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Fixed point theorems of contractive mappings in cone bmetric spaces and applications
Fixed Point Theory and Applications volume 2013, Article number: 112 (2013)
Abstract
In this paper we present some new examples in cone bmetric spaces and prove some fixed point theorems of contractive mappings without the assumption of normality in cone bmetric spaces. The results not only directly improve and generalize some fixed point results in metric spaces and bmetric spaces, but also expand and complement some previous results in cone metric spaces. In addition, we use our results to obtain the existence and uniqueness of a solution for an ordinary differential equation with a periodic boundary condition.
1 Introduction
Fixed point theory plays a basic role in applications of many branches of mathematics. Finding a fixed point of contractive mappings becomes the center of strong research activity. There are many works about the fixed point of contractive maps (see, for example, [1, 2]). In [2], Polish mathematician Banach proved a very important result regarding a contraction mapping, known as the Banach contraction principle, in 1922. In [3], Bakhtin introduced bmetric spaces as a generalization of metric spaces. He proved the contraction mapping principle in bmetric spaces that generalized the famous Banach contraction principle in metric spaces. Since then, several papers have dealt with fixed point theory or the variational principle for singlevalued and multivalued operators in bmetric spaces (see [4–6] and the references therein). In recent investigations, the fixed point in nonconvex analysis, especially in an ordered normed space, occupies a prominent place in many aspects (see [7–10]). The authors define an ordering by using a cone, which naturally induces a partial ordering in Banach spaces. In [7], Huang and Zhang introduced cone metric spaces as a generalization of metric spaces. Moreover, they proved some fixed point theorems for contractive mappings that expanded certain results of fixed points in metric spaces. In [10], Hussain and Shah introduced cone bmetric spaces as a generalization of bmetric spaces and cone metric spaces. They established some topological properties in such spaces and improved some recent results about KKM mappings in the setting of a cone bmetric space. Throughout this paper, we firstly offer some new examples in cone bmetric spaces, then obtain some fixed point theorems of contractive mappings without the assumption of normality in cone bmetric spaces. Furthermore, we support our results by an example. The results greatly generalize and improve the work of [3, 4, 7, 8] and [10]. As some applications, we show the existence and uniqueness of a solution for a firstorder ordinary differential equation with a periodic boundary condition.
Consistent with Huang and Zhang [7], the following definitions and results will be needed in the sequel.
Let E be a real Banach space and P be a subset of E. By θ we denote the zero element of E and by intP the interior of P. The subset P is called a cone if and only if:

(i)
P is closed, nonempty, and P\ne \{\theta \};

(ii)
a,b\in \mathbb{R}, a,b\ge 0, x,y\in P\Rightarrow ax+by\in P;

(iii)
P\cap (P)=\{\theta \}.
On this basis, we define a partial ordering ≤ with respect to P by x\le y if and only if yx\in P. We shall write x<y to indicate that x\le y but x\ne y, while x\ll y will stand for yx\in intP. Write \parallel \cdot \parallel as the norm on E. The cone P is called normal if there is a number K>0 such that for all x,y\in E, \theta \le x\le y implies \parallel x\parallel \le K\parallel y\parallel. The least positive number satisfying the above is called the normal constant of P. It is well known that K\ge 1.
In the following, we always suppose that E is a Banach space, P is a cone in E with intP\ne \mathrm{\varnothing} and ≤ is a partial ordering with respect to P.
Definition 1.1 ([7])
Let X be a nonempty set. Suppose that the mapping d:X\times X\to E satisfies:

(d1)
\theta <d(x,y) for all x,y\in X with x\ne y and d(x,y)=\theta if and only if x=y;

(d2)
d(x,y)=d(y,x) for all x,y\in X;

(d3)
d(x,y)\le d(x,z)+d(z,y) for all x,y,z\in X.
Then d is called a cone metric on X and (X,d) is called a cone metric space.
Definition 1.2 ([10])
Let X be a nonempty set and s\ge 1 be a given real number. A mapping d:X\times X\to E is said to be cone bmetric if and only if, for all x,y,z\in X, the following conditions are satisfied:

(i)
\theta <d(x,y) with x\ne y and d(x,y)=\theta if and only if x=y;

(ii)
d(x,y)=d(y,x);

(iii)
d(x,y)\le s[d(x,z)+d(z,y)].
The pair (X,d) is called a cone bmetric space.
Remark 1.3 The class of cone bmetric spaces is larger than the class of cone metric spaces since any cone metric space must be a cone bmetric space. Therefore, it is obvious that cone bmetric spaces generalize bmetric spaces and cone metric spaces.
We can present a number of examples, as follows, which show that introducing a cone bmetric space instead of a cone metric space is meaningful since there exist cone bmetric spaces which are not cone metric spaces.
Example 1.4 Let E={\mathbb{R}}^{2}, P=\{(x,y)\in E:x,y\ge 0\}\subset E, X=\mathbb{R} and d:X\times X\to E such that d(x,y)=({xy}^{p},\alpha {xy}^{p}), where \alpha \ge 0 and p>1 are two constants. Then (X,d) is a cone bmetric space, but not a cone metric space. In fact, we only need to prove (iii) in Definition 1.2 as follows:
Let x,y,z\in X. Set u=xz, v=zy, so xy=u+v. From the inequality
we have
which implies that d(x,y)\le s[d(x,z)+d(z,y)] with s={2}^{p}>1. But
is impossible for all x>z>y. Indeed, taking account of the inequality
we arrive at
for all x>z>y. Thus, (d3) in Definition 1.1 is not satisfied, i.e., (X,d) is not a cone metric space.
Example 1.5 Let X={l}^{p} with 0<p<1, where {l}^{p}=\{\{{x}_{n}\}\subset \mathbb{R}:{\sum}_{n=1}^{\mathrm{\infty}}{{x}_{n}}^{p}<\mathrm{\infty}\}. Let d:X\times X\to {\mathbb{R}}_{+},
where x=\{{x}_{n}\},y=\{{y}_{n}\}\in {l}^{p}. Then (X,d) is a bmetric space (see [5]). Put E={l}^{1}, P=\{\{{x}_{n}\}\in E:{x}_{n}\ge 0,\text{for all}n\ge 1\}. Letting the mapping \tilde{d}:X\times X\to E be defined by \tilde{d}(x,y)={\{\frac{d(x,y)}{{2}^{n}}\}}_{n\ge 1}, we conclude that (X,\tilde{d}) is a cone bmetric space with the coefficient s={2}^{\frac{1}{p}}>1, but it is not a cone metric space.
Example 1.6 Let X=\{1,2,3,4\}, E={\mathbb{R}}^{2}, P=\{(x,y)\in E:x\ge 0,y\ge 0\}. Define d:X\times X\to E by
Then (X,d) is a cone bmetric space with the coefficient s=\frac{6}{5}. But it is not a cone metric space since the triangle inequality is not satisfied. Indeed,
Definition 1.7 ([10])
Let (X,d) be a cone bmetric space, x\in X and \{{x}_{n}\} be a sequence in X. Then

(i)
\{{x}_{n}\} converges to x whenever, for every c\in E with \theta \ll c, there is a natural number N such that d({x}_{n},x)\ll c for all n\ge N. We denote this by {lim}_{n\to \mathrm{\infty}}{x}_{n}=x or {x}_{n}\to x (n\to \mathrm{\infty}).

(ii)
\{{x}_{n}\} is a Cauchy sequence whenever, for every c\in E with \theta \ll c, there is a natural number N such that d({x}_{n},{x}_{m})\ll c for all n,m\ge N.

(iii)
(X,d) is a complete cone bmetric space if every Cauchy sequence is convergent.
The following lemmas are often used (in particular when dealing with cone metric spaces in which the cone need not be normal).
Lemma 1.8 ([9])
Let P be a cone and \{{a}_{n}\} be a sequence in E. If c\in intP and \theta \le {a}_{n}\to \theta (as n\to \mathrm{\infty}), then there exists N such that for all n>N, we have {a}_{n}\ll c.
Lemma 1.9 ([9])
Let x,y,z\in E, if x\le y and y\ll z, then x\ll z.
Lemma 1.10 ([10])
Let P be a cone and \theta \le u\ll c for each c\in intP, then u=\theta.
Lemma 1.11 ([11])
Let P be a cone. If u\in P and u\le ku for some 0\le k<1, then u=\theta.
Lemma 1.12 ([9])
Let P be a cone and a\le b+c for each c\in intP, then a\le b.
2 Main results
In this section, we will present some fixed point theorems for contractive mappings in the setting of cone bmetric spaces. Furthermore, we will give examples to support our main results.
Theorem 2.1 Let (X,d) be a complete cone bmetric space with the coefficient s\ge 1. Suppose the mapping T:X\to X satisfies the contractive condition
where \lambda \in [0,1) is a constant. Then T has a unique fixed point in X. Furthermore, the iterative sequence \{{T}^{n}x\} converges to the fixed point.
Proof Choose {x}_{0}\in X. We construct the iterative sequence \{{x}_{n}\}, where {x}_{n}=T{x}_{n1}, n\ge 1, i.e., {x}_{n+1}=T{x}_{n}={T}^{n+1}{x}_{0}. We have
For any m\ge 1, p\ge 1, it follows that
Let \theta \ll c be given. Notice that \frac{{s}^{p}{\lambda}^{m+1}}{s\lambda}d({x}_{1},{x}_{0})+{s}^{p1}{\lambda}^{m}d({x}_{1},{x}_{0})\to \theta as m\to \mathrm{\infty} for any k. Making full use of Lemma 1.8, we find {m}_{0}\in \mathbb{N} such that
for each m>{m}_{0}. Thus,
for all m>{m}_{0} and any p. So, by Lemma 1.9, \{{x}_{n}\} is a Cauchy sequence in (X,d). Since (X,d) is a complete cone bmetric space, there exists {x}^{\ast}\in X such that {x}_{n}\to {x}^{\ast}. Take {n}_{0}\in \mathbb{N} such that d({x}_{n},{x}^{\ast})\ll \frac{c}{s(\lambda +1)} for all n>{n}_{0}. Hence,
for each n>{n}_{0}. Then, by Lemma 1.10, we deduce that d(T{x}^{\ast},{x}^{\ast})=\theta, i.e., T{x}^{\ast}={x}^{\ast}. That is, {x}^{\ast} is a fixed point of T.
Now we show that the fixed point is unique. If there is another fixed point {y}^{\ast}, by the given condition,
By Lemma 1.11, {x}^{\ast}={y}^{\ast}. The proof is completed. □
Example 2.2 Let X=[0,1], E={\mathbb{R}}^{2} and p>1 be a constant. Take P=\{(x,y)\in E:x,y\ge 0\}. We define d:X\times X\to E as
Then (X,d) is a complete cone bmetric space. Let us define T:X\to X as
Therefore,
Here 0\in X is the unique fixed point of T.
Theorem 2.3 Let (X,d) be a complete cone bmetric space with the coefficient s\ge 1. Suppose the mapping T:X\to X satisfies the contractive condition
where the constant {\lambda}_{i}\in [0,1) and {\lambda}_{1}+{\lambda}_{2}+s({\lambda}_{3}+{\lambda}_{4})<min\{1,\frac{2}{s}\}, i=1,2,3,4. Then T has a unique fixed point in X. Moreover, the iterative sequence \{{T}^{n}x\} converges to the fixed point.
Proof Fix {x}_{0}\in X and set {x}_{1}=T{x}_{0} and {x}_{n+1}=T{x}_{n}={T}^{n+1}{x}_{0}. Firstly, we see
It follows that
Secondly,
This establishes that
Adding up (2.1) and (2.2) yields
Put \lambda =\frac{{\lambda}_{1}+{\lambda}_{2}+s({\lambda}_{3}+{\lambda}_{4})}{2{\lambda}_{1}{\lambda}_{2}s({\lambda}_{3}+{\lambda}_{4})}, it is easy to see that 0\le \lambda <1. Thus,
Following an argument similar to that given in Theorem 2.1, there exists {x}^{\ast}\in X such that {x}_{n}\to {x}^{\ast}. Let c\gg \theta be arbitrary. Since {x}_{n}\to {x}^{\ast}, there exists N such that
Next we claim that {x}^{\ast} is a fixed point of T. Actually, on the one hand,
which implies that
On the other hand,
which means that
Combining (2.3) and (2.4) yields
Simple calculations ensure that
It is easy to see from Lemma 1.10 that d({x}^{\ast},T{x}^{\ast})=\theta, i.e., {x}^{\ast} is a fixed point of T. Finally, we show the uniqueness of the fixed point. Indeed, if there is another fixed point {y}^{\ast}, then
Owing to 0\le s({\lambda}_{3}+{\lambda}_{4})<1, we deduce from Lemma 1.11 that {x}^{\ast}={y}^{\ast}. Therefore, we complete the proof. □
Remark 2.4 Theorem 2.1 extends the famous Banach contraction principle to that in the setting of cone bmetric spaces.
Remark 2.5 Any fixed point theorem in the setting of a metric space, a bmetric space or a cone metric space cannot cope with Example 2.2. So, Example 2.2 shows that the fixed point theory of cone bmetric spaces offers independently a strong tool for studying the positive fixed points of some nonlinear operators and the positive solutions of some operator equations.
Remark 2.6 The main results are some valuable additions to the available references regarding cone bmetric spaces since we have known few fixed point theorems of contractive mappings in the setting of cone bmetric spaces.
3 Applications
In this section we shall apply Theorem 2.1 to the firstorder periodic boundary problem
where F:[h,h]\times [\xi \delta ,\xi +\delta ] is a continuous function.
Example 3.1 Consider the boundary problem (3.1) with the continuous function F and suppose F(x,y) satisfies the local Lipschitz condition, i.e., if x\le h, {y}_{1},{y}_{2}\in [\xi \delta ,\xi +\delta ], it induces
Set M={max}_{[h,h]\times [\xi \delta ,\xi +\delta ]}F(x,y) such that {h}^{2}<min\{\delta /{M}^{2},1/{L}^{2}\}, then there exists a unique solution of (3.1).
Proof Let X=E=C([h,h]) and P=\{u\in E:u\ge 0\}. Put d:X\times X\to E as d(x,y)=f(t){max}_{h\le t\le h}{x(t)y(t)}^{2} with f:[h,h]\to \mathbb{R} such that f(t)={e}^{t}. It is clear that (X,d) is a complete cone bmetric space.
Note that (3.1) is equivalent to the integral equation
Define a mapping T:C([h,h])\to \mathbb{R} by Tx(t)=\xi +{\int}_{0}^{t}F(\tau ,x(\tau ))\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau. If
then from
and
we speculate T:B(\xi ,\delta f)\to B(\xi ,\delta f) is a contractive mapping.
Finally, we prove that (B(\xi ,\delta f),d) is complete. In fact, suppose \{{x}_{n}\} is a Cauchy sequence in B(\xi ,\delta f). Then \{{x}_{n}\} is also a Cauchy sequence in X. Since (X,d) is complete, there is x\in X such that {x}_{n}\to x (n\to \mathrm{\infty}). So, for each c\in intP, there exists N, whenever n>N, we obtain d({x}_{n},x)\ll c. Thus, it follows from
and Lemma 1.12 that d(\xi ,x)\le \delta f, which means x\in B(\xi ,\delta f), that is, (B(\xi ,\delta f),d) is complete.
Owing to the above statement, all the conditions of Theorem 2.1 are satisfied. Hence, T has a unique fixed point x(t)\in B(\xi ,\delta f). That is to say, there exists a unique solution of (3.1). □
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Acknowledgements
The authors thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper. The research was supported by the National Natural Science Foundation of China (10961003) and partly supported by the Graduate Initial Fund of Hubei Normal University (2008D36).
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An erratum to this article can be found online at 10.1186/16871812201455.
An erratum to this article is available at http://dx.doi.org/10.1186/16871812201455.
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Huang, H., Xu, S. Fixed point theorems of contractive mappings in cone bmetric spaces and applications. Fixed Point Theory Appl 2013, 112 (2013). https://doi.org/10.1186/168718122013112
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DOI: https://doi.org/10.1186/168718122013112
Keywords
 cone bmetric space
 fixed point
 periodic boundary problem