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 [1]
In [1], 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.[1]). Let X ω be a complete modular metric space and f be a self-mapping on X satisfying the inequality
for all x, y ∈ X ω , where k ∈ [0, 1). Then, f has a unique fixed point in and the sequence{fn x} converges to x * .
Theorem 1.2 (Theorem 3.6 in Mongkolkeha et al.[1]). Let X ω be a complete modular metric space and f be a self mapping on X satisfying the inequality
for all x, y ∈ X ω , whereThen, f has a unique fixed point in and the sequence{fnx} 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
Thus, the modular metric space X ω = X. Now let f be a self-mapping on X defined by
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 [1].
Theorem 2.1. Let X ω be a complete modular metric space and f be a self mapping on X satisfying the inequality
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 inand the sequence {fnx0} 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
for all x, y ∈ X ω .whereSuppose that there exists x0 ∈ X such that ω λ (x 0 , fx0) < ∞ for all λ > 0. Then, f has a unique fixed point inand the sequence {fn x} converges to x *.
Proof (of Theorem 2.1). Let λ > 0 and observe that
for all
Assume m > n be two positive integers. Observe that
Since ω λ (x0, fx0) < ∞, we deduce that for any given ε > 0, ωλ(fmx0, fnx0) < ε for m > n > N with big enough. Thus, {fnx0} is Cauchy and hence it converges to some in essence of the completeness of X ω . Observe further that
Letting n → ∞ to obtain that for all λ > 0. Therefore, x* is a fixed point of f. Suppose also that Note that
which implies that 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 [1] to obtain the expected results.
References
Mongkolkeha C, Sintunavarat W, Kumam P: Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory and Applications 2011, 2011: 93. 10.1186/1687-1812-2011-93
Acknowledgements
The authors would like to thank Professor Rahim Alizadeh for questions and comments. Also, this work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No.55000613).
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The online version of the original article can be found at 10.1186/1687-1812-2011-93
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Mongkolkeha, C., Sintunavarat, W. & Kumam, P. Erratum to Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl. 2011, 2011:93. Fixed Point Theory Appl 2012, 103 (2012). https://doi.org/10.1186/1687-1812-2012-103
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DOI: https://doi.org/10.1186/1687-1812-2012-103