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.
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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)
\(d_{b}(x,y)\ge0\mbox{ and }d_{b}(x,y)= 0 \Leftrightarrow x=y\).
-
(2)
\(d_{b}(y,x)=d(x,y) \) (symmetry).
-
(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,
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 \(x_{1},x_{2},x_{3} \in\mathbb{A}\):
-
(BM1)
\(d_{b}(x_{1},x_{2})=0_{\mathbb{A}} \Leftrightarrow x_{1}=x_{2} \).
-
(BM2)
\(d_{b}\) is symmetric, that is, \(d_{b}(x_{1},x_{2})=d_{b}(x_{2},x_{1})\).
-
(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)
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)
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:
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]:
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)
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)
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)
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
Example 2.2
Let \(\mathbb{A}= \mathbb{R}^{2}\) and \(X=[0,\infty)\). Let ⪯ be the partial order on \(\mathbb{A}\) given by
Define
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 \(a=(\frac{1}{3},0)\):
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
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
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
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,
We claim that x is a fixed point of T. In fact, from the triangle inequality (BM3) and the contraction condition (1) we have:
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
Using the norm of \(\mathbb{A}\), we have
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.
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
where \(\pi_{h}\colon L^{2}(E)\rightarrow L^{2}(E)\) is the product operator given by
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
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
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
Then
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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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
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DOI: https://doi.org/10.1186/s13663-015-0486-z