 Research
 Open Access
\(C^{*}\)Valued Gcontractions and fixed points
 Dure Shehwar^{1} and
 Tayyab Kamran^{2}Email author
https://doi.org/10.1186/s1366001508279
© Shehwar and Kamran 2015
 Received: 29 June 2015
 Accepted: 15 September 2015
 Published: 29 September 2015
Abstract
Recently, Ma et al. introduced the notion of \(C^{*}\)valued metric spaces and extended the Banach contraction principle for selfmappings on \(C^{*}\)valued metric spaces. Motivated by the work of Jachymski, in this paper we extend and improve the result of Ma et al. by proving a fixed point theorem for selfmappings on \(C^{*}\)valued metric spaces satisfying the contractive condition for those pairs of elements from the metric space which form edges of a graph in the metric space. Our result generalizes and extends the main result of Jachymski and Ma et al. We also establish some examples to elaborate our new notions and to substantiate our result.
Keywords
 Banach Space
 Directed Graph
 Fixed Point Theorem
 Contraction Condition
 Cauchy Sequence
1 Preliminaries and introduction
In this section we recollect some basic definitions and notions and fix our terminology to be used throughout the paper.
Afterwards, many results appeared in the literature for fixed points of mappings on partially ordered metric spaces by various authors, see for example, Bashkar and Lakshmikanthm [3], Nieto and RodriguezLopez [4], Petrusel and Rus [5], and Nieto et al. [6, 7]. By using graph theory, Jachymski [8] unified and extended the results by the abovementioned authors.
 (\(\mathcal{P}\)):

for any \(\{x_{n}\}\) in X such that \(x_{n} \to x\) with \((x_{n+1},x_{n}) \in E(G)\) for all \(n\geq1 \) there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \((x,x_{n_{k}}) \in E(G)\).
 (\(\mathcal{P'}\)):

for any \(\{f^{n}x\}\) in X such that \(f^{n}x \to y\in X\) with \((f^{n+1}x,f^{n}x) \in E(G)\) there exist a subsequence \(\{f^{n_{k}}x\}\) of \(\{f^{n}x\}\) and \(n_{0}\in\mathbb{N}\) such that \((y,f^{n_{k}}x) \in E(G)\) for all \(k\geq n_{0}\).
Example 1.1
[20]
Let \(X=[0,1]\) endowed with usual metric \(d(x,y)=xy\). Consider a graph G consisting of \(V(G):=X\) and \(E(G):=\{(\frac{n}{n+1}, \frac{n+1}{n+2}):n\in\mathbb{N}\}\cup \{(\frac{x}{2^{n}}, \frac{x}{2^{n+1}}):n\in \mathbb{N},x\in[0,1]\}\cup\{(\frac{x}{2^{2n}}, 0):n\in \mathbb{N},x\in[0,1]\}\). Note that G does not satisfy property (\(\mathcal{P}\)) as \(\frac{n}{n+1}\to1\). By defining \(f:X\to X\) as \(fx=\frac{x}{2}\), G satisfies property (\(\mathcal{P'}\)). We have \(f^{n}x=\frac{x}{2^{n}}\to0\) as \(n\to\infty\).
Recently, Ma et al. [21] introduced the notion of \(C^{*}\)valued metric spaces and extended the Banach contraction principle for selfmappings on \(C^{*}\)valued metric spaces. Before giving the definition and result by Ma et al. [21] let us recall some notions from \(C^{*}\)algebra that may be found in [22, 23]. A ∗algebra \(\mathbb{A}\) is a complex algebra with conjugate linear involution ∗ such that for any \(x, y \in\mathbb{A}\), \(x^{**}= x\) and \((xy)^{*}=y^{*}x^{*}\). In addition, if \(\mathbb{A}\) is a Banach space and for \(x \in\mathbb{A}\), \(\ x^{*} x \= \ x \^{2}\), then \(\mathbb{A}\) is called a \(C^{*}\)algebra. The set \(\sigma(x)=\lbrace\lambda\in\mathbb{C}: \lambda Ix \mbox{ is not invertible}\rbrace\) is called the spectrum of an element \(x \in \mathbb{A}\). An element \(x \in\mathbb{A}\) is called a positive element of \(\mathbb{A}\) if x is selfadjoint i.e., \(x= x^{*}\) and \(\sigma(x)\subset[0, \infty)\). The set \(\mathbb{A}_{+}\) denotes the set of positive elements in \(\mathbb{A}\). We will write \(x \succeq y\) iff \(xy \in\mathbb{A}_{+}\). Each positive element x of a \(C^{*}\)algebra has a unique positive square root. If x and y are selfconjugate elements of a \(C^{*}\)algebra and \(\theta\preceq x \preceq y\) then \(\ x \\preceq\ y \\), where θ is the zero element of the \(C^{*}\)algebra \(\mathbb{A}\).
Definition 1.2
[21]
 (i)
\(d(x, y)\succeq\theta\), for all \(x, y \in X\);
 (ii)
\(d(x, y)=\theta \Leftrightarrow x=y\);
 (iii)
\(d(x, y)=d(y, x)\), for all \(x, y \in X\);
 (iv)
\(d(x, y)\preceq d(x, z)+ d(z, y)\), for all \(x, y, z \in X\).
The tuple \((X, \mathbb{A}, d)\) is called a \(C^{*}\)valued metric space. Let \(x\in(X,\mathbb{A},d)\). A sequence \(\lbrace x_{n} \rbrace\) in \((X,\mathbb{A},d)\) is said to be convergent with respect to \(\mathbb{A}\), if for any \(\epsilon>0\) there exists a positive integer N such that \(\d(x_{n},x) \\leqslant\epsilon\) for all \(n> N\). A sequence \(\lbrace x_{n} \rbrace\) is called a Cauchy sequence with respect to \(\mathbb{A}\) if for any \(\epsilon>0\) there exists a positive integer N such that \(\d(x_{n},x_{m}) \\leqslant\epsilon\) for all \(n, m > N\). If every Cauchy sequence with respect to \(\mathbb{A}\) is convergent, then \((X,\mathbb{A},d)\) is said to be a complete \(C^{*}\)valued metric space.
Definition 1.3
[21]
Theorem 1.4
[21]
If \((X,\mathbb{A},d)\) is a \(C^{*}\)algebra valued metric space and T satisfies (2), then T has a unique fixed point in X.
It is natural question to ask whether the mapping T, considered above, has a fixed point if the contraction condition holds for those pair of elements that form edges of the graph as defined by Jachymski [8]. In this paper, we give a positive answer to this question by introducing the notion of \(C^{*}\)valued Gcontractions and then proving a fixed point theorem for such contractions. We construct some examples to elaborate the generalities of our notion and result.
2 Main results
We begin this section by introducing the notion of a \(C^{*}\)valued Gcontraction which is weaker than the notion of a \(C^{*}\)valued contraction, Definition 1.3.
Definition 2.1
Remark 2.2
By taking \(G_{1}=(X, X \times X)\), we see that a \(C^{*}\)valued contraction is a \(C^{*}\)valued \(G_{1}\)contraction.
The following example shows that the converse of the above statement is not true in general.
Example 2.3
The following lemma is straightforward.
Lemma 2.4
The proof of (7) follows from the fact that \(\sum_{k=m}^{n}{\ x \}^{k}\) is a geometric series with the common ratio \(\ x \< 1\) and \(m\rightarrow\infty\) implies \(\ x \^{m}\rightarrow 0\).
Theorem 2.5
 (I)
if \((x, y) \in E(G)\) then \((Tx, Ty)\in E(G)\);
 (II)
there exists an \(x_{\circ}\in X\) such that \((x_{\circ}, Tx_{\circ}) \in E(G)\).
Proof
Remark 2.6
By taking \(G=(X, X \times X)\), we see that Theorem 1.4 is a special case of Theorem 2.5. Moreover, [8], Theorem 3.2, is a special case of Theorem 2.5 when \(\mathbb{A}=\mathbb{R}\).
Example 2.7
3 Conclusion
Declarations
Acknowledgements
The authors are grateful to the reviewers for their valuable suggestions and comments.
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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