# On Pata Convex-Type Contractive Mappings

Authors: Merve Aktay; Murat Özdemir
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this work, we introduce weak Pata convex contractions and weakE-Pata convex contractions via simulation functions in metric spaces to prove some fixed point results for such mappings. Also, we consider an example related to weak Pata convex contractions. Consequently, our results generalize and unify some results in the literature.


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## Body

## 1. Introduction and Preliminaries

It is well known that Banach [1] pioneered in fixed point theory by introducing a novel notion, namely, Banach contraction principle in 1922. After this date, several authors generalized and extended this principle. A generalization was given by Pata [2] known as Pata contraction. Recently, Pata contraction has been studied by many authors. Some of the studies were for Pata contraction presented by [3–13].Firstly, the concept ofϕ−weak contraction was given by Alber et al. [14]. Zhang et al. and Rhoades’s results [15, 16] extend previous results given by Alber et al., and they obtained fixed point results for single-valued mappings in Banach spaces, and Rhoades [15] got a unique common fixed point of such contractions, respectively.In 2012, Samet et al. [17] suggested a novel notion, the so-called α-admissible. Later, Karapinar et al. [18] presented triangular α-admissible mappings, and then, Arshad et al. [19] introduced α-orbital admissible and triangular α-orbital admissible mappings. Due to the importance, many authors studied such mappings. For more knowledge and different examples related to admissible mappings, one can see [20–25].Istratescu [26–28] gave the concept of contractions known as the convex contraction of order 2 and two-sided convex contraction mappings. Very recently on, Karapinar et al. [10] introduced the notion of α-almost Istratescu contraction of type E. Some notable generalizations related to Istratescu’s results were obtained by [29–35].In a recent work, Khojasteh et al. [36] introduced the notion of Z-contraction using simulation functions. Later, Karapinar [37] and Argoubi et al. [38] studied such contractions. After that, some new studies were obtained related to simulation functions in [39–44].The aim of this paper is to establish some fixed point results for weak Pata convex contractive mapping and weakE-Pata convex contractive mapping via α-admissible mappings by using simulation functions in metric spaces. Our results are generalization of recent fixed point results derived by Karapinar et al. ([10, 32, 45]), Alber et al. [14], Zhang et al. [16], Istratescu [26], Pata [2], and Banach [1] and some other related results in the literature.Firstly, we start this section by recalling some definitions related to our work.In the course of this manuscript,ℝ,ℕ denote the set of real numbers and the set of natural numbers, respectively. Let FixS=w∈W:Sw=w.Alber et al. [14] gave the definition of ϕ−weak contraction, stated below.Definition 1. See [14].
LetW,ρ be a metric space. A mapping S:W⟶W is called ϕ−weak contraction, if there exists a map ϕ:0,+∞⟶0,+∞ with ϕ0=0 and ϕw>0 for all w>0 such that
(1)ρSw,Sυ≤ρw,υ−ϕρw,υ,for all w,υ∈W.The concept ofϕ-weak contraction was generalized by Zhang et al. [16] as generalized ϕ-weak contraction.Definition 2. See [16].
LetW,ρ be a metric space. A mapping S:W⟶W is called generalized ϕ-weak contraction, if there exists a map ϕ:0,+∞⟶0,+∞ with ϕ0=0 and ϕw>0 for all w>0 such that
(2)ρSw,Sυ≤Mw,υ−ϕMw,υ,for all w,υ∈W, where
(3)Mw,υ=maxρw,υ,ρw,Sw,ρυ,Sυ,ρw,Sυ+ρυ,Sw2.Samet et al. [17] and Karapinar et al. [18] introduced the following concepts, respectively.Definition 3.
LetW,ρ be a metric space, S:W⟶W be a map, and α:W×W⟶0,+∞ be a function.
(i)
[17] If αw,υ≥1 implies αSw,Sυ≥1 for all w,υ∈W, then S is called α−admissible(ii)
[18] If Sisα−admissible and αw,z≥1 and αz,υ≥1 imply αw,υ≥1, then S is called triangular α−admissibleExample 4.
LetW=ℝ, the mappings S:W⟶W by
(4)Sw=w2+13,w∈0,1,12,w∉0,1,and α:W×W⟶0,+∞ by
(5)αw,υ=1,w,υ∈0,1,0,w,υ∉0,1.
Thus,S is a triangular α-admissible mapping.Khojasteh et al. [36] gave the simulation function and Z-contraction as follows.Definition 5. See [36].
A mappingζ:0,∞×0,∞⟶ℝ is called a simulation function if it satisfies the following conditions:
(ζ1) ζ0,0=0
(ζ2) ζw,υ<w−υ
(ζ3) if wn and υn are sequences in 0,∞ such that limn⟶+∞wn=limn⟶+∞vn>0, then limsupn⟶+∞ζwn,υn<0.Definition 6. See [36].
LetW,ρ be a metric space and S:W⟶W be a mapping. If there exists ζ∈Z such that
(6)ζρSw,Sυ,ρw,υ≥0,forallw,υ∈W,then, S is called Z−contraction with respect to ζ.(ζ1) condition was removed in the above definition of simulation function by Argoubi et al. [38] in 2015. Also, Z′ denotes the set of all simulation functions.Example 7. See [36, 42, 44].
Letζ:0,∞×0,∞⟶ℝ and φi:0,∞⟶0,∞,i=1,2,3 be continuous functions with φiw=0⇔t=0.
ζw,υ=φ1w−φ2v, for all w,v∈0,∞, where φ1w<w≤φ2v for all w>0.(7)ζw,υ=v−φ3w−w.
For the above examples and other examples related to simulation functions, one can see [36, 37, 42, 44] and references therein.The following two concepts were defined by Istratescu [26] as follows.Definition 8. See [26].
LetW,ρ be a metric space and S:W⟶W be a self-mapping. For all w,υ∈W, S is called convex contraction of order 2 if there exist d1,d2∈0,1 such that d1+d2<1 and
(8)ρS2w,S2υ≤d1ρSw,Sυ+d2ρw,υ.
S is called two-sided convex contraction mappings if there exist d1,d2,d3,d4∈0,1 such that d1+d2+d3+d4<1 and
(9)ρS2w,S2υ≤d1ρw,Sw+d2ρSw,S2w+d3ρυ,Sυ+d4ρSυ,S2υ.
In the course of this work,Ψ denotes the set of all increasing function ψ:0,1⟶0,∞, which vanishes with continuity at zero. For a random w0∈W, we denote w=ρw,w0,∀w∈W.Introducing a novel generalization of the Banach contraction principle, Pata [2] proved Theorem 9.Theorem 9. See [2].
LetW,ρ be a metric space and Λ≥0,ξ≥1 and ϑ∈0,ξ be fixed constants. ψ∈Ψ and S:W⟶W be functions. If for all w,υ∈W, the inequality
(10)ρSw,Sυ≤1−ερw,υ+Λεξψε1+w+υϑis satisfied for all ε∈0,1; then S has a unique fixed point, ω=Sω,ω∈W.
Pata-type contractions were studied by some authors. Karapinar et al. [11] introduced Pata-Ciric type contraction at a point. Alqahtani et al. [5] gave the α-Pata–Suzuki contraction and fixed point results for such contractions. After that, Karapinar and Himabindu [11] proved some common fixed point results for Pata–Suzuki Z-contraction.We recall here the following important Lemma10 that we will use to proof of our main results.Lemma 10. See [46].
LetW,ρ be a metric space and wn be a sequence in W such that ρwn+1,wn⟶0 as n⟶∞. If wn is not a Cauchy sequence, then there exist a ς>0 and subsequences wmj and wnj of wn such that limj⟶∞ρxmj+1,xnj+1=ς, limj⟶∞ρxmj,xnj=ς,limj⟶∞ρxmj+1,xnj=ς and limj⟶∞ρxmj,xnj+1=ς.


## 2. Main Results

The main objective of this work is to give some new fixed point theorems via a combination of convex contraction, weak contraction and Pata type contractive mappings by introducing the concept of weakE-Pata convex contractions and weak Pata convex contractions in metric spaces. We will use simulation functions and admissible mappings when combining these concepts. Also, we will give an example that supports our conclusion.In definitions and results in this paper,Λ≥0,ξ≥1, and ϑ∈0,ξ will be considered as fixed constants, and also, we will consider the following equations:
(11)EIw,υ=ρSw,Sυ+ρSw,S2w−ρSυ,S2υ,MIw,υ=maxρw,υ,ρSw,Sυ,ρw,Sw,ρυ,Sυ,ρSw,S2w,ρSυ,S2υ,PIw,υ=1+w+υ+Sw+Sυ+S2w+S2υϑ.At first, we begin our work by giving the following definitions.Definition 11.
LetW,ρ be a metric space. We say that S:W⟶W is weak Pata convex contractive mapping via simulation function if for all w,υ∈W, and ε∈0,1, there exist three functions ζ∈Z′,ψ∈Ψ, and α:W×W⟶0,+∞ such that S satisfies the inequality
(12)ζαw,υρS2w,S2υ,1−εMIw,υ−ϕMIw,υ+ΛεξψεPIw,υ≥0,where ϕ:0,+∞⟶0,+∞ is a continuous and nondecreasing function with ϕ0=0 and ϕw>0, for all w>0.Definition 12.
LetW,ρ be a metric space. We say that S:W⟶W is weak E-Pata convex contractive mapping via simulation function if for all w,υ∈W, and ε∈0,1, there exist three functions ψ∈Ψ,ζ∈Z′, and α:W×W⟶0,+∞ such that S satisfies the inequality
(13)ζαw,υρS2w,S2υ,1−εEIw,υ−ϕEIw,υ+ΛεξψεPIw,υ≥0,where ϕ:0,+∞⟶0,+∞ is a continuous and nondecreasing function with ϕ0=0 and ϕw>0, for all w>0.Now, we are in a position to present our main theorems.Theorem 13.
LetW,ρ be a complete metric space, α:W×W⟶0,+∞ and S:W⟶W be a weak E-Pata convex mapping via simulation function. Suppose that
(i)
S is triangular α-admissible(ii)
there existsw0∈W such that αw0,Sw0≥1(iii)
S is continuous(iv)
for allw,υ∈FixS, αw,υ≥1.
ThenS has a unique fixed point in W.Proof.
From hypothesis (ii) of the Theorem13, there exists w0∈W such that αw0,Sw0≥1. Firstly, we will show that αSnw0,Sn+1w0≥1 for all n∈ℕ. Since S is an α-admissible mapping, we have
(14)αw0,w1≥1=αw0,Sw0≥1⇒αSw0,S2w0≥1,αSw0,S2w0≥1⇒αS2w0,S3w0≥1.
By induction, we obtain that(15)αSnw0,Sn+1w0≥1,foralln∈ℕ.
Taking into account hypothesis (i) of the Theorem13, we have
(16)αSnw0,Sn+1w0≥1andαSn+1w0,Sn+2w0≥1⇒αSnw0,Sn+2w0≥1.
Again by induction, we obtain that(17)αSnw0,Smw0≥1,forallm>n≥0.
Now, we will show thatρSnw0,Sn+1w0 is a nonincreasing sequence. Since S is a weak E-Pata convex contractive mapping via simulation function, we have
(18)ζαw0,Sw0ρS2w0,S3w0,1−εEIw0,Sw0−ϕEIw0,Sw0+ΛεξψεPIw0,Sw0≥0,1−εEIw0,Sw0−ϕEIw0,Sw0+ΛεξψεPIw0,Sw0−αw0,Sw0ρS2w0,S3w0≥0.
From hypothesis (ii) of the Theorem13, we get
(19)ρS2w0,S3w0≤αw0,Sw0ρS2w0,S3w0≤1−εEIw0,Sw0−ϕEIw0,Sw0+ΛεξψεPIw0,Sw0=1−ερSw0,S2w0+ρSw0,S2w0−ρS2w0,S3w0−ϕρSw0,S2w0+ρSw0,S2w0−ρS2w0,S3w0+Λεξψε1+w0+Sw0+Sw0+S2w0+S2w0+S3w0ϑ≤1−ερSw0,S2w0+ρSw0,S2w0−ρS2w0,S3w0−ϕρSw0,S2w0+ρSw0,S2w0−ρS2w0,S3w01+w0+2Sw0+2S2w0+S3w0ϑ≤1−ερSw0,S2w0+ρSw0,S2w0−ρS2w0,S3w0−ϕρSw0,S2w0+ρSw0,S2w0−ρS2w0,S3w0+Kεξψε,for some K>0. If we assume that ρSw0,S2w0<ρS2w0,S3w0, then we have ρSw0,S2w0+ρSw0,S2w0−ρS2w0,S3w0=ρS2w0,S3w0. Hence, we have
(20)ρS2w0,S3w0≤1−ερS2w0,S3w0−ϕρS2w0,S3w0+Kεξψε.
The inequality (20) is true for all ε∈0,1. For ε=0, we obtain ρS2w0,S3w0<ρS2w0,S3w0 which is a contradiction. Therefore, we obtain
(21)ρS2w0,S3w0≤ρSw0,S2w0.
Analogously, asS is a weak E-Pata convex contractive mapping via simulation function, we have
(22)ζαSw0,S2w0ρS3w0,S4w0,1−εEISw0,S2w0−ϕEISw0,S2w0+ΛεξψεPISw0,S2w0≥0,1−εEISw0,S2w0−ϕEISw0,S2w0+ΛεξψεPISw0,S2w0−αSw0,S2w0ρS3w0,S4w0≥0.
Now, we can write(23)ρS3w0,S4w0≤αSw0,S2w0ρS3w0,S4w0≤1−ερS2w0,S3w0+ρS2w0,S3w0−ρS3w0,S4w0−ϕρS2w0,S3w0+ρS2w0,S3w0−ρS3w0,S4w0+Λεξψε1+Sw0+S2w0+S2w0+S3w0+S3w0+S4w0ϑ≤1−ερS2w0,S3w0+ρS2w0,S3w0−ρS3w0,S4w0−ϕρS2w0,S3w0+ρS2w0,S3w0−ρS3w0,S4w0+Kεξψε,for some K>0. In case that ρS2w0,S3w0<ρS3w0,S4w0; then we have ρS2w0,S3w0+ρS2w0,S3w0−ρS3w0,S4w0=ρS3w0,S4w0. So, we have
(24)ρS3w0,S4w0≤1−ερS3w0,S4w0−ϕρS3w0,S4w0+Kεξψε.
The inequality (24) is true for all ε∈0,1. For ε=0, we obtain ρS3w0,S4w0<ρS3w0,S4w0 which is again a contradiction. Therefore, we obtain
(25)ρS3w0,S4w0≤ρS2w0,S3w0.
By induction, sinceS is a weak E-Pata convex contractive mapping via simulation function, we have
(26)ζαSn−2w0,Sn−1w0ρSnw0,Sn+1w0,1−εEISn−2w0,Sn−1w0−ϕEISn−2w0,Sn−1w0+ΛεξψεPISn−2w0,Sn−1w0≥0,1−εEISn−2w0,Sn−1w0−ϕEISn−2w0,Sn−1w0+ΛεξψεPISn−2w0,Sn−1w0−αSn−2w0,Sn−1w0ρSnw0,Sn+1w0≥0.
We have that(27)ρSnw0,Sn+1w0≤αSn−2w0,Sn−1w0ρSnw0,Sn+1w0≤1−ερSn−1w0,Snw0+ρSn−1w0,Snw0−ρSnw0,Sn+1w0−ϕρSn−1w0,Snw0+ρSn−1w0,Snw0−ρSnw0,Sn+1w0+Λεξψε1+Sn−2w0+Sn−1w0+Sn−1w0+Snw0+Snw0+Sn+1w0ϑ≤1−ερSn−1w0,Snw0+ρSn−1w0,Snw0−ρSnw0,Sn+1w0−ϕρSn−1w0,Snw0+ρSn−1w0,Snw0−ρSnw0,Sn+1w0+Kεξψε,for some K>0. In case that ρSn−1w0,Snw0<ρSnw0,Sn+1w0; then we have
(28)ρSnw0,Sn+1w0<1−ερSnw0,Sn+1w0−ϕρSnw0,Sn+1w0+Kεξψε.
Again, the inequality (28) is true for all ε∈0,1 for ε=0; we obtain ρSnw0,Sn+1w0<ρSnw0,Sn+1w0 is again a contradiction. Therefore, we obtain
(29)ρSnw0,Sn+1w0≤ρSn−1w0,Snw0.
Consequently, we find that(30)ρSnw0,Sn+1w0≤ρSn−1w0,Snw0≤⋯≤ρS3w0,S4w0≤ρS2w0,S3w0≤ρSw0,S2w0.
If the pointw0∈W is taken as the starting point, the sequence wn is constructed by wn=Swn−1=Snw0, n≥1. If wn0+1=wn0 for any n0∈ℕ, then wn0 is a fixed point of S. As a result, supposing that wn0+1≠wn0 for all n0∈ℕ and let ρn=ρwn−1,wn. So, we get that ρn is a nonincreasing sequence. For this reason, there exists a δ≥0 such that
(31)limn⟶∞ρwn−1,wn=limn⟶∞ρn=δ.
We will demonstrate thatδ=0. For this, we should demostrate that the sequence wn is bounded. Since ρn is a nonincreasing sequence, we have
(32)ρn+1=ρwn,wn+1≤ρwn−1,wn≤⋯≤ρw3,w4≤ρw2,w3≤ρw1,w2=ρ2≤w1+w2.
By the triangle inequality, we have(33)wn=ρwn,w0≤ρwn,wn+1+ρwn+1,w2+ρw2,w0=ρn+1+ρwn+1,w2+w2≤ρ2+ρwn+1,w2+w2≤w1+2w2+ρwn+1,w2.
SinceS is a weak E-Pata convex contractive mapping, we have
(34)ζαwn,w0ρwn+1,w2,1−εEIwn−1,w0−ϕEIwn−1,w0+ΛεξψεPIwn−1,w0≥0,(35)1−εEIwn−1,w0−ϕEIwn−1,w0+ΛεξψεPIwn−1,w0−αwn,w0ρwn+1,w2≥0.
Together with (35), we obtain
(36)ρwn+1,w2≤αwn,w0ρwn+1,w2≤1−εEIwn−1,w0−ϕEIwn−1,w0+ΛεξψεPIwn−1,w0,where
(37)EIwn−1,w0=ρwn,w1+ρwn,wn+1−ρw1,w2≤ρwn,w0+ρw1,w0+ρwn,wn+1−ρw1,w2≤wn+w1+ρn+1−ρ2=wn+w1+ρ2−ρn+1≤wn+2w1+w2−ρn+1≤wn+2w1+w2,PIwn−1,w0=1+wn−1+w0+wn+w1+wn+1+w2ϑ≤1+w1+w2+wn+wn+w1+w1+w2+wn+w2ϑ=1+3w1+3w2+3wnϑ.
Now, we derive that(38)wn<w1+2w2+1−εwn+2w1+w2−ϕwn+2w1+w2+Λεξψε1+3w1+3w2+3wnϑ.
Usingϑ≤ξ, we get
(39)εwn<3−2εw1+3−εw2+Λεξψε1+3w1+3w2+3wnϑ≤3−2εw1+3−εw2+Λεξψε1+3w1+3w2+3wnξ=3−2εw1+3−εw2+Λεξψε1+3wnξ1+3w1+3w21+3wnξ≤3w1+3w2+Λεξψε3ξwnξ13wn+1ξ1+3w1+3w2ξ.
Conversely, we assume thatwn is not bounded sequence. So, there exists a subsequence wnj of wn such that limj⟶∞wnj=∞. If we take ε=εj=1+3w1+3w2/wnj in (39) inequality; then we have
(40)1≤Λ3ξεξwnξ1+3w1+3w2ξ13wnj+1ξψεj≤Λ3ξ1+3w1+3w2ξ1+3w1+3w2ξ13wnj+1ξψεj≤Λ3ξ1+3w1+3w22ξ13wnj+1ξψεj.
If we take limit in (40) inequality as j⟶∞, then we get
(41)Λ3ξ1+3w1+3w22ξ13wnj+1ξψεj⟶0,which is a contradiction. Therefore, we demonstrate that the sequencewn is bounded. So, there exists A>0 such that wn≤A for all n∈ℕ. Following this line of work, we demonstrate that δ=0. Since S is a weak E-Pata convex contractive mapping, we have
(42)ζαwn−1,wnρwn+1,wn+2,1−εEIwn−1,wn−ϕEIwn−1,wn+ΛεξψεPIwn−1,wn≥0,1−εEIwn−1,wn−ϕEIwn−1,wn+ΛεξψεPIwn−1,wn−αwn−1,wnρwn+1,wn+2≥0.
Sinceρn+1≤ρn for all n∈ℕ, we have
(43)EIwn−1,wn=ρwn,wn+1+ρwn,wn+1−ρwn+1,wn+2=2ρwn,wn+1−ρwn+1,wn+2=2ρn+1−ρn+2.
Since the sequencewn is bounded, we have
(44)PIwn−1,wn=Λεξψε1+wn−1+wn+wn+wn+1+wn+2+wn+3ϑ≤Λεξψε1+6Aϑ.
Now, we can write(45)ρn+2=ρwn+1,wn+2≤αwn−1,wnρwn+1,wn+2≤1−εEIwn−1,wn−ϕEIwn−1,wn+ΛεξψεPIwn−1,wn≤1−ε2ρn+1−ρn+2−ϕ2ρn+1−ρn+2+Λεξψε1+6Aϑ.
If we take the limit asn⟶∞ in (45) inequality, then we obtain
(46)δ≤1−εδ−ϕδ+Λεξψε1+6Aϑδ≤Λεξ−1ψε1+6Aϑ.
δ≤0 as ε⟶0, that is limn⟶∞ρwn+1,wn+2=δ=0. Now, we demonstrate that wn is a Cauchy sequence. On the contrary, assume that the sequence wn is not a Cauchy. From Lemma 10, there exist subsequence wmjand wnj with nj>mj>j such that limk⟶∞ρxmk−1,xnk+1=ς, limk⟶∞ρxmk−1,xnk=ς, limk⟶∞ρxmk,xnk=ς, limk⟶∞ρxmk+1,xnk+1=ς, and limk⟶∞ρxmk,xnk−1=ς. Since S is a weak E−Pata convex contractive mapping, we have
(47)ζαwnj−1,wmj−1ρwnj+1,wmj+1,1−εEIwnj−1,wmj−1−ϕEIwnj−1,wmj−1≥0,1−εEIwnj−1,wmj−1−ϕEIwnj−1,wmj−1−αwnj−1,wmj−1ρwnj+1,wmj+1≥0,where
(48)EIwnj−1,wmj−1=ρwnj,wmj+ρwnj,wnj+1−ρwmj,wmj+1,PIwnj−1,wmj−1=Λεξψε1+wnj−1+wmj−1+wnj+wmj+wnj+1+wmj+1ϑ=Λεξψε1+6Aϑ.
Now, we have(49)ς≤ρwnj+1,wmj+1≤αwnj−1,wmj−1ρwnj+1,wmj+1≤1−εEIwnj−1,wmj−1−ϕEIwnj−1,wmj−1+ΛεξψεPIwnj−1,wmj−1≤1−ερwnj,wmj+ρwnj,wnj+1−ρwmj,wmj+1−ϕρwnj,wmj+ρwnj,wnj+1−ρwmj,wmj+1+Λεξψε1+6Aϑ≤1−ερwnj,wmj+ρwnj,wnj+1−ρwmj,wmj+1+Λεξψε1+6Aϑ.
If we take the limit asj⟶∞, then we obtain
(50)ς≤1−ες+Kεψε,and so, we have
(51)ς≤Kψε,and thus, we get that ς=0, which is a contradiction. Therefore, we concluded that wn is a Cauchy sequence in W,ρ. By the completeness of W, the sequence wn is convergent to some ω∈W that is wn⟶ω as n⟶+∞. Since S is continuous, Swn⟶Sω as n⟶+∞. By the uniqueness of the limit, we obtain ω=Sω that is ω is a fixed point of S.
Next, we will demonstrate the uniqueness of the fixed point. Suppose thatΤ and ω are two fixed points of S. Since S satisfies the hypothesis (iv) of Theorem 13, S is an weak E-Pata convex contractive mapping; we have
(52)ρω,Τ≤αω,ΤρS2ω,S2Τ≤1−εEIω,Τ−ϕEIω,Τ+ΛεξψεPIω,Τ≤1−ερSω,SΤ+ρSω,S2ω−ρSΤ,S2Τ−ϕρSω,SΤ+ρSω,S2ω−ρSΤ,S2Τ+Λεξψε1+ω+Τ+Sω+SΤ+S2ω+S2Τϑ≤1−ερω,Τ+Λεξψε1+3ω+3Τϑ.
We obtain thatρω,Τ<Kψε for some K≥0, and so, we get ω=Τ. Hence, S has a unique fixed point in W, that is ω=Sω,ω∈W.
Following this line of work, Theorem14 does not require the continuity of S.Theorem 14.
LetW,ρ be a complete metric space, α:W×W⟶0,+∞ and S:W⟶W be a weak Pata-convex mapping. Suppose that
(i)
S is triangular α−admissible(ii)
there existsw0∈W such that αw0,Sw0≥1(iii)
S2 is continuous and for allω∈FixS2, αSω,ω≥1(iv)
for allw,ω∈FixS2, αw,ω≥1
Then,S has a unique fixed point in W.Proof.
Following the proof of Theorem13, we have already proved that wn is a Cauchy sequence in W. Since W is complete, we have wn⟶ω∈W as n⟶+∞. Taking into account hypothesis (iii) Theorem 14, we have limn⟶∞ρwn,S2ω=limn⟶∞ρS2wn−2,S2ω=0. In the uniqueness of the limit, we obtain that S2ω=ω. Next, we will prove that ω=Sω. On the contrary, we assume that ω is not fixed point of S. So, we have
(53)0<ρSω,ω=ρS2Sω,S2ω≤αSω,ωρS2Sω,S2ω≤1−εEISω,ω−ϕEISω,ω+ΛεξψεPISω,ω≤1−ερSω,S2ω+ρSω,S2ω−ρS2ω,S3ω−ϕρSω,S2ω+ρSω,S2ω−ρS2ω,S3ω+Λεξψε1+Sω+ω+Sω+S2ω+S3ω+S2ωϑ≤1−ερSω,ω−ϕρSω,ω+Kεξψε,for some K>0. We obtain
(54)ρSω,ω<1−ερSω,ω+Kεξψε.
Forε=0 in (54) which is a contradiction. Thus, we make an inference that Sω=ω, and so, ω is a fixed point of S. Following the proof of Theorem 13, the uniqueness of fixed point of S can be obtained.
Theorem15 is other fundamental result of our work.Theorem 15.
LetW,ρ be a complete metric space, α:W×W⟶0,+∞ and S:W⟶W be a weak Pata convex contractive mapping via simulation function. On the assumption that all of the Theorem 13 hypotheses are satisfied, then h has a unique fixed point.Proof.
In the proof of Theorem13, we have got that
(55)αSnw0,Sn+1w0≥1foralln∈ℕandαSnw0,Smw0≥1forallm>n≥0.
Settingℓ=minρw0,Sw0,ρSw0,S2w0 and now, we demonstrate that
ρSnw0,Sn+1w0 is a nonincreasing sequence. Since S is a weak Pata convex contractive mapping via simulation function, we have
(56)ζαw0,Sw0ρS2w0,S3w0,1−εMIw0,Sw0−ϕMIw0,Sw0+ΛεξψεPIw0,Sw0≥0.
Using hypothesis (ii) of the Theorem15, we get
(57)ρS2w0,S3w0≤αw0,Sw0ρS2w0,S3w0≤1−εMIw0,Sw0−ϕMIw0,Sw0+ΛεξψεPIw0,Sw0=1−εmaxρw0,Sw0,ρSw0,S2w0,ρw0,Sw0,ρSw0,S2w0,ρSw0,S2w0,ρS2w0,S3w0−ϕmaxρw0,Sw0,ρSw0,S2w0,ρw0,Sw0,ρSw0,S2w0,ρSw0,S2w0,ρS2w0,S3w0+Λεξψε1+w0+2Sw0+2S2w0+S3w0ϑ≤1−εmaxρw0,Sw0,ρSw0,S2w0,ρS2w0,S3w0−ϕmaxρw0,Sw0,ρSw0,S2w0,ρS2w0,S3w0+Kεξψε,for some K>0. Assuming that maxℓ,ρS2w0,S3w0=ρS2w0,S3w0, then we have ρSw0,S2w0<ρS2w0,S3w0. Thus, we have
(58)ρS2w0,S3w0≤1−ερS2w0,S3w0−ϕρS2w0,S3w0+Kεξψε,and since ρS2w0,S3w0≥ρS2w0,S3w0−ϕρS2w0,S3w0, we have
(59)ρS2w0,S3w0<1−ερS2w0,S3w0+Kεξψε.
The inequality (59) is true for all ε∈0,1. For ε=0, we obtain ρS2w0,S3w0<ρS2w0,S3w0 which is a contradiction. Hence, we obtain
(60)ρS2w0,S3w0≤ℓ.
Analogously, sinceS is a weak Pata convex contractive mapping via simulation function, we have
(61)ζαSw0,S2w0ρS3w0,S4w0,1−εMISw0,S2w0−ϕMISw0,S2w0+ΛεξψεPISw0,S2w0≥0,1−εMISw0,S2w0−ϕMISw0,S2w0+ΛεξψεPISw0,S2w0−αSw0,S2w0ρS3w0,S4w0≥0,and we can write that
(62)ρS3w0,S4w0≤αSw0,S2w0ρS3w0,S4w0≤1−εmaxρSw0,S2w0,ρS2w0,S3w0,ρSw0,S2w0,ρS2w0,S3w0,ρS2w0,S3w0,ρS3w0,S4w0−ϕmaxρSw0,S2w0,ρS2w0,S3w0,ρSw0,S2w0,ρS2w0,S3w0,ρS2w0,S3w0,ρS3w0,S4w0+Λεξψε1+Sw0+S2w0+S2w0+S3w0+S3w0+S4w0ϑ≤1−εmaxρSw0,S2w0,ρS2w0,S3w0,ρS3w0,S4w0+Kεξψεfor some K>0. In case that
(63)maxρSw0,S2w0,ρS2w0,S3w0,ρS3w0,S4w0=ρS3w0,S4w0,then we have
(64)ρS3w0,S4w0<1−ερS3w0,S4w0+Kεξψε.
The inequality (64) is true for all ε∈0,1. For ε=0, we obtain ρS3w0,S4w0<ρS3w0,S4w0 is again a contradiction. Therefore, we obtain
(65)ρS3w0,S4w0≤ρS2w0,S3w0≤ℓ.
Again, by induction, sinceS is a weak Pata convex contractive mapping via simulation function, we have
(66)ζαSn−2w0,Sn−1w0ρSnw0,Sn+1w0,1−εMISn−2w0,Sn−1w0−ϕMISn−2w0,Sn−1w0+ΛεξψεPISn−2w0,Sn−1w0≥0,1−εMISn−2w0,Sn−1w0−ϕMISn−2w0,Sn−1w0+ΛεξψεPISn−2w0,Sn−1w0−αSn−2w0,Sn−1w0ρSnw0,Sn+1w0≥0,and we have that
(67)ρSnw0,Sn+1w0≤αSn−2w0,Sn−1w0ρSnw0,Sn+1w0≤1−εmaxρSn−2w0,Sn−1w0,ρSn−1w0,Snw0,ρSn−2w0,Sn−1w0,ρSn−1w0,Snw0,ρSn−1w0,Snw0,ρSnw0,Sn+1w0ϕmaxρSn−2w0,Sn−1w0,ρSn−1w0,Snw0,ρSn−2w0,Sn−1w0,ρSn−1w0,Snw0,ρSn−1w0,Snw0,ρSnw0,Sn+1w0+Λεξψε1+Sn−2w0+Sn−1w0+Sn−1w0+Snw0+Snw0+Sn+1w0ϑ<1−εmaxρSn−2w0,Sn−1w0,ρSn−1w0,Snw0,ρSnw0,Sn+1w0−ϕmaxρSn−2w0,Sn−1w0,ρSn−1w0,Snw0,ρSnw0,Sn+1w0+Kεξψε,for some K>0. In case that maxρSn−2w0,Sn−1w0,ρSn−1w0,Snw0,ρSnw0,Sn+1w0=ρSnw0,Sn+1w0, then we have
(68)ρSnw0,Sn+1w0<1−ερSnw0,Sn+1w0+Kεξψε.
Again, the inequality (68) is true for all ε∈0,1 and for ε=0, we obtain ρSnw0,Sn+1w0<ρSnw0,Sn+1w0 is again a contradiction. Consequently, we can find that
(69)ρSnw0,Sn+1w0≤ρSn−1w0,Snw0≤⋯≤ρS3w0,S4w0≤ρS2w0,S3w0≤ρSw0,S2w0.
Starting at the pointw0∈W, the sequence wn is constructed by wn=Swn−1=Snw0, n≥1. If wn0+1=wn0 for any n0∈ℕ, then wn0 is a fixed point of S. Hereby, assume that wn0+1≠wn0 for all n0∈ℕ and let ρn=ρwn−1,wn. Therefore, we get that ρn is a nonincreasing sequence. Thereupon, there exists a δ≥0 such that
(70)limn⟶∞ρwn−1,wn=limn⟶∞ρn=δ.
We will demostrate thatδ=0. For this, we should demostrate that the sequence wn is bounded. Since ρn is a nonincreasing sequence, we have
(71)ρn+1=ρwn,wn+1≤ρwn−1,wn≤⋯≤ρw3,w4≤ρw2,w3≤ρw1,w2=ρ2≤w1+w2.
From the triangle inequality, we can write(72)wn=ρwn,w0≤ρwn,wn+1+ρwn+1,w2+ρw2,w0=ρn+1+ρwn+1,w2+w2≤ρ2+ρwn+1,w2+w2≤w1+2w2+ρwn+1,w2.
SinceS is a weak Pata convex contractive mapping via simulation function, we have
(73)ζαwn,w0ρwn+1,w2,1−εMIwn−1,w0−ϕMIwn−1,w0+ΛεξψεPIwn−1,w0≥0,1−εMIwn−1,w0−ϕMIwn−1,w0+ΛεξψεPIwn−1,w0−αwn,w0ρwn+1,w2≥0.
Together with (71), we obtain that
(74)ρwn+1,w2≤αwn,w0ρwn+1,w2≤1−εMIwn−1,w0−ϕMIwn−1,w0+ΛεξψεPIwn−1,w0.
From (71) and ρ2≤w1+w2, we have
(75)MIwn−1,w0=maxρwn−1,w0,ρwn,w1,ρwn−1,wn,ρw0,w1,ρwn,wn+1,ρw1,w2=maxρwn−1,w0,ρwn,w1,ρn,ρ1,ρn+1,ρ2≤w1+w2+wn,PIwn−1,w0=1+wn−1+w0+wn+w1+wn+1+w2ϑ≤1+3w1+3w2+3wnϑ.
Now, we derive that(76)εwn<2−εw1+3−εw2+Λεξψε1+3w1+3w2+3wnϑ≤2−εw1+3−εw2+Λεξψε1+3wnξ1+3w1+3w21+3wnξ≤2w1+3w2+Λεξψε3ξwnξ13wn+1ξ1+3w1+3w2ξ.
Contrarily, supposing thatwn is not bounded sequence. Thence, there exists a subsequence wnj of wn such that limj⟶∞wnj=∞. If we take ε=εj=1+3w1+3w2/wnj in (76) inequality, then we have
(77)1≤Λ3ξεξwnξ1+3w1+3w2ξ13wnj+1ξψεj≤Λ3ξ1+3w1+3w2ξ1+3w1+3w2ξ13wnj+1ξψεj≤Λ3ξ1+3w1+3w22ξ13wnj+1ξψεj.
If we take limit in (77) inequality as j⟶∞, then we get that
(78)Λ3ξ1+3w1+3w22ξ13wnj+1ξψεj⟶0is a contradiction. Next, we show that the sequencewn is bounded. So, there exists A>0 such that wn≤A for all n∈ℕ. Following this line of work, we will demonstrate that δ=0. Since S is a weak Pata convex contractive mapping via simulation function, we have
(79)ζαwn−1,wnρwn+1,wn+2,1−εMIwn−1,wn−ϕMIwn−1,wn+ΛεξψεPIwn−1,wn≥0,1−εMIwn−1,wn−ϕMIwn−1,wn+ΛεξψεPIwn−1,wn−αwn−1,wnρwn+1,wn+2≥0,where
(80)MIwn−1,wn=maxρwn−1,wn,ρwn,wn+1,ρwn−1,wn,ρwn,wn+1,ρwn,wn+1,ρwn+1,wn+2=maxρwn−1,wn,ρwn,wn+1,ρwn+1,wn+2≤w1+w2+wn.
Since the sequencewn is bounded, we have
(81)PIwn−1,wn=Λεξψε1+wn−1+wn+wn+wn+1+wn+2+wn+3ϑ≤Λεξψε1+6Aϑ.
Therefore, we have(82)ρn+2=ρwn+1,wn+2≤αwn−1,wnρwn+1,wn+2≤1−εMIwn−1,wn−ϕMIwn−1,wn+ΛεξψεPIwn−1,wn≤1−εmaxρwn−1,wn,ρwn,wn+1,ρwn+1,wn+2−ϕmaxρwn−1,wn,ρwn,wn+1,ρwn+1,wn+2+Λεξψε1+6Aϑ.
If the limit is taken asn⟶∞ in (82) inequality, then we get
(83)δ≤1−εδ−ϕδ+Λεξψε1+6Aϑδ≤Λεξ−1ψε1+6Aϑ.
δ≤0 as ε⟶0, that is limn⟶∞ρwn+1,wn+2=δ=0. Now, we demonstrate that wn is a Cauchy sequence. Contrarily, supposing that the sequence wn is not a Cauchy. From Lemma 10, we say that there exist subsequence wmjand wnj with nj>mj>j such that limk⟶∞ρxmk−1,xnk+1=ς, limk⟶∞ρxmk,xnk=ς, limk⟶∞ρxmk−1,xnk=ς, limk⟶∞ρxmk+1,xnk+1=ς, and limk⟶∞ρxmk,xnk−1=ς. Since S is a weak Pata convex contractive mapping, we have
(84)ζαwnj−1,wmj−1ρwnj+1,wmj+1,1−εMIwnj−1,wmj−1−ϕMIwnj−1,wmj−1+ΛεξψεPIwnj−1,wmj−1≥0,1−εMIwnj−1,wmj−1−ϕMIwnj−1,wmj−1+ΛεξψεPIwnj−1,wmj−1−αwnj−1,wmj−1ρwnj+1,wmj+1≥0where
(85)MIwnj−1,wmj−1=maxρwnj−1,wmj−1,ρwnj,wmj,ρwnj−1,wnj,ρwmj−1,wmj,ρwnj,wnj+1,ρwmj,wmj+1,PIwnj−1,wmj−1=Λεξψε1+wnj−1+wmj−1+wnj+wmj+wnj+1+wmj+1ϑ=Λεξψε1+6Aϑ.
Now, we can write(86)ς≤ρwnj+1,wmj+1≤αwnj−1,wmj−1ρwnj+1,wmj+1≤1−εMIwnj−1,wmj−1−ϕMIwnj−1,wmj−1+ΛεξψεPIwnj−1,wmj−1≤1−εmaxρwnj−1,wmj−1,ρwnj,wmj,ρwnj−1,wnj,ρwmj−1,wmj,ρwnj,wnj+1,ρwmj,wmj+1−ϕmaxρwnj−1,wmj−1,ρwnj,wmj,ρwnj−1,wnj,ρwmj−1,wmj,ρwnj,wnj+1,ρwmj,wmj+1+Λεξψε1+6Aϑ.
If we take the limit asj⟶∞, we get
(87)ς≤1−ες−ϕς+Kεψε≤1−ες+Kεψε,and so, we have
(88)ς≤Kψε,that is, we get ς=0 which is a contradiction. Therefore, we concluded that wn is a Cauchy sequence in W,ρ. By the completeness of W, the sequence wn is convergent to some ω∈W that is wn⟶ω as n⟶+∞. Since S is continuous, Swn⟶Sω as n⟶+∞. By the uniqueness of the limit, we obtain ω=Sω that is ω is a fixed point of S.
Now, we will demonstrate that the fixed point is unique. Assuming thatΤ and ω are two fixed points of S. From hypothesis (iv) of Theorem 15 and since S is an a weak Pata convex contractive mapping via simulation function, we have
(89)ζαω,ΤρS2ω,S2Τ,1−εMIω,Τ−ϕMIω,Τ+ΛεξψεPIω,Τ≥0,1−εMIω,Τ−ϕMIω,Τ+ΛεξψεPIω,Τ−αω,ΤρS2ω,S2Τ≥0,and so, we have
(90)ρω,Τ≤αω,ΤρS2ω,S2Τ≤1−εMIω,Τ−ϕMIω,Τ+ΛεξψεPIω,Τ≤1−εmaxρω,Τ,ρSω,SΤ,ρω,Sω,ρΤ,SΤ,ρSω,S2ω,ρSΤ,S2Τ−ϕmaxρω,Τ,ρSω,SΤ,ρω,Sω,ρΤ,SΤ,ρSω,S2ω,ρSΤ,S2Τ+Λεξψε1+ω+Τ+Sω+SΤ+S2ω+S2Τϑ≤1−ερω,Τ+Λεξψε1+3ω+3Τϑ.
We obtain thatρω,Τ<Kψε for some K≥0, and thus, we get ω=Τ. Hence, S has a unique fixed point in W.Example 16.
LetW,. the usual metric space where W=0,3/2. Let define the mappings S:W⟶W by
(91)Sw=w2+13,w∈0,1,12,w∈1,32,
ϕ:0,+∞⟶0,+∞ by ϕw=w/10 and α:W×W⟶0,+∞ by
(92)αw,υ=1,w,υ∈0,1,0,w,υ∉0,1.
It is easily seen thatS is a triangular α-admissible mapping, and also, S2w=w4+2w2+10/27,w∈0,3/2. Though the mapping, S is discontinuous in x=1 and S2 is continuous on W=0,3/2. Now, we want to demonstrate that S satisfies (11). For w,v∈0,1, we have
(93)ρS2w,S2v=w4+2w227−v4+2v227≤29w−v+12Sw−Sv=34827w−v+23Sw−Sv≤34maxw−v,Sw−hv≤34MIw,v.
Sinceϕw=w/10 and αw,υ=1, for w,υ∈0,1, we get that
(94)αw,υρS2w,S2v≤56910MIw,v=56MIw,v−ϕMIw,v.
For arbitraryε∈0,1, as one can see, the above inequality turns into the following inequality,
(95)αw,υρS2w,S2v≤1−εMIw,v−ϕMIw,v+34+ε−1MIw,v≤1−εMIw,v−ϕMIw,v+34+ε−11+w+υ+Sw+Sυ+S2w+S2υ.
Now, our goal is to show thatγ≥0 and Λ≥0 such that
(96)34+ε−11+w+υ+Sw+Sυ+S2w+S2υ≤Λεγ+11+w+υ+Sw+Sυ+S2w+S2υholds for all w,v∈0,1, and every 0≤ε≤1. We can find Λ≥0 such that
(97)Λ=3/4+ε−1εγ+1
holds for each0≤ε≤1 and some γ≥0. If we choose γ such that γ/γ+1>1−3/4, then
(98)Λ=γγγ+1γ+11−3/4γ.
Hence, we have that(99)αw,vρS2w,S2v≤1−εMIw,v−ϕMIw,v+Λεγ+11+w+υ+Sw+Sυ+S2w+S2υ.
Now, we can write(100)1−εMIw,v−ϕMIw,v+Λεγ+11+w+υ+Sw+Sυ+S2w+S2υ−αw,vρS2w,S2v≥0,and for ζ∈Z′, we have
(101)ζαw,vρS2w,S2v,1−εMIw,v−ϕMIw,v+Λεγ+11+w+υ+Sw+Sυ+S2w+S2υ≥0,which satisfies for each ε>0 and all w,v∈0,1. If ε=0, it can be seen that (11) is satisfied. Hence, all conditions of Theorem 15 are satisfied with ξ=ϑ=1 and ψε=εγ. By an application of Theorem 15, S has a unique fixed point in W=0,3/2.Suppose thatε=0 in Theorems 13 and 15; then we obtain the following corollaries.Corollary 17.
LetW,ρ be a complete metric space and ζ∈Z′ and S:W⟶W be two functions. If for all w,υ∈W, there exists a function, α:W×W⟶0,+∞ such that S satisfies the inequality either
(102)ζαw,υρS2w,S2υ,EIw,υ−ϕEIw,υ≥0orζαw,υρS2w,S2υ,MIw,υ−ϕMIw,υ≥0,where ϕ:0,+∞⟶0,+∞ is a continuous and nondecreasing function with ϕ0=0 and ϕw>0, for all w>0, and assuming that all of the hypotheses of Theorem 13 are satisfied, then S has a unique fixed point.Karapinar’s contractive conditions [10, 32, 45] are a special case of ours, and also, Corollary 17 generalizes the results of Samet [17] and Istratescu [26–28].Corollary 18.
LetW,ρ be a complete metric space and S:W⟶W be a function. If for all w,υ∈W, there exist two functions, α:W×W⟶0,+∞ such that S satisfies the inequality either
(103)αw,υρS2w,S2υ≤EIw,υ−ϕEIw,υorαw,υρS2w,S2υ≤MIw,υ−ϕMIw,υ,where ϕ:0,+∞⟶0,+∞ is a continuous and nondecreasing function with ϕ0=0 and ϕw>0, for all w>0, and assuming that all of the hypotheses of Theorem 13 are satisfied, then S has a unique fixed point.In comparison with recent results such as Alber et al. [14] and Zhang [16], our results are a generalization of them.Corollary 19.
LetW,ρ be a complete metric space and S:W⟶W be a function. If for all w,υ∈W, there exists a function α:W×W⟶0,+∞ such that S satisfies the inequality either
(104)αw,υρS2w,S2υ≤EIw,υorαw,υρS2w,S2υ≤MIw,υ,and assuming that all of the hypotheses of Theorem 13 are satisfied, then h has a unique fixed point.Puttingαw,υ=1 in Theorems 13 and 15, we can see the following results.Corollary 20.
LetW,ρ be a complete metric space and ζ∈Z′ and S:W⟶W be two functions. If for all w,υ∈W, and ε∈0,1, there exists a function ψ∈Ψ, such that S satisfies the inequality either
(105)ζρS2w,S2υ,1−εEIw,υ−ϕEIw,υ+ΛεξψεPIw,υ≥0orζρS2w,S2υ,1−εMIw,υ−ϕMIw,υ+ΛεξψεPIw,υ≥0,where ϕ:0,+∞⟶0,+∞ is a continuous and nondecreasing function with ϕ0=0 and ϕw>0, for all w>0, and assuming that all of the hypotheses of Theorem 13 are satisfied, then S has a unique fixed point.Corollary 21.
LetW,ρ be a complete metric space and S:W⟶W be a function. If for all w,υ∈W, and ε∈0,1, there exists a function ψ∈Ψ, such that S satisfies the inequality either
(106)ρS2w,S2υ≤1−εEIw,υ−ϕEIw,υ+ΛεξψεPIw,υorρS2w,S2υ≤1−εMIw,υ−ϕMIw,υ+ΛεξψεPIw,υ,where ϕ:0,+∞⟶0,+∞ is a continuous and nondecreasing function with ϕ0=0 and ϕw>0, for all w>0, and assuming that all of the Theorem 13 hypotheses are satisfied, then S has a unique fixed point.Assume now thatαw,υ=1 and ε=0 in Theorem 13 and Theorem 15; then we get the following corollaries.Corollary 22.
LetW,ρ be a complete metric space and ζ∈Z′, and S:W⟶W be two functions. If for all w,υ∈W, S satisfies the inequality either
(107)ζρS2w,S2υ,EIw,υ−ϕEIw,υ≥0orζρS2w,S2υ,MIw,υ−ϕMIw,υ≥0,where ϕ:0,+∞⟶0,+∞ is a continuous and nondecreasing function with ϕ0=0 and ϕs>0, for all s>0 and assume that S is continuous or S2 is continuous. Then, S has a unique fixed point that is ω=Sω,ω∈W.Corollary 23.
LetW,ρ be a complete metric space and S:W⟶W be a function. If for all w,υ∈W, S satisfies the inequality either
(108)ρS2w,S2υ≤EIw,υ−ϕEIw,υorρS2w,S2υ≤MIw,υ−ϕMIw,υ,where ϕ:0,+∞⟶0,+∞ is a continuous and nondecreasing function with ϕ0=0 and ϕw>0, for all w>0 and assume that S is continuous or S2 is continuous. Then, S has a unique fixed point that is ω=Sω,ω∈W.We derive that the main result of Pata [2] and Banach [1] can be expressed as a corollary of our main result.


## 3. Conclusion

We present the concept of weakE-Pata convex contractions and weak Pata convex contractions in metric spaces in this paper. After that, we investigate the existence of a fixed point for our novel type contraction and we state some consequences. Our results generalize and merge the results derived by Istratescu [26] and Pata [2] and some other related results in the literature. Besides the corollaries in this paper, to underline the novelty of our given results, we give an example that shows that Theorem 15 is a genuine generalization of Istratescu’s results [26]. Our novel concept allows for further studies and applications.



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