# Erratum to “Fixed-Point Theorems for Multivalued Mappings in Modular Metric Spaces”

Authors: Parin Chaipunya; Chirasak Mongkolkeha; Wutiphol Sintunavarat; Poom Kumam
Journal: Abstract and Applied Analysis
 (2012)

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## Body

## 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).
LetXω 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,y∈Xω, 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 Theorem1.1 in the single-valued case as follows.Example 1.2.
LetX:={0,1} and ω be given by
(1.2)ωλ(x,y)={∞,if0<λ<1andx≠y,0,ifλ≥1orx=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  existsx0∈Xωsuch  thatωλ(x0,y)<∞∀y∈Fx0,λ>0.For the proofs, take the initial pointx0 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.(H1)
There existsx⋆∈Xω such that ωλ(x⋆,y)<∞ for all y∈Fx⋆ andλ>0.(H2)
Ifx0∈Fix(F), then ωλ(x0,y)<∞ for all y∈Gx0 and λ>0.To prove, conditions (H1) and (H2) imply the existence of the fixed points of F and G. Take x0∈Fix(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:(H3)
There existsx0∈Xω such that ωλ(x0,y)<∞ for all y∈F1x0 and λ>0.(H4)
Ifxn∈Fix(Fn), then ωλ(xn,y)<∞ for all y∈Fn+1xn and λ>0.(H5)
There existsx∞∈Xω such that ωλ(x∞,y)<∞ for all y∈Fx∞ 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.



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*Source: 241919-2012-07-16.xml*