Brezis Pseudomonotonicity is Strictly Weaker than Ky–Fan Hemicontinuity
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Abstract
In 1968, H. Brezis introduced a notion of operator pseudomonotonicity which provides a unified approach to monotone and nonmonotone variational inequalities. A closely related notion is that of Ky–Fan hemicontinuity, a continuity property which arises if the famous Ky–Fan minimax inequality is applied to the variational inequality framework. It is clear from the corresponding definitions that Ky–Fan hemicontinuity implies Brezis pseudomonotonicity, but quite surprisingly, a recent publication by Sadeqi and Paydar (J Optim Theory Appl 165(2):344–358, 2015) claims the equivalence of the two properties. The purpose of the present note is to show that this equivalence is false; this is achieved by providing a concrete example of a nonlinear operator which is Brezis pseudomonotone but not Ky–Fan hemicontinuous.
Keywords
Brezis pseudomonotonicity Ky–Fan hemicontinuity Counterexample Variational inequality Equilibrium problemMathematics Subject Classification
46B 46T 47H 47J1 Introduction
Variational inequalities (VIs) are a prominent tool in applied mathematics. They have found numerous applications, including constrained optimization problems, Nash equilibrium problems, and several types of contact problems in mechanics. More details can be found, for instance, in [1, 2, 3, 4] and the references therein.
The study of variational inequalities can be divided into multiple facets: most commonly, one is interested in sufficient conditions for the existence of solutions, the design of suitable algorithms for their computation, or other properties of the solution set such as closedness or convexity. Throughout the last decades, various concepts have been developed in order to ascertain these properties, including the monotonicity of the variational operator (and multiple relaxed versions thereof) as well as several types of continuity (hemicontinuity, continuity on finitedimensional subspaces, etc.). In addition, one often requires suitable properties of the feasible set such as closedness or (weak) compactness.
One of the most general properties which can be used to tackle variational inequalities is that of (Brezis) pseudomonotonicity, a property which was introduced in [5]. (This should not be confused with pseudomonotonicity in the sense of Karamardian.) The attractive property of Brezis pseudomonotonicity is that it provides a unified approach to monotone and nonmonotone problems—indeed, it is best viewed as a hybrid combining elements of both monotonicity and continuity; see Definition 2.2.
Since its conception in 1968, the notion of pseudomonotonicity has occurred prominently in the works of Browder [6]; Brezis, Nirenberg, and Stampacchia [7]; Zeidler [8]; and Barbu and Precupanu [9]. The standard application of pseudomonotonicity was the construction of existence results for VIs, a topic which occurs in all these references and was also revisited in [10]. In addition, pseudomonotonicity has turned out to be quite useful when analyzing convergence of iterative algorithms for constrained minimization and variational or quasivariational inequalities; see [11, 12] for more details.
A different but related approach to the existence of solutions is given by the classical minimax inequality of Ky Fan [13]. An application of this result to the VI framework gives rise to a continuity property which is sometimes called Ky–Fan hemicontinuity. This property implies Brezis pseudomonotonicity (a fact which follows directly from the corresponding definitions, see below), but the latter appears to be more refined and convenient when dealing with infinitedimensional VIs. However, quite surprisingly, a 2015 publication by Sadeqi and Paydar [14] claims the equivalence of the two properties. The purpose of the present paper is to discuss this equivalence and provide a counterexample which shows that the two properties are in fact distinct. In addition, we also outline an error in the reference which may have led to the false result.
This paper is organized as follows. In Sect. 2, we give a brief summary of the properties in question, their consequences, and relations to other standard properties for VIs. Section 3 contains the main counterexample.
2 Hemicontinuity, Pseudomonotonicity, and Their Role in the Study of Variational Inequalities
Throughout this paper, X is a real Banach space with norm \(\Vert \cdot \Vert _X\) and continuous dual \(X^*\). The duality pairing between \(X^*\) and X is denoted by \(\langle \cdot ,\cdot \rangle \). We write \(\rightarrow \), \(\rightharpoonup \), and \(\rightharpoonup ^*\) for strong, weak, and weak\(^*\) convergence.
Definition 2.1
(Ky–Fan hemicontinuity) We say that \(F:X\rightarrow X^*\) is Ky–Fan hemicontinuous if, for every \(y\in X\), the function \(x\mapsto \langle F(x),xy \rangle \) is weakly sequentially lower semicontinuous.
The above is one of the two main properties which we will discuss in this paper. The second one was introduced by Brezis [5] and is given as follows.
Definition 2.2
It is clear from the above definitions that Brezis pseudomonotonicity is weaker than Ky–Fan hemicontinuity: the latter requires that (3) holds for all weakly convergent sequences \(\{x^k\}\subseteq X\) with limit \(x\in X\), whereas Brezis pseudomonotonicity only asserts this estimate for sequences which additionally satisfy the \(\limsup \)condition in (2).
The set of pseudomonotone operators is large and encompasses many practically relevant examples. Various sufficient conditions for pseudomonotonicity can be found in [8, 12]; in particular, F is pseudomonotone provided it is either (i) monotone and continuous, (ii) completely continuous, or (iii) the sum of two operators which are themselves pseudomonotone. Using results from differential calculus in Banach spaces, one can also give sufficient conditions for pseudomonotonicity in the special case where F is the Fréchet derivative of a realvalued functional; see [12].
The following is the basic existence result for VIs with pseudomonotone operators. Note that we call F bounded if it maps bounded sets in X to bounded sets in \(X^*\).
Proposition 2.1
(Pseudomonotone VIs [11, Corollary 4.2]) Let \({A\subseteq X}\) be a nonempty, convex, weakly compact set, and \(F:X\rightarrow X^*\) a bounded pseudomonotone operator. Then the variational inequality (1) admits a solution \(\hat{x}\in A\).

If \(F:X\rightarrow X^*\) is pseudomonotone, then the solution set of the VI (1) is always weakly sequentially closed. More generally, if \(\{x^k\}\) is a sequence of suitable “approximate” solutions of the VI, then every weak limit point of \(\{x^k\}\) belongs to its solution set; see [11].

If X is finitedimensional (without loss of generality, a Hilbert space), then an operator \(F:X\rightarrow X\) is bounded and pseudomonotone if and only if it is continuous. Thus, in this case, the study of pseudomonotone VIs subsumes the wellknown theory of finitedimensional VIs; see, for instance, [1].
Remark 2.1
(Sequences versus nets) Some authors define the aforementioned concepts on a general Hausdorff topological vector space (instead of a Banach space endowed with its weak topology). In that case, the properties ought to be formulated in terms of nets or filters instead of ordinary sequences; see, for instance, [7].
3 A Nonlinear Operator Which is Brezis Pseudomonotone But Not Ky–Fan Hemicontinuous
As we shall see below, Brezis pseudomonotonicity and Ky–Fan hemicontinuity are not equivalent. The particular example shown here involves the wellknown function spaces \(L^p(\varOmega )\), \(W^{k,p}(\varOmega )\), and \(W_0^{k,p}(\varOmega )\), with \(\varOmega \) a bounded finitedimensional domain, \(k\in \mathbb {N}\), and \(p\in [1,+\infty ]\); see, for instance, [15].
Example 3.1
An interesting question that remains is where the argumentation from [14] is incorrect. The following is a particular error which is contained in that paper.
Remark 3.1
4 Conclusions
The example in this paper shows that Brezis pseudomonotonicity and Ky–Fan hemicontinuity are distinct properties. In particular, the former is strictly weaker than the latter and therefore remains one of the most general properties which can be used to tackle variational inequalities.
Notes
Acknowledgements
The author would like to thank Daniel Wachsmuth for the basic idea of the counterexample and Ildar Sadeqi for the discussion leading to the creation of this paper. This research was supported by the German Research Foundation (DFG) within the priority program “Nonsmooth and Complementaritybased Distributed Parameter Systems: Simulation and Hierarchical Optimization” (SPP 1962) under Grant No. KA 1296/241.
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