FIXED POINT THEOREMS FOR ψ−CONTRACTIVE MAPPING IN C∗−ALGEBRA VALUED RECTANGULAR B-METRIC SPACES

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we present a new insight of C∗−algebra valued rectangular b−metric spaces in the perspective of the fixed point theory using contractive mapping. Using contractive mapping in the rectangular b−metric spaces, we discussed the existence and the uniqueness of the fixed point with mapping satisfying a contractive condition. As a result, we obtained an interesting and important result for the general case of C∗−algebra valued metric spaces. In particular, we study some fixed point theorems in the C∗−algebra valued rectangular b−metric spaces using a positive function.


INTRODUCTION
C * −algebra theory is a critical subject in functional analysis and operator theory that plays a central role in fixed point theory and applications.
In this context, several researchers have obtained fixed point results for mapping under multiple contractive conditions in the framework of different types of metric spaces.
In 1989 Bakhtin [5] introduced the concept of b−metric spaces. Later, Czerwik [6] extended the results of the renowned Banach fixed point theorem in the b−metric spaces.
In 2000, Branciari [4] introduced the notion of rectangular metric spaces where the triangle inequality of metric spaces was replaced by another inequality, the so-called rectangular inequality. In [8], George et al. established the concept of rectangular b−metric space which generalizes the concept of rectangular metric space and b−metric space.
Ma et al. [11] introduced the concept of C * −algebra valued metric space and studied some fixed point theorems. The notion of C * −algebra valued metric spaces generalized to that of C * −algebra rectangular b− metric space, where b− is an element of C * − algebra greater than I, and the triangle inequality is modified into Then, various fixed point theorems are obtained for self-map with contractive condition [9,10].
In this paper, inspired by the work done in [13], we introduce the notion of ψ−contractive mapping in C * −algebra valued rectangular b−metric and establish some new fixed point theorems. Moreover, an illustrative example is presented to support the obtained results.

PRELIMINARIES
Throughout this paper, we denote A by an unital (i.e. unity element I) C * -algebra with linear involution * , such that for all x, y ∈ A, (xy) * = y * x * and x * * = x.
We call an element x ∈ A a positive element, denote it by Using positive element, we can define a partial ordering on A h as follows: x y if and only if y − x θ where θ means the zero element in A.
We denote A + = {a ∈ A, θ a} and A = {a ∈ A, ab = ba; ∀b ∈ A} and |x| = (x * x) 1 2 Definition 2.2. [14] Let X be a non-empty set and b ∈ A such that b I. Suppose the mapping d : X × X → A + satisfies: for all x, y ∈ X and for all distinct points u, v ∈ X − {x, y}.
Then (X, A + , d) is called a C * -algebra valued rectangular b−metric space.
If for any ε > 0 there is N such that for all n > N , d(x n , x) ≤ ε, then {x n } ⊂ X is said to be convergent with respect to A and {x n } ⊂ X converges to x and x is the limit of {x n } ⊂ X. We denote it by lim n−→∞ x n = x.
If for any ε > 0 there is N such that for all n, m > N , d(x n , x m ) ≤ ε ,then {x n } n∈N is called a Cauchy sequence with respect to A.
We say (X, A + , d) is a complete C * -algebra valued rectangular b− metric space if every Cauchy sequence with respect to A is convergent.
It is obvious that if X is a Banach space, then (X, A + , d) is a complete C * -algebra valued For any A ∈ A we define its norm as , Let A be a C * -algebra and suppose that ϕ is a linear functional on A. Define ϕ * (a) = ϕ(a * ) for all a ∈ A.
Then ϕ * is also a linear function on A .
A linear function ϕ on A is called positive if ϕ(a * a) θ for all a ∈ A.
[15] Let A be a C * -algebra with 1 then a positive functional is bounded and Proposition 2.7.
[15] Let A be a C * -algebra with 1 and let ϕ be a bounded linear functional on A, such that ϕ(a) = ϕ a . There exists positive element a ∈ A such that ϕ is a positive linear functional.
Definition 2.8. [12] Let the function ψ : A + → A + be positive if having the following constraints: (i) ψ is continous and nondecrasing Definition 2.9. [12] Suppose that A and B are C * -algebra .
A mapping ψ : A → B is said to be C * -homomorphism if : Definition 2.10. [12] Let A and B be C * -algebra spaces and let ψ : A → B be a homomorphism, Lemma 2.11. [16] Let A and B be C * -algebra spaces and ψ : Corollary 2.12.

MAIN RESULTS
In this part, we give some fixed point theorems in C * -algebra valued rectangular b−metric space using a positive function.
Theorem 3.1. Let (X, A, d) be a complete C * -algebra valued rectangular b−metric space.
Then T has a unique fixed point.
Proof. : Let x n+1 = T x n .for each n ≥ 1, then: Then for m ≥ 1 and p ≥ 1 : Therefore x n is a Cauchy sequence with respect to A.By the completness of (X, A, d) there exists an x ∈ X such that lim n→∞ x n = lim n→∞ T x n−1 = x = T x.
Let y be another fixed point of T where: wich is a contradiction Hence d(x, y) = θ and x = y,wich implies that the fixed point is unique.

Lemma 3.2. [13]
Let (X, A, d) be a C * -algebra valued rectangular b−metric space such that d(x, y) ∈ A + , for all x, y ∈ X where x = y.
Let φ : A + −→ A + be a function with the following propreties: where a ∈ A + and x, y ∈ X. Let T : X −→ X be a mapping function: where ψ is a * −homomorphism and φ : A + −→ A + is a continuous function with the constraint ψ(a) ≺ φ (a). Then, T has a fixed point.
Proof. Let x 0 ∈ X, we define : We have d(x n+1 , x n ) d(x n , x n−1 ) then || d(x n+1 , x n ) || || d(x n , x n−1 ) || Hence, the sequence d(x n+1 , x n ) is norm decreasing,from the condition of the condition of the theorem we have d(x n+1 , x n ) −→ θ this implies || d(x n+1 , x n ) ||→ 0 Then for m ≥ 1 and p ≥ 1 : Then {x n } is a Cauchy sequence. Since X is complete, then there exists x ∈ X such that lim n→∞ x n = x. Due to the continuity of T , Example 3.4. Let X = [0, 2] and A = C with a norm || z ||=| z | be a C * − algebra.
The partial order ≤ with respect to the C * − algebra C i s the partial order in C, z 1 ≤ z 2 if Re(z 1 ) ≤ Re(z 2 ) and Im(z 1 ) ≤ Im(z 2 ) for any two elements z 1 , z 2 in C. Let ψ, φ : C + → C + such that they can defined as follows: for t = (x, y) ∈ C + , i f x ≤ 2 and y > 2 (x 2 , y 2 ) i f x > 2 and y > 2 and for s = (s 1 , s 2 ) ∈ C + with v = min{s 1 , s 2 }, Then ψ and φ have the propreties mentioned in definitions 2.8 and 2.9.
Let T : X → X be defined as follows : Then ,T has the required properties montioned in theorem 3.3 we show that 0 is a fixed point of T Theorem 3.5. Let (X, A, d) be a complete C * − algebra valued rectangular b−metric space.
Let T : X −→ X be a mapping function and : ψ and φ are * − homomorphisms and with the constraint ψ(a) < φ (a).
Then, T has a fixed point.
Proof. Let x 0 ∈ X and define Using a proprety of φ , we have Using the strongly monotone proprety of ψ, we have Therefore {d(x n+1 , x n )} is monotone decreasing sequence.
Hence by lemma 3.2 there exists u ∈ A + such that d(x n+1 , x n ) → u as n → ∞.
Next we show that {x n } is a Cauchy sequence .If {x n } is not a Cauchy sequence then by lemma 3.2 ,there exists c ∈ A such that ∀n 0 ∈ N, ∃n, m ∈ N with n > m ≥ n 0 φ (c) d(x n , x m ). Therefore there exists sequences {m k } and {n k } in N such that for all positive integers k, n k > m k > k and letting k → ∞ we have letting k → ∞ in above inequalities , we have Further, Letting k → ∞ in the above four inequalities we have Using (1), (2), (4), and (5) we have Clearly x m k x n k .Putting x = x n (k) ,y = x m (k) Letting k → ∞ in the above inequality using (2), (3) and(6) and the continuities of ψ and φ we have ψ(bφ (c)) ψ((a 1 + 2a 2 )bφ (c)) − φ (bφ (c)) that is ψ(bφ (c)) ψ(φ (c)) − φ (φ (c)) ,(since (a 1 + 2a 2 )b < 1) and ψ is strongly monotonic increasing. Which a contradiction by virtue of a proprety of φ . Hence {x n }is a Cauchy sequence. From the completness of X, there exists z ∈ X such that x n → z as n → ∞. Since T is continous and T x n → T z as n → ∞ that is lim n→∞ x n+1 = T z, that is z = T z. Hence z is a fixed point of T .
The partial order ≤ with respect to the C * − algebra C is the partial order in C, z 1 ≤ z 2 if Re(z 1 ) ≤ Re(z 2 ) and Im(z 1 ) ≤ Im(z 2 ) for any two elements z 1 , z 2 in C.
Then, (X, C, d) is a C * − algebra valued rectangular b− metric space where b = 1 with the required propreties of theorem 3.5.
Let T : X → X be defined as follows : Then ,T has the required properties mentioned in theorem 3.5.
Let a 1 = 1 2 ,a 2 = 1 8 and a 3 = 1 8 . It can be verified that ψ(d(T x, Ty)) ψ(M(x, y)) − φ (d(x, y)) for all x, y ∈ X with y x the conditions of theorem 3.5 are satisfied .Here it is seen that 0 is a fixed point of T