which is stated as follows.Theorem 2 (see [6]).
LetG,d be a complete metric space. A self map T:G⟶G be a Chatterjea type contraction. If(3)dTσ,Tυ≤ϱdσ,Tυ+dυ,Tσ,for all σ,υ∈G, where ϱ∈0,1/2, then T has a unique fixed point.In 2015, quasi-partial metric space was generalized by Gupta and Gautam [7] to quasi-partial b-metric space. In 2018, Karapinar [8] enriched the concept of interpolation and established a new contraction known as interpolative Kannan-type contractive mapping.Theorem 3 (see [8]).
Let (G, d) be a metric space; a self mapping T:G⟶G is said to be an interpolative Kannan-type contraction map if there exist a constant ϱ∈ (0, 1) and α∈ (0, 1) such that(4)dTσ,Tυ≤ϱdσ,Tσα⋅dυ,Tυ1−α,for allσ,υ∈G∖FixT,where FixT=z∈G:Tz=z.Numerous studies have been done on interpolation inequalities in the literature. Many new research papers based on the concept of interpolation came into existence (for example, see [9–11]). Noorwali [12], in the year 2019, extended the Kannan interpolative contraction by proving common fixed point result for two self maps. Several authors [9, 13–25] have contributed immensely to enrich this field.Definition 1 (see [26]).
A quasi-partial metric on a nonempty setG is a function qp:G×G⟶ℝ+ such that for all σ,υ,ϵ∈G:(1)
qpσ,σ=qpσ,υ=qpυ,υ implies σ=υ(2)
qpσ,σ≤qpσ,υ(3)
qpσ,σ≤qpυ,σ(4)
qpσ,υ≤qpσ,ϵ+qpϵ,υ−qpϵ,ϵ
The pairG,qp is called a quasi-partial metric space.Definition 2 (see [7]).
A quasi-partial b-metric on a nonempty setG is a function qpb:G×G⟶ℝ+ such that for some real number s≥1 and all σ,υ,ϵ∈G:(1)
qpbσ,σ=qpbσ,υ=qpbυ,υ implies σ=υ(2)
qpbσ,σ≤qpbσ,υ(3)
qpbσ,σ≤qpbυ,σ(4)
qpbσ,υ≤sqpbσ,ϵ+qpbϵ,υ−qpbϵ,ϵ
The pairG,qpb is called a quasi-partial b-metric space. The number s is called the coefficient of G,qpb. Let qpb be a quasi-partial b-metric on the set G. Then,(5)dqpbσ,υ=qpbσ,υ+qpbυ,σ−qpbσ,σ−qpbυ,υis a b−metric onG.Example 1.
LetG=0,∞ be endowed with metric qpb:G×G⟶0,∞ defined by qpbσ,υ=maxσ,υ+σ−υ. Dominance of (G, qpb) for the special case G=0,3 is visually checked in Figure 1.
Here,qpbσ,σ=qpbσ,υ=qpbυ,υ⇒σ=υ as σ=maxσ,σ+|σ−σ|=υ.
Again,qpbσ,σ≤qpbσ,υ as maxσ,σ+σ−σ≤maxσ,υ+σ−υ and similarly qpbσ,σ≤qυ,σ.
Also,qpbσ,υ+qpbϵ,ϵ≤qpbσ,ϵ+qpbϵ,σ.
Letσ,ϵ,υ∈G. If σ≤υ≤ϵ, then(6)maxσ,υ+σ−υ≤υ+σ−ϵ+ϵ−υ≤maxσ,ϵ+σ−ϵ+maxϵ,υ+ϵ−υ−ϵ.
Thus,G,qpb is a quasi-partial b-metric space.Figure 1
The graphical surface represents a 3D view of the functionqpbσ,υ=maxσ,υ+σ−υ for [0,3].Lemma 1 (see [23]).
LetG,qpb be a quasi-partial b-metric space. Then, the following holds:(i)
Ifqpb (σ, υ) =  0, then σ=υ(ii)
Ifσ=υ, then qpb (σ, υ) >rbin0 and qpb (υ, σ) >rbin0Definition 3 (see [7]).
LetG,qpb be a quasi-partial b-metric. Then,(1)
A sequenceσn⊂G converges to σ∈G if and only if qpbσ,σ=limn⟶∞qpbσ,σn.(2)
A sequenceσn⊂G is called a Cauchy sequence if and only if limn,m⟶∞qpbσn,σm exists.(3)
A quasi-partial b-metric spaceG,qpb is said to be complete if every Cauchy sequence σn⊂G converges with respect to τqpb to a point σ∈G such that(7)qpbσ,σ=limn,m⟶∞qpbσn,σm.(4)
A mappingf:G⟶G is said to be continuous at σ0∈G if, for every ε>0, there exists δ>0such that(8)fBσ0,δ⊂Bfσ0,ε.Lemma 2 (see [18]).
Let (G, qpb) be a quasi-partial b-metric space and (G, dqpb) be the corresponding b-metric space. Then, (G, dqpb) is complete if (G, qpb) is complete.Definition 4 (see [23]).
LetG,qpb be a quasi-partial b-metric space and H:G⟶G be a given mapping. Then, H is said to be sequentially continuous at z∈G if for each sequence σn in G converging to z, we have Hσn⟶Hz, that is, limn⟶∞qpbHσn,Hz=qpbHz,Hz. Similarly, let K:G⟶G be a given mapping. K is said to be sequentially continuous at z∈G if for each sequence σn in G converging to z, we have limn⟶∞qpbKz,Kσn=qpbKz,Kz.


## 2. Main Results

In this paper, the notion of interpolative Chatterjea type contraction is given in the setting of quasi-partial b-metric space.Definition 5.
LetG,qpb be a complete quasi-partial b-metric space. We say that the self mappings H,K:G⟶G are an interpolative Chatterjea contraction if there exists ϱ∈0,1/s,α∈0,1, s>1 such that(9)qpbHσ,Kυ≤ϱ1s2qpbσ,Kυα1s2qpbυ,Hσ1−α,for all σ,υ∈G such that Hσ≠σ whenever Kυ≠υ.Theorem 4.
LetG,qpb be a complete quasi-partial b-metric space and H and K be an interpolative Chatterjea type contraction. Then, K and H posses a common fixed point.Proof.
Letσ0∈G. Define the sequence σn by σ2n+1=Hσ2n, σ2n+2=Kσ2n+1 for all n=1,2,…. If there exist n∈0,1,2,…, such that σn=σn+1=σn+2, then σn is a common fixed point of K and H. Suppose that there are no such three consecutive identical terms in the sequence σn and that σ0≠σ1. Now, using equation (9), we deduce that(10)qpbσ2n+1,σ2n+2=qpbHσ2n,Kσ2n+1≤ϱ1s2qpbσ2n,Kσ2n+1α1s2qpbσ2n+1,Hσ2n1−α≤ϱ1s2qpbσ2n,σ2n+2α1s2qpbσ2n+1,σ2n+11−α.
By using Definition2,(11)qpbσ2n+1,σ2n+1≤qpbσ2n+1,σ2n+2.
Therefore,(12)qpbσ2n+1,σ2n+2≤ϱ1s2qpbσ2n,σ2n+2α1s2qpbσ2n+1,σ2n+21−α,qpbσ2n+1,σ2n+2α≤ϱ1s2qpbσ2n,σ2n+2α≤1s2sqpbσ2n,σ2n+1+qpbσ2n+1,σ2n+2−qpbσ2n+1,σ2n+1α,qpbσ2n+1,σ2n+2α≤ϱ1sqpbσ2n,σ2n+1+qpbσ2n+1,σ2n+2α.
Suppose that(13)qpbσ2n,σ2n+1≤qpbσ2n+1,σ2n+2,1sqpbσ2n,σ2n+1+qpbσ2n+1,σ2n+2≤qpbσ2n+1,σ2n+2.
By using equation (12),(14)qpbσ2n+1,σ2n+2α≤ϱqpbσ2n+1,σ2n+2α,which is a contradiction. Thus, we have(15)qpbσ2n+1,σ2n+2<qpbσ2n,σ2n+1,for alln≥1,1sqpbσ2n,σ2n+1+qpbσ2n+1,σ2n+2≤qpbσ2n,σ2n+1.
Using equation (12),(16)qpbσ2n+1,σ2n+2α≤ϱqpbσ2n,σ2n+1α,qpbσ2n+1,σ2n+2≤ϱ1/αqpbσ2n,σ2n+1,qpbσ2n+1,σ2n+2≤ϱqpbσ2n,σ2n+1.
Hence,(17)qpbσ2n+1,σ2n+2≤ϱqpbσ2n,σ2n+1≤ϱ2qpbσ2n−1,σ2n.
Therefore,(18)qpbσ2n+1,σ2n+2≤ϱ2n+1qpbσ0,σ1.
Similarly,(19)qpbσ2n+1,σ2n=qpbHσ2n,Kσ2n−1≤ϱ1s2qpbσ2n,Kσ2n−1α1s2qpbσ2n−1,Hσ2n1−α≤ϱqpbσ2n,σ2nα1s2qpbσ2n−1,σ2n+11−α≤ϱqpbσ2n,σ2n+1α.1s2qpbσ2n−1,σ2n+11−α≤ϱqpbσ2n,σ2n+1α.1s2.sqpbσ2n−1,σ2n+qpbσ2n,σ2n+1−qpbσ2n,σ2n1−α≤ϱqpbσ2n,σ2n+1α.1s2.sqpbqpbσ2n−1,σ2n+qpbσ2n,σ2n+11−α,qpbσ2n,σ2n+11−α≤ϱqpbσ2n−1,σ2n1−α,qpbσ2n,σ2n+1≤ϱqpbσ2n−1,σ2n.
Hence,(20)qpbσ2n,σ2n+1≤ϱqpbσ2n−1,σ2n≤ϱ2qpbσ2n−1,σ2n−2.
This yields that(21)qpbσ2n,σ2n+1≤ϱ2nqpbσ0,σ1.
From equations (18) and (21), we deduce that(22)qpbσn,σn+1≤ϱnqpbσ0,σ1.
To prove sequenceσn is Cauchy, let n,k∈N(23)qpbσn,σn+k≤sqpbσn,σn+1+s2qpbσn+1,σn+2+…+skqpbσn+k−1,σn+k≤sϱn+s2ϱn+1+…+skϱn+k−1qpbσ0,σ1≤sk∑i=nn+k−1ϱiqpbσ0,σ1≤sk∑i=n∞ϱiqpbσ0,σ1.
From (23),(24)qpbσn+m,σn+m+k≤sk∑i=m∞ϱiqpbσn,σn+1,limm⟶∞,n⟶∞qpbσn+m,σn+m+k≤sklimm⟶∞∑i=m∞limn⟶∞ϱiqpbσn,σn+1=0.
Therefore,(25)limn⟶∞qpbσn,σn+k=limm⟶∞,n⟶∞qpbσn+m,σn+m+k=0.
We conclude thatσn is a Cauchy sequence. Since G,qpb is complete, there exists z∈G such that limn⟶∞σn=z. Next, we shall prove that z is a common fixed point of K and H.(26)qpbHz,σ2n+2=qpbHz,Kσ2n+1≤ϱ1s2qpbz,kσ2n+1α1s2qpbσ2n+1,Hz1−α≤ϱqpbz,kσ2n+1αqpbσ2n+1,Hz1−α.
Lettingn⟶∞, we get Hz=z. Similarly,(27)qpbσ2n+1,Kz=qpbHσ2n,Kz≤ϱ1s2qpbσ2n,Kzα1s2qpbz,Hσ2n1−α≤ϱqpbσ2n,Kzαqpbz,Hσ2n1−α.
Lettingn⟶∞, we get Kz=z. Hence, K and H attain a common fixed point.
We can justify this result by illustrating it with an example.Example 2.
LetG=0,1,2,3. Define complete quasi-partial b-metric as qpbσ,υ=σ−υ2+σ, that is:
We define self mappingsH and K on G as H:01230010, K:01230001. Now, consider the value α=4/5 and ϱ=99/100.qpbσ,υ0123001491212526323312743Case 1.
Letσ,υ=1,1. Without loss of generality, we have(28)qpbHσ,Kυ≤ϱ1s2qpbσ,Kυα1s2qpbυ,Hσ1−α,qpbH1,K1=0≤ϱ1s2qpb1,K14/51s2qpb1,H11−4/5.
Dominance of right-hand side of equation (9) is easily visually checked in Figure 2. Thus, the inequality required in equation (9) holds for σ,υ=1,1.Figure 2
0 is the common fixed point ofH and K.Case 2.
Letσ,υ=0,0.(29)qpbHσ,Kυ≤ϱ1s2qpbσ,Kυα1s2qpbυ,Hσ1−α,qpbH0,K0=0≤ϱ1s2qpb0,K04/51s2qpb0,H01−4/5.
Dominance of right-hand side of equation (9) is easily visually checked in Figure 3. Thus, the inequality required in equation (9) holds for σ,υ=0,0.Figure 3
0 is the common fixed point ofH and K.Case 3.
Letσ,υ=3,3.(30)qpbHσ,Kυ≤ϱ1s2qpbσ,Kυα1s2qpbυ,Hσ1−α,qpbH3,K3=1≤ϱ1s2qpb3,K34/51s2qpb3,H31−4/5.
Dominance of right-hand side of equation (9) is easily visually checked in Figure 4. Thus, the inequality required in equation (9) holds for σ,υ=3,3.
Therefore, 0 is the common fixed point ofH and K in the setting of interpolative Chatterjea type contraction. Many more common fixed points can be obtained in similar manner. Hence, a fixed point exists but is not unique.Figure 4
0 is the common fixed point ofH and K.Open problem 1.
LetG,qpb be a complete quasi-partial b-metric space. Let us consider a family of self mappings Kn: G⟶G, n≥1, and s≥1 such that(31)qpbKiσ,Kjυ≤ϱi,j1s2qpbσ,Kiυα1s2qpbυ,Kjσ1−α.
What are the conditions onϱi,jandαi for Kn to have a common fixed point?


## 3. Modified Chatterjea Type Suzuki Contraction

In our next result, we present the existence of interpolative Chatterjea contraction via Suzuki map. First, let us recall w-admissible map.Definition 6.
Let us consider a self mapH defined on G and w:G×G⟶0,∞ be a map where G≠ϕ. H is said to be w-orbital admissible if for all σ∈G, we have(32)wσ,Hσ≥1⟶wHσ,H2σ≥1.
R Condition. If σn is a sequence in G such that wσn,σn+1≥1 for each n and σn⟶σ∈G as n⟶∞, then there exists σnk from σn such that wσnk,σ≥1, for each k.Definition 7.
Let (G, qpb) be a quasi-partial b-metric space. A self map H defined on G satisfies the C-condition if(33)12qpbσ,Hσ≤qpbσ,υ⇒qpbHσ,Hυ≤qpbσ,υ,for each σ, υ∈G.Definition 8 (see [15,22]).
LetΨ be the set of all nondecreasing self mappings ψ on 0,∞ such that(34)∑n=1∞ψnt<∞,for eacht>0.
Notice that forψ∈Ψ, we have ψ0=0 and ψt<t for all t>0.Definition 9.
Let us consider a quasi-partial b-metric space (G,qpb). A map H:G⟶G is said to be a w-ψ-interpolative Chatterjea contraction of Suzuki type if there exist ψ and a self-mapping, w:G×G⟶ [0, ∞) and a real number α∈ [0, 1), such that(35)12qpbσ,Hυ≤qpbσ,υ⇒wσ,υqpbHσ,Hυ≤ψqpbσ,Hυαqpbυ,Hσ1−α,for each σ, υ∈G.Theorem 5.
Let (G, qpb) be a complete quasi-partial b-metric space and H:G⟶G be a w-ψ-interpolative Chatterjea contraction of Suzuki type. Assume that H is w-orbital admissible mapping and w(σ0, Hσ0) ≥ 1 for some σ0∈G. Then, H has a fixed point in G provided that at least one of the following conditions holds:(1)
(G, qpb) is w-regular(2)
H is continuous(3)
ψH2 is continuous and w(σ, Hσ) ≥ 1 where σ∈ Fix ψH2Proof.
Let us consider an initial pointσ0∈G such that w(σ0, Hσ0) ≥ 1 and now construct a sequence σn by using Hnσ0 = σn for each positive integer n. If σn0 = σn0+1 for some n0∈N, we get σn0 = Hσn0 that means we get σn0 is a fixed point of H. Then, we can assume that σn≠σn+1 for each positive integer n. As H is w-orbital admissible, we have the condition w(σ0, Hσ0) = w(σ0, σ1) ≥ 1 implies that w(σ1, Hσ1) = w(σ1, σ2) ≥ 1. In continuation, we have(36)wσn,σn+1≥1.
Hence, choosingσ = σn−1 and ϵ = Hσn−1 in (54), we get(37)12qpbσ,ϵ=12qpbσn−1,Hσn−1=12qpbσn−1,σn≤qpbσn−1,σn,which implies(38)qpbσn,σn+1≤wσn−1,σnqpbHσn−1,Hσn≤ψqpbσn−1,Hσnα.qpbσn,Hσn−11−α=ψqpbσn−1,σn+1α.qpbσn,σn1−α<qpbσn−1,σnα.qpbσn,σn+11−α.
Hence, it follows that(39)qpbσn,σn+1α<qpbσn−1,σnα,or equivalently(40)qpbσn,σn+1<qpbσn−1,σn.
Thus, it follows that the sequenceqpbσn−1,σn is a nonincreasing sequence of positive terms and limn⟶∞qpbσn−1,σn = 1. From the above equations and the nondecreasing nature of function ψ, we obtain(41)qpbσn,σn+1≤ψqpbσn−1,σn≤ψ2qpbσn−2,σn−1≤⋯≤ψnqpbσ0,σ1.
Now, applying triangular inequality, for allj≥ 1, we get(42)qpbσn,σn+j≤sqpbσn,σn+1+s2qpbσn+1,σn+2+⋯+sjqpbσn+j−1,σn+j≤sψnqpbσ0,σ1+s2ψn+1qpbσ0,σ1+⋯+sjψn+j−1qpbσ0,σ1=∑m=n,r=1n+j−1srψmqpbσ0,σ1=Pn+j−1−Pn−1,where Pk=sk∑m=0kψmqpbσ0,σ1. However, the series ∑m=0∞ψmqpbσ0,σ1 is convergent, so there exists a positive real number P such that limk⟶∞Pk = P. Letting n and j⟶∞ in the above inequality, we get(43)qpbσn,σn+j⟶0.
Therefore,σn is a Cauchy sequence, and using the completeness property of qpb space, it follows that there exists w∈G such that(44)limn⟶∞σn=w,and we claim that w is the fixed point of H.
In case that the assumption (1) holds, we havewσn,t≥ 1 and we claim that(45)12qpbσn,Hσn≤qpbσn,w,or12qpbHσn,HHσn≤qpbHσn,w,for every n∈N. Suppose(46)12qpbσn,Hσn>qpbσn,w,12qpbHσn,HHσn>qpbHσn,w.
By triangular inequality inqpb space, we have(47)qpbσn,σn+1=qpbσn,Hσn+qpbw,w≤qpbσn,w+qpbw,Hσn<12qpbσn,Hσn+12qpbHσn,HHσn=12qpbσn,σn+1+12qpbσn+1,σn+2≤12qpbσn,σn+1+12qpbσn,σn+1=qpbσn,σn+1,which is a contradiction. Therefore, for every n∈N, either(48)12qpbσn,Hσn≤qpbσn,w,or12qpbHσn,HHσn≤qpbHσn,w,holds. If the first condition holds, we obtain(49)qpbσn+1,Hw≤wσn,wqpbHσn,Hw≤ψqpbσn,Hσnα.qpbw,Hw1−α=ψqpbσn,Hσn+1α.qpbw,Hw1−α<qpbσn,Hσn+1α.qpbw,Hw1−α.
If the second condition holds, we get(50)qpbσn+2,Hw≤wσn+1,wqpbH2σn,Hw≤ψqpbHσn,H2σnα.qpbw,Hw1−α=ψqpbσn+1,σn+2α.qpbw,Hw1−α<qpbσn+1,σn+2α.qpbw,Hw1−α.
Therefore, lettingn⟶∞, we get that qpbw,Hw=0, that is, w=Hw.
In case that assumption (2) holds, that is, the mappingH is continuous, we get(51)Hw=limn⟶∞Hσn=limn⟶∞Hσn+1=w.
If the last assumption (3) holds, we haveψH2w = ψlimn⟶∞H2σn =  ψlimn⟶∞σn+2=ψw and we show that, Hw=w. Suppose on the contrary, that Hw≠w since(52)12qpbHw,ψH2w=12qpbHw,ψw≤qpbHw,ψw.
By (54), we get(53)qpbw,Hw≤wHw,wqpbH2w,Hw≤ψqpbHw,H2wα.qpbw,Hw1−α=ψqpbσn+1,σn+2α.qpbw,Hw1−α<qpbHw,wα.qpbw,Hw1−α=qpbw,Hw,which is a contradiction. So, w=Hw, i.e., w is the fixed point of the mapping H.


## 4. Application

In this section, we have discussed the existence of the solution for the nonlinear matrix equation by using interpolative Chatterjea mapping in the notion of quasi-partial b-metric space. Consider a nonlinear matrix equation as(54)σ=Q+∑i=1nMi∗HσMi,(55)Q=σ−M1∗HσM1−…−Mn∗HσMn,where ℋn denotes the set of n×n Hermitian matrices and pn denotes the set of n×n positive definite matrices. Here, pn⊆ℋn, Q∈pn, and Mi is n×n matrices. Let us suppose a continuous order preserving map H:pn⟶pn with H0=0.Define a trace norm.tr on the set n that is a complete quasi-partial b-metric space and partially ordered with partial ordering ≺¯, where σ≺¯υ⇒υ≺¯σ.Lemma 3.
Ifσ,υ≽0 are n×n matrices, then(56)0≤trσ,υ≤υtrσ.Lemma 4.
Ifσ,υ≺¯In, then(57)σ<1.Theorem 6.
Let a nonlinear matrix equation (54) and suppose the following conditions hold:(1)
There existsQ∈pn with(58)Q=Q+∑i=1nMi∗HQMi.(2)
For allσ,υ∈pn,(59)σ≤υ⇒∑i=1nMi∗HσMi≤∑i=1nMi∗HυMi.(3)
There existsϱ∈0,1/2s for which ∑i=1nMi∗Mi<ϱIn and ∑i=1nMi∗HQMi>0 such that(60)σ≤υ⇒∑i=1nMi∗HσMi≤∑i=1nMi∗HυMi.(4)
There existsσ,υ∈pn and ϱ∈0,1/2s such that(61)Hσ−Hυtr≤1s2σ−Hυtrα1s2υ−Hυtr1−α.
Then, equation (54) owns a solution in pn⊆ℋn.Proof.
Consider a quasi-partial b-metricqpb:pn×pn⟶ℝ+ as(62)qpbσ,υ=σ−υtr+σtr.
Define a continuous mapH:pn⟶pn as(63)Hσ=Q+∑i=1nMi∗HσMi,for all σ∈pn. Then, the solution of the matrix equation (54) is the fixed point of the map H.
On taking,σ,υ∈pn with σ≺¯υ, then Hσ≺¯Hυ. Thus, for qpbσ,υ>0. Now, (1) ⟶ (4) of Theorem 6 yields the following:(64)qpbHσ,Hυ=Hσ−Hυtr=Hσ−Hυ1=Hσ−Hυ+Hσ=∑i=1nMi∗HσMi−∑Mi∗HυMi+Q+∑i=1nMi∗HσMi,(65)qpbσ,υ=σ−υtr⇒σ−υ1=σ−υ+σ,qpbσ,Hυ=σ−Hυtr⇒σ−Hυ1=σ−Hυ+σ,=σ−Q−∑i=1nMi∗HυMi+σ,(66)qpbυ,Hσ=υ−Hσtr⇒υ−Hσ=υ−Hσ+υ=σ−Q−∑i=1nMi∗HσMi+υ.
From equation (64) ⟶ (66) and (61), we have(67)∑i=1nMi∗MiHσ−Hυ+Q+∑i=1nMi∗HσMi≤ϱ1s2σ−Q−∑i=1nMi∗HυMi+σα1s2σ−Q−∑i=1nMi∗HσMi+υ1−α.
We deduce that(68)Hσ−Hυ≤ϱ1s2σ−Hυ1α1s2υ−Hυ1−α.
Thus,(69)qpbHσ,Hυ≤ϱ1s2qpbσ,Hυα1s2qpbυ,Hσ1−α.
Since∑i=1nMi∗HQMi>0, we conclude that Q≤HQ.Thus, by using Theorem 4, H owns a fixed point in pn.


## 5. Conclusion

The significant contribution of this manuscript is to provide the existence of common fixed points that are nonunique for interpolative Chatterjea contraction mappings in the realm of quasi-partial b-metric space. Further, we have given examples to show that the new results are applicable. In metric fixed point theory, interpolative contraction is a generalization of Kannan-type contraction. Additionally, these proposed contractions can be generalized in other well-known spaces and can give new fixed point results. The uniqueness property of obtained fixed points for these mappings and their application in the study of nonlinear integral equations will be an interesting concept for subsequent work.



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*Source: 2347294-2022-04-13.xml*