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Common fixed point theorems on quasi-cone metric space over a divisible Banach algebra

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.

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, [220].

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 , \(a \in \mathrm{R}\),

  1. (a)

    ;

  2. (b)

    , ;

  3. (c)

    ;

  4. (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

(1)

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. 1

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

  2. 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

  1. (a)

    for all and if and only if ,

  2. (b)

    for all ,

  3. (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

  1. (a)

    for all ,

  2. (b)

    if and only if ,

  3. (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

for all .

Definition 1.10

Suppose \((\mathrm{X}, \mathsf{q}, \mathcal{E})\) is a qCMS, and is a sequence in X. Then

  1. (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 .

  2. (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 \(m>p \geq N\).

  3. (c)

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

  4. (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

  1. (a)

    ψ is an increasing;

  2. (b)

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

  3. (c)

    for all ;

  4. (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 .

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

(2)

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,

(3)

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

(4)

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

(5)
(6)

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

(7)

Then, setting and , we get

and then

(8)

Also if we put and , then we have

Thus,

(9)

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 \(N=1\)),

$$ \Vert \kappa \Vert \leq \Vert \mathsf{e} \Vert =1, $$

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 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

$$\begin{aligned}& \mathsf{q}(1,3)=(4,2),\qquad \mathsf{q}(1,4)=(6,3),\qquad \mathsf{q}(1,5)=(8,4), \qquad \mathsf{q}(3,4)=(2,1), \\& \mathsf{q}(3,5)=(4,2),\qquad\mathsf{q}(4,5)=(2,1),\qquad \mathsf{q}(1,\mathcal{U}1)=(0,0),\qquad \mathsf{q}(3,\mathcal{U}3)=(0,0), \\& \mathsf{q}(4,\mathcal{U}4)=\biggl(1, \frac{1}{2}\biggr),\qquad\mathsf{q}(5, \mathcal{U}5)=(0,0),\qquad \mathsf{q}(1,\mathcal{V}1)=(0,0), \\& \mathsf{q}(3,\mathcal{V}3)=(0,0),\qquad \mathsf{q}(4,\mathcal{V}4)=(0,0), \qquad \mathsf{q}(5, \mathcal{V}5)=\biggl(1,\frac{1}{2}\biggr). \end{aligned}$$

Let \(\psi :\mathsf{P}^{1}{\mathcal{E}}\rightarrow \mathsf{P}^{1}{ \mathcal{E}}\), and \(k=(\frac{9}{8},\frac{9}{8})\).

Therefore:

  1. 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. 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. 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. 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. 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. 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

(10)

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

(11)
(12)

for all with , where

$$ \theta < \psi ^{-1}(\alpha _{2}), \qquad \theta \leq \psi ^{-1}(\beta ) < \psi ^{-1}(\alpha _{1})+\psi ^{-1}(\alpha _{2})+\psi ^{-1}(\alpha _{3}), $$

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

(13)

In the same way, for and , (12) becomes

or, equivalent

Thereupon,

which yields

(14)

(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

$$ \bigl\Vert \bigl[\bigl(\psi ^{-1}(\alpha _{1}) \bigr)^{-1}\psi ^{-1}(\beta )\bigr]^{n} \bigr\Vert \leq \bigl\Vert \bigl( \psi ^{-1}(\alpha _{1}) \bigr)^{-1}\psi ^{-1}(\beta ) \bigr\Vert ^{n} \rightarrow \theta , $$

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 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

(15)

for all with , where

$$ \theta < \psi ^{-1}(\alpha _{2}),\, \theta \leq \psi ^{-1}(\beta ) < \psi ^{-1}(\alpha _{1})+\psi ^{-1}(\alpha _{2})+\psi ^{-1}(\alpha _{3}), $$

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

(16)
(17)

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 Δ-symmetric, there exists an invertible element \(\Delta \in \mathcal{E}\) such that , for all \(m\in \mathbb{N}\) and then we have

Therefore,

(18)

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

(19)

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 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

(20)

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.

Availability of data and materials

The data and material used to support the findings of this study are included within the article.

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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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Keywords

  • Common fixed point
  • ψ-operator
  • Quasi-cone metric
  • Quasi-Banach-valued metric space