 Research
 Open Access
Dislocated cone metric space over Banach algebra and αquasi contraction mappings of Perov type
 Reny George^{1, 2},
 R Rajagopalan^{1},
 Hossam A Nabwey^{1, 3} and
 Stojan Radenović^{4, 5}Email author
https://doi.org/10.1186/s1366301706197
© The Author(s) 2017
 Received: 30 May 2017
 Accepted: 30 October 2017
 Published: 4 December 2017
Abstract
A dislocated cone metric space over Banach algebra is introduced as a generalisation of a cone metric space over Banach algebra as well as a dislocated metric space. Fixed point theorems for Perovtype αquasi contraction mapping, Kannantype contraction as well as Chatterjeetype contraction mappings are proved in a dislocated cone metric space over Banach algebra. Proper examples are provided to establish the validity of our claims.
Keywords
 fixed points
 cone metric space
 dislocated cone metric space
 αadmissible mapping
MSC
 47H10
 54H25
1 Introduction
Generalising the concept of cone metric space, Liu and Xu in [1] introduced a cone metric space over Banach algebra (in short CMSBA) and proved contraction principles in such a space. They replaced the usual real contraction constant with a vector constant and scalar multiplication with vector multiplication in their results and also furnished proper examples to show that their results were different from those in a cone metric space and a metric space. While studying the applications of topology in logic programming semantics, Hitzler and Seda [2] introduced a dislocated metric space as a generalisation of a metric space and discussed the associated topologies. Later George and Khan introduced a dislocated fuzzy metric space [3], and then various fixed point results were proved in dislocated spaces. For some details, refer to [4]. On the other hand, Perov [5] generalised the Banach contraction principle by replacing the contractive factor with a matrix convergent to zero. Cvetkovic and Rakocevic [6] introduced a Perovtype quasicontractive mapping replacing contractive factor with bounded linear operator with spectral radius less than one and obtained some interesting fixed point results in the setup of cone metric spaces.
In this work we introduce the concept of dislocated cone metric space over Banach algebra (in short dCMSBA) as a generalisation of CMSBA as well as a dislocated metric space and prove fixed point theorems for a Perovtype αquasi contraction mapping in dCMSBA and CMSBA. Simple examples are given to illustrate the validity and superiority of our results.
2 Preliminaries
 (i)
\((xy)z = x(yz)\);
 (iia)
\(x(y + z) = xy + xz\);
 (iib)
\((x + y)z = xz + yz\);
 (iii)
\(\alpha (xy) = (\alpha x)y = x(\alpha y)\).
A Banach algebra is a Banach space \(\mathcal{A}\) over \({K} \in \{ \mathbb{R},\mathbb{C}\}\) such that, for all \(x,y \in {\mathcal{A}}\), \(\Vert xy\Vert \leq \Vert x\Vert \Vert y\Vert \).
For a given cone \({P}\subset {\mathcal{A}}\) and \(x,y \in {\mathcal{A}}\), we say that \(x\preceq y\) if and only if \(yx\in {P}\). Note that ⪯ is a partial order relation defined on \(\mathcal{A}\). For more details on the basic concepts of Banach algebra, solid cone, unit element e, zero element θ, invertible elements in Banach algebra etc., the reader may refer to [1, 7].
In what follows \(\mathcal{A}\) will always denote a Banach algebra, P a solid cone in \(\mathcal{A}\) and e the unit element of \(\mathcal{A}\).
Definition 2.1
A sequence \({p_{n}}\) in a solid cone P of a Banach space is a csequence if, for each \(c\gg \theta \), there exists \(n_{0}\in \mathbb{N}\) such that \(p_{n}\ll c\) for all \(n\ge n_{0}\).
Lemma 2.2
([8])
Lemma 2.3
([9])
Lemma 2.4
([9])
Lemma 2.5
([7])
 (i)
If \(a,b,c\in E\) and \(a\preceq b\ll c\), then \(a\ll c\).
 (ii)
If \(\theta \preceq a\ll c\) for each \(c\gg \theta \), then \(a=\theta \).
Lemma 2.6
([9])
Let \(\{u_{n}\}\) be a sequence in \(\mathcal{A}\) with \(\{u_{n}\}\rightarrow \theta\) (\(n\rightarrow \infty \)). Then \(\{u_{n}\}\) is a csequence.
Lemma 2.7
([7])
Let \(\{u_{n}\}\) be a csequence in P. If \(\beta \in {P}\) is an arbitrarily given vector, then \(\{\beta u _{n}\}\) is a csequence.
Lemma 2.8
([8])
Let \(\alpha \in {\mathcal{A}}\) and \(r(\alpha )<1\), then \(\{\alpha^{n}\}\) is a csequence.
Definition 2.10
 (i)
T is an αadmissible mapping iff \(\alpha (x,y)\ge 1\) implies \(\alpha (Tx, Ty) \ge 1\), \(x,y\in X\).
 (ii)
T is an αdominated mapping iff \(\alpha (x,y)\ge 1\) implies \(\alpha (x, Tx) \ge 1\), \(x,y\in X\).
3 Main results
In this section first we introduce the definition of a dislocated cone metric space over Banach algebra (in short dCMSBA) and furnish examples to show that this concept is more general than that of CMSBA. We then define convergence and Cauchy sequence in a dCMSBA and then prove fixed point results in this space.
Definition 3.1
 (dCM1)
\(\theta \preceq d_{\mathrm {lc}}(x,y)\) and \(d_{\mathrm {lc}}(x,y)=\theta \) imply \(x=y\);
 (dCM2)
\(d_{\mathrm {lc}}(x,y)=d_{\mathrm {lc}}(y,x)\);
 (dCM3)
\(d_{\mathrm {lc}}(x,y)\preceq d_{\mathrm {lc}}(x,z)+ d_{\mathrm {lc}}(z,y)\).
Note that every metric space and CMSBA is a dCMSBA, but the converse is not necessarily true. Inspired by [1, 7, 10], we furnish the following examples which will establish our claim.
Example 3.2
Then \((\chi ,d_{\mathrm {lc}})\) is a dCMSBA over \(\mathcal{A}\) but not a CMSBA over Banach algebra \(\mathcal{A}\).
Example 3.3
Definition 3.4
 (i)
\(\{p_{n}\}\) converges to p if, for each \(c\in {\mathcal{A}}\) with \(\theta \ll c\), there exists \(n_{0}\in \mathbb{N}\) such that \(d_{\mathrm {lc}}(p_{n},p) \ll c\) for all \(n\ge n_{0}\). We write it as \(\operatorname{Lim}_{n\rightarrow \infty }p_{n} = p\).
 (ii)
\(\{p_{n}\}\) is a Cauchy sequence if and only if for each \(c\in {\mathcal{A}}\) with \(\theta \ll c\), there exists \(n_{0}\in \mathbb{N}\) such that \(d_{\mathrm {lc}}(p_{n},p_{m}) \ll c\) for all \(n,m\ge n_{0}\).
 (iii)
\((\chi ,d_{\mathrm {lc}})\) is a complete \(dCMS\) if and only if every Cauchy sequence in \((\chi ,d_{\mathrm {lc}})\) is convergent.
Proposition 3.5
 (i)
\(d_{\mathrm {lc}}(p_{n},p)\) is a csequence;
 (ii)
\(d_{\mathrm {lc}}(p_{n},p_{n+r})\) is a csequence.
In [11] Samet et al. introduced the concept of αadmissible mappings and proved fixed point theorems for alphapsi contractivetype mappings, which paved the way for proving new and existing results in fixed point theory. As in [11] and others, we give the following definitions.
Definition 3.6
 (i)
T is an αadmissible mapping iff \(\alpha (x,y)\ge 1\) implies \(\alpha (Tx, Ty) \ge 1\), \(x,y\in X\).
 (ii)
T is an αdominated mapping iff \(\alpha (x,y)\ge 1\) implies \(\alpha (x, Tx) \ge 1\), \(x,y\in X\).
 (iii)
α is a triangular function iff \(\alpha (x,y)\ge 1\), \(\alpha (y,z) \ge 1\) imply \(\alpha (x,z)\ge 1\), \(x,y,z\in X\).
 (iv)
\((\chi ,d_{\mathrm {lc}})\) is αregular iff for any sequence \(\{x_{p}\}\) in χ with \(\alpha (x_{p}, x_{p+1}) \ge 1\) and \(x_{p} \rightarrow x_{*}\) as \(p\rightarrow \infty \), then \(\alpha (x_{p}, x_{*}) \ge 1\)
 \((\mathbf{K})\) :

For all \(x\neq y\in \chi \), there exists \(w\in \chi \) such that \(\alpha (x,w) \ge 1\), \(\alpha (y,w) \ge 1\) and \(\alpha (w,Tw) \ge 1\).
We now introduce the following definitions.
Definition 3.7
 (i)
T is an αidentical function iff \(\alpha (Tx,Tx) \ge 1\) for all \(x\in \chi \).
 (ii)
T is weak semi αadmissible iff \(\alpha (x,y) \ge 1\) implies \(\alpha (x,T^{2}y) \ge 1\) for any \(x,y\in \chi \).
 (iii)
T satisfies condition \((\mathbf{G})\) iff \(\alpha (x,Tx)\ge 1\) and \(\alpha (y,Ty)\ge 1\) imply \(\alpha (x,y)\ge 1\) or \(\alpha (Tx,Ty) \ge 1\) for any \(x,y\in \chi \).
 (iv)
T satisfies condition \((\mathbf{G}')\) iff for all \(x\neq y\in \chi \) with \(\alpha (x,Tx) \ge 1\) and \(\alpha (y,Ty) \ge 1\), there exists \(w\in \chi \) such that \(\alpha (x,w) \ge 1\), \(\alpha (y,w) \ge 1\), \(\alpha (w, w) \ge 1\) and \(\alpha (w,Tw) \ge 1\).
Example 3.8
Example 3.9
Example 3.10
Example 3.11
Lemma 3.12
Let X be a nonempty set and \(T : X \rightarrow X\) and \(\alpha : X \times X \rightarrow [0, \infty )\) be mappings. Let \(\{x_{n}\}\) be the Picard sequence starting with \(x_{0}\). If \(\alpha (x_{0}, x_{0}) \ge 1\) and \(\alpha (x_{0}, Tx_{0}) \ge 1\), and if α is a triangular function and T is αadmissible, then for all \(n \ge 1\) and \(0\le p\le q\le n\), \(\alpha (x_{p}, x_{q}) \ge 1\).
Proof
For the proof, we will make use of the principle of mathematical induction.
As \(\alpha (x_{0}, x_{0}) \ge 1\), \(\alpha (x_{0}, x_{1}) \ge 1\) and T is αadmissible, \(\alpha (x_{1}, x_{1}) = \alpha (Tx_{0}, Tx_{0})\ge 1\), and so the result holds good for \(n=1\). Again, by αadmissibility of T, we get \(\alpha (x_{1}, x_{2}) \ge 1\), \(\alpha (x_{2}, x_{2}) \ge 1\), and then, since α is triangular, we get \(\alpha (x_{0}, x_{2}) \ge 1\). Thus the result holds good for \(n=2\). Suppose the result is true for \(n=r\), i.e. \(\alpha (x_{p}, x _{q}) \ge 1\) for all \(0\le p\le q\le r\). We will show that it is true for \(n=r+1\). It is enough to consider the case \(\alpha (x_{p}, x_{r+1})\), \(0\le p\le r+1\). By induction hypothesis and α admissibility of T, we have \(\alpha (x_{p}, x_{r+1}) \ge 1\) for all \(1\le p\le r+1\). Since \(\alpha (x_{0}, x_{1}) \ge 1\), by α admissibility of T and triangularity of function α, we get \(\alpha (x_{0}, x_{r+1}) \ge 1\), and thus the result is true for \(n=r+1\). Hence, by the principle of mathematical induction, the result is true for all n. □
Lemma 3.13
Let X be a nonempty set and \(T : X \rightarrow X\) and \(\alpha : X \times X \rightarrow [0, \infty )\) be mappings. Let \(\{x_{n}\}\) be the Picard sequence starting with \(x_{0}\) such that \(\alpha (x_{0}, x_{0}) \ge 1\) and \(\alpha (x_{0}, Tx_{0}) \ge 1\). If T is αadmissible and weak semi αadmissible, then for all \(n \ge 1\) and \(0\le p\le q\le n\), \(\alpha (x_{p}, x_{q}) \ge 1\).
Proof
As \(\alpha (x_{0}, x_{0}) \ge 1\), \(\alpha (x_{0}, x _{1}) \ge 1\) and T is αadmissible, \(\alpha (x_{1}, x_{1}) = \alpha (Tx_{0}, Tx_{0})\ge 1\), and so the result holds good for \(n=1\). Again, by αadmissibility of T, we get \(\alpha (x_{1}, x _{2}) \ge 1\), \(\alpha (x_{2}, x_{2}) \ge 1\). Since T is weak semi αadmissible and \(\alpha (x_{0}, x_{0}) \ge 1\), we get \(\alpha (x_{0}, x_{2}) \ge 1\). Thus the result holds good for \(n=2\). Suppose the result is true for \(n=r\), i.e. \(\alpha (x_{p}, x_{q}) \ge 1\) for all \(0\le p\le q\le r\). We will show that it is true for \(n=r+1\). It is enough to consider the case \(\alpha (x_{p}, x_{r+1})\), \(0\le p\le r+1\). By induction hypothesis and α admissibility of T, we have \(\alpha (x_{p}, x_{r+1}) \ge 1\) for all \(1\le p\le r+1\). If r is even, then using \(\alpha (x_{0}, x_{1}) \ge 1\) and repeatedly using weak semi α admissibility of T, we get α, we get \(\alpha (x_{0}, x_{r+1}) \ge 1\). If r is odd, then using \(\alpha (x _{0}, x_{0}) \ge 1\) and repeatedly using weak semi α admissibility of T, we get α, we get \(\alpha (x_{0}, x_{r+1}) \ge 1\). Thus the result is true for \(n=r+1\). Hence, by the principle of mathematical induction, the result is true for all n. □
Definition 3.14
Lemma 3.15
Proof
(i) \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{p1},x_{r}) \).
Case i(c): \(p = r+1\). In this case \(d_{\mathrm {lc}}(x_{p},x _{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{r},x_{r})\), and the result follows from (3.3) and (3.4).
(ii) \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{p1},x_{p}) \).
Case ii(b): \(2\le p\le r\). The result follows from (3.3).
Case ii(c1): \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{r},x_{r+1})\preceq \mu^{2}d_{\mathrm {lc}}(x_{r1},x_{r})\).
By (3.3) and (3.4), we get \(\mu^{2}d_{\mathrm {lc}}(x_{r1},x_{r}) \preceq \mu (e\mu )^{1}(d_{\mathrm {lc}}(x_{0},x_{1}) +d_{\mathrm {lc}}(x_{0},x_{0}))\).
Case ii(c2): \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{r},x_{r+1})\preceq \mu^{2}d_{\mathrm {lc}}(x_{r},x_{r+1})\).
Now, \(\mu d_{\mathrm {lc}}(x_{r},x_{r+1})\preceq \mu^{2}d_{\mathrm {lc}}(x_{r},x_{r+1})\) implies \(\mu (e\mu )d_{\mathrm {lc}}(x_{r},x_{r+1})\preceq \theta \). Note that \(r(\mu )<1\), and so \((e\mu )\) is invertible and \((e\mu )^{1} > e\). Therefore we get \(d_{\mathrm {lc}}(x_{r},x_{r+1})\preceq \theta \), i.e. \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \theta \preceq \mu (e\mu )^{1}(d_{\mathrm {lc}}(x _{0},x_{1}) +d_{\mathrm {lc}}(x_{0},x_{0})) \).
Case ii(c3): \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{r},x_{r+1})\preceq \mu^{2}d_{\mathrm {lc}}(x_{r1},x_{r+1})\)
 (i)
\(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu^{k} d_{\mathrm {lc}}(x_{p},x_{q})\) for some \(1\le p,q\le r\), \(2\le k\le r\), and the result follows from this by (3.3) and (3.4).
 (ii)
\(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu^{k} d_{\mathrm {lc}}(x_{r},x_{r+1})\), and the result follows by proceeding as in Case ii(c2).
 (iii)
\(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu^{r}d_{\mathrm {lc}}(x_{p},x_{r+1})\), which implies \((e\mu ) d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \theta \), and by the same argument as in Case ii(c2) the result follows.
Case ii(c4): \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{r},x_{r+1})\preceq \mu^{2}d_{\mathrm {lc}}(x_{r},x_{r})\). The result follows from (3.3) and (3.4).
(iii) \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{r},x_{r+1})= \mu d_{\mathrm {lc}}(Tx_{r1},Tx_{r})\). The result follows proceeding as in Case ii(c).
(iv) \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{r},x_{p})\). The result follows from (3.3) and (3.4).
(v) \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{p1},x_{r+1})\).
Using (3.1) and continuing in a similar manner as above, either we will get the desired result or we get \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{p1},x_{r+1})\preceq \mu ^{2} d_{\mathrm {lc}}(x_{p2},x_{r+1})\cdots \preceq \mu^{p1} d_{\mathrm {lc}}(x_{1},x _{r+1})\).
Theorem 3.16
 (i)
T is a Perovtype αquasi contraction mapping.
 (ii)
α is a triangular function or T is weak semi αadmissible.
 (iii)
T is αadmissible.
 (iv)
There exists \(x_{0} \in \chi \) such that \(\alpha (x_{0}, x_{0})\ge 1\) and \(\alpha (x_{0}, Tx_{0})\ge 1\).
 (v)
\((\chi ,d_{\mathrm {lc}})\) is αregular.
Proof
In Theorem 3.16 condition (v) can be replaced with another condition as in Popescu [12]. We have the following.
Theorem 3.17
 (i)
T is a Perovtype αquasi contraction mapping.
 (ii)
α is a triangular function or T is weak semi αadmissible.
 (iii)
T is αadmissible.
 (iv)
There exists \(x_{0} \in \chi \) such that \(\alpha (x_{0}, x_{0})\ge 1\) and \(\alpha (x_{0}, Tx_{0})\ge 1\).
 (v)
If \(\{x_{p}\}\) is a sequence in χ such that \(\alpha (x_{p}, x _{p+1}) \ge 1 \) for all p and \(x_{p} \rightarrow u\in \chi \) as \(p \rightarrow \infty \), then there exists a subsequence \(\{x_{p(k)} \}\) of \(\{x_{p}\}\) such that \(\alpha (x_{p(k)}, u)\ge 1\) for all k.
Proof
Theorem 3.18
Let \((\chi ,d_{\mathrm {lc}})\), T and α be as in Theorem 3.16. Suppose that all conditions of Theorem 3.16 or Theorem 3.17 are satisfied. If T is an αidentical function or if T is αdominated, then T has a fixed point \(x_{*} \in \chi \) and \(d_{\mathrm {lc}}(x_{*}, x_{*}) = \theta \). Further, if T satisfies condition \((\mathbf{G})\), then the fixed point is unique.
Proof
As in the proof of Theorem 3.16 or Theorem 3.17, we see that T has a fixed point \(x_{*} \in \chi \). If T is αidentical , then \(\alpha (x_{*}, x_{*}) = \alpha (Tx _{*}, Tx_{*})\ge 1\). If T is αdominated, then \(\alpha (x _{*}, x_{*}) = \alpha (x_{*}, Tx_{*})\ge 1\). Then from (3.1) we have \(d_{\mathrm {lc}}(x_{*},x_{*}) = d_{\mathrm {lc}}(Tx_{*},Tx_{*}) \preceq \mu \{d_{\mathrm {lc}}(x _{*},x_{*}), d_{\mathrm {lc}}(x_{*},x_{*}), d_{\mathrm {lc}}(x_{*},x_{*}), d_{\mathrm {lc}}(x_{*},x _{*}), d_{\mathrm {lc}}(x_{*},x_{*})\} = \mu d_{\mathrm {lc}}(x_{*},x_{*})\). Hence \({d_{\mathrm {lc}}(x_{*},x_{*}) = \theta }\).
Theorem 3.19
Let \((\chi ,d_{\mathrm {lc}})\), T and α be as in Theorem 3.16. Suppose that all conditions of Theorem 3.16 or Theorem 3.17 are satisfied. If T is an αidentical function or if T is αdominated, then T has a fixed point \(x_{*} \in \chi \) and \(d_{\mathrm {lc}}(x_{*}, x_{*}) = \theta \). Further, if T satisfies condition \((\mathbf{G}')\), then the fixed point is unique.
Proof
Remark 3.20
In Theorems 3.18 and 3.19 we can replace the requirement of condition \((\mathbf{G})\) or condition \((\mathbf{G}')\) with that of condition \((\mathbf{K})\). But as in Examples 3.8 and 3.9, there exist functions α and T such that T is αidentical and T satisfies condition \((\mathbf{G})\) and condition \((\mathbf{G}')\) but does not satisfy condition \((\mathbf{K})\). Hence our approach is new and justifiable.
Since every CMSBA is a dCMSBA and since in a cone metric space \((\chi , d_{c})\), \(d_{c}(x,y) = \theta \) for all \(x,y \in \chi \), we give the following generalised results which are easily deduced from our main results.
Theorem 3.21
 (i)
T is a Perovtype αquasi contraction mapping.
 (ii)
α is a triangular function.
 (iii)
T is αadmissible.
 (iv)
There exists \(x_{0} \in \chi \) such that and \(\alpha (x_{0}, Tx_{0}) \ge 1\).
 (v)
\((\chi ,d_{\mathrm {lc}})\) is αregular.
Theorem 3.22
 (i)
T is a Perovtype αquasi contraction mapping.
 (ii)
α is a triangular function.
 (iii)
T is αadmissible.
 (iv)
There exists \(x_{0} \in \chi \) such that and \(\alpha (x_{0}, Tx_{0}) \ge 1\).
 (v)
If \(\{x_{p}\}\) is a sequence in χ such that \(\alpha (x_{p}, x _{p+1}) \ge 1 \) for all p and \(x_{p} \rightarrow u\in \chi \) as \(p \rightarrow \infty \), then there exists a subsequence \(\{x_{p(k)} \}\) of \(\{x_{p}\}\) such that \(\alpha (x_{p(k)}, u)\ge 1\) for all k.
Theorem 3.23
Let \((\chi ,d_{c})\), T and α be as in Theorem 3.21. Suppose that all conditions of Theorem 3.21 or Theorem 3.22 are satisfied. If T is an αidentical or αdominated function, then T has a fixed point \(x_{*} \in \chi \). Further, if T satisfies condition \((\mathbf{G})\), then the fixed point is unique.
Theorem 3.24
Let \((\chi ,d_{c})\), T and α be as in Theorem 3.21. Suppose that all conditions of Theorem 3.21 or Theorem 3.22 are satisfied. If T is an αidentical or αdominated function, then T has a fixed point \(x_{*} \in \chi \). Further, if T satisfies condition \((\mathbf{G}')\), then the fixed point is unique.
Theorem 3.25
Proof
Corollary 3.26
Proof
The proof easily follows from Theorems 3.21 and 3.23 or Theorems 3.22 and 3.24 by taking \(\alpha (x,y)=1\) for all \(x,y \in \chi \). □
Corollary 3.27
(Theorem 9, [13])
Proof
Since every CMSBA is a dCMSBA, the proof follows from Corollary 3.26. □
Example 3.28
Theorem 3.29
Proof
Theorem 3.30
Theorem 3.31
4 Conclusion
In this paper we have introduced the concept of dislocated cone metric space over Banach algebra and proved some generalised fixed point theorems in such a space. Some new properties of mappings such as αidentical mappings, semi αadmissible mappings, mappings satisfying condition \((\mathbf{G})\) and condition \((\mathbf{G}')\) are also introduced. Our work is a generalisation of some work already done on metric spaces and cone metric spaces over Banach algebra. There is further scope for extending and generalising various fixed point theorems in the setting of a dislocated cone metric space over Banach algebra.
Declarations
Acknowledgements
This project is supported by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Al kharj, Kingdom of Saudi Arabia, under International Project Grant No. 2016/01/6714. The authors are thankful to the learned reviewers for their valuable suggestions which helped in bringing this paper in its present form.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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