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 Open Access
Common fixed point theorems for generalized kordered contractions and Bcontractions on noncommutative Banach spaces
 Qiaoling Xin^{1} and
 Lining Jiang^{1}Email author
https://doi.org/10.1186/s1366301503270
© Xin and Jiang 2015
 Received: 19 September 2014
 Accepted: 17 May 2015
 Published: 4 June 2015
Abstract
The paper introduces the concepts of the generalized kordered contraction and the kordered Bcontraction in noncommutative Banach spaces. Then some common fixed point theorems are given. As an application, the existence and uniqueness theorem for a common solution of integral equations is presented.
Keywords
 common fixed point
 noncommutative Banach space
 generalized kordered contraction
 kordered Bcontraction
MSC
 47H10
1 Introduction
The study of common fixed points of mappings satisfying certain contractive conditions has many applications and has been at the center of vigorous research activity. In 1976, Jungck [1] proved common fixed point theorems for commuting mappings in metric spaces, generalizing the Banach contraction principle. Later, Das and Naik [2] investigated the corresponding common fixed point result for Ćirić’s fixed point theorem [3]. Thereafter the concept of commuting mappings has been developed in various directions. One of such notions which is weaker than commuting is the concept of compatibility introduced by Jungck [4]. In common fixed point problems this concept and its generalizations have been used extensively (see [5–7]), e.g., Ćirić et al. [7] proved some common fixed point theorems for two pairs of weakly compatible mappings satisfying a generalized contraction condition on partial metric spaces. For a survey of common fixed point theory, its applications, comparison of different contractive conditions, and related results, one can refer to [8–10] and references contained therein.
Recently, Huang and Zhang [11] generalized the notion of metric spaces by substituting the set of real numbers with the ordered Banach space, and defined the concept of cone metric spaces. Based on the notion of cone metric spaces, several fixed point theorems were obtained for mappings satisfying certain contractive type condition; see for example [12–18]. Subsequently, Xin and Jiang [19] introduced noncommutative Banach spaces which generalize the concept of Banach spaces and established fixed point theorems for mappings with the kordered contractive condition.
The paper will give some common fixed theorems in the framework of noncommutative Banach spaces. In details, we firstly introduce the concept of generalized kordered contractions in noncommutative Banach spaces, and prove some common fixed point theorems for generalized kordered contractions which generalize the results in [19] (see Theorem 2.1). Then the notion of kordered Bcontractions is introduced in noncommutative Banach spaces and the corresponding common fixed point theorems are given. At the end, to see the applicability of our results, we give an existence and uniqueness theorem for a common solution of integral equations.
Now we recall the definition of noncommutative Banach spaces and some of their properties which will be needed in the sequel [19].
Definition 1.1
 (1)
for any \(x, y, z\in E\), we have \(d(xz, yz)=d(x,y)\);
 (2)there exists a binary continuous operationsuch that \(F(1,x)=x^{1}\) is exactly the inverse of x in the group E and \(F(0,x)=x^{0}=e\) is the unit in the group E, and that$$F\colon\mathbb{R}\times E\rightarrow E, \qquad (\alpha, x)\mapsto x^{\alpha}$$for \(m,n\in\mathbb{R}\), \(x\in E\);$$F(mn,x)=F\bigl(m,F(n,x)\bigr), \qquad F(m+n,x)=F(m,x)F(n,x) $$
 (3)for any \(x\in E\), there exists a constant \(M_{x}>0\) such that$$d\bigl(x^{\alpha}, e\bigr)\leq M_{x}\alpha, \quad \forall \alpha\in\mathbb{R}. $$
It can be shown that all Banach spaces and unitary groups of Hilbert spaces are all noncommutative Banach spaces. For more details, one can refer to [19].
 (1)
P is nonempty, closed, and \(P\neq\{e\}\);
 (2)
\(x, y\in P\) and \(\alpha, \beta\in\mathbb{R}^{+}\) imply \(x^{\alpha}y^{\beta}\in P\);
 (3)
\(P\cap P^{1}=\{e\}\) where \(P^{1}=\{x^{1}\colon x\in P\}\).
Lemma 1.1
 (1)
set \(x\lesssim y\), then \(x^{\alpha}\lesssim y^{\alpha}\) holds for all \(\alpha\in[0, 1]\);
 (2)
if \(x\in P\) and there exists \(\lambda\in[0, 1)\) such that \(x\lesssim x^{\lambda}\), then \(x=e\);
 (3)
if x and y are comparable, then \(xy^{1}\) and \(yx^{1}\) are comparable, and \(e\lesssim{\vee}(xy^{1},yx^{1})\);
 (4)
if x and y are comparable, then \(d({\vee}(xy^{1},yx^{1}),e)=d(x,y)\).
Proof
(1) Let \(x\lesssim y\), we have \(y^{\beta}x^{\beta}\in P\) for all \(\beta \in[0, 1]\). Since \(\alpha\beta\in[0, 1]\) for any \(\alpha\in[0, 1]\), we see \((y^{\alpha})^{\beta}(x^{\alpha})^{\beta}= y^{\alpha\beta}x^{\alpha\beta}\in P\), which means \(x^{\alpha}\lesssim y^{\alpha}\).
(2) From \(x\lesssim x^{\lambda}\), we know \(x^{\lambda1}\in P\). It follows from \(\frac{1}{1\lambda}>0\) and the definition of cone that \(x^{1}\in P\), which together with \(x\in P\) yields \(x=e\).
(3) One may suppose that \(x\lesssim y\), which means \(yx^{1}\in P\). For all \(\beta\in[0, 1]\), \((yx^{1})^{\beta}(xy^{1})^{\beta}= (yx^{1})^{\beta}((yx^{1})^{1})^{\beta}=(yx^{1})^{2\beta}\in P\), then \(xy^{1}\lesssim yx^{1}\). Furthermore \((yx^{1})^{\beta}e^{\beta}=(yx^{1})^{\beta}\in P\), which implies \(e\lesssim yx^{1}\), and therefore \(e\lesssim{\vee}(xy^{1},yx^{1})\).
In the rest of the paper, we always suppose that E is a noncommutative Banach space, P is a normal cone in E with the normal constant N and ≲ is a partial ordering with respect to P.
2 Common fixed point theorems for the generalized kordered contraction
In this section, we give the generalized kordered contraction in noncommutative Banach spaces, and prove some common fixed point theorems for the generalized kordered contraction which generalize the results in [19].
Definition 2.1
In particular, if \(A=B\), A is called the kordered contraction, which can also be found in [19].
Theorem 2.1
 (i)
A or B is continuous;
 (ii)
A is the generalized kordered contraction;
 (iii)
there is \(x_{0}\in E\) such that \(x_{0}\) and \(Ax_{0}\) are comparable.
Proof
Similarly, if B is continuous, again we have \(Ax^{*}=x^{*}\). Therefore, A and B have a common fixed point. □
The following theorem gives the sufficient condition for the uniqueness of a common fixed point of A and B in Theorem 2.1.
Theorem 2.2
In addition to the hypotheses of Theorem 2.1, assume that for all \(x,y\in E\), there exists \(w\in E\) depending on x and y such that w is comparable with x and y. Then A and B have a unique common fixed point.
Proof
The set of common fixed points of A and B is not empty due to Theorem 2.1; suppose now that x and y are two common fixed points of A and B, i.e., \(Ax=Bx=x\), \(Ay=By=y\). We distinguish two cases:
Corollary 2.1
 (i)there exists \(k\in(0,1)\) such that for all \(x,y\in E\), if x and y are comparable, then x and \(A^{p}y\), Ax and Ay are comparable, respectively, and, moreover,for any \(n,m,p\in\mathbb{N}\);$${\vee} \bigl(A^{n}x\bigl(A^{m}y\bigr)^{1},A^{m}y \bigl(A^{n}x\bigr)^{1} \bigr)\lesssim{\vee} \bigl(xy^{1},yx^{1} \bigr)^{k} $$
 (ii)
there is \(x_{0}\in E\) such that \(x_{0}\) and \(Ax_{0}\) are comparable;
 (iii)
for all \(x,y\in E\), there exists \(w\in E\) such that w is comparable with x and y.
Proof
It follows from Theorem 2.1 and Theorem 2.2 by putting \(A^{n}\equiv A\) and \(A^{m}\equiv B\). □
As a consequence of the previous corollary, we obtain a fixed point theorem for the kordered contraction on a noncommutative Banach space, which can also be seen in [19].
Corollary 2.2
 (i)there exists \(k\in(0,1)\) such that for all \(x,y\in E\), if x and y are comparable, then Ax and Ay are comparable, and, moreover,$${\vee} \bigl(Ax(Ay)^{1},Ay(Ax)^{1} \bigr)\lesssim{\vee} \bigl(xy^{1},yx^{1} \bigr)^{k}; $$
 (ii)
there is \(x_{0}\in E\) such that \(x_{0}\) and \(Ax_{0}\) are comparable;
 (iii)
for all \(x,y\in E\), there exists \(w\in E\) such that w is comparable with x and y.
3 Common fixed point theorems for the kordered Bcontraction
In the following we shall introduce the notion of kordered Bcontractions in the framework of noncommutative Banach spaces and prove some common fixed point theorems. Let us start with the following definition.
Definition 3.1
Theorem 3.1
 (i)
A and B commute;
 (ii)
A is the kordered Bcontraction;
 (iii)
there exists \(x_{0}\in E\) such that \(Ax_{0}\) and \(Bx_{0}\) are comparable;
 (iv)
if \(\{Bx_{n}\}_{n=0}^{\infty}\) is a sequence in E which has comparable adjacent terms and converges to some z in E, then Bz and \(B(Bz)\) are comparable.
Proof
Let \(x_{0}\in E\) be such that \(Ax_{0}\) and \(Bx_{0}\) are comparable and \(x_{1}\in E\) be chosen such that \(Bx_{1}=Ax_{0}\). This can be done since \(R(A)\subseteq R(B)\). Let \(x_{2}\in E\) be such that \(Bx_{2}=Ax_{1}\). Continuing this process, we can construct a sequence \(\{Bx_{n}\} _{n=0}^{\infty}\) in E such that \(Bx_{n+1}=Ax_{n}\) for all \(n\in\mathbb{N}\).
In addition to the hypotheses of Theorem 3.1, suppose that for all \(x,y\in E\), there exists \(z\in E\) depending on x and y such that Bz is comparable with Bx and By. Then A and B have a unique common fixed point. Indeed, assume that there exist \(x,y\in E\) which are two common fixed points of A and B. We claim that \(x=y\).
By assumption, there exists \(z\in E\) such that Bz is comparable with Bx and By. Define a sequence \(\{Bz_{n}\}_{n=0}^{\infty}\) such that \(z_{0}=z\) and \(Bz_{n}=Az_{n1}\) for all n.
Theorem 3.2
 (i)
A and B commute;
 (ii)there exist nonnegative real numbers s, k, and h with \(s>\max\{k+h,1+h\}\) such that for all \(\beta\in[0, 1]\), if x and y are comparable, then Ax and Ay, Ax, and Bx, Ay and By are comparable, respectively, and furthermore$${\vee} \bigl(Ax(Ay)^{1},Ay(Ax)^{1} \bigr)^{s\beta} \lesssim{\vee} \bigl(Ax(Bx)^{1},Bx(Ax)^{1} \bigr)^{k\beta} \vee \bigl(Ay(By)^{1},By(Ay)^{1} \bigr)^{h\beta}; $$
 (iii)
if \(\{Bx_{n}\}_{n=0}^{\infty}\) is a sequence in E which has comparable adjacent terms and converges to some z in E, then Bz and \(B(Bz)\) are comparable.
Proof
Remark 3.1
 (ii′):

there exist nonnegative real numbers s, k, h, and l with \(s>\max\{k+h+l, 1+l\}\) such that for all \(\beta\in[0, 1]\), if x and y are comparable, then Ax and Ay, Bx and By, Ax and Bx, Ay and By are comparable, respectively, and furthermore$$\begin{aligned}& {\vee} \bigl(Ax(Ay)^{1},Ay(Ax)^{1} \bigr)^{s\beta} \\& \quad \lesssim {\vee} \bigl(Bx(By)^{1},By(Bx)^{1} \bigr)^{k\beta} \vee \bigl(Ax(Bx)^{1},Bx(Ax)^{1} \bigr)^{h\beta} \\& \qquad {} \vee \bigl(Ay(By)^{1},By(Ay)^{1} \bigr)^{l\beta}. \end{aligned}$$
4 Existence of a common solution for a system of integral equations
For \(u,v\in\mathbb{R}\), if \(u\leq v\) or \(v\leq u\), we say that u and v are comparable.
 (i)
\(K_{1}\) or \(K_{2}\colon[0,1]\times[0,1]\times\mathbb{R}\rightarrow\mathbb {R}\) is continuous, and \(g\colon[0,1]\rightarrow\mathbb{R}\) is also continuous;
 (ii)there exist \(k\in(0,1)\) and a continuous function \(\varphi\colon [0,1]\times[0,1]\rightarrow\mathbb{R}^{+}\) such that if u and v are comparable, then \(K_{1}(t,s,u)\) and \(K_{2}(t,s,v)\) are comparable, furthermorefor \(t,s\in[0,1]\) and \(u,v\in\mathbb{R}\);$$\bigl\vert K_{1}(t,s,u)K_{2}(t,s,v)\bigr\vert \leq k \varphi(t,s)uv $$
 (iii)
\(\sup_{t\in[0,1]}\int_{0}^{1}\varphi(t,s)\, ds\leq1\);
 (iv)
there exists \(x_{0}\in C[0, 1]\) such that for any \(t,s\in[0, 1]\), \(x_{0}(t)\) and \(\int_{0}^{1}K_{1}(t, s,x_{0}(s))\, ds+g(t)\) are comparable.
Theorem 4.1
Under the assumptions (i)(iv), then the integral equations have a unique common solution x in \(C[0,1]\).
Proof
Hence all of the conditions of Theorem 2.1 and Theorem 2.2 are satisfied, and so A and B have a unique common fixed point, which is a unique common solution of integral equations. □
Notice that the example given above is in linear spaces. As to the noncommutative case, it is under consideration now.
Declarations
Acknowledgments
The authors thank referees for their careful reviewing of the manuscript and their remarkable suggestions which help to improve the quality of this paper. This work is supported financially by the NSFC (10971011, 11401022).
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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