Approximate Solutions of Variational Inequalities and the Ekeland Principle

Abstract

Let K be a closed convex subset of a real Banach space X, and let F be a map from X to its dual $X^{*}$. We study the so-called variational inequality problem: given $y\in X^{*,},$ does there exist $x_0\in K$ such that (in standard notation) $⟨F\left(x_0\right)-y,x-x_0⟩\ge 0$ for all $x\in K?$ After a short exposition of work in this area, we establish conditions on F sufficient to ensure a positive answer to the question of whether F is a gradient operator. A novel feature of the proof is the key role played by the well-known Ekeland variational principle.

1. Introduction

Let X be a real Banach space, and let F be a map of X into its dual $X^{*}$. We denote by $\langle .,.\rangle$ the canonical pairing between $X^{*}$ and X, so that $\langle y,x\rangle$ stands for the value $y\left(x\right)$ of the bounded linear form $y\in X^{*}$ at the point $x\in X$. Recall that given a closed convex subset K of X and given $y\in X^{*}$, a point $x_0\in K$ such that

$$\langle F\left(x_0\right)-y,x-x_0\rangle \ge 0\forall x\in K$$

(1)

is said to be a solution of the Variational Inequality given by F, K and y.

Classical references on the topic of Variational Inequality (V.I.) and their applications to partial differential equations and inequalities are, for instance, the book of Kinderlehrer and Stampacchia [1], as well as Chapter 2 of Lions’ book [2]. For a more recent reference, with special emphasis on the applications to concrete problems in mechanics, see [3].

In the first part of this paper (Section 2), which is of an introductory and expository nature, we merely comment on the definition just given and recall some known existence results for (1). We emphasize that no boundedness assumption on the (closed convex) set $K\subset X$ is made through the paper. In the second part (Section 3), we consider the special case in which F is a gradient operator, which allows us to prove the existence of solutions to (1) via a minimization process for the corresponding functional: we do this using the Ekeland Variational Principle (see, e.g., [4]), and extend in this way the surjectivity results for such operators obtained in [5,6].

Wishing to describe in more detail the content of the present paper, we start recalling a famous result due to Browder ([7], Theorem 11), that ensures the existence of a solution to (1) for any given $y\in X^{*}$ under the assumptions that $F:X\to X^{*}$ is continuous, monotone and coercive, with the latter condition meaning that

$${\displaystyle \frac{\langle F\left(x\right),x\rangle }{\left|\right|x\left|\right|}}\to +\infty \mathrm{as}\left|\right|x\left|\right|\to \infty .$$

For a coercive F, it is easily seen that it suffices to consider the case that $y=0$ in (1), and so we consider the equation

$$\langle F\left(x_0\right),x-x_0\rangle \ge 0\forall x\in K.$$

(2)

In the search of a solution $x_0\in K$ to (2), a fundamental step of Browder [7]—see also Lions’ book [2]—is the construction of an increasing sequence $\left(K_n\right)$ of compact convex subsets of K and a sequence $\left(x_n\right)$ of points of K, with $x_n\in K_n$ for each $n\in \mathbb{N}$, such that

$$\langle F\left(x_n\right),x-x_n\rangle \ge 0\forall x\in K_n.$$

(3)

The points $\left(x_n\right)$ satisfying (3) can be considered as approximate solutions of the V.I. (2), and this is, in fact, the terminology employed by Lions in his discussion of V.I. ([2], Chapter 2, Section 8). In this review paper, we wish to show that a somewhat similar, yet conceptually and constructively different, situation appears quite naturally when dealing with the case that F is a gradient operator, meaning that $F=f^{\prime }$ for some differentiable functional $f:X\to \mathbb{R}$. For such an F, the existence of a solution to (2) can be proved through a minimization procedure on K for the corresponding functional f: indeed, as is well known and as we shall briefly recall later, (2) merely expresses the condition that must be necessarily satisfied if the restriction of f to K has a minimum at the point $x_0\in K$. In turn, the Ekeland variational principle (see, e.g., [4]) is a powerful, and by now classical, tool available for minimization problems, and we use it here—as we did in [5,6] to prove surjectivity of F—to provide, among others, a simple alternative proof of two existence results of Browder [7] about the V.I. (1), the one cited above and a second one, having somewhat different assumptions and reported here as Theorem 3.

As we did in papers [5,6], the preliminary conditions that we impose on F for a fruitful use of the Ekeland principle are that $F:X\to X^{*}$ is continuous, bounded on bounded sets, and coercive. The novelty here with respect to the papers [5,6] is that using the Ekeland principle for the minimization of the functional f on the convex set K rather than on the whole of X, we find a sequence $\left({ϵ}_n\right)$ of reals, with ${ϵ}_n>0$ for all $n\in \mathbb{N}$ and ${ϵ}_n\to 0$ as $n\to \infty$, and a corresponding sequence $\left(x_n\right)$ of points of K, which is a minimizing sequence for f in K—in the sense that $f\left(x_n\right)\to {inf}_{x\in K}f\left(x\right)$—and is such that

$$\langle F\left(x_n\right),x-x_n\rangle \ge -{ϵ}_n\left|\right|x-x_n\left|\right|\forall x\in K,\forall n\in \mathbb{N}.$$

(4)

In our approach, the inequalities (4) are the approximate variational inequalities which replace those in Browder’s approach defined via (3). Then, we show that the addition to (4) of properties of F such as Monotonicity, ${\left(S\right)}_{+}$ property or, more generally, Pseudo-Monotonicity—each of these qualities of F will be carefully recalled, as due, in the course of the next Section—leads easily to the existence of an $x_0\in K$ satisfying (2). Moreover, $x_0$ will actually minimize f on K if, in addition, we know that $\left(x_n\right)$ converges strongly to $x_0$ and assume—as we do in this paper—the continuity of f.

Resuming, the ideas sketched above lead us to the following result (stated formally as Theorem 6): Suppose that X is a real, reflexive Banach space and let $F:X\to X^{*}$ be a gradient operator that is continuous and bounded on bounded subsets of X. Then, if F is pseudo-monotone and coercive, the V.I. (1) has a solution to any closed convex set $K\subset X$ and the point $y\in X^{*}$.

This paper is organized as follows. In the first part (Section 2), for the content of which we claim no real novelty, we introduce the reader interested in our theme by presenting a brief review of some relevant points concerning V.I., and propose an elementary proof based on the Contraction Mapping Theorem—of a basic existence result about (1) in the simplified context of Hilbert spaces and under the assumptions of Lipschitz continuity and strong Monotonicity of F. It is possible that such a proof already exist in the literature; however, we include it here both for completeness and because of our admittedly limited knowledge of the vast literature on the subject.

Some classical and more general existence results for (1) are then recalled and commented, concerning operators that are monotone or, more generally, satisfy a generalized Monotonicity condition in the language of Browder [7]. In particular, we close the Section stating an existence theorem that follows from more general results of Lions [2] and involves the class of pseudo-monotone operators.

In Section 3, whose content we believe to be new—though still presented in an expository style—we recall and use the Ekeland principle to state and prove our main result in this paper, namely Theorem 6 already highlighted above. Our statement turns out to be essentially identical, with the one by Lions just cited and regarding pseudo-monotone but not necessarily gradient operators. However, we feel that the appeal of our procedures stems from the simplicity and novelty of the proof and also from the proposed parallel with classical results for non-gradient operators that arise in the passage from approximate to exact solutions of a variational inequality, along the lines summarized above and that will be detailed in the proof of Theorem 6.

2. A View to Variational Inequalities

We begin this expository Section, recalling two simple general remarks about (1).

Remark 1

(Variational interpretation of (1)). Suppose that $F:X\to X^{*}$ is the gradient of a real functional f defined on X, meaning that

$$F\left(x\right)=f^{\prime }\left(x\right)$$

for all $x\in X$, with $f^{\prime }\left(x\right)$ standing for the Fréchet derivative of f at the point $x\in X$, that is an element of $X^{*}$. Then, for $y=0$, the V.I. (1) reads

$$\langle f^{\prime }\left(x_0\right),x-x_0\rangle \ge 0\forall x\in K$$

(5)

and expresses the condition that must be necessarily satisfied if the restriction $f_K\equiv f|K$ of f to the convex subset K has a local minimum at $x_0\in K$ (“Fermat’s constrained condition”). Indeed, recall that by the definition and properties of $f^{\prime }\left(x\right)$ we have

$$\langle f^{\prime }\left(x\right),v\rangle =\underset{t\to 0}{lim}{\displaystyle \frac{f(x+tv)-f\left(x\right)}{t}}\forall v\in X.$$

(6)

Next, let $\left|\right|.\left|\right|$ stand for the norm in X and for $r>0$, let $B(x_0,r)=\{x\in X:||x-x_0\left|\right|<r\}$ stand for the r-neighborhood of $x_0\in X$. Then, using the convexity of K and the assumption that $f\left(x\right)\ge f\left(x_0\right)$ for some $r>0$ and for every $x\in B(x_0,r)\cap K$, we obtain, in particular, that

$${\displaystyle \frac{f(x_0+t(x-x_0))-f\left(x_0\right)}{t}}\ge 0$$

(7)

for every $x\in K$ and every $t>0$ sufficiently small. Taking the limit as $t\to 0^{+}$ in (7) and looking at (6), we thus obtain (5).

Remark 2

(Relation with Surjectivity). It is immediate to check that, in the special case $K=X$, the existence of a solution $x_0$ to the V.I. (1) for a given y means that

$$F\left(x_0\right)=y.$$

(8)

Indeed, taking $x=x_0+tv$ in (1) with $t\in \mathbb{R}$ and $v\in X$ gives

$$t\langle F\left(x_0\right)-y,v\rangle \ge 0\forall t\in \mathbb{R}$$

whence taking $t=\pm 1$ we conclude that $\langle F\left(x_0\right)-y,v\rangle =0$ for every $v\in X$, which yields the conclusion (8). Therefore, an existence result for the V.I. (1) for every closed convex $K\subset X$ and every $y\in X^{*}$ will, in particular, imply the surjectivity of F.

Let us continue for a while the reasoning of Remark 2. We recall that if $K:X\to X$ is a contraction, that is,

$$\left|\right|K\left(x\right)-K\left(y\right)\left|\right|\le k\left|\right|x-y\left|\right|$$

(9)

for all $x,y\in X$ and for some constant k with $0\le k<1$, then $F=I-K$ is surjective (and in fact a homeomorphism of X onto itself): this is an easy consequence of the Contraction Mapping Theorem, for the equation $x-K\left(x\right)=y$ can be written as the fixed-point equation

$$x=K\left(x\right)+y\equiv K_y\left(x\right)$$

that has a unique solution by virtue of the assumption (9). Now observe that if $X=H$ is a Hilbert space with scalar product still denoted with $\langle .,.\rangle$, and if $K:H\to H$ is a contraction, then $F=I-K$ is strongly monotone, in the sense that

$$\langle F\left(x\right)-F\left(y\right),x-y\rangle \ge c\left|\right|x-{y\left|\right|}^2$$

(10)

for all $x,y\in H$ and for a positive constant c. Indeed, we have

$$\langle F\left(x\right)-F\left(y\right),x-y\rangle =\langle x-K\left(x\right)-(y-K(y\left)\right),x-y\rangle =$$

$$=\left|\right|x-{y\left|\right|}^2-\langle K\left(x\right)-K\left(y\right),x-y\rangle$$

(11)

However, the Schwarz inequality shows that

$$|\langle K(x)-K(y),x-y\rangle |\le \left|\right|K\left(x\right)-K\left(y\right)\left|\right|\left|\right|x-y\left|\right|$$

so that inserting this in (11) and using (9) we obtain (10) with $c=1-k>0$.

On the basis of these remarks, we can imagine that the surjectivity property (or, more strongly, the existence of a solution to the V.I. (1) for every $y\in X^{*}$) continues to hold for maps $F:X\to X^{*}$ that are strongly monotone in the general sense that satisfy (10) with $\langle .,.\rangle$ now expressing the duality pairing between X and $X^{*}$. For a continuous F this is really so, and it is interesting to see that for Lipschitz-continuous F and for the Hilbert space case, this can be proved again on the basis of the Contraction Mapping Theorem. Here follows the formal statement of this fact and its proof.

Theorem 1.

Let H be a real Hilbert space, and let $F:H\to H$ be Lipschitz continuous and strongly monotone. Then, for every closed convex $K\subset H$ and for every $y\in H$, the V.I. (1) given by F, K, and y has a unique solution.

The proof of Theorem 1 exploits, in addition to the Contraction Mapping Theorem, the basic result on the existence and uniqueness of the nearest point in a closed convex set to a given point of a Hilbert space, which we recall next.

Proposition 1.

Let H be a real Hilbert space, and let $K\subset H$ be closed and convex. Then, for every $y\in H$, there exists a unique $x_0\in K$ such that

$$\left|\right|y-x_0\left|\right|\le \left|\right|y-x\left|\right|\forall x\in K.$$

(12)

Moreover, $x_0$ is characterized by the property

$$\langle y-x_0,x-x_0\rangle \le 0\forall x\in K.$$

(13)

The vector $x_0=x_0\left(y\right)$ is called the projection of y onto K and is denoted with the symbol $P_K\left(y\right)$.

For a proof of Proposition 1 see, for instance, Theorem 5.2 in Brezis’ book [8]. Aside and independently from the core of the proof, which exploits the parallelogram law satisfied by the Hilbert norm, it is important to note the following facts, which can be checked by mere computation:

1. The properties (12) and (13) are a priori equivalent; in particular, (12) follows immediately from (13) on expanding $\left|\right|y-{x\left|\right|}^2=\left|\right|y-x_0+x_0-x{\left|\right|}^2$, while (13) is nothing but the necessary consequence—stated in general form as (5)—of the minimum property on K for the functional $f\left(x\right)=\left|\right|y-{x\left|\right|}^2$ stated by the inequality (12).

2. The map $P_K$ of H into K defined via Proposition 1 is Lipschitz continuous of constant 1, that is,

$$\left|\right|P_K\left(y\right)-P_K\left(z\right)\left|\right|\le \left|\right|y-z\left|\right|\forall y,z\in H.$$

(14)

To obtain (14), take $y,z\in H$ and write down the inequalities (13) for the corresponding projections $x_1\equiv P_K\left(y\right),x_2\equiv P_K\left(z\right)$; adding up these two inequalities and using the Schwarz inequality, the result easily follows.

Before going to the proof of Theorem 1, one more remark is useful. Writing (13) in the form

$$\langle x_0-y,x-x_0\rangle \ge 0\forall x\in K$$

(15)

and comparing it with the general form (1) of a Variational Inequality, we see that Theorem 1 is true in the special case that $F=I$, the identity map in H. It is thus a matter of extending this fact to Lipschitz continuous and strongly monotone maps that are quite close to the identity. We do this by employing for such operators the arguments given by Brezis (Theorem 5.6 in [8]) in the special case that F is the bounded linear operator associated with a coercived continuous bilinear form $a:H\times H\to \mathbb{R}$: in this case, Theorem 1 reduces to the well known Stampacchia Theorem, whose familiar form states that for each given $y\in X^{*}$ there is a unique $x_0\in K$ such that

$$a(x_0,x-x_0)\ge \langle y,x-x_0\rangle \forall x\in K.$$

To prove Theorem 1 in general, we start writing the conclusion of Proposition 1 in the form of an equivalence as follows:

$$[x_0\in K,\langle x_0-y,x-x_0\rangle \ge 0\forall x\in K]\iff x_0=P_K\left(y\right).$$

(16)

Now, let $ϵ$ be any positive constant and write the V.I. (1) in the form

$$\langle x_0-x_0+ϵ[F\left(x_0\right)-y],x-x_0\rangle \ge 0\forall x\in K$$

or else

$$\langle x_0-[(x_0+ϵy)-ϵF\left(x_0\right)],x-x_0\rangle \ge 0\forall x\in K.$$

(17)

Comparing this with (16), we conclude that the existence of a solution $x_0$ to (1) is equivalent to the existence of an $x_0$ such that

$$x_0=P_K[(x_0+ϵy)-ϵF\left(x_0\right)]$$

that is, to the existence of a fixed point for the map $S_y:K\to K$ defined putting

$$S_y\left(x\right)=P_K[(x+ϵy)-ϵF\left(x\right)]\forall x\in K.$$

(18)

It is now a matter to show that for some $ϵ>0$, $S_y$ is a contraction of the complete metric space K into itself. Here, the Lipschitz and strong Monotonicity assumptions on F come into play. Suppose indeed that

$$\left|\right|F\left(x\right)-F\left(y\right)\left|\right|\le k\left|\right|x-y\left|\right|$$

(19)

and that

$$\langle F\left(x\right)-F\left(y\right),x-y\rangle \ge c\left|\right|x-{y\left|\right|}^2$$

(20)

for all $x,y\in H$ and for some positive constant k and c. We first have, from (18) and (14),

$$\left|\right|S_y\left(x_1\right)-S_y\left(x_2\right)\left|\right|=\left|\right|P_K[(x_1+ϵy)-ϵF\left(x_1\right)]-P_K[(x_2+ϵy)-ϵF\left(x_2\right)]\left|\right|$$

$$\le \left|\right|(x_1+ϵy)-(x_2+ϵy)-ϵ[F\left(x_1\right)-F\left(x_2\right)]\left|\right|$$

$$=\left|\right|(x_1-x_2)-ϵ[F\left(x_1\right)-F\left(x_2\right)]\left|\right|$$

for every pair $x_1,x_2$ of points of K (and in fact of the entire space H). Therefore,

$$\left|\right|S_y\left(x_1\right)-S_y\left(x_2\right){\left|\right|}^2\le \left|\right|x_1-x_2{\left|\right|}^2-2ϵ\langle F\left(x_1\right)-F\left(x_2\right),x_1-x_2\rangle +{ϵ}^2\left|\right|F\left(x_1\right)-F\left(x_2\right){\left|\right|}^2$$

whence, using (19) and (20), we finally obtain

$$\left|\right|S_y\left(x_1\right)-S_y\left(x_2\right){\left|\right|}^2\le \left|\right|x_1-x_2{\left|\right|}^2-2ϵc\left|\right|x_1-x_2{\left|\right|}^2+{ϵ}^2{k}^2\left|\right|x_1-x_2{\left|\right|}^2=$$

$$=(1-2ϵc+{ϵ}^2{k}^2)\left|\right|x_1-x_2{\left|\right|}^2.$$

(21)

For $ϵ$ positive and sufficiently small (precisely, for $0<ϵ<2c/k^2$), the coefficient in the right-hand side of (21) becomes less than 1, thus proving the claim that $S_y$ is a contraction of K into itself and therefore possesses a unique fixed point, which is, by the discussion made before, the unique solution of the V.I. (1).

As to the existence of solutions, the main lines along which Theorem 1 has been extended to more general situations can roughly be resumed as follows:

(i)

From Hilbert spaces to reflexive Banach spaces;

(ii)

From strongly monotone to monotone and coercive maps;

(iii)

From Lipschitz continuous to merely continuous maps.

The properties mentioned in ii) above have not been defined so far and so we do this next. Henceforth, we consider mappings F from a real Banach space X to its dual $X^{*}$ and denote with $\langle .,.\rangle$, the canonical pairing between $X^{*}$ and X.

Definition 1.

Let F be a map of X into its dual $X^{*}$. F is said to be monotone if

$$\langle F\left(x\right)-F\left(y\right),x-y\rangle \ge 0\forall x,y\in X.$$

(22)

Definition 2.

Let F be a map of X into its dual $X^{*}$. F is said to be coercive if

$${\displaystyle \frac{\langle F\left(x\right),x\rangle }{\left|\right|x\left|\right|}}\to +\infty \mathit{as}\left|\right|x\left|\right|\to \infty .$$

(23)

It is clear that if $F:X\to X^{*}$ is strongly monotone (definition as in (10), with the scalar product replaced by the canonical pairing between X and $X^{*}$), then it is both monotone and coercive: as to coercivity, just note that (10) implies, in particular,

$$\langle F\left(x\right),x\rangle \ge {c\left|\right|x\left|\right|}^2+\langle F\left(0\right),x\rangle \forall x\in X$$

whence (23) follows at once noting that $|\langle F(x),y\rangle |\le \left|\right|F\left(x\right)\left|\right|\left|\right|y\left|\right|$ for all $x,y\in X$.

To appreciate the importance of the definitions just given, it will be enough to recall a famous surjectivity result (usually referred to as the Minty–Browder Theorem) for mappings from a reflexive Banach space to its dual.

Theorem 2.

Let X be a real, reflexive Banach space, and let F be a map of X into its dual $X^{*}$. Suppose that F is continuous, monotone, and coercive. Then, F is surjective.

For the statement and the proof of Theorem 2, we refer to Theorem 2 of Browder’s paper [7]. The continuity assumption on F can be weakened to that of hemicontinuity (see, for instance, Theorem 12.1 in Deimling [9] or Theorem 2.1 in Chapter 2 of Lions [2]), but for simplicity and uniformity of exposition, we shall keep assuming the continuity of F throughout this whole paper. We rather stress the point that Browder’s paper [7] contains a variant of Theorem 2, stated as Theorem 5 in [7], in which the same conclusion is reached on replacing the Monotonicity property of F with those of boundedness on bounded sets and $\left(S\right)$-property, the latter being defined as follows:

Definition 3.

A map $F:X\to X^{*}$ is said to be of type $\left(S\right)$ (or to satisfy condition $\left(S\right)$) if for every sequence $\left(x_n\right)\subset X$, which converges weakly to some $x\in X$ and is such that

$$\langle F\left(x_n\right)-F\left(x\right),x_n-x\rangle \to 0$$

(24)

we have that $\left(x_n\right)$ converges strongly to x.

It is easily seen that the condition $\left(S\right)$ is implied by the strong Monotonicity property defined via Equation (10); in fact, an even stronger property is implied by the latter, and is given in the next Definition [7].

Definition 4.

A map $F:X\to X^{*}$ is said to be of type ${\left(S\right)}_{+}$ if whenever $\left(x_n\right)$ converges weakly to $x\in X$ and

$$lim\; sup\langle F\left(x_n\right)-F\left(x\right),x_n-x\rangle \le 0,$$

(25)

we have that $\left(x_n\right)$ converges strongly to x.

To check that the implication $F:X\to X^{*}$ is strongly monotone ⇒F of type ${\left(S\right)}_{+}$ holds true, suppose that a sequence $\left(x_n\right)\subset X$ converges weakly to $x\in X$ and that (25) is satisfied. If F is strongly monotone, we also have

$$\langle F\left(x_n\right)-F\left(x\right),x_n-x\rangle \ge c\left|\right|x_n-x{\left|\right|}^2$$

(26)

for all $n\in \mathbb{N}$ and some $c>0$, whence we conclude that $\left(x_n\right)$ converges strongly to x.

For a rather complete discussion of the various “generalized Monotonicity conditions” such as $\left(S\right)$ and ${\left(S\right)}_{+}$, including the Pseudo-Monotonicity that we are going to define shortly, see, for instance, Section 6 of [10].

Let us now move—following Browder [7]—from the surjectivity property to the more general existence problem for Variational Inequalities. Here, we can see the importance of the property ${\left(S\right)}_{+}$ defined considering the following statement, which is a special case of Theorem 13 in [7]. Indeed, both in view of the comparison with our results in Section 3 and in order to reduce technicalities to the bare minimum, here and elsewhere, we suppose that the operators are defined on the whole of X and coercive on X, along with Definition 2.

Theorem 3.

Let X be a real, reflexive Banach space and let $F:X\to X^{*}$ be of type ${\left(S\right)}_{+}$. Suppose, moreover, that F is bounded on bounded sets, continuous and coercive. Also, let $K\subset X$ be closed and convex with $0\in K$. Then, given any $y\in X^{*}$, there exists $x_0\in K$ such that

$$\langle F\left(x_0\right)-y,x-x_0\rangle \ge 0\forall x\in K.$$

(27)

Remark 3.

The condition $0\in K$ is employed in the proof, together with the coercivity of F, in order to show that the approximating sequence $\left(x_n\right)\subset K$ to the solution $x_0$ of the Variational Inequality (27)—see the discussion below—is bounded and thus, by the reflexivity of X, contains a subsequence converging weakly to some $x_0\in K$. In detail, (28) shows (since by construction $0\in K_n\forall n\in \mathbb{N}$) that $\langle F\left(x_n\right),x_n\rangle \le 0\forall n\in \mathbb{N}$, forcing the sequence $\left(x_n\right)$ to stay bounded by virtue of the coercivity assumption (23).

As already recalled in the Introduction, a result similar to Theorem 3 is stated and proved by Browder himself ([7], Theorem 11) on replacing the ${\left(S\right)}_{+}$ assumption on F with that of Monotonicity and dropping the assumption that F maps bounded sets onto bounded sets. The proof of both results is based on the construction of (essentially) finite-dimensional approximations $x_n$ of the desired solution $x_0$ to the Variational Inequality given by F and K. In fact, one first proves an existence theorem for continuous mappings defined on compact convex subsets of a Banach space X ([7], Lemma 5). Then, one constructs an increasing sequence $\left(K_n\right)$ of compact convex subsets of K, whose union $K_0$ is dense in K, and a corresponding sequence $\left(x_n\right)$ with $x_n\in K_n$ for each $n\in \mathbb{N}$, such that

$$\langle F\left(x_n\right),x-x_n\rangle \ge 0\forall n\in \mathbb{N},\forall x\in K_n.$$

(28)

The approximate inequalities (28) then lead—using the weak convergence of (a subsequence of) $\left(x_n\right)$ to some $x_0\in K$ and either the Monotonicity of F or its ${\left(S\right)}_{+}$ property—to the exact inequality

$$\langle F\left(x_0\right),x-x_0\rangle \ge 0\forall x\in K,$$

thus proving Theorem 3 in the special case that $y=0$. The general case follows easily on considering, for each given $y\in X^{*}$, the map $F_y:X\to X^{*}$ defined putting

$$F_y\left(x\right)=F\left(x\right)-y\forall x\in X$$

that satisfies the same properties stated for F.

Let us add here an informal and qualitative remark concerning the two properties mentioned above in commenting on Browder’s work [7], namely the ${\left(S\right)}_{+}$ property and Monotonicity. While ${\left(S\right)}_{+}$ is implied by strong Monotonicity, as shown above, it appears to be somewhat independent from mere Monotonicity. Indeed, taking $F=const$, we have a trivial example of a monotone operator that is not of type ${\left(S\right)}_{+}$: to see this, just take a sequence $\left(x_n\right)\subset X$ converging weakly but not strongly (which of course requires that $dimX=\infty$). On the other hand, if $dimX<\infty$ then any continuous map $F:X\to X$ is of type ${\left(S\right)}_{+}$ without being necessarily monotone.

Pseudo-Monotone Operators

A “reunion” of the two properties, Monotonicity and ${\left(S\right)}_{+}$ (Definitions 1 and 4, respectively) is made possible in a sense by the introduction—due to Brezis [11]—of the idea of a Pseudo-Monotone Operator, such quality being defined as follows. Here and elsewhere, the symbol ⇀ will denote the weak convergence of a sequence, while the symbol → will denote its strong convergence.

Definition 5.

A map $F:X\to X^{*}$ is said to be Pseudo-Monotone if, for every sequence $\left(x_n\right)\subset X$, the conditions

$$\left\{\begin{array}{c}{x}_n\rightharpoonup x\hfill \\ \underset{n\to \infty }{lim\; sup}\langle F\left(x_n\right),x_n-x\rangle \le 0\hfill \end{array}\right.$$

(29)

imply that

$$\underset{n\to \infty }{lim\; inf}\langle F\left(x_n\right),x_n-z\rangle \ge \langle F\left(x\right),x-z\rangle$$

(30)

for every $z\in X$.

Indeed, assuming—as we are doing throughout the whole paper—that F is continuous, we have:

(a)

F Monotone and bounded on bounded sets ⇒F Pseudo-Monotone. For a proof of this implication, see, for instance, Chapter 2 Proposition 2.5 of [2].

(b)

On the other hand, it is immediate to check that F of type ${\left(S\right)}_{+}$⇒F is Pseudo-Monotone for, suppose that the conditions in (29) are satisfied. Then, the property ${\left(S\right)}_{+}$ implies that $x_n\to x$ and so—by the continuity of F—is $F\left(x_n\right)\to F\left(x\right)$. It then follows that for every $z\in X$,

$$\underset{n\to \infty }{lim}\langle F\left(x_n\right),x_n-z\rangle =\langle F\left(x\right),x-z\rangle .$$

More generally, the Pseudo-Monotonicity property will be satisfied by F provided that the conditions in (29) imply that $F\left(x_n\right)\to F\left(x\right)$.

As a final statement in this Section, we report a result that shows the relevance of Pseudo-Monotonicity in the context of Variational Inequalities and—in view of the remarks made above connecting this property with Monotonicity and ${\left(S\right)}_{+}$—marks a generalization of the two mentioned Theorems of Browder (Theorems 11 and 13 in [7]).

Theorem 4.

Let X be a real, reflexive Banach space, and let $F:X\to X^{*}$ be Pseudo-Monotone. Suppose, moreover, that F is bounded on bounded sets, continuous and coercive. Let $K\subset X$ be closed and convex with $0\in K$. Then, for every $y\in X^{*}$, the V.I. (27) has a solution $x_0\in K$.

Theorem 4 is a special case of more general results to be found in Lions’ book (see Theorems 8.1 and 8.2 in Chapter 2 of [2]), whose proofs follow essentially the same pattern as that of Browder’s Theorem 3. For an updated reference to the concept of a Pseudo-Monotone map and its use for the applications to existence results for nonlinear elliptic equations see, for instance, Chapter 10 of [12].

So far, we have merely reported a few statements and remarks about Variational Inequalities considered in their generality. In the next Section, we restrict our attention to the special case in which the involved operator $F:X\to X^{*}$ is a gradient: as recalled in the Introduction, this allows us to employ variational methods in order to obtain the existence results of the type discussed above. In fact, on the basis of Remarks 1 and 2, it is natural to see whether the method that we have employed in [5,6] for the study of surjectivity of F, namely the Ekeland Variational Principle and its consequences (see, e.g., [4]) can be used also in the context of Variational Inequalities, and this is precisely what we do in the next Section.

3. The Ekeland Principle on Convex Sets and Pseudo-Monotonicity

Let X be a real Banach space and let f be a $C^1$ functional defined on X. For $x_0\in X$, $f^{\prime }\left(x_0\right)$ is a bounded linear form on X, whose norm in $X^{*}$ is

$$\parallel f^{\prime }\left(x_0\right)\parallel =sup\left\{{\displaystyle \frac{|\langle f^{\prime }\left(x_0\right),v\rangle |}{\parallel v\parallel }}:v\in X,v\ne 0\right\}.$$

(31)

In the study of minimum problems for f, one standard way of using the Ekeland principle (see, e.g., [4]) is provided by the following statement.

Lemma 1.

Let f be a $C^1$ functional defined on X and suppose that f is bounded below on X. Let $m={inf}_{x\in X}f\left(x\right)$. Then, given any $ϵ>0$, there exists $x_{ϵ}\in X$ such that

$$\left\{\begin{array}{c}f\left(x_{ϵ}\right)<m+ϵ\hfill \\ \parallel f^{\prime }\left(x_{ϵ}\right)\parallel \le ϵ.\hfill \end{array}\right.$$

(32)

Before looking more closely at Lemma 1 and its proof, it can be noted at once that taking a sequence $\left({ϵ}_n\right)$ with ${ϵ}_n\to 0$ and putting $x_n\equiv x_{{ϵ}_n}$, it follows from (32) that

$$\left\{\begin{array}{c}f\left(x_n\right)\to m\hfill \\ f^{\prime }\left(x_n\right)\to 0,\hfill \end{array}\right.$$

(33)

and if we know in addition that $\left(x_n\right)$ converges through some subsequence, we can argue from (33) the existence of an $x_0\in X$ such that $f\left(x_0\right)=m$ and $f^{\prime }\left(x_0\right)=0$, thus giving an affirmative answer to the minimum problem for f on X. In turn, one way to obtain the desired information about $\left(x_n\right)$ is by knowing at least that it is bounded—this property is usually recovered if f is assumed to be coercive (see Definition 6 below)—and assuming, for instance, that $f^{\prime }:X\to X^{*}$ is locally proper, meaning that any bounded sequence $\left(x_n\right)\subset X$ such that $f^{\prime }\left(x_n\right)$ converges contains a convergent subsequence. In fact, this is the reasoning that we have followed in [5] to prove that $f^{\prime }$ is surjective.

Question: What can be said when X is replaced by a closed convex $K\subset X$?

For an answer to this question, it is useful to see how Lemma 1 follows the Ekeland Variational Principle. The following “weak form” of the latter (see, e.g., [4], Chapter 4, Theorem 4.1) is sufficient for our purposes:

Theorem 5

(Ekeland variational principle). Let $(X,d)$ be a complete metric space. Let $f:X\to \mathbb{R}$ be lower semicontinuous and bounded below. Put $m={inf}_{x\in X}f\left(x\right)$; then given any $ϵ>0$, there exists $x_{ϵ}\in X$ such that

$$\left\{\begin{array}{c}f\left(x_{ϵ}\right)<m+ϵ\hfill \\ f\left(x_{ϵ}\right)<f\left(x\right)+ϵd(x,x_{ϵ})\forall x\in X,x\ne x_{ϵ}.\hfill \end{array}\right.$$

(34)

To prove Lemma 1 on the basis of Theorem 5, let $v\in X$, $v\ne 0$ and take $x=x_{ϵ}+tv,t\ne 0$ in the second inequality in (34): this yields

$$f\left(x_{ϵ}\right)<f(x_{ϵ}+tv)+ϵ\left|t\right|\parallel v\parallel .$$

(35)

Thus, taking $t>0$, we obtain

$${\displaystyle \frac{f(x_{ϵ}+tv)-f\left(x_{ϵ}\right)}{t}}>-ϵ\parallel v\parallel$$

and therefore, letting $t\to 0^{+}$,

$$\langle f^{\prime }\left(x_{ϵ}\right),v\rangle \ge -ϵ\parallel v\parallel .$$

(36)

Similarly, taking $t<0$ in (35) yields

$${\displaystyle \frac{f(x_{ϵ}+tv)-f\left(x_{ϵ}\right)}{t}}<ϵ\parallel v\parallel$$

whence, letting $t\to 0^{-}$,

$$\langle f^{\prime }\left(x_{ϵ}\right),v\rangle \le ϵ\parallel v\parallel .$$

(37)

Using (31), (36) and (37) then yields the second inequality in (32).

Suppose now that K is a closed, convex subset of the Banach space X. Then, Lemma 1 should be replaced by the following statement.

Proposition 2.

Let f be a $C^1$ functional defined on X and let $K\subset X$ be closed and convex. Suppose that f is bounded below on K, and let $m={inf}_{x\in K}f\left(x\right)$. Then, given any $ϵ>0$, there exists $x_{ϵ}\in K$ such that

$$\left\{\begin{array}{c}f\left(x_{ϵ}\right)<m+ϵ\hfill \\ \langle f^{\prime }\left(x_{ϵ}\right),v-x_{ϵ}\rangle \ge -ϵ\parallel v-x_{ϵ}\parallel \forall v\in K.\hfill \end{array}\right.$$

(38)

Proof. 

Apply the Ekeland principle taking as a complete metric space K itself. Given $ϵ>0$, let $x_{ϵ}$ be as in the statement of Theorem 5. Moreover, let $v\in K,v\ne x_{ϵ}$, and for $t\in ]0,1]$ take $x=(1-t)x_{ϵ}+tv=x_{ϵ}+t(v-x_{ϵ})\in K$ in the second inequality in (34): this yields

$$f(x_{ϵ}+t(v-x_{ϵ}))-f\left(x_{ϵ}\right)>-ϵt\parallel v-x_{ϵ}\parallel$$

(39)

and thus, dividing both members of (39) by t and letting $t\to 0^{+}$, we obtain the second inequality in (38). □

As noted before when commenting on the statement of Lemma 1, it follows at once from Proposition 2 that taking a sequence $\left({ϵ}_n\right)$ with ${ϵ}_n>0$ and ${ϵ}_n\to 0$, and putting $x_n\equiv x_{{ϵ}_n}$, we obtain a minimizing sequence for f in K such that, in addition,

$$\langle f^{\prime }\left(x_n\right),v-x_n\rangle \ge -{ϵ}_n\parallel v-x_n\parallel \forall v\in K.$$

(40)

Now, the continuity of $f^{\prime }:X\to X^{*}$ will imply that, if $\left(x_n\right)$ converges through a subsequence to some $x_0$, then $x_0\in K$ is a minimum point for f in K and is also a solution of the variational inequality

$$\langle f^{\prime }\left(x_0\right),v-x_0\rangle \ge 0\forall v\in K.$$

(41)

However, at this stage, the convergence of (a subsequence of) $\left(x_n\right)$ is not granted even if we assume the local properness of $f^{\prime }$: indeed, the inequality (40)—that replaces the limit relation $f^{\prime }\left(x_n\right)\to 0$ appearing in (33) and holding in case $K=X$—does not necessarily yield the convergence of $f^{\prime }\left(x_n\right)$.

In the discussion that follows, we are going to see how a different condition on $f^{\prime }$, namely its Monotonicity or its ${\left(S\right)}_{+}$ property (Definitions 1 and 4, respectively), or more generally its Pseudo-Monotonicity (Definition 5), will guarantee the existence of a solution $x_0$ to the variational inequality (41) starting from the solutions $x_n$ of the approximate inequalities (40).

Before doing that, we have to satisfy ourselves that the Ekeland Variational Principle and its concrete application to our problem, namely Proposition 2, is being used correctly, and in particular, that the relevant functional is bounded below on K. In our simplified discussion, we will, in fact, give conditions on f, implying that it is bounded below on the whole of X. In this respect, we first need to complete the definition of coercivity: this was given in Definition 2 for operators acting from X to $X^{*}$, and we give it next for functionals defined on X.

Definition 6.

Let X be a real Banach space. A functional $f:X\to \mathbb{R}$ is said to be coercive if

$$f\left(x\right)\to +\infty \mathit{as}\left|\right|x\left|\right|\to \infty .$$

(42)

The two Propositions below are taken from [6]—where they appear as Lemma 1 and Proposition 2, respectively—to which we refer also for their proof.

Proposition 3.

Let $F:X\to X^{*}$ be a gradient operator with potential f. Suppose that F is continuous, coercive and bounded on bounded subsets of X. Then, $f:X\to \mathbb{R}$ is coercive and bounded on bounded subsets of X.

Proposition 4.

If $f:X\to \mathbb{R}$ is continuous, coercive and bounded on bounded subsets of X, then f is bounded below on X, and any minimizing sequence is necessarily bounded.

We are now ready to state and prove our main result.

Theorem 6.

Let X be a real, reflexive Banach space, and let $F:X\to X^{*}$ be a gradient operator that is continuous and bounded on bounded subsets of X. Suppose, moreover, that:

(i)

F is coercive;

(ii)

F is Pseudo-Monotone.

Then, for every $K\subset X$, K closed and convex, and for every $y\in X^{*}$, there exists $x_0\in K$ such that

$$\langle F\left(x_0\right)-y,x-x_0\rangle \ge 0\forall x\in K.$$

(43)

Remark 4.

If the assumption (ii) above is strengthened to either “F is of type ${\left(S\right)}_{+}$” or “F is Monotone”, and if f is the potential of F ($F=f^{\prime }$), then in addition, $x_0$ minimizes in K the functional $f_y$ defined putting

$$f_y\left(x\right)=f\left(x\right)-\langle y,x\rangle (x\in X).$$

(44)

We are unable to say whether this conclusion can be drawn in the more general case that F is merely Pseudo-Monotone.

Proof of Theorem 6. 

We find it convenient to divide the proof of Theorem 6 into two parts as follows. □

• Part A: Approximate Variational Inequalities through the EVP

Let X, F and K be as in Theorem 6. For a given $y\in X^{*}$, consider the map $F_y:X\to X^{*}$ defined putting

$$F_y\left(x\right)=F\left(x\right)-y(x\in X),$$

and check that $F_y$ satisfies the same assumptions as F. Indeed, $F_y$ is plainly continuous and bounded on bounded subsets of X if F is as well. Moreover, if f is the potential of F, then the functional $f_y$ defined in (44) is the potential of $F_y$, so that $F_y$ is a gradient is F is as well. It is also coercive if F is as well, by virtue of the inequality

$${\displaystyle \frac{\langle F_y\left(x\right),x\rangle }{\left|\right|x\left|\right|}}={\displaystyle \frac{\langle F\left(x\right),x)\rangle }{\left|\right|x\left|\right|}}-{\displaystyle \frac{\langle y,x\rangle }{\left|\right|x\left|\right|}}\ge {\displaystyle \frac{\langle F\left(x\right),x\rangle }{\left|\right|x\left|\right|}}-\left|\right|y\left|\right|.$$

The verification that the Pseudo-Monotonicity of F implies that $F_y$ is likewise straightforward.

Thus, if we assume the validity of Theorem 6 for the case $y=0$, it follows that there exists an $x_0\in K$ such that

$$\langle F_y\left(x_0\right),x-x_0\rangle \ge 0\forall x\in K$$

which is nothing but the desired inequality (43).

It is, therefore, enough to prove Theorem 6 for the case $y=0$. In turn, as F is a gradient, in order to find a solution $x_0\in K$ of the V.I.

$$\langle F\left(x_0\right),x-x_0\rangle \ge 0(x\in K)$$

it will be enough to prove that the potential f of F is bounded below on K and achieves its infimum on K (Remark 1).

By the assumptions on F and Propositions 3 and 4, we know that f is bounded below on the whole of X and also that any minimizing sequence is necessarily bounded; a fortiori, the same conclusions will hold if we consider the restriction of f to a given closed and convex subset K of X.

Now, applying the Ekeland Variational Principle to f on K, namely using Proposition 2—see, in particular, the comments following it and leading to Equation (40)—we find a sequence $\left({ϵ}_n\right)$ with ${ϵ}_n>0$ and ${ϵ}_n\to 0$ and a minimizing sequence $\left(x_n\right)\subset K$, such that

$$\langle F\left(x_n\right),v-x_n\rangle \ge -{ϵ}_n\parallel v-x_n\parallel \forall v\in K.$$

(45)

It is now convenient to put ${ϵ}_n^{\prime }={ϵ}_n\parallel v-x_n\parallel (n\in \mathbb{N})$, noting that ${ϵ}_n^{\prime }\ge 0$ for every n and ${ϵ}_n^{\prime }\to 0$ as $n\to \infty$ by virtue of the boundedness of the sequence $\left(x_n\right)$.

• Part B: From Approximate to Exact inequalities

Our proof of Theorem 6 will now be completed starting from the Approximate Variational Inequalities

$$\langle F\left(x_n\right),x-x_n\rangle \ge -{ϵ}_n^{\prime }(n\in \mathbb{N},x\in K)$$

(46)

that hold in the closed convex $K\subset X$, and showing that our Pseudo-Monotonicity assumption on F ensures that the approximate inequalities (46) lead, in the limit as $n\to \infty$, to the exact desired inequality

$$\langle F\left(x_0\right),x-x_0\rangle \ge 0(x\in K)$$

(47)

holding for some $x_0\in K$.

The first assertion to be made is that using the boundedness of $\left(x_n\right)$ and the reflexivity of X, we can assume—passing if necessary to a subsequence—that $\left(x_n\right)$ converges weakly to some $x_0$; then $x_0\in K$ since K, being closed and convex, is weakly closed.

Now the approximate inequalities (46) can be equivalently written as

$$\langle F\left(x_n\right),x_n-x\rangle \le {ϵ}_n^{\prime }\forall n\in \mathbb{N},\forall x\in K$$

(48)

and thus imply that

$$\underset{n\to \infty }{lim\; sup}\langle F\left(x_n\right),x_n-x\rangle \le 0\forall x\in K$$

(49)

whence in particular, since $x_0\in K$,

$$\underset{n\to \infty }{lim\; sup}\langle F\left(x_n\right),x_n-x_0\rangle \le 0.$$

(50)

Resuming, the approximate inequalities (46) imply not only the particular inequality (50) involving the weak limit $x_0$ of the sequence $\left(x_n\right)$, but also the general inequality (49), which implies a fortiori that

$$\underset{n\to \infty }{lim\; inf}\langle F\left(x_n\right),x_n-x\rangle \le 0\forall x\in K.$$

(51)

The Pseudo-Monotonicity condition satisfied by F (see Definition 5) now implies that

$$0\ge \langle F\left(x_0\right),x_0-x\rangle \forall x\in K,$$

(52)

which is precisely (47). This ends the proof of Theorem 6.

Remark 5.

It is interesting to see how our proof of Theorem 6 based on the approximate inequalities (46) becomes immediate if one strengthens the assumption of Pseudo-Monotonicity on F to that of being of type ${\left(S\right)}_{+}$, and more transparent anyway if one strengthens the assumption of Pseudo-Monotonicity to that of Monotonicity (see the remarks following Definition 5).

1. The desired inequality (47) follows immediately from (46) if F is of type ${\left(S\right)}_{+}$. Indeed, as we have seen above in the proof of Theorem 6, (46) imply (50) so that, looking back at Definition 4, we conclude that $x_n\to x_0$ (strongly). The continuity of F then implies that $F\left(x_n\right)\to F\left(x_0\right)$, and using this in (46) yields (47).

2. The desired inequality (47) also follows from (46) if F is monotone: indeed, if

$$\langle F\left(x\right)-F\left(y\right),x-y\rangle \ge 0\forall x,y\in X,$$

(53)

then (46) implies that

$$\langle F\left(x\right),x-x_n\rangle \ge -{ϵ}_n^{\prime }\forall n\in \mathbb{N},\forall x\in K$$

(54)

whence, letting $n\to \infty$, we obtain

$$\langle F\left(x\right),x-x_0\rangle \ge 0\forall x\in K.$$

(55)

However, this implies (47) by virtue of the convexity of K and the continuity of F: to see this, just take any $v\in K$, using (55) with $x=(1-t)x_0+tv$ ($0<t<1$) and finally let $t\to 0$.