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

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.

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, where Then, 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.

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 in and 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 .where Suppose that there exists x₀ ∈ X such that ω_λ(x₀, fx₀) < ∞ for all λ > 0. Then, f has a unique fixed point in and 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(f^mx₀, 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.

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).

1. Mongkolkeha, C, Sintunavarat, W, Kumam, P: Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory and Applications. 2011, 93 (2011). BioMed Central Full Text