AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 241919 10.1155/2012/241919 241919 Erratum Erratum to “Fixed-Point Theorems for Multivalued Mappings in Modular Metric Spaces” Chaipunya Parin Mongkolkeha Chirasak Sintunavarat Wutiphol Kumam Poom Department of Mathematics Faculty of Science King Mongkut’s University of Technology Thonburi Bangkok 10140 Thailand kmutt.ac.th 2012 16 7 2012 2012 31 05 2012 18 06 2012 2012 Copyright © 2012 Parin Chaipunya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. On the Results in the Original Paper

In the original paper, the authors have studied and introduced some fixed-point theorems in the framework of a modular metric space. We will first state the main result and then discuss some small gap herewith.

Theorem 1.1 (see the original paper).

Let Xω be a complete modular metric space and F a multivalued self-mapping on Xω satisfying the inequality (1.1)Ωλ(Fx,Fy)kωλ(x,y), for  all x,yXω, where  k[0,1). Then,  F  has  a  fixed  point  in  Xω.

We now claim that the conditions in this theorem are not sufficient to guarantee the existence of the fixed points. We state a counterexample to Theorem 1.1 in the single-valued case as follows.

Example 1.2.

Let X:={0,1} and ω be given by (1.2)ωλ(x,y)={,if0<λ<1  andxy,0,ifλ1orx=y. Thus, the modular metric space Xω=X. Now let f be a self-mapping on X defined by (1.3)f(0)=1,f(1)=0. Then, f satisfies the hypothesis of Theorem 1.1 with any k[0,1) but it possesses no fixed point after all. Notice that this gap flaws the theorem only when is involved. Also, this gap is also found along the rest of the paper.

2. Revised Theorems

In this section, we will now give the corrections of our theorems in the original paper. Every theorem in Section 3 the original paper can be corrected by adding the following condition: (2.1)there  exists  x0Xω  such  that  ωλ(x0,y)<yFx0,   λ>0.

For the proofs, take the initial point x0 satisfying the condition (2.1). The rest of the proofs run the same lines.

Further, Theorems 4.1 and 4.3 in Section 4  of the original paper must be corrected by adding the following two conditions.

There exists xXω such that ωλ(x,y)< for all yFx and  λ>0.

If x0Fix(F), then ωλ(x0,y)< for all yGx0 and λ>0.

To prove, conditions (H1) and (H2) imply the existence of the fixed points of F and G. Take x0Fix(F) and follow the proof lines in the original paper to obtain the results.

Similarly Corollaries 4.2 and 4.4 in Section 4 of the original paper, must be corrected by adding the following three conditions:

There exists x0Xω such that ωλ(x0,y)< for all yF1x0 and λ>0.

If xnFix(Fn), then ωλ(xn,y)< for all yFn+1xn and λ>0.

There exists xXω such that ωλ(x,y)< for all yFx and λ>0.

For the proof, conditions (H3) and (H4) guarantee the existence of fixed points of each Fn, while condition (H5) implies the existence of fixed points of F. The rest of the proofs are as illustrated in the original paper.