# Correction to: An order theoretic fixed point theorem with application to multivalued variational inequalities with nonsmooth bifunctions

## 1 Correction to: J. Fixed Point Theory Appl. (2019) 21:5 https://doi.org/10.1007/s11784-018-0639-x

*C*in a poset

*D*equals an increasing sequence \((c_n)\). Unfortunately, there are simple counter-examples. Consider, e.g., the set

*C*is a well-ordered subset of \([0,1]\subset \mathbb {R}\), there is no increasing sequence \((c_n)\subset \mathbb {R}\) such that \(C = \{c_n: n\in \mathbb {N}\}\).

- 1.
The claim “(C1) and (C2) are equivalent” stated after Theorem 2.10 is false, and thus, it should be deleted. Furthermore, Theorem 2.10 only simplifies if

*D*has property (C1). - 2.
The claim “

*C*equals an increasing sequence \((c_n)\)” in the proof of Theorem 2.11 is false. However, the proof will be correct if we do not start with a general chain \(C\subset S_+ = \{x\in D: \{x\} \le ^*S(x)\}\), but with an increasing sequence \((c_n) \subset S_+\). Then, it is seen exactly like in the original paper that all assumptions of Corollary 0.1 below are satisfied, whence Theorem 2.11 holds true.

*D*is said to have Property (C1) if, for each nonempty well-ordered subset

*C*of

*D*such that each increasing sequence in

*C*has an upper bound in

*D*, there is an increasing sequence in

*C*that has the same upper bounds as

*C*.

First, let us prove the following corollary of Theorem 2.10 of the original paper:

### Corollary 0.1

*D*be a poset satisfying Property (C1), and assume that \(S:D \rightarrow \mathcal {P}_\emptyset (D)\) satisfies the following hypotheses:

- (i)
The set \(S_+ = \{x\in D: \{x\}\le ^*S(x)\}\) is nonempty.

- (ii)
For all increasing sequences \((c_n)\subset S_+\) and \((s_n)\subset D\) such that, for all

*n*, \(c_n\le s_n \in S(c_n)\), there is an upper bound \(s'\in S_+\) of \((s_n)\).

*S*has a maximal fixed point which is also a maximal element of \(S_+\).

### Proof

Let \(C\subset S_+\) be a nonempty well-ordered chain and \(s:C\rightarrow D\) an increasing function such that \(c\le s(c) \in S(c)\) for all \(c\in C\). Then, for every increasing sequence \((s_n)\subset s(C)\subset D\), let \(c_n\) be the smallest element of \(\{c\in C: s(c) = s_n\}\), such that \((c_n)\subset C\subset S_+\) is an increasing sequence with \(c_n \le s_n \in S(c_n)\). From the assumptions, \((s_n)\) has an upper bound \(s'\in S_+ \subset D\), and since *s*(*C*) is well-ordered and due to (C1), we can assume that \((s_n)\) has the same upper bounds as *s*(*C*), so that \(s'\in S_+\) is an upper bound of *s*(*C*). This shows that all assumptions of Theorem 2.10 are satisfied, and thus, *S* has a maximal fixed point which is also a maximal element of \(S_+\). \(\square \)

Second, let us give a sufficient condition for Property (C1) to hold:

### Proposition 0.2

Let *D* be a bounded subset of a reflexive ordered Banach space *E*. Then, *D* has Property (C1).

### Proof

Let *C* be a nonempty well-ordered chain in *D*. Then, from [1, Prop. 1.3.6], we know that there is an increasing sequence \((c_n)\) in *C* converging weakly to the supremum of *C*. Then, for each upper bound *b* of \((c_n)\), it follows \(\sup C \le b\), since the set \(\{x\in D: x\le b\}\) is weakly closed. Consequently, *C* and \((c_n)\) have the same upper bounds, whence *D* has property (C1). \(\square \)

Since Theorem 2.11 holds true, the rest of the paper is not effected.

## Notes

## Reference

- 1.Heikkilä, S., Lakshmikantham, V.: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations. Marcel Dekker Inc., New York (1994)zbMATHGoogle Scholar