# New Results on Perov-Interpolative Contractions of Suzuki Type Mappings

Authors: Erdal Karapınar; Andreea Fulga; Seher Sultan Yesilkaya
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we introduce some common fixed point theorems for interpolative contraction operators using Perov operator which satisfy Suzuki type mappings. Further, some results are given. These results generalize several new results present in the literature.


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## Body

## 1. Introduction

Banach [1] introduced the Banach contraction principle that generalized in various wide directions by many researchers. One of the generalizations was supposed by Kannan [2] in 1968 and later with other researchers such as C′iric′ Reich Rus. In 2018, Karapınar [3] adopted the interpolative approach to define the generalized Kannan type contraction on a complete metric space and proved the following.A mappingF:P⟶P on P,d a complete metric space such that
(1)dFx,Fz≤kdx,Fxα·dz,Fz1−α,where k∈0,1 and α∈0,1, for each x,z∈P\FixF. Then, F has a unique fixed point in P. Afterward, this concept has been extended in different aspects for example [4–14] and also see e.g. [15–19].On the other hand, Perov [20, 21] gave a characterization of Banach contraction principle in the framework vector-valued metric space.For a nonempty setP, a function d:P×P⟶ℝk is called a vector-valued metric on P if the followings are fulfilled:
(1)
dx,z≥0 for all x,z∈P; if dx,z=0, then x=z(2)
dx,z=dz,x for all x,z∈P(3)
dx,z≤dx,t+dt,z for all x,z∈P(4)
where0≔0,0,⋯,0⏟k−times. We mention that, for x,y∈ℝk, that is, x=xii=1k and y=yii=1k(2)x≤y⇔xi≤yiforeachi∈1,2,3,⋯,k.The notationsMmmℝ (respectively, Mmmℝ0+) denote the collection of all square matrices of real numbers (respectively, nonnegative real numbers) where ℝ (respectively, ℝ0+≔0,∞) is the set of real numbers (respectively, nonnegative real numbers). Furthermore, ℂ denotes the complex numbers, as usual.A matrixA∈Mmmℝ converges to zero if and only if its spectral radius is strictly less than 1, that is, ρA<1, see, e.g., [22]. It is equivalent to saying that all the eigenvalues of A are in the open unit disc, that is, λ<1, for every λ∈ℂ with detA−λI=0, where I denotes the unit matrix of Mmmℝ.Theorem 1 (see, e.g., [22, 23]).
LetA∈Mmmℝ0+. Then, the following assertions are equivalent:
(i)
A converges to zero(3)ρA<1;(ii)
An⟶0 as n⟶∞(iii)
The matrixI−A is nonsingular and(4)I−A−1=I+A+⋯+An+⋯;(iv)
The matrixI−A is nonsingular and I−A−1 has nonnegative elements(v)
Anq and qAn are convergent towards zero as n⟶∞, for each q∈ℝmNote also that ifA,B∈Mmmℝ0+ with A≤B (in the component-wise meaning), then, ρB<1 implies ρA<1.Theorem 2 (Perov Cauchy, perov1966certain).
LetP,d be a complete vector-valued metric space and the operator F:P⟶P with the property that there exists a matrix A∈Mmmℝ0+ convergent towards zero such that
(5)dFx,Fz≤Adx,z,forallx,z∈P.
Then,F possesses a unique fixed point.Ali et al. [24] defined Λ admissible that a generalized of α-admissible given by Samet et al. [25].Definition 3 (see [24]).
LetP≠∅, Λ:P×P⟶Mmmℝ+. A mapping F:P⟶P
is said to beΛ-admissible if
(6)Λx,z≥IimpliesΛFx,Fz≥I,forallx,z∈P,where I is the m×m identity matrix and the inequality between matrices mans entrywise inequality.We define some related toΛ-admissible the following concept of admissibility using the above definition and given by some authors [25–30].Definition 4.
LetP≠∅, Λ:P×P⟶Mmmℝ+, and F:P⟶P be a mapping. We say that F is an Λ-orbital admissible mapping if
(7)Λx,Fx≥IimpliesΛFx,F2x≥I,forallx,z∈P.
Moreover, anΛ-orbital admissible mapping F is said to be triangular Λ-orbital admissible if for all x,z∈P, we have
(8)Λx,z≥IandΛz,Fz≥I,impliesΛx,Fz≥I.Definition 5.
For a nonempty setP, let F,G:P⟶P and Λ:P×P⟶Mmmℝ+ be mappings. We say that F,G is a generalized Λ-admissible pair if for all x,z∈P, we have
(9)Λx,z≥I⇒ΛFx,Gz≥IandΛGz,Fx≥I.Lemma 6.
LetP≠∅ and F:P⟶P be a triangular Λ-admissible map. Suppose that there exists x0∈P such that Λx0,Fx0≥I. Identify a sequence xn using xn+1=Fxn. Thus, we have Λxn,xm≥I for all m,n∈ℕ∪0 with n<m.Lemma 7.
LetP≠∅ and F,G:P⟶P be a triangular Λ-admissible mapping. Assume that there exists x0∈P such that Λx0,Fx0≥I. Identify sequence x2r+1=Fx2r, and x2r+2=Gx2r+1, where r=0,1,2,⋯. So, we have Λxn,xm≥I for all m,n∈ℕ∪0 with n<m.Recently, one of the interesting generalizations was given by Suzuki [31, 32] which characterizes the completeness of underlying metric spaces. Suzuki introduced and generalized versions of Banach’s and Edelstein’s basic results. In addition, Popescu [33] has modified the nonexpansiveness situation with the weaker C-condition presented by Suzuki. As stated, the existence of fixed points of maps satisfying the C-condition has been extensively studied; see [34–38]. Karapnar [39] investigated the definition of a nonexpansive mapping satisfying the C-condition.Definition 8.
A mappingF on a metric space P,d satisfies the C-condition if
(10)12dx,Fx≤dx,z⇒dFx,Fz≤dx,z,for each x,z∈P.Theorem 9.
LetP,d be a compact metric space and F:P⟶P be a mapping satisfying condition C for all x,z∈P. Then, F has a unique fixed point.


## 2. Main Results

For the rest of the paper, we use the following notation:PF=P\FixF, where FixF=x∈P∣Fx=x.Definition 10.
LetP,d be a complete vector-valued metric space and Λ:P×P⟶Mmmℝ+, F,G:P⟶P be mappings. We say that F,G forms a pair of Perov-interpolative C′iric′-Reich-Rus contractions of Suzuki type, if there exist A,B∈Mmmℝ+ converges towards zero, (where B=Aq, q>1) such that
(11)12mindx,Fx,dz,Gz≤dx,z⇒Λx,zdFx,Gz≤Adx,zβ·dx,Fxα·dz,Gz1−α−β,
for eachx,z∈PF×PG, where β≥0, α>0 are such that β+α<1.Theorem 11.
LetP,d be a complete vector-valued metric space and F,G:P⟶P be two mappings such that F,G is a pair of Perov-interpolative C′iric′-Reich-Rus contractions of Suzuki type. Assume that
(i)
F,G is a generalized Λ-admissible pair(ii)
There existsx0∈P such that Λx0,Fx0≥I and ΛFx0,x0≥I(iii)
F and G are continuous mappings
Then,F and G have a common fixed point.Proof.
Letx0∈P such that Λx0,Fx0≥I. We define the sequence xr in P as following
(12)x2r+1=Fx2randx2r+2=Gx2r+1,
for everyr∈ℕ. From (i) and (ii), Λx0,Fx0≥I and ΛFx0,x0≥I, and then Λx1,x2=ΛFx0,Gx1≥I and Λx2,x1=ΛGx1,Fx0≥I. Similarly, we get Λx2,x3=ΛFx1,Gx2≥I and Λx3,x2=ΛGx2,Fx1≥I. Repeating this process, we write
(13)Λxm,xm+1≥IandΛxm+1,xm≥I,
for everym∈ℕ.
On the other hand, we have(14)12mindx2r,Fx2r,dx2r+1,Gx2r+1=12mindx2r,x2r+1,dx2r+1,x2r+2≤dx2r,x2r+1.
So, since the mappingsF,G:P⟶P forms a pair of Perov-interpolative contractions of Suzuki type, we get
(15)dx2r+1,x2r+2=IdFx2r,Gx2r+1≤Λx2r,x2r+1dFx2r,Gx2r+1≤Adx2r,x2r+1β·dx2r,Fx2rα·dx2r+1,Gx2r+11−α−β=Adx2r,x2r+1β+α·dx2r+1,x2r+21−α−β.
Therefore, it follows that(16)dx2r+1,x2r+2α+β≤Adx2r,x2r+1β+α,or equivalent
(17)dx2r+1,x2r+2≤Aqdx2r,x2r+1,
whereq=1/β+α>1. Then,
(18)dx2r+1,x2r+2≤Bdx2r,x2r+1whereAq=B.
Lettingx=x2r and z=x2r−1, since
(19)12mindx2r,Fx2r,dx2r−1,Gx2r−1=12mindx2r,x2r+1,dx2r−1,x2r≤dx2r,x2r−1,similarly, we get
(20)dx2r,x2r+1≤Bdx2r−1,x2r.
Thus, combining (18) and (20), we have that
(21)dx2r+1,x2r+2≤B2rdx1,x2,(22)dx2r,x2r+1≤B2rdx0,x1.
Take into account (21) and (22), we obtain that for each m∈ℕ(23)dxm,xm+1≤Bmwx,
wherewx=maxdx0,x1,dx1,x2. By this way, using triangular inequality and (23), for p≥0, we get
(24)dxm,xm+p≤dxm,xm+1+dxm+1,xm+2+⋯+dxm+p−1,xm+p=∑m=lm+p−1dxi,xi+1≤Bm∑i=0∞Biwx=BmI+B+⋯+Bm+⋯wx.
BecauseB is convergent to zero, we attain that I−B is nonsingular and
(25)I−B−1=I+B+⋯+Bm+⋯.
Therefore,(26)dxm,xm+pBmI−B−1wx,(27)dxm,xm+p⟶0asm⟶∞.
So, the sequencexm is a fundamental (Cauchy), and using the completeness of the space P,d, there exists t∈P such that
(28)limm⟶∞dxm,t=0.
We claim that the pointt is a common fixed point of F and G. If (iii.) is provide, that is, the mapping F and G are continuous, we have
(29)limr⟶∞dx2r+1,Ft=limr⟶∞dFx2r,Ft=0,then, Ft=t. Also, similarly, we get
(30)limr⟶∞dx2r+2,Gt=limr⟶∞dGx2r+1,Gt=0.
Gt=t. Therefore, t is a common fixed point of F and G. The proof is complete.☐In the following theorem, we remove the assumption of the continuity of the mappingsF and G.Theorem 12.
Besides the hypothesis (i) and (ii) of Theorem11, if we assume that the condition:
(i)
Ifxr is a sequence in P such that xr⟶x∈P as r⟶∞ and, there exists a subsequence xrk of xr such that Λxrk,x≥IandΛx,xrk≥I,forallk
holds, then, the mappingsF and G have a common fixed point.Proof.
From Theorem11, the sequence xr defined by (12) is a Cauchy sequence and converges to some t∈P. Similarly, using (13) and the condition HΛ, there exists a subsequence xrk of xr such that Λx2rk,t≥I and Λt,x2rk−1≥I for all k. We claim that for all k≥0(31)12mindx2rk,Fx2rk,dt,Gt≤dx2rk,t.
Supposing on the contrary,(32)12mindx2rk,Fx2rk,dt,Gt>dx2rk,t,we get
(33)12mindx2rk,x2rk+1,dt,Gt>dx2rk,t,
and lettingk⟶∞, we acquire that a contradiction. Therefore, the condition (31) holds, and from (11), we obtain
(34)dx2rk+1,Gt=IdFx2rk,Gt≤Λx2rk,tdFx2rk,Gt≤Adx2rk,tβ·dx2rk,Fx2rkα·dt,Gt1−α−β=Adx2rk,tβ·dx2rk,x2rk+1α·dt,Gt1−α−β.
On the takingk tend to infinity, it follows that we get Gt=t. Similarly, we assert that, for all k≥0(35)12mindt,Ft,dx2rk−1,Gx2rk−1≤dt,x2rk−1,
Supposing on the contrary,(36)12mindt,Ft,dx2rk−1,Gx2rk−1>dt,x2rk−1,and, so
(37)12mindt,Ft,dx2rk−1,x2rk>dt,x2rk−1,
takingk⟶∞, we obtain that a contradiction. Hence, condition (35) is true and from (11), we obtain
(38)dFt,x2rk=IdFt,Gx2rk−1≤Λt,x2rk−1dFt,Gx2rk−1≤Adt,x2rk−1β·dt,Ftα·dx2rk−1,Gx2rk−11−α−β=Adt,x2rk−1β·dt,Ftα·dx2rk−1,x2rk1−α−β.
Lettingk tend to infinity, it follows that we acquire Ft=t. Thus, t is a common fixed point of F and G.☐Corollary 13.
LetP,d be a complete vector-valued metric space and F,G:P⟶P be two continuous mappings such that
(39)12mindx,Fx,dz,Gz≤dx,z⇒dFx,Gz≤Adx,zβ·dx,Fxα·dz,Gz1−α−β,
for eachx,z∈PF×PG, where A,Aq∈Mmmℝ+, q>1, converges towards zero and β≥0,α>0, are such that β+α<1. Then, F and G have a common fixed point.Corollary 14.
LetP,d be a complete vector-valued metric space and the mappings Λ:P×P⟶Mmmℝ+ and F,G:P⟶P such that
(40)Λx,zdFx,Gz≤Adx,zβ·dx,Fxα·dz,Gz1−α−β,
for eachx,z∈PF×PG, where A,Aq∈Mmmℝ+, q>1, converges towards zero and the constants β≥0,α>0, are such that β+α<1. Assume that
(i)
F,G is a generalized Λ-admissible pair(ii)
There existsx0∈P such that Λx0,Fx0≥I and ΛFx0,x0≥I(iii)
The conditionHΛ holds or F and G are continuous mappings
Then,F and G have a common fixed point.LettingF=G in Theorem 11, we obtain the next results.Corollary 15.
LetP,d be a generalized metric spaces and Λ:P×P⟶Mmmℝ+. Let F:P⟶P be a Λ-orbital admissible mapping such that
(41)12dx,Fx≤dx,z⇒Λx,zdFx,Fz≤Adx,zβ·dx,Fxα·dz,Fz1−α−β,
for eachx,z∈PF, where A,Aq∈Mmmℝ+, q>1, converges towards zero, and β,α are constants, such that β≥0, α>0, and β+α<1. Assume that
(i)
F is a triangular Λ-orbital admissible(ii)
There existsx0∈P such that Λx0,Fx0≥I and ΛFx0,x0≥I(iii)
EitherF is continuous, or the condition HΛ holds
Then,F has a fixed point.Definition 16.
LetP,d be a vector-valued metric space and Λ:P×P⟶Mmmℝ+, F,G:P⟶P, be mappings. We say that F,G:P⟶P are Perov-interpolative Kannan contractions of Suzuki type, if there exist a real number α∈0,1 and A,C∈Mmmℝ+ converges towards zero, where A1/a=C, such that
(42)12mindx,Fx,dz,Gz≤dx,z⇒Λx,zdFx,Gz≤Adx,Fxα·dz,Gz1−α,
for eachx,z∈PF×PG.Theorem 17.
LetP,d be a complete vector-valued metric space and F,G:P⟶P be Perov-interpolative Kannan contractions of Suzuki type. Assume that
(i)
F,G is a generalized Λ-admissible pair(ii)
There existsx0∈P such that Λx0,Fx0≥I and ΛFx0,x0≥I(iii)
Either,F and G are continuous mappings or, the condition HΛ holds
Then,F and G have a common fixed point.Proof.
Takingβ=0 in Theorem 11.☐Remark 18.
Ifm=1 and A=κ∈0,1 in the above Theorems, then, we find the concept of the usual metric spaces and interpolative Kannan contraction of Suzuki type and interpolative Ćirić–Reich–Rus contraction of Suzuki type.Example 19.
LetP=0,2, d:P×P⟶0,+∞, where dx,z=x−zx−z, and two mappings F,G:P⟶P, defined as
(43)Fx=13,ifx∈0,1,x4,ifx∈1,2,respectively,
(44)Gx=13,ifx∈0,1,x8,ifx∈1,2.
We chooseβ=1/3, α=1/3 and A=1/2001/2. Let also Λ:P×P⟶M22ℝ+, where
(45)Λx,z=2002ifx,z∈0,1,1001ifx=0,z=2,0000otherwise.
Then, we have to check that (11) holds. We have to examine the following cases:
(1)
x,z∈0,1. Let U1=1/2×1/2n:n∈2,3,4,⋯ and U2=1/2n:n∈2,3,4,⋯×1/2. For x,z∈A1∪A2(46)12mindx,Fx,dz,Gz≤112112<n−12nn−12n=dx,zΛx,zdFx,Gz≤Adx,z1/3dx,Fx1/3dz,Gz1/3,
and sincedFx,Gz=0, the inequality (11) holds.
(2)
x,z∈1,2. Similarly, since Λx,z=0, the relation (11) holds(3)
Forx=0 and z=2(47)12mind0,13,d2,14=12min1313,7474=1616<22=d0,2⇒Λ0,2dF0,G2=112112<7481/37481/3=Ad0,21/3·d0,F01/3·d2,G21/3.
Then, (11) holds. Consequently, the assumptions of Theorem 11 being satisfied, it follows that the mappings F and G have a fixed point, which is x=1/3.



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