1 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 [5–10].

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.

2 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.

3 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. □