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Common fixed point theorems on quasi-cone 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 self-mappings defined on quasi-cone 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: quasi-metric, symmetric, dislocated metric, b-metric, 2-metric, D-metric, S-metric, G-metric, partial metric, ultra-metric, etc. We shall focus on cone metric space or more precisely, Banach-valued metric space. The idea of Banach-valued 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 Banach-valued 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, quasi-cone 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 non-zero 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
![](http://media.springernature.com/lw292/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ1_HTML.png)
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 non-empty 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 non-empty 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 quasi-cone 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 quasi-cone metric space over Banach algebra, in short, qCMS.
A quasi-cone metric space is called Δ-symmetric, if there exists an invertible element \(\Delta \in \mathcal{E}\) such that
![](http://media.springernature.com/lw212/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equa_HTML.png)
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 bi-Cauchy 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 ψ-operator 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
![](http://media.springernature.com/lw242/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equb_HTML.png)
which yields
![](http://media.springernature.com/lw179/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equc_HTML.png)
Hence,
![](http://media.springernature.com/lw305/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equd_HTML.png)
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
![](http://media.springernature.com/lw208/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Eque_HTML.png)
which yields
![](http://media.springernature.com/lw144/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equf_HTML.png)
Then we obtain
![](http://media.springernature.com/lw271/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equg_HTML.png)
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 Δ-symmetric 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
![](http://media.springernature.com/lw177/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ2_HTML.png)
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
![](http://media.springernature.com/lw168/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equh_HTML.png)
Let \(m>p \geq N\). Thereupon,
![](http://media.springernature.com/lw366/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equi_HTML.png)
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
![](http://media.springernature.com/lw206/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equj_HTML.png)
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,
![](http://media.springernature.com/lw338/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equk_HTML.png)
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,
![](http://media.springernature.com/lw254/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ3_HTML.png)
for all \(m>p>p_{0}\). But, since the space \((\mathrm{X}, \mathsf{q}, \mathcal{E}_{d})\) is supposed to be Δ-symmetric, we have
![](http://media.springernature.com/lw149/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ4_HTML.png)
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 Δ-symmetric 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 ψ-operator and \(\mathcal{U}, \mathcal{V}: \mathrm{X}\rightarrow \mathrm{X}\) are mappings satisfying the conditions
![](http://media.springernature.com/lw284/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ5_HTML.png)
![](http://media.springernature.com/lw284/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ6_HTML.png)
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
![](http://media.springernature.com/lw314/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ7_HTML.png)
Then, setting and
, we get
![](http://media.springernature.com/lw287/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equl_HTML.png)
and then
![](http://media.springernature.com/lw389/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ8_HTML.png)
Also if we put and
, then we have
![](http://media.springernature.com/lw287/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equm_HTML.png)
Thus,
![](http://media.springernature.com/lw377/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ9_HTML.png)
Moreover, applying \(\psi ^{-1}\) in (8), (9) and keeping in mind the properties of the operator \(\psi ^{-1}\), it follows
![](http://media.springernature.com/lw349/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equn_HTML.png)
or by simplifying, we obtain
![](http://media.springernature.com/lw286/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equo_HTML.png)
Denoting \(\kappa =\psi ^{-1}(k)-\mathsf{e}\), the above inequalities pass into
![](http://media.springernature.com/lw177/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equp_HTML.png)
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
![](http://media.springernature.com/lw331/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equr_HTML.png)
and from (c), Definition 1.11,
![](http://media.springernature.com/lw310/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equs_HTML.png)
Moreover, by Definition 1.9 and Remark 1.13, we have
![](http://media.springernature.com/lw375/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equt_HTML.png)
and from Lemma 1.6 we obtain . Therefore,
. Similarly, choosing in (6)
and
, and taking into account the properties of ψ we have
![](http://media.springernature.com/lw475/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equu_HTML.png)
which leads us to
![](http://media.springernature.com/lw414/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equv_HTML.png)
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
![](http://media.springernature.com/lw253/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equw_HTML.png)
be a 2-symmetric quasi-metric 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 Δ-symmetric 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 ψ-operator and \(\mathcal{U}: \mathrm{X}\rightarrow \mathrm{X}\) is a mapping satisfying the condition
![](http://media.springernature.com/lw284/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ10_HTML.png)
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 Δ-symmetric 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 ψ-operator and \(\mathcal{U}, \mathcal{V}: \mathrm{X}\rightarrow \mathrm{X}\) are mappings satisfying the conditions
![](http://media.springernature.com/lw437/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ11_HTML.png)
![](http://media.springernature.com/lw437/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ12_HTML.png)
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
![](http://media.springernature.com/lw456/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equaf_HTML.png)
or
![](http://media.springernature.com/lw434/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equag_HTML.png)
Taking into account the properties of \(\psi ^{-1}\), we have
![](http://media.springernature.com/lw489/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equah_HTML.png)
and moreover
![](http://media.springernature.com/lw466/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equai_HTML.png)
Therefore, since the Banach algebra is divisible, we get
![](http://media.springernature.com/lw481/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equaj_HTML.png)
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
![](http://media.springernature.com/lw212/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ13_HTML.png)
In the same way, for and
, (12) becomes
![](http://media.springernature.com/lw456/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equak_HTML.png)
or, equivalent
![](http://media.springernature.com/lw554/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equal_HTML.png)
Thereupon,
![](http://media.springernature.com/lw453/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equam_HTML.png)
which yields
![](http://media.springernature.com/lw464/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ14_HTML.png)
(here we took into account that the Banach algebra is divisible). Now, by (13) and (14) we have
![](http://media.springernature.com/lw177/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equan_HTML.png)
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
![](http://media.springernature.com/lw432/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equao_HTML.png)
for \(m\geq m_{2}\). Moreover, applying \(\psi ^{-1}\) in the above inequalities,
![](http://media.springernature.com/lw250/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equap_HTML.png)
which are equivalent (since the Banach algebra is divisible) with
![](http://media.springernature.com/lw446/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equaq_HTML.png)
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,
![](http://media.springernature.com/lw492/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equar_HTML.png)
Thus,
![](http://media.springernature.com/lw445/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equas_HTML.png)
and we obtain
![](http://media.springernature.com/lw481/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equat_HTML.png)
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
![](http://media.springernature.com/lw312/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equav_HTML.png)
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 Δ-symmetric 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 ψ-operator and \(\mathcal{U}: \mathrm{X}\rightarrow \mathrm{X}\) is a mapping satisfying the condition
![](http://media.springernature.com/lw438/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ15_HTML.png)
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 Δ-symmetric 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 ψ-operator and \(\mathcal{U}, \mathcal{V}: \mathrm{X}\rightarrow \mathrm{X}\) are mappings satisfying the conditions
![](http://media.springernature.com/lw437/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ16_HTML.png)
![](http://media.springernature.com/lw437/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ17_HTML.png)
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
![](http://media.springernature.com/lw456/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equax_HTML.png)
Moreover, by applying \(\psi ^{-1}\), and taking into account the properties of it,
![](http://media.springernature.com/lw466/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equay_HTML.png)
and using the triangle inequality we get
![](http://media.springernature.com/lw300/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equaz_HTML.png)
and then
![](http://media.springernature.com/lw426/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equba_HTML.png)
which is equivalent with
![](http://media.springernature.com/lw536/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbb_HTML.png)
Further, since the qCMS is Δ-symmetric, there exists an invertible element \(\Delta \in \mathcal{E}\) such that , for all \(m\in \mathbb{N}\) and then we have
![](http://media.springernature.com/lw478/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbc_HTML.png)
Therefore,
![](http://media.springernature.com/lw494/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ18_HTML.png)
On the other hand, with and
, the inequality (17) becomes
![](http://media.springernature.com/lw456/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbd_HTML.png)
Applying \(\psi ^{-1}\) and keeping in mind its properties we get
![](http://media.springernature.com/lw442/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Eqube_HTML.png)
Therefore, since
![](http://media.springernature.com/lw304/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbf_HTML.png)
we have
![](http://media.springernature.com/lw442/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbg_HTML.png)
Thus,
![](http://media.springernature.com/lw462/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbh_HTML.png)
and
![](http://media.springernature.com/lw481/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ19_HTML.png)
Consequently, from (18) we conclude that
![](http://media.springernature.com/lw177/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbi_HTML.png)
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
![](http://media.springernature.com/lw544/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbj_HTML.png)
which becomes (by applying \(\psi ^{-1}\))
![](http://media.springernature.com/lw430/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbk_HTML.png)
But since and also
![](http://media.springernature.com/lw265/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbl_HTML.png)
we get
![](http://media.springernature.com/lw473/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbm_HTML.png)
Thereupon,
![](http://media.springernature.com/lw466/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbn_HTML.png)
which (by taking into account Lemma 1.6) shows us that .
Now, similarly, by (19), we have
![](http://media.springernature.com/lw464/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbo_HTML.png)
which is equivalent with
![](http://media.springernature.com/lw559/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbp_HTML.png)
Moreover, by using the triangle inequality,
![](http://media.springernature.com/lw474/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbq_HTML.png)
then
![](http://media.springernature.com/lw531/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equbr_HTML.png)
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 Δ-symmetric 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 ψ-operator and \(\mathcal{U}: \mathrm{X}\rightarrow \mathrm{X}\) is satisfying the condition
![](http://media.springernature.com/lw438/springer-static/image/art%3A10.1186%2Fs13662-021-03464-z/MediaObjects/13662_2021_3464_Equ20_HTML.png)
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 quasi-cone metric space over a divisible Banach algebra. Adv Differ Equ 2021, 306 (2021). https://doi.org/10.1186/s13662-021-03464-z
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DOI: https://doi.org/10.1186/s13662-021-03464-z