$\alpha$-Admissible mapping in $C^{*}$-algebra-valued b-metric spaces and fixed point theorems

1.   Introduction

In 2014, Ma et al ^[18] introduced the concept of $C^*$-algebra-valued metric spaces by replacing the range of $\mathbb{R}$ with an unital $C^*$-algebra. Later in 2015, Ma et al ^[19] introduced the nation of $C^*$-algebra-valued metric spaces as a generalization of $C^*$-algebra-valued metric space. They proved some Banach fixed point theorems. Several research are obtained some results in Banach and common fixed point theorems in $C^*$-algebra-valued metric spaces (see ^{[2,3,7,10,13,14,25,26,31,34]}. The notion of $C^*$-algebra-valued partial metric space and $C^*$-algebra-valued partial b-metric spaces are introduced in ^[8,22] and proved fixed point results as analogous of Banach contraction principle.

In ^[27] introduced the study of fixed point for the $\alpha$-admissibility of mappings and generalized several known results of metric spaces see also ^[28]. Later on, many authors proved $\alpha$-admissible mappings theorems with various contraction condition see ^{[1,5,9,12,17,20,21,30,32,33,35,36]}. The aim of this paper is generalizing some results of metric spaces and $C^*$-algebra b-valued metric spaces.

We start with some definition and results about $C^*$-algebra b-valued metric spaces. Suppose that $A$ is a unital $C^*$-algebra with a unit $I$. Set $A_h = \{x \in A : x = x^*\}$. An element $x \in A$ is a positive element, if $x = x^*$ and $\sigma(x) \subset \mathbb{R}^+$ is the spectrum of $x$. We define a partial ordering $\preceq$ on $A_h$ as $x \preceq y$ if $0_A \preceq y-x$, where $0_A$ means the zero element in $A$ and we let $A^+$ denote the $\{x \in A : x \succeq 0_A \}$ and $\vert x \vert = (x^* x)^{\frac{1}{2}}$.

On the other hand, ^[27] introduced the study of fixed point for the $\alpha$-admissibility of mappings and generalized several know results of metric spaces.

Throughout this paper, we use the concept of $\alpha$-admissibility of mappings defined on $C^*$-algebra b-valued metric spaces and we defined the generalized Lipschitz contractions on such spaces. The aim of this paper is generalizing some results of metric space and $C^*$-algebra b-valued metric spaces.

Lemma 1.1. Suppose that $A$ is a unital $C^*$-algebra with unit $I_A$. The following are holds.

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

(2) For any $x \in A$ and $a, b \in A^+$, such that $a \preceq b$, we have $x^* a x$ and $x^* b x$ are positive element and $x^* a x \preceq x^* b x$.

(3) If $0_A \preceq a \preceq b$ then $\Vert a\Vert \leq \Vert b \Vert$.

(4) If $a, b \in A^+$ and $ab = ba$, then $a.b \succeq 0_A$.

(5) Let $A^{\prime}$ denote the set $\{ a \in A : ab = ba \, \, \, \forall \, \, b \in A \}$ and let $a \in A^{\prime}$, if $b, c \in A$ with $b \succeq c \succeq 0_A$ and $1-a \in (A^{\prime})^{+}$ is an invertible element, then $(I_A - a)^{-1}b \prec (I_A - a)^{-1}c$.

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

Definition 1.2. Let $X$ be a non-empty set and $b \succeq I_A$, $b \in A^{\prime }$, suppose the mapping

$d_A : X\times X \rightarrow A$, satisfies:

(1) $d_A(x, y) \succeq 0_A$ for all $x, y \in X$ and $d_A(x, y) = 0_A \Leftrightarrow x = y$.

(2) $d_A(x, y) = d_A(y, x)$ for all $x, y \in X$.

(3) $d_A(x, z) \preceq b[d_A(x, y) + d_A(y, z)]$ for all $x, y, z \in X$, where $0_A$ is zero-element in $A$ and $I_A$ is the unit element in $A$. Then $d_A$ is called a $C^*$-algebra valued b-metric on $X$ and $(X, A, d_A)$ is called $C^{*}$-algebra- valued b-metric space.

Example 1.3. Let $X$ be a Banach space, $d_A : X \times X \rightarrow A$ given by $d_A(x, y) = \Vert x-y \Vert ^p \cdot a$, for all $x, y \in X$, $a \in A^+$, $a \succeq 0$ and $p > 1$.

Its easy to variety that $(X, A, d_A)$ is a $C^*$-algebra -valued b-metric space.

Using the inequality $(a+b)^p \leq 2^p (a^p +b^p)$ for all $a, b \geq 0$, $p > 1$, we have

for $x, y, z \in X$, which implies that

In the next we give a counter example, show that in general, a $C^*$-algebra valued b-metric space in not necessary a $C^*$-algebra valued metric space.

Example 1.4. Let $X = \mathbb{R}$ and $A = M_2 (\mathbb{R})$. Define

$x, y \in \mathbb{R}$, $k > 0$, it is clear that $(X, A, d_A)$ is a $C^*$-algebra valued b-metric space by using the same argument in example 1.3 when $p = 2$

Now, $d_A(0, 1) = \begin{pmatrix} 1 & 0\\ 0 & k.1 \end{pmatrix}$, $d_A(1, 2) = \begin{pmatrix} 1 & 0\\ 0 & k.1 \end{pmatrix}$, $d_A(0, 2) = \begin{pmatrix} 4 & 0\\ 0 & k.4 \end{pmatrix}$.

Its obvious that $d_A(0, 2) \succeq d_A(0, 1) + d_A(1, 2)$.

So $(X, A, d_A)$ is not a $C^*$-algebra valued metric space

Definition 1.5. Let $(X, A, d_A)$ be a $C^{*}$-algebra- valued b-metric space, $x\in X$, and $\{x_n \}^\infty _{n = 1}$ be a sequence in $X$, then

(i) $\{x_n \}^\infty _{n = 1}$ convergent to $x$ whenever, for every $c \in A$ with $c \succ 0_A$, there is a natural number $N \in \mathbb{N}$ such that

for all $n > N$. We denote this by $\lim \limits _{n \rightarrow \infty} x_n = x$ or $x_n \rightarrow x$ as $n \rightarrow +\infty$.

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

for all $n, m > N$.

Lemma 1.6. (i) $\{x_n \}^\infty _{n = 1}$ is a convergence in $X$. If for any element $\epsilon > 0$ there is $N \in \mathbb{N}$ such that for all $n > N$, $\Vert d(x_n, x)\Vert \leq \epsilon$.

(ii) $\{x_n \}^\infty _{n = 1}$ is a Cauchy sequence in $X$, for any $\epsilon > 0$ there $N \in \mathbb{N}$ such that

$\Vert d_A(x_n, x_m)\Vert \leq \epsilon$, for all $n, m > N$. We say that $(X, A, d_A)$ is a complete $C^{*}$-algebra- valued b-metric space if every Cauchy sequence is convergent with respect to $A$.

Example 1.7. Let $X = \mathbb{R}$ and $A = M_n(\mathbb{R})$ the set of all $n \times n$-matrices with entries in $\mathbb{R}$. Define

where $a = (a_{ij})^{n}_{i, j = 1}$, $b = (b_{ij})^{n}_{i, j = 1}$ are two $n\times n$-matrices, $a_{ij}, b_{ij} \in \mathbb{R}$ for all $i, j = 1, ..., n, \, \, \lambda_i \geq 0$ for $i = 1, ..., n$ are positive real numbers.

One can define a partial ordering on ($\preceq _{M_n(\mathbb{R})})$ on $M_n(\mathbb{R})$ as following $a \preceq _{M_n(\mathbb{R})} b$ if and only if $a_{ij} \leq b_{ij} \forall i, j = 1, ..., n$. And an element $a \succeq_{M_n(\mathbb{R})} 0$ is positive in $M_n(\mathbb{R})$ if and only if $a_{ij} \geq 0$ for all $i, j = 1, ..., n$. $(X, M_n(\mathbb{R}), d_{M_n(\mathbb{R})})$ is $C^{*}$-algebra- valued b-metric space.

One can prove that

for all $a, b, c \in M_n(\mathbb{R})$.

We need only to use the following inequality in $\mathbb{R}$

where $b = 2^p I_{M_{n}(\mathbb{R})} \succeq I_{M_{n}(\mathbb{R})}$ $\forall p \geq 1$, where $I_{M_{n}(\mathbb{R})}$is the unit element in $M_{n}(\mathbb{R}).$

Remark 1.8. In the above example the inequality $\vert x-z\vert^p \leq |x-y|^p + |y-z|^p$ it is impossible for $x > y > z$. Then the $(X, A, d)$ is not a $C^*$-algebra valued metric space.

It is useful to discuss the relation between $C^*$-algebra valued metric spaces and lattices-valued metric spaces. To classify $C^*$-algebra-valued-metric spaces and its relation with lattices, we have to discuss the concept of quantale which introduced by Mulvey ^[23]. A quantale $Q$ is a complete lattice together with an associative multiplication $\circledast : Q \times Q \rightarrow Q$ such that $a \circledast (\vee_i b_i) = \vee_i(a \circledast b_i)$, and $\vee_i(a_i) \circledast b = \vee_i(a_i \circledast b)$ for all $a_i, b_i, a, b \in Q$, $i \in I$, $I$ is an index set the quantale is said to be unit, if it has a unital 'e' satisfy $a \circledast e = e \circledast a$ for all $a \in Q$. And $Q$ is called an involuative quantale with relation $* :Q \rightarrow Q$ satisfy $(a^*)^* = a$, $(a \circledast b)^* = b^* \circledast a^*$ and $(\vee_i a_i)^* = \vee_i a^*_i$ for all $a_i, b_i, a \in Q$.

The top element of $Q$ is denoted by $1$ and the bottom element denoted by $0$. A typical example of quantale is given by $End(S)$, the set of all sublattices of Endomorphisms of the complete lattices S is a unital quantale with join calculated by point wise $(\vee_i f_i) (x) = \vee_if_i(x)$ and multiplication as composition $(f\circledast g)(x) = (f \circ y)(x)$.

And it is unit identity is $Id_S$. An element $a \in Q$ is said to be right-sided if $a \circledast 1 \leq a$, for all $a \in Q$, denote by $R(Q)$ the set of all right-sided elements. Similarly, an element $a \in Q$ is said to be left- side if $1\circledast a \leq a$ for all $a \in Q$, $L(Q)$ denote the set of left-sided elements.if $a\in Q$ is right-sided elements and left-sided elements it is said to be 2-sided elements and the set of 2-sided-elements denoted by $I(Q)$. Any two sided-elements a is distributive in the sense that $a \wedge \vee_i b_i = \vee_i(a \wedge b_i)$.

A quantale is commutative if it is commutative under the multiplication. If the quntale is commutative then $Q \cong I(Q)$. If $A$ is a $C^*$-algebra and $R(A)$ is the lattice of all closed right ideals of $A$, then $R(A)$ is a quantale and the multiplication of closed right ideals obtained by taking the topological closure of the usual product of ideals, simply, $I \circledast J = \overline{IJ}$ for any two ideals $I, J \in R(A)$.

By Gelfand duality theorem ^[11]. Any commutative $C^*$-algebra is isomorphic to the set of all continuous functions of the compact Hausdorf topological space. So, in this case $R(A)$ is isomorphic to the lattice $O(\widehat{A})$ of all open sublattices of $\widehat{A}$, where $\widehat{A}$ is the topological space determined by $A$, the spectrum of $A$. Therefore, commutative $C^*$-algebra classify by commutative quantales as given in ^[6]. $A$ is a commutative $C^*$-algebra if and only if $R(A)$ is commutative quantale.

On the other hand 'Sherman ^[29]' show that if $A_{sa}$ is the space of self-.adjoint elements of a $C^*$-algebra $A$ with the canonical order $\preceq$ given by $a \preceq b$ if and only if $b-a \succeq 0$ is positive. Then $A_{sa}$ is a lattice ordered if and only if $A$ is commutative. Therefore, the $C^*$-algebra valued metric space in commutative case coinside with the commutative quantale -valued-metric space, with a suitable metric For a non-commutative $C^*$-algebra $A$ with unit, by $Max A$ is meant. The set of all subspace of $A$ together with the multiplication defined by $M \circledast N = \overline{MN}$ to be the closure of product liner subspace, for each $M, N \in Max A$, and the join $“\vee''$ defined by $\vee_i M_i = \overline{\sum\limits_i M_i}$, and the involution $M^* = \{a^* : a \in M \}$ and the unit of $Max A$ is given by the identity. So, $Max A$ is defined $A$ unital involutive quantale. In the case the non-commutative $C^*$-algebra is classify by $Max A$, following ^[15,16]. If $A$ and $B$ are two unital $C^*$-algebras. Then $A$ and $B$ are isomorphic if and only if $Max A$ and $Max B$ are isomorphic as a unital involutive quantale. So, $C^*$-algebra valued metric spaces are classify by the unital involutive quantale-valued metric space.

2.   Main results

In 2012 Samet et al ^[27], introduced the concept of $\alpha$-admissible mapping as follows.

Definition 2.1. Let $T : X \rightarrow X$ be self map and $\alpha : X \times X\rightarrow [0, +\infty)$. Then $T$ is called $\alpha$-admissible if for all $x, y \in X$ with $\alpha (x, y) \geq 1$ implies $\alpha (Tx, Ty) \geq 1$.

Next, we introduced an analogue definition of $\alpha$-admissible for a unital $C^*$-algebra.

Definition 2.2. Let X be a non-empty set and $\alpha_{A}: X \times X \rightarrow (A^{\prime})^{+}$ be a function, we say that the self map $T$ is $\alpha_{A}$ - admissible if $(x, y) \in X \times X$, $\alpha_{A}(x, y) \succeq I_A$ $\Rightarrow \alpha_{A}(T x, Ty)\succeq I_A$, where $I_A$ the unity of $A$.

Definition 2.3. Let $(X, A, d)$ be a complete $C^{*}$-algebra- valued b-metric space, the mapping $T : X \rightarrow X$ is said to be generalised Lipschitz condition if there exist $a \in A$ such that $\Vert a\Vert < 1$ and

for all $x, y \in X$ with $\alpha_A(x, y) \succeq I_A$.

Example 2.4. Let $X = \mathbb{R}$ and $A = M_n(\mathbb{R})$ as given in example (1.7), define $T : X \rightarrow X$, by $Tx = \dfrac{x}{2}$, and $\alpha_{M_{n}(\mathbb{R})} : X\times X \rightarrow {M_{n}(\mathbb{R})}^+$, given by $\alpha_{M_{n}(\mathbb{R})}(x, y) = I_{M_{n}(\mathbb{R})}$ and $\alpha_{M_{n}(\mathbb{R})}(Tx, Ty) = \alpha_{M_{n}(\mathbb{R})}(\frac{x}{2}, \frac{y}{2}) = I_{M_{n}(\mathbb{R})}$ thus T is $\alpha_{M_{n}(\mathbb{R})}-$admissible, where ${M_{n}(\mathbb{R})^+}$ is the set of all positive elements

and $a = \dfrac{I_{M_{n}(\mathbb{R})}}{(\sqrt{2})^p}$, $a^* = \dfrac{I_{M_{n}(\mathbb{R})}}{(\sqrt{2})^p}$, so $T$ satisfy the generalised Lipschitz condition.

Theorem 2.5. Let $(X, A, d_A)$ be a complete $C^{*}$-algebra- valued b-metric space, with $b\succeq I_A$, $b \in A^\prime$, $\Vert b \Vert\Vert a \Vert^2 < 1$ suppose that $T : X \rightarrow X$, be a generalised Lipschitz contraction satisfies the following conditions:

(i) $T$ is $\alpha_{A}$-admissible.

(ii) There exists $x_{0} \in X$ such that $\alpha_{A}(x_0, Tx_0) \succeq I_A$.

(iii) $T$ is continuous.

Then $T$ has a fixed point.

Proof: let $x_{0} \in X$ such that $\alpha_{A}(x_0, Tx_0) \succeq I_A$ and define a sequence $\{x_n\}_{n = 0}^{\infty}\subseteq X$ such that $x_n = Tx_{n-1}$ for all $n \in \mathbb{N}$. If $x_n = x_{n+1}$ for some $n \in \mathbb{N}$, then $x = x_n$ is a fixed point for $T$.

Assume that $x_n \neq x_{n+1}$ for all $n \in \mathbb{N}$, since $T$ is $\alpha_A$-admissible, we have

By induction we get

Since $T$ is generalised Lipschitz condition, then

Denote that $d_0 : = d_A(x_0, x_1)$ in $A$, notice that in $C^*$-algebra, if $a, b \in A^+$ and $0_A \preceq a \preceq b$, then for any $x \in A$ both $x^*ax$ and $x^*bx$ are positive elements and

Now, for $m \geq 1$, $p \geq 1$ it following that

with the condition $\Vert b\Vert \Vert a \Vert ^2 < 1$ and at $m \rightarrow + \infty$.

It implies that $\lbrace x_n \rbrace^\infty _{n = 0}$ is a Cauchy sequence. By completeness of X that exists $x \in X$ such that $x_n \rightarrow x$ as $n \rightarrow +\infty$.

Since $T$ is continuous and the lim is unique it follows $x_{n+1} = Tx_n \rightarrow Tx$ as $n \rightarrow +\infty$ such that $x = \lim \limits_{n \rightarrow \infty} x_{n +1} = \lim \limits_{n \rightarrow \infty} Tx_n = Tx$, so, $Tx = x$ is a fixed point for T.

Now, we replace the assumption of continuoity of $T$ in the above theorem by another condition.

Theorem 2.6. Let $(X, A, d_A)$ be a complete $C^{*}$-algebra- valued b-metric space, with $b \succeq I_A$.

Let $T : X \rightarrow X$ be generalized Lipschitz condition as in (2.5) and the following conditions are satisfies:

(i) T is $\alpha_A$-admissible.

(ii) There exists $x_0 \in X$ such that $\alpha_A (x_0, Tx_0) \succeq I_A$.

(iii) If $\lbrace x_n \rbrace^\infty _{n = 0}$ is a sequence in X such that $\alpha_A(x_n, x_{n+1} \succeq I_A$ for all $n \in \mathbb{N}$ and $x_n \rightarrow x \in X$, as $n \rightarrow +\infty$, then $\alpha_A(x_n, x) \succeq I_A$ for all $n \in \mathbb{N}$. Then $T$ has a fixed point in $X$.

Proof: From theorem 2.5, we Know that $\lbrace x_n \rbrace ^\infty _{n = 0}$ is a Couchy sequence in $(X, A, d_A)$, then there exists $x \in X$ such that $x_n \rightarrow x$ as $n \rightarrow +\infty$.

On the other hand from equation(3.1) and by hypothesis (iii), we have $d_A(x_n, x) \succeq I_A$, for all $n \in \mathbb{N}$, since $T$ is generalized Lipschitz Contraction using 2.2 we get

To prove the uniqueness of the fixed point of generalized Lipschitz mapping we have to consider the following property.

(H): For all $x, y \in X$, there exists $z \in X$ such that $d_A(x, z) \succeq I_A$ and $d_A(y, z) \succeq I_A$.

Theorem 2.7. Adding condition (H) to the hypothesis of theorem (2.5) we obtain the uniqueness of the fixed point of T.

Proof: Suppose that $x$ and $y$ are two fixed points of T from (H), there exists $z \in X$ such that

Since $T$ is $\alpha_A$-admissible, from (2.2) we have

Since T is generalized Lipschitz contraction, so by using (2.4), we have

Since $\Vert b \Vert \Vert a \Vert^2 < 1$, $\Vert a \Vert < 1$, we have $\Vert a \Vert ^{2n} \rightarrow 0_A$ as $n \rightarrow +\infty$ and $d_A(x, T^n z) \rightarrow 0_A$, $T^n z = x$ as $n \rightarrow +\infty$.

Similarly we get $T^n z = y$ as $n \rightarrow + \infty$, there for by uniqueness of the limit, we obtain $x = y$. This complete the proof.

3.   Common fixed point theorems

Now, we give a common fixed point theorems for two mappings satisfy a common $\alpha_A$-admissible.

Definition 3.1. let $(T, S) : X \rightarrow X$ be a continuous self mappings on X. $\alpha_A : X \times X \rightarrow A^+$. $(T, S)$ are said to be common $\alpha_A$-admissible if for any $x_0 \in X$,

Theorem 3.2. Let $(X, A, d_A)$ be complete $C^{*}$-algebra- valued b-metric space and $T, S : X \rightarrow X$, such that

and $\Vert a \Vert < 1,$ $\Vert b \Vert. \Vert a \Vert ^2 < 1$ and the following conditions are satisfies:

(i) $(T, S)$ are common $\alpha_A$-admissible.

(ii) The exists $x_0 \in X$ such that

(iii) T and S are continuous and have a common fixed point in X.

Proof: Let $x_0 \in X$ and construct a sequence $\lbrace x_n\rbrace \subseteq X$ such that $Tx_{2n} = x_{2n+1}$, $Sx_{2n +1} = x_{2n+2}$ form (3.1), we get

by induction, we have $\alpha_A(x_{2n}, x_{2n+1}) \succeq I_A$, for all $n \in \mathbb{N}$.

by induction, we obtain

Similarly,

Now, we can obtain for any $n \in \mathbb{N}$

Then for $p \in \mathbb{N}$, $p \geq 1$, $m \geq 1$, and applying the triangle inequality, we have

by similar calculation as theorem (2.5), we get

as $n \rightarrow +\infty$, where $I_A$ is the unitary in $A$, $d_0 : = d_A(x_0, x_1), \, \, \, b \in (A^+)^\prime$.

So, $\{x_n\}$ is a Cauchy sequence in $X$.

The completion of $(X, A, d_A)$ implies that there exists $x \in X$ such that $\lim\limits_{n\rightarrow \infty} x_n = x$

Now, we using triangle inequality and (3.1), we set

Since $\Vert a \Vert < 1$, we have a contradiction $\Rightarrow d_A(x, Sx) = 0_A \Rightarrow Sx = x,$ similarly, we get $Tx = x$, so, $S$ and $T$ have a common fixed point.

In the following, we will show that the uniquely of common fixed point in $X$, for that assume that is another fixed point $y \in X$ such that $Ty = y = y$.

Since $x$ satisfy property H, and $(T, S)$ are $\alpha_A$ -admissible, we have

So, $d_A(x, S^n z) = 0_A$ this implies that $S^n z = x$.

Similarly, we get $S^n z = y$ Thus $x$ is a unique common fixed point.

Theorem 3.3. Let $(X, A, d_A)$ be a complete $C^{*}$-algebra- valued b-metric space, suppose that two mappings $T, S : X \rightarrow X$, satisfy

where $a \in A$, with $\Vert b \Vert \Vert a \Vert^2 < 1$ and $\Vert a \Vert < 1$.

If $R(T) \subseteq R(S)$ and $R(S)$ is complete in X, T and S are weakly compatible, such that the following holds

(i) $(T, S)$ are common $\alpha_A$-admissible.

(ii) There is $x_0 \in X$ such that $\alpha_A(x_0, y) \succeq I_A\Rightarrow \alpha_A(Tx_0, Sy) \succeq I_A$.

(iii) T and S are continuous.

(iv) $X$ has a property (H), they T and S have a unique common fixed point in X.

Proof: Let $x_0 \in X$, choose $x_1 \in X$, such that $Sx_1 = Sx_0$, which can be done since $R(T) \subseteq R(S)$. Let $x_0 \in X$ such that $Sx_2 = Tx_1$.

Repeating the process, we have a sequence $\lbrace Sx_n\rbrace^\infty _{n = 1}$ in X satisfying $Sx_n = Tx_{n-1}$.

Then, since $(T, S)$ are $\alpha_A$-admissible, we get

Now,

For $m \geq 1$, $p \geq 1$.

Using similar calculation as in theorem 2.5, we get

Where $d_0 = d_A(Sx_0, Sx_1)$.

So, $\{Sx_n \}^\infty _{n = 1}$ is a Cauchy sequence in $R(S)$ and is complete in $X$, there exists $x \in X$ such that $\lim\limits_{n \rightarrow +\infty} Sx_n = Sx$.

Also,

So, $Sx_n \rightarrow Tx$ as $n \rightarrow +\infty$. Hens $Sx_n = Tx = Sx$, so $x$ is coincidence common fixed point in X.

Morovere of $y$ is another common fixed point such that $Ty = Sy = y$, so

Since $\Vert a \Vert < 1$, so we yet $d_A(Sx, Sy) = 0_A \Rightarrow Sx = Sy$.

So $S$, $T$ have coincidence fixed point is unique $Sx = Tx = x$.

Since $\{Sx_n \}^\infty _{n = 1}$ is a sequence in X, convergent to $Sx$ and $Sy$ respectively,

$Sx = \lim\limits_{n \rightarrow +\infty} Sx_n = Tx$, since the lim is unique, so $Tx = Sx = x$, so $S$ and $T$ have a common fixed point in X.\\ Since X has a property (H) and $(S, T)$ are $\alpha_A$-admissible, we get

Similarly $T^nz = y$, so $x = y$ and this complete the proof.

4.   Application

We introduce a non-trivial example satisfy the theorem 2.5.

Example 4.1. Let $X = [0, 1]$, $A = M_2(\mathbb{R})$, $p > 1$ and $k > 0$ is a constant, we define $d_A = X \times X \rightarrow A$ as $d_A(x, y) = \begin{pmatrix} \vert x-y \vert^p & 0\\ 0 & k\vert x-y \vert^p \end{pmatrix}$ for all $x, y \in X$. Then $(X, A, d_A)$ is $C^*$-algebra valued b-metric space. Define $T: X \rightarrow X$ as $Tx = x^2$, then

Define, $\alpha_A : X \times X \rightarrow A$, by $\alpha _{A}(x, y) = \left \{ \begin{array}{ll} \begin{pmatrix} x & 0\\ 0 & y \end{pmatrix} & {if }~~x = y = 1 \\ 0 & {otherwise, } \end{array} \right.$

it is clear that $\alpha _{A}(x, y) = \left \{ \begin{array}{ll} I_{M_2(\mathbb{R})} & {if }~~x = y, \\ 0 & {otherwise, } \end{array} \right.$

$\alpha _{A}(Tx, Ty) = \left \{ \begin{array}{ll} \begin{pmatrix} x^2 & 0\\ 0 & y^2 \end{pmatrix} & {if }~~x = y = 1 \\ 0 & {otherwise, } \end{array} \right.$

$\alpha_A(x, y) = I_{M_2(\mathbb{R})} \Rightarrow \alpha_A(Tx, Ty) = I_{M_2(\mathbb{R})}$

So, $\alpha_A(x, y)d_A(Tx, Ty) \leq (\sqrt{2})^p)d_A(x, y)(\sqrt{2})^p$. So, it is satisfy the conditions of theorem 2.5, and then T has a fixed point $0 \in X$.

As an application, we use the $C^*$-algebra-valued b-metric space to study the existence and uniqueness of the system of matrix equations in ^[4] by using theorem 2.5.

Example 4.2. Application: Suppose that $M_n(\mathbb{C})$ the set of all $m\times n$ matrices with complex entries. $M_n(\mathbb{C})$ is a $C^*$-algebra with the operator norm. Let $B_1$, $B_2$, ..., $B_n \in M_n(\mathbb{C})$ are diagonal matrices which satisfy $\sum\limits_{k = 1}^n |B_k|^2 < 1$.

Let $A = (a_{ij})_{1\leq i, j \leq n} \in M_n(\mathbb{C})$ and $C = (c_{ij})_{1\leq i, j \leq n} \in M_n(\mathbb{C})^+$, where $M_n(\mathbb{C})^+$ denote the set of all positive definite matrices "hermitian and the eigenvalues are non-negative". Then the matrix equations

has a unique solution.

Proof: Set $\alpha = \sum\limits_{k = 1}^n| B_k|^2$, clear if $\alpha = 0$, then the equations has a unique solution in $M_n(\mathbb{C})$. Without loss of generality, suppose that $\alpha > 0$. For $A, D \in M_n(\mathbb{C})$ and $p\geq 1$, define $d_{M_n(\mathbb{C})} : M_n(\mathbb{C}) \times M_n(\mathbb{C}) \rightarrow M_n(\mathbb{C})^+$ as

$d_{M_n(\mathbb{C})}(A, D)$ = $diag (\lambda_1\vert a_{11}-d_{11}, ..., \lambda_n\vert a_{nn}-d_{nn} \vert^p)$, $\lambda_1, ..., \lambda_n > 0$, then $(M_n(\mathbb{C}), d_{M_n(\mathbb{C})})$ is a $C^*$-algebra valued b-metric space and is complete since the set $M_n(\mathbb{C})$ is complete (the proof is given in the example 1.7). Consider the map $T = (T_{ii}) : M_n(\mathbb{C}) \rightarrow M_n(\mathbb{C})$ defined by

$T_{ii}(a_{ij})_{1\leq i, j \leq n} = \sum\limits_{k = 1}^n B_k^*(a_{ii})B_k+c_{ii}$. Define $\alpha_{M_n(\mathbb{C})} : M_n(\mathbb{C}) \times M_n(\mathbb{C}) \rightarrow M_n(\mathbb{C})^+$,

$\alpha_{M_n(\mathbb{C})}(A, B) = I_{M_n(\mathbb{C})}$, clear that $T$ is $\alpha_{M_2(\mathbb{C})}$ admissible. Then

Therefore, $T$ satisfy the condition of theorem 2.5 and has a fixed point. So the matrix equations (4.1) has a solution on $M_n(\mathbb{C})$. Moreover $\alpha_{M_n(\mathbb{C})}$ is satisfy the condition (H), so the system of matrix equations have a unique hermitian matrix solution A.

5.   Conclusions

In this paper, we define a new version of $\alpha_A$-admissible in the case of self map $T : A\rightarrow A$ and $\alpha_A$-admissible in two self mappings $(T, S)$. We prove the principal Banach fixed point theorem and two common fixed point theorems in the $C^{*}$-algebra- valued b-metric space, which generalized the given results in ^{[18,19,26,27]}. Moreover, we introduced an application to show that the useful of $C^{*}$-algebra- valued b-metric space to study the existence and unique of system matrix equations.

Acknowledgments

The first author thanks the south valley university, Egypt, for partially supporting the study. The second author thanks the Jazan university, Saudi Arabia, for supporting the study.

Conflict of interest

The authors of this current research declaring that this study has been done without any competing intersts.