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- Open Access
\(C^{*}\)-Valued contractive type mappings
- Samina Batul^{1} and
- Tayyab Kamran^{2}Email author
https://doi.org/10.1186/s13663-015-0393-3
© Batul and Kamran 2015
- Received: 20 April 2015
- Accepted: 30 July 2015
- Published: 19 August 2015
Abstract
In this paper we generalize the notion of \(C^{*}\)-valued contraction mappings, recently introduced by Ma et al., by weakening the contractive condition introduced by them. Using the new notion of \(C^{*}\)-valued contractive type mappings, we establish a fixed point theorem for such mappings. Our result generalizes the result by Ma et al. and those contained therein except for the uniqueness.
Keywords
- C∗-algebra
- contractions
- fixed point theorems
- orbits
1 Introduction and preliminaries
Recently, Ma et al. [3] introduced the notion of \(C^{*}\)-valued metric spaces and, analogous to the Banach contraction principle, established a fixed point theorem for \(C^{*}\)-valued contraction mappings. In this paper, we first introduce the notion of continuity in the context of \(C^{*}\)-valued metric spaces and show that a \(C^{*}\)-valued contraction map is continuous with respect to our notion of continuity. Then we introduce a \(C^{*} \)-valued contractive type condition and establish a fixed point theorem analogous to the results presented in [2]. We also show that a \(C^{*}\)-valued contractive type map need not be continuous in the context of \(C^{*}\)-valued metric.
We now recollect some basic definitions, notations, and results that will be used subsequently. For details, we refer to [4, 5]. An algebra \(\mathbb{A}\) together with a conjugate linear involution map \(*: \mathbb{A}\rightarrow\mathbb{A}\), defined by \(a\mapsto a^{*}\) such that for all \(a,b \in\mathbb{A}\) we have \((ab)^{*}=b^{*}a^{*}\) and \((a^{*})^{*}=a \), is called a ∗-algebra. Moreover, if \(\mathbb{A}\) contains an identity element \(1_{\mathbb{A}}\), then the pair \((\mathbb{A},*)\) is called a unital ∗-algebra. A unital ∗-algebra \((\mathbb{A},*)\) together with a complete sub multiplicative norm satisfying \(\|a^{*} \|=\|a \|\) for all \(a\in\mathbb {A}\) is called a Banach ∗-algebra. A \(C^{*}\)-algebra is a Banach ∗-algebra \((\mathbb{A}, *)\) such that \(\|a^{*}a \|=\|a \|^{2}\) for all \(a\in\mathbb{A} \). An element \(a \in\mathbb{A}\) is called a positive element if \(a=a^{*}\) and \(\sigma(a)\subset\mathbb{R_{+}}\), where \(\sigma(a)=\{\lambda \in\mathbb{R} : \lambda I-a \mbox{ is non-invertible} \}\). If \(a\in\mathbb{A}\) is positive, we write it as \(a \succeq 0_{\mathbb{A}}\). Using positive elements, one can define a partial ordering on \(\mathbb{A}\) as follows: \(b \succeq a\) if and only if \(b-a\succeq0_{\mathbb{A}}\). Each positive element a of a \(C^{*}\)-algebra \(\mathbb{A}\) has a unique positive square root. Subsequently, \(\mathbb{A}\) will denote a unital \(C^{*}\)-algebra with the identity element \(1_{\mathbb{A}}\). Further, \(\mathbb{A}_{+}\) is the set \(\{ a\in\mathbb{A}:a\succeq0_{\mathbb{A}} \}\) of positive elements of \(\mathbb{A}\) and \((a^{*}a)^{1/2}=| a |\). Using the concept of positive elements in \(\mathbb{A}\), a \(C^{*}\)-algebra-valued metric space is defined in the following way.
Definition 1.1
[3]
- (i)
\(0_{\mathbb{A}} \preceq d(x,y)\) for all \(x, y\in X\) and \(d(x,y)=0_{\mathbb{A}} \Leftrightarrow x=y \),
- (ii)
\(d(x,y)=d(y,x)\) \(\forall x,y \in X\),
- (iii)
\(d(x,y)\preceq d(x,z)+d(z,y)\) \(\forall x,y,z \in X\).
A sequence \(\{ x_{n} \}\) in \((X,\mathbb{A},d)\) is said to converge to \(x \in X\) with respect to \(\mathbb{A}\) if for any \(\epsilon>0\) there exists \(N \in \mathbb{N}\) such that \(\|d(x_{n},x) \| < \epsilon\) for all \(n> N\). We write it as \(\lim_{n\rightarrow\infty} x_{n}=x\). A sequence \(\{ x_{n} \}\) is called a Cauchy sequence with respect to \(\mathbb{A}\) if for any \(\epsilon>0\) there exists \(N \in \mathbb{N}\) such that \(\|d(x_{n},x_{m}) \| < \epsilon\) for all \(n, m > N\). The triplet \((X,\mathbb{A},d)\) is said to be a complete \(C^{*}\)-valued metric space if every Cauchy sequence with respect to \(\mathbb{A}\) is convergent. Now we state the definition and result from [3], for convenience.
Definition 1.2
[3]
Theorem 1.3
[3]
Let \((X,\mathbb{A},d)\) be a \(C^{*}\)-algebra-valued complete metric space and \(T:X \rightarrow X \) satisfy (3), then T has a unique fixed point in X.
From now on, we call a \(C^{*}\)-algebra-valued metric and a \(C^{*}\)-algebra-valued metric space simply a \(C^{*}\)-valued metric and a \(C^{*}\)-valued metric space, respectively.
2 Main results
We begin this section by introducing the notion of continuity in the context of \(C^{*}\)-valued metric spaces.
Definition 2.1
Let \((X,\mathbb{A},d)\) be a \(C^{*}\)-valued metric space. A mapping \(T:X \rightarrow X\) is said to be continuous at \(x_{0}\) with respect to \(\mathbb{A}\) if given any \(\epsilon>0\) there exists \(\delta>0\) such that \(\|d(Tx,Tx_{0}) \| < \epsilon\) whenever \(\|d(x,x_{0}) \| < \delta\). T is said to be continuous on X with respect to \(\mathbb{A}\) if it is continuous for every \(x\in X\).
Example 2.2
Remark 2.3
Note that every continuous self-map is continuous with respect \(\mathbb{A}=\mathbb{R}\) and a \(C^{*}\)-valued contraction map is continuous with respect to the \(C^{*}\)-algebra \(\mathbb{A}\).
Definition 2.4
Remark 2.5
If \(\mathbb{A}=\mathbb{R}\), then our definition coincides with the usual definition of T-orbitally lower semicontinuous as defined by [2].
Example 2.6
Definition 2.7
Remark 2.8
A \(C^{*}\)-valued contraction mapping is a \(C^{*}\)-valued contractive type mapping, but the converse is not true as shown in the following example.
Example 2.9
Before giving our main result, we prove the following lemma which is essentially extracted from the proof of Theorem 1.3.
Lemma 2.10
Proof
We are now ready to state and prove our main result.
Theorem 2.11
- (A1)
\(\exists x_{0} \in X \) such that the sequence \({T^{n} x}\) converges to \(x_{0}\),
- (A2)
\(d(T^{n}x,x_{0}) \preceq\frac{\|a\| ^{2n}}{1-\|a\|} \|d(x,Tx)^{\frac{1}{2}}\|^{2} 1_{\mathbb{A}}\),
- (A3)
\(x_{0}\) is a fixed point of T if and only if \(G(x)=d(x,Tx)\) is T-orbitally lower semicontinuous at \(x_{0}\) with respect to \(\mathbb{A}\).
Proof
If \(\mathbb{A} = \{0_{\mathbb{A}}\}\), then there is nothing to prove. Assume that \(\mathbb{A}\ne\{0_{\mathbb{A}}\}\).
Remark 2.12
The following example shows that our result properly generalizes Theorem 1.3.
Example 2.13
3 Application
In this section we provide the existence result for an integral equation as an application of \(C^{*}\)-valued contractive type mappings on complete \(C^{*}\)-valued metric spaces. Let E be a Lebesgue measurable set, \(X=L^{\infty}(E)\), and \(H=L^{2}(E)\). We denote the set of all bounded linear operators on a Hilbert space H by \(L(H)\). With the usual operator norm, \(L(H)\) is a \(C^{*}\)-algebra. For \(S, T \in X\), define \(d:X \times X \rightarrow L(H)\) by \(d(T,S)=\pi_{|T-S|}\), where \(\pi_{h}:H\rightarrow H\) is the multiplication operator given by \(\pi_{h}(\phi)=h\cdot\phi\) for \(\phi\in H\). Then \((X,L(H),d)\) is a complete \(C^{*}\)-valued metric space [3].
Example 3.1
- (1)
\(K:E \times E \times\mathbb{R} \rightarrow\mathbb{R} \), and let T be a self-mapping on X,
- (2)there exists a continuous function \(\phi: E\times E \rightarrow\mathbb{R}\) and \(\alpha\in(0,1)\) such that for every \(x\in X\), \(y \in\mathcal{O}_{T}(x)\), and \(t, s \in E\), we have$$ \bigl|K\bigl(t,s, x(s)\bigr) - K\bigl(t, s, y(s)\bigr)\bigr| \le\alpha\bigl|\phi(t,s) \bigl(x(s)-y(s)\bigr)\bigr|. $$(12)
- (3)
\(\sup_{t\in E} \int_{E} |\phi(t,s)|\,ds \le1\).
Proof
Here \((X,L(H),d)\) is a complete \(C^{*}\)-valued metric space with respect to \(L(H)\).
Declarations
Acknowledgements
The authors are thankful to the reviewers for their useful comments and suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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
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