# Some Results on Generalized Pata–Suzuki Type Contractive Mappings

Authors: Merve Aktay; Murat Özdemir
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this work, we establish new fixed point theorems for generalized Pata–Suzuki type contraction viaα-admissible mapping in metric spaces and to prove some fixed point results for such mappings. Moreover, we give an example to illustrate our main result. Consequently, the results presented in this paper generalize and improve the corresponding results of the literature.


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## Body

## 1. Introduction and Preliminaries

The Banach contraction principle was introduced by Banach [1] which is one of the earlier and the most important result in fixed point theory. Because of its importance, over the years, many authors extended and generalized the Banach contraction principle in many directions.The concept of almost contraction was given by Berinde and Pacurar [2]. Also this concept was compared with other contractions and proved some fixed point theorems by Berinde ([2–4]).C-condition was introduced by Suzuki ([5–7]). Suzuki [8] proved generalized versions of Edelstein theorem [9] for compact metric spaces.Firstly, Samet et al. [10] introduced an interesting contraction which is called α−ϱ− contractive and gave α−admissible mappings and investigated the existence and uniqueness of such mappings in the setting of complete metric spaces. Later, in addition to Samet et al. [10], some different fixed point theorems were introduced by Karapinar and Samet [11] and Babu et al. [12].Recently, one of the famous generalizations of the Banach contraction principle for the existence and uniqueness of fixed points for self mappings on metric spaces is the theorem by Pata [13]. Since Pata’s fixed point theorem, some authors have studied this theorem in several ways (see [14–18]).Using the ideas of Pata [13], we give a generalization of Pata-type contractions. The purpose of this paper is to introduce almost α,ϱ−Pata–Suzuki type contractive mapping and to prove some fixed point results for such mappings with admissibility condition. These results are generalizations of various results in the literature.Now, we give some definitions and fundamental results.LetE,ρ be a metric space; a point u∈E is said to be a fixed point of h:E⟶E if u=hu. The set of all fixed points of f is denoted by Fixh.Berinde and Pacurar [2] introduced the concept of almost contraction as a generalization of Banach contraction principle.Definition 1.
(see [2]). Let E,ρ be a metric space and h:E⟶E be a mapping which is called an almost contraction if there exists a constant ϑ∈0,1 and L≥0 such that(1)ρht,hs≤ϑt,s+Lρs,ht,for all t,s∈E.
In 2011, Pata [13] proved the following result.Theorem 1 (see [13]).
LetE,ρ be a complete metric space and Ψ denote the class of all increasing function ψ:0,1⟶0,∞, which vanishes with continuity at zero. For an arbitrary t0∈E, we denote t=ρt,t0,∀t∈E. Let Λ≥0,ξ≥1, and β∈0,ξ be fixed constants and ψ∈Ψ and f:E⟶E be a function. If for all t,s∈E, the following inequality is satisfied:(2)ρht,hs≤1−ɛρt,s+Λɛξψɛ1+t+sβ,for all ɛ∈0,1, and then h has a unique fixed point u=hu.Suzuki ([5–7]) introduced C−condition as follows.Definition 2.
LetE,ρ be a metric space and h be a given self mapping on E. h is said to satisfy C-condition if for all t,s∈E(3)12ρt,ht≤ρt,simpliesρht,hs≤ρt,s.
Samet et al. [10] gave admissibility condition and the definition of α−ϱ−contractive mapping as follows.Definition 3.
(see [10]). Let E,ρ be a metric space, h:E⟶E be a map, and α:E×E⟶0,+∞ be a function.(i)
Ifαt,s≥1 implies αht,hs≥1 for all t,s∈E, then h is said to be α-admissible(ii)
Ifh is α-admissible and αt,z≥1 and αt,s≥1 imply αt,s≥1, then h is said to be triangular α-admissible.
ϒ denotes the family of nondecreasing functions ϱ:0,+∞⟶0,+∞ such that ∑n=1+∞ϱnt<+∞ for each t>0, where ϱn is the n−th iterate of ϱ.Remark 1.
Every functionϱ∈ϒ satisfies limn⟶∞ϱnt=0 and ϱt<t for all t>0.Definition 4.
(see [10]). Let E,ρ be a metric space and h:E⟶E be a mapping. We say that h is an α−ϱ−contractive mapping if there exist two functions α:E×E⟶0,+∞ and ϱ∈ϒ such that αt,sρht,hs≤ϱρt,s for all t,s∈E.


## 2. Almostα,ϱ−Pata–Suzuki Type Contraction

In this section, we introduce concept of almostα,ϱ−Pata–Suzuki type contractions in metric spaces. We establish some fixed point results for such contractions on metric spaces.The following lemma is necessary in our Theorem’s proofs.Lemma 1.
LetE,ρ be a metric space and tn be a sequence in E such that ρtn+1,tn⟶0 as n⟶∞. If tn is not a Cauchy sequence, then there exist a ς>0 and sequences of positive integers mk and nk with mk>nk>k such that ρtmk,tnk≥ς and ρtmk−1,tnk≤ς and limk⟶∞ρtmk−1,tnk+1=ç, limk⟶∞ρtmk,tnk=ς, and limk⟶∞ρtmk−1,tnk=ς.
From Lemma1, we obtain limk⟶∞ρtmk+1,tnk+1=ς and limk⟶∞ρtmk,tnk−1=ς.
LetE,ρ be a complete metric space. Along this paper, Ψ denotes the class of all increasing function ψ:0,1⟶0,∞, which vanishes with continuity at zero. For an arbitrary t0∈E, we denote t=ρt,t0,∀t∈E.
First, we begin with the following definition.Definition 5.
LetΛ≥0,ξ≥1, and β∈0,ξ be fixed constants, and ψ∈Ψ and h:E⟶E be two functions. There exist two functions ϱ∈ϒ, α:E×E⟶0,+∞ and L≥0 such that, if for all t,s∈E, and ɛ∈0,1, h satisfies the following inequality:(4)12ρt,ht≤ρt,s,which implies(5)αt,sρht,hs≤1−ɛϱMt,s+LNt,s+Λɛξψɛ1+t+s+ht+hsβ,where(6)Mt,s=maxρt,s,ρt,ht+ρs,hs2,ρt,hs+ρs,ht2,Nt,s=minρt,ht,ρs,hs,ρt,hs,ρs,ht.
Then, we say thath is an almost α,ϱ−Pata–Suzuki contractive mapping.
Now, we are ready to prove our first result.Theorem 2.
LetE,ρ be a complete metric space. h:E⟶E be an almost α,ϱ−Pata–Suzuki contractive mapping. Assume that(i)
h is triangular α−admissible(ii)
There existsx0∈E such that αt,ht0≥1(iii)
h is continuous(iv)
For allu,v∈Fixh, αu,v≥1
Then,h has a unique fixed point, that is, u=hu, u∈E.Proof.
From the hypothesis (ii) of Theorem2, there exists t0∈E such that αt0,ht0≥1. Starting at the point t0∈E, the sequence tn is constructed by tn=htn−1,n≥1. If tn0+1=tn0 for any n0∈ℕ, then xn0 is a fixed point of h. Consequently, assume that tn0+1≠tn0 for all n0∈ℕ. First of all, we show that αtn,tn+1≥1 for all n∈ℕ. Since h is an α−admissible mapping, we have(7)αt0,t1≥1=αt0,ht0≥1impliesαt1,t2≥1,αt1,t2≥1impliesαt2,t3≥1.
By induction, we obtain(8)αtn,tn+1≥1,for alln∈ℕ.
From the hypothesis (i) of Theorem2, we have(9)αtn,tn+1≥1,αtn+1,tn+2≥1implyαtn,tn+2≥1.
Hence, by induction, we obtain(10)αtn,tm≥1,for allm>n≥0.
Now, we will show thatρtn+1,tn is a decreasing sequence. Using Remark 1, since 1/2ρtn−1,tn≤ρtn−1,tn, we obtain(11)ρtn+1,tn≤1−ɛϱmaxρtn,tn−1,ρtn+1,tn+ρtn,tn−12,ρtn,t+ρtn−1,tn+12+Lminρtn+1,tn,ρtn,tn−1,ρtn,tn,ρtn−1,tn+1+Λɛξψɛ1+tn−1+tn+tn+tn+1β≤1−ɛϱmaxρtn,tn−1,ρtn+1,tn+ρtn,tn−12+Kɛξψɛ<1−ɛmaxρtn,tn−1,ρtn+1,tn+ρtn,tn−12+Kɛξψɛ,for some K>0. If ρtn,tn−1<ρtn+1,tn, then we get ρtn+1,tn≤1−ɛρtn+1,tn+Kɛξψɛ. In this way, we obtain ρtn+1,tn=0, a contraction. Thus, we have(12)ρtn+1,tn<ρtn,tn−1<⋯<ρt1,t=t1,that is, ρtn+1,tn is a decreasing sequence. Since ρtn+1,tn is decreasing, it is convergent to a nonnegative real number. Let limn⟶∞tn,tn+1=ρ. Now, we will show that the sequence tn is bounded. From triangle inequality, we have(13)tn=ρt,t0≤ρtn,tn+1+ρtn+1,t1+ρt1,t0.
We assert a claim that(14)12ρtn,tn+1≤ρtn,t0or12ρtn−1,tn≤ρtn−1,t0.
On the contrary, suppose that(15)12ρtn,tn+1>ρtn,t0,12ρtn−1,tn>ρtn−1,t0.
From triangular inequality, we have(16)ρtn−1,tn≤ρtn−1,t0+ρt0,tn<12ρtn−1,tn+ρtn,tn+1≤ρtn−1,tn,which is a contradiction. Thus, (14) is satisfied. Since h is an almost α,ϱ−-Pata-Suzuki contractive mapping and using (10) and Remark 1, we obtain(17)ρt1,tn+1≤αt0,tnρht0,htn≤1−ɛϱmaxρtn,t0,ρtn,tn+1+ρt0,t12,ρtn,t1+ρt0,tn+12+Lminρtn,tn+1,ρt0,t1,ρtn,t1,ρt0,tn+1+Λɛξψɛ1+tn+0+tn+1+t1β≤1−εϱmaxρtn,t0,ρtn,tn+1+ρt0,t12,ρtn,t0+ρt1,t0+ρtn+1,tn+ρtn,t02+Lminρtn,tn+1,ρt0,t1,ρtn,t0+ρt0,t1,ρt0,tn+1+Λɛξψɛ1+2tn+2t1β≤1−ɛϱmaxtn,t1,tn+t1+Lmint1,tn+t1,tn+1+Λɛξψɛ1+2tn+2t1β≤1−ɛϱtn+t1+Ltn+1+Λɛξψɛ1+2tn+2t1β<1−ɛtn+t1+Ltn+1+Λɛξψɛ1+2tn+2t1β.
Usingβ≤ξ, we get(18)tn<1−ɛtn+t1+Lt1+2t1+Λɛξψɛ1+2tn+2tn+1ξ,ɛtn<kɛξψɛtnξ+l,for some k,l>0. By the same reason as in [13], the sequence tn is bounded. Let limn⟶∞ρtn,tn+1=ρ. Using Remark 1 and from (8), we have(19)ρtn,tn+1≤αtn−1,tnρtn,tn+1≤1−ɛϱmaxρtn,tn−1,ρtn+1,t+ρtn,tn−12,ρtn,tn+ρtn−1,tn+12+Lminρtn+1,tn,ρtn,tn−1,ρtn,tn,ρtn−1,tn+1+Λɛξψɛ1+tn+tn−1+tn+1+tnβ≤1−ɛϱmaxρtn,tn−1,ρtn+1,tn+ρtn,tn−12+Kɛξψɛ,<1−ɛmaxρtn,tn−1,ρtn+1,tn+ρtn,tn−12+Kɛξψɛ,for some K>0. Taking limit as n⟶∞, we obtain ρ≤Kɛξψɛ and thus ρ=0.
Next, we demonstrate thattn is a Cauchy sequence. Assume that the sequence tn is not a Cauchy sequence. From Lemma 1, there exist subsequence tmk and tnk with nk>mk>k such that limk⟶∞ρtmk−1,tnk+1=ς, limk⟶∞ρtmk,tnk=ς, limk⟶∞ρtmk−1,tnk=ς, limk⟶∞ρtmk+1,tnk+1=ς, and limk⟶∞ρtmk,tnk−1=ς. We assert that(20)12ρtmk−1,tmk≤ρtmk−1,tnk−1.
If we assume that(21)12ρtmk−1,tmk>ρtmk−1,tnk−1,then we obtain a contradiction. If we take limit k⟶∞ in inequality (21), we obtain contradiction. Since h is an almost α,ϱ−Pata–Suzuki contractive mapping, we have(22)ς≤ρtmk,tnk=ρhtmk−1,htnk−1≤αtmk,tnkρhtmk−1,htnk−1≤1−ɛϱmaxρtmk−1,tnk−1,ρtmk−1,tmk+ρtnk−1,tnk2,ρtnk−1,tmk+ρtmk−1,tnk2+Lminρtmk−1,tmk,ρtnk−1,tnk,ρtnk−1,tmk,ρtmk−1,tnk+Λɛξψɛ1+tmk+tnk+tnk+1+tmk+1β<1−ɛmaxρtmk−1,tnk−1,ρtmk−1,tmk+ρtnk−1,tnk2,ρtnk−1,tmk+ρtmk−1,tnk2+Lminρtmk−1,tmk,ρtnk−1,tnk,ρtnk−1,tmk,ρtmk−1,tnk+Λɛξψɛ1+tmk+tnk+tnk+1+tmk+1β.
Taking the limit ask⟶∞, we obtain(23)ς≤1−ɛς+Kɛψɛ,and then(24)ς≤Kψɛ,that is, ς=0, which is a contradiction. Hence, the sequence tn is a Cauchy sequence in E,ρ. By the completeness of E, the sequence tn is convergent to some u∈E, that is, tn⟶u as n⟶+∞. Since h is continuous, htn⟶hu as n⟶+∞. By the uniqueness of the limit, we obtain u=hu; that is, u is a fixed point of h.
Now, we observe that fixed point ofh is unique. Assume that u and v are fixed points of h. Since h satisfies the hypothesis (iv) of Theorem 2, h is an almost α,φ−Pata–Suzuki contractive mapping, Remark 1, and 1/2ρu,hu≤ρu,v, we have(25)ρhu,hv≤αu,vρhu,hv≤1−ɛϱmaxρu,v,ρu,hu+ρv,hv2,ρu,hv+ρv,hu2+Lminρu,hu,ρv,hv,ρu,hv,ρv,hu+Kɛψɛ<1−ɛmaxρu,v,ρu,hu+ρv,hv2,ρu,hv+ρv,hu2+Lminρu,hu,ρv,hv,ρu,hv,ρv,hu+Kɛψɛ.
Hence, we obtain thatρu,v≤Kψɛ, and thus u=v. Hence, h has a unique fixed point in E.
Now, we prove the following theorem without the assumption of continuity ofh.Theorem 3.
LetE,ρ be a complete metric space. h:E⟶E be an almost α,φ−Pata–Suzuki contractive mapping. Assume that(i)
h is triangular α−admissible(ii)
There existst0∈E such that αt0,ht0≥1(iii)
Iftn is a sequence in E such that αtn,tn+1≥1 for all n and tn⟶u∈E as n⟶+∞, then αtn,u≥1 for all n(iv)
For allu,v∈Fixh, αu,v≥1
Then,h has a unique fixed point, that is, u=hu,u∈E.Proof.
Following the proof of Theorem2, we have already known that tn is a Cauchy sequence in E. Since E is complete, we have tn⟶u∈E as n⟶+∞. Now, we prove that u=hu. From (8) and hypothesis (iii) of Theorem 3, we have αtn,u≥1 for all n. For all n≥1, we assert that(26)12ρtn,tn+1≤ρtn,uor12ρtn−1,tn≤ρtn−1,u.
We assume that(27)12ρtn,tn+1>ρtn,u,12ρtn−1,tn>ρtn−1,u.
Sinceρtn+1,tn is a decreasing sequence,(28)ρtn−1,tn≤ρtn−1,u+ρu,tn<12ρtn−1,tn+ρtn,tn+1≤ρtn−1,tn,which is a contradiction. Thus, (26) is held, and we have(29)ρhu,u=ρhu,tn+1+ρtn+1,u≤αtn,uρhu,tn+1+ρtn+1,u≤1−ɛϱmaxρu,tn,ρu,hu+ρtn,tn+12,ρu,tn+1+ρtn,hu2+Lminρu,hu,ρtn,tn+1,ρu,tn+1,ρtn,hu+Λɛξψɛ1+tn+u+hu+tn+1β+dtn+1,u<maxρu,tn,ρu,tu+ρtn,tn+12,ρu,tn+1+ρtn,hu2+Lminρu,hu,ρtn,tn+1,ρu,tn+1,ρtn,hu+Kɛξψɛ+ρtn+1,u,for some K>0. Taking the limit as n⟶∞, we obtain(30)ρhu,u≤12ρhu,u+Kɛξψɛ.
Thus, we make an inference thathu=u and that is u is a fixed point of h. Similar to the proof of Theorem 2 the uniqueness of fixed point of h can be obtained.
We give an example related to our main theorem.Example 1.
LetE=0,1 with the usual metric and define the mapping h:E⟶E by ht=t/3. Let ϱ:0,+∞⟶0,+∞ be defined as ϱt=t/2 and α:E×E⟶0,+∞, αt,s=1t,s∈0,10t,s∉0,1. It is easy to get h is triangular α−admissible. Our aim is to prove that h satisfies (5). For x,y∈0,1,(31)12ρt,ht=12t−t3≤ρt,s,which implies(32)ρht,hs=αt,sρht,hs=t3−s3=13ρt,s.
Sinceρt,s≤Mt,s, we obtain(33)ρht,hs≤13Mt,s=23ϱMt,s,and for L=0, we can write(34)ρht,hs≤23ϱMt,s+LNt,s.
For arbitraryɛ∈0,1, we can write inequality (34) as follows:(35)ρht,hs≤1−ɛϱMt,s+LNt,s+23+ɛ−1Mt,s≤1−ɛϱMt,s+LNt,s+23+ɛ−11+t+s+ht+hs.
Our aim is to prove thatγ≥0 and Λ≥0 such that(36)23+ɛ−11+t+s+ht+hs≤Λɛγ+11+t+s+ht+hs,which satisfies for all t,s∈0,1, and every 0≤ɛ≤1. We can find Λ≥0 such that(37)Λ=2/3+ɛ−1ɛγ+1,which satisfies for each 0≤ɛ≤1 and some γ≥0. If we choose γ such that γ/γ+1>1−2/3, then(38)Λ=γγγ+1γ+11−2/3γ.
Thus, we have that(39)ρt,ht=t−t3≤ρt,s,which implies(40)αt,sρht,hs≤1−ɛϱMt,s+LNt,s+Λɛγ+11+t+s+ht+hs,which satisfies for all t,s∈0,1 and each ɛ>0. If ɛ=0, it can be seen that (5) is satisfied. Hence, the conditions of Theorem 2 are satisfied with ψɛ=ɛγ,ξ=β=1. By an application of Theorem 2, h has a unique fixed point in E=0,1. It is seen that u=0 is the unique fixed point of h in E.
The following result is directly obtained from Theorem2 by taking ɛ=0.Corollary 1.
Leth:E⟶E be a function. There exist two functions φ∈ϒ and α:E×E⟶0,+∞ and L≥0 such that if for all t,s∈E, h satisfies the inequality(41)12ρt,ht≤ρt,s,which implies(42)αt,sρht,hs≤ϱMt,s+LNt,s,where(43)Mt,s=maxρt,s,ρt,ht+ρs,hs2,ρt,hs+ρs,ht2,Nt,s=minρt,ht,ρs,hs,ρt,ht,ρs,ht,and assume that(i)
h is triangular α−admissible(ii)
There existst0∈E such that αt0,ht0≥1(iii)
h is continuous(iv)
For allu,v∈Fixh, αu,v≥1
Then,h has a unique fixed point, that is, u=hu,u∈E.Corollary1 generalizes the results of Samet [10] and Karapınar [11].Corollary 2.
LetΛ≥0,ξ≥1 and β∈0,ξ be fixed constants and h:E⟶E be a function. There exist two functions ψ∈Ψ and α:E×E⟶0,+∞ and L≥0 such that if for all t,s∈E, and ɛ∈0,1, h satisfies the inequality(44)12ρt,ht≤ρt,s,which implies(45)αt,sρht,hs≤1−ɛMt,s+LNt,s+Λɛξψɛ1+t+s+ht+hsβ,where(46)Mt,s=maxρt,s,ρt,ht+ρs,hs2,ρt,hs+ρs,ht2,Nt,s=minρt,ht,ρs,hs,ρt,hs,ρs,hy,and assume that(i)
h is triangular α−admissible(ii)
There existst0∈E such that αt0,ht0≥1(iii)
h is continuous(iv)
For allu,v∈Fixh, αu,v≥1Then,h has a unique fixed point that is u=hu,u∈E.If we takeL=0 in Theorem 2, then we say that h is α,ϱ−Pata–Suzuki contractive mapping and get the following corollary.Corollary 3.
LetΛ≥0,ξ≥1, and β∈0,ξ be fixed constants; ψ∈Ψ and h:E⟶E be two functions. There exist two functions ϱ∈ϒ and α:E×E⟶0,+∞ such that if for all t,s∈E, and ɛ∈0,1, h satisfies the inequality(47)12ρt,ht≤ρt,s,which implies(48)αt,sρht,hs≤1−ɛϱMt,s+Λɛξψɛ1+t+s+ht+hsβ,where(49)Mt,s=maxρt,s,ρt,ht+ρs,hs2,ρt,hs+ρs,ht2,and assume that(i)
h is triangular α−admissible(ii)
There existst0∈E such that αt0,ht0≥1(iii)
h is continuous(iv)
For allu,v∈Fixh, αu,v≥1
Then,h has a unique fixed point, that is, u=hu,u∈E.If we takeαt,s=1 for all t,s∈E, Mt,s=ρt,s and Nt,s=ρs,ht, in Theorem 2, then we get the following corollary.Corollary 4.
LetE,ρ be a complete metric space, Λ≥0,ξ≥1, and β∈0,ξ be fixed constants, and ψ∈Ψ,ϱ∈ϒ, and h:E⟶E be a function. There exist a function φ∈ϒ and L≥0 such that if for all t,s∈E,h satisfies the inequality(50)12ρt,ht≤ρt,s,which implies(51)ρht,hs≤1−ɛϱρt,s+Lρs,ht+Λɛξψɛ1+t+s+ht+hsβ,is satisfied for all ɛ∈0,1, and then h has a unique fixed point u=hu,u∈E.If we takeαt,s=1 for all t,s∈E, Mt,s=ρt,s and Nt,s=ρs,ht, ϱt<t, for all t>0, in Theorem 2, then we get the following corollary.Corollary 5.
LetE,ρ be a complete metric space, Λ≥0,ξ≥1, and β∈0,ξ be fixed constants, and ψ∈Ψ and h:E⟶E be a function. There exists L≥0 such that if for all t,s∈E, and ɛ∈0,1, h satisfies the inequality(52)12ρt,ht≤ρt,s,which implies(53)ρht,hs≤1−ɛρt,s+Lρs,ht+Λɛξψɛ1+t+s+ht+hsβ,is satisfied for all ɛ∈0,1, and then h has a unique fixed point u=hu,u∈E.Corollary 6.
LetE,ρ be a complete metric space, Λ≥0,ξ≥1, and β∈0,ξ be fixed constants, and ψ∈Ψ and h:E⟶E be a function. There exists L≥0 such that if for all t,s∈E, and ɛ∈0,1, h satisfies the inequality(54)ρht,hs≤1−ɛρt,s+Lρs,ht+Λɛξψɛ1+t+s+ht+hsβ,which is satisfied for all ɛ∈0,1, and then h has a unique fixed point u=hu,u∈E.Corollary6 generalizes the results of Pata [13] and Banach [1].



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