Erratum to “Fixed-Point Theorems for Multivalued Mappings in Modular Metric Spaces”
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 be a complete modular metric space and a multivalued self-mapping on satisfying the inequality for all , where . Then, has a fixed point in .
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 and be given by Thus, the modular metric space . Now let be a self-mapping on defined by Then, satisfies the hypothesis of Theorem 1.1 with any 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:
For the proofs, take the initial point 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 exists such that for all and. (H2)If , then for all and . To prove, conditions (H1) and (H2) imply the existence of the fixed points of and . Take 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 exists such that for all and . (H4)If , then for all and . (H5)There exists such that for all and .For the proof, conditions (H3) and (H4) guarantee the existence of fixed points of each , while condition (H5) implies the existence of fixed points of . The rest of the proofs are as illustrated in the original paper.