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Dislocated cone metric space over Banach algebra and αquasi contraction mappings of Perov type
Fixed Point Theory and Applicationsvolume 2017, Article number: 24 (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.
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.
Preliminaries
A linear space \(\mathcal{A}\) over \({K} \in \{\mathbb{R},\mathbb{C} \}\) is an algebra if for each ordered pair of elements \(x,y \in {\mathcal{A}}\), a unique product \(xy\in {\mathcal{A}}\) is defined such that for all \(x,y,z \in {\mathcal{A}}\) and scalar α:

(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])
For \(x\in \mathcal{A}\), \(\lim_{n\rightarrow \infty }\Vert x^{n}\Vert ^{\frac{1}{n}}\) exists and the spectral radius \(r(x)\) satisfies
If \(r(x)< \vert \lambda \vert \), then \(\lambda ex\) is invertible in \(\mathcal{A}\); moreover,
where λ is a complex constant.
Lemma 2.3
([9])
Let \(x\in \mathcal{A}\). If the spectral radius \(r(x)\) of x is less than 1, i.e.
then \((ex)\) is invertible. Actually,
Lemma 2.4
([9])
Let \(a,b\in \mathcal{A}\). If a commutes with b, then
Lemma 2.5
([7])
Let E be a Banach space.

(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.
Remark 2.9
For more on csequences, see [7, 8].
Definition 2.10
Let X be any nonempty set and \(T : X \rightarrow X\) and \(\alpha : X \times X \rightarrow [0, \infty )\) be mappings. Then

(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 \(\alpha (x, Tx) \ge 1\), \(x,y\in X\).
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
Let χ be a nonempty set and \(d_{\mathrm {lc}}:\chi \times \chi \rightarrow {\mathcal{A}}\) be such that for all \(x,y,z\in \chi \),

(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)
\(d_{\mathrm {lc}}(x,y)\preceq d_{\mathrm {lc}}(x,z)+ d_{\mathrm {lc}}(z,y)\).
Then \(d_{\mathrm {lc}}\) is called a dislocated cone metric on χ and \((\chi,d_{\mathrm {lc}})\) is called a dislocated cone metric space over Banach algebra (in short dCbMSBA).
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
Let \(\mathcal{A} = \{a = (a_{i,j})_{3\times 3} : a_{i,j}\in \mathbb{R}, 1 \le i,j \le 3\}\), \(\Vert a\Vert = \sum_{1\le i,j\le 3}\vert a_{i,j}\vert \), \(P = \{a \in {\mathcal{A}} : a_{i,j}\ge 0, 1\le i,j\le 3\}\) be a cone in \(\mathcal{A}\). Let \(\chi = R^{+} \cup \{0\}\). Let \(d_{\mathrm {lc}}\colon \chi \times \chi \rightarrow {\mathcal{A}}\) be given by
Then \((\chi ,d_{\mathrm {lc}})\) is a dCMSBA over \(\mathcal{A}\) but not a CMSBA over Banach algebra \(\mathcal{A}\).
Example 3.3
Let \(\chi = \mathbb{R}\) and let \(\mathcal{A} = C_{R^{2}}(\chi )\). For \(\alpha = (f,g)\) and \(\beta = (u,v)\) in \(\mathcal{A}\), we define \(\alpha .\beta = (f.u,g.v)\) and \(\Vert \alpha \Vert =\max(\Vert f\Vert , \Vert g\Vert )\), where \(\Vert f\Vert = \sup_{x\in \chi }\vert f(x)\vert \). Then \(\mathcal{A}\) is a Banach algebra with unit \(e=(1,1)\), zero element \(\theta =(0,0)\) and \(P=\{(f,g)\in {\mathcal{A}}:f(t)\geq 0,g(t)\ge 0, t\in \chi \}\) is a nonnormal cone in \(\mathcal{A}\). Consider \(d_{\mathrm {lc}}\colon \chi \times \chi \rightarrow {\mathcal{A}}\) given by
Clearly \((\chi ,d_{\mathrm {lc}})\) is a dCMSBA over \(\mathcal{A}\) but not a CMSBA over Banach algebra \(\mathcal{A}\).
For any \(a\in \chi \), the open sphere with centre a and radius \(\lambda > \theta \) is given by
Let \(\mathcal {U} = \{Y\subseteq \chi : \forall {x} \in Y, \exists {r}>\theta \hbox{ such that } B_{r}(x) \subseteq Z\}\). Then \(\mathcal {U}\) defines the dislocated cone metric topology for the dCMSBA \((\chi ,d_{\mathrm {lc}})\).
Definition 3.4
Let \((\chi ,d_{\mathrm {lc}})\) be a dCMSBA over \(\mathcal{A}\), \(p\in \chi \) and \(\{p_{n}\}\) be a sequence in χ.

(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
Let \((\chi ,d_{\mathrm {lc}})\) be a dCMSBA over \(\mathcal{A}\), P be a soloid cone and \(\{p_{n}\}\) be a sequence in χ. If \(\{p_{n}\}\) converges to \(p\in \chi \), then

(i)
\(d_{\mathrm {lc}}(p_{n},p)\) is a csequence;

(ii)
\(d_{\mathrm {lc}}(p_{n},p_{n+r})\) is a csequence.
Proof
Follows from Definitions 2.1,3.1 and 3.4(i). □
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
Let \((\chi ,d_{\mathrm {lc}})\) be a dCMSBA, \(T : X \rightarrow X\) and \(\alpha : X \times X \rightarrow [0, \infty )\) be mappings. Then

(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 \(\alpha (x_{p}, x_{*}) \ge 1\)
However, for proving the uniqueness of the fixed point, different hypotheses were used by different authors. In the sequel Popescu [12] considered the following condition:
 \((\mathbf{K})\) :

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
Let \((\chi ,d_{\mathrm {lc}})\) be a dCMSBA, \(T : X \rightarrow X\) and \(\alpha : X \times X \rightarrow [0, \infty )\) be mappings. Then

(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
Let \(\chi = [0,\infty ]\), \(Tx = x^{2}\) for all \(x \in X\). Let
Then T is an αidentical function and T satisfies conditions \((\mathbf{G})\) and \((\mathbf{G}')\), but T does not satisfy condition \((\mathbf{K})\) and T is not αdominated.
Example 3.9
Let \(\chi = [n,n]\) for some \(n\in \mathbb{N}\), \(Tx = x\) for all \(x \in \chi \). Let
Then T is an αidentical function and T satisfies conditions \((\mathbf{G})\) and \((\mathbf{G}')\), but T is not αdominated and does not satisfy condition \((\mathbf{K})\).
Example 3.10
Let \(A = [n,0]\), \(B = [0, n]\) and \(\chi = A\cup B\) for some \(n\in \mathbb{N}\). Let \(Tx = x\) for all \(x \in \chi \) and
Then α is not triangular and T is not αidentical, but T is weak semi αadmissible and αdominated. T does not satisfy condition \((\mathbf{G})\) but satisfies condition \((\mathbf{G}')\).
Example 3.11
Let \(A = [n,0)\), \(B = (0, n]\) and \(\chi = A\cup \{0\}\cup B\) for some \(n\in \mathbb{N}\). Let \(Tx = \frac{x^{2}}{n}\) for all \(x \in \chi \) and
Then α is triangular and T is αidentical, but T is not weak semi αadmissible and not αdominated. T satisfies conditions \((\mathbf{G})\) and \((\mathbf{G}')\) but does not satisfy condition \((\mathbf{K})\).
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
Let \((\chi ,d_{\mathrm {lc}})\) be a dCMSBA, \(T\colon \chi \rightarrow \chi \) and \(\alpha : X \times X \rightarrow [0, \infty )\) be mappings. Then T is a Perovtype αquasi contraction mapping iff there exists \(\mu \in P\) such that \(0\le r(\mu )< 1\), and for all \(u,v\in \chi \) with \(\alpha (u,v)\ge 1\),
where \(\varphi (u,v) \in \{ d_{\mathrm {lc}}(u,v), d_{\mathrm {lc}}(u,Tu), d_{\mathrm {lc}}(v,Tv), d _{\mathrm {lc}}(u,Tv), d_{\mathrm {lc}}(v,Tu)\}\).
Lemma 3.15
Let T be an αadmissible Perovtype αquasi contraction mapping in a dCMSBA \((\chi ,d_{\mathrm {lc}})\), where \(\alpha : X \times X \rightarrow [0, \infty )\), let \(x_{p}\) be the iterative sequence defined by \(x_{p+1}=Tx_{p}\) for some arbitrary \(x_{0}\in \chi \) and all \(p \in \mathbb{N}\) such that \(\alpha (x_{0}, x_{0}) \ge 1\) and \({\alpha (x_{0}, Tx_{0}) \ge 1}\). If α is a triangular function or T is weak semi αadmissible, then for all \(n\ge 1\), \(p,q\in \mathbb{N}\), we have \(\alpha (x_{p}, x_{q}) \ge 1\) for \(0\le p\le q\le n\) and for all \(1\le p\le q\le n\)
Proof
Let \(d_{\mathrm {lc}}(x_{p},x_{p+1})=d_{p}\) and \(d_{\mathrm {lc}}(x_{p},x _{p})=d_{p,p}\). For the proof, we will make use of the principle of mathematical induction. Note that by Lemma 3.12 or Lemma 3.13 the case may be \(\alpha (x_{p}, x_{q}) \ge 1\) for all \(0\le p\le q\). For \(n=1\), \(p=q=1\), since \(\alpha (x_{0}, x_{0}) \ge 1\), \(\alpha (x_{0}, x_{1}) \ge 1\) and \(\alpha (x_{1}, x_{1}) \ge 1\), from (3.1) we have
and thus the result holds good. Now suppose (3.2) is true for \(n=r\), i.e.
We will show that (3.2) is true for \(n=r+1\). It is enough to consider the case \(1\le p \le r+1\) and \(q=r+1\). Note that
and
Since \(\alpha (x_{p1},x_{r}) \ge 1\), from (3.1) we have
We will analyse each term on the righthand side of the above inequality as follows.
(i) \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{p1},x_{r}) \).
Case i(a): \(p = 1\).
Case i(b): \(2\le p\le r\). By (3.3) and (3.4) we get
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(a): \(p = 1\).
Case ii(b): \(2\le p\le r\). The result follows from (3.3).
Case ii(c): \(p = r+1\).
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): \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{r},x_{r+1})\preceq \mu^{2}d_{\mathrm {lc}}(x_{r1},x_{r+1})\)
Again by (3.1)
Continuing this process we will at most arrive at the following :

(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 \(d_{\mathrm {lc}}(x_{p},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{p1},x_{r+1})\preceq \mu ^{2} d_{\mathrm {lc}}(x_{p2},x_{r+1})\cdots \preceq \mu^{p1} d_{\mathrm {lc}}(x_{1},x _{r+1})\).
Now
If \(d_{\mathrm {lc}}(x_{1},x_{r+1}) \preceq \mu \{ d_{\mathrm {lc}}(x_{0},x_{r}) \hbox{ or } d_{\mathrm {lc}}(x_{0},x_{1}) \hbox{ or } d_{\mathrm {lc}}(x_{r},x_{r+1}) \hbox{ or } d_{\mathrm {lc}}(x_{r},x_{1})\}\), then the result follows by proceeding as in Case i(a) or ii(a) or ii(c) or by (3.3) and (3.4). If \(d_{\mathrm {lc}}(x_{1},x_{r+1}) \preceq \mu d_{\mathrm {lc}}(x_{0},x_{r+1})\), then
Thus (3.2) is true for \(n=r+1\), and hence by the principle of mathematical induction it is true for all n. □
Theorem 3.16
Let \((\chi ,d_{\mathrm {lc}})\) be a complete dCMSBA, \(T\colon \chi \rightarrow \chi \) and \(\alpha : X \times X \rightarrow [0, \infty )\) be mappings such that

(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.
Then T has a fixed point.
Proof
Consider the iterative sequence defined by \(x_{p+1}=Tx _{p}\) for all \(p \in \mathbb{N}\). Let \(d_{\mathrm {lc}}(x_{p},x_{p+1})=d_{p}\) and \(d_{\mathrm {lc}}(x_{p},x_{p})=d_{p,p}\). Note that \(d_{p,p}\preceq 2d_{p1}\) and \(d_{p,p}\preceq 2d_{p+1}\). We will show that \(\{x_{p}\}\) is a Cauchy sequence. For \(1< p< q\), let \(\Gamma_{p,q} = \{d_{\mathrm {lc}}(x_{i},x_{j}) : p \le i \le j \le q\}\). Then, using (3.1) and by the same argument as that in the proof of Lemma 12 in [13], we can find \(u_{1}\in \Gamma_{p1,q}, u_{2}\in \Gamma_{p2,q},\ldots, u_{p1} \in \Gamma_{1,q}\) satisfying
By Lemma 3.15, \(u_{p1}\preceq \mu (e\mu )^{1}(d_{\mathrm {lc}}(x_{0},x _{1})+ d_{\mathrm {lc}}(x_{0},x_{0}))\). Therefore \(d_{\mathrm {lc}}(x_{p},x_{q}) \preceq \mu^{p} (e\mu )^{1}(d_{\mathrm {lc}}(x_{0},x_{1})+ d_{\mathrm {lc}}(x_{0},x_{0}))\). Since \(r(\mu )< 1\), by Lemmas 2.8 and 2.7, we see that \(\mu^{p} (e \mu )^{1}(d_{\mathrm {lc}}(x_{0},x_{1})+ d_{\mathrm {lc}}(x_{0},x_{0}))\) is a csequence. Now let \(\theta \ll c\) be arbitrary in \(\mathcal {A}\). By Definition 2.1, there exists a natural number \(p_{0}\) such that \(\mu^{p} (e \mu )^{1}(d_{\mathrm {lc}}(x_{0},x_{1})+ d_{\mathrm {lc}}(x_{0},x_{0})) \ll c\) for all \(p \ge p_{0}\). Thus we get \(d_{\mathrm {lc}}(x_{p},x_{q}) \ll c\) for all \(p \ge p_{0}\). Therefore \(\{x_{p}\}\) is a Cauchy sequence, and by the completeness of \((\chi ,d_{\mathrm {lc}})\) there exists \(x_{*}\in \chi \) such that \(\lim_{n\rightarrow \infty }x_{p} = x_{*}\). By Proposition 3.5 and Lemma 2.6, \(d_{\mathrm {lc}}(x_{p},x_{*})\rightarrow \theta \), \(d_{\mathrm {lc}}(x _{p1},x_{p})\rightarrow \theta \) and \(d_{\mathrm {lc}}(x_{p1},x_{*})\rightarrow \theta \). Since \((\chi ,d_{\mathrm {lc}})\) is αregular, \(\alpha (x_{p1},x _{*}) \ge 1\), and so by (3.1)
By Proposition 3.5 and Lemma 2.6, \(d_{\mathrm {lc}}(x_{p},x_{*}) \rightarrow \theta \) and \(d_{\mathrm {lc}}(x_{p1},x_{p})\rightarrow \theta \). Thus \(d_{\mathrm {lc}}(x_{*}, Tx_{*}) \preceq \theta \) and so \(Tx_{*} = x_{*}\). □
In Theorem 3.16 condition (v) can be replaced with another condition as in Popescu [12]. We have the following.
Theorem 3.17
Let \((\chi ,d_{\mathrm {lc}})\) be a complete dCMSBA, \(T\colon \chi \rightarrow \chi \) and \(\alpha : X \times X \rightarrow [0, \infty )\) be mappings such that

(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.
Then T has a fixed point.
Proof
Proceeding as in the proof of Theorem 3.16, the Picard sequence \(\{x_{p}\}\) starting with \(x_{0}\) converges to \(x_{*} \in \chi \). By (v) there exists a subsequence \(\{x_{p(k)}\}\) of \(\{x_{p}\}\) such that \(\alpha (x_{p(k)}, u)\ge 1\) for all k. Thus we have
By Proposition 3.5 and Lemma 2.6, \(d_{\mathrm {lc}}(x_{p(k)},x_{*}) \rightarrow \theta \) and \(d_{\mathrm {lc}}(x_{p(k)1},x_{p(k)})\rightarrow \theta \). Thus \(d_{\mathrm {lc}}(x_{*},Tx_{*}) \preceq \theta \), and so \(Tx_{*} = x_{*}\). □
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 \({d_{\mathrm {lc}}(x_{*},x_{*}) = \theta }\).
Now suppose \(y_{*}\) is another fixed point of T. Then as above \(\alpha (y_{*}, y_{*}) \ge 1\) and \(d_{\mathrm {lc}}(y_{*}, y_{*}) = \theta \). Since T satisfies condition \((\mathbf{G})\), we have \(\alpha (x_{*}, y_{*}) \ge 1\), and then by (3.1)
Thus \(d_{\mathrm {lc}}(x_{*},y_{*})\preceq \theta \) and so \(x_{*}=y_{*}\). □
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
As in the proof of Theorem 3.18, we see that T has a fixed point \(x_{*} \in \chi \) and \(d_{\mathrm {lc}}(x_{*}, x_{*}) = \theta \), and if \(y_{*}\) is another fixed point of T, then \(\alpha (y_{*}, y_{*}) \ge 1\) and \(d_{\mathrm {lc}}(y_{*},y_{*}) = \theta \). Since T satisfies condition \((\mathbf{G}')\), there exists \(w\in \chi \) such that \(\alpha (x_{*}, w) \ge 1\), \(\alpha (y_{*}, w) \ge 1\), \(\alpha (w, w) \ge 1\) and \(\alpha (w, Tw) \ge 1\). By Theorem 3.16 the sequence \(\{T^{n}w\}\) will converge to a fixed point say \(w_{*}\) of T. Since T is αadmissible, we get \(\alpha (x_{*}, T^{n}w) \ge 1\) and \(\alpha (y_{*}, T^{n}w) \ge 1\), and then by (3.1) we have
Then, as \(n\rightarrow \infty \), using Proposition 3.5 and Lemma 2.6, we get \(d_{\mathrm {lc}}(x_{*},w_{*})\preceq \theta \) and so \(x_{*}=w_{*}\). Similarly, we can show that \(y_{*}=w_{*}\). Therefore \(x_{*}=y_{*}\). □
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
Let \((\chi ,d_{c})\) be a complete CMSBA, \(T\colon \chi \rightarrow \chi \) and \(\alpha : X \times X \rightarrow [0, \infty )\) be mappings such that

(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.
Then T has a fixed point.
Theorem 3.22
Let \((\chi ,d_{c})\) be a complete CMSBA, \(T\colon \chi \rightarrow \chi \) and \(\alpha : X \times X \rightarrow [0, \infty )\) be mappings such that

(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.
Then T has a fixed point.
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
Let \((\chi ,d_{c})\) be a complete CMSBA, \(T\colon \chi \rightarrow \chi \), A and B be nonempty subsets of χ such that \(\chi = A\cup B\) and \(T(A)\subseteq T(B)\) and \(T(B)\subseteq T(A)\). If there exists \(\mu \in P\) such that \(0\le r(\mu )< 1\), and
for all \(u\in A\), \(v\in B\) and \(\varphi (u,v) \in \{ d_{\mathrm {lc}}(u,v), d_{\mathrm {lc}}(u,Tu), d_{\mathrm {lc}}(v,Tv), d_{\mathrm {lc}}(u,Tv), d_{\mathrm {lc}}(v,Tu)\}\), then T has a unique fixed point in \(A\cap B\).
Proof
Let
Then T is an αadmissible, weak semi αadmissible and αdominated function and satisfies condition \((\mathbf{G}')\). Hence, by Theorem 3.24, T has a unique fixed point \(x_{*}\) in χ. Since T is αdominated, \(\alpha (x_{*}, x_{*})= \alpha (x_{*}, Tx_{*})\ge 1\). This is possible iff \(x_{*}\in A\cap B\). □
Corollary 3.26
Let \((\chi ,d_{\mathrm {lc}})\) be a complete dCMSBA and \(T\colon \chi \rightarrow \chi \) be a mapping. If there exists \(\mu \in P\) such that \(0\le r( \mu )< 1\), and
for all \(u,v\in \chi \) and \(\varphi (u,v) \in \{ d_{\mathrm {lc}}(u,v), d_{\mathrm {lc}}(u,Tu), d_{\mathrm {lc}}(v,Tv), d_{\mathrm {lc}}(u,Tv), d_{\mathrm {lc}}(v,Tu)\}\), then T has a unique fixed point.
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])
Let \((\chi ,d_{c})\) be a complete CMSBA and \(T\colon \chi \rightarrow \chi \) be a mapping. If there exists \(\mu \in P\) such that \(0\le r(\mu )< 1\), and
for all \(u,v\in \chi \) and \(\varphi (u,v) \in \{ d_{c}(u,v), d_{c}(u,Tu), d_{c}(v,Tv), d_{c}(u,Tv), d_{c}(v,Tu)\}\), then T has a unique fixed point.
Proof
Since every CMSBA is a dCMSBA, the proof follows from Corollary 3.26. □
Example 3.28
Let \(\chi = [0,\infty )\) and \(\mathcal{A}\) be as in Example 3.3 and \(d_{\mathrm {lc}}(x,y)(t) = (\vert x  y\vert (1+t^{2}) , \vert x  y\vert (1+t ^{2}))\). Let \(T : \chi \rightarrow \chi \) be given by
and
Then α is a triangular function, T is αadmissible, αidentical and satisfies condition \((\mathbf{G})\). Also, for all \(\alpha (x,y) \ge 1\), we see that \(d_{\mathrm {lc}}(Tx,Ty) \preceq q.d_{\mathrm {lc}}(x,y)\), where \(q = \frac{1}{2}\). Thus T satisfies all conditions of Theorems 3.18 and 3.19 but does not satisfy conditions of Corollary 3.27. Further 0 is a unique common fixed point of T.
Theorem 3.29
Let \((\chi ,d_{\mathrm {lc}})\) be a complete dCMSBA and \(T\colon \chi \rightarrow \chi \) be a mapping. Let \(\alpha : X \times X \rightarrow [0, \infty )\) be a mapping satisfying conditions (iii), (iv) and (v) of Theorem 3.16 or 3.17. If there exist \(\lambda , \mu , \nu \in {P}\) such that λ commutes with \(\mu +3\nu \), \(\mu + \nu \) commutes with \(\mu +3\nu \), \(r(\lambda +\mu +\nu )+r(\mu +3 \nu ) < 1\) and for all \(x,y\in \chi \) with \(\alpha (x,y) \ge 1\)
then T has a fixed point. Further, if T is an αidentical function or if T is αdominated, then T has a fixed point \(x_{*} \in \chi \) and \(d_{\mathrm {lc}}(x_{*}, x_{*}) = \theta \). Moreover, if T satisfies condition \((\mathbf{G})\) or \((\mathbf{G}')\), then the fixed point is unique.
Proof
Consider the iterative sequence defined by \(x_{p+1}=Tx_{p}\) for all \(p \in \mathbb{N}\). Let \(d_{\mathrm {lc}}(x_{p},x_{p+1})=d _{p}\) and \(d_{\mathrm {lc}}(x_{p},x_{p})=d_{p,p}\). Note that \(d_{p,p}\preceq 2d _{p1}\) and \(d_{p,p}\preceq 2d_{p+1}\). By Lemma 3.12 or 3.13, \(\alpha (x_{p}, x_{p+1}) \ge 1\), and therefore using (3.9) we have
i.e.
or
Since λ commutes with \(\mu +3\nu \), \(\mu +\nu \) commutes with \(\mu +3\nu \), simple calculations show that \((\lambda +\mu +\nu )(e \mu 3\nu )^{1}=(e\mu 3\nu )^{1}(\lambda +\mu +\nu )\), and using Lemma 2.4 again by simple calculations we have \(r(\beta ) = r( \lambda +\mu +\nu )(e\mu 3\nu )^{1} < 1\). Therefore \(e\beta \) is invertible and \((e\beta )^{1} = \sum_{i=0}^{\infty }(\beta )^{i}\). Hence
Also using Lemma 2.8 and Lemma 2.7, \(\beta^{p}(e\beta )^{1}d _{0}\) is a csequence. Thus by Definition 2.1 for any \(c\in A\) with \(\theta \ll c\), there is \(N_{1}\in \mathbb{N}\) satisfying \(n>N_{1}\) implies
Thus \(\{x_{p}\}\) is a Cauchy sequence, and since \((\chi ,d_{\mathrm {lc}})\) is complete, we have \(u\in \chi \) such that
By Proposition 3.5 and Lemma 2.6, \(d_{\mathrm {lc}}(x_{p},u)\rightarrow \theta \), \(d_{\mathrm {lc}}(x_{p1},x_{p})\rightarrow \theta \) and \(d_{\mathrm {lc}}(x _{p1},u)\rightarrow \theta \). Since \((\chi ,d_{\mathrm {lc}})\) is αregular, \(\alpha (x_{p1},u) \ge 1\) and so by (3.1). Since \(d_{p} \neq d_{q}\) whenever \(p \neq q\), there exists \(k\in \mathbb{N}\) such that \(d_{\mathrm {lc}}(u,Tu) \neq \{d_{k}, d_{k+1}, \ldots \}\). Then, for any \(p > k\),
i.e.
By Lemma 2.6, \(\nu (e\mu \nu )^{1}d_{\mathrm {lc}}(u,x_{p})\rightarrow \theta \), \((e\mu \nu )^{1}(\lambda +\nu )d_{\mathrm {lc}}(u,x_{p1})\rightarrow \theta \) and \((e\mu \nu )^{1}\mu ( d_{\mathrm {lc}}(x_{p1},x_{p})+d_{\mathrm {lc}}(u,Tu)) \rightarrow \theta \). Hence \(d_{\mathrm {lc}}(u,Tu)\rightarrow \theta \). Thus \(Tu=u\).
If T is an αidentical function or if T is αdominated, then proceeding as in the proof of Theorem 3.18 and using (3.9), we get \(d_{\mathrm {lc}}(u,u)=\theta \). Now suppose that there exists \(u^{*}\) such that \(Tu^{*}=u^{*}\). Then, as above, \(d_{\mathrm {lc}}(u^{*},u^{*})=\theta \). If T satisfies condition \((\mathbf{G})\), we have \(\alpha (u, u_{*}) \ge 1\), and then by (3.9)
As above, by Lemma 2.8 and Lemma 2.7, \((\lambda + 2\nu )^{n} d_{\mathrm {lc}}(u,u^{*})\) is a csequence, and so by Lemma 2.6 \((\lambda + 2\nu )^{n} d_{\mathrm {lc}}(u,u^{*})\rightarrow \theta \) as \(n \rightarrow \infty \). Thus \(u=u^{*}\).
If T satisfies condition \((\mathbf{G}')\), then there exists \(w\in \chi \) such that \(\alpha (u, w) \ge 1\), \(\alpha (u_{*}, w) \ge 1\), \(\alpha (w, w) \ge 1\) and \(\alpha (w, Tw) \ge 1\). Then, by replacing \(x_{0}\) with w in condition (iv) and proceeding as above, the sequence \(\{T^{n}w\}\) will converge to a fixed point say \(w_{*}\) of T and \(d_{\mathrm {lc}}(w^{*},w^{*}) = \theta \). Since T is αadmissible, we get \(\alpha (u, T^{n}w) \ge 1\) and \(\alpha (u_{*}, T^{n}w) \ge 1\), and then by (3.9) we have
Then, as \(n\rightarrow \infty \), using Proposition 3.5 and Lemma 2.6, we get \(d_{\mathrm {lc}}(u,w_{*})\preceq \theta \) and so \(u=w_{*}\). Similarly, we can show that \(u_{*}=w_{*}\). Therefore \(u=u_{*}\). □
Theorem 3.30
Let \((\chi ,d_{\mathrm {lc}})\) be a complete dCMSBA and \(T\colon \chi \rightarrow \chi \) be a mapping. If there exists \(\lambda \in P\) such that \(0\le r(2\lambda )< 1\), and
for all \(x,y\in \chi \) with \(\alpha (x,y)\ge 1\), then T has a unique fixed point.
Proof
Note that (3.12) implies (3.9). Hence the result follows from Theorem 3.29. □
Theorem 3.31
Let \((\chi ,d_{\mathrm {lc}})\) be a complete dCMSBA and \(T\colon \chi \rightarrow \chi \) be a mapping. If there exists \(\lambda \in P\) such that \(0\le r(4\lambda )< 1\), and
for all \(x,y\in \chi \) with \(\alpha (x,y)\ge 1\), then T has a unique fixed point.
Proof
Note that (3.13) implies (3.9). Hence the result follows from Theorem 3.29. □
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.
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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.
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MSC
 47H10
 54H25
Keywords
 fixed points
 cone metric space
 dislocated cone metric space
 αadmissible mapping