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Common fixed point theorems on quasicone metric space over a divisible Banach algebra
Advances in Difference Equations volume 2021, Article number: 306 (2021)
Abstract
In this manuscript, we investigate the existence and uniqueness of a common fixed point for the selfmappings defined on quasicone metric space over a divisible Banach algebra via an auxiliary mapping ϕ.
1 Introduction and preliminaries
The notion of metric has been extended in several ways by changing the axioms of the metric notion: quasimetric, symmetric, dislocated metric, bmetric, 2metric, Dmetric, Smetric, Gmetric, partial metric, ultrametric, etc. We shall focus on cone metric space or more precisely, Banachvalued metric space. The idea of Banachvalued metric space was considered by several authors in distinct periods of the last century. This notion became popular and raised interest among researchers after the paper of Huang and X. Zhang [1] in 2007. Since then, a number of authors got the characterization of several known fixed point theorems in the context of Banachvalued metric space, such as, [2–20].
In this paper, we consider common fixed point theorems in the framework of the refined cone metric space, namely, quasicone metric space.
In what follows, we shall recall the basic notions and notations as well as the fundamental results.
Definition 1.1
([21])
Suppose \(\mathcal{E}\) is a real Banach algebra, that is, for , \(a \in \mathrm{R}\),

(a)
;

(b)
, ;

(c)
;

(d)
.
If Banach algebra \(\mathcal{E}\) with unit element e, i.e. multiplicative identity e, is with for , then \(\ \mathsf{e}\ = 1\).
An element is said to be invertible if there exists such that . Moreover, if every nonzero element of \(\mathcal{E}\) has an inverse in \(\mathcal{E}\), then \(\mathcal{E}\) is called a divisible Banach algebra.
Proposition 1.2
([22])
Let \(\mathcal{E}\) be a Banach algebra, an element in \(\mathcal{E}\) and the spectral radius of . If then is invertible in \(\mathcal{E}\) and
Remark 1.3
([2])
for all in a Banach algebra \(\mathcal{E}\).
Let \((\mathcal{E}, \\cdot \)\) be a real algebra and P a closed subset of \(\mathcal{E}\).
The set P is a cone if the following conditions hold:
 \((c_{1})\):

P is nonempty and \(\mathsf{P}\neq \{ \theta \} \);
 \((c_{2})\):

for all and \(\mathsf{a}_{1},\mathsf{a}_{2}\in (0,\infty )\);
 \((c_{3})\):

\(\mathsf{P}\cap (\mathsf{P})= \{ \theta \} \).
Moreover, for a given cone \(\mathsf{P}\subseteq \mathcal{E}\) we can consider a partial ordering ≤ such that if and only if . We write for and indicates that and . The cone P is called normal if there exists a constant \(N > 0\) such that implies , for and is called solid if \(\textit{intP} \neq \emptyset \).
Definition 1.4
([3])
Let be a sequence in a solid cone P. We say that is a sequence, if for any with there exists \(m_{0}\in \mathbb{N}\) such that for all \(m>m_{0}\).
Lemma 1.5
([3])
If is a sequence in a solid cone P and κ is arbitrary (but given) in P, then is also a sequence.
Lemma 1.6
([4])
On a real Banach algebra \(\mathcal{E}\) with a solid cone \(\mathcal{E}\), the following statements hold:

1
\(\varsigma \ll \omega \) if ;

2
\(\varsigma =\theta \) if \(\varsigma \ll \omega \) for every \(\omega \gg \theta \).
Let \(\mathcal{E}\) be a Banach algebra and \(\mathsf{P}\subset \mathcal{E}\) be a cone. Then is an invertible element in P for any with .
Definition 1.7
([23])
Suppose \(\mathcal{E}\) is a Banach algebra with unit e and \(\mathsf{P}\subseteq \mathcal{E}\) is a cone. P is called algebra cone if \(\mathsf{e}\in \mathsf{P}\) and for , .
In what follows we consider that \(\mathcal{E}\) (\(\mathcal{E}_{d}\)) represents a real (divisible) Banach algebra with a unit e and θ be its zero element, P is a solid cone in \(\mathcal{E}\), \(\mathsf{P}_{\mathcal{E}_{d}}\) a normal algebra cone in \(\mathcal{E}_{d}\) with a normal constant N and X is a nonempty set.
Definition 1.8
(see [24])
A mapping \(\mathsf{d}: \mathrm{X}\times \mathrm{X}\rightarrow \mathcal{E}\) is a cone metric on X if

(a)
for all and if and only if ,

(b)
for all ,

(c)
,
for all . The pair \((\mathrm{X}, \mathsf{d})\) is said to be a cone metric space over Banach algebra, in short, CMS.
Definition 1.9
(see [25])
A mapping \(\mathsf{q}: \mathrm{X}\times \mathrm{X}\rightarrow \mathcal{E}\) is said to be a quasicone metric if

(a)
for all ,

(b)
if and only if ,

(c)
,
for all . The triplet \((\mathrm{X}, \mathsf{q}, \mathcal{E})\) is said to be a quasicone metric space over Banach algebra, in short, qCMS.
A quasicone metric space is called Δsymmetric, if there exists an invertible element \(\Delta \in \mathcal{E}\) such that
for all .
Definition 1.10
Suppose \((\mathrm{X}, \mathsf{q}, \mathcal{E})\) is a qCMS, and is a sequence in X. Then

(a)
(bi)converges to if for \(c\in \mathcal{E}\) with \(\theta \ll c\), there is a natural number N satisfying and for \(m\geq N\). We denote or .

(b)
is a (l)(left)Cauchy ((r)(right)Cauchy)) if for \(c\in \mathcal{E}\) with \(\theta \ll c\), there exists a natural number N satisfying (respectively, for \(m>p \geq N\).

(c)
is a biCauchy if for \(c\in \mathcal{E}\) with \(\theta \ll c\), there exists a natural number N satisfying for \(m,p \geq N\).

(d)
\((\mathrm{X}, \mathsf{q}, \mathcal{E})\) is (l)complete ((r)complete) if every (l)Cauchy((r)Cauchy) sequence is (bi)convergent and is complete if it is (l) and (r)complete.
Definition 1.11
We say that the mapping \(\psi : \mathsf{P}_{\mathcal{E}_{d}} \rightarrow \mathsf{P}_{ \mathcal{E}_{d}}\) is a ψoperator if

(a)
ψ is an increasing;

(b)
ψ is a continuous bijection and has an inverse mapping \(\psi ^{1} \) which is also continuous and increasing;

(c)
for all ;

(d)
for all .
Remark 1.12
By Definition 1.11, the part of (c), we can obtain for all . In fact, note that for all and \(\psi ^{1}\) is also a continuous and increasing operator, then
which yields
Hence,
Since \(\psi : \mathsf{P}_{\mathcal{E}_{d}} \rightarrow \mathsf{P}_{ \mathcal{E}_{d}}\) is a continuous bijection, thus , for all .
Remark 1.13
By Definition 1.11, the part of (d), we can obtain , for .
Indeed, from for and \(\psi ^{1} : \mathsf{P}_{\mathcal{E}}\rightarrow \mathsf{P}_{\mathcal{E}}\) is also continuous, we get
which yields
Then we obtain
Thanks to that \(\psi : \mathsf{P}_{\mathcal{E}}\rightarrow \mathsf{P}_{\mathcal{E}}\) is a continuous bijection, , for all .
Remark 1.14
For example, let \(E_{d}=\mathrm{R}\) be a divisible Banach algebra, be a normal cone in \(\mathcal{E}_{d}\), suppose \(\psi : \mathsf{P}_{\mathcal{E}_{d}} \rightarrow \mathsf{P}_{ \mathcal{E}_{d}}\), defined by and then , for all .
2 Main results
Lemma 2.1
Let \((\mathrm{X}, \mathsf{q}, \mathcal{E}_{d})\) be a Δsymmetric qCMS over a divisible Banach algebra, a sequence in X. If there exists \(\kappa \in \mathsf{P}_{\mathcal{E}_{d}}\), with \(\rho (\kappa )<1\) such that
for all \(m\in \mathbb{N}\), then is a (bi)Cauchy sequence.
Proof
First of all, we remark that, successively applying Eq. (2), we have
Let \(m>p \geq N\). Thereupon,
Now, since \(\rho (\kappa )<1\), and taking into account Proposition 1.2, we see that \((\mathsf{e}\kappa )\) is an invertible element and \((\mathsf{e}\kappa )^{1}=\sum_{j=0}^{\infty }\kappa ^{j}\) and the above inequality becomes
For a given , with , we choose \(\delta >0\) such that . (Here \(N_{\delta }(\theta )= \{ \omega \in \mathcal{E}_{d}:\\omega \\}< \delta \} \).) Letting \(p_{0}\in \mathbb{N}\) such that for all \(p\geq p_{0}\) we get , for all \(p\geq p_{0}\). Therefore,
Then by (b) in Definition 1.10 it follows that the sequence is (l)Cauchy. On the other hand, from Definition 1.4, we see that the sequence is convergent and moreover in view of Lemma 1.5 the sequence , where \(\Delta \in \mathsf{P}_{\mathcal{E}_{d}}\), is also a sequence, that is,
for all \(m>p>p_{0}\). But, since the space \((\mathrm{X}, \mathsf{q}, \mathcal{E}_{d})\) is supposed to be Δsymmetric, we have
and taking Lemma 1.5 into account we get , for all \(m>p\geq p_{0}\), which means the sequence is (r)Cauchy. Obviously, in view of statement \((c)\) in Definition 1.10, it follows that is a (bi)Cauchy sequence. □
Let (\(\mathcal{E}_{d}\)) be a real (divisible) Banach algebra with a unit e and θ be its zero element and \(\mathsf{P}^{1}_{\mathcal{E}_{d}}\) be a normal algebra cone with constant \(N=1\) in \(\mathcal{E}_{d}\).
Theorem 2.2
Let \((\mathrm{X}, \mathsf{q}, \mathcal{E}_{d})\) be a complete Δsymmetric qCMS over \(\mathcal{E}_{d}\) and \(\mathsf{P}^{1}_{\mathcal{E}_{d}}\). Suppose that \(\psi : \mathsf{P}^{1}_{\mathcal{E}_{d}} \rightarrow \mathsf{P}^{1}_{ \mathcal{E}_{d}}\) is a ψoperator and \(\mathcal{U}, \mathcal{V}: \mathrm{X}\rightarrow \mathrm{X}\) are mappings satisfying the conditions
for all with , where \(\psi (e) \leq k < \psi (2e)\) in \(\mathsf{P}_{\mathcal{E}^{1}_{d}}\). Then \(\mathcal{U}\) and \(\mathcal{V}\) have a common fixed point.
Proof
Let be an arbitrary point and the sequence defined by
Then, setting and , we get
and then
Also if we put and , then we have
Thus,
Moreover, applying \(\psi ^{1}\) in (8), (9) and keeping in mind the properties of the operator \(\psi ^{1}\), it follows
or by simplifying, we obtain
Denoting \(\kappa =\psi ^{1}(k)\mathsf{e}\), the above inequalities pass into
for any positive integer m. Now, by hypothesis \(\psi (\mathsf{e})\leq k<\psi (2\mathsf{e})\) it follows that \(\theta \leq \psi ^{1}(k)\mathsf{e}<\mathsf{e}\) and since the cone \(\mathsf{P}_{\mathcal{E}_{d}}\) is normal (with \(N=1\)),
and then \(\rho (\kappa )<1\). Thereupon, by Lemma 2.1 we see that the sequence is (bi)Cauchy. Further, we can find such that the sequence converges to . That is, for every there exists \(m_{1}\in \mathbb{N}\) such that and , for \(m\geq m_{1}\). Thus, replacing in (5) by and ω by we have
and from (c), Definition 1.11,
Moreover, by Definition 1.9 and Remark 1.13, we have
and from Lemma 1.6 we obtain . Therefore, . Similarly, choosing in (6) and , and taking into account the properties of ψ we have
which leads us to
Consequently, . □
Example 2.3
Let \(\mathcal{E}_{d}=\mathbb{R}^{2}\), and for any we define the multiplication as . Then \(\mathcal{E}_{d}\) is a Banach algebra with a unit \(\mathsf{e}=(1,1)\). Let \(\mathrm{X}= \{ 1,3,4,5 \} \) and \(\mathsf{q}:\mathrm{X}\times \mathrm{X}\rightarrow \mathcal{E}\) defined by
be a 2symmetric quasimetric on X. Consider also the mappings \(\mathcal{U},\mathcal{V}:\mathrm{X}\rightarrow \mathrm{X}\) defined by \(\mathcal{U}1=1\), \(\mathcal{U}3=3\), \(\mathcal{U}4=3\), \(\mathcal{U}5=5\) and \(\mathcal{V}1=1\), \(\mathcal{V}3=3\), \(\mathcal{V}4=4\), \(\mathcal{V}5=4\). Then we have
Let \(\psi :\mathsf{P}^{1}{\mathcal{E}}\rightarrow \mathsf{P}^{1}{ \mathcal{E}}\), and \(k=(\frac{9}{8},\frac{9}{8})\).
Therefore:

1
, \(\omega =3\)
$$\begin{aligned}& \psi \bigl(\mathsf{q}(1,\mathcal{U}1)\bigr)+\psi \bigl(\mathsf{q}(3, \mathcal{V}3)\bigr)=(0,0) \leq \biggl(\frac{9}{2},\frac{9}{4} \biggr)=k\psi \bigl(\mathsf{q}(1,3)\bigr), \\& \psi \bigl(\mathsf{q}(1,\mathcal{V}1)\bigr)+\psi \bigl(\mathsf{q}(3, \mathcal{U}3)\bigr)=(0,0) \leq \biggl(\frac{9}{2},\frac{9}{4} \biggr)=k\psi \bigl(\mathsf{q}(1,3)\bigr) \end{aligned}$$ 
2
, \(\omega =4\)
$$\begin{aligned}& \psi \bigl(\mathsf{q}(1,\mathcal{U}1)\bigr)+\psi \bigl(\mathsf{q}(4, \mathcal{V}4)\bigr)=(0,0) \leq \biggl(\frac{27}{4},\frac{27}{8} \biggr)=k\psi \bigl(\mathsf{q}(1,4)\bigr) , \\& \psi \bigl(\mathsf{q}(1,\mathcal{V}1)\bigr)+\psi \bigl(\mathsf{q}(4, \mathcal{U}4)\bigr)=\biggl(1, \frac{1}{2}\biggr)\leq \biggl( \frac{27}{4},\frac{27}{8}\biggr)=k\psi \bigl(\mathsf{q}(1,4) \bigr) \end{aligned}$$ 
3
, \(\omega =5\)
$$\begin{aligned}& \psi \bigl(\mathsf{q}(1,\mathcal{U}1)\bigr)+\psi \bigl(\mathsf{q}(5, \mathcal{V}5)\bigr)=\biggl(1, \frac{1}{2}\biggr)\leq \biggl(9, \frac{9}{2}\biggr)=k\psi \bigl(\mathsf{q}(1,5)\bigr), \\& \psi \bigl(\mathsf{q}(1,\mathcal{V}1)\bigr)+\psi \bigl(\mathsf{q}(5, \mathcal{U}5)\bigr)=(0,0) \leq \biggl(9,\frac{9}{2}\biggr)=k\psi \bigl( \mathsf{q}(1,5)\bigr) \end{aligned}$$ 
4
, \(\omega =4\)
$$\begin{aligned}& \psi \bigl(\mathsf{q}(3,\mathcal{U}3)\bigr)+\psi \bigl(\mathsf{q}(4, \mathcal{V}4)\bigr)=(0,0) \leq \biggl(\frac{9}{4},\frac{9}{8} \biggr)=k\psi \bigl(\mathsf{q}(3,4)\bigr), \\& \psi \bigl(\mathsf{q}(3,\mathcal{V}3)\bigr)+\psi \bigl(\mathsf{q}(4, \mathcal{U}4)\bigr)=\biggl(1, \frac{1}{2}\biggr)\leq \biggl( \frac{9}{4},\frac{9}{8}\biggr)=k\psi \bigl(\mathsf{q}(3,4) \bigr) \end{aligned}$$ 
5
, \(\omega =5\)
$$\begin{aligned}& \psi \bigl(\mathsf{q}(3,\mathcal{U}3)\bigr)+\psi \bigl(\mathsf{q}(5, \mathcal{V}5)\bigr)=\biggl(1, \frac{1}{2}\biggr)\leq \biggl( \frac{9}{2},\frac{9}{4}\biggr)=k\psi \bigl(\mathsf{q}(3,5) \bigr), \\& \psi \bigl(\mathsf{q}(3,\mathcal{V}3)\bigr)+\psi \bigl(\mathsf{q}(5, \mathcal{U}5)\bigr)=(0,0) \leq \biggl(\frac{9}{2},\frac{9}{4} \biggr)=k\psi \bigl(\mathsf{q}(3,5)\bigr) \end{aligned}$$ 
6
, \(\omega =5\)
$$\begin{aligned}& \psi \bigl(\mathsf{q}(4,\mathcal{U}4)\bigr)+\psi \bigl(\mathsf{q}(5, \mathcal{V}5)\bigr)=(2,1) \leq \biggl(\frac{9}{2},\frac{9}{4} \biggr)=k\psi \bigl(\mathsf{q}(3,5)\bigr), \\& \psi \bigl(\mathsf{q}(4,\mathcal{V}4)\bigr)+\psi \bigl(\mathsf{q}(5, \mathcal{U}5)\bigr)=(0,0) \leq \biggl(\frac{9}{2},\frac{9}{4} \biggr)=k\psi \bigl(\mathsf{q}(3,5)\bigr). \end{aligned}$$
Consequently, the assumptions of Theorem 2.2 are verified and the mappings \(\mathcal{U}\), \(\mathcal{V}\) have 2 common fixed points, these being , .
Corollary 2.4
Let \((\mathrm{X}, \mathsf{q}, \mathcal{E}_{d})\) be a complete Δsymmetric qCMS over \(\mathcal{E}_{d}\) and \(\mathsf{P}^{1}_{\mathcal{E}_{d}}\). Suppose that \(\psi : \mathsf{P}^{1}_{\mathcal{E}_{d}} \rightarrow \mathsf{P}^{1}_{ \mathcal{E}_{d}}\) is a ψoperator and \(\mathcal{U}: \mathrm{X}\rightarrow \mathrm{X}\) is a mapping satisfying the condition
for all with , where \(\psi (e) \leq k < \psi (2e)\) in \(\mathsf{P}_{\mathcal{E}^{1}_{d}}\). Then \(\mathcal{U}\) has a fixed point.
Proof
Put \(\mathcal{U}=\mathcal{V}\) in Theorem 2.2. □
Theorem 2.5
Let \((\mathrm{X}, \mathsf{q}, \mathcal{E}_{d})\) be a complete Δsymmetric qCMS over \(\mathcal{E}_{d}\) and \(\mathsf{P}^{1}_{\mathcal{E}_{d}}\). Suppose that \(\psi : \mathsf{P}^{1}_{\mathcal{E}_{d}} \rightarrow \mathsf{P}^{1}_{ \mathcal{E}_{d}}\) is a ψoperator and \(\mathcal{U}, \mathcal{V}: \mathrm{X}\rightarrow \mathrm{X}\) are mappings satisfying the conditions
for all with , where
in \(\mathsf{P}^{1}_{\mathcal{E}_{d}}\). Then \(\mathcal{U}\) and \(\mathcal{V}\) have a common fixed point. Moreover, if \(\psi ^{1}(\beta )<\psi ^{1}(\alpha _{1})\) then the common fixed point is unique.
Proof
Let be the sequence in X defined by (7). Letting and in (11) we have
or
Taking into account the properties of \(\psi ^{1}\), we have
and moreover
Therefore, since the Banach algebra is divisible, we get
If we denote \(\kappa =(\psi ^{1}(\alpha _{1})+\psi ^{1}(\alpha _{3}))^{1}(\psi ^{1}( \beta )\psi ^{1}(\alpha _{2}))\), we can easily see that \(\theta \leq \kappa <\mathsf{e}\) and
In the same way, for and , (12) becomes
or, equivalent
Thereupon,
which yields
(here we took into account that the Banach algebra is divisible). Now, by (13) and (14) we have
for all \(m\in \mathbb{N}\), where \(\theta \leq \kappa <\mathsf{e}\). Then, by using Lemma 2.1, we see that the sequence is (bi)Cauchy and since the qCMS \((\mathrm{X}, \mathsf{q}, \mathcal{E}_{d})\) is complete, we can have such that converges to . Thus, there exists \(m_{2}\in \mathbb{N}\) such that for any we have , and also , , for any \(m\geq m_{1}\). Hence, by (11), respectively, (12) we have
for \(m\geq m_{2}\). Moreover, applying \(\psi ^{1}\) in the above inequalities,
which are equivalent (since the Banach algebra is divisible) with
for all \(m\geq m_{2}\) and any . Therefore, by Lemma 1.6, it follows that and also , which means that is a common fixed point of the mappings \(\mathcal{V}\), \(\mathcal{U}\).
Finally, considering the additional hypothesis, we will prove the uniqueness of the common fixed point. Supposing, on the contrary, that there exists another point, let us say \(\omega _{*}\in \mathrm{X}\) different from , such that , we have, by (11), for example,
Thus,
and we obtain
for any \(n\in \mathbb{N}\). Further, since \((\psi ^{1}(\alpha _{1}))^{1}\psi ^{1}(\beta )<\mathsf{e}\), we get
as \(n\rightarrow \infty \), which means that for any we can have \(n_{0}\in \mathbb{N}\) such that
Thereby, by Lemma 1.6 it follows that , and is the unique fixed point of the mappings \(\mathcal{U}\) and \(\mathcal{V}\). □
Corollary 2.6
Let \((\mathrm{X}, \mathsf{q}, \mathcal{E}_{d})\) be a complete Δsymmetric qCMS over \(\mathcal{E}_{d}\) and \(\mathsf{P}^{1}_{\mathcal{E}_{d}}\). Suppose that \(\psi : \mathsf{P}^{1}_{\mathcal{E}_{d}} \rightarrow \mathsf{P}^{1}_{ \mathcal{E}_{d}}\) is a ψoperator and \(\mathcal{U}: \mathrm{X}\rightarrow \mathrm{X}\) is a mapping satisfying the condition
for all with , where
in \(\mathsf{P}^{1}_{\mathcal{E}_{d}}\). Then \(\mathcal{U}\) has a fixed point. Moreover, if \(\psi ^{1}(\beta )<\psi ^{1}(\alpha _{1})\) then the fixed point is unique.
Proof
Put \(\mathcal{U}=\mathcal{V}\) in Theorem 2.5. □
Theorem 2.7
Let \((\mathrm{X}, \mathsf{q}, \mathcal{E}_{d})\) be a complete Δsymmetric qCMS over \(\mathcal{E}_{d}\) and \(\mathsf{P}^{1}_{\mathcal{E}_{d}}\) be a normal algebra cone in \(\mathcal{E}_{d}\). Suppose that \(\psi : \mathsf{P}^{1}_{\mathcal{E}_{d}} \rightarrow \mathsf{P}^{1}_{ \mathcal{E}_{d}}\) is a ψoperator and \(\mathcal{U}, \mathcal{V}: \mathrm{X}\rightarrow \mathrm{X}\) are mappings satisfying the conditions
for all with , where \(\theta \leq \psi ^{1}(\beta )+(\Delta \mathsf{e})\psi ^{1}( \alpha _{3})\leq \psi ^{1}(\alpha _{1}) \) in \(\mathsf{P}^{1}_{\mathcal{E}_{d}}\). Then \(\mathcal{U}\) and \(\mathcal{V}\) have a common fixed point.
Proof
Let be the sequence in X defined by (7). Letting and , by (16), we have
Moreover, by applying \(\psi ^{1}\), and taking into account the properties of it,
and using the triangle inequality we get
and then
which is equivalent with
Further, since the qCMS is Δsymmetric, there exists an invertible element \(\Delta \in \mathcal{E}\) such that , for all \(m\in \mathbb{N}\) and then we have
Therefore,
On the other hand, with and , the inequality (17) becomes
Applying \(\psi ^{1}\) and keeping in mind its properties we get
Therefore, since
we have
Thus,
and
Consequently, from (18) we conclude that
for any \(m\in \mathbb{N}\), where \(\kappa =(\psi ^{1}(\alpha _{1})+\psi ^{1}(\alpha _{3}))^{1}(\psi ^{1}( \beta )+\Delta \psi ^{1}(\alpha _{3}))<\mathsf{e}\). In this case we get \(\rho (\kappa )<1\) and taking into account Lemma 2.1 we can conclude that the sequence is Cauchy and moreover convergent to an element . Therefore, for any , there exists \(m_{1}\in \mathbb{N}\) such that , . We claim that is a fixed point of mappings \(\mathcal{V}\) and \(\mathcal{U}\). Indeed, from (16) and (17) we have
which becomes (by applying \(\psi ^{1}\))
But since and also
we get
Thereupon,
which (by taking into account Lemma 1.6) shows us that .
Now, similarly, by (19), we have
which is equivalent with
Moreover, by using the triangle inequality,
then
Thus, and is a common fixed point of the mappings \(\mathcal{V}\) and \(\mathcal{U}\). □
Corollary 2.8
Let \((\mathrm{X}, \mathsf{q}, \mathcal{E}_{d})\) be a complete Δsymmetric qCMS over \(\mathcal{E}_{d}\) and \(\mathsf{P}^{1}_{\mathcal{E}_{d}}\) be a normal algebra cone in \(\mathcal{E}_{d}\). Assume \(\psi : \mathsf{P}^{1}_{\mathcal{E}_{d}} \rightarrow \mathsf{P}^{1}_{ \mathcal{E}_{d}}\) is a ψoperator and \(\mathcal{U}: \mathrm{X}\rightarrow \mathrm{X}\) is satisfying the condition
for all with , where \(\theta \leq \psi ^{1}(\beta )+(\Delta \mathsf{e})\psi ^{1}( \alpha _{3})\leq \psi ^{1}(\alpha _{1}) \) in \(\mathsf{P}^{1}_{\mathcal{E}_{d}}\). Then \(\mathcal{U}\) has a fixed point.
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Fulga, A., Afshari, H. & Shojaat, H. Common fixed point theorems on quasicone metric space over a divisible Banach algebra. Adv Differ Equ 2021, 306 (2021). https://doi.org/10.1186/s1366202103464z
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DOI: https://doi.org/10.1186/s1366202103464z