Chatterjea and $\stackrel{\mathbf{`}}{\mathbf{C}}$iri$\stackrel{\mathbf{`}}{\mathbf{C}}$ -Type Fixed-Point Theorems Using (α − ψ) Contraction on C^*-Algebra-Valued Metric Space

Abstract

In the present paper, we provide and verify several results obtained by using the Chatterjea and $\stackrel{`}{\mathrm{C}}$iri$\stackrel{`}{\mathrm{c}}$ fixed-point theorems by using $(\alpha -\psi )$-contractive mapping in $C^{*}$-algebra-valued metric space. We provide some examples and an application to illustrate our results. Our study extends and generalizes the results of several studies in the literature.

1. Introduction

The Banach contraction principle [1] is one of the most important tools of analysis and has many significant applications in various fields of science. It has been improved in many ways and generalized by many researchers. A map $T:\Omega \to \Omega$, where $(\Omega ,d)$ is a complete metric space, is said to be a contraction map if there exists $\lambda \in (0,1)$, such that for all $\mu ,\nu \in \Omega$

$$\begin{array}{c}\hfill d(T\mu ,T\nu )\le \lambda d(\mu ,\nu ).\end{array}$$

(1)

This result was introduced by Banach in 1922. Kannan [2] in 1968 proved that, if $(\mathsf{\Omega },d)$ is a complete metric space and $T:\Omega \to \Omega$ is a map satisfying

$$\begin{array}{c}\hfill d(T\mu ,T\nu )\le \lambda \left(d\right(T\mu ,\mu )+d(T\nu ,\nu \left)\right),\end{array}$$

(2)

where $\lambda \in (0,\frac{1}{2})$ for all $\mu ,\nu \in \Omega$, then there is a unique fixed point on T. Later, in 1972, Chatterjea [3] proved that if $(\Omega ,d)$ is a complete metric space and $T:\Omega \to \Omega$ is a mapping that exists $\lambda \in (0,\frac{1}{2})$, such that $\mu ,\nu \in \Omega$, the inequality

$$\begin{array}{c}\hfill d(T\mu ,T\nu )\le \lambda \left(d\right(T\mu ,\nu )+d(T\nu ,\mu \left)\right)\end{array}$$

(3)

is satisfied; thus, T has a unique fixed point.

$\stackrel{`}{\mathrm{C}}$iri$\stackrel{`}{\mathrm{c}}$ [4] in 1974 introduced an interesting general contraction condition. If there exists $\lambda \in (0,1)$, such that for all $\mu ,\nu \in \Omega$, and $T:\Omega \to \Omega$ is a map satisfying

$$\begin{array}{c}\hfill d(T\mu ,T\nu )\le \lambda ·max\left\{d\right(\mu ,\nu ),d(T\mu ,\mu \left)d\right(T\nu ,\nu ),d(T\mu ,\nu ),d(T\nu ,\mu \left)\right\},\end{array}$$

(4)

then T has a unique fixed point.

On the other hand, Samet et al. [5,6] studied $\alpha$-$\psi$-contractive mappings in metric spaces. Many researchers have established related studies to $\alpha$-admissible and $\alpha -\psi$-contractive mappings and related fixed-point theorems (see [7,8,9,10,11,12,13,14,15]).

Recently, Ma et al. [10] introduced the more generalized notion 0f a $C^{*}$-algebra-valued metric space by replacing real numbers with the positive cone of $C^{*}$-algebra. This line of research was continued in [16,17,18,19,20,21,22], where several other fixed-point results were obtained in the framework of $C^{*}$-algebra-valued metric space.

Throughout this paper, we suppose that A is a unital $C^{*}$-algebra with a unit $I_A$. We mean that a unital $C^{*}$-algebra is a complex Banach algebra A with an involution map $*:A\to A$, $a\to a^{*}$, such that ${\left(a^{*}\right)}^{*}=a$, ${\left(ab\right)}^{*}=a^{*}{b}^{*}$, ${(a+b)}^{*}=a^{*}+b^{*}$ and ${\left(\lambda a\right)}^{*}=\overline{\lambda }{a}^{*}$ for $a,b,A,\lambda \in \mathbb{C}$, such that $\parallel a^{*}{a\parallel =\parallel a\parallel }^2$. Set $A_h=\{a\in A:a=a^{*}\}$. An element $a\in A$ is a positive element if $a=a^{*}$ and $\sigma \left(a\right)\subset {\mathbb{R}}^{+}$, where $\sigma \left(a\right)$ is the spectrum of a. We define a partial ordering ⪯ on A as $a⪯b$ if $0_A⪯b-a$, where $0_A$ means the zero element in A, and we let $A^{+}$ denote the $\{a\in A:a⪰0_A\}$ and $\left|a\right|={\left(a^{*}a\right)}^{\frac{1}{2}}$.

The results described in this article extend some fixed-point theorems in $C^{*}$-algebra-valued metric spaces. $C^{*}$-algebras are considered typical examples of quantum spaces and non-commutative spaces. They play an important role in the non-commutative geometry project introduced by Alain Connes [23]. Thus, the theory of metric space-valued $C^{*}$-algebras should apply to many problems in quantum spaces, such as matrices and bounded linear operators on Hilbert spaces. Therefore, $C^{*}$-algebras and their metric provide a non-commutative version of ordinary metric spaces.

2. Preliminaries

In this section, we introduce some basic notions which will be used in the following work.

Lemma 1.

Suppose that A is a unital $C^{*}$-algebra with unit $I_A$. The following holds.

(1) If $a\in A$, with $\parallel a\parallel <\frac{1}{2}$, then $1-a$ is invertible and ${\parallel a(1-a)}^{-1}\parallel <1$.

(2) If $a,b\in A^{+}$ and $ab=ba$, then $a·b⪰0_A$.

(3) Let $a\in A^{\prime }$. If $b,c\in A$ with $b⪰c⪰0_A$ and $1-a\in {\left(A^{\prime }\right)}^{+}$ is an invertible element, then ${(I_A-a)}^{-1}b⪰{(I_A-a)}^{-1}c$, where $A^{\prime }=\{b\in A:ab=ba\forall a\in A\}$.

We refer to [24] for more $C^{*}$algebra details.

Definition 1.

[10] Let Ω be a non-empty set. Suppose the mapping $d_A:\Omega \times \Omega \to A$ satisfies:

(1) $d_A(\mu ,\nu )⪰0_A$ for all $\mu ,\nu \in \Omega$ and $d_A(\mu ,\nu )=0_A\iff \mu =\nu$.

(2) $d_A(\mu ,\nu )=d_A(\nu ,\mu )$ for all $\mu ,\nu \in \Omega$.

(3) $d_A(\mu ,\xi )⪯d_A(\mu ,\nu )+d_A(\nu ,\xi )$ for all $\mu ,\nu ,\xi \in \Omega$.

Then, $d_A$ is called a $C^{*}$-algebra-valued metric on Ω and $(\Omega ,A,d_A)$ is called $C^{*}$-algebra-valued metric space.

Example 1.

Let Ω be a Banach space and $d_A:\Omega \times \Omega \to A$ given by $d_A(\mu ,\nu )=\parallel \mu -\nu \parallel ·a$, for all $\mu ,\nu \in \Omega$, which should be where $a\in A^{+}$, $a⪰0$.

It is easy to verify that $(\Omega ,A,d_A)$ is a $C^{*}$-algebra-valued metric space.

Example 2.

Let $\Omega =\mathbb{C}$ and $A=M_n\left(\mathbb{C}\right)$. It is obvious that A is a $C^{*}$-algebra with the matrix norm and the involution given by $*:M_n\left(\mathbb{C}\right)\to M_n\left(\mathbb{C}\right)$, ${\left(Z_{ij}\right)}_{1\le i,j\le n}\to {\left(Z_{ij}\right)}_{1\le i,j\le n}^{*}={\left(\overline{Z_{ji}}\right)}_{1\le i,j\le n}$, where $\overline{Z_{ij}}$ is the conjugate of $Z_{ij}$, $Z_{ij}\in \mathbb{C}$. Define a mapping $d_A:\Omega \times \Omega \to A$, by:

$$\begin{array}{ccc}\hfill d(Z_1,Z_2)& =& diag\left(e^{i{\theta }_1}\right|Z_1-Z_2|,\dots ,e^{i{\theta }_k}|Z_1-Z_2|,\dots e^{i{\theta }_n}|Z_1-Z_2\left|\right)\hfill \\ \\ & =& \left(\begin{array}{ccccc}{e}^{i{\theta }_1}|Z_1-Z_2|& \cdots & 0& \cdots & 0\\ ⋮& \ddots & ⋮& & ⋮\\ 0& \cdots & e^{i{\theta }_k}|Z_1-Z_2|& \cdots & 0\\ ⋮& & ⋮& \ddots & ⋮\\ 0& \cdots & 0& \cdots & e^{i{\theta }_n}|Z_1-Z_2|\end{array}\right),\hfill \end{array}$$

for all $Z_1,Z_2\in \mathbb{C}$, $i=\sqrt{-1}$, $k=1,\dots ,n$, ${\theta }_k\in [0,\frac{\pi }{2}]$. Then, $(\Omega ,A,d_A)$ is a $C^{*}$-algebra-valued metric space. It is clear that it is a generalization of the complex-valued metric space given in [25], when $A=\mathbb{C}.$

Definition 2.

Let $(\Omega ,A,d_A)$ be a $C^{*}$-algebra-valued metric space, $\mu \in \Omega$, and ${\left\{{\mu }_n\right\}}_{n=1}^{+\infty }$ be a sequence in Ω. Then,

(i) ${\left\{{\mu }_n\right\}}_{n=1}^{+\infty }$ convergent to μ whenever, for every $\epsilon \in A$ with $\epsilon \succ 0_A$, there is a natural number $N\in \mathbb{N}$, such that

$$d_A({\mu }_n,\mu )\prec \epsilon ,$$

for all $n>N$. We denote this by $\underset{n\to \infty }{lim}{\mu }_n=\mu$ or ${\mu }_n\to \mu$ as $n\to +\infty$.

(ii) ${\left\{{\mu }_n\right\}}_{n=1}^{+\infty }$ is said to be a Cauchy sequence whenever, for every $\epsilon \in A$ with $\epsilon \succ 0_A$, there is a natural number $N\in \mathbb{N}$, such that

$$d_A({\mu }_n,{\mu }_m)\prec \epsilon ,$$

for all $n,m>N$.

Lemmaμ2.

(i) ${\left\{{\mu }_n\right\}}_{n=1}^{+\infty }$ is convergent in Ω if, for any element $ϵ>0$, there is $N\in \mathbb{N}$, such that for all $n>N$, $\parallel d({\mu }_n,\mu )\parallel \le ϵ$.

(ii) ${\left\{{\mu }_n\right\}}_{n=1}^{+\infty }$ is a Cauchy sequence in Ω if, for any $ϵ>0$ there is $N\in \mathbb{N}$, such that

$\parallel d_A({\mu }_n,{\mu }_m)\parallel \le ϵ$, for all $n,m>N$. We say that $(\Omega ,A,d_A)$ is a complete $C^{*}$-algebra-valued metric space if every Cauchy sequence is convergent with respect to A.

Example 3.

Let Ω be a compact Hausdorff space. We denote by $C(\Omega )$ the algebra of all complex-valued continuous functions on Ω with pointwise addition and multiplication. The algebra $C(\Omega )$ with the involution defined by $f^{*}\left(\mu \right)=\overline{f\left(\mu \right)}$ for each $f\in C(\Omega ),\mu \in \Omega$ and with the norm ${\parallel f\parallel }_{\infty }=sup\left\{\right|f\left(\mu \right)|,\mu \in \Omega \}$ is a commutative $C^{*}$-algebra where unit $I_{C(\Omega )}$ is the constant function. Let $C^{+}(\Omega )=\{f\in C(\Omega ):\overline{f\left(\mu \right)}=f\left(\mu \right),f\left(\mu \right)\ge 0\}$ denote the positive cone of$C(\Omega )$, with partial order relation $f\le g$ if and only if $f\left(\mu \right)\le g\left(\mu \right)$. Put $d_{C(\Omega )}:C(\Omega )\times C(\Omega )\to C(\Omega )$ as $d_{C(\Omega )}(f,g)={sup}_{\mu \in \Omega }\left\{\right|f\left(\mu \right)-g\left(\mu \right)\left|\right\}.I_{C(\Omega )}$. It is clear that $(C(\Omega ),C(\Omega ),d_{C(\Omega )})$ is a complete $C^{*}$-algebra-valued metric space.

Definition 3.

[6] Let $T:\Omega \to \Omega$ be a self map and $\alpha :\Omega \times \Omega \to [0,+\infty )$. Then, T is called α-admissible if for all $\mu ,\nu \in \Omega$ and $\alpha (\mu ,\nu )\ge 1$ implies $\alpha (T\mu ,T\nu )\ge 1$.

Definition 4.

Let Ω be a non-empty set and ${\alpha }_A:\Omega \times \Omega \to {\left(A^{+}\right)}^{\prime }$ be a function. We say that the self map T is ${\alpha }_A$ -admissible if for all $(\mu ,\nu )\in \Omega \times \Omega$, ${\alpha }_A(\mu ,\nu )⪰I_A$$\Rightarrow {\alpha }_A(T\mu ,T\nu )⪰I_A$, where $I_A$ is the unit of A.

Definition 5.

Let $(\Omega ,A,d_A)$ be a $C^{*}$-algebra-valued metric space and $T:\Omega \to \Omega$ be a mapping. We say that T is an ${\alpha }_A$-${\psi }_A$-contractive mapping if there exist two functions ${\alpha }_A:\Omega \times \Omega \to A_{+}$ and ${\psi }_A\in {\Psi }_A$, such that

$$\begin{array}{c}\hfill {\alpha }_A(\mu ,\nu )d_A(T\mu ,T\nu )⪯{\psi }_A\left(d_A(\mu ,\nu )\right),\end{array}$$

for all $\mu ,\nu \in \Omega .$

Definition 6.

Suppose that A and B are $C^{*}$-algebras. A mapping $\psi :A\to B$ is said to be a $C^{*}$-homomorphism if:

(a) $\psi ({\lambda }_1{a}_1+{\lambda }_2{a}_2)={\lambda }_1\psi \left(a_1\right)+{\lambda }_2\psi \left(a_2\right)$ for all ${\lambda }_1,{\lambda }_2\in \mathbb{C}$ and $a_1,a_2\in A$;

(b) $\psi \left(a_1{a}_2\right)=\psi \left(a_1\right)\psi \left(a_2\right)$, $\forall a_1,a_2\in A$;

(c) $\psi \left(a^{*}\right)=\psi {\left(a\right)}^{*}$, $\forall a\in A$; and

(d) ψ maps the unit in A to the unit in B.

Definition 7.

If $\psi :A\to B$ is a linear mapping in $C^{*}$-algebra, it is said to be positive if $\psi \left(A^{+}\right)\subseteq B^{+}$. In this case, $\psi \left(A_h\right)\subseteq B_h$, and the restriction map $\psi :A_h\to B_h$ increases. Every $C^{*}$-homomorphism is contractive and hence bounded and every *-homomorphism is positive.

Definition 8.

Let ${\Psi }_A$ be the set of positive functions ${\psi }_A:A^{+}\to A^{+}$ satisfying the following conditions:

(a) ${\psi }_A\left(a\right)$ is continuous and non-decreasing, ${\psi }_A\left(a\right)\prec a$;

(b) ${\psi }_A\left(a\right)=0$ iff $a=0$; and

(c) ${\displaystyle \sum _{n=1}^{\infty }}{\psi }_A^n\left(a\right)<\infty$, $\underset{n\to \infty }{lim}{\psi }_A^n\left(a\right)=0$ for each $a\succ 0$, where ${\psi }_A^n$ is the nth-iterate of ${\psi }_A$.

3. Main Results

In this section, we give some types of Chatterjea and $\stackrel{`}{\mathrm{C}}$iri$\stackrel{`}{\mathrm{c}}$ fixed-point theorems in a $C^{*}$-algebra-valued metric space using $(\alpha -\psi )$-contraction.

Theorem 1.

(Chatterjea Type) Let $(\Omega ,A,d_A)$ be a complete $C^{*}$-algebra-valued metric space and $T:\Omega \to \Omega$, be a mapping satisfying:

$$\begin{array}{c}\hfill {\alpha }_A(\mu ,\nu )d_A(T\mu ,T\nu )⪯{\psi }_A\left(\frac{d_A(T\mu ,\nu )+d_A(T\nu ,\mu )}{2}\right),\end{array}$$

(5)

$for\mu ,\nu \in \Omega$, where

$$\begin{array}{c}\hfill {\alpha }_A:\Omega \times \Omega \to A^{+}and{\psi }_A\in {\Psi }_A,{\psi }_A\prec \frac{1}{2}·I_A\end{array}$$

and the following conditions hold:

(a) T is ${\alpha }_A$-admissible;

(b) There exists ${\mu }_0\in \Omega$, such that ${\alpha }_A({\mu }_0,T{\mu }_0)⪰I_A$; and

(c) T is continuous.

Then, T has a fixed point in Ω.

Proof. 

Let ${\mu }_0\in \Omega$, such that ${\alpha }_A({\mu }_0,T{\mu }_0)⪰I_A$, and define the sequence ${\left\{{\mu }_n\right\}}_{n=0}^{+\infty }$ in $\Omega$, such that ${\mu }_{n+1}=T{\mu }_n$ for all $n\in \mathbb{N}$. If ${\mu }_n={\mu }_{n+1}$ for some $n\in \mathbb{N}$, then ${\mu }_n$ is a fixed point for T.

Suppose that ${\mu }_n\ne {\mu }_{\mu +1}$ for all $n\in \mathbb{N}$. Because T is ${\alpha }_A$-admissible, we obtain

$${\alpha }_A({\mu }_0,{\mu }_1)={\alpha }_A({\mu }_0,T{\mu }_0)⪰I_A\Rightarrow$$

$$\begin{array}{c}\hfill {\alpha }_A(T{\mu }_0,T^2{\mu }_0)={\alpha }_A({\mu }_1,{\mu }_2)⪰I_A.\end{array}$$

(6)

By induction, we have ${\alpha }_A({\mu }_n,{\mu }_{n+1})⪰I_A$ for all $n\in \mathbb{N}$.

By using inequalities (5) and (6), we have

$$\begin{array}{ccc}\hfill d_A({\mu }_n,{\mu }_{n+1})& =& d_A(T{\mu }_{n-1},T{\mu }_n)\hfill \\ & ⪯& {\alpha }_A({\mu }_{n-1},{\mu }_n)d_A(T{\mu }_{n-1},T{\mu }_n)\hfill \\ & ⪯& {\psi }_A\left({\displaystyle \frac{d_A(T{\mu }_{n-1},{\mu }_n)+d_A(T{\mu }_n,{\mu }_{n-1})}{2}}\right)\hfill \\ & =& {\psi }_A\left({\displaystyle \frac{d_A({\mu }_n,{\mu }_n)+d_A(T{\mu }_n,{\mu }_{n-1})}{2}}\right)\hfill \\ & =& {\psi }_A(\left({\displaystyle \frac{d_A({\mu }_n,{\mu }_n))+{\psi }_A(d_A({\mu }_{n+1},{\mu }_{n-1})}{2}}\right).\hfill \end{array}$$

Because ${\phi }_A\left(0\right)=0$, we obtain

$$\begin{array}{c}\hfill d_A({\mu }_n,{\mu }_{n+1})⪯{\psi }_A\left({\displaystyle \frac{d_A({\mu }_{n+1},{\mu }_{n-1})}{2}}\right).\end{array}$$

(7)

Applying triangular inequality in (7), we have

$$d_A({\mu }_n,{\mu }_{n+1})⪯{\psi }_A{\displaystyle \frac{(d_A({\mu }_{n+1},{\mu }_n)+d_A({\mu }_n,{\mu }_{n-1}))}{2}}.$$

Because ${\psi }_A$ is additive, we have

$$\begin{array}{ccc}\hfill d_A({\mu }_n,{\mu }_{n+1})& ⪯& \frac{{\psi }_A(d_A({\mu }_{n+1},{\mu }_n)}{2}+\frac{{\psi }_A\left(d_A({\mu }_n,{\mu }_{n-1})\right)}{2}.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill (\frac{1}{2}-{\psi }_A)\left(d_A({\mu }_n,{\mu }_{n+1})\right)& ⪯& \frac{1}{2}{\psi }_A\left(d_A({\mu }_n,{\mu }_{n-1})\right),\hfill \end{array}$$

and we have

$$\begin{array}{ccc}\hfill d_A({\mu }_n,{\mu }_{n+1})& ⪯& \frac{1}{2}\left({\psi }_A{(\frac{1}{2}-{\psi }_A)}^{-1}\right)\left(d_A({\mu }_n,{\mu }_{n-1})\right).\hfill \end{array}$$

Putting $\frac{1}{2}{\psi }_A{(\frac{1}{2}-{\psi }_A)}^{-1}={\varphi }_A$ by induction, we have

$$d_A({\mu }_n,{\mu }_{n+1})⪯{\varphi }_A^n\left(d_A({\mu }_0,{\mu }_1)\right),$$

for all $n\in \mathbb{N}$. Let $n,m\in \mathbb{N}$ with $m>n$. We obtain

$$\begin{array}{ccc}\hfill d_A({\mu }_n,{\mu }_m)⪯\sum _{k=n}^{m-1}{\varphi }_A^k\left(d_A({\mu }_0,{\mu }_1)\right)& \to & 0_A(asn\to +\infty ).\hfill \end{array}$$

Therefore, we can prove that $\left\{{\mu }_n\right\}$ is a Cauchy sequence in the $C^{*}$-algebra metric space $(\Omega ,A,d_A)$.

Because $(\Omega ,A,d_A)$ is complete, there exists $\mu \in \Omega$, such that ${\mu }_n\to \mu$ as $n\to +\infty$. From the continuity of T, it follows that ${\mu }_{n+1}=T{\mu }_n\to T\mu$ is as $n\to +\infty$.

By continuity of this limit, we have $T\mu =\mu$—that is, $\mu$ is a fixed point of T.

The proof of the uniqueness is as follows. If $\nu (\ne \mu )$ is another fixed point of T, then

$$\begin{array}{ccc}\hfill 0_A⪯d_A(\mu ,\nu )& =& d_A(T\mu ,T\nu )\hfill \\ & ⪯& {\alpha }_A(\mu ,\nu )d_A(T\mu ,T\nu )\hfill \\ & ⪯& {\psi }_A\frac{(d_A(T\mu ,\nu )+d_A(T\nu ,\mu ))}{2}\hfill \\ & =& {\psi }_A{\displaystyle \frac{(d_A(\mu ,\nu )+d_A(\mu ,\nu ))}{2}}\hfill \\ & =& I_A{\psi }_A\left(d_A(\mu ,\nu )\right),{\psi }_A\left(a\right)\prec aforanya\in A,.\hfill \end{array}$$

This implies that

$$0_A⪯d_A(\mu ,\nu )\prec d_A(\mu ,\nu ),$$

which gives a contradiction, and we can obtain $\mu =\nu$. This completes the proof. □

Corollary 1.

Let $(\Omega ,A,d)$ be a complete $C^{*}$-algebra-valued metric space. Suppose $T:\Omega \to \Omega$ satisfies for all $\mu ,\nu \in \Omega$

$$d_A(T\mu ,T\nu )\le \mathbb{A}(d_A(T\mu ,\nu )+d_A(T\nu ,\mu )),$$

where $\mathbb{A}\in {\left(A^{\prime }\right)}^{+}$ and $\parallel \mathbb{A}\parallel \le \frac{1}{2}$. Then, there exists a unique fixed point T in Ω [10].

Proof. 

This is an immediate consequence of Theorem 1, with ${\alpha }_A(\mu ,\nu )=Id$, ${\psi }_A\left(a\right)=\mathbb{A}a,$, where $a\in A$, $\mathbb{A}\in {\left(A^{\prime }\right)}^{+}$. □

Theorem 2.

(Banach-Chatterjea Type) Let $(\Omega ,A,d_A)$ be a complete $C^{*}$-algebra-valued metric space and $T:\Omega \to \Omega$ be a mapping satisfying

$$\begin{array}{ccc}\hfill {\alpha }_A(\mu ,\nu )d_A(T\mu ,T\nu )& ⪯& {\displaystyle \frac{{\psi }_A(d_A(\mu ,\nu )+(d_A(T\mu ,\nu )+d_A(T\nu ,\mu )))}{3}},{\psi }_A\prec \frac{1}{3}·I_A\hfill \end{array}$$

(8)

$for\mu ,\nu \in \Omega$, where the following conditions hold:

(i) T is ${\alpha }_A$-admissible;

(ii) there exists ${\mu }_0\in \Omega$, such that ${\alpha }_A({\mu }_0,T{\mu }_0)⪰I_A$; and

(iii) T is continuous.

Then, T has a fixed point in Ω.

Proof. 

Following the first part of the proof in the Theorem 1, we obtain

$$\begin{array}{c}\hfill {\alpha }_A({\mu }_n,{\mu }_{n+1})⪰I_{A}foralln\in \mathbb{N}.\end{array}$$

(9)

By using inequalities (8) and (9), we have

$$\begin{array}{ccc}\hfill d_A({\mu }_n,{\mu }_{n+1})& =& d_A(T{\mu }_{n-1},T{\mu }_n)\hfill \\ & ⪯& {\alpha }_A({\mu }_{n-1},{\mu }_n)\left(d_A(T{\mu }_{n-1},T{\mu }_n)\right)\hfill \\ & ⪯& \frac{1}{3}{\psi }_A(d_A({\mu }_{n-1},{\mu }_n)+d_A(T{\mu }_{n-1},{\mu }_n)+d_A(T{\mu }_n,{\mu }_{n-1}))\hfill \\ & =& \frac{1}{3}{\psi }_A(d_A({\mu }_{n-1},{\mu }_n)+d_A({\mu }_n,{\mu }_n)+d_A({\mu }_{n+1},{\mu }_{n-1}))\hfill \\ & =& \frac{1}{3}{\psi }_A(d_A({\mu }_{n-1},{\mu }_n)+d_A({\mu }_{n+1},{\mu }_{n-1})).\hfill \end{array}$$

By using triangular inequality, we obtain

$$\begin{array}{ccc}\hfill d_A({\mu }_n,{\mu }_{n+1})& ⪯& \frac{1}{3}{\psi }_A(d_A({\mu }_{n-1},{\mu }_n)+d_A({\mu }_{n-1},{\mu }_n)+d_A({\mu }_n,{\mu }_{n+1}))\hfill \\ & =& \frac{2}{3}{\psi }_A\left(d_A({\mu }_{n-1},{\mu }_n)\right)+\frac{1}{3}{\psi }_A\left(d_A({\mu }_n,{\mu }_{n+1})\right).\hfill \end{array}$$

Thus, we have

$$\begin{array}{ccc}\hfill (1-\frac{1}{3}{\psi }_A)\left(d_A({\mu }_n,{\mu }_{n+1})\right)& ⪯& \frac{2}{3}{\psi }_A\left(d_A({\mu }_{n-1},{\mu }_n)\right).\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}\hfill d_A({\mu }_n,{\mu }_{n+1})& ⪯& \frac{2}{3}{\psi }_A{(1-\frac{1}{3}{\psi }_A)}^{-1}\left(d_A({\mu }_{n-1},{\mu }_n)\right).\hfill \end{array}$$

Putting ${\varphi }_A=\frac{2}{3}{\psi }_A{(1-\frac{1}{3}{\psi }_A)}^{-1}$, we obtain

$$\begin{array}{c}\hfill d_A({\mu }_n,{\mu }_{n+1})⪯{\varphi }_A^n\left(d_A({\mu }_0,{\mu }_1)\right)\end{array}$$

for $m\ge n$. Thus, we obtain

$$\begin{array}{ccc}\hfill d_A({\mu }_n,{\mu }_m)& ⪯& \sum _{k=n}^{m-1}{\varphi }_A^k\left(d_A({\mu }_0,{\mu }_1)\right)\hfill \\ & \to & 0as(n\to +\infty ).\hfill \end{array}$$

Thus, $\left\{{\mu }_n\right\}$ is a Cauchy sequence in $\Omega$ with respect to $(\Omega ,A,d_A)$.

Because $(\Omega ,A,d_A)$ is a complete $C^{*}$-algebra-valued metric space, we conclude that $\left\{{\mu }_n\right\}$ is a convergence sequence, and so $\left\{{\mu }_n\right\}\to \mu$ as $n\to +\infty$ and $T\mu =\mu$ as $n\to +\infty$. Therefore, $\mu$ is a fixed point of T.

To prove the uniqueness, we suppose that $(\nu \ne \mu )$ is another fixed point of T. Thus,

$$\begin{array}{ccc}\hfill 0_A⪯d_A(\mu ,\nu )& =& d_A(T\mu ,T\nu )\hfill \\ & ⪯& {\alpha }_A(\mu ,\nu ){\psi }_A\left(d_A(T\mu ,T\nu )\right)\hfill \\ & ⪯& \frac{1}{3}{\psi }_A(d_A(\mu ,\nu )+d_A(T\mu ,\nu )+d_A(T\nu ,\mu ))\hfill \\ & ⪯& {\displaystyle \frac{1}{3}}{\psi }_A(d_A(\mu ,\nu )+d_A(\mu ,\nu )+d_A(\mu ,\nu ))\hfill \\ & ⪯& {\psi }_A\left(d_A(\mu ,\nu )\right)\prec d_A(\mu ,\nu ).\hfill \end{array}$$

This is a contradiction, so $d_A(\mu ,\nu )=0_A$ and $\mu =\nu$. □

Corollary 2.

Let $(\Omega ,d)$ be a complete real-valued metric space. Suppose $T:\Omega \to \Omega$ satisfies for all $\mu ,\nu \in \Omega$

$$d(T\mu ,T\nu )\le k\left(d\right(\mu ,\nu )+d(T\mu ,\nu )+d(T\nu ,\mu \left)\right),$$

where $k\in (0,\frac{1}{3})$. Then, T has a unique fixed point in Ω.

Proof. 

This is an immediate consequence of Theorem 2, with $\mathbb{A}=\mathbb{R}$ and ${\alpha }_A(\mu ,\nu )=I$ and ${\psi }_A\left(t\right)=kt,$$t\in \mathbb{R}$. □

Theorem 3.

(Ćirić Contraction Type) Let $(\Omega ,A,d_A)$ be a complete $C^{*}$-algebra-valued metric space and $T:\Omega \to \Omega$ be a mapping satisfying

$$\begin{array}{cc}\hfill {\alpha }_A(\mu ,\nu )d_A(T\mu ,T\nu )⪯& {\psi }_A\left(M_A(\mu ,\nu )\right)\end{array}$$

(10)

$$\begin{array}{ccc}\hfill M_A(\mu ,\nu )& =& {\displaystyle \frac{I_A}{3}}[d_A(\mu ,\nu )+(d_A(T\mu ,\mu )+d_A(T\nu ,\nu ))+(d_A(T\mu ,\nu )+d_A(T\nu ,\mu )],{\psi }_A\prec \frac{1}{2}.I_A\hfill \end{array}$$

$for\mu ,\nu \in \Omega$, where the following conditions hold:

(i) T is ${\alpha }_A$-admissible;

(ii) there exists ${\mu }_0\in \Omega$, such that ${\alpha }_A({\mu }_0,T{\mu }_0)⪰I_A$; and

(iii) T is continuous.

Then, T has a fixed point in Ω.

Proof. 

Following the first part of the proof in the Theorem 1, we obtain

$$\begin{array}{c}\hfill {\alpha }_A({\mu }_n,{\mu }_{n+1})⪰I_{A}foralln\in \mathbb{N}.\end{array}$$

(11)

By using (10) and (11), we have

$$\begin{array}{ccc}\hfill d_A({\mu }_n,{\mu }_{n+1})& =& d_A(T{\mu }_{n-1},T{\mu }_n)\hfill \\ & ⪯& {\alpha }_A({\mu }_{n-1},{\mu }_n)d_A(T{\mu }_{n-1},T{\mu }_n)\hfill \\ & ⪯& {\psi }_A\left(M_A({\mu }_{n-1},{\mu }_n)\right).\hfill \end{array}$$

(12)

On the other hand, we have

$$\begin{array}{ccc}\hfill M_A({\mu }_{n-1},{\mu }_n)& =& \frac{1}{3}(d_A({\mu }_{n-1},{\mu }_n)+d_A(T{\mu }_{n-1},{\mu }_{n-1})+d_A(T{\mu }_n,{\mu }_n)\hfill \\ & +& d_A(T{\mu }_{n-1},{\mu }_n)+d_A(T{\mu }_n,{\mu }_{n-1}))·I_A\hfill \\ \hfill So,M_A({\mu }_{n-1},{\mu }_n)& =& \frac{1}{3}{I}_A(d_A({\mu }_n,{\mu }_{n-1})+d_A({\mu }_n,{\mu }_{n-1})+d_A({\mu }_n,{\mu }_{n+1})\hfill \\ & +& d_A({\mu }_n,{\mu }_n)+d_A({\mu }_{n+1},{\mu }_{n-1})).\hfill \end{array}$$

Because $d_A(\mu ,\mu )=0$, we obtain

$$\begin{array}{c}\hfill M_A({\mu }_{n-1},{\mu }_n)⪯\frac{1}{3}{I}_A(d_A({\mu }_n,{\mu }_{n-1})+d_A({\mu }_n,{\mu }_{n-1})+d_A({\mu }_n,{\mu }_{n+1})+d_A({\mu }_{n+1},{\mu }_{n-1})).\end{array}$$

$$\begin{array}{c}\hfill So,M_A({\mu }_{n-1},{\mu }_n)⪯\frac{1}{3}{I}_A(2d_A({\mu }_n,{\mu }_{n-1})+d_A({\mu }_n,{\mu }_{n+1})+d_A({\mu }_{n+1},{\mu }_{n-1}))).\end{array}$$

By using triangular inequality, we obtain

$$\begin{array}{ccc}\hfill d_A({\mu }_n,{\mu }_{n+1})& ⪯& \frac{1}{3}{\psi }_A{I}_A[2d_A({\mu }_n,{\mu }_{n-1})+d_A({\mu }_n,{\mu }_{n+1})+d_A({\mu }_n,{\mu }_{n+1})+d_A({\mu }_n,{\mu }_{n-1})].\hfill \\ \hfill d_A({\mu }_n,{\mu }_{n+1})& ⪯& \frac{1}{3}{\psi }_A{I}_A[3d_A({\mu }_{n-1},{\mu }_n)+2d_A({\mu }_n,{\mu }_{n+1})].\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill (1-\frac{2}{3}{\psi }_A)\left(d_A({\mu }_n,{\mu }_{n+1})\right)& ⪯& {\psi }_A{I}_A\left(d_A({\mu }_{n-1},{\mu }_n)\right)\hfill \\ \hfill d_A({\mu }_n,{\mu }_{n+1})& ⪯& {\psi }_A{(1-\frac{2}{3}{\psi }_A)}^{-1}{I}_A\left(d_A({\mu }_{n-1},{\mu }_n)\right).\hfill \end{array}$$

Putting ${\varphi }_A={\psi }_A{(1-\frac{2}{3}{\psi }_A)}^{-1}$, $\parallel {\psi }_A\parallel <\frac{1}{2}$; then, we obtain

$$\begin{array}{c}\hfill d_A({\mu }_n,{\mu }_{n+1})⪯{\varphi }_A^n\left(d_A({\mu }_0,{\mu }_1)\right).\end{array}$$

(13)

Let $n,m\in \mathbb{N}$, such that $m>n$. We thus obtain

$$\begin{array}{ccc}\hfill d_A({\mu }_n,{\mu }_m)& ⪯& \sum _{k=n}^{m-1}{\varphi }_A^k\left(d_A({\mu }_0,{\mu }_1)\right)\hfill \\ & \to & 0as(n\to +\infty ).\hfill \end{array}$$

Thus, $\left\{{\mu }_n\right\}$ is a Cauchy sequence and ${\mu }_n\to \mu$ as $n\to +\infty$. Thus, we obtain $T\mu =\mu$ as a fixed point of T.

To prove the uniqueness, we suppose that $(\nu \ne \mu )$ is another fixed point of T. Thus,

$$\begin{array}{ccc}\hfill 0_A⪯d_A(\mu ,\nu )& =& d_A(T\mu ,T\nu )\hfill \\ & ⪯& {\alpha }_A(\mu ,\nu )d_A(T\mu ,T\nu )\hfill \\ & ⪯& \frac{1}{3}{\psi }_A(d_A(\mu ,\nu )+d_A(T\mu ,\nu )+d_A(T\nu ,\mu )+d_A(T\mu ,\mu )+d_A(T\nu ,\nu ))·I_A\hfill \\ & =& {\displaystyle \frac{1}{3}}{\psi }_A(d_A(\mu ,\nu )+d_A(\mu ,\nu )+d_A(\nu ,\mu )+d_A(\mu ,\mu )+d_A(\nu ,\nu ))·I_A\hfill \\ & =& {\displaystyle \frac{1}{3}}{\psi }_A\left(3d_A(\mu ,\nu )\right)·I_A,\hfill \\ \hfill so,0_A⪯d_A(\mu ,\nu )& ⪯& {\psi }_A\left(d_A(\mu ,\nu )\right).\hfill \end{array}$$

Because ${\psi }_A\left(a\right)\prec a$, this implies that $0⪯d_A(\mu ,\nu )\prec d_A(\mu ,\nu ),$, which gives a contradiction. Then, we obtain $\mu =\nu$. □

Example 4.

Let Ω be a Banach space and $d_A:\Omega \times \Omega \to A$ be defined as $d_A(\mu ,\nu )=\parallel \mu -\nu \parallel ·I_A$ for all $\mu ,\nu \in \Omega$. $I_A$ is the unit of A because Ω is a Banach space. Then, $(\Omega ,A,d_A)$ is a complete $C^{*}$-algebra-valued metric space. Define $T:\Omega \to \Omega$ as $T\mu =2\mu$ and define ${\psi }_A:A^{+}\to A^{+}$ as ${\psi }_A\left(a\right)=3a{I}_A$ for all $a\in A^{+}$, where $A^{+}$ is the positive cone of A. Additionally, ${\alpha }_A:\Omega \times \Omega \to A^{+}$ is defined by ${\alpha }_A(\mu ,\nu )=I_A$, where

$$\begin{array}{c}\hfill {\alpha }_A(T\mu ,T\nu )={\alpha }_A(2\mu ,2\nu )=2{\alpha }_A(\mu ,\nu )=2I_A⪰I_A.\end{array}$$

Now,

$$\begin{array}{ccc}\hfill d_A(T\mu ,T\nu )& =& \parallel T\mu -T\nu \parallel ·I_A=\parallel 2\mu -2\nu \parallel ·I_A\hfill \\ & =& \parallel 2\mu -2\nu +\nu -\nu +\mu -\mu \parallel ·I_A\hfill \\ & =& \parallel (2\mu -\nu )-(2\nu -\mu )-(\mu -\nu )\parallel ·I_A\hfill \\ & ⪯& (\parallel 2\mu -\nu \parallel +\parallel 2\nu -\mu \parallel +\parallel \mu -\nu \parallel )·I_A\hfill \\ & ⪯& (\parallel T\mu -\nu \parallel +\parallel T\nu -\mu \parallel +\parallel \mu -\nu \parallel )·I_A\hfill \\ & =& (d_A(T\mu ,\nu )+d_A(T\nu ,\mu )+d_A(\mu ,\nu ))\hfill \\ & ⪯& \frac{1}{3}{\psi }_A(d_A(T\mu ,\nu )+d_A(T\nu ,\mu )+d_A(\mu ,\nu )).\hfill \end{array}$$

Applying ${\alpha }_A(\mu ,\nu )$, we obtain

$$\begin{array}{ccc}\hfill {\alpha }_A(\mu ,\nu )d_A(T\mu ,T\nu )& ⪯& \frac{1}{3}{\psi }_A(d_A(T\mu ,\nu )+d_A(T\nu ,\mu )+d_A(\mu ,\nu )).\hfill \end{array}$$

This satisfies the conditions in Theorem 2. Then, T has a fixed point of Ω.

We introduce a numerical example, assuming that the metric space is valued-non-commutative $C^{*}$-algebra $M_2\left(\mathbb{R}\right)$

Example 5.

Let $\Omega =\mathbb{R}$ and $A=M_2\left(\mathbb{R}\right)$, where $M_2\left(\mathbb{R}\right)$ is the set of all $2\times 2$ matrices entries in $\mathbb{R}$. It is obvious that $M_2\left(\mathbb{R}\right)$ is a $C^{*}$-algebra with matrix norm and involution $*:M_2\left(\mathbb{R}\right)\to M_2\left(\mathbb{R}\right)$ given by $*:a\to a^t$, where $a^t$ is the transpose of a, $a\in M_2\left(\mathbb{R}\right)$. Define

$$\begin{array}{c}\hfill d_A(\mu ,\nu )=\left(\begin{array}{cc}|\mu -\nu |& 0\\ 0& k|\mu -\nu |\end{array}\right),\end{array}$$

for all $\mu ,\nu \in \Omega$, $k>0$. It is clear that $(\Omega ,A,d_A)$ is $C^{*}$-algebra-valued metric space. To verify the contraction conditions in Theorem 3, we take $\mu =1$, $\nu =2$, $k=3$.

Additionally, we define $T:\Omega \to \Omega$ by $T\left(\mu \right)=2\mu$ and ${\alpha }_A:\Omega \times \Omega \to M_2{\left(\mathbb{R}\right)}^{+}$ by

$$\begin{array}{c}\hfill {\alpha }_A(\mu ,\nu )=2\left(\begin{array}{cc}|\mu -\nu |& 0\\ 0& |\mu -\nu |\end{array}\right),\end{array}$$

and ${\psi }_A:M_2{\left(\mathbb{R}\right)}^{+}\to M_2{\left(\mathbb{R}\right)}^{+}$, by ${\psi }_A\left(a\right)=3a$, for $a\in M_2{\left(\mathbb{R}\right)}^{+}$, $\mu ,\nu \in Z$, where $M_2{\left(\mathbb{R}\right)}^{+}$ is the set of positive matrices of $M_2\left(\mathbb{R}\right)$.

Now, by simple calculation, we obtain

$$\begin{array}{c}\hfill d_A(\mu ,\nu )=d_A(1,2)=\left(\begin{array}{cc}1& 0\\ 0& 3\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill d_A(T\mu ,T\nu )=d_A(2,4)=\left(\begin{array}{cc}2& 0\\ 0& 6\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill d_A(T\mu ,\nu )=d_A(2,2)=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill d_A(T\nu ,\mu )=d_A(4,1)=\left(\begin{array}{cc}3& 0\\ 0& 9\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill d_A(T\mu ,\mu )=d_A(4,1)=\left(\begin{array}{cc}1& 0\\ 0& 3\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill d_A(T\nu ,\nu )=d_A(4,1)=\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\alpha }_A(\mu ,\nu )=\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right).\end{array}$$

Thus, we calculate the right hand side of the inequality (10) in Theorem 3 as

$$\begin{array}{ccc}\hfill M_A(\mu ,\nu )& =& \frac{1}{3}(d_A(\mu ,\nu )+d_A(T\mu ,\nu )+d_A(T\nu ,\mu )+d_A(T\mu ,\mu )+d_A(T\nu ,\nu ))\hfill \\ & =& \frac{1}{3}\left(\begin{array}{cc}7& 0\\ 0& 21\end{array}\right).\hfill \end{array}$$

Therefore, $\psi (M_A(\mu ,\nu )=\left(\begin{array}{cc}7& 0\\ 0& 21\end{array}\right)$.

On the other hand, the left hand side of the inequality (10) in Theorem 3 is given by ${\alpha }_A(\mu ,\nu )d_A(T\mu ,T\nu )=\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right)·\left(\begin{array}{cc}2& 0\\ 0& 6\end{array}\right)=\left(\begin{array}{cc}4& 0\\ 0& 12\end{array}\right)$.

Hence, it is obvious that T is ${\alpha }_A-{\psi }_A$-admissible and, because $\left(\begin{array}{cc}2& 0\\ 0& 12\end{array}\right)\le \left(\begin{array}{cc}7& 0\\ 0& 21\end{array}\right),$ we can obtain

$${\alpha }_A(\mu ,\nu )d_A(T\mu ,T\nu )\le {\psi }_A\left(M_2(\mu ,\nu )\right).$$

Thus, all conditions of Theorem 3 are satisfied. Therefore, there exists a unique fixed point of T, and the zero matrix is the fixed point of $T\in \Omega$.

We discuss a numerical example that satisfies the conditions of Theorem 3, where the metric space in this example is valued-commutative $C^{*}$-algebra ${\mathbb{C}}^2.$

Example 6.

Let $\Omega =[0,\infty )$ and $A={\mathbb{C}}^2=\mathbb{C}\oplus \mathbb{C}$, the set of direct sum of two copies of complex numbers. ${\mathbb{C}}^2$ with the vector addition and pointwise multiplication defined by $(Z_1,Z_2)+(W_1,W_2)=(Z_1+W_1,Z_2+W_2)$, and $(Z_1,Z_2)·(W_1,W_2)=(Z_1·W_1,Z_2·W_2)$, for all $Z_1,Z_2,W_1,W_2\in \mathbb{C}$, is a $C^{*}$-algebra with the maximum norm given by $\parallel (Z_1,Z_2)\parallel =max\left\{\right|Z_1|,|Z_2\left|\right\}$, and involution $*:{\mathbb{C}}^2\to {\mathbb{C}}^2$ given by ${(Z_1,Z_2)}^{*}=(\overline{Z_1},\overline{Z_2})$, for all $Z_1,Z_2\in \mathbb{C}$. Define a partial order ⪯ on ${\mathbb{C}}^2:(Z_1,Z_2)⪯(W_1,W_2)$ if and only if

(a) $Re\left(Z_1\right)\le Re\left(W_1\right)$, Im $W_1\le$ Im $W_1$, and

(b) $Re\left(Z_2\right)\le Re\left(W_2\right)$, Im $W_2\le$ Im $W_2$.

Thus, $(W_1,W_2)-(Z_1,Z_2)⪰0$ iff $(Z_1,Z_2)⪯(W_1,W_2)$. Additionally, $(Z_1,Z_2)⪰0$ if $Z_1⪰0$ and $Z_2⪰0$. In addition, $Re\left(Z_1\right)\ge 0$, Im$Z_1\ge 0$ and $Re\left(Z_2\right)\ge 0$, Im$Z_2\ge 0$

Let ${\mathbb{C}}_{+}^2$ be the set of all positive element in ${\mathbb{C}}^2$. Suppose $\Omega =[0,\infty )$ and $d_A:\Omega \times \Omega \to {\mathbb{C}}^2$ be a mapping defined by $d_A\left(\right|\mu -\nu |+i|\mu -\nu |,|\mu -\nu |+2i|\mu -\nu \left|\right)$ for all $\mu ,\nu \in \Omega$ and $i=\sqrt{-1}$.

It is clear that $(\Omega ,A,d_A)$ is $C^{*}$-algebra-valued metric space.

Now, define $T:\Omega \to \Omega$ by $T\mu =e^{\mu }$ and ${\alpha }_A:\Omega \times \Omega \to {\mathbb{C}}_{+}^2$ as ${\alpha }_A(\mu ,\nu )=I_A$. In addition, assume ${\psi }_A:{\mathbb{C}}_{+}^2\to {\mathbb{C}}_{+}^2$ defined by ${\psi }_A\left(a\right)=3a$∀ a $\in {\mathbb{C}}_{+}^2$.

To verify the contraction conditions in Theorem 3, we take $\mu =1$, $\nu =2$. By calculation, one can obtain the following:

$$\begin{array}{ccc}\hfill d_A(\mu ,\nu )& =& d(1,2)=(1+i,1+2i),\hfill \\ \hfill d_A(T\mu ,T\nu )& =& d_A(e,e^2),\hfill \\ & \simeq & (4.670+4.670i,4.670+9.340i),\hfill \\ \hfill d_A(T\mu ,\nu )& =& d_A(e,2),\hfill \\ & \simeq & (0.718+0.718i,0.718+1.436i),\hfill \\ \hfill d_A(T\nu ,\mu )& =& d_A(e^2,1),\hfill \\ & \simeq & (6.389+6.389i,6.389+12.778i),\hfill \\ \hfill d_A(T\mu ,\mu )& =& d_A(e,1),\hfill \\ & \simeq & (1.718+1.718i,1.718+3.436i),\hfill \\ \hfill d_A(T\nu ,\nu )& =& d_A(e^2,2),\hfill \\ & \simeq & (5.389+5.389i,5.389+10.778i),\hfill \\ \hfill {\alpha }_A(\mu ,\nu )& =& {\alpha }_A(1,2)=(1,2).\hfill \end{array}$$

We calculate the right-hand side of the inequality (10) in the Theorem 3 and obtain

$${\psi }_A\left(M_A(\mu ,\nu )\right)\simeq (15.214+15.214i,15.214+30.428i).$$

On the other hand, the left-hand side of the inequality (10) in the Theorem 3 gives

$${\alpha }_A(\mu ,\nu )d_A(T\mu ,T\nu )\simeq (4.670+4.670i,4.670+18.680i).$$

It is clear that ${\alpha }_A(\mu ,\nu )d_A(T\mu ,T\nu )\le {\psi }_A\left(M_A(\mu ,\nu )\right)$, and this satisfies all conditions of the Theorem 3.

In the following, we provide an application scenario with which to study the existence and uniqueness of the solution of a system of matrix equations. The existence and uniqueness of the solution of the linear matrix equations are very interesting and important in linear systems.

Here, we are interested in using $C^{*}$-algebra-valued metric spaces to find a positive definite hermitian solution for a system of matrix equations with complex entries.

The proof is based on the positive cones and the linear continuous operator mapping a cone into itself.

4. Application

Suppose that $M_n\left(\mathbb{C}\right)$ is the set of all $n\times n$ matrices with complex entries. Additionally, $M_n{\left(\mathbb{C}\right)}^{+}$ is the set of all positive definite matrices of $M_n\left(\mathbb{C}\right)$. $M_n\left(\mathbb{C}\right)$ is a Banach space with matrix norm and $M_n\left(\mathbb{C}\right)$ is also a $C^{*}$-algebra with matrix norm and the involution $*:M_n\left(\mathbb{C}\right)\to M_n\left(\mathbb{C}\right)$, $Z\to Z^{*}$, where $Z={\left(Z_{ij}\right)}_{1\le i,j\le n}\in M_n\left(\mathbb{C}\right)$ and $Z^{*}={\left(Z_{ij}\right)}_{1\le i,j\le n}^{*}={\left(\overline{Z_{ji}}\right)}_{1\le i,j\le n}\in M_n\left(\mathbb{C}\right).$

Let $A_1,A_2,\dots ,A_n\in M_n\left(\mathbb{C}\right)$, $Z,W\in M_n\left(\mathbb{C}\right)$. Additionally, $Q\in M_n{\left(\mathbb{C}\right)}^{+}$. Then, the matrix equation

$$\begin{array}{c}\hfill Z-\sum _{k=1}^n{A}_k^{*}Z{A}_k=Q\end{array}$$

(14)

has a unique solution.

Proof. 

For $Z,W\in M_n\left(\mathbb{C}\right)$, define $d_{M_n\left(\mathbb{C}\right)}:M_n\left(\mathbb{C}\right)\times M_n\left(\mathbb{C}\right)\to M_n\left(\mathbb{C}\right)$, as

$$d_{M_n\left(\mathbb{C}\right)}(Z,W)=\parallel Z-W\parallel ·I_{d_{M_n\left(\mathbb{C}\right)}}.$$

Then, $(M_n\left(\mathbb{C}\right),d_{M_n\left(\mathbb{C}\right)})$ is a $C^{*}$-algebra-valued metric space and is complete, because the set $M_n\left(\mathbb{C}\right)$ is complete. Consider $T:M_n\left(\mathbb{C}\right)\to M_n\left(\mathbb{C}\right)$, defined by $T\left(Z\right)=\sum _{k=1}^n{A}_k^{*}Z{A}_k+Q$. Additionally, ${\psi }_{M_n\left(\mathbb{C}\right)}\left(Z\right)=3Z·I_{M_n\left(\mathbb{C}\right)}.$ Define

$${\alpha }_{M_n\left(\mathbb{C}\right)}:M_n\left(\mathbb{C}\right)\times M_n\left(\mathbb{C}\right)\to M_n{\left(\mathbb{C}\right)}^{+}$$

$${\alpha }_{M_n\left(\mathbb{C}\right)}(Z,W)=I_{M_n\left(\mathbb{C}\right)}.$$

It is clear that T is ${\alpha }_{M_n\left(\mathbb{C}\right)}-{\psi }_{M_n\left(\mathbb{C}\right)}$ admissible. Then,

$$\begin{array}{ccc}\hfill d_{M_n\left(\mathbb{C}\right)}(TZ,TW)& =& \parallel TZ-TW\parallel ·I_{M_n\left(\mathbb{C}\right)}\hfill \\ & =& \parallel (\sum _{k=1}^n{A}_k^{*}Z{A}_k+Q)-(\sum _{k=1}^n{A}_k^{*}W{A}_k+Q)\Vert ·I_{M_n\left(\mathbb{C}\right)}\hfill \\ & =& \Vert ((\sum _{k=1}^n{A}_k^{*}Z{A}_k+Q)-W)-((\sum _{k=1}^n{A}_k^{*}W{A}_k+Q)-Z)-(Z-W)\Vert ·I_{M_n(\mathbb{C})}\hfill \\ & ⪯& \Vert ((\sum _{k=1}^n{A}_k^{*}Z{A}_k+Q)-W)\Vert ·I_{M_n\left(\mathbb{C}\right)}+((\sum _{k=1}^n{A}_k^{*}W{A}_k+Q)-Z)\parallel ·\parallel I_{M_n(\mathbb{C})}\hfill \\ & +& \Vert (Z-W)\parallel ·I_{M_n(\mathbb{C})}\hfill \\ & =& \parallel TZ-W\parallel ·I_{M_n(\mathbb{C})}+\parallel TW-Z\parallel ·I_{M_n(\mathbb{C})}+\parallel (Z-W)\parallel ·I_{M_n(\mathbb{C})}\hfill \\ & =& d_{M_n(\mathbb{C})}(TZ,W)+d_{M_n(\mathbb{C})}(TW,Z)+d_{M_n(\mathbb{C})}(Z,W)\hfill \\ & ⪯& \frac{1}{3}{\psi }_{M_n\left(\mathbb{C}\right)}(d_{M_n\left(\mathbb{C}\right)}(TZ,W)+d_{M_n\left(\mathbb{C}\right)}(TW,Z)+d_{M_n\left(\mathbb{C}\right)}(Z,W)).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill {\alpha }_A(Z,W)d_{M_n\left(\mathbb{C}\right)}(TZ,TW)& ⪯& \frac{1}{3}{\psi }_{M_n\left(\mathbb{C}\right)}(d_{M_n\left(\mathbb{C}\right)}(TZ,W)+d_{M_n\left(\mathbb{C}\right)}(TW,Z)+d_{M_n\left(\mathbb{C}\right)}(Z,W)).\hfill \end{array}$$

This satisfies the conditions of Theorem 2. Thus, the system of matrix Equation (14) has a unique hermitian matrix solution. □

5. Conclusions

In this paper, we provide some results obtained for the Chatterjea and $\stackrel{`}{\mathrm{C}}$iri$\stackrel{`}{\mathrm{c}}$ fixed-point theorems by using ${\alpha }_A$-${\psi }_A$-contractive mapping in a $C^{*}$-algebra-valued metric space. Furthermore, illustrated examples and an application scenario are introduced. It is worth mentioning that these results generalize and extend some results described in [1,2,3,5,9,23,24,26,27,28,29,30].