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

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 .