Fixed point theorems of order-Lipschitz mappings in Banach algebras
- Shujun Jiang^{1} and
- Zhilong Li^{2, 3}Email authorView ORCID ID profile
https://doi.org/10.1186/s13663-016-0519-2
© Jiang and Li 2016
Received: 22 October 2015
Accepted: 2 March 2016
Published: 11 March 2016
Abstract
In this paper, by introducing the concept of Picard-completeness and using the sandwich theorem in the sense of w-convergence, we first prove some fixed point theorems of order-Lipschitz mappings in Banach algebras with non-normal cones which improve the result of Sun’s since the normality of the cone was removed. Moreover, we reconsider the case with normal cones and obtain a fixed point theorem under the assumption relating to the spectral radius, which partially improves the results of Krasnoselskii and Zabreiko’s. In addition, we present some suitable examples which show the usability of our theorems.
Keywords
MSC
1 Introduction
Theorem 1
(see [2])
Remark 1
The normality of the cone is essential for ensuring that the sandwich theorem holds in the sense of norm-convergence which plays an important role in the proof of Theorem 1. However, if P is non-normal then the sandwich theorem does not hold in the sense of norm-convergence, and consequently, the method used in the proof of Theorem 1 may become invalid.
Krasnoselskii and Zabreiko [3] considered order-Lipschitz mappings in Banach spaces restricted with linear bounded mappings (i.e., k, l are linear bounded mappings), and proved the following fixed point theorem by using the Banach contraction principle.
Theorem 2
(see [3])
Let P be a normal solid cone of a Banach space \((E,\Vert \cdot \Vert )\) and \(T:E\rightarrow E\) an order-Lipschitz mapping with \(k=l\), where \(k:P\rightarrow P\) is a linear bounded mapping. If \(\Vert k\Vert <1\), then T has a unique fixed point \(x^{*}\in E\). And for each \(x_{0}\in E\), let \(\{x_{n}\}\) be the Picard iterative sequence, then we must have \(x_{n}\overset{\Vert \cdot \Vert }{\rightarrow} x^{*}\).
To our knowledge, in all the works concerned with fixed points of order-Lipschitz mappings, the involving cone is necessarily assumed to be normal. In this paper, we shall remove the normality of the cone in Theorem 1 and extend Theorems 1 and 2 to Banach algebras. From Remark 1 we know that the method in [2] is not applicable for the case with non-normal cones, and so we need to find a new way to solve it. By introducing the concept of Picard-complete and using the sandwich theorem in the sense of w-convergence established in [4], we first prove some fixed point theorems of order-Lipschitz mappings in Banach algebras with non-normal cones. Motivated by [3], we reconsider the case with normal cones, and we obtain a fixed point theorem of order-Lipschitz mappings in Banach algebras under the assumption that \(r(k)<1\) by showing that there exists some \(n_{0}\) such that \(T^{n_{0}}\) is a Banach contraction in \((E,\Vert \cdot \Vert _{0})\), where \(\Vert \cdot \Vert _{0}\) is a newly introduced norm which is equivalent to \(\Vert \cdot \Vert \); see Lemma 5. In addition, some suitable examples are presented to show the usability of our theorems.
2 Preliminaries and lemmas
A Banach space \((E,\Vert \cdot \Vert )\) is called a Banach algebra [5] if there exists a multiplication in E such that, for each \(x, y, z \in E\) and \(a\in \mathbb{R}\), the following conditions are satisfied: (I) \((xy)z = x(yz)\); (II) \(x(y + z) = xy + xz\) and \((x + y)z = xz + yz\); (III) \(a(xy) = (ax)y = x(ay)\); (IV) \(\Vert xy\Vert \leq \Vert x\Vert \Vert y\Vert \). If there exists some \(e\in E\) such that \(ex =xe=x\) for each \(x\in E\) then e is called a unit (i.e., a multiplicative identity) of E. A nonempty closed subset P of a Banach space \((E,\Vert \cdot \Vert )\) is a cone [5, 6] if it is such that the following conditions are satisfied: (V) \(ax+by\in P\) for each \(x,y\in P\) and each \(a,b\geq0\); (VI) \(P\cap(-P)=\{\theta\}\), where θ is the zero element of E. A nonempty closed subset P of a Banach algebra \((E,\Vert \cdot \Vert )\) is a cone [1, 5] if it is such that (V) and (VI) are satisfied and (VII) \(\{e\}\subset P\) and \(P^{2}=PP\subset P\).
Each cone P of a Banach space E determines a partial order ⪯ on E by \(x\preceq y\Leftrightarrow y-x\in P \) for each \(x,y\in X\). For each \(u_{0},v_{0}\in E\) with \(u_{0}\preceq v_{0}\), we set \([u_{0},v_{0}]=\{u\in E:u_{0}\preceq u\preceq v_{0}\}\), \([u_{0},+\infty)=\{x\in E:u_{0}\preceq x\}\) and \((-\infty,v_{0}]=\{x\in E:x\preceq v_{0}\}\). A cone P is solid [5, 6] if \(\operatorname {\mathsf {int}}P\neq\O\), where int P denotes the interior of P. For each \(x,y\in E\) with \(y-x\in \operatorname {\mathsf {int}}P\), we write \(x\ll y\).
Definition 1
- (i)
the sequence \(\{x_{n}\}\) is w-convergent [4, 7] if for each \(\epsilon\in \operatorname {\mathsf {int}}P\), there exist some positive integer \(n_{0}\) and \(x\in E\) such that \(x-\epsilon\ll x_{n}\ll x+\epsilon\) for each \(n\geq n_{0}\) (denote \(x_{n}\overset{w}{\rightarrow} x\) and x is called a w-limit of \(\{ x_{n}\}\));
- (ii)
the sequence \(\{x_{n}\}\) is w-Cauchy if for each \(\epsilon\in \operatorname {\mathsf {int}}P\), there exists some positive integer \(n_{0}\) such that \(-\epsilon\ll x_{n}-x_{m}\ll\epsilon\) for each \(m,n\geq n_{0}\), i.e., \(x_{n}-x_{m}\overset{w}{\rightarrow} \theta\) (\(m,n\rightarrow\infty\));
- (iii)
the subset D is w-closed if for each \(\{x_{n}\}\subset D\), \(x_{n}\overset{w}{\rightarrow}x\) implies \(x\in D\).
Lemma 1
Let P be a solid cone of a Banach space E, \(\{x_{n}\}\) a w-convergent sequence of E and \(u_{0},v_{0}\in E\) with \(u_{0}\preceq v_{0}\). Then \(\{x_{n}\}\) has a unique w-limit, and the partial order intervals \([u_{0},v_{0}]\), \([u_{0},+\infty)\) and \((-\infty,v_{0}]\) are w-closed.
Proof
Suppose that there exists \(x,y\in E\) such that \(x_{n}\overset{w}{\rightarrow} x\) and \(x_{n}\overset{w}{\rightarrow} y\). From Definition 1 we find that, for each \(\epsilon\in \operatorname {\mathsf {int}}P\), there exists a positive integer \(n_{0}\) such that \(x-\epsilon\ll x_{n}\ll x+\epsilon\) and \(y-\epsilon\ll x_{n}\ll y+\epsilon \) for each \(n\geq n_{0}\). This forces that \(x-y-2\epsilon\ll x_{n}-x_{n}=\theta\ll x-y+2\epsilon\) for each \(n\geq n_{0}\). So we have \(-2\epsilon\ll x-y\ll2\epsilon\), which together with the arbitrary property of ϵ implies that \(x=y\). This shows that \(\{x_{n}\}\) has a unique w-limit.
A cone P of a Banach space E is normal if there is some positive number N such that \(x, y \in E\) and \(\theta\preceq x\preceq y\) implies that \(\Vert x\Vert \leq N\Vert y\Vert \), and the minimal N is called a normal constant of P. Note that an equivalent condition of a normal cone is that \(\inf\{\Vert x+y\Vert :x,y\in P \text{ and } \Vert x\Vert =\Vert y\Vert =1\}>0\), then it is not hard to conclude that a cone P is non-normal if and only if there exist \(\{u_{n}\},\{v_{n}\}\subset P\) such that \(u_{n}+v_{n}\overset{\Vert \cdot \Vert }{\rightarrow} \theta\nRightarrow u_{n} \overset{\Vert \cdot \Vert }{\rightarrow }\theta \). This implies that the sandwich theorem does not hold in the sense of norm-convergence. Recently, without using the normality of P Li and Jiang [4] proved the following sandwich theorem in the sense of w-convergence, which is very important for our further discussions.
Lemma 2
(see [4])
Let P be a solid cone of a Banach space \((E,\Vert \cdot \Vert )\) and \(\{x_{n}\},\{y_{n}\},\{z_{n}\}\subset E\) with \(x_{n}\preceq y_{n}\preceq z_{n}\) for each n. If \(x_{n}\overset{w}{\rightarrow}z\) and \(z_{n}\overset{w}{\rightarrow}z\), then \(y_{n}\overset{w}{\rightarrow }z\).
Lemma 3
(see [7])
Let P be a solid cone of a Banach space \((E,\Vert \cdot \Vert )\) and \(x_{n}\subset E\). Then \(x_{n}\overset{\Vert \cdot \Vert }{\rightarrow} x\) implies \(x_{n}\overset {w}{\rightarrow} x\). Moreover, if P is normal then \(x_{n}\overset{w}{\rightarrow} x \Leftrightarrow x_{n}\overset{\Vert \cdot \Vert }{\rightarrow} x\).
Lemma 4
Let P be a solid cone of a Banach space \((E,\Vert \cdot \Vert )\). Then there is \(\tau>0\) such that, for each \(x\in E\), there exist \(y,z\in P\) with \(\Vert y\Vert \leq\tau \Vert x\Vert \) and \(\Vert z\Vert \leq\tau \Vert x\Vert \) such that \(x=y-z\).
Lemma 5
Proof
Let P be a cone of a Banach space E and \(T:E\rightarrow E\). For each \(x_{0}\in E\), set \(O(T,x_{0})=\{x_{n}\}\), where \(\{x_{n}\}\) is the Picard iterative sequence (i.e., \(x_{n}=T^{n}x_{0}\) for each n).
Definition 2
Let P be a solid cone of a Banach space \((E,\Vert \cdot \Vert )\), \(x_{0}\in E\) and \(T:E\rightarrow E\). If the Picard iterative sequence \(O(T,x_{0})\) is w-convergent provided that it is w-Cauchy, then T is said to be Picard-complete at \(x_{0}\). Moreover, if T is Picard-complete at each \(x\in E\), then T is said to be Picard-complete on E.
Remark 2
- (i)
If \(O(T,x_{0})\) is w-convergent then T is certainly Picard-complete at \(x_{0}\).
- (ii)
If P is normal then each mapping \(T:E\rightarrow E\) is Picard-complete on E by Lemma 3.
3 Fixed point theorems
We first state and prove a fixed point result of order-Lipschitz mappings in Banach algebras with non-normal cones as follows.
Theorem 3
Let P be a solid cone of a Banach algebra \((E,\Vert \cdot \Vert )\) and \(u_{0},v_{0}\in E\) with \(u_{0}\preceq v_{0}\). Assume that \(T:[u_{0},v_{0}]\rightarrow E\) is a nondecreasing order-Lipschitz mapping with \(k,l\in P\) such that (2) is satisfied. If \(r(k)<1\) and T is Picard-complete at \(u_{0}\) and \(v_{0}\), then T has a unique fixed point \(x^{*}\in[u_{0},v_{0}]\). And for each \(x_{0}\in[u_{0},v_{0}]\), we have \(x_{n}\overset{w}{\rightarrow} x^{*}\), where \(\{x_{n}\}=O(T,x_{0})\).
Proof
Example 1
Let \(E=C_{\mathbb{R}}^{1}[0,1]\) be endowed with the norm \(\Vert u\Vert =\Vert u\Vert _{\infty}+\Vert u'\Vert _{\infty}\) and \(P=\{u\in E:u(t)\geq0, \forall t\in[0,1]\}\), where \(\Vert u\Vert _{\infty}=\max_{t\in[0,1]}u(t)\) for each \(u\in C_{\mathbb{R}}[0,1]\). Define a multiplication in E by \((xy)(t)=x(t)y(t)\) for each \(x,y\in E\) and \(t\in[0,1]\). Clearly, \((E,\Vert \cdot \Vert )\) is a Banach algebra with a unit \(e(t)\equiv1\) and P is a non-normal solid cone.
Let \(Tx=x^{2}\), \(u_{0}=\theta\) and \(v_{0}(t)\equiv a\), where \(a\in[0,\frac {1}{2})\). Clearly, \(Tu_{0}\preceq u_{0}\) and \(Tv_{0}\preceq v_{0}\). For each \(x,y\in[u_{0},v_{0}]\) with \(y\preceq x\), we have \(0\leq(Tx)(t)-(Ty)(t)=x^{2}(t)-y^{2}(t)=(x(t)+y(t))(x(t)-y(t))\leq k(x(t)-y(t))\) for each \(t\in[0,1]\), where \(k=2ae\). This shows that \(T:[u_{0},v_{0}]\rightarrow E\) is a nondecreasing order-Lipschitz mapping with \(r(k)=2a<1\). Let \(\{u_{n}\}\) and \(\{v_{n}\}\) be the Picard iterative sequences of \(u_{0}\) and \(v_{0}\), then \(u_{n}=\theta\) and \(v_{n}(t)\equiv a^{2^{n}}\) for each n, and so \(\Vert u_{n}\Vert \equiv0\) and \(\Vert v_{n}\Vert =a^{2^{n}}\) for each n, which forces that \(u_{n}\overset{\Vert \cdot \Vert }{\rightarrow}\theta\) and \(v_{n}\overset{\Vert \cdot \Vert }{\rightarrow}\theta\). This together with (i) of Remark 2 and Lemma 3 implies that T is Picard-complete at \(u_{0}\) and \(v_{0}\). Hence by Theorem 3, T has a unique fixed point in \([u_{0},v_{0}]\).
However, Theorems 1 and 2 are not applicable here since P is non-normal.
In analogy to the proof of Theorem 3, we can prove the following fixed point theorem of order-Lipschitz mappings in Banach space.
Theorem 4
Let P be a solid cone of a Banach space \((E,\Vert \cdot \Vert )\) and \(u_{0},v_{0}\in E\) with \(u_{0}\preceq v_{0}\). Assume that \(T:[u_{0},v_{0}]\rightarrow E\) is a nondecreasing order-Lipschitz mapping with \(k\in[0,1)\) such that (2) is satisfied. If T is Picard-complete at \(u_{0}\) and \(v_{0}\), then T has a unique fixed point \(x^{*}\in[u_{0},v_{0}]\). And for each \(x_{0}\in[u_{0},v_{0}]\), we have \(x_{n}\overset{w}{\rightarrow} x^{*}\), where \(\{x_{n}\}=O(T,x_{0})\).
Corollary 1
Let P be a solid cone of a Banach space \((E,\Vert \cdot \Vert )\) and \(u_{0},v_{0}\in E\) with \(u_{0}\preceq v_{0}\). Assume that \(T:[u_{0},v_{0}]\rightarrow E\) is an order-Lipschitz mapping with \(l\in[0,+\infty)\) and \(k\in[0,1)\) such that (2) is satisfied. If A is Picard-complete at \(u_{0}\) and \(v_{0}\), where \(Ax=\frac{Tx+lx}{1+l}\) for each \(x\in E\), then T has a unique fixed point \(x^{*}\in[u_{0},v_{0}]\).
Proof
Remark 3
By (ii) of Remark 2, Theorem 1 immediately follows from Corollary 1, which indeed improves Theorem 1 since the normality of P has been removed.
Motivated by [3], we reconsider the case with normal cones, and we obtain the following fixed point result.
Theorem 5
Let P be a normal solid cone of a Banach algebra \((E,\Vert \cdot \Vert )\) and \(T:E\rightarrow E\) an order-Lipschitz mapping with \(l=k\in P\). If \(r(k)<1\), then T has a unique fixed point \(x^{*}\in E\). And for each \(x_{0}\in E\), we have \(x_{n}\overset{\Vert \cdot \Vert }{\rightarrow} x^{*}\), where \(\{x_{n}\} =O(T,x_{0})\).
Proof
Remark 4
Theorem 5 partially improves Theorem 2 since the norm condition \(\Vert k\Vert <1\) is replaced by the spectral radius condition \(r(k)<1\).
The following example will show Theorem 5 is more applicable than many other fixed point results.
Example 2
In the case that \(c>1-\frac{1}{a}\), we get \(\Vert k\Vert =\frac {1}{a}+c>1\), and hence Theorem 2 is not applicable even taking k as a linear bounded mapping.
In the case that \(c>2\), we get \(\Vert Tx-Ty\Vert \geq c\vert x_{1}-y_{1}\vert >2\vert x_{1}-y_{1}\vert \geq \vert x_{1}-y_{1}\vert +\vert x_{2}-y_{2}\vert =\Vert x-y\Vert \) for each \(x,y\in E\) with \(\vert x_{1}-y_{1}\vert \geq \vert x_{2}-y_{2}\vert \), and hence the Banach contraction principle is not applicable.
In the case that \(c>1\), we get \(\arctan(b+x_{2})-\arctan(b+y_{2})+c(x_{1}-y_{1})\geq c(x_{2}-y_{2})>x_{2}-y_{2}\) for each \(x,y\in E\) with \(y\preceq x\) and \(x_{1}-y_{1}\geq x_{2}-y_{2}\). This implies that there does not exist \(l\in[0,1)\) such that \(Tx-Ty\preceq l(x-y)\). Consequently, Theorem 1 is not applicable.
Remark 5
The normality of P is essential for the completeness of \((E,\Vert \cdot \Vert _{0})\) (see Lemma 5), which leads to that the Banach contraction principle is applicable in Theorem 5. Naturally, one may wonder whether the normality of P in Theorem 5 could be removed by the method used in Theorem 1 or other methods.
Declarations
Acknowledgements
The work was supported by the National Natural Science Foundation of China (11161022, 11561026, 71462015), the Natural Science Foundation of Jiangxi Province (20142BCB23013, 20143ACB21012, 20151BAB201003, 20151BAB201023), the Natural Science Foundation of Jiangxi Provincial Education Department (KJLD14034, GJJ150479).
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.
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