# The Banach contraction principle in $$C^{*}$$-algebra-valued b-metric spaces with application

## Abstract

We introduce the notion of a $$C^{*}$$-algebra-valued b-metric space. We generalize the Banach contraction principle in this new setting. As an application of our result, we establish an existence result for an integral equation in a $$C^{*}$$-algebra-valued b-metric space.

## Introduction

The Banach contraction principle [1], also known as the Banach fixed point theorem, is one of the main pillars of the theory of metric fixed points. According to this principle, if T is a contraction on a Banach space X, then T has a unique fixed point in X. Many researchers investigated the Banach fixed point theorem in many directions and presented generalizations, extensions, and applications of their findings. Among them, Bakhtin [2] introduced a prominent generalization of the idea of a metric space, which is later used by Czerwick [3, 4]. They introduced and used the concept of real-valued b-metric space to establish certain fixed point results. The idea clearly is an extension of the metric space as follows from the following definition.

### Definition 1.1

([5])

Let X be a nonempty set, and $$b \in\mathbb{R}$$ be such that $$b \geq1$$. A b-metric on X is a real-valued mapping $$d_{b}\colon X \times X \rightarrow\mathbb{R}$$ that satisfies the following conditions for all $$x,y,z \in X$$:

1. (1)

$$d_{b}(x,y)\ge0\mbox{ and }d_{b}(x,y)= 0 \Leftrightarrow x=y$$.

2. (2)

$$d_{b}(y,x)=d(x,y)$$ (symmetry).

3. (3)

$$d_{b}(y,z)\le b [d_{b}(y,x)+d_{b}(x,z)]$$.

By a b-metric space with coefficient b we mean the pair $$(X, d_{b})$$.

For recent development on b-metric spaces, we refer to [510].

Recently, Ma et al. [11] presented their work on the extension of Banach contraction principle for $$C^{*}$$-algebra-valued metric spaces. Later, Batul and Kamran [12] introduced the notion of a $$C^{*}$$-valued contractive type mapping and established a fixed point result in this setting. Motivated by the ideas and results presented in [11, 12], in this paper, we will introduce a new notion of $$C^{*}$$-algebra-valued b-metric space and establish a fixed point result in such spaces.

We now recollect some basic definitions, notation, and results. The details on $$C^{*}$$-algebras are available in [13, 14].

An algebra $$\mathbb{A}$$, together with a conjugate linear involution map $$a\mapsto a^{*}$$, is called a -algebra if $$(ab)^{*}=b^{*}a^{*}$$ and $$(a^{*})^{*}=a$$ for all $$a,b \in\mathbb{A}$$. Moreover, the pair $$(\mathbb{A},*)$$ is called a unital -algebra if $$\mathbb{A}$$ contains the identity element $$1_{\mathbb{A}}$$. By a Banach -algebra we mean a complete normed unital -algebra $$(\mathbb{A},*)$$ such that the norm on $$\mathbb{A}$$ is submultiplicative and satisfies $$\|a^{*} \|=\|a \|$$ for all $$a\in\mathbb{A}$$. Further, if for all $$a\in\mathbb{A}$$, we have $$\|a^{*}a \|=\|a \|^{2}$$ in a Banach -algebra $$(\mathbb{A}, *)$$, then $$\mathbb{A}$$ is known as a $$C^{*}$$-algebra. A positive element of $$\mathbb{A}$$ is an element $$a \in\mathbb{A}$$ such that $$a=a^{*}$$ and its spectrum $$\sigma(a)\subset\mathbb{R_{+}}$$, where $$\sigma(a)=\lbrace\lambda \in\mathbb{R} : \lambda1_{\mathbb{A}}\mbox{-}a \mbox{ is noninvertible}\rbrace$$. The set of all positive elements will be denoted by $$\mathbb{A}_{+}$$. Such elements allow us to define a partial ordering ‘’ on the elements of $$\mathbb{A}$$. That is,

$$b \succeq a \quad\mbox{if and only if}\quad b-a \in\mathbb{A}_{+}.$$

If $$a\in\mathbb{A}$$ is positive, then we write $$a \succeq 0_{\mathbb{A}}$$, where $$0_{\mathbb{A}}$$ is the zero element of $$\mathbb{A}$$. Each positive element a of a $$C^{*}$$-algebra $$\mathbb{A}$$ has a unique positive square root. From now on, by $$\mathbb{A}$$ we mean a unital $$C^{*}$$-algebra with identity element $$1_{\mathbb{A}}$$. Further, $$\mathbb{A}_{+} = \lbrace a\in\mathbb{A}:a\succeq0_{\mathbb{A}} \rbrace$$ and $$(a^{*}a)^{1/2}=\vert a \vert$$. Using the concept of positive elements in $$\mathbb{A}$$, a $$C^{*}$$-algebra-valued metric d on a nonempty set X is defined in [11] as a mapping $$d\colon X\times X \rightarrow\mathbb{A}_{+}$$ that satisfies, for all $$x_{1},x_{2},x_{3} \in X$$, (i) $$d(x_{1},x_{2})=0_{\mathbb{A}} \Leftrightarrow x_{1}=x_{2}$$, (ii) $$d(x_{1},x_{2})=d(x_{2},x_{1})$$, and (iii) $$d(x_{1},x_{2})\preceq d(x_{1},x_{3})+d(x_{3},x_{2})$$. The triplet $$(X,\mathbb{A},d)$$ is then called a $$C^{*}$$-algebra-valued metric space.

## Main results

In this section, we extend Definition 1.1 to introduce the notion b-metric space in the setting of $$C^{*}$$-algebras as follows.

### Definition 2.1

Let $$\mathbb{A}$$ be a $$C^{*}$$-algebra, and X be a nonempty set. Let $$b \in\mathbb{A}$$ be such that $$\|b \| \geq1$$. A mapping $$d_{b}\colon X \times X \rightarrow\mathbb{A}_{+}$$ is said to be a $$C^{*}$$-algebra-valued b-metric on X if the following conditions hold for all $$x_{1},x_{2},x_{3} \in\mathbb{A}$$:

1. (BM1)

$$d_{b}(x_{1},x_{2})=0_{\mathbb{A}} \Leftrightarrow x_{1}=x_{2}$$.

2. (BM2)

$$d_{b}$$ is symmetric, that is, $$d_{b}(x_{1},x_{2})=d_{b}(x_{2},x_{1})$$.

3. (BM3)

$$d_{b}(x_{1},x_{2})\preceq b [d_{b}(x_{1},x_{3})+d_{b}(x_{3},x_{2})]$$.

The triplet $$(X,\mathbb{A}, d_{b})$$ is called a $$C^{*}$$-algebra-valued b-metric space with coefficient b.

### Remark 2.1

Note that:

1. (1)

If we take $$\mathbb{A}=\mathbb{R}$$, then the new notion of $$C^{*}$$-algebra-valued b-metric space becomes equivalent to Definition 1.1 of the real b-metric space.

2. (2)

If we take $$b=1_{\mathbb{A}}$$ in Definition 2.1, then $$d_{b}$$ becomes the usual $$C^{*}$$-algebra-valued metric as defined in [11].

Thus, the class of ordinary $$C^{*}$$-algebra-valued metric spaces is clearly smaller than the class of $$C^{*}$$-algebra-valued b-metric spaces. In fact, there are $$C^{*}$$-algebra-valued b-metric spaces that are not $$C^{*}$$-algebra-valued metric spaces, as illustrated by the following example.

### Example 2.1

Let $$X=\ell_{p}$$ be the set of sequences $$\{x_{n}\}$$ in $$\mathbb{R}$$ such that $$\sum_{n=1}^{\infty}|x_{n}|^{p} < \infty$$ and $$0< p<1$$. Let $$\mathbb {A}=M_{2}(\mathbb{R})$$. For $$x=x_{n}, y=y_{n} \in\ell_{p}$$, define $$d_{b}:X \times X \rightarrow \mathbb{A}$$ as follows:

$$d_{b}(x,y) = \begin{pmatrix} (\sum_{n=1}^{\infty}|x_{n}-y_{n}|^{p} )^{\frac{1}{p}} & 0 \\ 0 & (\sum_{n=1}^{\infty}|x_{n}-y_{n}|^{p} )^{\frac{1}{p}} \end{pmatrix}.$$

Then one can show that $$d_{b}$$ is a $$C^{*}$$-algebra-valued b-metric space with coefficient $$b =\bigl( {\scriptsize\begin{matrix}{} 2^{\frac{1}{p}} & 0 \cr 0 & 2^{\frac{1}{p}} \end{matrix}}\bigr)$$ such that $$\|b\|=2^{\frac{1}{p}}$$. The claim follows from the following observation in [4]:

$$\Biggl(\sum_{n=1}^{\infty}|x_{n}-z_{n}|^{p} \Biggr)^{\frac{1}{p}} \le 2^{\frac{1}{p}} \Biggl[ \Biggl(\sum _{n=1}^{\infty}|x_{n}-y_{n}|^{p} \Biggr)^{\frac {1}{p}} + \Biggl(\sum_{n=1}^{\infty}|y_{n}-z_{n}|^{p} \Biggr)^{\frac{1}{p}} \Biggr].$$

Note that here $$d_{b}$$ is not a usual $$C^{*}$$-algebra-valued metric on X.

From now on, we call a $$C^{*}$$-algebra-valued b-metric space simply a $$C^{*}$$-valued b-metric, and the triplet $$(X,\mathbb{A},d_{b})$$ is then called a $$C^{*}$$-valued b-metric space. Given $$(X,\mathbb{A},d_{b})$$, the following are natural deductions from the corresponding notions in $$C^{*}$$-valued metric spaces.

1. (1)

A sequence $$\lbrace x_{n} \rbrace$$ in X is said to be convergent to a point $$x \in X$$ with respect to the algebra $$\mathbb{A}$$ if and only if for any $$\epsilon>0$$, there is an $$N \in\mathbb{N}$$ such that $$\|d_{b}(x_{n},x) \| < \epsilon$$ for all $$n> N$$. Symbolically, we then write $$\lim_{n\rightarrow \infty} x_{n}=x$$.

2. (2)

If for any $$\epsilon>0$$, there exists $$N \in\mathbb{N}$$ such that $$\|d_{b}(x_{n},x_{m}) \| < \epsilon$$ for all $$n, m > N$$, then the sequence $$\lbrace x_{n} \rbrace$$ is called a Cauchy sequence with respect to $$\mathbb{A}$$.

3. (3)

If every Cauchy sequence in X is convergent with respect to $$\mathbb{A}$$, then the triplet $$(X,\mathbb{A},d)$$ is called a complete $$C^{*}$$-valued b-metric space.

### Definition 2.2

Let $$(X,\mathbb{A}, d_{b})$$ be a $$C^{*}$$-valued b-metric space. A contraction on X is a mapping $$T\colon X \rightarrow X$$ if there exists an $$a\in\mathbb{A}$$ with $$\| a \| < 1$$ such that

$$d_{b}(Tx,Ty)\preceq a^{*}d_{b}(x,y)a \quad\mbox{for all } x,y \in X.$$
(1)

### Example 2.2

Let $$\mathbb{A}= \mathbb{R}^{2}$$ and $$X=[0,\infty)$$. Let be the partial order on $$\mathbb{A}$$ given by

\begin{aligned}& (a_{1},b_{1})\preceq(a_{2},b_{2}) \quad\Leftrightarrow\quad a_{1} \leq a_{2} \mbox{ and } b_{1} \leq b_{2}. \end{aligned}

Define

$$d_{b}\colon X \times X \rightarrow\mathbb{A},\qquad d_{b}(x,y)= \bigl((x-y)^{2},0\bigr).$$

Then $$d_{b}$$ is $$C^{*}$$-valued b-metric with coefficient $$(2,0)$$, and with this $$d_{b}$$, the triplet $$(X,\mathbb{A},d_{b})$$ becomes a $$C^{*}$$-valued b-metric. Consider $$T\colon X \rightarrow X$$ given by $$Tx=\frac{x}{3}+5$$; then T is a contraction on X with $$a=(\frac{1}{3},0)$$:

\begin{aligned}[b] d_{b}(Tx,Ty)= \bigl((Tx-Ty)^{2},0 \bigr) = \biggl( \biggl(\frac{x}{3}-\frac{y}{3} \biggr)^{2},0 \biggr) = \biggl(\frac{1}{3},0 \biggr)d_{b}(x,y) \biggl( \frac{1}{3},0 \biggr). \end{aligned}

### Theorem 2.1

Consider a complete $$C^{*}$$-valued b-metric space $$(X,\mathbb{A},d_{b})$$ with coefficient b. Let $$T\colon X \rightarrow X$$ be a contraction with the contraction constant a such that $$\| b\| \|a \|^{2} < 1$$. Then T has a unique fixed point in X.

### Proof

If $$\mathbb{A} = \{0_{\mathbb{A}}\}$$, then there is nothing to prove. Assume that $$\mathbb{A}\ne\{0_{\mathbb{A}}\}$$.

Choose $$x_{0} \in X$$ and define inductively a sequence $$\{x_{n}\}$$ by the iterative scheme as

$$x_{n+1}=Tx_{n}.$$

Then it follows that $$x_{n}=T^{n}x_{0}$$ for $$n=0,1,2, \ldots$$ . From the contraction condition (1) on T it follows that

\begin{aligned} d_{b}(x_{n},x_{n+1}) =& d_{b}(Tx_{n-1},Tx_{n}) \\ \preceq& a^{*}d_{b}(x_{n-1},x_{n})a \\ =& a^{*}d_{b}(Tx_{n-2},Tx_{n-1})a \\ \preceq& \bigl(a^{*}\bigr)^{2}d_{b}(x_{n-2},x_{n-1})a^{2} \\ \preceq& \bigl(a^{*}\bigr)^{3}d_{b}(x_{n-3},x_{n-2})a^{3} \preceq \bigl(a^{*}\bigr)^{n}d_{b}(x_{0},x_{1})a^{n}= \bigl(a^{*}\bigr)^{n}Da^{n}, \end{aligned}

where $$D=d_{b}(x_{0},x_{1})$$.

Now suppose that $$m>n$$; then the triangle inequality (BM3) for the b-metric $$d_{b}$$ implies

\begin{aligned} d_{b}(x_{n},x_{m}) \preceq& b d(x_{n},x_{n+1}) + b^{2}d(x_{n+1},x_{n+2})+ \cdots+ b^{m-n-1}d(x_{m-2},x_{m-1}) \\ &{}+ b^{m-n-1}d(x_{m-1},x_{m}) \\ \preceq& b\bigl(a^{*}\bigr)^{n}Da^{n} +b^{2} \bigl(a^{*}\bigr)^{n+1}Da^{n+1} + \cdots+ b^{m-n-1} \bigl(a^{*}\bigr)^{m-2}Da^{m-2} \\ &{}+ s^{m-n-1}\bigl(a^{*}\bigr)^{m-1}Da^{m-1} \\ =& b\bigl[\bigl(a^{*}\bigr)^{n}Da^{n} +b\bigl(a^{*} \bigr)^{n+1}Da^{n+1} + \cdots+ b^{m-n-2}\bigl(a^{*} \bigr)^{m-2}Da^{m-2}\bigr] \\ &{}+ b^{m-n-1}\bigl(a^{*}\bigr)^{m-1}Da^{m-1} \\ =& b\sum_{k=n}^{m-2}b^{k-n} \bigl(a^{*}\bigr)^{k}Da^{k} + b^{m-n-1}\bigl(a^{*} \bigr)^{m-1}Da^{m-1} \\ =& b\sum_{k=n}^{m-1}b^{k-n} \bigl(a^{*}\bigr)^{k}D^{\frac{1}{2}}D^{\frac{1}{2}}a^{k} + b^{m-n-1}\bigl(a^{*}\bigr)^{m-1}D^{\frac{1}{2}}D^{\frac{1}{2}}a^{m-1} \\ =& b\sum_{k=n}^{m-1}b^{k-n} \bigl(D^{\frac{1}{2}}a^{k}\bigr)^{*} \bigl(D^{\frac{1}{2}}a^{k} \bigr) + b^{m-n-1}\bigl(D^{\frac{1}{2}}a^{m-1}\bigr)^{*} \bigl(D^{\frac{1}{2}}a^{m-1}\bigr) \\ =& b\sum_{k=n}^{m-1}b^{k-n}\bigl|D^{\frac{1}{2}}a^{k}\bigr|^{2} + b^{m-n-1}\bigl|D^{\frac {1}{2}}a^{m-1}\bigr|^{2} \\ \preceq& \Biggl\| b\sum_{k=n}^{m-1}b^{k-n}\bigl|D^{\frac{1}{2}}a^{k}\bigr|^{2} \Biggr\| 1_{\mathbb{A}} + \bigl\| b^{m-n-1}\bigl|D^{\frac{1}{2}}a^{m-1}\bigr|^{2} \bigr\| 1_{\mathbb{A}} \\ \preceq& \|b\|\sum_{k=n}^{m-1} \bigl\| b^{k-n}\bigr\| \bigl\| D^{\frac{1}{2}}\bigr\| ^{2} \bigl\| a^{k} \bigr\| ^{2} 1_{\mathbb{A}} + \bigl\| b^{m-n-1}\bigr\| \bigl\| D^{\frac{1}{2}} \bigr\| ^{2} \bigl\| a^{m-1}\bigr\| ^{2} 1_{\mathbb{A}} \\ \preceq& \|b\|\sum_{k=n}^{m-1} \|b\|^{k-n} \bigl\| D^{\frac{1}{2}}\bigr\| ^{2} \bigl\| a^{k} \bigr\| ^{2} 1_{\mathbb{A}} + \|b\|^{m-n-1} \bigl\| D^{\frac{1}{2}} \bigr\| ^{2} \bigl\| a^{m-1}\bigr\| ^{2} 1_{\mathbb{A}} \\ \preceq& \|b\|^{1-n} \bigl\| D^{\frac{1}{2}}\bigr\| ^{2}\sum _{k=n}^{m-1}\|b\|^{k} \bigl\| a^{2}\bigr\| ^{k} 1_{\mathbb{A}} + \|b\|^{-n}\|b \|^{m-1} \bigl\| D^{\frac{1}{2}}\bigr\| ^{2} \bigl\| a^{m-1} \bigr\| ^{2} 1_{\mathbb{A}} \\ \preceq& \|b\|^{1-n} \bigl\| D^{\frac{1}{2}}\bigr\| ^{2}\sum _{k=n}^{m-1}\bigl(\|b\| \bigl\| a^{2}\bigr\| \bigr)^{k} 1_{\mathbb{A}} + \|b\|^{-n}\bigl\| D^{\frac{1}{2}} \bigr\| ^{2}\bigl(\|b\| \bigl\| a^{2}\bigr\| \bigr)^{m-1} 1_{\mathbb{A}} \\ \longrightarrow& 0_{\mathbb{A}} \quad\mbox{as } m, n \rightarrow\infty, \end{aligned}

which follows from the observation that the summation in the first term is a geometric series, and $$\|b\|\|a^{2}\| < 1$$ implies that both $$(\|b\| \|a^{2}\|)^{m-1} \rightarrow0$$ and $$(\|b\| \|a^{2}\|)^{n-1} \rightarrow0$$. This proves that $$\{x_{n}\}$$ is a Cauchy sequence in X with respect to $$\mathbb{A,}$$ and from the completeness of $$(X, \mathbb{A}, d)$$ it follows that $$x_{n} \rightarrow x \in X$$, that is,

$$\lim_{n\rightarrow\infty} x_{n} = \lim_{n\rightarrow\infty} Tx_{n-1} = x .$$

We claim that x is a fixed point of T. In fact, from the triangle inequality (BM3) and the contraction condition (1) we have:

\begin{aligned} 0_{\mathbb{A}} \preceq& d(Tx,x) \\ \preceq& b\bigl[d(Tx,Tx_{n})+d(Tx_{n},x)\bigr] \\ \preceq& b a^{*}d(x,x_{n})a + d(x_{n-1},x) \longrightarrow 0_{\mathbb{A}} \quad\mbox{as } n\rightarrow\infty. \end{aligned}

This shows that $$Tx=x$$.

To prove that x is the unique fixed point, we suppose that $$y\in X$$ is another fixed point of T. Then again from the contraction condition (1) we have

\begin{aligned} 0_{\mathbb{A}} \preceq d(x,y) = d(Tx,Ty) \preceq a^{*} d(x,y) a. \end{aligned}

Using the norm of $$\mathbb{A}$$, we have

\begin{aligned} 0\le\bigl\| d(x,y)\bigr\| \le\bigl\| a^{*} d(x,y) a\bigr\| \le\bigl\| a^{*}\bigr\| \bigl\| d(x,y)\bigr\| \|a\| =\|a\| ^{2} \bigl\| d(x,y)\bigr\| . \end{aligned}

The above inequality holds only when $$d(x,y) = 0_{\mathbb{A}}$$. Hence, $$x=y$$. □

### Example 2.3

The mapping T of Example 2.2 satisfies the hypothesis of Theorem 2.1, and T has unique fixed point $$x=1.5$$ in X.

### Remark 2.2

Theorem 2.1 generalizes the following results.

1. (1)

By taking $$\mathbb{A} =\mathbb{R}$$, the $$C^{*}$$-valued b-metric becomes simply the b-metric, and we immediately get the Banach contraction principle in b-metric spaces from Theorem 2.1.

2. (2)

Taking $$b=1$$, [11], Theorem 2.1, becomes a special case of Theorem 2.1.

## Application

As an application of the fixed point theorem for contractions on a $$C^{*}$$-valued complete b-metric space, we provide an existence result for a class of integral equations.

### Example 3.1

Let E be a Lebesgue-measurable set and $$X=L^{\infty}(E)$$. Consider the Hilbert space $$L^{2}(E)$$. Let the set of all bounded linear operators on $$L^{2}(E)$$ be denoted by $$BL(L^{2}(E))$$. Note that $$BL(L^{2}(E))$$ is a $$C^{*}$$-algebra with usual operator norm. For $$S, T \in X$$, define

$$d_{b}\colon X \times X \rightarrow BL\bigl(L^{2}(E)\bigr),\qquad d_{b}(T,S)=\pi_{(T-S)^{2}},$$

where $$\pi_{h}\colon L^{2}(E)\rightarrow L^{2}(E)$$ is the product operator given by

$$\pi_{h}(f)=h\cdot f \quad\mbox{for } f \in L^{2}(E).$$

Working in the same lines as in [11], Example 2.1, we can show that $$(X,BL(L^{2}(E)),d_{b})$$ is a complete $$C^{*}$$-valued b-metric space. With these settings, suppose that there exist a continuous function $$f \colon E\times E \rightarrow\mathbb{R}$$ and a constant $$0< \alpha<1$$ such that for all $$x, y \in X$$ and $$u,v \in E$$, we have

$$\bigl|K\bigl(u,v, x(v)\bigr) - K\bigl(u, v, y(v)\bigr)\bigr| \le\alpha\bigl|f(u,v) \bigl(x(v)-y(v)\bigr)\bigr|,$$
(2)

where K is a function from $$E \times E \times\mathbb{R}$$ to $$\mathbb{R}$$, and $$\sup_{t\in E} \int_{E} |f (u,v)|\,dv \le1$$. Then the integral equation

$$x(u)= \int_{E} K\bigl(u,v,x(v)\bigr)\,dv,\quad u\in E$$

has a unique solution.

### Proof

Here $$(X,BL(L^{2}(E)),d_{b})$$ is a $$C^{*}$$-valued complete b-metric space with respect to $$BL(L^{2}(E))$$.

Let

$$T\colon X\rightarrow X,\quad Tx(u)= \int_{E} K\bigl(u,v,x(v)\bigr)\,dv,\quad u\in E.$$

Then

\begin{aligned} \bigl\| d(Tx,Ty)\bigr\| =& \|\pi_{(Tx-Ty)^{2}}\| \\ =& \sup_{\|g\|=1} \langle\pi_{(Tx-Ty)^{2}}g,g\rangle\quad \mbox{for every } g\in L^{2}(E) \\ =& \sup_{\|g\|=1} \int_{E} (Tx-Ty)^{2}g(u)\overline{g(u)}\,dv \\ =& \sup_{\|g\|=1} \int_{E} \biggl[ \int_{E} \bigl(K\bigl(u,v,x(v)\bigr)-K\bigl(u,v,y(v)\bigr) \bigr)\,dv \biggr]^{2} g(u)\overline{g(u)}\,du \\ \le& \sup_{\|g\|=1} \int_{E} \biggl[ \int_{E} \bigl(K\bigl(u,v,x(v)\bigr)-K\bigl(u,v,y(v)\bigr) \bigr)\,dv \biggr]^{2}\bigl|g(u)\bigr|^{2}\,du \\ \le& \sup_{\|g\|=1} \int_{E} \alpha^{2} \biggl[ \int_{E} \bigl(f(u,v) \bigl(x(v)-y(v)\bigr)\bigr)\,dv \biggr]^{2}\bigl|g(u)\bigr|^{2}\,du \\ \le& \alpha^{2} \sup_{\|g\|=1} \int_{E} \biggl[ \int_{E} \bigl|f(u,v)\bigr|\,dv \biggr]^{2}\bigl|g(u)\bigr|^{2}\,du \cdot\bigl\| (x-y)^{2}\bigr\| _{\infty}\\ \le& \alpha^{2} \sup_{t \in E} \int_{E} \bigl|f(u,v)\bigr|^{2}\,dv \cdot\sup _{\|g\|=1} \int_{E} \bigl|g(u)\bigr|^{2}\,du \cdot\bigl\| (x-y)^{2} \bigr\| _{\infty}\\ \le& \alpha^{2}\bigl\| (x-y)^{2}\bigr\| _{\infty}\\ = & \|a\| \bigl\| d(x,y)\bigr\| . \end{aligned}

Setting $$a= \alpha I_{2}$$, we have $$a\in BL(L^{2}(E))_{+}$$ and $$\|a\|=\alpha^{2} <1$$. Thus, all the conditions of Theorem 2.1 hold, and hence the conclusion. □

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The authors contributed equally to this work. All authors read and approved the final manuscript.

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Kamran, T., Postolache, M., Ghiura, A. et al. The Banach contraction principle in $$C^{*}$$-algebra-valued b-metric spaces with application. Fixed Point Theory Appl 2016, 10 (2016). https://doi.org/10.1186/s13663-015-0486-z