# On Quasi-Pseudometric Type Spaces

Authors: Eniola Funmilayo Kazeem; Collins Amburo Agyingi; Yaé Ulrich Gaba
Journal: Chinese Journal of Mathematics
 (2014)

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## Abstract

We introduce the concept of a quasi-pseudometric type space and prove some fixed point theorems. Moreover, we connect this concept to the existing notion of quasi-cone metric space.


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## Body

## 1. Introduction

Cone metric spaces were introduced in [1] and many fixed point results concerning mappings in such spaces have been established. In [2], Khamsi connected this concept with a generalised form of metric that he namedmetric type. Recently in [3], Shadda and Md Noorani discussed the newly introduced notion of quasi-cone metric spaces and proved some fixed point results of mappings on such spaces. Basically, cone metric spaces are defined by substituting, in the definition of a metric, the real line by a real Banach space that we endowed with a partial order. The fact that the introduced order is not linear does not allow us to always compare any two elements and then gives rise to a kind of duality in the definition of the induced topology, hence the convergence in such space. We introduce a quasi-pseudometric type structure and show that some proofs follow closely the classical proofs in the quasi-pseudometric case but generalize them.


## 2. Preliminaries

In this section, we recall some elementary definitions from the asymmetric topology which are necessary for a good understanding of the work below.Definition 1.
LetX be a nonempty set. A function d:X×X→[0,∞) is called aquasi-pseudometric on X if(i)
d(x,x)=0∀x∈X,(ii)
d(x,z)≤d(x,y)+d(y,z)∀x,y,z∈X.Moreover, if d(x,y)=0=d(y,x)⇒x=y, then d is said to be a T0-quasi-pseudometric. The latter condition is referred to as the T0-condition.Remark 2.
(i) Letd be a quasi-pseudometric on X; then the map d-1 defined by d-1(x,y)=d(y,x) whenever x,y∈X is also a quasi-pseudometric on X, called theconjugate of d. In the literature, d-1 is also denoted by dt or d-.
(ii) It is easy to verify that the functionds defined by ds≔d∨d-1, that is, ds(x,y)=max⁡{d(x,y),d(y,x)}, defines ametric on X whenever d is a T0-quasi-pseudometric.Let(X,d) be a quasi-pseudometric space. Then for each x∈X and ϵ>0, the set
(1)Bd(x,ϵ)={y∈X:d(x,y)<ϵ}
denotes the open ϵ-ball at x with respect to d. It should be noted that the collection
(2){Bd(x,ϵ):x∈X,ϵ>0}
yields a base for the topology τ(d) induced by d on X. In a similar manner, for each x∈X and ϵ≥0, we define
(3)Cd(x,ϵ)={y∈X:d(x,y)≤ϵ},
known as the closed ϵ-ball at x with respect to d.Also the collection(4){Bd-1(x,ϵ):x∈X,ϵ>0}
yields a base for the topology τ(d-1) induced by d-1 on X. The set Cd(x,ϵ) is τ(d-1)-closed but not τ(d)-closed in general.The balls with respect tod are often calledforward balls and the topology τ(d) is calledforward topology, while the balls with respect to d-1 are often calledbackward balls and the topology τ(d-1) is calledbackward topology.Definition 3.
Let(X,d) be a quasi-pseudometric space. The convergence of a sequence (xn) to x with respect to τ(d), called d-convergence orleft-convergence and denoted by xn→dx, is defined in the following way:
(5)xn⟶dx⟺d(x,xn)⟶0.
Similarly, the convergence of a sequence(xn) to x with respect to τ(d-1), called d-1-convergence orright-convergence and denoted by xn→d-1x, is defined in the following way:
(6)xn→d-1x⟺d(xn,x)⟶0.
Finally, in a quasi-pseudometric space(X,d), we will say that a sequence (xn)ds-converges to x if it is both left and right convergent to x, and we denote it as xn→dsx or xn→x when there is no confusion. Hence
(7)xn⟶dsx⟺xn⟶dx,xn⟶d-1x.Definition 4.
A sequence(xn) in a quasi-pseudometric (X,d) is called (a)
leftK-Cauchy with respect to d if, for every ϵ>0, there exists n0∈ℕ such that(8)∀n,k:n0≤k≤nd(xk,xn)<ϵ;
(b)
right
K
-Cauchy with respect to d if, for every ϵ>0, there exists n0∈ℕ such that
(9)
∀
n
,
k
:
n
0
≤
k
≤
n
d
(
x
n
,
x
k
)
<
ϵ
;
(c)
d
s-Cauchy if, for every ϵ>0, there exists n0∈ℕ such that
(10)∀n,k≥n0d(xn,xk)<ϵ.Remark 5.
(i) A sequence is leftK-Cauchy with respect to d if and only if it is right K-Cauchy with respect to d-1.
(ii) A sequence isds-Cauchy if and only if it is both left and right K-Cauchy.Definition 6.
A quasi-pseudometric space(X,d) is calledleft-complete provided that any left K-Cauchy sequence is d-convergent.Definition 7.
A quasi-pseudometric space(X,d) is calledright-complete provided that any right K-Cauchy sequence is d-convergent.Definition 8.
AT0-quasi-pseudometric space (X,d) is calledbicomplete provided that the metric ds on X is complete.We now recall some known definitions, notations, and results concerning cones in Banach spaces.Definition 9.
LetE be a real Banach space with norm ∥·∥ and let P be a subset of E. Then P is called a cone if and only if (1)
P is closed and nonempty and P≠{θ}, where θ is the zero vector in E;(2)
for anya,b≥0, and x,y∈P, one has ax+by∈P;(3)
forx∈P, if -x∈P, then x=θ.Given a coneP in a Banach space E, one defines on E a partial order ⪯ with respect to P by
(11)x⪯y⟺y-x∈P.
We also write x≺y whenever x⪯y and x≠y, while x≪y will stand for y-x∈Int(P) (where Int(P) designates the interior of P).The coneP is callednormal if there is a number C>0, such that for all x,y∈E, one has
(12)θ⪯x⪯y⟹∥x∥≤C∥y∥.
The least positive number satisfying this inequality is called thenormal constant of P. Therefore, one will then say that P is a K-normal cone to indicate the fact that the normal constant is K.Definition 10 (compare [3]).
LetX be a nonempty set. Suppose the mapping q:X×X→E satisfies (q1)
θ⪯q(x,y) for all x,y∈X;(q2)
q(x,y)=θ=q(y,x) if and only if x=y;(q3)
q(x,z)⪯q(x,y)+q(y,z) for all x,y,z∈X.Then, q is called aquasi-cone metric on X and (X,q) is called aquasi-cone metric space.Definition 11 (compare [3]).
A sequence in a quasi-cone metric space(X,q) is called(a)
Q-Cauchy orbi-Cauchy if, for every c∈X with c≫θ, there exists n0∈ℕ such that
(13)∀n,m≥n0q(xn,xm)≪c;(b)
left (right) Cauchy if, for every c∈X with c≫θ, there exists n0∈ℕ such that
(14)∀n,m:n0≤m≤nq(xm,xn)≪chh(q(xn,xm)≪cresp.).Remark 12.
A sequence isQ-Cauchy if and only if it is both left and right Cauchy.We also recall the following lemma, which we take from [4] and we give the proof as it is.Lemma 13 (compare [4, Lemma 2]).
Let(X,q) be a cone metric space. Then for each c∈E,c≫θ, there exists σ>0 such that x≪c whenever ∥x∥<σ,x∈E.Proof.
Sincec≫θ, then c∈Int(P). Hence, find σ>0 such that {x∈E:∥x-c∥<σ}⊂Int(P). Now if ∥x∥<σ then ∥(c-x)-c∥=∥-x∥=∥x∥<σ and hence (c-x)∈Int(P).Remark 14.
Although the lemma is stated for a cone metric space, it remains valid for a quasi-cone metric space.


## 3. Some First Results

Definition 15.
(1) In a quasi-cone metric space(X,q), one says that the sequence (xn)left-converges to x∈X if for every c∈E with θ≪c there exists N such that, for all n>N, q(xn,x)≪c.
(2) Similarly, in a quasi-cone metric space(X,q), one says that a sequence (xn)right- converges to x∈X if for every c∈E with θ≪c there exists N such that, for all n>N, q(x,xn)≪c.
(3) Finally, in a quasi-cone metric space(X,q), one says that the sequence (xn)converges to x∈X if for every c∈E with θ≪c there exists N such that, for all n>N,  q(xn,x)≪c and q(x,xn)≪c.Definition 16.
A quasi-cone metric space(X,q) is called (1)
left-complete (resp.,right-complete) if every left Cauchy (resp., right Cauchy) sequence in X left (resp., right) converges,(2)
bicomplete if every Q-Cauchy sequence converges.Remark 17.
A quasi-cone metric space(X,q) is bicomplete if and only if it is left-complete and right-complete.Definition 18.
Let(X,q) be a quasi-cone metric space. A function f:X→X is said to belipschitzian if there exists some κ∈ℝ such that
(15)q(f(x),f(y))⪯κq(x,y),∀x,y∈X.
The smallest constant which satisfies the above inequality is called thelipschitizian constant of f and is denoted by Lip(f). In particular f is said to becontractive if Lip(f)∈[0,1) andexpansive if Lip(f)=1.Lemma 19.
Let(X,d) be a quasi-pseudometric space. If a sequence (xn)  ds-converges to x, then it is ds-Cauchy.Proof.
Since(xn)  ds-converges to x, for every ϵ>0, there exist N1 such that d(x,xk)<ϵ/2 for any k≥N1 and N2 such that d(xm,x)<ϵ/2 for any m≥N2. Hence for any n,p≥max⁡{N1,N2}, d(xn,xp)≤d(xn,x)+d(x,xp)<ϵ.Lemma 20.
Let(X,q) be a quasi-cone metric space and P a K-normal cone. Let (xn) be a sequence in X. Then (xn) converges to x if and only if q(xn,x)→θ  (n→∞) and q(x,xn)→θ  (n→∞).Proof.
Suppose(xn) converges to x. For every real ϵ>0, choose c∈E with θ≪c and K∥c∥<ϵ. Then there exists N>0 such that for all n>N  q(xn,x)≪c and q(x,xn)≪c. This implies that when n>N, ∥q(xn,x)∥≤K∥c∥<ϵ and ∥q(x,xn)∥≤K∥c∥<ϵ. This means that q(xn,x)→θ and q(x,xn)→θ.
Conversely, suppose thatq(xn,x)→θ  (n→∞) and q(x,xn)→θ  (n→∞). For any c∈E with θ≪c, there is σ>0 such that ∥x∥<σ implies that x≪c. For this σ, there exist N1 and N2 such that ∥q(xn,x)∥<σ for any n>N1 and ∥q(x,xn)∥<σ for any n>N2. Hence, for n>max⁡{N1,N2}, c-q(xn,x)∈Int(P) and c-q(x,xn)∈Int(P). Therefore (xn) converges to x.Remark 21.
In fact, a sequence(xn) left-converges (resp., right-converges) to x if and only if q(xn,x)→θ (resp., q(x,xn)→θ) (n→∞).Lemma 22.
Let(X,q) be a quasi-cone metric space and let (xn) be a sequence in X. If (xn) converges to x, then (xn) is a bi-Cauchy sequence.Proof.
For anyc∈E with θ≪c, there exists N>0 such that, for all m,n>N,  q(xn,x)≪c/2 and q(x,xm)≪c/2. Hence
(16)q(xn,xm)⪯q(xn,x)+q(x,xm)≪c.
Therefore, (xn) is a bi-Cauchy sequence.Lemma 23.
Let(X,q) be a quasi-cone metric space, P a K-normal cone, and (xn) a sequence in X. Then (xn) is a bi-Cauchy sequence if and only if q(xn,xm)→θ as n,m→∞.Proof.
Suppose that(xn) is a bi-Cauchy sequence. For every real ϵ>0, choose c∈E with θ≪c and K∥c∥<ϵ. Then there exists N such that, for all n,m>N,  q(xn,xm)≪c. Therefore, whenever n,m>N, ∥q(xn,xm)∥≤K∥c∥<ϵ. This means that q(xn,xm)→θ as n,m→∞.
Conversely, suppose thatq(xn,xm)→θ as n,m→∞. For any c∈E with θ≪c, there is σ>0 such that ∥x∥<σ implies that x≪c. For this σ, there exist N such that ∥q(xn,xm)∥<σ for any n,m>N1. Hence c-q(xn,xm)∈Int(P). Therefore (xn) is a bi-Cauchy sequence.


## 4. First Fixed Points Results

Theorem 24.
Let(X,q) be a bicomplete quasi-cone metric space and P a K-normal cone. Suppose that a mapping T:X→X satisfies the contractive condition
(17)q(Tx,Ty)⪯kq(x,y)∀x,y∈X,
where k∈[0,1). Then T has a unique fixed point. Moreover for any x∈X, the orbit {Tnx,n≥0} converges to the fixed point.Proof.
Take an arbitraryx0∈X and denote xn=Tnx0. Then
(18)q(xn,xn+1)=q(Txn-1,Txn)⪯kq(xn-1,xn)gggggggggggggggg⪯k2q(xn-2,xn-1)⪯⋯⪯knq(x0,x1).
Similarly,
(19)q(xn+1,xn)⪯knq(x1,x0).
So for n<m,
(20)q(xn,xm)⪯q(xn,xn+1)+q(xn+1,xn+2)+⋯+q(xm-1,xm)⪯(kn+kn+1+⋯+km-1)q(x0,x1)⪯kn1-kq(x0,x1).
It entails that ∥q(xn,xm)∥≤K(kn/(1-k))∥q(x0,x1)∥→0 as n,m→∞.
Similarly forn>m(21)q(xn,xm)⪯kn1-kq(x1,x0).
It entails that ∥q(xn,xm)∥≤K(kn/(1-k))∥q(x1,x0)∥→0 as n,m→∞. Hence (xn) is a bi-Cauchy sequence. Since (X,q) is bicomplete, there exists x*∈X such that (xn) converges to x*.
Moreover since(22)q(Tx*,x*)⪯q(Tx*,Txn)+q(Txn,x*)⪯kq(x*,xn)+q(xn+1,x*),q(x*,Tx*)⪯q(x*,Txn)+q(Txn,Tx*)⪯q(x*,xn+1)+kq(xn,x*),
we have that
(23)∥q(Tx*,x*)∥≤K(k∥q(x*,xn)∥+∥q(xn+1,x*)∥)⟶0,∥q(x*,Tx*)∥≤K(k∥q(xn,x*)∥+∥q(x*,xn+1)∥)⟶0.
Hence∥q(Tx*,x*)∥=0=∥q(x*,Tx*)∥. This implies, using property (q2), that Tx*=x*. So x* is a fixed point.
Ifz* is another fixed point of T, then
(24)q(x*,z*)=q(Tx*,Tz*)⪯kq(x*,z*),q(z*,x*)=q(Tz*,Tx*)⪯kq(z*,x*).
Hence, ∥q(x*,z*)∥=0=∥q(z*,x*)∥ and x*=z*. Therefore the fixed point is unique.Corollary 25.
Let(X,q) be a bicomplete quasi-cone metric space and P a K-normal cone. For c∈E with 0≪c and x0∈X, set B(x0,c)={x∈X:q(x0,x)⪯c}. Suppose the mapping T:X→X satisfies the contractive condition
(25)q(Tx,Ty)⪯kq(x,y),∀x,y∈B(x0,c),
where k∈[0,1) is a constant and q(x0,Tx0)⪯(1-k)c. Then T has a unique fixed point in B(x0,c).Proof.
We only need to prove thatB(x0,c) is bicomplete and Tx∈B(x0,c) for all x∈B(x0,c).
Suppose(xn) is a bi-Cauchy sequence in B(x0,c). Then (xn) is also a bi-Cauchy sequence in X. By the bicompleteness of X, there is x∈X such that (xn) converges to x. We have
(26)q(x0,x)⪯q(x0,xn)+q(xn,x)⪯q(xn,x)+c.
Since(xn) converges to x, q(xn,x)→θ.  Hence q(x0,x)⪯c and x∈B(x0,c). Therefore, B(x0,c) is bicomplete.
For everyx∈B(x0,c),
(27)q(x0,Tx)⪯q(x0,Tx0)+q(Tx0,Tx)⪯(1-k)c+kq(x0,x)⪯(1-k)c+kc=c.
Hence Tx∈B(x0,c).Remark 26.
A weaker version of this corollary is actually sufficient. Indeed, it is enough to consider(X,q) as a left-complete quasi-cone metric space with the same assumption. In this case, we would just have to prove that B(x0,c) is left-complete and Tx∈B(x0,c) for all x∈B(x0,c).Corollary 27.
Let(X,q) be a bicomplete quasi-cone metric space and P a  K-normal cone. Suppose a mapping T:X→X satisfies for some positive integer n,
(28)q(Tnx,Tny)⪯kq(x,y),∀x,y∈X,
where k∈[0,1) is a constant. Then T has a unique fixed point in X.Proof.
From Theorem24, Tn has a unique fixed point x*. But Tn(Tx*)=T(Tnx*)=Tx*, so Tx* is also a fixed point of Tn. Hence Tx*=x*, x* is a fixed point of T. Since the fixed point of T is also a fixed point of Tn, the fixed point of T is unique.Theorem 28.
Let(X,q) be a quasi-cone metric space over the Banach space E with the K-normal cone P. The mapping Q:X×X→[0,∞) defined by Q(x,y)=∥q(x,y)∥ satisfies the following properties:(Q1)
Q(x,x)=0 for any x∈X;(Q2)
Q(x,y)≤K(Q(x,z1)+Q(z1,z2)+⋯+Q(zn,y)), for any points x,y,zi∈X,i=1,2,…,n.Proof.
The proof of(Q1) is immediate by property (q2) of the quasi-cone metric. In order to prove (Q2), consider x,y,z1,…,zn as points in X. Using property (q3), we get
(29)q(x,y)⪯(q(x,z1)+q(z1,z2)+⋯+q(zn,y)).
Since P is K-normal
(30)∥q(x,y)∥≤K(∥q(x,z1)+q(z1,z2)+⋯+q(zn,y)∥),
which implies that
(31)∥q(x,y)∥≤K(∥q(x,z1)∥+∥q(z1,z2)∥+⋯+∥q(zn,y)∥).
This completes the proof.We are therefore led to the following definition.Definition 29.
LetX be a nonempty set, and let the function D:X×X→[0,∞) satisfy the following properties: (D1)
D(x,x)=0 for any x∈X;(D2)
D(x,y)≤α(D(x,z1)+D(z1,z2)+⋯+D(xn,y)) for any points x,y,zi∈X,i=1,2,…,n and some constant α>0.Then (X,D,α) is called a quasi-pseudometric type space. Moreover, if D(x,y)=0=D(y,x)⇒x=y, then D is said to be a T0-quasi-pseudometric type space. The latter condition is referred to as the T0-condition.Remark 30.
(i) LetD be a quasi-pseudometric type on X; then the map D-1 defined by D-1(x,y)=D(y,x) whenever x,y∈X is also a quasi-pseudometric type on X, called the conjugate of D. We will also denote D-1 by Dt or D-.(ii)
It is easy to verify that the functionDs defined by Ds≔D∨D-1, that is, Ds(x,y)=max⁡{D(x,y),D(y,x)}, defines ametric-type (see [2]) on X whenever D is a T0-quasi-pseudometric type.(iii)
If we substitute the property(D1) by the following property,(D3)
D(x,y)=0⇔x=y,we obtain aT0-quasi-pseudometric type space directly. For instance, this could be done if the map D is obtained from quasi-cone metric.The concepts ofleftK-Cauchy,rightK-Cauchy, Ds-Cauchy, andconvergence for a quasi-pseudometric type space are defined in a similar way as defined for a quasi-pseudometric space. Moreover, for α=1, we recover the classical quasi-pseudometric; hence quasi-pseudometric type generalizes quasi-pseudometric.Definition 31.
A quasi-pseudometric type space(X,D,α) is calledleft-complete provided that any left K-Cauchy sequence is D-convergent.Definition 32.
AT0-quasi-pseudometric type space (X,D,α) is calledbicomplete provided that the metric type space (X,Ds) is complete.Definition 33.
Let(X,D,α) be a quasi-pseudometric type space. A function f:X→X is calledlipschitzian if there exists some λ≥0 such that
(32)D(fx,fy)≤λD(x,y)∀x,y∈X.
The smallest constant λ will be denoted by Lip(f).Definition 34.
Let(X,D,α) be a quasi-pseudometric type space. A function f:X→X is called D-sequentially continuous if, for any D-convergent sequence (xn) with xn→Dx, the sequence (fxn)  D-converges to fx; that is, (fxn)→Dfx.


## 5. Some Fixed Point Results

In [2], Khamsi proved the following.Theorem 35.
Let(X,d) be a complete metric type space. Let T:(X,d)→(X,d) be a map such that Tn is lipschitzian for all n≥0 and ∑n=0∞Lip(Tn)<∞. Then T has a unique fixed point ω∈X. Moreover for any x∈X, the orbit {Tnx,n≥0} converges to ω.We state here an analogue of Khamsi's theorem.Theorem 36.
Let(X,D,α) be a bicomplete quasi-pseudometric type. Let T:(X,D,α)→(X,D,α) be a map such that Tn is lipschitzian for all n≥0 and ∑n=0∞Lip(Tn)<∞. Then T has a unique fixed point ω∈X. Moreover for any x∈X, the orbit {Tnx,n≥0} converges to ω.Proof.
We just have to prove thatT:(X,Ds)→(X,Ds) is a map such that Tn is lipschitzian for all n≥0.
Indeed, sinceT:(X,D,α)→(X,D,α) is a map such that Tn is lipschitzian for all n≥0, then
(33)D(Tnx,Tny)≤Lip(Tn)D(x,y)∀x,y∈X.
Since for anyx,y∈X, we have
(34)D-1(Tnx,Tny)=D(Tny,Tnx)≤Lip(Tn)D(y,x)hhhhhhhhh∀n≥0,
that is,
(35)D-1(Tnx,Tny)≤Lip(Tn)D-1(x,y),
we see that T:(X,D-1,α)→(X,D-1,α) is a map such that Tn is lipschitzian for all n≥0.
Therefore(36)D(Tnx,Tny)≤Lip(Tn)D(x,y)≤Lip(Tn)Ds(x,y),D-1(Tnx,Tny)≤Lip(Tn)D-1(x,y)≤Lip(Tn)Ds(x,y),
for all x,y∈X and for all n≥0. Hence
(37)Ds(Tnx,Tny)≤Lip(Tn)Ds(x,y),
for all x,y∈X and for all n≥0, so, T:(X,Ds)→(X,Ds) is a map such that Tn is lipschitzian for all n≥0.
By assumption,(X,D,α) is bicomplete; hence (X,Ds) is complete. Therefore, by Theorem 35, T has a unique fixed point ω∈X and for any x∈X, the orbit {Tnx,n≥0} converges to ω.The connection between a quasi-cone metric space and a quasi-pseudometric type space is given by the following corollary.Corollary 37.
Let(X,q) be a bicomplete quasi-cone metric space over the Banach space E with the K-normal cone P. Consider Q:X×X→[0;∞) defined by Q(x,y)=∥q(x,y)∥. Let T:X→X be a contraction with constant 0<κ<1. Then
(38)Q(Tnx,Tny)≤KκnQ(x,y),
for any x,y∈X and n≥0. Hence Lip(Tn)≤Kκn, for any n≥0. Therefore ∑n=0∞Lip(Tn) is convergent, which implies that T has a fixed point ω and any orbit converges to ω.Proof.
It is enough to prove that the metric type space(X,Qs) is complete. Let (xn) be a Qs-Cauchy sequence. Therefore lim⁡n,m→∞Qs(xn,xm)=0, which implies that the sequence (xn) is bi-Cauchy in (X,q). Since (X,q) is bicomplete, there exists x*∈X such that q(xn,x*)→θ and q(x*,xn)→θ. Hence xn→Qsx*.
Moreover, sinceT is a contraction with constant κ, we have that
(39)q(Tnx,Tny)⪯κq(Tn-1x,Tn-1y)⪯⋯⪯κnq(x,y)hhhhhhhhhhforanyx,y∈X,n≥0.
Hence Lip(Tn)≤Kκn, for any n≥0.


## 6. More Fixed Point Results

We begin with the following lemmas.Lemma 38.
Let(yn) be a sequence in a quasi-pseudometric type space (X,D,α) such that
(40)D(yn,yn+1)≤λD(yn-1,yn),
for some λ>0 with λ<1/α. Then (yn) is left K-Cauchy.Proof.
Letm<n∈ℕ. From the condition (Qb) in the definition of a quasi-pseudometric type, we can write
(41)D(ym,yn)≤α(D(ym,ym+1)+D(ym+1,yn))≤αD(ym,ym+1)+α2D(ym+1,ym+2)+D(ym+2,yn)⋮≤αD(ym,ym+1)+α2D(ym+1,ym+2)+⋯+αn-m-1D(yn-2,yn-1)+αn-mD(yn-1,yn).
From (40) and λ<1/α, the above becomes
(42)D(ym,yn)≤(αλm+α2λm+1+⋯+αn-m+1λn-1)D(y0,y1)≤αλm(1+αλ+⋯+(αλ)n-1)D(y0,y1)≤αλm1-αλD(y0,y1)⟶0asm⟶∞.
It follows that (yn) is left K-Cauchy.Similarly, we have the following.Lemma 39.
Let(yn) be a sequence in a quasi-pseudometric type space (X,D,α) such that
(43)D-1(yn,yn+1)≤λD-1(yn-1,yn),
for some λ>0 with λ<1/α. Then (yn) is right K-Cauchy.Theorem 40.
Let(X,D,α) be a Hausdorff left-complete T0-quasi-pseudometric type space, and let f:X→X be a D-sequentially continuous function such that for some λ>0 with λ/(1-λ)<1/α,
(44)D(fx,fy)≤λ(D(x,fx)+D(y,fy))∀x,y,z∈X.
Then f has a unique fixed point z and for every x0∈X, the sequence (fn(x0))  D-converges to z.Proof.
Take an arbitraryx0∈X and denote yn=fn(x0). Then
(45)D(yn,yn+1)=D(fyn-1,fyn)≤λ(D(yn-1,fyn-1)+D(yn,fyn))≤λ(D(yn-1,yn)+D(yn,yn+1)),
which implies that
(46)D(yn,yn+1)≤λ1-λD(yn-1,yn).
Hence, sinceλ/(1-λ)<1/α, by Lemma 38 we have that (yn) is left K-Cauchy and since (X,D,α) is left-complete and f  D-sequentially continuous, there exists y* such that yn→Dy* and yn+1→Dfy*. Since X is Hauforff, the limit is unique, hence y*=fy*.
For uniqueness, assume by contradiction that there exists another fixed pointz*. Then
(47)D(y*,z*)=D(fy*,fz*)≤λ(D(y*,fy*)+D(z*,fz*))=0,D(z*,y*)=D(fz*,fy*)≤λ(D(z*,fz*)+D(y*,fy*))=0.
Hence D(y*,z*)=0=D(z*,y*) and using the T0-condition, we conclude that y*=z*.Theorem 41.
Let(X,D,α) be a Hausdorff left-complete T0-quasi-pseudometric type space, and let f:X→X be a D-sequentially continuous function such that for some λ>0 with λα2/(1-λα)<1,
(48)D(fx,fy)≤λ(D(fx,y)+D(x,fy))∀x,y∈X.
Then f has a unique fixed point z and for every x0∈X, the sequence (fn(x0))  D-converges to z.Proof .
Take an arbitraryx0∈X and denote yn=fn(x0). Then
(49)D(yn,yn+1)=D(fyn-1,fyn)≤λ(D(fyn-1,yn)+D(yn-1,fyn))≤λD(yn-1,yn+1)≤λα(D(yn-1,yn)+D(yn,yn+1)),
which implies that
(50)D(yn,yn+1)≤λα1-λαD(yn-1,yn).
Hence, sinceλα2/(1-λα)<1/α, by Lemma 38, we have that (yn) is left K-Cauchy and since (X,D,α) is left-complete and f  D-sequentially continuous, there exists y* such that yn→Dy* and yn+1→Dfy*. Since X is Hauforff, the limit is unique, hence y*=fy*.
For uniqueness, assume by contradiction that there exists another fixed pointz*. Then
(51)D(y*,z*)=D(fy*,fz*)≤λ(D(fy*,z*)+D(y*,fz*)),D(z*,y*)=D(fz*,fy*)≤λ(D(fz*,y*)+D(z*,fz*)).
Hence D(y*,z*)=0=D(z*,y*) and using the T0-condition, we conclude that y*=z*.Theorem 42.
Let(X,D,α) be a Hausdorff left-complete T0-quasi-pseudometric type space and let f:X→X be a D-sequentially continuous function such that for some λ>0 with λ<1/α<1/α and any γ>0,
(52)D(fx,fy)≤λD(x,y)+γD(fx,y)∀x,y∈X.
Then f has a unique fixed point z and for every x0∈X, the sequence (fn(x0))  D-converges to z.Corollary 43.
Let(X,D,α) be a Hausdorff left-complete T0-quasi-pseudometric type space and let f:X→X be a D-sequentially continuous function such that for some λ1,λ3,λ4,λ5>0 with (λ1+λ3+αλ5)/(1-λ4-αλ5)<1/α and any λ2>0(53)D(fx,fy)≤λ1D(x,y)+λ2D(fx,y)+λ3D(x,fx)+λ4D(y,fy)+λ5D(x,fy),
for all  x,y∈X. Then f has a unique fixed point z and for every x0∈X, the sequence (fn(x0))  D-converges to z.



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*Source: 198685-2014-04-15.xml*