On Fixed Points in the Setting of C ∗ -Algebra-Valued Controlled F c -Metric Type Spaces
aa r X i v : . [ m a t h . G M ] S e p On Fixed Points in the Setting of C ∗ -Algebra-Valued Controlled F c -MetricType Spaces G. Kalpana , ∗ and Z. Sumaiya Tasneem Department of Mathematics,SSN College of Engineering,Kalavakkam, Chennai-603 110, India. ∗ Correspondence: [email protected]@ssn.edu.in
Abstract
In the present article, we first examine the conception of C ∗ -algebra-valued controlled F c -metric type spaces as a generalization of F -cone metric spaces over banach algebra. Further,we prove some fixed point theorem with different contractive conditions in the frameworkof C ∗ -algebra-valued controlled F c -metric type spaces. Secondly, we furnish an example bymeans of the acquired result. Keywords and Phrases: C ∗ -algebra; C ∗ -algebra-valued controlled F c -metric type spaces;Contractive mapping; Fixed point theorem. Article Type:
Research Article.
The conception of b -metric space was initiated by Bakhtin [10] as a generalization of metricspaces. In 1994, Matthews [12] proposed the concept of partial metric spaces where the self-distance of any point need not be zero. Tayyab Kamran et al. [11] introduced a new type of met-ric spaces, namely extended b-metric spaces by replacing the constant s by a function θ ( x, y )depending on the parameters of the left-hand side of the triangle inequality. Nabil Mlaiki etal. [23] proved banach contraction principle in the setting of controlled metric type spaces whichis a generalization of extended b -metric space. For more engrossing results in extended b -metricspaces, the reader may refer to [3–9]. In [2], Aiman Mukheimer have recently examined the hy-pothesis of extended partial S b -metric spaces.On the other hand, Fernandez et al. [1] established the notion of F -cone metric space overbanach algebra and investigated the existence and uniqueness of the fixed point under the samemetric. In [24], Ma initiated the concept of C ∗ -algebra-valued metric spaces where the set of realnumbers is replaced by the set of all positive elements of a unital C ∗ -algebra. For further probeson C ∗ -algebra, we refer to [13–22].As noted above, a vigorous research on fixed point results in C ∗ -algebra-valued metricspaces, extended b -metric spaces and controlled metric type spaces has been developed in the1ast few years, we focus our study on the concept of C ∗ -algebra-valued controlled F c -metric typespaces in the present paper and prove fixed point theorem with disparate contractive condition. To start with, we recollect some necessary definitions which will be utilized in the main theorem.Throughout this paper, A denotes an unital C ∗ -algebra. Set A h = { z ∈ A : z = z ∗ } . We callan element z ∈ A a positive element, denote it by θ A (cid:22) z, if z ∈ A h and σ ( z ) ⊆ [0 , ∞ ), where θ A is a zero element in A and σ ( z ) is the spectrum of z . There is a natural partial ordering on A h given by z (cid:22) w if and only if θ A (cid:22) w − z . We denote A + and A ′ I as { z ∈ A : θ A (cid:22) z } and theset { z ∈ A : zw = wz, ∀ w ∈ A } and | z | = ( z ∗ z ) respectively. Definition 2.1. [1] Let X be a nonempty set. A function F : X → A is called F -cone metricon X if for any α, β, γ, δ ∈ X , the following conditions hold: . α = β = γ if and only if F ( α, α, α ) = F ( β, β, β ) = F ( γ, γ, γ ) = F ( α, β, γ ) ; . θ (cid:22) F ( α, α, α ) (cid:22) F ( α, α, β ) (cid:22) F ( α, β, γ ) for all α, β, γ ∈ X with α = β = γ ; . F ( α, β, γ ) (cid:22) s [ F ( α, α, δ ) + F ( β, β, δ ) + F ( γ, γ, δ )] − F ( δ, δ, δ ) .Then the pair ( X, F ) is called an F -cone metric space over Banach Algebra A . The number s ≥ is called the coefficient of ( X, F ) . Definition 2.2. [16] Let X be a nonempty set and A ∈ A ′ such that A (cid:23) I A . Suppose themapping S b : X × X × X → A satisfies: . θ A (cid:22) S b ( α, β, γ ) for all α, β, γ ∈ X with α = β = γ = α ;2 . S b ( α, β, γ ) = θ A if and only if α = β = γ ;3 . S b ( α, β, γ ) (cid:22) A [ S b ( α, α, δ ) + S b ( β, β, δ ) + S b ( γ, γ, δ )] for all α, β, γ, δ ∈ X. Then S b is said to be C ∗ -algebra-valued S b -metric on X and ( X, A , S b ) is said to be a C ∗ -algebra-valued S b -metric space. Definition 2.3. [23] Given a non-empty set X and δ : X × X → [1 , ∞ ) . A function d : X × X → [0 , ∞ ) is called a controlled metric type if: . d ( α, β ) = 0 if and only if α = β ; . d ( α, β ) = d ( β, α ) ; . d ( α, β ) ≤ δ ( α, γ ) d ( α, γ ) + δ ( γ, β ) d ( γ, β ) , for all α, β, γ ∈ X .The pair ( X, d ) is called a controlled metric type space. In this main segment, as a generalization of F -cone metric space over Banach algebra, we intro-duce the notion of C ∗ -algebra valued controlled F c -metric type spaces and furnish an exampleof the underlying spaces.Hereinafter, A ′ I will denote the set { z ∈ A : zw = wz, ∀ w ∈ A and z (cid:23) I A } respectively.2 efinition 3.1. Let X be a nonempty set and C : X × X × X → A ′ I . Suppose the mapping F c : X × X × X → A satisfies: . ̟ = ¯ ν = ¯ ς if and only if F c ( ̟, ̟, ̟ ) = F c (¯ ν, ¯ ν, ¯ ν ) = F c (¯ ς, ¯ ς, ¯ ς ) = F c ( ̟, ¯ ν, ¯ ς ) ; . θ A (cid:22) F c ( ̟, ̟, ̟ ) (cid:22) F c ( ̟, ̟, ¯ ν ) (cid:22) F c ( ̟, ¯ ν, ¯ ς ) ; . F c ( ̟, ¯ ν, ¯ ς ) (cid:22) C ( ̟, ̟, ¯ α ) F c ( ̟, ̟, ¯ α ) + C (¯ ν, ¯ ν, ¯ α ) F c (¯ ν, ¯ ν, ¯ α ) + C (¯ ς, ¯ ς, ¯ α ) F c (¯ ς, ¯ ς, ¯ α ) − F c ( ¯ α, ¯ α, ¯ α ) for all ̟, ¯ ν, ¯ ς, ¯ α ∈ X. Then F c is called a C ∗ -algebra-valued controlled F c - metric type on X and ( X, A , F c ) is a C ∗ -algebra-valued controlled F c -metric type spaces. Remark 3.2. If C ( ̟, ̟, ¯ α ) = C (¯ ν, ¯ ν, ¯ α ) = C (¯ ς, ¯ ς, ¯ α ) = C ( ̟, ¯ ν, ¯ ς ) for all ̟, ¯ ν, ¯ ς, ¯ α ∈ X , thenwe get F c ( ̟, ¯ ν, ¯ ς ) (cid:22) C ( ̟, ¯ ν, ¯ ς )[ F c ( ̟, ̟, ¯ α ) + F c (¯ ν, ¯ ν, ¯ α ) + F c (¯ ς, ¯ ς, ¯ α )] − F c ( ¯ α, ¯ α, ¯ α ) . In this case, F c is called a C ∗ -algebra-valued extended F c -metric on X and ( X, A , c ) is called a C ∗ -algebra-valued extended F c -metric space. Remark 3.3.
In a C ∗ -algebra-valued controlled F c -metric type space ( X, A , F c ) , if ̟, ¯ ν, ¯ ς ∈ X and F c ( ̟, ¯ ν, ¯ ς ) = θ , then ̟ = ¯ ν = ¯ ς , but the converse need not be true. Definition 3.4. A C ∗ -algebra-valued controlled F c -metric type space ( X, A , F c ) is said to besymmetric if it satisfies, F c ( ̟, ̟, ¯ ν ) = F c (¯ ν, ¯ ν, ̟ ) , for all ̟, ¯ ν ∈ X. Example 3.5.
Let X = { , , , . . . } and A = R . If α, β ∈ A with ̟ = ( ̟ , ̟ ) , ¯ ν =(¯ ν , ¯ ν ) , then the addition, multipilcation and scalar multipilcation can be defined as follows: ̟ + ¯ ν = ( ̟ + ¯ ν , ̟ + ¯ ν ) , k̟ = ( k̟ , k̟ ) , ̟ ¯ ν = ( ̟ ¯ ν , ̟ ¯ ν ) Now define the metric F c : X × X × X → A and the control function C : X × X × X → A ′ I as: F c ( ̟, ¯ ν, ¯ ς ) = (cid:16)
12 ( | ̟ + ¯ ς | + | ¯ ν + ¯ ς | ) ,
12 ( | ̟ + ¯ ς | + | ¯ ν + ¯ ς | ) (cid:17) and C ( ̟, ¯ ν, ¯ ς ) = (cid:16) | ̟ + ¯ ν − ¯ ς + 1 | , | ̟ + ¯ ν − ¯ ς + 1 | (cid:17) . It is easy to verify that F c is a C ∗ -algebra-valued controlled F c -metric type space. Indeed for ̟ = 1 , ¯ ν = 2 , ¯ ς = 3 and ¯ α = 0 , we have F c (1 , ,
3) = (20 . , . ≻ (1 , ,
1) + (4 ,
4) + (9 , − (0 ,
0) = (14 , C (1 , , F c (1 , ,
0) + F c (2 , ,
0) + F c (3 , , − F c (0 , , . Hence F c is not a C ∗ -algebra-valued extended F c -metric space. efinition 3.6. A sequence { ̟ n } in a C ∗ -algebra-valued controlled F c -metric type space is saidto be:(i) convergent sequence ⇐⇒ ∃ ̟ ∈ X such that F c ( ̟ n , ̟ n , ̟ ) → θ A as n → ∞ and we denoteit by lim n →∞ ̟ n = ̟ ;(ii) Cauchy sequence ⇐⇒ F c ( ̟ n , ̟ n , ̟ m ) → θ A as n, m → ∞ . Definition 3.7. A C ∗ -algebra-valued controlled F c -metric type space ( X, A , F c ) is said to becomplete if every Cauchy sequence is convergent in X with respect to A . Theorem 3.8.
Let ( X, A , F c ) be a complete symmetric C ∗ -algebra-valued controlled F c -metrictype space and suppose T : X → X is a mapping satisfying the following condition: F c ( T ̟, T ̟, T ¯ ν ) (cid:22) P ∗ F c ( ̟, ̟, ¯ ν ) P + Q ∗ F c ( ̟, ̟, T ̟ ) Q + R ∗ F c (¯ ν, ¯ ν, T ¯ ν ) R, ∀ ̟, ¯ ν, ∈ X, (1) where P, Q, R ∈ A with k P k , k Q k , k R k ≥ satisfying k P k + k Q k + k R k < and for ̟ ∈ X , choose ̟ n = T n ̟ assume that sup m ≥ lim i →∞ k C ( ̟ i +1 , ̟ i +1 , ̟ i +2 ) C ( ̟ i +1 , ̟ i +1 , ̟ m ) k < − k R k k P k + k Q k . (2) In addition, for each ̟ ∈ X , suppose that lim n →∞ k C ( ̟, ̟, ̟ n ) k and lim n →∞ k C ( ̟ n , ̟ n , ̟ ) k (3) exist and are finite. Then T has a unique fixed point in X .Proof. Let ̟ ∈ X be arbitrary and define the iterative sequence { ̟ n } by: ̟ n +1 = T ̟ n = . . . = T n +1 ̟ , n = 1 , , . . . . (4)If follows from (1) and (4) that F c ( ̟ n , ̟ n , ̟ n +1 ) = F c ( T ̟ n − , T ̟ n − , T ̟ n ) (cid:22) P ∗ F c ( ̟ n − , ̟ n − , ̟ n ) P + Q ∗ F c ( ̟ n − , ̟ n − , T ̟ n − ) Q + R ∗ F c ( ̟ n , ̟ n , T ̟ n ) R ⇐⇒ k F c ( ̟ n , ̟ n , ̟ n +1 ) k ≤ k P ∗ F c ( ̟ n − , ̟ n − , ̟ n ) P + Q ∗ F c ( ̟ n − , ̟ n − , T ̟ n − ) Q + R ∗ F c ( ̟ n , ̟ n , T ̟ n ) R k≤ k P ∗ F c ( ̟ n − , ̟ n − , ̟ n ) P k + k Q ∗ F c ( ̟ n − , ̟ n − , T ̟ n − ) Q k + k R ∗ F c ( ̟ n , ̟ n , T ̟ n ) R k = ( k P k + k Q k ) k F c ( ̟ n − , ̟ n − , ̟ n ) k + k R k k F c ( ̟ n , ̟ n , ̟ n +1 ) k ∴ k F c ( ̟ n , ̟ n , ̟ n +1 ) k ≤ k P k + k Q k − k R k k F c ( ̟ n − , ̟ n − , ̟ n ) k (5)4ccordingly we get k F c ( ̟ n , ̟ n , ̟ n +1 ) k ≤ k S k k F c ( ̟ n − , ̟ n − , ̟ n ) k = k S ∗ S kk F c ( ̟ n − , ̟ n − , ̟ n ) k≤ k S ∗ kk F c ( ̟ n − , ̟ n − , ̟ n ) kk S k⇐⇒ F c ( ̟ n , ̟ n , ̟ n +1 ) (cid:22) S ∗ F c ( ̟ n − , ̟ n − , ̟ n ) S, (6)where k S k = k P k + k Q k −k R k <
1. Recursively, we find that F c ( ̟ n , ̟ n , ̟ n +1 ) (cid:22) ( S ∗ ) n F c ( ̟ n − , ̟ n − , ̟ n ) S n (7)For any n ≥ q ≥
1, we have F c ( ̟ n , ̟ n , ̟ n + q ) (cid:22) C ( ̟ n , ̟ n , ̟ n +1 ) F c ( ̟ n , ̟ n , ̟ n +1 ) + C ( ̟ n , ̟ n , ̟ n +1 ) F c ( ̟ n , ̟ n , ̟ n +1 )+ C ( ̟ n + q , ̟ n + q , ̟ n +1 ) F c ( ̟ n + q , ̟ n + q , ̟ n +1 ) − F c ( ̟ n +1 , ̟ n +1 , ̟ n +1 ) (cid:22) C ( ̟ n , ̟ n , ̟ n +1 ) F c ( ̟ n , ̟ n , ̟ n +1 ) + C ( ̟ n + q , ̟ n + q , ̟ n +1 ) F c ( ̟ n +1 , ̟ n +1 , ̟ n + q ) (cid:22) C ( ̟ n , ̟ n , ̟ n +1 ) F c ( ̟ n , ̟ n , ̟ n +1 ) + C ( ̟ n + q , ̟ n + q , ̟ n +1 ) (cid:2) C ( ̟ n +1 , ̟ n +1 , ̟ n +2 ) F c ( ̟ n +1 , ̟ n +1 , ̟ n +2 ) + C ( ̟ n + q , ̟ n + q , ̟ n +2 ) F c ( ̟ n +2 , ̟ n +2 , ̟ n + q ) (cid:3) − F c ( ̟ n +2 , ̟ n +2 , ̟ n +2 )...= 2 C ( ̟ n , ̟ n , ̟ n +1 ) F c ( ̟ n , ̟ n , ̟ n +1 )+2 n + q − X i = n +1 C ( ̟ i , ̟ i , ̟ i +1 ) F c ( ̟ i , ̟ i , ̟ i +1 ) i Y j = n +1 C ( ̟ n + q , ̟ n + q , ̟ j )+ n + q − Y i = n +1 C ( ̟ n + q , ̟ n + q , ̟ i ) F c ( ̟ n + q − , ̟ n + q − , ̟ n + q ) (cid:22) C ( ̟ n , ̟ n , ̟ n +1 ) F c ( ̟ n , ̟ n , ̟ n +1 )+2 n + q − X i = n +1 C ( ̟ i , ̟ i , ̟ i +1 ) F c ( ̟ i , ̟ i , ̟ i +1 ) i Y j = n +1 C ( ̟ n + q , ̟ n + q , ̟ j ) (cid:22) C ( ̟ n , ̟ n , ̟ n +1 )( S ∗ ) n S S n +2 n + q − X i = n +1 C ( ̟ i , ̟ i , ̟ i +1 )( S ∗ ) i S S i i Y j =1 C ( ̟ n + q , ̟ n + q , ̟ j )5 2 (cid:16) S C ( ̟ n , ̟ n , ̟ n +1 ) S n (cid:17) ∗ ( S C ( ̟ n , ̟ n , ̟ n +1 ) S n (cid:17) +2 n + q − X i = n +1 (cid:16) S (cid:2) C ( ̟ i , ̟ i , ̟ i +1 ) i Y j =1 C ( ̟ n + q , ̟ n + q , ̟ j ) (cid:3) S i (cid:17) ∗ (cid:16) S (cid:2) C ( ̟ i , ̟ i , ̟ i +1 ) i Y j =1 C ( ̟ n + q , ̟ n + q , ̟ j ) (cid:3) S i (cid:17) = 2 | S C ( ̟ n , ̟ n , ̟ n +1 ) S n | +2 n + q − X i = n +1 (cid:12)(cid:12)(cid:12) S (cid:2) C ( ̟ i , ̟ i , ̟ i +1 ) i Y j =1 C ( ̟ n + q , ̟ n + q , ̟ j ) (cid:3) S i (cid:12)(cid:12)(cid:12) (cid:22) k S k h k C ( ̟ n , ̟ n , ̟ n +1 ) k k S k n I A + k C ( ̟ i , ̟ i , ̟ i +1 ) i Y j =1 C ( ̟ n + q , ̟ n + q , ̟ j ) kk S k i I A i where I A is the unit element in A and c ( ̟ , ̟ , ̟ ) = S for some S ∈ A . Let Y m = P mi =1 k S k i k C ( ̟ i , ̟ i , ̟ i +1 ) i Q j =1 C ( ̟ n + q , ̟ n + q , ̟ j ) k . Consequently the above inequality implies, F c ( ̟ n , ̟ n , ̟ n + q ) ≤ k S k h k C ( ̟ n , ̟ n , ̟ n +1 ) k k S k n + ( Y n + q − − Y n ) i I A (8)The ratio test jointly with (2) implies that the limit of the sequence { Y n } exists and so { Y n } isCauchy. Letting n → ∞ in the inequality above, we getlim n →∞ F c ( ̟ n , ̟ n , ̟ n + q ) = θ A . (9)Wherefore the sequence { ̟ n } is Cauchy with respect to A . Since ( X, A , F c ) is a complete C ∗ -algebra-valued controlled F c -metric type space, there exists a point ̟ ∈ X such thatlim n →∞ F c ( ̟ n , ̟ n , ̟ ) = θ A . (10)Consider, F c ( ̟, ̟, ̟ n +1 ) (cid:22) C ( ̟, ̟, ̟ n ) F c ( ̟, ̟, ̟ n ) + C ( ̟ n +1 , ̟ n +1 , ̟ n ) F c ( ̟ n , ̟ n , ̟ n +1 ) − F c ( ̟ n , ̟ n , ̟ n ) ⇐⇒ k F c ( ̟, ̟, ̟ n +1 ) k ≤ k C ( ̟, ̟, ̟ n ) kk F c ( ̟, ̟, ̟ n ) k + k C ( ̟ n +1 , ̟ n +1 , ̟ n ) kk F c ( ̟ n , ̟ n , ̟ n +1 ) k It yields from (14) and (10) that lim n →∞ k F c ( ̟, ̟, ̟ n +1 ) k = 0 . (11)6ence k F c ( ̟, ̟, T ̟ ) k ≤ k C ( ̟, ̟, ̟ n +1 ) kk F c ( ̟, ̟, ̟ n +1 ) k + k C ( T ̟, T ̟, ̟ n +1 ) kk F c ( ̟ n +1 , ̟ n +1 , T ̟ ) k = 2 k C ( ̟, ̟, ̟ n +1 ) kk F c ( ̟, ̟, ̟ n +1 ) k + k C ( T ̟, T ̟, ̟ n +1 ) kk F c ( T n +1 ̟, T n +1 ̟, T ̟ ) k Regarding (11), we get k F c ( ̟, ̟, ̟ n +1 ) k → n → ∞ . Since T n → x and from continuity of T , we acquire T n +1 → T x i.e., k F c ( T n +1 ̟, T n +1 ̟, T ̟ ) k → , as n → ∞ . Thuslim n →∞ k F c ( ̟, ̟, T ̟ ) k = 0 ⇐⇒ lim n →∞ F c ( ̟, ̟, T ̟ ) = θ A . Hence
T ̟ = ̟ i.e., ̟ is a fixed point of T . Now to prove uniqueness, let ¯ ν = ̟ be anotherfixed point of T . Taking the expression (1) into account, we have F c ( ̟, ̟, ¯ ν ) = F c ( T ̟, T ̟, T ¯ ν ) (cid:22) P ∗ F c ( ̟, ̟, ¯ ν ) P + Q ∗ F c ( ̟, ̟, T ̟ ) Q + R ∗ F c (¯ ν, ¯ ν, T ¯ ν ) R = P ∗ F c ( ̟, ̟, ¯ ν ) P + Q ∗ F c ( ̟, ̟, ̟ ) Q + R ∗ F c (¯ ν, ¯ ν, ¯ ν ) R (cid:22) P ∗ F c ( ̟, ̟, ¯ ν ) P + Q ∗ F c ( ̟, ̟, ¯ ν ) Q + R ∗ F c (¯ ν, ¯ ν, ̟ ) R k F c ( ̟, ̟, ¯ ν ) k ≤ ( k P k + k Q k ) F c ( ̟, ̟, ¯ ν ) + k R k k F c (¯ ν, ¯ ν, ̟ ) kk F c ( ̟, ̟, ¯ ν ) k ≤ k R k (1 − k P k − k Q k ) k F c (¯ ν, ¯ ν, ̟ ) k < k F c (¯ ν, ¯ ν, ̟ ) k = k F c ( ̟, ̟, ¯ ν ) k which is a contradiction. Hence the fixed point is unique.In Theorem (3.8), if we take Q = R = θ , then the above theorem reduces to a Banachcontraction principle, which can be stated as follows: Corollary 3.9.
Let ( X, A , F c ) be a complete C ∗ -algebra-valued controlled F c -metric type spaceand suppose T : X → X is a mapping satisfying the following condition: F c ( T ̟, T ̟, T ¯ ν ) (cid:22) P ∗ F c ( ̟, ̟, ¯ ν ) P, ∀ ̟, ¯ ν, ∈ X, (12) where P ∈ A with ≤ k P k < and for ̟ ∈ X , choose ̟ n = T n ̟ assume that sup m ≥ lim i →∞ k C ( ̟ i +1 , ̟ i +1 , ̟ i +2 ) C ( ̟ i +1 , ̟ i +1 , ̟ m ) k < k P k . (13) In addition, for each ̟ ∈ X , suppose that lim n →∞ k C ( ̟, ̟, ̟ n ) k and lim n →∞ k C ( ̟ n , ̟ n , ̟ ) k (14) exist and are finite. Then T has a unique fixed point in X . xample 3.10. Let X = [0 , and A = M ( R ) be the set of all × matrices under usualaddition, multiplication and scalar multiplication. Define F c : X × X × X → A as follows: F c ( ̟, ¯ ν, ¯ ς ) = (cid:18) max { ̟, ¯ ς } + max { ¯ ν, ¯ ς } max { ̟, ¯ ς } + max { ¯ ν, ¯ ς } (cid:19) Hence ( X, A , F c ) is a C ∗ -algebra-valued controlled F c -metric type space with C ( ̟, ¯ ν, ¯ ς ) = 2 + max { ̟, ¯ ν, ¯ ς } . Now for any A ∈ A , we define its norm as k A k = max ≤ i ≤ {| a i |} . Let T : X → X bedefined as T ̟ = ̟ . Then F c ( T ̟, T ̟, T ¯ ν ) = F c ( ̟ , ̟ , ¯ ν (cid:18) max { ̟ , ¯ ν }
00 2 max { ̟ , ¯ ν } (cid:19) = P ∗ F c ( ̟, ̟, ¯ ν ) P where P = √ √ ! with k P k = √ < . Now consider C ( ̟ i +1 , ̟ i +1 , ̟ i +2 ) = C ( T i +1 ̟, T i +1 ̟, T i +2 ̟ )= C ( ̟ i +1 , ̟ i +1 , ̟ i +2 )= (cid:18) max { ̟ i +1 , ̟ i +1 , ̟ i +2 }
00 2 + max { ̟ i +1 , ̟ i +1 , ̟ i +2 } (cid:19) Similarly, C ( ̟ i +1 , ̟ i +1 , ̟ m ) = (cid:18) max { ̟ i +1 , ̟ i +1 , ̟ m }
00 2 + max { ̟ i +1 , ̟ i +1 , ̟ m } (cid:19) Thus lim i →∞ k C ( ̟ i +1 , ̟ i +1 , ̟ i +2 ) C ( ̟ i +1 , ̟ i +1 , ̟ m ) k = lim i →∞ (cid:13)(cid:13)(cid:13)(cid:18) (2 + ̟ i +1 )(2 + max ( ̟ i +1 , ̟ m ) 00 (2 + ̟ i +1 )(2 + max ( ̟ i +1 , ̟ m ) (cid:19)(cid:13)(cid:13)(cid:13) = lim i →∞ (2 + ̟ i +1 )(2 + max ( ̟ i +1 , ̟ m ) = 4 + 2 ̟ m and sup m ≥ lim i →∞ k C ( ̟ i +1 , v i +1 , ̟ i +2 ) C ( ̟ i +1 , ̟ i +1 , ̟ m ) k = 4 + 2 ̟ < k P k . Thus T satisfies all the conditions of Theorem (3.8), hence it has a unique fixed point which is ̟ = 0 . Conclusion
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