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

*Correspondence: hojat.afshari@yahoo.com 2Department of Mathematics, Basic Science Faculty, University of Bonab, Bonab, Iran Full list of author information is available at the end of the article 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 .

In this paper, we consider common fixed point theorems in the framework of the refined cone metric space, namely, quasi-cone metric space.
An element v ∈ E is said to be invertible if there exists v -1 ∈ E such that vv -1 = v -1 v = e. Moreover, if every non-zero element of E has an inverse in E, then E is called a divisible Banach algebra. Proposition 1.2 ([22]) Let E be a Banach algebra, v an element in E and ρ(v) the spectral radius of v . If ρ(v) < 1 then (e -v) is invertible in E and ( 1 ) Let (E, · ) be a real algebra and P a closed subset of E. The set P is a cone if the following conditions hold: (c 1 ) P is non-empty and P = {θ }; (c 2 ) a 1 v + a 2 ω ∈ P for all v, ω ∈ P and a 1 , a 2 ∈ (0, ∞); (c 3 ) P ∩ (-P) = {θ }. Moreover, for a given cone P ⊆ E we can consider a partial ordering Let E be a Banach algebra and P ⊂ E be a cone. Then (e -v) is an invertible element in P for any v ∈ P with ρ(v) < 1.

Definition 1.7 ([23]
) Suppose E is a Banach algebra with unit e and P ⊆ E is a cone. P is called algebra cone if e ∈ P and for v, ω ∈ P, vω ∈ P.
In what follows we consider that E (E d ) represents a real (divisible) Banach algebra with a unit e and θ be its zero element, P is a solid cone in E, P E d a normal algebra cone in E d with a normal constant N and X is a non-empty set. Definition 1.8 (see [24]) A mapping d : for all v, ω, η ∈ X. The pair (X, d) is said to be a cone metric space over Banach algebra, in short, CMS. Definition 1.9 (see [25]) A mapping q : X × X → E is said to be a quasi-cone metric if The triplet (X, q, E) is said to be a quasi-cone metric space over Banach algebra, in short, qCMS.
A quasi-cone metric space is called -symmetric, if there exists an invertible element ∈ E such that for all v, ω ∈ X.

Definition 1.11
We say that the mapping ψ : ψ is a continuous bijection and has an inverse mapping ψ -1 which is also continuous and increasing; Remark 1.12 By Definition 1.11, the part of (c), we can obtain ψ - is also a continuous and increasing operator, then Hence, Remark 1.13 By Definition 1.11, the part of (d), we can obtain ψ - Indeed, from ψ(vω) = ψ(v)ψ(ω) for v, ω ∈ P E and ψ -1 : P E → P E is also continuous, we get Then we obtain Thanks to that ψ :

Main results
for all m ∈ N, then {v m } is a (bi)-Cauchy sequence.
Proof First of all, we remark that, successively applying Eq. (2), we have Now, since ρ(κ) < 1, and taking into account Proposition 1.2, we see that (eκ) is an invertible element and (eκ) -1 = ∞ j=0 κ j and the above inequality becomes Then by (b) in Definition 1.10 it follows that the sequence {v m } is (l)-Cauchy. On the other hand, from Definition 1.4, we see that the sequence and taking Lemma 1.5 into account we get q(v p , v m ) c, for all m > p ≥ p 0 , which means the sequence {v m } is (r)-Cauchy. Obviously, in view of statement (c) in Definition 1.10, it follows that {v m } is a (bi)-Cauchy sequence.
Let (E d ) be a real (divisible) Banach algebra with a unit e and θ be its zero element and P 1 E d be a normal algebra cone with constant N = 1 in E d .
Then U and V have a common fixed point.

Corollary 2.4
Let (X, q, E d ) be a complete -symmetric qCMS over E d and P 1 E d . Suppose that ψ : P 1 E d → P 1 E d is a ψ-operator and U : X → X is a mapping satisfying the condition Then U and V have a common fixed point. Moreover, if ψ -1 (β) < ψ -1 (α 1 ) then the common fixed point is unique.
Proof Let {v m } be the sequence in X defined by (7). Letting v = v 2m and ω = v 2m+1 in (11) we have Taking into account the properties of ψ -1 , we have and moreover Therefore, since the Banach algebra is divisible, we get If we denote κ = (ψ -1 (α 1 ) + ψ -1 (α 3 )) -1 (ψ -1 (β)ψ -1 (α 2 )), we can easily see that θ ≤ κ < e and In the same way, for v = v 2m-1 and ω = v 2m , (12) becomes or, equivalent Thereupon, (here we took into account that the Banach algebra is divisible). Now, by (13) and (14) we have for all m ∈ N, where θ ≤ κ < e. Then, by using Lemma 2.1, we see that the sequence {v m } is (bi)-Cauchy and since the qCMS (X, q, E d ) is complete, we can have v * ∈ X such that {v m } converges to v * . Thus, there exists m 2 ∈ N such that for any c θ we have q(v 2m , v * ) c, q(v 2m-1 , v * ) c and also q(v 2m , v 2m+1 ) c, q(v 2m+1 , v 2m+2 ) c, for any m ≥ m 1 . Hence, by (11), respectively, (12) we have for m ≥ m 2 . Moreover, applying ψ -1 in the above inequalities, which are equivalent (since the Banach algebra is divisible) with for all m ≥ m 2 and any c θ . Therefore, by Lemma 1.6, it follows that q(v * , Vv * ) = v * and also q(v * , Vv * ) = v * , which means that v * is a common fixed point of the mappings V, 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 ω * ∈ X different from v * , such that Vv * = v * = Uv * , we have, by (11), for example, Thus, and we obtain for any n ∈ N. Further, since (ψ -1 (α 1 )) -1 ψ -1 (β) < e, we get as n → ∞, which means that for any c θ we can have n 0 ∈ N such that Thereby, by Lemma 1.6 it follows that q(v * , ω * ) = θ , and v * is the unique fixed point of the mappings U and V.

Corollary 2.6
Let (X, q, E d ) be a complete -symmetric qCMS over E d and P 1 E d . Suppose that ψ : P 1 E d → P 1 E d is a ψ-operator and U : X → X is a mapping satisfying the condition Then U and V have a common fixed point.
Thus, q(v * , Uv * ) = θ and v * is a common fixed point of the mappings V and U .