# On Somew−Interpolative Contractions of Suzuki-Type Mappings in Quasi-Partial b-Metric Space

Authors: Pragati Gautam; Santosh Kumar; Swapnil Verma; Soumya Gulati
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, our focus is to acquaint with the Suzuki-type mappings to establish some fixed point results using the new w-interpolative approach. We present some results for interpolative contraction operators via the w-admissible maps which satisfy the Kannan, Ćirić–Reich–Rus, and Hardy–Rogers contractions in quasi-partial b-metric space. Further, the outcomes so obtained are affirmed with relevant examples.


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## Body

## 1. Introduction

In the early20th century, Fréchet [1], a French mathematician, initiated the concept of metric space, and due to its efficiency and practicable implementations, the idea has been upgraded, improved, and generalized by many authors. In 1922, Banach [2] discovered a remarkable result, that is, Banach contraction principle, which holds a significant position in the field of nonlinear analysis. Later, Karapinar [3] introduced quasi-partial metric spaces which were followed by the discovery of b−metric spaces in 1993, by Czerwik [4]. Gupta and Gautam [5] generalized quasi-partial metric to quasi-partial b−metric space and proved some fixed point results for such spaces. After all these classical results, Suzuki [6] introduced a new type of mappings which generalized the well-known Banach contraction principle.In 2014, the notion of w-orbital admissible maps was introduced by Popescu [7] which is a refinement of the concept of α-admissible maps of Samet et al. [8].SupposeS is a self-map defined on G and w:G×G⟶0,∞ is a mapping where G is nonempty. The mapping S is said to be w-orbital admissible if for all η∈G, we have(1)wη,Sη≥1⟶wSη,S2η≥1.If the continuity of the involved contractive mappings is removed, we necessarily need (G, qpb) to be w-regular, i.e., if ηn is a sequence in (G, qpb) such that ηn⟶t∈G as n⟶∞ and wηn,ηn+1 ≥ 1 for each n, then we have wηn,t ≥1.We show that the condition of w-regularity holds in quasi-partial b-metric space by using w-admissibility condition. In our earlier work [9], we have shown that w-admissibility holds in quasi-partial b-metric space, i.e., wηn,Sηn≥1; then, as n⟶∞, we get ηn⟶t, and hence we get the condition for w-regularity.Throughout the paper,R+, N, and ϕ stand for the set of positive real numbers, natural numbers, and an empty set, respectively. Let Ψ be the set of all nondecreasing self-mappings ψ on 0,∞ such that ∑r=1∞ψrη<∞ for every η>0. Notice that for Ψ∈ψ, we have ψ0=0 and ψη<η for all η>0 (see [10]).


## 2. Preliminaries

Definition 1 (see [5]).
A functionqpb:G×G⟶R+ is said to be a quasi-partial b-metric on a nonempty set G if it satisfies the properties(1)
qpbη,η=qpbη,ζ=qpbζ,ζ implies η=ζ;(2)
qpbη,η≤qpbζ,η;(3)
qpbη,η≤qpbη,ζ;(4)
qpbη,ζ≤sqpbη,σ+qpbσ,ζ−qpbσ,σ,where s is called the coefficient of G,qpb such that s≥1 for all η,ζ,σ∈G.Definition 2 (see [5]).
SupposeG,qpb is a quasi-partial b-metric space. Then,(1)
A sequenceηn is called a Cauchy sequence if and only if limn,m⟶∞qpbηn,ηm and limn,m⟶∞qpbηm,ηn exist finitely.(2)
A sequenceηn⊂G converges to η∈G if and only if qpbη,η = limn⟶∞qpbη,ηn=limn⟶∞qpbηn,η.(3)
G,qpb is said to be complete if every Cauchy sequence ηn⊂G converges with respect to τqpb to a point η∈G that holds(2)qpbη,η=limn,m⟶∞qpbηn,ηm=limn,m⟶∞qpbηm,ηn.(4)
A mappingf:G⟶G is said to be continuous at η0∈G if for every ε>0, there exists δ>0 such that fBη0,δ⊂Bfη0,ε.Definition 3 (see [9]).
SupposeG,qpb is a quasi-partial b-metric space. A self-map S:G⟶G is known as a w-interpolative Ćirić–Reich–Rus contraction if there exist λ∈ψ and a map w:G×G⟶0,∞ with real numbers α,β>0, satisfying α+β<1, that holds(3)wη,ζqpbSη,Sζ≤λqpbη,ζβ.qpbη,Sηα.qpbζ,Sζ1−α−β,for all η, ζ∈G.Definition 4 (see [11]).
SupposeG,qpb is a quasi-partial b-metric. Define a self-mapping S:G⟶G and a map w:G×G⟶0,∞ where λ∈ψ that holds(4)wη,ζqpbSη,Sζ≤λqpbη,ζα·qpbη,Sζβ·qpbζ,Sζγ·12sqpbη,Sζ+qpbζ,Sη1−α−β−γ,for all η, ζ∈G and real numbers α, β, γ>0 that satisfy the condition α+β+γ<1. Such a mapping is known as w-interpolative Hardy–Rogers-type contraction.Definition 5 (see [10]).
A mappingS is said to satisfy C-condition on (G, qpb), if it satisfies(5)12qpbη,Sη≤qpbη,ζ⇒qpbSη,Sζ≤qpbη,ζ,for all η, ζ∈G.
Throughout the paper,qpb and Cqpb denote the quasi-partial b-metric space and complete quasi-partial b-metric space, respectively. One can see for more related point results in [12–15] and the references therein.


## 3. Main Results

We now define the main results for Suzuki-type mappings using the notion of w-interpolation (see16, 17) and the fact that the condition of w-regularity holds in quasi-partial b−metric space (see 18–20]).Definition 6.
LetG,qpb be a quasi-partial b-metric space and there exists a self-map w:G×G⟶ [0, ∞) with a real number α∈ [0, 1). A self-map S:G⟶G is said to be a w-ψ-interpolative Kannan contraction of Suzuki type if there exist ψ∈Ψ that satisfies(6)12qpbη,Sη≤qpbη,ζ⇒wη,ζqpbSη,Sζ≤ψqpbη,Sηαqpbζ,Sζ1−α,for all η, ζ∈G.Theorem 1.
SupposeG,qpb is a Cqpb and S:G⟶G is a w-ψ-interpolative Kannan contraction of Suzuki type. Let S be a w-orbital admissible map and wη0,Sη0≥ 1 for some η0∈G. Then, S possesses a fixed point in G if any of the following conditions hold:(1)
(G,qpb) is w-regular.(2)
S is continuous.(3)
ψS2 is continuous and wη,Sη≥ 1 when η∈FixS2.Proof.
Letη0∈G with the condition wη0,Sη0≥ 1 and ηn be the sequence such that Snη0 = ηn for each positive integer n. For some η0∈N, we have ηn0=ηn0+1. Hence, we get ηn0 = Sηn0, so ηn0 is a fixed point of S. Hence, the proof is complete.
On the contrary, takeηn≠ηn+1 for every positive integer n. As S is w-orbital admissible, we have the condition w(η0, S η0) = w(η0, η1) ≥ 1 which implies that w(η1, Sη1) = w(η1, η2) ≥ 1. Proceeding in a similar way, we get(7)wηn,ηn+1≥1.
Hence, choosingη = ηn−1 and ζ = Sηn−1 in (6) gives(8)12qpbη,ζ=12qpbηn−1,Sηn−1=12qpbηn−1,ηn≤qpbηn−1,ηn,which implies(9)qpbηn,ηn+1≤wηn−1,ηnqpbSηn−1,Sηn≤ψqpbηn−1,Sηn−1α.qpbηn,Sηn1−α=ψqpbηn−1,ηnα.qpbηn,ηn+11−α<qpbηn−1,ηnα.qpbηn,ηn+11−α.
Hence, we have(10)qpbηn,ηn+1α<qpbηn−1,ηnα,which equivalently can be written as(11)qpbηn,ηn+1<qpbηn−1,ηn.
Thus, we get thatqpbηn−1,ηn is a nonincreasing sequence of positive terms, so there exists l≥ 0 such that limn⟶∞qpbηn−1,ηn = l. On the other hand, from the above equations and the nondecreasing nature of function ψ, we obtain(12)qpbηn,ηn+1≤ψqpbηn−1,ηn≤ψ2qpbηn−2,ηn−1≤…≤ψnqpbη0,η1.
By triangular inequality, for allj≥1, we get(13)qpbηn,ηn+j≤sqpbηn,ηn+1+s2qpbηn+1,ηn+2+⋯+sjqpbηn+j−1,ηn+j≤sψnqpbη0,η1+s2ψn+1qpbη0,η1+⋯+sjψn+j−1qpbη0,η1=∑m=n,r=1n+j−1srψmqpbη0,η1=Pn+j−1−Pn−1,where Pk=sk∑m=0kψmqpbη0,η1. But, the series ∑m=0∞ψmqpbη0,η1 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(14)qpbηn,ηn+j⟶0.
Hence,ηn is a Cauchy sequence and using the completeness property of qpb space, it shows that there exists t∈G such that(15)limn⟶∞ηn=t.
Also, we claim thatS possesses a fixed point as t.
In the case when assumption (1) holds true, we havewηn,t≥ 1 and we claim that(16)12qpbηn,Sηn≤qpbηn,tor12qpbSηn,SSηn≤qpbSηn,t,for every n∈N. On the contrary, if the above condition is not true, then by triangular inequality in qpb space, we have(17)qpbηn,ηn+1=qpbηn,Sηn+qpbt,t≤qpbηn,t+qpbt,Sηn<12qpbηn,Sηn+12qpbSηn,SSηn=12qpbηn,ηn+1+12qpbηn+1,ηn+2≤12qpbηn,ηn+1+12qpbηn,ηn+1=qpbηn,ηn+1,which is a contradiction, and hence our claim is proved. If the first condition holds, we obtain(18)qpbηn+1,St≤wηn,tqpbSηn,St≤ψqpbηn,Sηnα.qpbt,St1−α=ψqpbηn,Sηn+1α.qpbt,St1−α<qpbηn,Sηn+1α.qpbt,St1−α.
If the second condition holds, we get(19)qpbηn+2,St≤wηn+1,tqpbS2ηn,St≤ψqpbSηn,S2ηnα.qpbt,St1−α=ψqpbηn+1,ηn+2α.qpbt,St1−α<qpbηn+1,ηn+2α.qpbt,St1−α.
Therefore, lettingn⟶∞, we get qpbt,St=0, that is, t=St.
In the case when assumption (2) holds, we have that the mappingS is continuous, so we get(20)St=limn⟶∞Sηn=limn⟶∞Sηn+1=t.
In the case when assumption (3) holds, we haveψS2t = ψlimn⟶∞S2ηn = ψlimn⟶∞ηn+2=ψt and we prove that St=t. On the contrary, take St≠t; then,(21)12qpbSt,ψS2t=12qpbSt,ψt≤qpbSt,ψt.
By (6), we get(22)qpbt,St≤wSt,tqpbS2t,St≤ψqpbSt,S2tα.qpbt,St1−α=ψqpbηn+1,ηn+2α.qpbt,St1−α<qpbSt,tα.qpbt,St1−α=qpbt,St.
Hence, it is a contradiction. Thus,t=St, that is, t is a fixed point of the mapping S.Example 1.
LetG = [0, π/4] and define qpb:G×G⟶ [0, ∞) such that(23)qpbη,ζ=Sinη+Sinζ.
Define a self-mappingS:G⟶G as(24)Sη=π/9,η∈0,π/8,η,η∈π/8,π/4.
Also, definew:G×G⟶ [0, ∞) such that(25)wη,ζ=2,η=π/9andζ=π/5,0,otherwise.
Chooseα=1/3 and define the function ψ∈Ψ as ψη=2η/3. The only case we need to verify is when η = π/9 and ζ = π/5; as for the remaining cases, we have wη,ζ=0, which clearly implies that inequality (6) holds. So, when η = π/9 and ζ = π/5, we get(26)12qpbπ9,Sπ9=12qpbπ9,π9=0.342≤0.929=qpbπ9,π5,which implies(27)wπ9,π5qpbπ9,π5≤ψqpbπ9,π91/3qpbπ5,π52/3.
Hence, all the assumptions of Theorem1 are satisfied, which follows that the mapping S owns a fixed point, that is, η=π/9, as shown in Figure 1.Figure 1
The graphical surface represents a 3-D view of the function,qpbη,ζ=Sinη+Sinζ. Clearly, the fixed point of the map S is π/9.Corollary 1.
LetG,qpb be a C qpb and S be a self-map on G, satisfying(28)12qpbη,Sη≤qpbη,ζ⇒qpbSη,Sζ≤ψqpbη,Sηαqpbζ,Sζ1−α,for all η, ζ∈G, where α∈α∈ [0, 1). Then, S owns a fixed point in G.Proof.
For the proof, takewη,ζ  = 1 in Theorem 1.Corollary 2.
LetG,qpb be a C qpb and S be a self-map on G, satisfying(29)12qpbη,Sη≤qpbη,ζ⇒qpbSη,Sζ≤gqpbη,Sηαqpbζ,Sζ1−α,for all η, ζ∈G, where α∈ [0, 1). Then, S owns a fixed point in G.Proof.
For the proof, takeψη=ηg, with g∈ [0, 1) and η> 0 in Corollary 1.Definition 7.
SupposeG,qpb is a quasi-partial b-metric space. Define a self-mapping S:G⟶G such that there exist ψ∈Ψ satisfying(30)12qpbη,Sη≤qpbη,ζ,wη,ζqpbSη,Sζ≤ψqpbη,ζβqpbη,Sηα.qpbζ,Sζ1−α−β,for all η, ζ∈G and real numbers α, β> 0 that satisfy α + β< 1. Such a mapping is called w-ψ-interpolative Ćirić–Reich–Rus contraction of Suzuki type (see 21–23).Theorem 2.
SupposeG,qpb is a C qpb and S:G⟶G is a w-ψ-interpolative Ćirić–Reich–Rus contraction of Suzuki type. Let S be a w-orbital admissible map and w(η0, Sη0) ≥ 1 for some η0∈G. Then, S possesses a fixed point in G if any of the conditions hold:(1)
(G,qpb) is w-regular.(2)
S is continuous.(3)
ψS2 is continuous and wSη,η≥ 1 when η∈FixS2.Proof.
Letη0∈G with the condition wη0,Sη0≥ 1 and ηn be the sequence such that Snη0 = ηn for each positive integer n. Assume that for some η0∈N, we have the condition ηn0=ηn0+1. Hence, we get ηn0 = Sηn0, which implies ηn0 is the fixed point of S. So, the proof is complete.
On the contrary, takeηn≠ηn+1 for each positive integer n. Since S is w-orbital admissible, we have w(η0, S η0) = w(η0, η1) ≥ 1, which implies that w(η1, S η1) = w(η1, η2) ≥ 1. Proceeding similarly,(31)wηn,ηn+1≥1.
Hence, choosingη = ηn−1 and ζ = Sηn−1 in (30), we get(32)12qpbη,ζ=12qpbηn−1,Sηn−1=12qpbηn−1,ηn≤qpbηn−1,ηn,which implies(33)qpbηn,ηn+1≤wηn−1,ηnqpbSηn−1,Sηn≤ψqpbηn−1,ηnβ.qpbηn−1,Sηn−1α.qpbηn,Sηn1−α−β=ψqpbηn−1,ηnβ.qpbηn−1,ηnα.qpbηn,ηn+11−α−β.
Then, usingψη<η for every η> 0, we get(34)qpbηn,ηn+1≤qpbηn−1,ηnβ+α.qpbηn,ηn+11−β−α,which equivalently can be written as(35)qpbηn,ηn+1α+β<qpbηn−1,ηnα+β.
So, we get(36)qpbηn,ηn+1<qpbηn−1,ηn.
Hence, it shows thatqpbηn−1,ηn is a decreasing sequence. Eventually, we have(37)qpbηn,ηn+1≤ψqpbηn−1,ηn.
In a similar way, we get(38)qpbηn,ηn+1≤ψnqpbη0,η1.
Sinceηn is a fundamental sequence, applying triangular inequality, we get(39)qpbηn,ηn+l≤sqpbηn,ηn+1+s2qpbηn+1,ηn+2+⋯+slqpbηn+l−1,ηn+l≤sψnqpbη0,η1+s2ψn+1qpbη0,η1+⋯+slψn+l−1qpbη0,η1=∑k=n,r=1∞srψkqpbη0,η1.
Takingn⟶∞, we deduce that ηn is a fundamental sequence in qpb and by the completeness property of qpb, there exists t∈G satisfying(40)limn⟶∞qpbηn,t=0.
If assumption (1) holds, then we havewηn,t≥ 1 and we claim that(41)12qpbηn,Sηn≤qpbηn,tor12qpbSηn,SSηn≤qpbSηn,t,for every n∈N. On the contrary, suppose the above inequality does not hold; then, by triangular inequality in qpb, we have(42)qpbηn,ηn+1=qpbηn,Sηn+qpbt,t≤qpbηn,t+qpbt,Sηn<12qpbηn,Sηn+12qpbSηn,SSηn=12qpbηn,ηn+1+12qpbηn+1,ηn+2≤12qpbηn,ηn+1+12qpbηn,ηn+1=qpbηn,ηn+1.
Hence, the contradiction occurs. Therefore, for everyn∈ N, our claim holds. If the first condition holds, we obtain(43)qpbηn+1,St≤wηn,tqpbSηn,St≤ψqpbηn,tβ.qpbηn,Sηnα.qpbt,St1−α−β=ψqpbηn,tβ.qpbηn,Sηn+1α.qpbt,St1−α−β<qpbηn,tβ.qpbηn,Sηn+1α.qpbt,St1−α−β.
If the second condition holds true, clearlyt is the fixed point of S in a similar manner. Furthermore, if the w-regular condition is removed and S is a continuous map, we get a fixed point in G because(44)t=limn⟶∞ηn+1=limn⟶∞Sηn=Slimn⟶∞ηn=St.
Finally, if the last condition holds, i.e.,ψS2 is continuous, we easily obtain ψS2=ψt. Suppose on the contrary that St≠t, since wSη,η≤ 1 for any η∈ Fix S2 and(45)12qpbSt,ψS2t=12qpbSt,ψt≤qpbSt,ψt.
We have(46)qpbt,St=qpbS2t,St≤wSt,tqpbS2t,St≤ψqpbSt,tα.qpbSt,S2tβ.qpbt,St1−α−β<qpbSt,tα.qpbSt,tβ.qpbt,St1−α−β=qpbt,St,which is a contradiction. So, t=St, that is, the mapping S owns a fixed point t.Example 2.
SupposeG = 0,1/4,1/3,1/2,1 and define qpb:G×G⟶ [0, ∞) such that(47)qpbη,ζ=η+ζ.
Let the transformationS:G⟶G maps as follows:(48)S0=S14=S1=12,S13=13,S12=0.
Also, definew:G×G⟶ [0, ∞) such that(49)wη,ζ=0.2,η,ζ=0,1,0,13,1,0,0,otherwise.
Chooseα=β=1/3 and define the function ψ∈Ψ as ψη=η/2. We need to check when wη,ζ=1. So, the following cases occur.Case 1.
Whenη,ζ = (0, 1),(50)12qpb0,S0=12qpb0,12=14≤1=qpb0,1,which implies(51)w0,1qpb12,12≤ψqpb0,11/3qpb0,121/3.qpb1,121/3.Case 2.
Whenη,ζ = (0, 1/3),(52)12qpb0,S0=12qpb0,12=14≤13=qpb0,13,which implies(53)w0,13qpb12,13≤ψqpb0,131/3qpb0,121/3.qpb13,131/3.Case 3.
Whenη,ζ = (1, 0),(54)12qpb1,S1=12qpb1,12=34≤1=qpb1,0,which implies(55)w1,0qpb12,12≤ψqpb1,01/3qpb1,131/2.qpb0,121/3.
Hence, all the assumptions of Theorem2 are satisfied, and it follows that the mapping S owns a fixed point, that is, η=1/3, as shown in Figure 2.Figure 2
The 3-D plane in yellow represents the quasi-partial b-metric space defined by the function,qpbη,ζ=η+ζ. Clearly, the fixed point of S is 1/3.Definition 8.
LetG,qpb be a quasi-partial b-metric space. Define a self-mapping S:G⟶G with ψ∈Ψ satisfying(56)12qpbη,Sη≤qpbη,ζ,qpbSη,Sζ≤ψqpbη,ζβmiddot;qpbη,Sηα.qpbζ,Sζ1−α−β,for all η, ζ∈G and α, β> 0 with the condition α + β< 1. Such a mapping is called a ψ-interpolative Ćirić–Reich–Rus contraction of Suzuki type.Corollary 3.
Suppose (G,qpb) is a C qpb and S is a ψ-interpolative Ćirić–Reich–Rus contraction of Suzuki type. Then, S owns a fixed point in G.Proof.
For the proof, takewη,ζ = 1 in Theorem 2.Corollary 4.
Suppose (G, qpb) is a C qpb and S is an interpolative Ćirić–Reich–Rus contraction of Suzuki type if there exist g ∈ [0, 1) and positive reals α, β> 0, with α + β< 1 such that(57)12qpbη,Sη≤qpbη,ζ⇒qpbSη,Sζ≤gqpbη,ζβqpbη,Sηαqpbζ,Sζ1−α−β,for all η, ζ∈ G. Then, S possesses a fixed point in G.Proof.
In Theorem2, it is sufficient to put ψη=gη, for all η> 0 and g∈0,1, for the proof.Definition 9.
LetG,qpb be a qpb space and define a map w:G×G⟶ [0, ∞). A mapping S:G⟶G is w-ψ-interpolative Hardy–Rogers contraction of Suzuki type if there exists ψ∈Ψ with real numbers α, β, γ> 0, holding α + β + γ< 1 such that(58)12qpbη,Sη≤qpbη,ζ,wη,ζqpbSη,Sζ≤ψqpbη,ζαmiddot;qpbη,Sηβ.qpbζ,Sζγ.12vqpbζ,Sη+qpbη,Sζ1−α−β−γ,for all η, ζ∈G and v≥1 (see [24]).Theorem 3.
Suppose (G,qpb) is a C qpb and S:G⟶G is a w−ψ-interpolative Hardy–Rogers contraction of Suzuki type. Let S be a w-orbital admissible mapping and w(η0, S η0) ≥ 1 for some η0∈ G. Then, S possesses a fixed point in G if any of the conditions hold:(1)
(G,qpb) is w-regular.(2)
S is continuous.(3)
ψS2 is continuous and w(Sη, η) ≥ 1 when η∈ Fix S2.Proof.
Letη0∈G with the condition w(η0, Sη0) ≥ 1 and ηn be the sequence such that Snη0 = ηn for each positive integer n. Assume that for some η0∈N, we have the condition ηn0=ηn0+1. Hence, we get ηn0 = Sηn0 which implies ηn0 is a unique fixed point of S. Hence, the proof is complete.
Now, considerηn≠ηn+1 for each positive integer n. As S is w-orbital admissible, we have the condition w(η0, Sη0) = w(η0, η1) ≥ 1 which implies that w(η1, Sη1) = w(η1, η2) ≥ 1. Proceeding in a similar way, we get(59)wηn,ηn+1≥1.
Choosingη =  ηn−1 and ζ = Sηn−1 in (58), we get(60)12qpbη,ζ=12qpbηn−1,Sηn−1=12qpbηn−1,ηn≤qpbηn−1,ηn,which implies(61)qpbηn,ηn+1≤wηn−1,ηnqpbSηn−1,Sηn≤ψqpbηn−1,ηnα.qpbηn−1,Sηn−1β.qpbηn,Sηnγ.12vqpbηn,Sηn−1+qpbηn−1,Sηn1−α−β−γ=ψqpbηn−1,ηnα.qpbηn−1,ηnβ.qpbηn,ηn+1γ.12vqpbηn,ηn+qpbηn−1,ηn+11−α−β−γ.
Then, usingψη<η for every η> 0, we get(62)qpbηn,ηn+11−γ≤qpbηn−1,ηnβ+α.12vqpbηn,ηn+qpbηn−1,ηn+11−α−β−γ,which is equivalent to(63)qpbηn,ηn+1<qpbηn−1,ηn.
Hence,qpbηn−1,ηn is a decreasing sequence. Eventually, we have(64)qpbηn,ηn+1≤ψqpbηn−1,ηn.
In a similar way, we get(65)qpbηn,ηn+1≤ψnqpbη0,η1.
Sinceηn is a fundamental sequence, applying triangular inequality,(66)qpbηn,ηn+l≤sqpbηn,ηn+1+s2qpbηn+1,ηn+2+⋯+slqpbηn+l−1,ηn+l≤sψnqpbη0,η1+s2ψn+1qpbη0,η1+⋯+slψn+l−1qpbη0,η1=∑k=n,r=1∞srψkqpbη0,η1.
Takingn⟶∞, we deduce that ηn is a fundamental sequence in qpb space and by the completeness property of qpb, there exists t∈G satisfying(67)limn⟶∞qpbηn,t=0.
Now we show thatt is the fixed point of S. If assumption (1) holds true, then we have wηn,t≥ 1 and we claim that(68)12qpbηn,Sηn≤qpbηn,tor12qpbSηn,SSηn≤qpbSηn,t,for every n∈N. Suppose the above condition does not hold; then, by triangular inequality in qpb, we have(69)qpbηn,ηn+1=qpbηn,Sηn+qpbt,t≤qpbηn,t+qpbt,Sηn<12qpbηn,Sηn+12qpbSηn,SSηn=12qpbηn,ηn+1+12qpbηn+1,ηn+2≤12qpbηn,ηn+1+12qpbηn,ηn+1=qpbηn,ηn+1,which is a contradiction. Therefore, for every n∈ N, either(70)12qpbηn,Sηn≤qpbηn,tor12qpbSηn,SSηn≤qpbSηn,tholds. If the first condition holds, we obtain(71)qpbηn+1,St≤wηn,tqpbSηn,St≤ψqpbηn,tα.qpbηn,Sηnβ.qpbt,Stγ.12vqpbηn,ηn+qpbt,St1−α−β−γ=ψqpbηn,tα.qpbηn,Sηnβ.qpbt,Stγ.12vqpbηn,ηn+qpbt,St1−α−β−γ<qpbηn,tα.qpbηn,ηn+1β.qpbt,Stγ.12vqpbt,ηn+1+qpbηn,St1−α−β−γ.
If assumption (2) holds, we get thatt is the fixed point of S in a similar manner. Furthermore, if the w-regular condition is removed and S is continuous, then we get that S owns a unique fixed point in G because(72)t=limn⟶∞ηn+1=limn⟶∞Sηn=Slimn⟶∞ηn=St.
Finally, if the last condition holds, i.e.,ψS2 is continuous, we easily obtain ψS2=ψt. Suppose on the contrary that St≠t, since wSη,η≤ 1 for any η∈ Fix S2 and(73)12qpbSt,ψS2t=12qpbSt,ψt≤qpbSt,ψt.
We have(74)qpbt,St=qpbS2t,St≤wSt,tqpbS2t,St≤ψqpbSt,tα.qpbSt,S2tβ.qpbt,Stγ.12vqpbt,St+qpbt,St1−α−β−γ<qpbSt,tα.qpbSt,tβ.qpbt,Stγ.12qpbt,St+qpbt,St1−α−β−γ=qpbt,St,which is a contradiction. So, t=St, which implies that t is a fixed point of the map S.Definition 10.
LetG,qpb be a quasi-partial b-metric space. A mapping S:G⟶G is said to be a ψ-interpolative Hardy–Rogers contraction of Suzuki type if there exist ψ and α, β, γ> 0, with the condition α + β + γ< 1 such that(75)12qpbη,Sη≤qpbη,ζ,qpbSη,Sζ≤ψqpbη,ζαqpbη,Sηβ.qpbζ,Sζγ.12vqpbζ,Sη+qpbη,Sζ1−α−β−γ,for all η, ζ∈G.Corollary 5.
Let (G,qpb) be a C qpb and S be a ψ-interpolative Hardy–Rogers contraction of Suzuki type. Then, the mapping S possesses a fixed point in G.Proof.
For the proof, takewη,ζ = 1 in Theorem 3.Example 3.
LetG  = [0, 3] and define qpb:G×G⟶ [0, ∞) such that(76)qpbη,ζ=η+ζ+η.
Let the mappingS:G⟶G be defined as(77)Sη=2,η∈0,123,η∈1,2,η,η∈2,3.
Also, definew:G×G⟶ [0, ∞) such that(78)wη,ζ=0.1,η=0andζ=3,0,otherwise.
Chooseα=β=γ=1/4 and define the function ψ∈Ψ as ψη=η/3. For η, ζ∈ [0, 2), we have wη,ζ=0, which clearly implies that inequality (58) holds. As per the definition of function w, the only case left is when we have η = 0 and ζ = 3 as wη,ζ=1, so(79)12qpb0,S0=12qpb0,2=1≤3=qpb0,3,which implies(80)w0,3qpb2,3≤ψqpb0,31/4qpb0,21/4qpb3,31/412vqpb3,2+qpb0,31/4,where we assume v=1. Hence, the assumptions of Theorem 3 are satisfied, so the mapping S owns a fixed point, that is, η=2, as shown in Figure 3.Figure 3
The graphical surface clearly shows that 2 is the fixed point of the mapS.


## 4. Conclusion and Future Aspects

The paper propounds the idea of using interpolation in noteworthy Suzuki-type mappings in the quasi-partial b-metric space. The incentive behind the paper was to introduce new concepts on completeness ofw-ψ-interpolative Kannan, Ćirić–Reich–Rus, and Hardy–Rogers contractions of Suzuki-type mappings in quasi-partial b-metric space. Further, some fixed point results are obtained and are validated by illustrative examples. Interpolation is a noble concept which can be utilized to obtain different interpolative contraction of Suzuki-type mappings in other well-known spaces in the future. We are certain that the paper is a significant improvement of the known results in the existing fixed point literature.



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