Skip to content
• Erratum
• Open Access

# Erratum to: Fixed point theorems of contractive mappings in cone b-metric spacesand applications

Fixed Point Theory and Applications20142014:55

https://doi.org/10.1186/1687-1812-2014-55

• Received: 26 November 2013
• Accepted: 26 November 2013
• Published:

The original article was published in Fixed Point Theory and Applications 2013 2013:112

## Correction

In this note we correct some errors that appeared in the article (Huang and Xu in FixedPoint Theory Appl. 2013:112, 2013) by modifying some conditions in the main theorems andexamples.

After examining the proofs of the main results in , we can find that there is something wrong with the proof of the Cauchy sequencein [, Theorem 2.1]. This leads to subsequent errors in Theorem 2.3 andrelated examples in . We also find that it is not rigorous to use the corresponding lemmas, and so theproof is inaccurate. The detailed reasons are given in the following.

On p.5 in , we conclude that
$\frac{{s}^{p}{\lambda }^{m+1}}{s-\lambda }d\left({x}_{1},{x}_{0}\right)+{s}^{p-1}{\lambda }^{m}d\left({x}_{1},{x}_{0}\right)\to \theta$
as $m\to \mathrm{\infty }$ for any $p\ge 1$. This is incorrect. Indeed, note that taking$\lambda =\frac{1}{\sqrt{s}}>\frac{1}{s}$ and $p=m+1$ leads to
$\frac{{s}^{p}{\lambda }^{m+1}}{s-\lambda }d\left({x}_{1},{x}_{0}\right)+{s}^{p-1}{\lambda }^{m}d\left({x}_{1},{x}_{0}\right)=\frac{{s}^{\frac{m+2}{2}}}{{s}^{\frac{3}{2}}-1}d\left({x}_{1},{x}_{0}\right)+{s}^{\frac{m}{2}}d\left({x}_{1},{x}_{0}\right)↛\theta$

as $m\to \mathrm{\infty }$. Therefore, it is impossible to utilize [, Lemma 1.8, 1.9] and demonstrate that $\left\{{x}_{n}\right\}$ is a Cauchy sequence.

In this note, we would like to slightly modify only one of the used conditions to achieveour claim.

The following theorem is a modification to [, Theorem 2.1]. The proof is the same as that in  except the proof of the Cauchy sequence. We will attain the desired goal by usingthe new modified condition $\lambda \in \left[0,\frac{1}{s}\right)$ instead of $\lambda \in \left[0,1\right)$.

Theorem 2.1 Let $\left(X,d\right)$ be a complete cone b-metric space with the coefficient $s\ge 1$. Suppose that the mapping $T:X\to X$ satisfies the contractive condition

where $\lambda \in \left[0,\frac{1}{s}\right)$ is a constant. Then T has a unique fixed point in X. Furthermore, the iterative sequence $\left\{{T}^{n}x\right\}$ converges to the fixed point.

Proof In order to show that $\left\{{x}_{n}\right\}$ is a Cauchy sequence, we only need the following calculations.For any $m\ge 1$, $p\ge 1$, it follows that
$\begin{array}{rcl}d\left({x}_{m},{x}_{m+p}\right)& \le & s\left[d\left({x}_{m},{x}_{m+1}\right)+d\left({x}_{m+1},{x}_{m+p}\right)\right]\\ \le & sd\left({x}_{m},{x}_{m+1}\right)+{s}^{2}\left[d\left({x}_{m+1},{x}_{m+2}\right)+d\left({x}_{m+2},{x}_{m+p}\right)\right]\\ \le & sd\left({x}_{m},{x}_{m+1}\right)+{s}^{2}d\left({x}_{m+1},{x}_{m+2}\right)+{s}^{3}d\left({x}_{m+2},{x}_{m+3}\right)\\ +\cdots +{s}^{p-1}d\left({x}_{m+p-2},{x}_{m+p-1}\right)+{s}^{p-1}d\left({x}_{m+p-1},{x}_{m+p}\right)\\ \le & s{\lambda }^{m}d\left({x}_{1},{x}_{0}\right)+{s}^{2}{\lambda }^{m+1}d\left({x}_{1},{x}_{0}\right)+{s}^{3}{\lambda }^{m+2}d\left({x}_{1},{x}_{0}\right)\\ +\cdots +{s}^{p-1}{\lambda }^{m+p-2}d\left({x}_{1},{x}_{0}\right)+{s}^{p}{\lambda }^{m+p-1}d\left({x}_{1},{x}_{0}\right)\\ =& s{\lambda }^{m}\left[1+s\lambda +{s}^{2}{\lambda }^{2}+\cdots +{\left(s\lambda \right)}^{p-1}\right]d\left({x}_{1},{x}_{0}\right)\le \frac{s{\lambda }^{m}}{1-s\lambda }d\left({x}_{1},{x}_{0}\right).\end{array}$
Let $\theta \ll c$ be given. Notice that $\frac{s{\lambda }^{m}}{1-s\lambda }d\left({x}_{1},{x}_{0}\right)\to \theta$ as $m\to \mathrm{\infty }$ for any p. Making full use of [, Lemma 1.8], we find ${m}_{0}\in \mathbb{N}$ such that
$\frac{s{\lambda }^{m}}{1-s\lambda }d\left({x}_{1},{x}_{0}\right)\ll c$
for each $m>{m}_{0}$. Thus,
$d\left({x}_{m},{x}_{m+p}\right)\le \frac{s{\lambda }^{m}}{1-s\lambda }d\left({x}_{1},{x}_{0}\right)\ll c$

for all $m\ge 1$, $p\ge 1$. So, by [, Lemma 1.9], $\left\{{x}_{n}\right\}$ is a Cauchy sequence in $\left(X,d\right)$. The proof is completed. □

As is indicated in the reviewer’s comments, [, Example 2.2] is too trivial. Therefore, [, Example 2.2] is withdrawn. Now we give another example as follows.

Example 2.2 Let $X=\left[0,0.48\right]$, $E={\mathbb{R}}^{2}$ and let $1\le p\le 6$ be a constant. Take $P=\left\{\left(x,y\right)\in E:x,y\ge 0\right\}$. We define $d:X×X\to E$ as
Then $\left(X,d\right)$ is a complete cone b-metric space with$s={2}^{p-1}$. Let us define $T:X\to X$ as
Thus, for all $x,y\in X$, we have
$\begin{array}{rcl}d\left(Tx,Ty\right)& =& \left({|Tx-Ty|}^{p},{|Tx-Ty|}^{p}\right)\\ =& \frac{1}{{2}^{p}}\left({|\left(cos\frac{x}{2}-cos\frac{y}{2}\right)-\left(|x-\frac{1}{2}|-|y-\frac{1}{2}|\right)|}^{p},\\ {|\left(cos\frac{x}{2}-cos\frac{y}{2}\right)-\left(|x-\frac{1}{2}|-|y-\frac{1}{2}|\right)|}^{p}\right)\\ \le & \frac{1}{{2}^{p}}\left({\left(|cos\frac{x}{2}-cos\frac{y}{2}|+|x-y|\right)}^{p},{\left(|cos\frac{x}{2}-cos\frac{y}{2}|+|x-y|\right)}^{p}\right)\\ \le & \frac{1}{{2}^{p}}\left({\left(\frac{|x+y|}{8}|x-y|+|x-y|\right)}^{p},{\left(\frac{|x+y|}{8}|x-y|+|x-y|\right)}^{p}\right)\\ \le & {0.56}^{p}\left({|x-y|}^{p},{|x-y|}^{p}\right)<\frac{1}{{2}^{p-1}}\left({|x-y|}^{p},{|x-y|}^{p}\right).\end{array}$

Hence, by Theorem 2.1, there exists ${x}_{0}\in X$ (in fact, it satisfies $0.472251591454<{x}_{0}<0.472251591479$) such that ${x}_{0}$ is the unique fixed point of T.

For the same reason, we need to use the new condition ${\lambda }_{1}+{\lambda }_{2}+s\left({\lambda }_{3}+{\lambda }_{4}\right)<\frac{2}{1+s}$ instead of the original condition ${\lambda }_{1}+{\lambda }_{2}+s\left({\lambda }_{3}+{\lambda }_{4}\right) in [, Theorem 2.3]. The correct statement is as follows.

Theorem 2.3 Let $\left(X,d\right)$ be a complete cone b-metric space with the coefficient $s\ge 1$. Suppose that the mapping $T:X\to X$ satisfies the contractive condition

where the constant ${\lambda }_{i}\in \left[0,1\right)$ and ${\lambda }_{1}+{\lambda }_{2}+s\left({\lambda }_{3}+{\lambda }_{4}\right)<\frac{2}{1+s}$, $i=1,2,3,4$. Then T has a unique fixed point in X. Moreover, the iterative sequence $\left\{{T}^{n}x\right\}$ converges to the fixed point.

Proof Following an identical argument that is given in [, Theorem 2.3] except substituting $0\le \lambda \le 1$ for $0\le \lambda \le \frac{1}{s}$ in line 26 of p.6 in , we obtain the proof of Theorem 2.3. □

In addition, based on the changes of Theorem 2.1, we need to change the condition${h}^{2} into ${h}^{2} for [, Example 3.1]. Let us give the corrected example.

We now apply Theorem 2.1 to the first-order periodic boundary problem
$\left\{\begin{array}{c}\frac{\mathrm{d}x}{\mathrm{d}t}=F\left(t,x\left(t\right)\right),\hfill \\ x\left(0\right)=\xi ,\hfill \end{array}$
(2.1)

where $F:\left[-h,h\right]×\left[\xi -\delta ,\xi +\delta \right]$ is a continuous function.

Example 2.4 Consider boundary problem (2.1) with the continuous function F,and suppose that $F\left(x,y\right)$ satisfies the local Lipschitz condition, i.e., if$|x|\le h$, ${y}_{1},{y}_{2}\in \left[\xi -\delta ,\xi +\delta \right]$, it induces
$|F\left(x,{y}_{1}\right)-F\left(x,{y}_{2}\right)|\le L|{y}_{1}-{y}_{2}|.$

Set $M={max}_{\left[-h,h\right]×\left[\xi -\delta ,\xi +\delta \right]}|F\left(x,y\right)|$ such that ${h}^{2}, then there exists a unique solution of (2.1).

Proof Let $X=E=C\left(\left[-h,h\right]\right)$ and $P=\left\{u\in E:u\ge 0\right\}$. Put $d:X×X\to E$ as $d\left(x,y\right)=f\left(t\right){max}_{-h\le t\le h}{|x\left(t\right)-y\left(t\right)|}^{2}$ with $f:\left[-h,h\right]\to \mathbb{R}$ such that $f\left(t\right)={e}^{t}$. It is clear that $\left(X,d\right)$ is a complete cone b-metric space with$s=2$.

Note that (2.1) is equivalent to the integral equation
$x\left(t\right)=\xi +{\int }_{0}^{t}F\left(\tau ,x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau .$
Define a mapping $T:C\left(\left[-h,h\right]\right)\to \mathbb{R}$ by $Tx\left(t\right)=\xi +{\int }_{0}^{t}F\left(\tau ,x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau$. If
$x\left(t\right),y\left(t\right)\in B\left(\xi ,\delta f\right)\triangleq \left\{\phi \left(t\right)\in C\left(\left[-h,h\right]\right):d\left(\xi ,\phi \right)\le \delta f\right\},$
then from
$\begin{array}{rcl}d\left(Tx,Ty\right)& =& f\left(t\right)\underset{-h\le t\le h}{max}{|{\int }_{0}^{t}F\left(\tau ,x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau -{\int }_{0}^{t}F\left(\tau ,y\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau |}^{2}\\ =& f\left(t\right)\underset{-h\le t\le h}{max}{|{\int }_{0}^{t}\left[F\left(\tau ,x\left(\tau \right)\right)-F\left(\tau ,y\left(\tau \right)\right)\right]\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau |}^{2}\\ \le & {h}^{2}f\left(t\right)\underset{-h\le \tau \le h}{max}{|F\left(\tau ,x\left(\tau \right)\right)-F\left(\tau ,y\left(\tau \right)\right)|}^{2}\\ \le & {h}^{2}{L}^{2}f\left(t\right)\underset{-h\le \tau \le h}{max}{|x\left(\tau \right)-y\left(\tau \right)|}^{2}\\ =& {h}^{2}{L}^{2}d\left(x,y\right),\end{array}$
and
$d\left(Tx,\xi \right)=f\left(t\right)\underset{-h\le t\le h}{max}{|{\int }_{0}^{t}F\left(\tau ,x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau |}^{2}\le {h}^{2}f\underset{-h\le \tau \le h}{max}{|F\left(\tau ,x\left(\tau \right)\right)|}^{2}\le {h}^{2}{M}^{2}f\le \delta f,$

we speculate that $T:B\left(\xi ,\delta f\right)\to B\left(\xi ,\delta f\right)$ is a contractive mapping.

Finally, we prove that $\left(B\left(\xi ,\delta f\right),d\right)$ is complete. In fact, suppose that $\left\{{x}_{n}\right\}$ is a Cauchy sequence in $B\left(\xi ,\delta f\right)$. Then $\left\{{x}_{n}\right\}$ is also a Cauchy sequence in X. Since$\left(X,d\right)$ 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 $d\left({x}_{n},x\right)\ll c$. Thus, it follows from
$d\left(\xi ,x\right)\le d\left({x}_{n},\xi \right)+d\left({x}_{n},x\right)\le \delta f+c$

and Lemma 1.12 in  that $d\left(\xi ,x\right)\le \delta f$, which means $x\in B\left(\xi ,\delta f\right)$, that is, $\left(B\left(\xi ,\delta f\right),d\right)$ is complete. □

Owing to the above statement, all conditions of Theorem 2.1 are satisfied. HenceT has a unique fixed point $x\left(t\right)\in B\left(\xi ,\delta f\right)$. That is to say, there exists a unique solution of (2.1).

Remark 2.5 Theorem 2.1 and Theorem 2.3 generalize and improve thecorresponding results in .

## Authors’ Affiliations

(1)
School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China
(2)
Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China

## References

Advertisement 