# Fixed Point Theorems for Cyclic Contractions inC⁎-Algebra-Valued b-Metric Spaces

Authors: Chakkrid Klin-eam; Prondanai Kaskasem
Journal: Journal of Function Spaces
 (2016)

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## Abstract

We study fundamental properties ofC⁎-algebra-valued b-metric space which was introduced by Ma and Jiang (2015) and give some fixed point theorems for cyclic mapping with contractive and expansive condition on such space analogous to the results presented in Ma and Jiang, 2015.


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## Body

## 1. Introduction

Firstly, we begin with the basic concept ofC∗-algebras. A real or a complex linear space A is algebra if vector multiplication is defined for every pair of elements of A satisfying two conditions such that A is a ring with respect to vector addition and vector multiplication and for every scalar α and every pair of elements x,y∈A, α(xy)=(αx)y=x(αy). A norm · on A is said to be submultiplicative if ab≤ab for all a,b∈A. In this case (A,·) is called normed algebra. A complete normed algebra is called Banach algebra. An involution on algebra A is conjugate linear map a↦a∗ on A such that a∗∗=a and (ab)∗=b∗a∗ for all a,b∈A. (A,∗) is called ∗-algebra. A Banach ∗-algebra A is ∗-algebra A with a complete submultiplicative norm such that a∗=a for all a∈A. C∗-algebra is Banach ∗-algebra such that a∗a=a2. There are many examples of C∗-algebra, such as the set of complex numbers, the set of all bounded linear operators on a Hilbert space H, L(H), and the set of n×n-matrices, Mn(C). If a normed algebra A admits a unit I, aI=Ia=a for all a∈A, and I=1, we say that A is a unital normed algebra. A complete unital normed algebra A is called unital Banach algebra. For properties in C∗-algebras, we refer to [1–3] and the references therein.Let(X,d) be a complete metric space. The well-known Banach’s contraction principle, which appeared in the Ph.D. dissertation of S. Banach in 1920, runs as follows: a mapping T:X→X is said to be a contraction if there exists r∈[0,1) such that(1)dTx,Ty≤rdx,y∀x,y∈X.Then, T has a unique fixed point in X which was published in 1922 [4]. Banach’s contraction principle has become one of the most important tools used for the existence of solutions of many nonlinear problems in many branches of science and has been extensively studied in many spaces which are more general than metric space by serveral mathematictians; see, for example, quasimetric spaces [5, 6], dislocated metric spaces [7], dislocated quasimetric spaces [8], G-metric spaces [9–11], b-metric spaces [12–14], metric-type spaces [15, 16], metric-like spaces [17], b-metric-like spaces (or dislocated b-metric spaces) [18, 19], quasi b-metric spaces [20], and dislocated quasi-b-metric spaces [21]. Note that the Banach contraction principle requires that mapping T satisfies the contractive condition that each point of X×X and ranges of T are positive real numbers. Consider the operator equation(2)X-∑n=1∞Ln∗XLn=Q,where {L1,L2,…,Ln} is subset of the set of linear bounded operators on Hilbert space H, X∈L(H), and Q∈L(H)+ is positive linear bounded operators on Hilbert space H. Then, we convert the operator equation to the mapping F:L(H)→L(H) which is defined by(3)FX=∑n=1∞Ln∗XLn+Q.Observe that the range of mapping F is not real numbers but it is linear bounded operators on Hilbert space H. Therefore, the Banach contraction principle can not be applied with this problem. Afterward, does such mapping have a fixed point which is equivalent to the solution of operator equation? In 2014, Ma et al. [22] introduced new spaces, called C∗-algebra-valued metric spaces, which are more general than metric space, replacing the set of real numbers by C∗-algebras, and establish a fixed point theorem for self-maps with contractive or expansive conditions on such spaces, analogous to the Banach contraction principle. As applications, existence and uniqueness results for a type of integral equation and operator equation are given and were able to solve the above problem if L1,L2,…,Ln∈L(H) satisfy ∑n=1∞‍Ln2<1.Later, many authors extend and improve the result of Ma et al. For example, in [23], Batul and Kamran generalized the notation of C∗-valued contraction mappings by weakening the contractive condition introduced by Ma et al. (the mapping is called C∗-valued contractive type mappings) and establish a fixed point theorem for such mapping which is more generalized than the result of Ma et al.; in [24], Shehwar and Kamran extend and improve the result of Ma et al. [22] and Jachymski [25] by proving a fixed point theorem for self-mappings on C∗-valued metric spaces satisfying the contractive condition for those pairs of elements from the metric space which form edges of a graph in the metric space. In 2015, Ma and Jiang [26] introduced a concept of C∗-algebra-valued b-metric spaces which generalize an ordinary C∗-algebra-valued metric space and give some fixed point theorems for self-map with contractive condition on such spaces. As applications, existence and uniqueness results for a type of operator equation and an integral equation are given.Generally, in order to use the Banach contraction principle, a self-mappingT must be Lipschitz continuous, with the Lipschitz constant r∈[0,1). In particular, T must be continuous at all elements of its domain. That is one major drawback. Next, many authors could find contractive conditions which imply the existence of fixed point in complete metric space but not imply continuity. We refer to [27, 28] (Kannan-type mappings) and [29] (Chatterjea-type mapping).Theorem 1 (see [27]).
If(X,d) is a complete metric space and mapping T:X→X satisfies(4)dTx,Ty≤rdx,Tx+dy,Ty,where 0≤r<1/2 and x,y∈X, then T has a unique fixed point.Theorem 2 (see [29]).
If(X,d) is a complete metric space and mapping T:X→X satisfies(5)dTx,Ty≤rdx,Ty+dy,Tx,where 0≤r<1/2 and x,y∈X, then T has a unique fixed point.In 2003, Kirk et al. [30] introduced the following notation of a cyclic representation and characterized the Banach contraction principle in context of a cyclic mapping as follows.Theorem 3.
LetA1,A2,…,Am be nonempty closed subsets of a complete metric space (X,d). Assume that a mapping T:⋃i=1mAi→⋃i=1mAi satisfies the following conditions: (i)
T(Ai)⊆Ai+1 for all 1≤i≤m and Am+1=A1.(ii)
There existsk∈[0,1) such that d(Tx,Ty)≤kd(x,y) for all x∈Ai and y∈Ai+1 for 1≤i≤m. Then, T has a unique fixed point.In 2011, Karapinar and Erhan [31] introduced Kannan-type cyclic contraction and Chatterjea-type cyclic contraction. Moreover, they derive some fixed point theorems for such cyclic contractions in complete metric spaces as follows.Theorem 4 (fixed point theorem for Kannan-type cyclic contraction).
LetA and B be nonempty subsets of metric spaces (X,d) and a cyclic mapping T:A∪B→A∪B satisfies(6)dTx,Ty≤kdx,Tx+dy,Ty,∀x∈A,y∈B,where 0≤k<1/2. Then, T has a unique fixed point in A∩B.Theorem 5 (fixed point theorem for Chatterjea-type cyclic contraction).
LetA and B be nonempty subsets of a metric spaces (X,d) and a cyclic mapping T:A∪B→A∪B satisfies(7)dTx,Ty≤kdx,Ty+dy,Tx,∀x∈A,y∈B,where 0≤k<1/2. Then, T has a unique fixed point in A∩B.The purpose of this paper is to study fundamental properties ofC∗-algebra-valued b-metric space which was introduced by Ma and Jiang [26] and give some fixed point theorems for cyclic mapping with contractive and expansive condition on such space analogous to the results presented in [26].


## 2. Preliminaries

In this section, we recollect some basic notations, defintions, and results that will be used in main result. Firstly, we begin with the concept ofb-metric spaces.Definition 6 (see [12, 13]).
LetX be a nonempty set. A mapping d:X×X→R is called b-metric if there exists a real number b≥1 such that, for every x,y,z∈X, we have (i)
d(x,y)≥0,(ii)
d(x,y)=0 if and only if x=y,(iii)
d(x,y)=d(y,x),(iv)
d(x,z)≤b[d(x,y)+d(y,z)].In this case, the pair (X,d) is called a b-metric space.The class ofb-metric spaces is larger than the calass of metric spaces, since a b-metric space is a metric when b=1 in the fourth condition in the above definition. There exist many examples in some work showing that the class of b-metric is efficiently larger than those metric spaces (see also [12, 14, 32, 33]).Example 7 (see [12]).
The setlp(R) with 0<p<1, where lp(R)≔{{xn}⊆R:∑n=1∞‍|xn|p<∞}, together with the function d:lp(R)×lp(R)→R,(8)dx,y=∑n=1∞xn-ynp1/p,where x={xn},y={yn}∈lp(R), is a b-metric space with coefficient b=21/p>1. Observe that the result holds for the general case lp(X) with 0<p<1, where X is a Banach space.Example 8 (see [12]).
The spaceLp(0<p<1) of all real functions x(t), t∈[0,1], such that ∫01‍|x(t)|pdt<∞, together with the function(9)dx,y=∫01xt-ytpdt1/p,∀x,y∈Lp0,1,is a b-metric space with b=21/p.Example 9 (see [33]).
Let(X,d1) be a metric space and d2(x,y)=(d1(x,y))p, where p>1 is natural numbers. Then, d2 is a b-metric with b=2p-1.The notation convergence, compactness, closedness, and completeness inb-metric space are given in the same way as in metric space.Next, we give concept of spectrum of element inC∗-algebra A.Definition 10 (see [3]).
We say thata∈A is invertible if there is an element b∈A such that ab=ba=I. In this case, b is unique and written a-1. The set(10)Inv⁡A=a∈A∣ais  invertibleis a group under multiplication. We define spectrum of an element a to be the set(11)σa=σAa=λ∈C∣λI-a∉Inv⁡A.Theorem 11 (see [3]).
LetA be a unital Banach algebra and let a be an element of A such that a<1. Then, I-a∈Inv⁡A and(12)I-a-1=∑n=0∞an.Theorem 12 (see [3]).
LetA be a unital C∗-algebra with a unit I, then (1)
I∗=I,(2)
For anya∈Inv⁡(A), (a∗)-1=(a-1)∗.(3)
For anya∈A, σ(a∗)=σ(a)∗={λ¯∈C:λ∈σ(a)}.All over this paper,A means a unital C∗-algebra with a unit I. R is set of real numbers and R+ is the set of nonnegetive real numbers. Mn(R) is n×n matrix with entries R.Definition 13 (see [3]).
The set of hermitain elements ofA is denoted by Ah; that is, Ah={x∈A:x=x∗}. An element x in A is positive element which is denoted by θ⪯x, where θ means the zero element in A if and only if x∈Ah and σ(x) is a subset of nonnegative real numbers. We define a partial ordering Ah by using definition of positive element as x⪯y if and only if y-x⪰θ. The set of positive elements in A is denoted by A+={x∈A:x⪰θ}.The following are definitions and some properties of positive element of aC∗-algebra A.Lemma 14 (see [3]).
The sum of two positive elements in aC∗-algebra is a positive element.Theorem 15 (see [3]).
If a is an arbitrary element of aC∗-algebra A, then a∗a is positive.We summarise some elementary facts aboutA+ in the following results.Theorem 16 (see [3]).
LetA be a C∗-algebra: (1)
The setA+ is closed cone in A [a cone C in a real or complex vector space is a subset closed under addition and under scalar multiplication by R+].(2)
The setA+ is equal to {a∗a:a∈A}.(3)
Ifθ⪯a⪯b, then a≤b.(4)
IfA is unital and a and b are positive invertible elements, then a⪯b⇒θ⪯b-1⪯a-1.Theorem 17 (see [3]).
LetA be a C∗-algebra. If a,b∈A+ and a⪯b, then for any x∈A both x∗ax and x∗bx are positive elements and x∗ax⪯x∗bx.Lemma 18 (see [3]).
Suppose thatA is a unital C∗-algebra with a unit I: (1)
Ifa∈A+ with a<1/2, then I-a is invertible and a(I-a)-1<1.(2)
Suppose thata,b∈A with a,b⪰θ and ab=ba; then, ab⪰θ.(3)
DefineA′={a∈A:ab=ba,∀b∈A}. Let a∈A'; if b,c∈A with b⪰c⪰θ and I-a∈A+′ is invertible operator, then(13)I-a-1b⪰I-a-1c.Definition 19 (see [34]).
A matrixA∈Mn(C) is Hermitian if A=A∗, where A∗ is a conjugate transpose matrix of A. A Hermitian matrix A∈Mn(C) is positive definite if x∗Ax>0 for all nonzero x∈Cn, and it is positive semidefinite if x∗Ax≥0 for all nonzero x∈Cn.In 2014, Ma et al. [22] introduced the concept of C∗-algebra-valued metric space by using the concept of positive elements in A. The following is definition of C∗-algebra-valued metric.Definition 20 (see [22]).
LetX be a nonempty set. A mapping d:X×X→A is called C∗-algebra-valued metric on X if it satisfies the following conditions: (1)
d(x,y)⪰θ for all x,y∈X.(2)
d(x,y)=θ if and only if x=y.(3)
d(x,y)=d(y,x) for all x,y∈X.(4)
d(x,y)⪯d(x,z)+d(z,y) for all x,y,z∈X. Then, d is called a C∗-algebra-valued metric on X and (X,A,d) is called a C∗-algebra-valued metric space.We know that range of mappingd in metric space is the set of real numbers which is C∗-algebra; then, C∗-algebra-valued metric space generalizes the concept of metric spaces, replacing the set of real numbers by A+. In such paper, Ma et al. state the notation of convergence, Cauchy sequence, and completeness in C∗-algebra-valued metric space. For detail, a sequence {xn} in a C∗-algebra-valued metric space (X,A,d) is said to converge to x∈X with respect to A if for any ɛ>0 there is N∈N such that d(xn,x)<ɛ for all n≥N. We write it as limn→∞xn=x. A sequence {xn} is called a Cauchy sequence with respect to A if for any ɛ>0 there is N∈N such that d(xm,xn)<ɛ for all n,m≥N. The (X,A,d) is said to be a complete C∗-algebra-valued metric space if every Cauchy sequence with respect to A is convergent. Moreover, they introduce definition of contractive and expansive mapping and give some related fixed point theorems for self-maps with C∗-algebra-valued contractive and expansive mapping, analogous to Banach contraction principle. The following is the definition of contractive mapping and the related fixed point theorem.Definition 21 (see [22]).
Suppose that(X,A,d) is a C∗-algebra-valued metric space. A mapping T:X→X is called C∗-algebra-valued contractive mapping on X, if there is an λ∈A with λ<1 such that (14)dTx,Ty⪯λ∗dx,yλ,∀x,y∈X.Theorem 22 (see [22]).
If(X,A,d) is a complete C∗-algebra-valued metric space and T:X→X satisfies Defintion 21, then T has a unique fixed point in X.In the same way, the concept of expansive mapping is defined in the following way.Definition 23 (see [22]).
LetX be a nonempty set. A mapping T is a C∗-algebra-valued expansive mapping on X, if T:X→X satisfies (1)
T(X)=X,(2)
d(Tx,Ty)⪰λ∗d(x,y)λ, for all x,y∈X, where λ∈A is an invertible element and λ-1<1.The following is the related fixed point theorem forC∗-algebra-valued expansive mapping.Theorem 24 (see [22]).
Let(X,A,d) be a complete C∗-algebra-valued metric space. If a T:X→X satisfies Defintion 23, then T has a unique fixed point in X.


## 3. Fundamental Properties ofC∗-Algebra-Valued b-Metric Spaces

In this section, we begin with the concept ofC∗-algebra-valued b-metric space which was introduced by Ma and Jiang [26] as follows.Definition 25 (see [26]).
LetX be a nonempty set. A mapping d:X×X→A is called C∗-algebra-valued b-metric on X if there exists b∈A′ such that b⪰I satisfies following conditions: (1)
d(x,y)⪰θ for all x,y∈X.(2)
d(x,y)=θ if and only if x=y.(3)
d(x,y)=d(y,x) for all x,y∈X.(4)
d(x,y)⪯b[d(x,z)+d(z,y)] for all x,y,z∈X.Then, (X,d,A) is called a C∗-algebra-valued b-metric space.Remark 26.
Ifb=I, then a C∗-algebra-valued b-metric spaces are C∗-algebra-valued metric spaces. In particular, if A is set of real numbers and b=1, then the C∗-algebra-valued b-metric spaces is the metric spaces.Definition 27 (see [26]).
Let(X,A,d) be a C∗-algebra-valued b-metric space. A sequence {xn} in (X,A,d) is said to converge to x if and only if for any ɛ>0 there exists N∈N such that, for all n≥N, d(xn,x)≤ɛ. Then, {xn} is said to be convergent with respect to A and x is called limit point of {xn}. We denote it by limn→∞xn=x.
A sequence{xn} is called a Cauchy seqeunce with respect to A if and only if for any ɛ>0 there exists N∈N such that, for all n,m≥N, d(xn,xm)≤ɛ.
We say(X,A,d) is a complete C∗-algebra-valued b-metric space if every Cauchy sequence with respect to A is convergent sequence with respect to A.The following is an example of completeC∗-algebra-valued b-metric space.Example 28 (see [26]).
LetX=R and let A=Mn(R). Define (15)dx,y=diag⁡x-yp,α1x-yp,α2x-yp,…,αn-1x-yp=x-yp00⋯00α1x-yp0⋯000α2x-yp⋯⋮⋮⋮⋮⋱⋮000⋯αn-1x-yp,where x,y∈R and αi>0 for all i=1,2,…,n-1 are constants and p is a natural number such that p≥2.  A norm · on A is defined by(16)A=maxi,j⁡aij1/p,where A=(aij)n×n∈A. The involution is given by A∗=(A¯)T, conjugate transpose of matrix A:(17)A∗=a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮an1an2⋯ann∗=a11¯a21¯⋯an1¯a12¯a22¯⋯an2¯⋮⋮⋱⋮a1n¯a2n¯⋯ann¯=a11a21⋯an1a12a22⋯an2⋮⋮⋱⋮a1na2n⋯ann.It is easy to verify d is a C∗-algebra-valued b-metric space and (X,M2(R),d) is a complete C∗-algebra-valued b-metric space be completeness of R.Proof.
An elementA∈A=Mn(R) is positive element; denote it by(18)A⪰θ,iffAis  positive  semidefinite.We define a partial ordering ⪯ on A as follows: (19)A⪯Biffθ⪯B-A,where θ mean the zero matrix in Mn(R). Firstly, it clears that ⪯ is partially order relation. Next, we show that d is a C∗-algebra-valued b-metric space. Let x,y,z∈X. It is easy to see that d satifies conditions (1), (2), and (3) of Definition 25. We will only show condition (4) where d(x,y)⪯b[d(x,z)+d(z,y)] with(20)b=2p-10⋯002p-1⋯0⋮⋮⋱⋮00⋯2p-1n×n.Since function f(x)=|x|p is convex function for all p≥2 and x∈R, this implies that(21)a+c2p=12a+1-12cp≤12ap+1-12cp=12ap+cpand hence |a+c|p≤2p-1(|a|p+|c|p) for all a,c∈R. We substitute a=x-y and c=y-z; then,(22)x-zp=x-y+y-zp≤2p-1x-yp+y-zp.Hence, setting M0=(|x-y|p+|y-z|p) and M1=|x-z|p, we obtain that(23)2p-1M0-M100⋯00α12p-1M0-M10⋯000α22p-1M0-M1⋯⋮⋮⋮⋮⋱⋮000⋯αn-12p-1M0-M1=2p-1M000⋯00α12p-1M00⋯000α22p-1M0⋯⋮⋮⋮⋮⋱⋮000⋯αn-12p-1M0-M100⋯00α1M10⋯000α2M1⋯⋮⋮⋮⋮⋱⋮000⋯αn-1M1=2p-10⋯002p-1⋯0⋮⋮⋱⋮00⋯2p-1M000⋯00α1M00⋯000α2M0⋯⋮⋮⋮⋮⋱⋮000⋯αn-1M0-M100⋯00α1M10⋯000α2M1⋯⋮⋮⋮⋮⋱⋮000⋯αn-1M1=bdx,y+dy,z-dx,zimplies that each eigenvalue of b[d(x,z)+d(z,y)]-d(x,y) is nonnegative. Since each eigenvalue of a positive semidefinite matrix is a nonnegative real number, we have that b[d(x,z)+d(z,y)]-d(x,y) is positive semidefinite; that is, b[d(x,z)+d(z,y)]-d(x,y)⪰θ, that is, d(x,y)⪯b[d(x,z)+d(z,y)], where b=2p-1I∈A′ and b⪰I by 2p-1>1. But |x-y|p≤|x-z|p+|z-y|p is impossible for all x,y,z∈R. Hence, (X,Mn(R),d) is C∗-algebra-valued b-metric spaces but not C∗-algebra-valued metric spaces.
Finally, we show that(X,A,d) is a complete C∗-algebra-valued b-metric space. Suppose that {xn} is a Cauchy sequence with respect to A. Then, for any ɛ>0, there exists N∈N such that d(xm,xn)≤ɛ for all m,n≥N; that is,(24)max⁡xm-xnp1/p,α1xm-xnp1/p,α2xm-xnp1/p,…,αn-1xm-xnp1/p≤ɛfor all m,n≥N. Therefore, (25)xm-xn=xm-xnp1/p≤max⁡xm-xnp1/p,α1xm-xnp1/p,α2xm-xnp1/p,…,αn-1xm-xnp1/p≤ɛfor all m,n≥N. Hence, {xn} is a Cauchy sequnce in R. By completeness of R, there exists x∈R such that limn→∞xn=x; that is, limn→∞|xn-x|=0. Then, we have that (26)dxn,x=max⁡xn-xp1/p,α1xn-xp1/p,α2xn-xp1/p,…,αn-1xn-xp1/pconverges to 0 as n→∞. Therefore, {xn} is convergent with respect to A and {xn} converging to x, so (X,A,d) is a complete C∗-algebra-valued b-metric space.Next, we disscus some fundamental properties ofC∗-algebra-valued b-metric spaces.Theorem 29.
Let(X,A,d) be C∗-algebra-valued b-metric space. If {xn} is a convergent sequence with respect to A, then {xn} is Cauchy sequence with respect to A.Proof.
Assume that{xn} is a convergent sequence with respect to A; then, there exists a x∈X such that limn→∞xn=x. Let ɛ>0, there is N∈N such that, for all n≥N,(27)dxn,x≤ɛ2b.For m,n∈N, we get that(28)dxm,xn⪯bdxm,x+dx,xn. By Theorem 16, for m,n≥N, we have(29)dxm,xn≤bdxm,x+dx,xn≤bdxm,x+dx,xn≤bdxm,x+bdx,xn≤bɛ2b+bɛ2b=ɛ.This implies that {xn} is Cauchy sequence with respect to A.Definition 30.
A subsetS of a C∗-algebra-valued b-metric space (X,A,d) is bounded with respect to A if there exists x¯∈X and a nonnegetive real number M such that(30)dx,x¯≤M,∀x∈X.Theorem 31.
Let(X,A,d) be a C∗-algebra-valued b-metric space and let {xn} be a sequence in X and x∈X. Then, (1)
xn→x if and only if d(xn,x)→θ,(2)
a convergent sequence inX is bounded with respect to A and its limit is unique,(3)
a Cauchy sequence inX is bounded with respect to A.Proof.
(1) Assume thatxn→x. Let ɛ>0 is given. Then, there exists N0∈N such that(31)dxn,x-θ=dxn,x≤ɛ.This implies that d(xn,x)→θ. Conversely, assume that d(xn,x)→θ. Then, for any ɛ>0, there exists N1∈N such that(32)dxn,x-θ≤ɛ⟹dxn,x≤ɛ;that is, xn→x.
(2) Let{xn} be a convergent sequence with respect to A. Suppose that xn→x. Then, taking ɛ=1, we can find N∈N such that(33)dxn,x≤1,∀n≥N.Let K=max⁡{d(x1,x),d(x2,x),…,d(xN,x)}. Setting M=max⁡{1,K}. This implies that(34)dxn,x≤M,∀n∈N.Next, suppose that xn→x and xn→y. Consider, d(x,y)⪯b[d(x,xn)+d(xn,y)]; by Theorem 16, we have(35)dx,y≤bdxn,x+dxn,y.From (1), letting n→∞, we obtian that dx,y=0; that is x=y.
(3) Assume that{xn} is a Cauchy sequence with respect to A. In particular, ɛ=1; there exists N1∈N such that(36)dxm,xn≤1∀m,n≥N1.Let K=max{d(x1,xN1), d(x2,xN1),…,d(xN1-1,xN1)}. Then,(37)dxn,xN1≤K∀n<N1.Set M=max⁡{1,K}. Then, we get that(38)dxn,xN1≤M∀n∈N.Theorem 32.
Let{xn} be a convergent sequence in a C∗-algebra-valued b-metric space (X,A,d) and limn→∞xn=x. Then, every subsequence {xnk} of  {xn} is convergent and has the same limit x.Proof.
Letɛ>0 be given. Then, there exists N∈N such that(39)dxn,x≤ɛ,∀n≥N.Since n1<n2<⋯<nk<⋯ is an increasing sequence of natural numbers, it is easily proved (by induction) that nk≥k. Hence, if k≥N, we also have nk≥k≥N so that(40)dxnk,x≤ɛ,∀nk≥N.Therefore, subsequence {xnk} also converges to x.Theorem 33.
Let(X,A,d) be a C∗-algebra-valued b-metric space. Then, every subsequence of a Cauchy sequence is Cauchy sequence.Proof.
Let{xnk} be a subsequence of Cauchy sequence {xn} in a C∗-algebra-valued b-metric space. Then, for every ɛ>0, there is N∈N such that, for all r,s≥N, we have d(xr,xs)≤ɛ. Similar to the facts in proof of previous theorem, we have nr≥r≥N and ns≥s≥N. Hence, we obtain that d(xnr,xns)≤ɛ. Therefore, {xnk} is Cauchy sequence.Theorem 34.
Let(X,A,d) be a C∗-algebra-valued b-metric space and let {xn} be a Cauchy sequence with respect to A. If {xn} contains its convergent subsequence, then {xn} is convergent sequence.Proof.
Letɛ>0. Since {xn} is a Cauchy sequence with respect to A, there exists an N0∈N such that(41)dxm,xp≤12bɛ,∀m,p≥N0.Let {xnk} be a convergent subsequence of {xn} and xnk→x(k→∞). Then, there exists N1∈N such that(42)dxnk,x≤12bɛ,∀nk≥N1. Let N=max⁡{N0,N1}. For n,k≥N, we have(43)dxn,x⪯bdxn,xnk+dxnk,x. By Theorem 16, we also have (44)dxn,x≤bdxn,xnk+dxnk,x≤bdxn,xnk+bdxnk,x≤bɛ2b+ɛ2b≤ɛ.Therefore, xn→x as n→∞.Theorem 35.
Let(X,A,d) be a C∗-algebra-valued b-metric space. Suppose that {xn} and {yn} are convergent with respect to A and converge to x and y, respectively. Then, d(xn,yn) converges to b2d(x,y).Proof.
Letɛ>0. Since xn→x and yn→y, there exist N0,N1∈N such that(45)dxn,x≤ɛ2b,∀n≥N0,dyn,y≤ɛ2b2,∀n≥N1. Since d(xn,yn)⪯bd(xn,x)+b2d(x,y)+b2d(y,yn), by Theorem 16, we have(46)dxn,yn-b2dx,y≤bdxn,x+b2dy,yn≤ɛ. Therefore, d(xn,yn)→b2d(x,y).Theorem 36.
Let(X,A,d) be a C∗-algebra-valued b-metric space. Suppose that {xn} and {yn} are convergent with respect to A and converge to x and y, respectively. Then,(47)1b2dx,y≤liminfn→∞dxn,yn≤limsupn→∞dxn,yn≤b2dx,y.In particular, if x=y, then we have limn→∞d(xn,yn)=0. Moreover, for any z∈X, we have(48)1bdx,z≤liminfn→∞dxn,z≤limsupn→∞dxn,z≤bdx,z.Proof.
By defintion ofC∗-algebra-valued b-metric space, it easy to see that(49)dx,y⪯bdx,xn+b2dxn,yn+b2dyn,y,dxn,yn⪯bdxn,x+b2dx,y+b2dy,yn.Using Theorem 16, we have(50)dx,y≤bdx,xn+b2dxn,yn+b2dyn,y,dxn,yn≤bdxn,x+b2dx,y+b2dy,yn.Taking the lower limit as n→∞ in the first inequality and the upper limit as n→∞ in the second inequality, this completes the first result. In particular, if x=y, we have(51)dxn,yn≤bdxn,x+b2dy,yn. Taking the limit as n→∞ in this inequality, we obtain that limn→∞d(xn,yn)=0. Since(52)dx,z⪯bdx,xn+dxn,z,dxn,z⪯bdxn,x+dx,z,by Theorem 16, we have(53)dx,z≤bdx,xn+bdxn,z,dxn,z≤bdxn,x+bdx,z.Again taking the lower limit as n→∞ in the first inequality and the upper limit as n→∞ in the second inequality, we obtain the second desired result.Definition 37.
Let(X,A,d) be a C∗-algebra-valued b-metric space. A subset F of (X,A,d) is called a closed set if a sequence {xn} in X and xn→x with respect to A imply x∈F.


## 4. Fixed Point Theorems for Cyclic Contractions

Theorem 38.
LetA and B be nonempty closed subset of a complete C∗-algebra-valued b-metric space (X,A,d). Assume that T:A∪B→A∪B is cyclic mapping that satisfies(54)dTx,Ty⪯λ∗dx,yλ,∀x∈A,∀y∈B,where λ∈A with λ<1/b. Then, T has a unique fixed point in A∩B.Proof.
Letx0 be any point in A. Since T is cyclic mapping, we have Tx0∈B and T2x0∈A. Using the contractive condition of mapping T, we get(55)dTx0,T2x0=dTx0,TTx0⪯λ∗dx0,Tx0λ. For all n∈N, we have(56)dTnx0,Tn+1x0⪯λ∗ndx0,Tx0λn=λ∗nβλn, where β=d(x0,Tx0). Consider, for any m,n∈N such that m≤n; then, (57)dTmx0,Tnx0⪯bdTmx0,Tm+1x0+dTm+1x0,Tnx0⪯bdTmx0,Tm+1x0+b2dTm+1x0,Tm+2x0+dTm+2x0,Tnx0⪯⋯⪯bdTmx0,Tm+1x0+b2dTm+1x0,Tm+2x0+⋯+bn-mdTn-1x0,Tnx0⪯bλ∗mβλm+b2λ∗m+1βλm+1+⋯+bn-mλ∗n-1βλn-1=∑k=mn-1bk-m+1λ∗kβλk.
From Theorem16, we have (58)dTmx0,Tnx0≤∑k=mn-1bk-m+1λ∗kβλk≤∑k=mn-1bk-m+1λ∗kβλk≤∑k=mn-1bk-m+1λ∗kβλk≤β∑k=mn-1bk-m+1λk2≤β∑k=mn-1bk-m+1λ2k≤β∑k=mn-1b2kλ2k≤β∑k=m∞bλ2k=βbλ2m1-bλ.Since 0≤λ<1/b, we have β(bλ)2m/(1-(bλ))→0 as m→∞. Therefore, {Tnx0} is Cauchy sequence with respect to A. By the completeness of (X,A,d), there exists an element x∈X such that x=limn→∞Tnx0.
Since{T2nx0} is a sequence in A and {T2n-1x0} is a sequence in B, we obtain that both sequences converge to the same limit x. Since A and B are closed set, this implies that x∈A∩B.
Next, we will complete the proof by showing thatx is a unique fixed point of T. Since(59)θ⪯dTx,x⪯bdTx,T2nx0+dT2nx0,x⪯bλ∗dx,T2n-1x0λ+dT2nx0,x by Theorem 16, we obtain that(60)0≤dTx,x≤bλ2dx,T2n-1x0+bdT2nx0,x⟶0n⟶∞. We have Tx=x; that is, x is a fixed point of T.
Suppose thaty is fixed point of T and y≠x. Since (61)θ⪯dx,y=dTx,Ty⪯λ∗dx,yλ, we have (62)dx,y≤λ∗dx,yλ≤λ∗dx,yλ=λ2dx,y<dx,y.This is a contradiction. Therefore, x=y which implies that the fixed point is unique.Example 39.
LetX be a set of real numbers and A=M2×2(R) with A=maxi,j|aij|, where aij are entries of the matrix A∈M2×2(R). Then, (X,A,d) is a C∗-algebra-valued b-metric space with b=2002, where the involution is given by A∗=(A¯)T,(63)dx,y=x-y200x-y2 and partial ordering on A is given as(64)a11a12a21a22⪯b11b12b21b22⟺aij≤bij∀i,j=1,2,3,4.
Define a mappingT:X→X by(65)
T
x
=
-
x
+
1
/
3
3
sin
⁡
1
x
-
1
3
;
x
∈
∞
,
-
1
3
-
1
3
;
x
∈
-
1
3
,
0
-
1
2
;
x
∈
0
,
+
∞
.It is clear that T is not continuous at all elements of X. Therefore, Theorem 22 cannot imply the existence of fixed point of mapping T.
Suppose thatA=[-1/2,-1/3] and B=[-1/3,0]. Firstly, we will show that T:A∪B→A∪B is cyclic mapping. Let x∈B; that is, -1/3≤x≤0. Then, Tx=-1/3∈A. Again, let y∈A; that is, -1/2≤x≤-1/3. Indeed, we consider(66)-12≤x≤-13⟹-16≤x+13≤0⟹-118≤x+1/33≤0⟹0≤-x+1/33≤118⟹0≤-x+1/33sin⁡1x≤118sin⁡1x≤118⟹-13≤-x+1/33sin1x-13≤118-13≤0;this implies that Tx∈[-1/3,0]=B. For any x∈A and y∈B, since -1/2≤x≤-1/3 and -1/3≤y, we have 1/9≤-x/3≤1/6 and -1/9≤y/3. Hence, we obtain that(67)0≤-x3-19≤-x3+y3.Next, we consider (68)Tx-Ty2=-x+1/33sin⁡1x-13--132=-x+1/33sin⁡1x2≤-x+1/332=-x3-192≤-x3+y32≤19x-y2. Then, we have(69)dTx,Ty=Tx-Ty200Tx-Ty2⪯19x-y20019x-y2=130013x-y200x-y2130013=λ∗dx,yλ,where λ=1/3001/3. Then, λ=1/3<1/2=1/b. Thus, T satisfies contraction of Theorem 38 implying that T has a unique fixed point in A∩B; that is, {-1/3}=F(T).Corollary 40.
Suppose that(X,A,d) is a C∗-algebra-valued b-metric space. Assume that T:X→X is called a C∗-algebra-valued b-contractive mapping on X; that is, T satisfies(70)dTx,Ty⪯λ∗dx,yλ,∀x,y∈X, where λ∈A with λ<1/b. Then, T has a unique fixed point in X.Proof.
PuttingA=B=X, by Theorem 38, this implies that T has a unique fixed point in A∩B=X.Theorem 41.
Suppose that(X,A,d) is a complete C∗-algebra-valued b-metric space. Assume that a mapping T:X→X satisfies (1)
T(X)=X;(2)
d(Tx,Ty)⪰λ∗d(x,y)λ for all x,y∈X,where λ∈A is an invertible element and λ-1<1/b such that T is a C∗-algebra-valued b-expansive mapping on X. Then, T has a unique fixed point in X.Proof.
We will begin to prove this theorem by showing thatT is injective. Let x,y be an element in X such that x≠y; that is, d(x,y)≠0. Assume that Tx=Ty. We have(71)θ=dTx,Ty⪰λ∗dx,yλ=λ∗dx,y1/2dx,y1/2λ=dx,y1/2λ∗dx,y1/2λ⪰θ.This implies that λ∗d(x,y)λ=θ. Since λ is invertible, we have d(x,y)=θ which leads to contradiction. Thus, T is injective. By the first condition of mapping T, we obtain that T is bijective which implies that T is invertibe and T-1 is bijective.
Next, we will show thatT has a unique fixed point in X. In fact, since T is C∗-algebra-valued b-expansive and invertible mapping, we substitute x,y with T-1x, T-1y in the second condition of T, respectively, which implies that(72)dTT-1x,TT-1y⪰λ∗dT-1x,T-1yλ,∀x,y∈X. That is(73)dx,y⪰λ∗dT-1x,T-1yλ,∀x,y∈X.Since d(x,y) and λ∗d(T-1x,T-1y)λ are positive elements in A, λ∗d(T-1x,T-1y)⪯λd(x,y) and λ-1∈A. By condition (2) of Theorem 12 and Theorem 17, we have (74)dT-1x,T-1y=λλ-1∗dT-1x,T-1yλλ-1=λ-1∗λ∗dT-1x,T-1yλλ-1⪯λ-1∗dx,yλ-1.Therefore, T-1 is b-contractive mapping. Using Corollary 40, there exists a unique x such that T-1x=x, which means it has a unique fixed point x∈X such that Tx=T(T-1x)=(TT-1)x=Ix=x.Theorem 42 (cyclic Kannan-type).
LetA and B be nonempty closed subset of a complete C∗-algebra-valued b-metric space (X,A,d). Assume that T:A∪B→A∪B is cyclic mapping that satisfies(75)dTx,Ty⪯λdx,Tx+dy,Ty,∀x∈A,∀y∈B,where λ∈A+′ with λ<1/2b. Then, T has a unique fixed point in A∩B.Proof.
Without loss of generality, we can assume thatλ≠θ. Since λ∈A+′ and θ⪯d(x,Tx)+d(y,Ty), by the second condition of Lemma 18, we have θ⪯λ{d(x,Tx)+d(y,Ty)}.
Letx0 be any element in A. Since T is cyclic mapping, we have Tx0∈B and T2x0∈A. Consider(76)dTx0,T2x0=dTx0,TTx0⪯λdx0,Tx0+dTx0,T2x0=λdx0,Tx0+λdTx0,T2x0;that is,(77)I-λdTx0,T2x0⪯λdx0,Tx0. Since λ∈A+′ and λ<1/2b<1/2, by the first condition of Lemma 18, we have that I-λ is invertible and (I-λ)-1λ<1. From the third condition of Lemma 18, we have(78)dTx0,T2x0⪯I-λ-1λdx0,Tx0. Similarly, we get that(79)dT2x0,T3x0⪯I-λ-1λdTx0,T2x0.Since (I-λ)-1λ∈A+′ and θ⪯(I-λ)-1λd(x0,Tx0)-d(Tx0,T2x0), the second condition of Lemma 18, we have(80)θ⪯I-λ-1λI-λ-1λdx0,Tx0-dTx0,T2x0;that is,(81)I-λ-1λdTx0,T2x0⪯I-λ-1λ2dx0,Tx0. Hence,(82)dT2x0,T3x0⪯I-λ-1λdTx0,T2x0⪯I-λ-1λ2dx0,Tx0.Continuing this process, we have(83)dTnx0,Tn+1x0⪯I-λ-1λndx0,Tx0=αnβ,where α=(I-λ)-1λ and β=d(x0,Tx0). Next, we will show that {Tnx0} is Cauchy sequence with respect to A. Consider for any m,n∈N and m≤n that we have (84)dTmx0,Tnx0⪯bdTmx0,Tm+1x0+b2dTm+1x0,Tm+2x0+⋯+bn-mdTn-1x0,Tnx0⪯bαmβ+b2αm+1β+⋯+bn-mαn-1β=∑k=mn-1bk-m+1αkβ.From Theorem 16, we get that (85)dTmx0,Tnx0≤∑k=mn-1bk-m+1αkβ≤∑k=mn-1bk-m+1αkβ≤∑k=mn-1bk-m+1αkβ≤∑k=mn-1bkαkβ=β∑k=mn-1bαk≤β∑k=m∞bαk=βbαm1-bα.Consider(86)bα=bλI-λ-1≤bλI-λ-1=bλ∑i=0∞λi≤bλ∑i=0∞λi<b12b11-λ<1211-1/2=1.
Therefore,β(bα)2m/(1-(bα))→0 as m→∞. Therefore, {Tnx0} is Cauchy sequence with respect to A. By the completeness of (X,A,d), there exists an element x∈X such that x=limn→∞Tnx0.
Since{T2nx0} is a sequence in A and {T2n-1x0} is a sequence in B, we obtain that both sequences converge to the same limit x. Since A and B are closed set, this implies x∈A∩B. Next, we will show that x is a unique fixed point of T. Consider (87)dTx,x⪯bdTx,T2nx0+dT2nx0,x=bdTx,TT2n-1x0+bdT2nx0,x⪯bλdx,Tx+dT2n-1x0,T2nx0+bdT2nx0,x⪯bλdx,Tx+b2λdT2n-1x0,x+b2λdx,T2nx0+bdT2nx0,x;by Theorem 16 and submultiplicative, we obtian that(88)dTx,x≤bλdx,Tx+b2λdT2n-1x0,x+b2λdx,T2nx0+bdT2nx0,x.Letting n→∞, we get that(89)dTx,x≤bλdx,Tx, and so(90)dTx,x≤b12bdx,Tx<12dx,Tx. This implies that dTx,x=0; that is, d(Tx,x)=θ and so Tx=x. That is, x is fixed point of T. Now if y is another fixed point of T and y≠x, then(91)θ⪯dx,y=dTx,Ty⪯λdx,Tx+,dy,Ty=λdx,x+dy,y=θ, which leads to contradiction. Therefore, x=y; we complete the proof.Example 43.
LetX=[-1,1] and A=M2×2(R) with A=maxi,j|aij| where aij are entries of the matrix A∈M2×2(R). Then, (X,A,d) is a C∗-algebra-valued b-metric space with b=2002, where the involution is given by A∗=(A¯)T:(92)dx,y=x-y200x-y2, and partial ordering on A is given as(93)a11a12a21a22⪯b11b12b21b22⟺aij≤bij∀i,j=1,2,3,4.
Suppose thatA=[-1,0] and B=[0,1]. Define a mapping T:A∪B→A∪B by Tx=-x/4. Firstly, we will show that T is cyclic mapping. Let x be an element in A; that is, -1≤x≤0. Then, 0≤-x/4≤1 implies Tx∈B. Similarly, let y∈B, so 0≤y≤1. Then, -1/4≤-y/4≤0. Hence, Ty∈A.
For anyx∈A and y∈B, we consider(94)Tx-Ty2=-x4--y42=116x-y2≤116x+y2≤1162x2+2y2=2252516x2+2516y2=225x+x42+y+y42=225x-Tx2+y-Ty2.Then, we have (95)dTx,Ty=Tx-Ty200Tx-Ty2⪯225x-Tx2+y-Ty200225x-Tx2+y-Ty2=22500225x-Tx2+y-Ty200x-Tx2+y-Ty2=λdx,Tx+dy,Ty,where λ=2/25002/25. Then, λ=2/25<1/4=1/2b. Thus, T satisfies contraction of Theorem 42 implying that T has a unique fixed point in A∩B; that is, {0}=F(T).Theorem 44 (cyclic Chatterjea-type).
LetA and B be nonempty closed subset of a complete C∗-algebra-valued b-metric space (X,A,d). Assume that T:A∪B→A∪B is cyclic mapping that satisfies(96)dTx,Ty⪯λdy,Tx+dx,Ty,∀x∈A,∀y∈B, where λ∈A+′ with λ<1/2b2. Then, T has a unique fixed point in A∩B.Proof.
Without loss of generality, we can assume thatλ≠θ. Since λ∈A+′ and θ⪯d(y,Tx)+d(x,Ty), by the second condition of Lemma 18, we have θ⪯λ{d(y,Tx)+d(x,Ty)}.
Letx0 be any element in A, Since T is cyclic mapping, we have Tx0∈B and T2x0∈A. Consider(97)dTx0,T2x0=dTx0,TTx0⪯λdTx0,Tx0+dx0,T2x0⪯bλdx0,Tx0+dTx0,T2x0; that is,(98)I-bλdTx0,T2x0⪯bλdx0,Tx0. Since λ∈A+′ and b∈A+′, from the second condition of Lemma 18, we get that bλ∈A+′. Since bλ<b(1/2b2)≤1/2 and bλ∈A+′, by the first condition of Lemma 18, we have (I-bλ)-1∈A+′ and (bλ)(I-bλ)-1∈A+′ with (bλ)(I-bλ)-1<1. From the third condition of Lemma 18, we have(99)dTx0,T2x0⪯bλI-bλ-1dx0,Tx0. Similarly, we get that(100)dT2x0,T3x0⪯bλI-bλ-1dTx0,T2x0.Since (bλ)(I-bλ)-1∈A+′ and θ⪯(bλ)(I-bλ)-1d(x0,Tx0)-d(Tx0,T2x0), the second condition of Lemma 18, we have(101)θ⪯bλI-bλ-1bλI-bλ-1dx0,Tx0-dTx0,T2x0; that is,(102)bλI-bλ-1dTx0,T2x0⪯bλI-bλ-12dx0,Tx0. Hence,(103)dT2x0,T3x0⪯bλI-bλ-1dTx0,T2x0⪯bλI-bλ-12dx0,Tx0. Continuing this process, we have(104)dTnx0,Tn+1x0⪯bλI-bλ-1ndx0,Tx0=ωnβ,where ω=(bλ)(I-bλ)-1 and β=d(x0,Tx0). Next, we will show that {Tnx0} is Cauchy sequence with respect to A. Consider for any m,n∈N and m≤n; we have(105)dTmx0,Tnx0⪯bdTmx0,Tm+1x0+b2dTm+1x0,Tm+2x0+⋯+bn-mdTn-1x0,Tnx0⪯bωmβ+b2ωm+1β+⋯+bn-mωn-1β=∑k=mn-1bk-m+1ωkβ.From Theorem 16, we get that (106)dTmx0,Tnx0≤∑k=mn-1bk-m+1ωkβ≤∑k=mn-1bk-m+1ωkβ≤∑k=mn-1bk-m+1ωkβ≤∑k=mn-1bkωkβ=β∑k=mn-1bωk=β∑k=m∞bωk=βbωm1-bω.Consider (107)bω=bbλI-bλ-1≤bbλI-bλ-1=bbλ∑i=0∞bλi≤bbλ∑i=0∞bλi<bb2b211-bλ<1211-1/2=1.
Therefore,β(bω)2m/(1-(bω))→0 as m→∞. Therefore, {Tnx0} is Cauchy sequence with respect to A. By the completeness of (X,A,d), there exists an element x∈X such that x=limn→∞Tnx0.
Since{T2nx0} is a sequence in A and {T2n-1x0} is a sequence in B, we obtain that both sequences converge to the same limit x. Since A and B are closed set, this implies x∈A∩B.
Next, we will complete the proof by showing thatx is a unique fixed point of T. Since (108)dTx,x⪯bdTx,T2nx0+dT2nx0,x=bdTx,TT2n-1x0+bdT2nx0,x⪯bλdx,T2nx0+dT2n-1x0,Tx+bdT2nx0,x=bλdx,T2nx0+bλdT2n-1x0,Tx+bdT2nx0,x⪯bλdx,T2nx0+b2λdT2n-1x0,x+b2λdx,Tx+bdT2nx0,x, by Theorem 16, we have(109)dTx,x≤bλdx,T2nx0+b2λdT2n-1x0,x+b2λdx,Tx+bdT2nx0,x.Letting n→∞, we get that(110)dTx,x≤b2λdx,Tx, and so(111)dTx,x≤b212b2dx,Tx<12dx,Tx.This implies that d(Tx,x)=0; that is, d(Tx,x)=θ and so Tx=x. That is, x is fixed point of T. Now if y is another fixed point of T and y≠x, then(112)θ⪯dx,y=dTx,Ty⪯λdy,Tx+dx,Ty=2λdx,y.
From Theorem16, we get that(113)dx,y≤2λdx,y≤2λdx,y<212b2dx,y≤dx,y,which leads to a contradiction. Therefore, x=y which implies that the fixed point is unique.Example 45.
LetX=[0,1] and A=M2×2(R) with A=maxi,j|aij|, where aij are entries of the matrix A∈M2×2(R). Then, (X,A,d) is a C∗-algebra-valued b-metric space with b=2002, where the involution is given by A∗=(A¯)T:(114)dx,y=x-y200x-y2,and partial ordering on A is given as(115)a11a12a21a22⪯b11b12b21b22⟺aij≤bij∀i,j=1,2,3,4.
Suppose thatA=[0,1] and B=[0,1/2]. Define a mapping T:A∪B→A∪B by Tx=x/5. Firstly, we will show that T is cyclic mapping. Let x∈A; that is, 0≤x≤1. Then, 0≤x/5≤1/5 implies Tx∈B. Similarly, let y∈B, so 0≤y≤1/2. Then, 0≤y/5≤1/10. Hence, Ty∈A.
Now, we will show thatT satisfies the contraction of Theorem 44. Consider (116)x-y5=166x-y5=16x-y5+x5-y and so (117)x-y52=16x-y5+x5-y2=136x-y5+x5-y2≤1362x-y52+2x5-y2=118x-Ty2+Tx-y2.
Then, we have(118)dTx,Ty=Tx-Ty200Tx-Ty2⪯118x-Ty2+Tx-y200118x-Ty2+Tx-y2=11800118x-Ty2+Tx-y200x-Ty2+Tx-y2=λdx,Ty+dy,Tx,where λ=1/18001/18. Then, λ=1/18<1/8=1/2b2. Thus, T satisfies contraction of Theorem 44 implying that T has a unique fixed point in A∩B.



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