• Erratum
• Open Access

# Erratum to Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl. 2011, 2011:93

Fixed Point Theory and Applications20122012:103

https://doi.org/10.1186/1687-1812-2012-103

• Accepted: 21 June 2012
• Published:

The original article was published in Fixed Point Theory and Applications 2011 2011:93

## Abstract

This article is written due to a small gap in our published paper. In this erratum, we point out and fix the problem to set our existed results at the best of their perfection.

## 1. On the results in 

In , the authors have studied and introduced some fixed point theorems in the frame-work of a modular metric space. We shall first state their results and then discuss some small gap herewith.

Theorem 1.1 (Theorem 3.2 in Mongkolkeha et al.). Let X ω be a complete modular metric space and f be a self-mapping on X satisfying the inequality
${\omega }_{\lambda }\left(fx,\phantom{\rule{2.77695pt}{0ex}}fy\right)\le k{\omega }_{\lambda }\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right),$

for all x, y X ω , where k [0, 1). Then, f has a unique fixed point in ${x}_{*}\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}{X}_{\omega }$ and the sequence{f n x} converges to x * .

Theorem 1.2 (Theorem 3.6 in Mongkolkeha et al.). Let X ω be a complete modular metric space and f be a self mapping on X satisfying the inequality
${\omega }_{\lambda }\left(fx,\phantom{\rule{2.77695pt}{0ex}}fy\right)\le k\left[{\omega }_{2\lambda }\left(x,\phantom{\rule{2.77695pt}{0ex}}fx\right)+{\omega }_{2\lambda }\left(y,\phantom{\rule{2.77695pt}{0ex}}fy\right)\right],$

for all x, y X ω , where$k\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}\left[0,\phantom{\rule{0.3em}{0ex}}\frac{1}{2}\right).$Then, f has a unique fixed point in ${x}_{*}\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}{X}_{\omega }$ and the sequence{f n x} converges to x *.

We now claim that the conditions in the above theorems are not sufficient to guarantee the existence and uniqueness of the fixed points. We state a counterexample to Theorem 1.1 in the following:

Example 1.3. Let X := {0, 1} and ω be given by
${\omega }_{\lambda }\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)=\left\{\begin{array}{c}\hfill \infty ,\phantom{\rule{1em}{0ex}}\mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}0<\lambda <1\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}x\ne y,\phantom{\rule{1em}{0ex}}\hfill \\ \hfill 0,\phantom{\rule{1em}{0ex}}\mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}\lambda \ge 1\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{or}}\phantom{\rule{2.77695pt}{0ex}}x=y.\phantom{\rule{1em}{0ex}}\hfill \end{array}\right\$
Thus, the modular metric space X ω = X. Now let f be a self-mapping on X defined by
$\left\{\begin{array}{c}\hfill f\left(0\right)=1,\hfill \\ \hfill f\left(1\right)=0.\hfill \end{array}\right\$

Then, f is satisfies the inequality (1.1) with any k [0, 1) but it possesses no fixed point after all.

Notice that this gap flaws the theorems only when is involved.

## 2. Revised theorems

In this section, we shall now give the corrections to our theorems in .

Theorem 2.1. Let X ω be a complete modular metric space and f be a self mapping on X satisfying the inequality
${\omega }_{\lambda }\left(fx,\phantom{\rule{2.77695pt}{0ex}}fy\right)\le k{\omega }_{\lambda }\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right),$

for all x, y X ω , where k [0, 1). Suppose that there exists x0 X such that ω λ (x0, fx0) < ∞ for all λ > 0. Then, f has a unique fixed point in${x}_{*}\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}{X}_{\omega }$and the sequence {f n x0} converges to x *.

Theorem 2.2. Let X ω be a complete modular metric space and f be a self-mapping on X satisfying the inequality
${\omega }_{\lambda }\left(fx,\phantom{\rule{2.77695pt}{0ex}}fy\right)\le k\left[{\omega }_{2\lambda }\left(x,\phantom{\rule{2.77695pt}{0ex}}fx\right)+{\omega }_{2\lambda }\left(y,\phantom{\rule{2.77695pt}{0ex}}fy\right)\right],$

for all x, y X ω .where$k\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}\left[0,\phantom{\rule{0.3em}{0ex}}\frac{1}{2}\right).$Suppose that there exists x0 X such that ω λ (x 0 , fx0) < ∞ for all λ > 0. Then, f has a unique fixed point in${x}_{*}\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}{X}_{\omega }$and the sequence {f n x} converges to x *.

Proof (of Theorem 2.1). Let λ > 0 and observe that

${\omega }_{\lambda }\left({f}^{n}{x}_{0},\phantom{\rule{2.77695pt}{0ex}}{f}^{n+1}{x}_{0}\right)\le k{\omega }_{\lambda }\left({f}^{n-1}{x}_{0},\phantom{\rule{2.77695pt}{0ex}}{f}^{n}{x}_{0}\right)\le \cdots \le {k}^{n}{\omega }_{\lambda }\left({x}_{0},\phantom{\rule{2.77695pt}{0ex}}f{x}_{0}\right)<\infty ,$ for all $n\in ℕ$

Assume m > n be two positive integers. Observe that
$\begin{array}{l}{\omega }_{\lambda }\left({f}^{m}{x}_{0},\phantom{\rule{1em}{0ex}}{f}^{n}{x}_{0}\right)\le {\omega }_{\lambda }\left({f}^{n}{x}_{0},\phantom{\rule{1em}{0ex}}{f}^{n+1}{x}_{0}\right)+\left({f}^{n+1}{x}_{0},\phantom{\rule{1em}{0ex}}{f}^{n+2}{x}_{0}\right)+\cdots +{\omega }_{\lambda }\left({f}^{m-1}{x}_{0},\phantom{\rule{1em}{0ex}}{f}^{m}{x}_{0}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\le \left({k}^{n}+{k}^{n+1}+\cdots +{k}^{m-1}\right){\omega }_{\lambda }\left({x}_{0},\phantom{\rule{1em}{0ex}}f{x}_{0}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\le \left({k}^{n}+{k}^{n+1}+\cdots \right){\omega }_{\lambda }\left({x}_{0},\phantom{\rule{1em}{0ex}}f{x}_{0}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}=\frac{{k}^{n}}{1-k}{\omega }_{\lambda }\left({x}_{0},\phantom{\rule{1em}{0ex}}f{x}_{0}\right).\end{array}$
Since ω λ (x0, fx0) < ∞, we deduce that for any given ε > 0, ωλ(f m x0, f n x0) < ε for m > n > N with $N\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}ℕ$ big enough. Thus, {f n x0} is Cauchy and hence it converges to some ${x}_{*}\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}{X}_{\omega }$ in essence of the completeness of X ω . Observe further that
${\omega }_{\lambda }\left({x}_{*},\phantom{\rule{2.77695pt}{0ex}}f{x}_{*}\right)\le {\omega }_{\lambda }\left({x}_{*},\phantom{\rule{2.77695pt}{0ex}}{f}^{n}{x}_{0}\right)+k{\omega }_{\lambda }\left({f}^{n-1}{x}_{0},\phantom{\rule{2.77695pt}{0ex}}{x}_{*}\right).$
Letting n to obtain that ${\omega }_{\lambda }\left({x}_{*},\phantom{\rule{2.77695pt}{0ex}}f{x}_{*}\right)=0$ for all λ > 0. Therefore, x * is a fixed point of f. Suppose also that ${y}_{*}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}f{y}_{*}.$ Note that
${\omega }_{\lambda }\left({x}_{*},\phantom{\rule{2.77695pt}{0ex}}{y}_{*}\right)={\omega }_{\lambda }\left(f{x}_{*},\phantom{\rule{2.77695pt}{0ex}}f{y}_{*}\right)\le k{\omega }_{\lambda }\left({x}_{*},\phantom{\rule{2.77695pt}{0ex}}{y}_{*}\right),$

which implies that ${\omega }_{\lambda }\left({x}_{*},\phantom{\rule{2.77695pt}{0ex}}f{x}_{*}\right)=0$ for all λ > 0. Therefore, the theorem is proved.    □

For the proofs of the remaining theorem, take the idea of the above correction and combine with the proof aforementioned in  to obtain the expected results.

## Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, Bangkok, 10140, Thailand

## References 