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Generalized αψ contractive mappings in quasimetric spaces and related fixedpoint theorems
Journal of Inequalities and Applications volume 2014, Article number: 36 (2014)
Abstract
In this paper, we characterize αψ contractive mappings in the setting of quasimetric spaces and investigate the existence and uniqueness of a fixed point of such mappings. We notice that by using our result some fixedpoint theorems in the context of Gmetric space can be deduced.
MSC:46T99, 47H10, 54H25, 46J10.
1 Introduction and preliminaries
Let Ψ be the family of functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ satisfying the following conditions:
(${\psi}_{1}$) ψ is nondecreasing;
(${\psi}_{2}$) ${\sum}_{n=1}^{+\mathrm{\infty}}{\psi}^{n}(t)<\mathrm{\infty}$ for all $t>0$, where ${\psi}^{n}$ is the n th iterate of ψ.
These functions are known in the literature as (c)comparison functions. One can easily deduce that if ψ is a (c)comparison function, then $\psi (t)<t$ for any $t>0$.
Definition 1 Let X be a nonempty and let $d:X\times X\to [0,\mathrm{\infty})$ be a function which satisfies:
(d1) $d(x,y)=0$ if and only if $x=y$,
(d2) $d(x,y)\le d(x,z)+d(z,y)$. Then d called a quasimetric and the pair $(X,d)$ is called a quasimetric space.
Remark 2 Any metric space is a quasimetric space, but the converse is not true in general.
Now, we give convergence and completeness on quasimetric spaces.
Definition 3 Let $(X,d)$ be a quasimetric space, $\{{x}_{n}\}$ be a sequence in X, and $x\in X$. The sequence $\{{x}_{n}\}$ converges to x if and only if
Definition 4 Let $(X,d)$ be a quasimetric space and $\{{x}_{n}\}$ be a sequence in X. We say that $\{{x}_{n}\}$ is leftCauchy if and only if for every $\epsilon >0$ there exists a positive integer $N=N(\epsilon )$ such that $d({x}_{n},{x}_{m})<\epsilon $ for all $n\ge m>N$.
Definition 5 Let $(X,d)$ be a quasimetric space and $\{{x}_{n}\}$ be a sequence in X. We say that $\{{x}_{n}\}$ is rightCauchy if and only if for every $\epsilon >0$ there exists a positive integer $N=N(\epsilon )$ such that $d({x}_{n},{x}_{m})<\epsilon $ for all $m\ge n>N$.
Definition 6 Let $(X,d)$ be a quasimetric space and $\{{x}_{n}\}$ be a sequence in X. We say that $\{{x}_{n}\}$ is Cauchy if and only if for every $\epsilon >0$ there exists a positive integer $N=N(\epsilon )$ such that $d({x}_{n},{x}_{m})<\epsilon $ for all $m,n>N$.
Remark 7 A sequence $\{{x}_{n}\}$ in a quasimetric space is Cauchy if and only if it is leftCauchy and rightCauchy.
Definition 8 Let $(X,d)$ be a quasimetric space. We say that

(1)
$(X,d)$ is leftcomplete if and only if each leftCauchy sequence in X is convergent.

(2)
$(X,d)$ is rightcomplete if and only if each rightCauchy sequence in X is convergent.

(3)
$(X,d)$ is complete if and only if each Cauchy sequence in X is convergent.
Definition 9 Let $(X,d)$ be a quasimetric space and $T:X\to X$ be a given mapping. We say that T is an αψ contractive mapping if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that
Remark 10 We easily see that any contractive mapping, that is, a mapping satisfying the Banach contraction, is an αψ contractive mapping with $\alpha (x,y)=1$ for all $x,y\in X$ and $\psi (t)=kt$, $k\in (0,1)$.
Definition 11 Let $T:X\to X$ and $\alpha :X\times X\to [0,\mathrm{\infty})$. We say that T is α admissible if for all $x,y\in X$ we have
2 Main results
We start this section by the following definition, which is a characterization of αψ contractive mappings [1] in the context of a quasimetric space.
Definition 12 (cf. [2])
Let $(X,d)$ be a quasimetric space and $T:X\to X$ be a given mapping. We say that T is an αψ contractive mapping if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$, we have
Theorem 13 Let $(X,d)$ be a complete quasimetric space. Suppose that $T:X\to X$ is a αψ contractive mapping which satisfies

(i)
T is α admissible;

(ii)
there exists ${x}_{0}\in X$ such that $\alpha (T{x}_{0},{x}_{0})\ge 1$ and $\alpha ({x}_{0},T{x}_{0})\ge 1$;

(iii)
T is continuous.
Then T has a fixed point.
Proof By (ii), there exists ${x}_{0}\in X$ such that $\alpha (T{x}_{0},{x}_{0})\ge 1$ and $\alpha ({x}_{0},T{x}_{0})\ge 1$. Let us define a sequence $\{{x}_{n}\}$ in X by ${x}_{n+1}=T{x}_{n}$ for all $n\in \mathbb{N}$. If ${x}_{{n}_{0}}={x}_{{n}_{0}+1}$ for some ${n}_{0}$, then it is evident that ${x}_{{n}_{0}}$ is a fixed point of T. Consequently, throughout the proof, we suppose that ${x}_{n}\ne {x}_{n+1}$ for all $n\in \mathbb{N}$. Regarding the assumption (i), we derive
Recursively, we get
Taking (2.1) and (2.3) into account, we find that
for all $n\ge 1$. Inductively, we obtain
By using the triangular inequality and (2.5), for all $k\ge 1$, we get
Letting $n\to \mathrm{\infty}$ in the above inequality, we derive ${\sum}_{p=n}^{\mathrm{\infty}}{\psi}^{n}(d({x}_{1},{x}_{0}))\to 0$. Hence, $d({x}_{n+k},{x}_{n})\to 0$ as $n\to \mathrm{\infty}$. Therefore, $\{{x}_{n}\}$ is a leftCauchy sequence in $(X,d)$.
Analogously, we deduce that $\{{x}_{n}\}$ is a rightCauchy sequence in $(X,d)$. Indeed, by assumption (i), we obtain
Recursively, we find that
By combining (2.1) with (2.8), we find
for all $n\ge 1$. By iteration, we have
Due to the triangular inequality, together with (2.10), for all $k\ge 1$, we get
Consequently, $\{{x}_{n}\}$ is a rightCauchy sequence in $(X,d)$. By Remark 7, we deduce that $\{{x}_{n}\}$ is a Cauchy sequence in complete quasimetric space $(X,d)$. It implies that there exists $u\in X$ such that
Then, by using the property (d1) together with the continuity of T, we obtain
and
Thus, we have
Keeping (2.12) and (2.15) in mind together with the uniqueness of the limit, we conclude that $u=Tu$, that is, u is a fixed point of T. □
Theorem 14 Let $(X,d)$ be a complete quasimetric space. Suppose that $T:X\to X$ is an αψ contractive mapping which satisfies:

(i)
T is α admissible;

(ii)
there exists ${x}_{0}\in X$ such that $\alpha (T{x}_{0},{x}_{0})\ge 1$;

(iii)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n+1},{x}_{n})\ge 1$ for all n and ${x}_{n}\to x\in X$ as $n\to \mathrm{\infty}$, then there exists a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that $\alpha (x,{x}_{n(k)})\ge 1$ for all k.
Then T has a fixed point.
Proof Following the lines of the proof of Theorem 13, we know that the sequence $\{{x}_{n}\}$ defined by ${x}_{n+1}=T{x}_{n}$, for all $n\ge 0$, converges for some $u\in X$. From (2.3) and condition (iii), there exists a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that $\alpha (u,{x}_{n(k)})\ge 1$ for all k. Applying (2.1), for all k, we get
Letting $k\to \mathrm{\infty}$ in the above equality, we obtain
Thus, we have $d(Tu,u)=0$, that is, $Tu=u$. □
Definition 15 (cf. [3])
Let $(X,d)$ be a quasimetric space and $T:X\to X$ be a given mapping. We say that T is a generalized αψ contractive mapping of type A if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$ and we have
where
Definition 16 Let $(X,d)$ be a quasimetric space and $T:X\to X$ be a given mapping. We say that T is a generalized αψ contractive mapping of type B if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$ and we have
where
Theorem 17 Let $(X,d)$ be a complete quasimetric space. Suppose that $T:X\to X$ is a generalized αψ contractive mapping of type A and satisfies

(i)
T is α admissible;

(ii)
there exists ${x}_{0}\in X$ such that $\alpha (T{x}_{0},{x}_{0})\ge 1$ and $\alpha ({x}_{0},T{x}_{0})\ge 1$;

(iii)
T is continuous.
Then T has a fixed point.
Proof By assumption (ii), there exists ${x}_{0}\in X$ such that $\alpha (T{x}_{0},{x}_{0})\ge 1$ and $\alpha ({x}_{0},T{x}_{0})\ge 1$. We construct a sequence $\{{x}_{n}\}$ in X in the following way:
If ${x}_{{n}_{0}}={x}_{{n}_{0}+1}$ for some ${n}_{0}$, then it is clear that ${x}_{{n}_{0}}$ is a fixed point of T. Hence, we assume that ${x}_{n}\ne {x}_{n+1}$ for all $n\in \mathbb{N}$. Due to assumption (i), we have
If we continue in this way, we obtain
From (2.18) and (2.23), for all $n\ge 1$, we derive
where
Since ψ is a nondecreasing function, (2.24) implies that
for all $n\ge 1$. We shall examine two cases. Suppose that $d({x}_{n+1},{x}_{n})>d({x}_{n},{x}_{n1})$. Since $d({x}_{n+1},{x}_{n})>0$, we obtain
a contradiction. Therefore, we find that $max\{d({x}_{n},{x}_{n1}),d({x}_{n+1},{x}_{n})\}=d({x}_{n},{x}_{n1})$. Since $\psi \in \mathrm{\Psi}$, (2.26) yields
for all $n\ge 1$. Recursively, we derive
Together with (2.29) and the triangular inequality, for all $k\ge 1$, we get
Therefore, $\{{x}_{n}\}$ is a leftCauchy sequence in $(X,d)$.
Analogously, we shall prove that $\{{x}_{n}\}$ is a rightCauchy sequence in $(X,d)$. Again by the assumption (i), we find that
Recursively, we obtain
From (2.18) and (2.32), for all $n\ge 1$, we deduce that
where
Since ψ is nondecreasing function, the inequality (2.33) turns into
for all $n\ge 1$. We shall examine three cases.
Case 1. Assume that $max\{d({x}_{n1},{x}_{n}),d({x}_{n},{x}_{n1}),d({x}_{n+1},{x}_{n})\}=d({x}_{n+1},{x}_{n})$. Since $d({x}_{n+1},{x}_{n})>0$ we get
a contradiction.
Case 2. Suppose that $max\{d({x}_{n1},{x}_{n}),d({x}_{n},{x}_{n1}),d({x}_{n+1},{x}_{n})\}=d({x}_{n1},{x}_{n})$. Since $\psi \in \mathrm{\Psi}$, from (2.34) we find that
for all $n\ge 1$. Inductively, we get
By using the triangular inequality and taking (2.38) into consideration, for all $k\ge 1$, we get
Case 3. Assume that $max\{d({x}_{n1},{x}_{n}),d({x}_{n},{x}_{n1}),d({x}_{n+1},{x}_{n})\}=d({x}_{n},{x}_{n1})$. Regarding $\psi \in \mathrm{\Psi}$ and (2.35), we obtain
for all $n\ge 1$. From (2.18) and (2.23), for all $n\ge 1$, we derive
where
Since ψ is a nondecreasing function, (2.24) implies that
for all $n\ge 1$.
We shall examine two cases. Suppose that $d({x}_{n},{x}_{n1})>d({x}_{n1},{x}_{n2})$. Since $d({x}_{n},{x}_{n1})>0$, we obtain
a contradiction. Therefore, we find that $max\{d({x}_{n1},{x}_{n2}),d({x}_{n},{x}_{n1})\}=d({x}_{n1},{x}_{n2})$. Since $\psi \in \mathrm{\Psi}$, (2.43) yields
for all $n\ge 1$. Recursively, we derive
If we combine the inequalities (2.40) with (2.46), we derive
Together with (2.47) and the triangular inequality, for all $k\ge 1$, we get
Therefore, by (2.39) and (2.48), we conclude that $\{{x}_{n}\}$ is a rightCauchy sequence in $(X,d)$.
From Remark 7, $\{{x}_{n}\}$ is a Cauchy sequence in complete quasimetric space $(X,d)$. This implies that there exists $u\in X$ such that
Then, using property (d1) and the continuity of T, we obtain
and
Thus, we have
It follows from (2.49) and (2.52) that $u=Tu$, that is, u is a fixed point of T. □
Corollary 18 Let $(X,d)$ be a complete quasimetric space. Suppose that $T:X\to X$ is a generalized αψ contractive mapping of type A and satisfies:

(i)
T is α admissible;

(ii)
there exists ${x}_{0}\in X$ such that $\alpha (T{x}_{0},{x}_{0})\ge 1$ and $\alpha ({x}_{0},T{x}_{0})\ge 1$;

(iii)
T is continuous.
Then T has a fixed point.
The proof is evident due to Theorem 17. Indeed, ψ is nondecreasing and, hence,
where $M(x,y)$ and $N(x,y)$ are defined as in Definition 15 and Definition 16. The rest follows from Theorem 17.
In the following theorem we are able to remove the continuity condition for the αψ contractive mappings of type B.
Theorem 19 Let $(X,d)$ be a complete quasimetric space. Suppose that $T:X\to X$ is a generalized αψ contractive mapping of type B which satisfies:

(i)
T is α admissible;

(ii)
there exists ${x}_{0}\in X$ such that $\alpha (T{x}_{0},{x}_{0})\ge 1$;

(iii)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n+1},{x}_{n})\ge 1$ for all n and ${x}_{n}\to x\in X$ as $n\to \mathrm{\infty}$, then there exists a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that $\alpha (x,{x}_{n(k)})\ge 1$ for all k.
Then T has a fixed point.
Proof Following the lines in the proof of Theorem 17, we know that the sequence $\{{x}_{n}\}$ defined by ${x}_{n+1}=T{x}_{n}$ for all $n\ge 0$, converges for some $u\in X$. From (2.23) and condition (iii), there exists a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that $\alpha (u,{x}_{n(k)})\ge 1$ for all k. Applying (2.20), for all k, we get
Also, using (2.21) we find
Taking the limit as $k\to \mathrm{\infty}$ in the above equality, we obtain
Assume that $d(Tu,u)>0$. From (2.55), for k large enough, we have $N(u,{x}_{n(k)})>0$, which implies that $\psi (N(u,{x}_{n(k)}))<N(u,{x}_{n(k)})$. Then, from (2.53), we have
Taking the limit as $k\to \mathrm{\infty}$ in the above equality, we get
which is a contradiction. Therefore, we find $d(Tu,u)=0$, that is, $Tu=u$. □
3 Consequences: fixedpoint result on Gmetric spaces
In this section, we note that some existing fixedpoint results in the context of Gmetric spaces are consequences of our main theorems. For the sake of completeness, we recollect some basic definitions and crucial results on the topic in the literature. For more details, see e.g. [4–6].
Definition 20 Let X be a nonempty set, $G:X\times X\times X\to {\mathbb{R}}^{+}$ be a function satisfying the following properties:
(G1) $G(x,y,z)=0$ if $x=y=z$,
(G2) $0<G(x,x,y)$ for all $x,y\in X$ with $x\ne y$,
(G3) $G(x,x,y)\le G(x,y,z)$ for all $x,y,z\in X$ with $y\ne z$,
(G4) $G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots $ (symmetry in all three variables),
(G5) $G(x,y,z)\le G(x,a,a)+G(a,y,z)$ (rectangle inequality) for all $x,y,z,a\in X$.
Then the function G is called a generalized metric, or, more specifically, a Gmetric on X, and the pair $(X,G)$ is called a Gmetric space.
Note that every Gmetric on X induces a metric ${d}_{G}$ on X defined by
For a better understanding of the subject we give the following examples of Gmetrics.
Example 21 Let $(X,d)$ be a metric space. The function $G:X\times X\times X\to [0,+\mathrm{\infty})$, defined by
for all $x,y,z\in X$, is a Gmetric on X.
Example 22 Let $X=[0,\mathrm{\infty})$. The function $G:X\times X\times X\to [0,+\mathrm{\infty})$, defined by
for all $x,y,z\in X$, is a Gmetric on X.
Definition 23 Let $(X,G)$ be a Gmetric space, and let $\{{x}_{n}\}$ be a sequence of points of X. We say that $\{{x}_{n}\}$ is Gconvergent to $x\in X$ if
that is, for any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G(x,{x}_{n},{x}_{m})<\epsilon $, for all $n,m\ge N$. We call x the limit of the sequence and write ${x}_{n}\to x$ or ${lim}_{n\to +\mathrm{\infty}}{x}_{n}=x$.
Proposition 24 Let $(X,G)$ be a Gmetric space. The following are equivalent:

(1)
$\{{x}_{n}\}$ is Gconvergent to x,

(2)
$G({x}_{n},{x}_{n},x)\to 0$ as $n\to +\mathrm{\infty}$,

(3)
$G({x}_{n},x,x)\to 0$ as $n\to +\mathrm{\infty}$,

(4)
$G({x}_{n},{x}_{m},x)\to 0$ as $n,m\to +\mathrm{\infty}$.
Definition 25 Let $(X,G)$ be a Gmetric space. A sequence $\{{x}_{n}\}$ is called a GCauchy sequence if, for any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G({x}_{n},{x}_{m},{x}_{l})<\epsilon $ for all $m,n,l\ge N$, that is, $G({x}_{n},{x}_{m},{x}_{l})\to 0$ as $n,m,l\to +\mathrm{\infty}$.
Proposition 26 Let $(X,G)$ be a Gmetric space. Then the following are equivalent:

(1)
the sequence $\{{x}_{n}\}$ is GCauchy,

(2)
for any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G({x}_{n},{x}_{m},{x}_{m})<\epsilon $, for all $m,n\ge N$.
Definition 27 A Gmetric space $(X,G)$ is called Gcomplete if every GCauchy sequence is Gconvergent in $(X,G)$.
For more details of Gmetric space, we refer e.g. to [7–9].
Theorem 28 Let $(X,G)$ be a Gmetric space. Let $d:X\times X\to [0,\mathrm{\infty})$ be the function defined by $d(x,y)=G(x,y,y)$. Then

(1)
$(X,d)$ is a quasimetric space;

(2)
$\{{x}_{n}\}\subset X$ is Gconvergent to $x\in X$ if and only if $\{{x}_{n}\}$ is convergent to x in $(X,d)$;

(3)
$\{{x}_{n}\}\subset X$ is GCauchy if and only if $\{{x}_{n}\}$ is Cauchy in $(X,d)$;

(4)
$(X,G)$ is Gcomplete if and only if $(X,d)$ is complete.
Every quasimetric induces a metric, that is, if $(X,d)$ is a quasimetric space, then the function $\delta :X\times X\to [0,\mathrm{\infty})$ defined by
is a metric on X.
As an immediate consequence of the definition above and Theorem 28, the following theorem is obtained.
Theorem 29 Let $(X,G)$ be a Gmetric space. Let $d:X\times X\to [0,\mathrm{\infty})$ be the function defined by $d(x,y)=G(x,y,y)$. Then

(1)
$(X,d)$ is a quasimetric space;

(2)
$\{{x}_{n}\}\subset X$ is Gconvergent to $x\in X$ if and only if $\{{x}_{n}\}$ is convergent to x in $(X,\delta )$;

(3)
$\{{x}_{n}\}\subset X$ is GCauchy if and only if $\{{x}_{n}\}$ is Cauchy in $(X,\delta )$;

(4)
$(X,G)$ is Gcomplete if and only if $(X,\delta )$ is complete.
Now, we state the characterization of Definition 9 and Definition 11 in the context of Gmetric space.
Definition 30 (See e.g. [10, 11])
Let $(X,G)$ be a Gmetric space and $T:X\to X$ be a given mapping. We say that T is a βψ contractive mapping of type I if there exist two functions $\beta :X\times X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$, we have
Definition 31 (See e.g. [10, 11])
Let $T:X\to X$ and $\beta :X\times X\times X\to [0,\mathrm{\infty})$. We say that T is β admissible if for all $x,y\in X$ we have
Lemma 32 Let $T:X\to X$ where X is nonempty set. It is clear that the selfmapping T is β admissible if and only if T is α admissible.
Proof It is sufficient to let $\alpha (x,y)=\beta (x,y,y)$. □
Theorem 33 Let $(X,G)$ be a complete Gmetric space. Suppose that $T:X\to X$ is a βψ contractive mapping which satisfies:

(i)
T is β admissible;

(ii)
there exists ${x}_{0}\in X$ such that $\beta (T{x}_{0},{x}_{0},{x}_{0})\ge 1$ and $\beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1$;

(iii)
T is continuous.
Then T has a fixed point.
Proof Consider the quasimetric $d(x,y)=G(x,y,y)$ for all $x,y\in X$. Due to Lemma 32 and (3.2), we have
Then the result follows from Theorem 13. □
Definition 34 Let $(X,G)$ be a Gmetric space and $T:X\to X$ be a given mapping. We say that T is a generalized βψ contractive mapping of type A if there exist two functions $\beta :X\times X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$ we have
where
Definition 35 Let $(X,G)$ be a Gmetric space and $T:X\to X$ be a given mapping. We say that T is a generalized βψ contractive mapping of type B if there exist two functions $\beta :X\times X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$ we have
where
Remark 36 It is simple to see that every βψ contractive mapping is a generalized βψ contractive mapping of type A.
Similarly, every βψ contractive mapping is a generalized βψ contractive mapping of type B.
Theorem 37 Let $(X,G)$ be a complete Gmetric space. Suppose that $T:X\to X$ is a generalized βψ contractive mapping of type A and satisfies:

(i)
T is β admissible;

(ii)
there exists ${x}_{0}\in X$ such that $\beta (T{x}_{0},{x}_{0},{x}_{0})\ge 1$ and $\beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1$;

(iii)
T is continuous.
Then T has a fixed point.
Proof Consider the quasimetric $d(x,y)=G(x,y,y)$ for all $x,y\in X$. From Lemma 32 together with (3.5) and (3.6), we deduce that
Then the result follows from Theorem 17. □
Theorem 38 Let $(X,G)$ be a complete Gmetric space. Suppose that $T:X\to X$ is a βψ contractive mapping which satisfies:

(i)
T is β admissible;

(ii)
there exists ${x}_{0}\in X$ such that $\beta (T{x}_{0},{x}_{0},{x}_{0})\ge 1$;

(iii)
if $\{{x}_{n}\}$ is a sequence in X such that $\beta ({x}_{n+1},{x}_{n},{x}_{n})\ge 1$ for all n and ${x}_{n}\to x\in X$ as $n\to \mathrm{\infty}$, then there exists a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that $\beta (x,{x}_{n(k)},{x}_{n(k)})\ge 1$ for all k.
Then T has a fixed point.
Proof Consider the quasimetric $d(x,y)=G(x,y,y)$ for all $x,y\in X$. By Lemma 32 and (3.2), we find that
Then the result follows from Theorem 14. □
Theorem 39 Let $(X,G)$ be a complete Gmetric space. Suppose that $T:X\to X$ is a generalized βψ contractive mapping of type B which satisfies:

(i)
T is β admissible;

(ii)
there exists ${x}_{0}\in X$ such that $\beta (T{x}_{0},{x}_{0},{x}_{0})\ge 1$;

(iii)
if $\{{x}_{n}\}$ is a sequence in X such that $\beta ({x}_{n+1},{x}_{n},{x}_{n})\ge 1$ for all n and ${x}_{n}\to x\in X$ as $n\to \mathrm{\infty}$, then there exists a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that $\beta (x,{x}_{n(k)},{x}_{n(k)})\ge 1$ for all k.
Then T has a fixed point.
Proof Consider the quasimetric $d(x,y)=G(x,y,y)$ for all $x,y\in X$. Regarding Lemma 32, (3.7), and (3.8), we derive
Then the result follows from Theorem 19. □
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Acknowledgements
The authors thank the Visiting Professor Programming at King Saud University for funding this work. The authors thank the anonymous referees for their remarkable comments, suggestion, and ideas that helped to improve this paper.
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Bilgili, N., Karapınar, E. & Samet, B. Generalized αψ contractive mappings in quasimetric spaces and related fixedpoint theorems. J Inequal Appl 2014, 36 (2014). https://doi.org/10.1186/1029242X201436
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Keywords
 fixed point
 αψ contractive mappings
 quasimetric spaces