# Fixed Point Theorems for Generalizedθ-ϕ-Contractions in G-Metric Spaces

Authors: Dingwei Zheng
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We introduce the notion of generalizedθ-ϕ contraction and establish some new fixed point theorems for this contraction in the setting of complete G-metric spaces. The results presented in the paper improve, extend, and unify some known results. Finally, we give an example to illustrate them.


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## Body

## 1. Introduction and Preliminaries

In 2006, Mustafa and Sims [1] introduced the notion of G-metric space and studied the properties of it. Subsequently, many authors studied the fixed point theory in the setting of complete G-metric spaces and obtained some fixed point theorems for different contractions (see [1–10]). In 2015, Agarwal et al. [11] presented a self-contained account of the fixed point theory (techniques and results) in G-metric spaces. The book [11] contains almost all the research findings that relate to basic fixed point theorems, common fixed point theorems, and coupled fixed point theorems in G-metric spaces and partially ordered G-metric spaces (see [11] and the references therein).In 2014, Jleli and Samet [12] introduced a new type of contraction called θ-contraction. Later, many authors have studied θ-contraction deeply (for example, see [13, 14]). Just recently, Zheng et al. [15] introduced the notion of θ-ϕ contraction in metric spaces which generalized θ-contraction and other contractions (see [12, 15] and the references therein).Inspired by [12, 15], we introduce the notion of generalized θ-ϕ contraction and establish some new fixed point theorems for this contraction in the setting of complete G-metric spaces. The results presented in the paper improve and extend the corresponding results of Agarwal et al. [11], Mustafa [4], Mustafa et al. [5], Mustafa and Sims [6], and Shatanawi [9]. Also, we give an example to illustrate them.According to [12, 15], denote by Θ the set of functions θ:(0,∞)→(1,∞) satisfying the following conditions:(Θ1)
θ is nondecreasing.(Θ2)
For each sequence{tn}⊂(0,∞), limn→∞θtn=1 if and only if limn→∞tn=0+.(Θ3)
θ is continuous on (0,∞).And byΦ the set of functions ϕ:[1,∞)→[1,∞) satisfies the following conditions:(Φ1)
ϕ:[1,∞)→[1,∞) is nondecreasing.(Φ2)
For eacht>1, limn→∞ϕn(t)=1.(Φ3)
ϕ is continuous on [1,∞).Lemma 1 (see [15]).
Ifϕ∈Φ, then ϕ1=1 and ϕ(t)<t for each t>1.Now we recall some basic definitions and give some lemmas that will be used in the paper.Definition 2 (see [1, 11]).
AG-metric space is a pair (X,G), where X is a nonempty set and G:X×X×X→[0,+∞) is a function such that, for all x,y,z,a∈X, the following conditions are fulfilled:(G1)
G(x,y,z)=0 if x=y=z.(G2)
G(x,x,y)>0 for all x,y∈X with x≠y.(G3)
G(x,x,y)≤G(x,y,z) for all x,y,z∈X with z≠y.(G4)
G(x,y,z)=G(x,z,y)=G(y,z,x)=⋯ (symmetry in all 3).(G5)
G(x,y,z)≤G(x,a,a)+G(a,y,z) (rectangle inequality).
In such a case, the functionG is called a G-metric.Example 3 (see [1, 11]).
IfX is a nonempty subset of R, then the function G:X×X×X→[0,+∞), given by G(x,y,z)=x-y+y-z+z-x for all x,y,z∈X, is a G-metric on X.Example 4 (see [1, 11]).
LetX=0,∞ be the interval of nonnegative real numbers and let G be defined by (1)Gx,y,z=0,ifx=y=z,max⁡x,y,z,otherwise.
ThenG is a complete G-metric on X.Definition 5 (see [1, 11]).
Let(X,G) be a G-metric space; let x∈X and xn⊆X be a sequence. We say that(i)
{xn}  G-converges to x, and we write xn→x if limn,m→∞G(xn,xm,x)=0; that is, for all ε>0 there exists n0∈N satisfying G(xn,xm,x)≤ε for all n,m∈N such that n,m≥n0;(ii)
xn is G-Cauchy if limn,m,k→∞G(xn,xm,xk)=0; that is, for all ε>0 there exists n0∈N satisfying G(xn,xm,xk)≤ε for all n,m,k∈N such that n,m,k≥n0;(iii)
X,G is complete if every G-Cauchy sequence in X is G-convergent in X.Lemma 6 (see [1, 11]).
Let(X,G) be a G-metric space, let x∈X and xn⊆X be a sequence. Then the following conditions are equivalent.(a)
{xn}  G-converges to x.(b)
limn→∞G(xn,xn,x)=0.(c)
limn→∞G(xn,x,x)=0.(d)
limn,m→∞,m≥nG(xn,xm,x)=0.(e)
limn,m→∞,m>nG(xn,xm,x)=0.Lemma 7 (see [1, 11]).
Let(X,G) be a G-metric space and xn⊆X be a sequence. Then the following conditions are equivalent.(a)
{xn} is G-Cauchy.(b)
limn,m→∞G(xn,xm,xm)=0.(c)
limn,m→∞,m≥nG(xn,xm,xm)=0.(d)
limn,m→∞,m>nG(xn,xm,xm)=0.(e)
limn,m→∞G(xn,xn,xm)=0.Lemma 8 (see [11]).
Letxn be an asymptotically regular sequence in a G-metric space (X,G) and suppose that {xn} is not Cauchy. Then there exist a positive real number ε>0 and two subsequences {xn(k)} and {xm(k)} of {xn} such that, for all k∈N,(2)k≤nk<mk<nk+1,Gxnk,xmk-1,xmk-1≤ε<Gxnk,xmk,xmk,and also, for all given p1,p2,p3∈Z,(3)limk→∞⁡Gxnk+p1,xmk+p2,xmk+p3=ε.Lemma 9 (see [11]).
Let(X,G) be a G-metric space; then
G
(
x
,
y
,
y
)
≤
2
G
(
y
,
x
,
x
) for all x,y∈X.


## 2. Main Results

Based on the functionsθ∈Θ and ϕ∈Φ, we give the following definition.Definition 10.
Let(X,G) be a G-metric space. A mapping T:X→X is said to be a generalized θ-ϕ contraction if there exist θ∈Θ and ϕ∈Φ such that, for any x,y,z∈X,(4)GTx,Ty,Tz≠0⟹θGTx,Ty,Tz≤ϕθNx,y,z,where(5)Nx,y,z=max⁡Gx,y,z,Gx,Tx,Tx,Gy,Ty,Ty,Gz,Tz,Tz,12Gx,Ty,Ty,12Gy,Tz,Tz,12Gz,Tx,Tx,13Gx,Ty,Ty+Gy,Tz,Tz+Gz,Tx,Tx.Theorem 11.
Let(X,G) be a complete G-metric space and let T:X→X be a generalized θ-ϕ contraction. Then T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Proof.
Letx0∈X be an arbitrary point. We define the sequence xn in X by xn+1=Txn, for all n∈N. If xn+1=xn for some n∈N, then x∗=xn is a fixed point for T. Next, we assume that xn+1≠xn for all n∈N. Then Gxn,xn+1,xn+1>0 for all n∈N. Applying inequality (4) with x=xn, y=xn+1, z=xn+1, we obtain(6)θGTxn,Txn+1,Txn+1≤ϕθNxn,xn+1,xn+1,where(7)Nxn,xn+1,xn+1=max⁡Gxn,xn+1,xn+1,Gxn,xn+1,xn+1,Gxn+1,xn+2,xn+2,Gxn+1,xn+2,xn+212Gxn,xn+2,xn+2,12Gxn+1,xn+2,xn+2,12Gxn+1,xn+1,xn+1,13Gxn,xn+2,xn+2+Gxn+1,xn+1,xn+1+Gxn+1,xn+2,xn+2=max⁡Gxn,xn+1,xn+1,Gxn+1,xn+2,xn+2,12Gxn,xn+2,xn+2,13Gxn,xn+2,xn+2+Gxn+1,xn+2,xn+2=max⁡Gxn,xn+1,xn+1,Gxn+1,xn+2,xn+2.byG5.
IfN(xn,xn+1,xn+1)=G(xn+1,xn+2,xn+2), then it follows from (4) that(8)θGxn+1,xn+2,xn+2=θGTxn,Txn+1,Txn+1≤ϕθNxn,xn+1,xn+1=ϕθGxn+1,xn+2,xn+2<θGxn+1,xn+2,xn+2,by Lemma1,which is a contradiction. Hence, for ∀n∈N, (9)Nxn,xn+1,xn+1=Gxn,xn+1,xn+1.
Thus, (4) becomes(10)θGTxn,Txn+1,Txn+1≤ϕθGxn,xn+1,xn+1.
Repeating this process, we get(11)θGxn,xn+1,xn+1=θGTxn-1,Txn,Txn≤ϕθGxn-1,xn,xn≤ϕ2θGxn-2,xn-1,xn-1≤ϕ3θGxn-3,xn-2,xn-2≤⋯≤ϕnθGx0,x1,x1.
By the definition ofθ and (Φ2), we have(12)limn→∞⁡ϕnθGx0,x1,x1=1.
By(Θ2), we obtain(13)limn→∞Gxn,xn+1,xn+1=0.
Thus,xn is an asymptotically regular sequence.
In what follows, we shall prove thatxn is a Cauchy sequence in X.
Suppose, on the contrary, that, by Lemma8, there exist a positive real number ε0>0 and two subsequences xnk and xmk of xn such that, for all k∈N,(14)k≤nk<mk<nk+1,Gxnk,xmk-1,xmk-1≤ε0<Gxnk,xmk,xmk,and also, for all given p1=p2=p3∈Z,(15)limk→∞⁡Gxnk,xmk,xmk=limk→∞⁡Gxnk+p1,xmk+p2,xmk+p3=ε0.
Pickk large enough, by (13), (15), and Lemma 9,(16)Nxnk,xmk,xmk=maxGxnk,xmk,xmk,Gxnk,xnk+1,xnk+1,Gxmk,xmk+1,xmk+1,12Gxnk,xnk+2,xnk+2,12Gxmk,xmk+1,xmk+1,12Gxmk,xnk+1,xnk+1,13Gxnk,xnk+1,xnk+1+Gxmk,xmk+1,xmk+1+Gxmk,xnk+1,xnk+1=maxGxnk,xmk,xmk,12Gxmk,xnk+1,xnk+1≤maxGxnk,xmk,xmk,Gxnk+1,xnk+1,xmk⟶ε0,ask⟶∞.
Using the contractivity condition (4),(17)θGxnk+1,xmk+1,xmk+1=θGTxnk,Txmk,Txmk≤ϕθNxnk,xmk,xmk.
Passing to limit ask→∞, then we get
θ
ε
0
≤
ϕ
θ
ε
0
. By Lemma 1, ϕθε0<θε0, then θε0≤ϕθε0<θε0, which is a contradiction. Thus, xn is a Cauchy sequence in X.
Taking into account the fact that(X,G) is complete, there exists x∗∈X such that xn converges to x∗. In particular,(18)limn→∞⁡Gxn,x∗,x∗=0.Using the fact that G is continuous on each variable,(19)Gx∗,Tx∗,Tx∗=limn→∞⁡Gxn+1,Tx∗,Tx∗.
We claim thatx∗ is a fixed point of T. Suppose, on the contrary, if x∗≠Tx∗, then by (18), (19),(20)Nxn,x∗,x∗=max⁡Gxn,x∗,x∗,Gxn,xn+1,xn+1,Gx∗,Tx∗,Tx∗,Gx∗,Tx∗,Tx∗,12Gxn,Tx∗,Tx∗,12Gx∗,Tx∗,Tx∗,12Gx∗,xn+1,xn+1,13Gxn,Tx∗,Tx∗+Gx∗,Tx∗,Tx∗+Gx∗,xn+2,xn+2⟶Gx∗,Tx∗,Tx∗,asn→∞.
Using the contractivity condition (4),(21)θGxn+1,Tx∗,Tx∗=θGTxn,Tx∗,Tx∗≤ϕθNxn,x∗,x∗.
Passing to limit asn→∞, then we have(22)θGx∗,Tx∗,Tx∗≤ϕθGx∗,Tx∗,Tx∗.
By Lemma1, ϕθGx∗,Tx∗,Tx∗<θGx∗,Tx∗,Tx∗. Then(23)θGx∗,Tx∗,Tx∗≤ϕθGx∗,Tx∗,Tx∗<θGx∗,Tx∗,Tx∗,which is a contradiction. As a consequence, we conclude that Tx∗=x∗.
Now, we will prove thatT has at most one fixed point. Suppose, on the contrary, that there exists another distinct fixed point y∗ of T such that Tx∗=x∗≠Ty∗=y∗. Therefore, G(Tx∗,Ty∗,Ty∗)=G(x∗,y∗,y∗)>0, and N(x∗,y∗,y∗)=G(x∗,y∗,y∗), and then by (4)(24)θGx∗,y∗,y∗=θGTx∗,Ty∗,Ty∗≤ϕθNx∗,y∗,y∗=ϕθGx∗,y∗,y∗,and by Lemma 1, θGx∗,y∗,y∗≤ϕθGx∗,y∗,y∗<θGx∗,y∗,x∗, which is a contradiction. Therefore, the fixed point of T is unique.Theorem 12.
LetX,G be a complete G-metric space and let T:X→X be a self-mapping. Assume that there exist θ∈Θ and ϕ∈Φ such that, for any x,y∈X,(25)GTx,Ty,Ty≠0⟹θGTx,Ty,Ty≤ϕθNx,y,y,where(26)Nx,y,y=maxGx,y,y,Gx,Tx,Tx,Gy,Ty,Ty,12Gy,Tx,Tx,13Gx,Ty,Ty+Gy,Ty,Ty+Gy,Tx,Tx.
ThenT has a unique fixed point x∗∈X such that the sequence Tnx converges to x∗ for every x∈X.The following Theorem13 is the main result of [5].Theorem 13 (see [5]).
LetX,G be a complete G-metric space and let T:X→X be a self-mapping which satisfies the following condition, for all x,y∈X,(27)GTx,Ty,Ty≤max⁡aGx,y,y,bGx,Tx,Tx+2Gy,Ty,Ty,bGx,Ty,Ty+Gy,Ty,Ty+Gy,Tx,Tx,where 0≤a<1 and 0≤b<1/3. Then T has a unique fixed point x∗∈X such that the sequence Tnx converges to x∗ for every x∈X.Proof.
Letλ=max⁡{a,3b}; then 0≤λ<1. And let θt=et, ϕ(t)=tλ; then θ∈Θ and ϕ∈Φ. Since (28)maxaGx,y,y,bGx,Tx,Tx+2Gy,Ty,Ty,bGx,Ty,Ty+Gy,Ty,Ty+Gy,Tx,Tx≤λmax⁡Gx,y,y,13Gx,Tx,Tx+2Gy,Ty,Ty,13Gx,Ty,Ty+Gy,Ty,Ty+Gy,Tx,Tx≤λmax⁡Gx,y,y,Gx,Tx,Tx,Gy,Ty,Ty,13Gx,Ty,Ty+Gy,Ty,Ty+Gy,Tx,Tx≤λNx,y,y.Therefore,(29)θGTx,Ty,Ty=eGTx,Ty,Ty≤eλNx,y,y=eNx,y,yλ=ϕθNx,y,y.
From Theorem12, we can see that T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.The following Theorem14 is the main result of [6].Theorem 14 (see [6]).
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping which satisfies the following condition, for all x,y∈X,(30)GTx,Ty,Tz≤kmax⁡Gx,y,z,Gx,Tx,Tx,Gy,Ty,Ty,Gz,Tz,Tz,Gx,Ty,Ty,Gy,Tz,Tz,Gz,Tx,Tx,where 0≤k<1/2. Then T has a unique fixed point x∗∈X such that the sequence Tnx converges to x∗ for every x∈X.Proof.
Letλ=2k; then 0≤λ<1. And let θ(t)=et, ϕ(t)=tλ; then θ∈Θ and ϕ∈Φ. Since (31)kmax⁡Gx,y,z,Gx,Tx,Tx,Gy,Ty,Ty,Gz,Tz,Tz,Gx,Ty,Ty,Gy,Tz,Tz,Gz,Tx,Tx=λmax⁡12Gx,y,z,12Gx,Tx,Tx,12Gy,Ty,Ty,12Gz,Tz,Tz,12Gx,Ty,Ty,12Gy,Tz,Tz,12Gz,Tx,Tx≤λNx,y,z,therefore,(32)θGTx,Ty,Tz=eGTx,Ty,Tz≤eλNx,y,z=eNx,y,zλ=ϕθNx,y,z.
From Theorem11, we can see that T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Theorem 15.
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping. Assume that there exist θ∈Θ and ϕ∈Φ such that, for any x,y,z∈X,(33)GTx,Ty,Tz≠0⟹θGTx,Ty,Tz≤ϕθGx,y,z.Then T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Theorem 16 (see [4]).
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping such that there exists λ∈[0,1) satisfying, for any x,y,z∈X, (34)GTx,Ty,Tz≤λGx,y,z.
ThenT has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Proof.
Letθt=et, ϕt=tλ; then θ∈Θ and ϕ∈Φ.
G
T
x
,
T
y
,
T
z
≤
λ
G
(
x
,
y
,
z
) is equivalent to eGTx,Ty,Ty≤eλGx,y,z=eGx,y,zλ; that is, θ(G(Tx,Ty,Tz))≤ϕ(θ(G(x,y,z))).
From Theorem15, we can see that T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Corollary 17.
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping. Assume that there exist θ∈Θ and ϕ∈Φ such that, for any x,y∈X, (35)GTx,Ty,Ty≠0⟹θGTx,Ty,Ty≤ϕθGx,y,y.Then T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Corollary 18.
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping such that there exists λ∈[0,1) satisfying, for any x,y∈X, (36)GTx,Ty,Ty≤λGx,y,y.
ThenT has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Corollary 19.
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping. Assume that there exist θ∈Θ and ϕ∈Φ such that, for any x,y,z∈X, (37)GTx,Ty,Ty≠0⟹θGTx,Ty,Tz≤ϕθGx,Tx,Tx+Gy,Ty,Ty+Gz,Tz,Tz3.Then T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Corollary 20.
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping such that there exists λ∈[0,1) satisfying, for any x,y,z∈X, (38)GTx,Ty,Tz≤λGx,Tx,Tx+Gy,Ty,Ty+Gz,Tz,Tz3.Then T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Corollary 21.
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping. Assume that there exist θ∈Θ and ϕ∈Φ such that, for any x,y,z∈X, GTx,Ty,Ty≠0, we have(39)θGTx,Ty,Ty≤ϕθmax⁡Gx,Tx,Tx+Gy,Ty,Ty2,Gy,Ty,Ty+Gz,Tz,Tz2,Gx,Tx,Tx+Gz,Tz,Tz2.
ThenT has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Corollary 22.
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping which satisfies the following condition, for all x,y,z∈X,(40)GTx,Ty,Tz≤kmax⁡Gx,Tx,Tx+Gy,Ty,Ty,Gy,Ty,Ty+Gz,Tz,Tz,Gx,Tx,Tx+Gz,Tz,Tz,where 0≤k<1/2. Then T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Corollary 23.
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping. Assume that there exist θ∈Θ and ϕ∈Φ such that, for any x,y∈X, (41)GTx,Ty,Ty≠0⟹θGTx,Ty,Ty≤ϕθGx,Tx,Tx+Gy,Ty,Ty2.Then T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Corollary 24.
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping such that there exists λ∈[0,1) satisfying, for any x,y∈X, (42)GTx,Ty,Ty≤λGx,Tx,Tx+Gy,Ty,Ty2.Then T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Theorem 25 (see [9]).
Let(X,G) be a complete G-metric space and let T:X→X be a mapping such that, for all x,y,z∈X, (43)GTx,Ty,Tz≤φGx,y,z,where φ:[0,∞)→[0,∞) is an increasing continuous function such that limn→∞φnt=0 for t>0.
ThenT has a unique fixed point x∗∈X and for every x∈X the sequence {Tnx} converges to x∗.Proof.
Letθt=et for all t∈[0,+∞), and ϕ(t)=eφln⁡t for all t∈[1,+∞).
Obviously,θ∈Θ, ϕ∈Φ.
By the definition ofϕ, we have ϕ(et)=eφ(t).(44)θGTx,Ty,Tz=eGTx,Ty,Tz≤eφGx,y,z=ϕeGx,y,z=ϕθGx,y,z.Therefore, from Theorem 15, T has a unique fixed point x∗∈X and for every x∈X the sequence {Tnx} converges to x∗.Remark 26.
In [9], the function φ is not required to be continuous. But due to Theorem 1 of [16] and item 4.2.3 of [11], we can suppose that φ is continuous.
F
alt
=
{
φ
:
[
0
,
∞
)
→
[
0
,
∞
)
∣
φ is continuous and nondecreasing, φ(t)=0⇔t=0}, and
F
alt
′
=
{
ω
:
[
0
,
∞
)
→
[
0
,
∞
)
∣
ω is lower semicontinuous, ω(t)=0⇔t=0}.Theorem 27 (see [11]).
Let(X,G) be a complete G-metric space and let T:X→X be a self-mapping. Assume that there exist φ∈Falt and ω∈F′alt such that, for any x,y∈X,(45)φGTx,Ty,Ty≤φGx,y,y-ωGx,y,y.Then T has a unique fixed point x∗∈X such that the sequence {Tnx} converges to x∗ for every x∈X.Proof.
Due to Theorem1 of [16] and item 4.2.3 of [11], the condition where there exist φ∈Falt and ω∈Falt′ such that, for any x, y∈X, φ(G(Tx,Ty,Ty))≤φ(G(x,y,y))-ω(G(x,y,y)) is equivalent to the condition where there exist a∈[0,1) and ψ∈Falt such that, for any x, y∈X, ψGTx,Ty,Ty≤aψGx,y,y. Let θt=eψt,ϕt=ta; then θ∈Θ and ϕ∈Φ. ψGTx,Ty,Ty≤aψGx,y,y is equivalent to eψGTx,Ty,Ty≤eaψGx,y,y; that is, θGTx,Ty,Ty≤ϕθGx,y,y.
From Theorem15, T has a unique fixed point x∗∈X and for every x∈X the sequence {Tnx} converges to x∗.Remark 28.
According to fixed point theory of metric spaces, we divide contractions into different type in the setting ofG-metrics. Then Theorem 16 and Corollary 18 belong to Banach type, Corollaries 19–24 Kannan type [17], Theorem 25 Browder type [18], and Theorem 27 Choudhury type. To some extent, our results unify them.


## 3. Example

In this section, we give an example to illustrate our results.Example 29.
LetX=0,±1,±2,… be endowed with the G-metric Gx,y,z=x-y+y-z+z-x for all x,y,z∈X. Then X,G is a complete G-metric space. Define the mapping T:X→X by (46)Tx=0,ifx=0;-n-1,ifx=n;n-1,ifx=-n.
At first, we observe that Theorems16, 25, and 27 cannot be applied since for all x=n>y=z=m>2, G(Tx,Ty,Tz)=G(Tn,Tm,Tm)=2n-2m=G(n,m,m).
And Theorem13 cannot be applied too. In fact, let x=n, y=z=0; then G(Tx,Ty,Tz)=GTn,T0,T0=2n-2, while(47)max⁡aGx,y,y,bGx,Tx,Tx+2Gy,Ty,Ty,bGx,Ty,Ty+Gy,Ty,Ty+Gy,Tx,Tx=max⁡aGn,0,0,bGn,-n-1,-n-1+0,bGn,0,0+G0,T0,T0+G0,-n-1,-n-1=max⁡2an,b4n-2≤max⁡2an,4n-23,GTx,Ty,Ty≤max⁡aGx,y,y,bGx,Tx,Tx+2Gy,Ty,Ty,bGx,Ty,Ty+Gy,Ty,Ty+Gy,Tx,Tx,is equivalent to 2n-2≤max2an,4n-2/3.
Since2n-2≤max⁡{2an,4n-2/3} for all n∈N, we have a=1, which yields a contradiction since a<1.
By the same way, we can see that Theorem14 cannot be applied.
Now, let the functionθ:(0,∞)→(1,∞) be defined by (48)θt=5t.
And defineϕ:[1,∞)→[1,∞) by (49)ϕt=1,if1≤t≤2;t-1,ift≥2.
Obviously,θ∈Θ, ϕ∈Φ.
In what follows, we prove thatT is some θ-ϕ Kannan-type contraction; that is, T satisfies the condition of Corollary 23.
We consider three cases.Case 1 (x=n≥1, y=0 or x=-nn≥1, y=0).
In this case, we have(50)GTx,Ty,Ty=2n-1,Gx,Tx,Tx=22n-1,Gy,Ty,Ty=0,θGTx,Ty,Ty=θ2n-2=52n-2,ϕθGx,Tx,Tx+Gy,Ty,Ty2=ϕθGx,Tx,Tx2=ϕθ2n-1=ϕ52n-1=52n-1-1=5×52n-1-1≥52n-1=θGTx,Ty,Ty.Case 2 (x=n>y=m≥1 or x=-n<y=-m≤-1).
In this case, we have(51)GTx,Ty,Ty=2n-2m,Gx,Tx,Tx=4n-2,Gy,Ty,Ty=4m-2,ϕθGx,Tx,Tx+Gy,Ty,Ty2=ϕθ2n+2m-2=ϕ52n+2m-2=54m-2×52n-2m-1≥52n-2m=θGTx,Ty,Ty.Case 3 (x=n, y=-m,n>m≥1 or x=-n,y=m,n>m≥1).
In this case, we have(52)GTx,Ty,Ty=2n+2m-4,Gx,Tx,TX=4n-2,Gy,Ty,Ty=4m-2,ϕθGx,Tx,Tx+Gy,Ty,Ty2=ϕθ2n+2m-2=ϕ52n+2m-2=25×52n+2m-4-1≥52n+2m-4=θGTx,Ty.
Therefore, we have for allx,y∈X(53)θGTx,Ty,Ty≤ϕθGx,Tx,Tx+Gy,Ty,Ty2.
Thus,T is a θ-ϕ Kannan-type contraction.
So all the hypotheses of Corollary23 are satisfied, thus T has a fixed point. In this example x=0 is the fixed point.



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*Source: 1418725-2018-01-09.xml*