Dislocated cone metric space over Banach algebra and α-quasi contraction mappings of Perov type

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 Perov-type α-quasi contraction mapping, Kannan-type contraction as well as Chatterjee-type 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 [] introduced a cone metric space over Banach algebra (in short CMS-BA) 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 [] 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 [], and then various fixed point results were proved in dislocated spaces. For some details, refer to []. On the other hand, Perov [] generalised the Banach contraction principle by replacing the contractive factor with a matrix convergent to zero. Cvetkovic and Rakocevic [] introduced a Perov-type quasi-contractive 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 dCMS-BA) as a generalisation of CMS-BA as well as a dislocated metric space and prove fixed point theorems for a Perov-type α-quasi contraction mapping in dCMS-BA and CMS-BA. Simple examples are given to illustrate the validity and superiority of our results.

Preliminaries
A linear space A over K ∈ {R, C} is an algebra if for each ordered pair of elements x, y ∈ A, a unique product xy ∈ A is defined such that for all x, y, z ∈ A and scalar α: (i) (xy)z = x(yz); (iia) x(y + z) = xy + xz; (iib) (x + y)z = xz + yz; (iii) α(xy) = (αx)y = x(αy). A Banach algebra is a Banach space A over K ∈ {R, C} such that, for all x, y ∈ A, xy ≤ x y . For a given cone P ⊂ A and x, y ∈ A, we say that x y if and only if yx ∈ P. Note that is a partial order relation defined on 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 [, ].
In what follows A will always denote a Banach algebra, P a solid cone in A and e the unit element of A.
Definition . A sequence p n in a solid cone P of a Banach space is a c-sequence if, for each c θ , there exists n  ∈ N such that p n c for all n ≥ n  .

Lemma . ([]
) For x ∈ A, lim n→∞ x n  n exists and the spectral radius r(x) satisfies If r(x) < |λ|, then λex is invertible in A; moreover, where λ is a complex constant.
If the spectral radius r(x) of x is less than , i.e. Definition . Let X be any nonempty set and T : X → X and α :

Main results
In this section first we introduce the definition of a dislocated cone metric space over Banach algebra (in short dCMS-BA) and furnish examples to show that this concept is more general than that of CMS-BA. We then define convergence and Cauchy sequence in a dCMS-BA and then prove fixed point results in this space.
Definition . Let χ be a nonempty set and d lc : . Then d lc is called a dislocated cone metric on χ and (χ, d lc ) is called a dislocated cone metric space over Banach algebra (in short dCbMS-BA).
Note that every metric space and CMS-BA is a dCMS-BA, but the converse is not necessarily true. Inspired by [, , ], we furnish the following examples which will establish our claim.

Example . Let
x + y x+ y x+ y x + y x + y x + y x + y x + y x + y Then (χ, d lc ) is a dCMS-BA over A but not a CMS-BA over Banach algebra A.
For any a ∈ χ , the open sphere with centre a and radius λ > θ is given by Then U defines the dislocated cone metric topology for the dCMS-BA (χ, d lc ).
Definition . Let (χ, d lc ) be a dCMS-BA over A, p ∈ χ and {p n } be a sequence in χ .
(i) {p n } converges to p if, for each c ∈ A with θ c, there exists n  ∈ N such that d lc (p n , p) c for all n ≥ n  . We write it as Lim n→∞ p n = p.
However, for proving the uniqueness of the fixed point, different hypotheses were used by different authors. In the sequel Popescu [] considered the following condition: We now introduce the following definitions.
Then T is an α-identical function and T satisfies conditions (G) and (G ), but T does not satisfy condition (K) and T is not α-dominated.
Then T is an α-identical function and T satisfies conditions (G) and (G ), but T is not α-dominated and does not satisfy condition (K).

Example . Let
Then α is not triangular and T is not α-identical, but T is weak semi α-admissible and α-dominated. T does not satisfy condition (G) but satisfies condition (G ).

Example . Let
Then α is triangular and T is α-identical, but T is not weak semi α-admissible and not α-dominated. T satisfies conditions (G) and (G ) but does not satisfy condition (K).
Lemma . Let X be a nonempty set and T : X → X and α : Proof For the proof, we will make use of the principle of mathematical induction.
and so the result holds good for n = . Again, by α-admissibility of T, we get α(x  , x  ) ≥ , α(x  , x  ) ≥ , and then, since α is triangular, we get α(x  , x  ) ≥ . Thus the result holds good for n = . Suppose the result is true for n = r, i.e. α(x p , x q ) ≥  for all  ≤ p ≤ q ≤ r. We will show that it is true for n = r + . It is enough to consider the case α(x p , x r+ ),  ≤ p ≤ r + . By induction hypothesis and α admissibility of T, we have α(x p , x r+ ) ≥  for all  ≤ p ≤ r + . Since α(x  , x  ) ≥ , by α admissibility of T and triangularity of function α, we get α(x  , x r+ ) ≥ , and thus the result is true for n = r + . Hence, by the principle of mathematical induction, the result is true for all n.
Lemma . Let X be a nonempty set and T : X → X and α : Thus the result holds good for n = . Suppose the result is true for n = r, i.e. α(x p , x q ) ≥  for all  ≤ p ≤ q ≤ r. We will show that it is true for n = r + . It is enough to consider the case α(x p , x r+ ),  ≤ p ≤ r + . By induction hypothesis and α admissibility of T, we have α(x p , x r+ ) ≥  for all  ≤ p ≤ r + . If r is even, then using α(x  , x  ) ≥  and repeatedly using weak semi α admissibility of T, we get α, we get α(x  , x r+ ) ≥ . If r is odd, then using α(x  , x  ) ≥  and repeatedly using weak semi α admissibility of T, we get α, we get α(x  , x r+ ) ≥ . Thus the result is true for n = r + . Hence, by the principle of mathematical induction, the result is true for all n.
Definition . Let (χ, d lc ) be a dCMS-BA, T : χ → χ and α : X × X → [, ∞) be mappings. Then T is a Perov-type α-quasi contraction mapping iff there exists μ ∈ P such that  ≤ r(μ) < , and for all u, v ∈ χ with α(u, v) ≥ , , let x p be the iterative sequence defined by x p+ = Tx p for some arbitrary x  ∈ χ and all p ∈ N such that α( For the proof, we will make use of the principle of mathematical induction. Note that by Lemma . or Lemma . the case may be α( and thus the result holds good. Now suppose (.) is true for n = r, i.e.
We will show that (.) is true for n = r + . It is enough to consider the case  ≤ p ≤ r +  and q = r + . Note that We will analyse each term on the right-hand side of the above inequality as follows.
Case i(b):  ≤ p ≤ r. By (.) and (.) we get Case i(c): p = r + . In this case d lc (x p , x r+ ) μd lc (x r , x r ), and the result follows from (.) and (.).

Case ii(c-):
Note that r(μ) < , and so (eμ) is invertible and (eμ) - > e. Therefore we get d lc ( Continuing this process we will at most arrive at the following : (iii) d lc (x p , x r+ ) μd lc (x r , x r+ ) = μd lc (Tx r- , Tx r ). The result follows proceeding as in Case ii(c).
(v) d lc (x p , x r+ ) μd lc (x p- , x r+ ). Using (.) and continuing in a similar manner as above, either we will get the desired result or we get d lc ( If d lc (x  , x r+ ) μ{d lc (x  , x r ) or d lc (x  , x  ) or d lc (x r , x r+ ) or d lc (x r , x  )}, then the result follows by proceeding as in Case i(a) or ii(a) or ii(c) or by (.) and (.). If d lc (x  , x r+ ) μd lc (x  , x r+ ), then Thus (.) is true for n = r + , and hence by the principle of mathematical induction it is true for all n.
(ii) α is a triangular function or T is weak semi α-admissible.

Then T has a fixed point.
Proof Consider the iterative sequence defined by x p+ = Tx p for all p ∈ N. Let d lc (x p , x p+ ) = d p and d lc (x p , x p ) = d p,p . Note that d p,p d p- and d p,p d p+ . We will show that {x p } is a Cauchy sequence. For  < p < q, let p,q = {d lc (x i , x j ) : p ≤ i ≤ j ≤ q}. Then, using (.) and by the same argument as that in the proof of Lemma  in [], we can find u  ∈ p-,q , u  ∈ p-,q , . . . , u p- ∈ ,q satisfying ). Since r(μ) < , by Lemmas . and ., we see that c for all p ≥ p  . Therefore {x p } is a Cauchy sequence, and by the completeness of (χ, d lc ) there exists x * ∈ χ such that lim n→∞ x p = x * . By Proposition . and Lemma .
In Theorem . condition (v) can be replaced with another condition as in Popescu []. We have the following. Let (χ, d lc ) be a complete dCMS-BA, T : χ → χ and α : X × X → [, ∞) be mappings such that (i) T is a Perov-type α-quasi contraction mapping.

Theorem .
(ii) α is a triangular function or T is weak semi α-admissible.
(v) If {x p } is a sequence in χ such that α(x p , x p+ ) ≥  for all p and x p → u ∈ χ as p → ∞, then there exists a subsequence {x p(k) } of {x p } such that α(x p(k) , u) ≥  for all k. Then T has a fixed point.
Proof Proceeding as in the proof of Theorem ., the Picard sequence {x p } starting with x  converges to x * ∈ χ . By (v) there exists a subsequence {x p(k) } of {x p } such that α(x p(k) , u) ≥  for all k. Thus we have By Proposition . and Lemma ., d lc (x p(k) , x * ) → θ and d lc (x p(k)- , x p(k) ) → θ . Thus d lc (x * , Tx * ) θ , and so Tx * = x * . Proof As in the proof of Theorem . or Theorem ., we see that T has a fixed point Now suppose y * is another fixed point of T. Then as above α(y * , y * ) ≥  and d lc (y * , y * ) = θ . Since T satisfies condition (G), we have α(x * , y * ) ≥ , and then by (.) Thus d lc (x * , y * ) θ and so x * = y * . Let (χ, d lc ), T and α be as in Theorem .. Suppose that all conditions of Theorem . or Theorem . are satisfied. If T is an α-identical function or if T is α-dominated, then T has a fixed point x * ∈ χ and d lc (x * , x * ) = θ . Further, if T satisfies condition (G ), then the fixed point is unique.

Theorem .
Proof As in the proof of Theorem ., we see that T has a fixed point x * ∈ χ and d lc (x * , x * ) = θ , and if y * is another fixed point of T, then α(y * , y * ) ≥  and d lc (y * , y * ) = θ . Since T satisfies condition (G ), there exists w ∈ χ such that α(x * , w) ≥ , α(y * , w) ≥ , α(w, w) ≥  and α(w, Tw) ≥ . By Theorem . the sequence {T n w} will converge to a fixed point say w * of T. Since T is α-admissible, we get α(x * , T n w) ≥  and α(y * , T n w) ≥ , and then by (.) we have Then, as n → ∞, using Proposition . and Lemma ., we get d lc (x * , w * ) θ and so x * = w * . Similarly, we can show that y * = w * . Therefore x * = y * .
Remark . In Theorems . and . we can replace the requirement of condition (G) or condition (G ) with that of condition (K). But as in Examples . and ., there exist functions α and T such that T is α-identical and T satisfies condition (G) and condition (G ) but does not satisfy condition (K). Hence our approach is new and justifiable.
Since every CMS-BA is a dCMS-BA and since in a cone metric space (χ, d c ), d c (x, y) = θ for all x, y ∈ χ , we give the following generalised results which are easily deduced from our main results.
(v) (χ, d lc ) is α-regular. Then T has a fixed point.
Corollary . Let (χ, d lc ) be a complete dCMS-BA and T : χ → χ be a mapping. If there exists μ ∈ P such that  ≤ r(μ) < , and