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

$$\begin{aligned} h^{\infty } (u) + \langle a - A^{\top } x_0, u \rangle > 0, \quad \forall ~ u \in K^{\infty } \backslash \{0\}, \end{aligned}$$
(1)

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,

$$\begin{aligned} h^{\infty } (u) + \langle a, u \rangle > 0,\,\, ~ \forall ~ u \in (K^{\infty } \cap \mathrm{Ker}\,A) \backslash \{0\} ~ \Longrightarrow ~ S(A; h; K) \ne \emptyset \mathrm{~ and ~ compact.} \end{aligned}$$
(2)

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.