Abstract
We correct the proofs of a previous publication.
Avoid common mistakes on your manuscript.
1 Introduction
We correct the proofs of [1, Corollary 3.1 and Theorem 3.2].
2 The Corrected Proofs
In the proof of [1, Corollary 3.1], we say “Since assumption (Th) holds immediately for \(T(x) = Ax + a\)”. This is not correct, as shown in the example given in the paper itself [1, page 127]. So, given a matrix \(A \in \mathbb {R}^{n \times n}\), a vector \(a \in \mathbb {R}^{n}\), and a closed and convex set \(K \subset {\mathbb R}^n\), we consider the following assumption:
(Ah): The pair (A, h) has the MVIP on K, with MVIP as defined in [1, Definition 3.1].
Hence, [1, Corollary 3.1] should be rewritten as follows:
Corollary 2.1
Let A be a K-copositive matrix and \(a \in \mathbb {R}^{n}\) such that assumption (Ah) holds. If there exists \(x_0 \in K\) such that
then S(A; h; K) is nonempty and compact.
In its proof, we replace “Since assumption (Th) holds immediately for \(T(x) = Ax + a\)” by “By assumption (Ah), assumption (Th) holds for \(T(x) = Ax + a\)”, and the proof follows.
Analogously, since the proof of [1, Theorem 3.2] is based on [1, Corollary 3.1], we rewrite this theorem as follows:
Theorem 2.1
Let \(h: \mathbb {R}^{n} \rightarrow \mathbb {R}\) be a function, and K a nonempty, closed and convex set from \(\mathbb {R}^{n}\). Suppose that assumptions (A0), (h0) and (Ah) hold. Then,
Finally, in the remainder of the paper, whenever [1, Theorem 3.2] is used, assumption (Ah) should be added.
3 Conclusions
We have corrected the proofs of a published paper.
Reference
Iusem, A., Lara, F.: Existence results for noncoercive mixed variational inequalities in finite dimensional spaces. J. Optim. Theory Appl. 183, 122–138 (2019)
Acknowledgements
This research was partially supported by Conicyt—Chile throughout Fondecyt Iniciación 11180320 (F. Lara).
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Iusem, A., Lara, F. A Note on “Existence Results for Noncoercive Mixed Variational Inequalities in Finite Dimensional Spaces”. J Optim Theory Appl 187, 607–608 (2020). https://doi.org/10.1007/s10957-020-01722-w
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DOI: https://doi.org/10.1007/s10957-020-01722-w
Keywords
- Asymptotic analysis
- Asymptotic functions
- Noncoercive Optimization
- Variational Inequalities
- Equilibrium Problems