Open Access

The Banach contraction principle in \(C^{*}\)-algebra-valued b-metric spaces with application

  • Tayyab Kamran1, 2,
  • Mihai Postolache3,
  • Adrian Ghiura3Email author,
  • Samina Batul4 and
  • Rashid Ali4
Fixed Point Theory and Applications20162016:10

Received: 2 October 2015

Accepted: 7 December 2015

Published: 13 January 2016


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.

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


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.

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

Example 2.2

Let \(\mathbb{A}= \mathbb{R}^{2}\) and \(X=[0,\infty)\). Let be the partial order on \(\mathbb{A}\) given by
$$\begin{aligned}& (a_{1},b_{1})\preceq(a_{2},b_{2}) \quad\Leftrightarrow\quad a_{1} \leq a_{2} \mbox{ and } b_{1} \leq b_{2}. \end{aligned}$$
$$d_{b}\colon X \times X \rightarrow\mathbb{A},\qquad d_{b}(x,y)= \bigl((x-y)^{2},0\bigr). $$
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.


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


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

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

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. □


Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

Department of Mathematics, Quaid-i-Azam University
Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology H-12
Department of Mathematics and Informatics, University Politehnica of Bucharest
Department of Mathematics, Capital University of Science and Technology


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