Research  Open  Published:
Some new theorems of expanding mappings without continuity in cone metric spaces
Fixed Point Theory and Applicationsvolume 2013, Article number: 3 (2013)
The Erratum to this article has been published in Fixed Point Theory and Applications 2014 2014:178
Abstract
In this paper, fixed point theorems for one mapping and common fixed point theorems for two mappings satisfying generalized expansive conditions are obtained. The mappings are not necessarily continuous and the cone is not normal. These results improve and generalize several wellknown comparable results in (Aage and Salunke, in Acta Mathematica Sinica, English Series 27(6):11011106, 2011). Moreover, examples are given to support our new results.
MSC:54H25, 47H10.
1 Introduction and preliminaries
Recently, Huang and Zhang [1] introduced the concept of a cone metric space as a generalization of a metric space. They proved the properties of sequences in cone metric spaces and obtained various fixed point theorems for contractive mappings. Afterwords, Abbas and Jungck [2] established the common fixed points for two mappings without exploiting the notion of continuity. Since then, common fixed point theorems in cone metric spaces have been proved for mappings satisfying different contractive conditions by many authors (see [3–8]). But there are few results about expanding mappings. Chintaman and Jagannath [9] introduced several meaningful fixed point theorems for one expanding mapping. However, the mapping depended strongly on continuity. In this paper, we delete the continuity of the mappings and obtain some fixed point theorems for one expanding mapping and introduce common fixed point theorems for two expanding mappings, which satisfy generalized expansive conditions in nonnormal cone metric spaces. These results improve and generalize some important known results in [9, 10].
We recall some definitions of cone metric spaces and some of their properties [1]. Let E be a real Banach space and P be a subset of E. θ denotes the zero element of E and intP denotes 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)
$x\in P$ and $x\in P\Rightarrow x=\theta $.
Given a cone $P\subset E$, we define a partial ordering ≤ with respect to P by $x\le y$ if and only if $yx\in P$. We will write $x<y$ if $x\le y$ and $x\ne y$, while $x\ll y$ will stand for $yx\in intP$. A cone P is called normal if there is a number $K>0$ such that for all $x,y\in P$,
The least positive number satisfying the above inequality is called the normal constant of P.
Definition 1.1 ([1])
Let X be a nonempty set. Suppose that the mapping $d:X\times X\to E$ satisfies the following:

(d1)
$\theta \le d(x,y)$ for all $x,y\in X$ 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. It is clear that the cone metric space is more general than a metric space.
Definition 1.2 ([1])
Let $(X,d)$ be a cone metric space. Then we say that $\{{x}_{n}\}$ is

(i)
a Cauchy sequence if for every $c\in E$ with $c\gg \theta $, there is N such that for all $n,m>N$, $d({x}_{n},{x}_{m})\ll c$;

(ii)
a convergent sequence if for every $c\in E$ with $c\gg \theta $, there is N such that for all $m>N$, $d({x}_{m},x)\ll c$ for some fixed x in X.
A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
Lemma 1.3 ([11])
The limit of a convergent sequence in a cone metric space is unique.
2 Main results
In this section, we prove some fixed point theorems for expanding mappings without continuity in the following theorems.
Theorem 2.1 Let $(X,d)$ be a complete cone metric space. Suppose the mapping $f:X\to X$ is onto and satisfies
for all $x,y\in X$, where ${a}_{i}$ ($i=1,2,3,4,5$) satisfies ${a}_{1}+{a}_{2}+{a}_{3}>1$ and ${a}_{3}\le 1+{a}_{4}$. Then f has a fixed point.
Proof Since f is an onto mapping, for each ${x}_{0}\in X$, there exists $f{x}_{1}={x}_{0}$. Continuing this process, we can define $\{{x}_{n}\}$ by ${x}_{n}=f{x}_{n+1}$, $n=0,1,2,\dots $ . Without loss of generality, we suppose that ${x}_{n1}\ne {x}_{n}$ for all $n\ge 1$. According to (2.1), we have
By $d({x}_{n+1},{x}_{n1})\ge d({x}_{n+1},{x}_{n})d({x}_{n1},{x}_{n})$, the above inequality implies that
Let $h=\frac{1{a}_{3}+{a}_{4}}{{a}_{1}+{a}_{2}+{a}_{4}}$. By ${a}_{3}\le 1+{a}_{4}$ and ${a}_{1}+{a}_{2}+{a}_{3}>1$, we know ${a}_{1}+{a}_{2}+{a}_{4}>1{a}_{3}+{a}_{4}\ge 0$ and $h\in [0,1)$. Hence, we get
So, by the triangle inequality, for any $n>m$, we see
Thus, as $h\in [0,1)$, we can choose a natural number ${N}_{0}$ such that $\frac{{h}^{m}}{1h}d({x}_{1},{x}_{0})\ll c$ for each $c\gg \theta $ and $m>{N}_{0}$. Hence, we see
Therefore, $\{{x}_{n}\}$ is a Cauchy sequence in $(X,d)$.
Since X is complete, there exists $q\in X$ such that $f{x}_{n+1}={x}_{n}\to q$ as $n\to \mathrm{\infty}$. Consequently, we can find a $p\in X$ such that $fp=q$. Now, we show that $p=q$. Substituting $x=p$, $y={x}_{n+1}$ in (2.1), we get
For the second and fourth term on the righthand side, we have $d(p,q)\ge d(p,{x}_{n+1})d(q,{x}_{n+1})$ and $d(p,{x}_{n})\ge d(p,{x}_{n+1})d({x}_{n},{x}_{n+1})$. For the lefthand side, $d(q,{x}_{n})\le d(q,{x}_{n+1})+d({x}_{n+1},{x}_{n})$. It follows that
Now, we have
If $1+{a}_{2}{a}_{5}>0$ for each $c\gg \theta $, we can choose a natural number ${N}_{1}$ such that $d({x}_{n+1},{x}_{n})\ll \frac{c}{2}$ and $d(q,{x}_{n+1})\ll \frac{({a}_{1}+{a}_{2}+{a}_{4})c}{2(1+{a}_{2}{a}_{5})}$ for $n\ge {N}_{1}$. Thus, we obtain
If $1+{a}_{2}{a}_{5}\le 0$ for $n\ge {N}_{1}$,
Therefore, ${x}_{n+1}\to p$. From Lemma 1.3, we see $p=q$. The conclusion is true. □
Taking some particular value of ${a}_{i}$ ($i=1,2,3,4,5$) in Theorem 2.1, we obtain several new results in the following.
Corollary 2.2 Let $(X,d)$ be a complete cone metric space. Suppose the mapping $f:X\to X$ is onto and satisfies
for all $x,y\in X$, where $p\le 1$ and $k+l+p>1$. Then f has a fixed point.
Corollary 2.3 Let $(X,d)$ be a complete cone metric space. Suppose the mapping $f:X\to X$ is onto and satisfies
for all $x,y\in X$, where k, l are constants and $k>1$. Then f has a fixed point.
Remark 2.4 Obviously, in our theorem and its corollaries above, we delete the continuity of the mappings which is essential in the results of [9]. Moreover, in Corollary 2.2 we delete $k\ge 1$, $l>1$, which is essential in Theorem 2.6 in [9]. In Corollary 2.3 we delete $l\ge 0$, which is essential in Theorem 2.5 in [9]. Theorem 2.3 in [9] is a special case of Theorem 2.1 with ${a}_{1}={a}_{4}={a}_{5}=0$, ${a}_{2}={a}_{3}=K$, and f is continuous.
Now, we introduce some common fixed point theorems for two expanding mappings which satisfy generalized expansive conditions without continuity of the mappings.
Theorem 2.5 Let $(X,d)$ be a complete cone metric space. Suppose mappings $f,g:X\to X$ are onto and satisfy
for all $x,y\in X$, where ${a}_{i}$ ($i=1,2,3,4,5$) satisfies ${a}_{1}+{a}_{2}+{a}_{3}>1$ and ${a}_{2}\le 1+{a}_{5}$, ${a}_{3}\le 1+{a}_{4}$. Then f and g have a common fixed point.
Proof Suppose ${x}_{0}$ is an arbitrary point in X. Since f, g are onto, there exist ${x}_{1},{x}_{2}\in X$ such that ${x}_{0}=g{x}_{1}$, ${x}_{1}=f{x}_{2}$. Continuing this process, we can define $\{{x}_{n}\}$ by ${x}_{2n}=g{x}_{2n+1}$, ${x}_{2n+1}=f{x}_{2n+2}$, $n=0,1,2,\dots $ . By (2.2), we have
Since $d({x}_{2n+2},{x}_{2n})\ge d({x}_{2n+2},{x}_{2n+1})d({x}_{2n},{x}_{2n+1})$, the above inequality implies that
Similarly, it can be shown that
which also implies that
Let $M=\frac{1{a}_{3}+{a}_{4}}{{a}_{1}+{a}_{2}+{a}_{4}}$, $N=\frac{1{a}_{2}+{a}_{5}}{{a}_{1}+{a}_{3}+{a}_{5}}$. From ${a}_{1}+{a}_{2}+{a}_{3}>1$ and ${a}_{2}\le 1+{a}_{5}$, ${a}_{3}\le 1+{a}_{4}$, we see ${a}_{1}+{a}_{2}+{a}_{4}>1{a}_{3}+{a}_{4}\ge 0$ and ${a}_{1}+{a}_{3}+{a}_{5}>1{a}_{2}+{a}_{5}\ge 0$. Thus, $h=MN\in [0,1)$. Now, by induction we have
and
Hence, for any $n>m$, we deduce
In an analogous way, we gain
and
Thus, for $n>m>0$,
where ${\lambda}_{m}\to 0$ as $m\to \mathrm{\infty}$.
For each $c\gg \theta $, choose $\delta >0$ such that $cx\in intP$, where $\parallel x\parallel <\delta $, i.e., $x\ll c$. For this δ, we can choose a natural number ${N}_{2}$ such that $\parallel {\lambda}_{m}d({x}_{1},{x}_{0})\parallel <\delta $ for $m>{N}_{2}$. Thus, we get
Therefore, $\{{x}_{n}\}$ is a Cauchy sequence in $(X,d)$.
As X is complete, there exists $q\in X$ such that ${x}_{n}\to q$ as $n\to \mathrm{\infty}$. It is equivalent to ${x}_{2n}=g{x}_{2n+1}\to q$, ${x}_{2n+1}=f{x}_{2n+2}\to q$ as $n\to \mathrm{\infty}$. Since f, g are onto, there exist $u,p\in X$ such that $fu=gp=q$. Now, we show that $u=p=q$. By (2.2), we have
that is,
From the fact that $d(p,q)\ge d(p,{x}_{2n+2})d(q,{x}_{2n+2})$, $d(p,{x}_{2n+1})\ge d(p,{x}_{2n+2})d({x}_{2n+1},{x}_{2n+2})$ and $d({x}_{2n+1},q)\le d({x}_{2n+1},{x}_{2n+2})+d({x}_{2n+2},q)$, we get
Now, we have
For each $c\gg \theta $, we can choose a natural number ${N}_{3}$ such that $d({x}_{2n+1},{x}_{2n+2})\ll \frac{c}{2}$ and $d({x}_{2n+2},q)\ll \frac{({a}_{1}+{a}_{3}+{a}_{5})c}{4}$ for $n\ge {N}_{3}$. Hence, we obtain $d(p,{x}_{2n+2})\ll \frac{c}{2}+\frac{c}{2}=c$, i.e., ${x}_{2n+2}\to p$. By Lemma 1.3, we know $p=q$, $gq=q$. Similarly, we also have
As in the previous proof, it is not difficult to get $q=u$, i.e., $fq=q$. Therefore, $fq=gq=q$. □
Corollary 2.6 Let $(X,d)$ be a complete cone metric space. Suppose mappings $f,g:X\to X$ are onto and satisfy
for all $x,y\in X$, where $\beta \le 1+\gamma $ and $\alpha +2\beta >1$. Then f and g have a common fixed point.
Corollary 2.7 Let $(X,d)$ be a complete cone metric space. Suppose mappings $f,g:X\to X$ are onto and satisfy
for all $x,y\in X$, where $k>1$ is a constant. Then f and g have a unique common fixed point.
Corollary 2.8 Let $(X,d)$ be a complete cone metric space. Suppose the mapping $f:X\to X$ is onto and satisfies
for all $x,y\in X$, where p, q are positive integers and $k>1$ is a constant. Then f has a unique fixed point.
Proof Let $f={f}^{p}$, $g={f}^{q}$. Since f is an onto mapping, $f={f}^{p}$, $g={f}^{q}$ are onto mappings, the conditions of Corollary 2.7 are satisfied. □
Remark 2.9 In Corollary 2.8, we obtain Corollary 2.2 in [9] when we take $p=q$.
Now, we present the following examples. In Example 1, we gain a fixed point for one expanding mapping of the situation when Corollary 2.2 can be applied, while the results in [9] cannot. In Example 2, we obtain the common fixed point for two expanding mappings in a cone metric space.
Example 1 Let $X=[1,+\mathrm{\infty})$, $E={C}_{\mathbb{R}}^{2}([0,1])$ with $\parallel x\parallel ={\parallel x\parallel}_{\mathrm{\infty}}+{\parallel {x}^{\prime}\parallel}_{\mathrm{\infty}}$ and $P=\{x\in E:x(t)\ge 0,t\in [0,1]\}$ (this cone is not normal). Define $d:X\times X\to E$ by $d(x,y)=xy\phi $, where $\phi :[0,1]\to \mathbb{R}$ such that $\phi (t)={e}^{t}$. Consider the mapping
which implies that f is onto in X. Taking $k=2$, $l=\frac{1}{4}$, $p=\frac{1}{4}$, for $1\le x<2$, all the conditions of Corollary 2.2 are fulfilled. Indeed, since $0<\frac{1}{2}x+\frac{1}{2}y+\frac{3}{2}<2x+2y1$, we have
For $x\ge 2$, since $f(x)$ is increasing in x, we have
Therefore, we can apply Corollary 2.2 and conclude that f has a (unique) fixed point 0 in X. Since f is not continuous in X and $l<1$, Theorem 2.6 in [9] is not applicable. Hence, our theorems have improved and generalized the main results in [9].
Example 2 Let $X=\{1,2,3\}$ and $d:X\times X\to {\mathbb{R}}^{2}$ be defined by $d(x,y)=(0,0)$ for $x=y$ and
Then $(X,d)$ is a complete cone metric space. Further, define mappings $f,g:X\to X$ as follows:
which implies that f, g are onto in X. Note that
for all $x,y\in X$ by taking ${a}_{1}=\frac{1}{7}$, ${a}_{2}=\frac{2}{7}$, ${a}_{3}=\frac{11}{7}$, ${a}_{4}=\frac{5}{7}$, ${a}_{5}=\frac{3}{7}$. Thus, all the conditions of Theorem 2.5 are fulfilled. Then f and g have a unique common fixed point 1 in X.
Remark 2.10 Obviously, in the above two examples, we obtain the (common) fixed point which essentially needs the structure of a cone metric and not an ordinary metric on X. Then the results in a metric space in [10] cannot be applied to these examples.
References
 1.
Huang LG, Zhang X: Cone metric space and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 2007, 332: 1468–1476. 10.1016/j.jmaa.2005.03.087
 2.
Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J. Math. Anal. Appl. 2008, 341: 416–420. 10.1016/j.jmaa.2007.09.070
 3.
Abbas M, Rhoades BE: Fixed and periodic point results in cone metric spaces. Appl. Math. Lett. 2009, 22: 511–515. 10.1016/j.aml.2008.07.001
 4.
Radenovic S: Common fixed points under contractive conditions in cone metric spaces. Comput. Math. Appl. 2009, 58: 1273–1278. 10.1016/j.camwa.2009.07.035
 5.
Altun I, Damjanovic B, Djoric D: Fixed point and common fixed point theorems on ordered cone metric spaces. Appl. Math. Lett. 2010, 23: 310–316. 10.1016/j.aml.2009.09.016
 6.
Rezapour S, Hamlbarani R: Some note on the paper “Cone metric spaces and fixed point theorems of contractive mappings”. J. Math. Anal. Appl. 2008, 345: 719–724. 10.1016/j.jmaa.2008.04.049
 7.
Han Y, Xu SY: New common fixed point results for four maps on cone metric spaces. Appl. Math. 2011, 2: 1114–1118. 10.4236/am.2011.29153
 8.
Bari CD, Vetro P: φ pairs and common fixed points in cone metric spaces. Rend. Circ. Mat. Palermo 2008, 57: 279–285. 10.1007/s1221500800209
 9.
Aage CT, Salunke JN: Some fixed point theorems for expansion onto mappings on cone metric spaces. Acta Math. Sin. Engl. Ser. 2011, 27(6):1101–1106. 10.1007/s1011401196069
 10.
Wang SZ, Li BY, Gao ZM, Lseki K: Some fixed points of expansion mappings. Math. Jpn. 1984, 29: 631–636.
 11.
Zhang X: Common fixed point theorems of Lipschitz type mappings in cone metric spaces. Acta Math. Sin. 2010, 53(6):1139–1148. (in Chinese)
Acknowledgements
The authors thank the editor and the referees for their valuable comments and suggestions which improve greatly the quality for this paper. The research was supported by the Foundation of Education Ministry, Hubei Province, China (No: D20102502).
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
An erratum to this article can be found online at 10.1186/168718122014178.
An erratum to this article is available at http://dx.doi.org/10.1186/168718122014178.
Rights and permissions
About this article
Received
Accepted
Published
DOI
Keywords
 cone metric space
 expanding mapping
 common fixed point