Existence and Relaxation Results for Second Order Multivalued Systems

We consider nonlinear systems driven by a general nonhomogeneous differential operator with various types of boundary conditions and with a reaction in which we have the combined effects of a maximal monotone term A(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(x)$\end{document} and of a multivalued perturbation F(t,x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F(t,x,y)$\end{document} which can be convex or nonconvex valued. We consider the cases where D(A)≠RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D(A)\neq \mathbb{R}^{N}$\end{document} and D(A)=RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D(A)= \mathbb{R}^{N}$\end{document} and prove existence and relaxation theorems. Applications to differential variational inequalities and control systems are discussed.

A multifunction F : → P f (X) is said to be "measurable", if for all u ∈ X, the function ω → d(u, F (ω)) = inf[ u − x X : x ∈ F (ω)] is -measurable. A measurable multifunction F : → P f (X) is graph measurable and the converse is true for P f (X)-valued multifunctions, if there is a σ -finite, complete measure μ defined on .
Let ( , , μ) be a σ -finite measure space and X a separable Banach space. For any 1 ≤ p ≤ ∞ and for any multifunction F : → 2 X \ {∅}, we define the set If F (·) is graph measurable and ω → inf{ x X : x ∈ F (ω)} belongs in L p ( ), then S p F = ∅. This set is decomposable in the sense that for all triples (B, f 1 , f 2 ) ∈ × S p F × S p F , we have Here by χ B we denote the characteristic function for the set B ∈ , that is, Since χ B c = 1 − χ B , the above definition of decomposability formally looks like that of convexity. Only now the coefficients are not constants in [0, 1], but functions with values in [0, 1]. Nevertheless decomposable sets exhibit properties similar to those of convex sets (see Fryszkowski [6] and Hu-Papageorgiou [11]). Suppose that Y , V are Hausdorff topological spaces and G : Y → 2 V \ {∅} a multifunction. We introduce the following notions: When F is single valued, then both notions in (a) and (b) coincide with that of continuity. Upper semicontinuity implies closedness, while the converse is true if G(·) is locally compact, that is, for every y ∈ Y , there is a neighborhood W of y such that ∪ y ∈W G(y ) ∈ P k (V ).
If V is a metric space with metric d V , then G(·) is lsc if and only if for every v ∈ V the function y → d V (v, G(y)) is upper semicontinuous as an R + -valued function.
For a metric space V with metric d V , on P f (V ) we can define a generalized metric, known as the "Hausdorff metric" by We know that if V is complete, then so is (P f (V ), h). A multifunction G : Y → P f (V ) is said to be "h-continuous", if it is continuous from Y into (P f (V ), h). We say that G(·) is "continuous", if it is both usc and lsc. The two notions are in general distinct and they coincide if G(·) is P k (V )-valued.
For a Banach space X and C ⊆ X nonempty, we define Let Y , V be two Banach spaces and ξ : Y → V . We say that ξ(·) is "completely continuous", if y n w − → y in Y implies ξ(y n ) → ξ(y) in V (so it is sequentially continuous from Y with the weak topology into V with the strong topology). A multifunction G : Y → 2 V \ {∅} is "compact", if it is usc and maps bounded sets in Y into relatively compact sets in V .
We will use two results from multivalued analysis. The first is the multivalued analog of the Leray-Schauder Alternative Principle due to Bader [1]. So, assume that Y , V are Banach spaces, N : Y → P wkc (V ) is usc from Y into V w = the Banach space V endowed with the weak topology and ξ : V → Y is completely continuous. Let L = ξ • N . The result of Bader [1] (Theorem 8) asserts the following: Theorem 1 If Y , V , L are as above and L(·) is compact, then one of the following statements holds: The next theorem is an extension of the celebrated Michael Selection Theorem (see [11], p. 92) to multifunctions with decomposable values. The result is a powerful illustration that decomposability is a good substitute for convexity and it is due to Bressan-Colombo [2] and Fryszkowski [5].
Theorem 2 If ( , , μ) is a finite measure space, X is a separable Banach space, Y is a separable metric space and N : Y → P f (L 1 ( , X)) is a lsc multifunction with decomposable values, then there exists a continuous map e : Y → L 1 ( , X) such that e(y) ∈ N(y) for all y ∈ Y . Now let E be a reflexive Banach space and E * its topological dual. By ·, · we denote the duality brackets for the pair (E, We say that A(·) is "maximal monotone", if Gr A is not properly included in the graph of another monotone map, that is, If A(·) is maximal monotone, then GrA is closed in E × E * w and in E w × E * . Here by E w (resp. by E * w ), we denote the space E (resp. E * ) equipped with the weak topology. Also we set D(A) = {u ∈ E : A(u) = ∅} (the domain of A).
Suppose that E = H = a Hilbert space and identify H with its dual (that is, H = H * , by the Riesz-Frechet Theorem). Let A : H → 2 H be a maximal monotone map. We introduce the following single-valued maps: .

Remark 1
We know that in a Hilbert space every closed, convex set has the best approximation property. So, in part (e) of the above proposition, the metric projection proj(u, D(A)) is well-defined.
We will also use an extension of the notion of maximal monotone operator. So, as before E is a reflexive Banach space with E * its topological dual. A multivalued map L : E → P wkc (E * ) is said to be "pseudomonotone" if it maps bounded sets to bounded sets and has the following property "for every sequences If L(·) is maximal monotone and D(L) = E, then L(·) is pseudomonotone (for details we refer to Gasiński-Papageorgiou [7], Sect. 3.2).
Finally we mention that by · we denote the norm of the Sobolev space W 1,p ( ). Recall that Now, we are ready to introduce the hypotheses on the data of problem (1). First for the map a(·) in the differential operator.
H 0 : a : R N → R N has the form a(y) = a 0 (|y|)y for all y ∈ R N , with a 0 (t) > 0 for all t > 0 and (i) a(·) is continuous, strictly monotone; (ii) there exists c 0 > 0 such that

Remark 2
Hypothesis H 0 (i) implies that a(·) is maximal monotone. It is worth pointing out that no global growth condition is imposed on a(·). So, the map is very general. Hypotheses H 0 imply that a : R N → R N is a homeomorphism and we have |a −1 (y)| → +∞ as |y| → +∞. The restriction 2 ≤ p is needed since we will not assume that D(A) = R N . When D(A) = R N , then we can have 1 < p. Such kind of maps a(·), were first used by Manásevich-Mawhin [16].
This map corresponds to the (p, q)-Laplacian.

Remark 3
We stress that we do not assume that D(A) = R N . This way our setting covers problems with unilateral constraints. Also this fact leads to the restriction 2 ≤ p.
In what follows by λ 1 we denote the principal eigenvalue of the eigenvalue problem − |u | p−2 u = λ|u| p−2 u, u ∈ BC.
These eigenvalue problems are discussed in Manásevich-Mawhin [17] and we have λ 1 > 0 for the Dirichlet and mixed problems, λ 1 = 0 for the Neumann and periodic problems.
For the convex problem our hypotheses on the multivalued perturbation F (t, x, y) are the following: For the nonconvex problem, the hypotheses on the multivalued perturbation are the following: x, y) is lsc; (iii) same as hypothesis H 1 (iii); (iv) same as hypothesis H 1 (iv).

Remark 4
Now the measurability hypothesis (see H 2 (i)) is stronger. This is always the case for nonconvex problems.

Convex Problem
In this section we consider the case when F is convex-valued.

Proposition 2
If hypotheses H 0 hold, then the map a : D( a) ⊆ L p → L p is maximal monotone.
Let A λ : L p → L p be defined by Since A λ (·) is Lipschitz continuous, A λ (0) = 0 and 2 ≤ p, we see that A λ is well-defined and we have: If hypotheses H 1 hold and λ > 0, then the map A λ : L p → L p is monotone, continuous (hence maximal monotone).

Remark 5
It is at this point that we use the restriction 2 ≤ p. If D(A) = R N , then we do not need to consider the approximate boundary value problem (2 λ ) and so we do not need the restriction 2 ≤ p.
Given h ∈ L p , we consider the following boundary value problem (3 λ )
Proof Let ϕ p : L p → L p be the map defined by Evidently D(V λ ) = D( a) and from Theorem 3.2.41, p. 328, of Gasiński-Papageorgiou [7], we have that V λ (·) is maximal monotone. Let (·, ·) pp denote the duality brackets for the pair (L p , L p ). We have (performing integration by parts and since A λ (0) = 0) Then Corollary 3.2.31, p. 319, of Gasiński-Papageorgiou [7] implies that In fact, on account of the strict monotonicity of ϕ p (·) this solution is unique.
In what follows X stands for one of the following spaces for vector-valued functions: for the Dirichlet problem for the periodic problem for the mixed problem Consider the map ξ λ : with u λ being the unique solution of (3 λ ) guaranteed by Proposition 4.

Proposition 5
If hypotheses H 0 , H 1 hold and λ > 0, then the solution map ξ λ : L p → X is completely continuous.
Proof Consider a sequence h n w − → h in L p and let u n = ξ λ (h n ), n ∈ N, and u = ξ λ (h). We have (acting with u n and performing integration by parts), Exploiting the fact that W 1,p → C(T , R N ) compactly, we infer that for all n ∈ N, From (5) we have We let k( u n ) ∈ L p be defined by Then we have with Then by Proposition 2.2 of Manásevich-Mawhin [16], we have with σ : (4) and (7) we see that (9)).
Recall that a −1 , seen as a map from C(T , R N ) into C(T , R N ), maps bounded sets to bounded sets. Therefore from (8) we see that Therefore from (6) and (10) it follows Now suppose that the boundary condition BC is Neumann or mixed. Then a( u n (0)) = 0 for all n ∈ N, Then again we infer that (11) holds. Using (11) and recalling that W 1,p → C(T , R N ) compactly, we infer that Then (4) and (12) imply that So, by passing to a subsequence if necessary, we may assume that Note that Recall that a(·) is maximal monotone (see Proposition 2). Therefore Gr a ⊆ L p × L p w is closed, with L p w being the Lebesgue space L p (T , R N ) equipped with the weak topology (see Gasiński-Papageorgiou [7], Proposition 3.2.15, p. 308). Then on account of (13), we have So, for the original sequence we have Next we introduce the multivalued map N : X → 2 L p defined by with N F (·) being the multivalued Nemyckii map

Proposition 6 If hypotheses H 2 hold, then the multifunction N(·) has values in P wkc (L p )
and it is usc from X with the norm topology into L p with the weak topology (denoted by L p w ).
Proof Note that hypotheses H 2 (i), (ii), do not imply graph measurability of F (see Hu-Papageorgiou [11], p. 226). So, it is not immediately clear that N(·) has nonempty values.
To show this, we argue as follows. Let u ∈ X. Then we can find two sequences {s n } n∈N , {r n } n∈N of simple functions such that We consider the multifunction G n : T → P kc (R N ) defined by G n (t) = F (t, s n (t), r n (t)) for all t ∈ T , all n ∈ N.
Therefore we have that Moreover, from hypothesis H 2 (iv) and the definition of N(·), it is clear that N(·) is P wkc (L p )-valued. Hypothesis H 2 (iv) shows that N(·) is locally compact into L p w (see Sect. 2) and so in order to prove the desired upper semicontinuity of N(·), it suffices to show that N(·) is closed (that is, Gr N ⊆ X × L p w is closed). To this end, let {u n } n∈N ⊆ X and { f n } n∈N ⊆ L p be two sequences which satisfy We (17)).
Consider the map L λ : Evidently a fixed point of L λ (·) is a solution of problem (2 λ ). To produce such a fixed point, we will use Theorem 1.
Proof We consider the set Then Acting with u ∈ X, performing integration by parts and using hypothesis H 0 (ii) and the fact that (18)).
On account of hypotheses H 2 (iii), (iv), given ε > 0, we can find k ε ∈ L 2 ⊆ L p (recall p ≥ 2) such that We return to (19) and use (20). Then Choosing ε ∈ (0, c 8 ), we conclude that Then as in the proof of Proposition 5, we obtain that K ⊆ X is bounded (in fact relatively compact).
So, we can apply Theorem 1 and find u λ ∈ W 1,p such that Finally we will let λ → 0 + to produce a solution for problem (1). To perform this passage to the limit as λ → 0 + , we will need the next lemma, which can be found in a more general form in Gasiński-Papageorgiou [9], Lemma 2.3. Consider the "lifting" of A(·) on (L p , L p ), that is, the operator A : L p → 2 L p defined by Now we are ready for the existence result for the convex problem. Proof Let λ n → 0 + and let u n = u λn ∈ X be a solution of problem (2 λn ), n ∈ N (see Proposition 7). We have Acting with u n ∈ X, performing integration by parts and using hypothesis H 0 (ii) and the fact that (A λn (x), x) R N ≥ 0 for all x ∈ R N , all n ∈ N (see hypotheses H 1 ), we obtain So, we may assume that (recall that W 1,p → C(T , R N ) compactly).
On account of Proposition 1, t → A λn ( u n (t)) is Lipschitz continuous on T and so by Rademacher's theorem (see , p. 56), it is differentiable almost everywhere. On (21) we act with A λn ( u n ) and have b 0 (−a( u n ) , A λn ( u n )) R N dt + A λn ( u n ) 2 2 ≤ b 0 |f n ||A λn (u n )|dt for all n ∈ N. (23) Performing integration by parts and using the fact that A λn (0) = 0, we obtain b 0 (−a( u n ) , A λn ( u n )) By the chain rule (see Leoni [14], Corollary 3.52, p. 97), we have d dt A λn ( u n (t)) = A λn ( u n (t)) u n (t) for a.a. t ∈ T .
So, we may assume that and f n w − → f in L 2 as n → +∞.
Then as before (see the proof of Proposition 6), using (29) and hypothesis H 2 (ii), we show that f ∈ N F ( u).
We know that ( J λn ( u n ), A λn ( u n )) ∈ Gr A for all n ∈ N (see Proposition 1).
Since A(·) is maximal monotone (see Lemma 1), from (28) and (31) we infer that which is what we wanted. So, we conclude that u ∈ X is a solution of problem (1).

Nonconvex Problem
In this section, we prove an existence theorem for the "nonconvex problem", that is, the multivalued perturbation F (t, x, y) has nonconvex values (hypotheses H 3 ).
Proof We consider the multifunction N F : X → 2 L p (recall that 2 ≤ p) defined by for all u ∈ X.
Hypotheses H 3 (i), (iv) imply that N F (·) has values in P f (L p ).
Claim: N F (·) is lsc. From Sect. 2 (see also Proposition 2.26, p. 45, of Hu-Papageorgiou [11]), we know that in order to prove the Claim, it suffices to show that for every g ∈ L p , the function is upper semicontinuous (as an R + -valued function). We have (see Theorem 3.24, p. 183, of Hu-Papageorgiou [11]).
Let d g (u) = d(g, N F (u)) p . We need to show that for all μ ≥ 0, the set So, let {u n } n∈N ⊆ U μ and assume that u n → u in X. We have d g (u n ) ≥ μ for all n ∈ N.
We have u n → u and u n → u in C(T , R N ).

g(t), F (t, u n (t), u n (t))) p dt
This proves the Claim. Since N F (·) has decomposable values in P f (L p ) and it is lsc (see the Claim), we can use Theorem 2 and produce a continuous map e : X → L p such that e(u) ∈ N F (u) for all u ∈ X.
Then we consider the following boundary value problem Reasoning as in the "convex" case (see Sect. 3). We show that problem (34) admits a solution u ∈ X. Evidently u ∈ X is also a solution of (1).

Relaxation
We introduce the following two sets: S c = solution set of the "convex" problem, S = solution set of the "nonconvex" problem.
Our aim in this section, is to determine conditions which guarantee that S · = S c . Such a result is known in the literature as "relaxation theorem" and has important consequences in many applied areas such as control theory and game theory. Our solution to this fundamental problem is partial. We prove a relaxation theorem only for the Dirichlet problem, under stronger conditions on the map a(·). When we try to extend the result to other boundary conditions, we encounter serious technical difficulties and it is an interesting open problem whether it is possible to have a relaxation theorem for the Neumann, periodic and mixed problems. Also, we will consider multivalued perturbations F which are independent of the derivative u . So, now the problem under consideration is the following a(u (t)) ∈ A(u(t)) + F (t, u(t)) for a.a. t ∈ T , The condition on the data of problem (35) are the following: H 4 : a : R N → R N has the form a(y) = a 0 (|y|)y for all y ∈ R N , with a 0 (t) > 0 for all t > 0 and (i) a(·) is continuous and there exists c > 0 such that c|y − y | 2 ≤ (a(y) − a(y ), y − y ) R N for a.a. t ∈ T , all y, y ∈ R N ; (ii) there exists c 0 > 0 such that c 0 |y| p ≤ (a(y), y) R N for all y ∈ R N , 2 ≤ p.
The hypotheses on the multivalued perturbation F (t, x) are the following: The "convex problem" is obtained by replacing F with conv F . We start by mentioning that the set S c ⊆ C 1 is closed. This follows easily from the proof of the "convex" existence theorem. This observation is valid for the more general setting of Sect. 3.
In what follows the solution sets S c and S refer to problem (35). .
Consider the multifunction K n : W 1,p → 2 L p defined by F (t, y(t))), for a.a. t ∈ T .
Since L p is reflexive, we see that K n (y) ∈ P wkc (L p ) for all y ∈ W 1,p .
Lemma 3.9, p. 239, of Hu-Papageorgiou [11], implies that K n (·) is lsc. Also, it has decomposable values. Therefore So, we can apply Theorem 2 and find a continuous map β n : W 1,p → L p such that β n (y) ∈ K n (y) · p for all y ∈ W 1,p .
We consider the following auxiliary Dirichlet problem We know that problem (37) admits a solution v n ∈ C 1 (T , R N ) (see the proof of Theorem 4).
In (37) we take inner product of both sides with −v n (t), integrate over T = [0, b], perform integration by parts and use the fact that (A(x), x) R N ≥ 0 for all x ∈ R N . Then for all n ∈ N (see hypothesis H 4 (ii)).
Hypotheses H 5 (iii), (iv) imply that given ε > 0, we can find γ ε ∈ L 1 (T ) such that Using (39) in (38), we obtain Choosing ε ∈ (0, c 10 ), we conclude that This proves the Claim. On account of the Claim we may assume that Performing integration by parts and using hypothesis H 4 (i) we have Moreover, using hypothesis H 5 (ii), we obtain |v n − u | 2 ds (using Jensen's inequality).
Since v n ∈ S for all n ∈ N (see (37)), we conclude that When dom A = R N and we restrict ourselves to the Dirichlet, Neumann and periodic problems, we can relax some of the hypotheses on the data of (1). More precisely, the new hypotheses are the following: H 6 : a : R N → R N is a map such that a(0) = 0 and (i) a(·) is continuous and strictly monotone; (ii) |a(y)| ≤ c 12 [1 + |y| p−1 ] for all y ∈ R N , some c 12 > 0; (iii) c 0 |y| p ≤ (a(y), y) R N for all y ∈ R N , some c 0 > 0 and 1 < p < +∞.

Remark 6
We do not require that a(y) = a 0 (|y|)y and we remove the restriction p ≥ 2. So, our formulation includes the singular vectorial p-Laplacian |y| p−2 y, 1 < p < 2. Another example which is covered by H 6 but not by H 0 , is the map a(y) = |y| p−2 y + proj(y, C), with C ∈ P f c (R N ), 0 ∈ C, proj(y, C) being the metric projection and 1 < p < +∞.

Remark 7
Even with this more restrictive condition on A(·), we continue to include in our framework many important classes of systems, such as gradient systems with a potential which is in general nonsmooth. In this case A = ∂ϕ with ∂ϕ being the convex subdifferential of a continuous convex function ϕ(·).
For the "convex" and "nonconvex" problems, we can improve the growth hypotheses.
is a multifunction such that hypotheses H 8 (i), (ii), (iii) are the same as the corresponding hypotheses H 2 (i), (ii), (iii) and (iv) for every r > 0, there exist γ r ∈ L p (T ) and c r > 0 such that for a.a. t ∈ T , all |x| ≤ r, all y ∈ R N .
is a multifunction such that hypotheses H 9 (i), (ii), (iii) are the same as the corresponding hypotheses H 3 (i), (ii), (iii) and (iv) for every r > 0, there exist γ r ∈ L p (T ) and c r > 0 such that for the Dirichlet problem, V = W 1,p for the Neumann problem and V = W 1,p per = {u ∈ W 1,p : u(0) = u(b)} for the periodic problem. Then let ξ : V → V * be defined by This map is continuous, monotone, thus maximal monotone too. As before A : L p → 2 L p is the "lifting" of A(·) on the pair (L p , L p ) with D( A) = {u ∈ L p : S p A(u(·)) = ∅}. We know from Lemma 1 that A is maximal monotone. Moreover, in the present setting with D(A) = R N , we have C(T , R N ) ⊆ D( A). To see this note that the map A(·) has values in P kc (R N ) and it is usc (see , Proposition 3.2.14, p. 308). So, A(·) maps compact sets to compact sets (see Hu-Papageorgiou [11], Corollary 2.20, p. 42). Let u ∈ C(T , R N ). Then |A(u(t))| ≤ c 13 for some c 13 > 0, all t ∈ T .
We can easily check that the multifunction t → A(u(t)) has closed graph. Using the Yankov-von Neumann-Aumann selection theorem, we can find a measurable map g : T → R N such that Therefore C(T , R N ) ⊆ D( A). As before N F : V → 2 V * is defined by N F (u) = S p F (·,u(·),u (·)) . From Sects. 3 and 4, we know that N F (u) ∈ P wkc (L p ) (convex case, hypotheses H 8 ), N F (u) ∈ P f (L p ) (nonconvex case, hypotheses H 9 ).
We introduce the multivalued map L : V → 2 V * \ {∅} defined by First we deal with the convex problem. H 6 , H 7 , H 8 hold, then L(·) is pseudomonotone.

Proposition 8 If hypotheses
Proof Evidently L(·) is bounded (that is, maps bounded sets to bounded sets). Consider For every n ∈ N, we have u * n = ξ(u n ) + g n + f n with g n ∈ A(u n ), f n ∈ N F (u n ).
Recall that V → C(T , R N ) compactly. So, from (47) it follows that we can find c 14 For every n ∈ N, we define η n (t) = (a(u n (t)) − a(u (t)), u n (t) − u (t)) R N .
It follows that for a.a. t ∈ T , {u n (t)} n∈N ⊆ R N is bounded. Hence by passing to a subsequence (depending a priori on t ∈ T ), we can say that From (51) in the limit as n → +∞, we obtain So, for the initial sequence we have From (47) we have Then from (52) and (53) it follows that u n → u in L p , (by Vitali's Theorem or see , Problem 1.23, p. 38). Hence We know that the sequences {g n } n∈N , {f n } n∈N ⊆ L p are bounded. So, for at least a subsequence we have We have (u n , g n ) ∈ Gr A for all n ∈ N.
This proves that L(·) is pseudomonotone.
Choosing ε ∈ (0, c 17 ), we infer that Now we are ready for the existence theorem for the convex problem.
Using hypotheses H 9 we can have an existence theorem for the nonconvex problem.
Proof We consider the multifunction N F : V → 2 L p defined by for all u ∈ V .
From the proof of Theorem 4, we know that N F (·) has decomposable values in P f (L p ) and it is lsc. So, invoking Theorem 2, we can find a continuous map e : V → L p such that e(u) ∈ N F (u) for all u ∈ X.
Then we consider the following boundary value problem Reasoning as in the proof of Theorem 6, we show that problem (62) admits a solution u ∈ C 1 .
We can have relaxation theorems. First we will consider problem (35), but now our hypotheses on a(·) are less restrictive and so we can incorporate in our analysis more differential operators.

Remark 8
Note that hypothesis H 10 (ii) is weaker than the corresponding hypothesis H 4 (ii), since now the strong monotonicity condition is local. This fits in the present framework more differential operators (see the examples below).

Example 3
The following maps satisfy hypotheses H 10 : Proof Evidently S c is closed in C 1 . Also, keeping the notation introduced in the proof of Theorem 5, we have that v n → u in W 1,2 as n → +∞ (recall u ∈ S c , see the proof of Theorem 5). We have From (63) and since W 1,p → C(T , R N ) compactly, we have |A(v n (t))| ≤ c 20 for some c 20 > 0, all t ∈ T , all n ∈ N (see hypotheses H 7 ).
Also from (63) and hypothesis H 11 (iv), we have that Then, from (65), (66) and (67) it follows that From (68) and reasoning as in the proof of Proposition 5, we obtain that {v n } n∈N ⊆ C 1 is relatively compact.
On account of (64), we have v n → u in C 1 with v n ∈ S (see (65)), If we strengthen the conditions on A(·), we can have relaxation for the case where F is also dependent on u and the boundary condition is always Dirichlet.
So, the new conditions on the data are: H 12 : A : R N → R N is locally Lipschitz and strictly monotone.
Hypotheses H 13 (i), (ii) imply that t → F (t, y(t), y (t)) is graph measurable and so Then the Yankov-von Neumann-Aumann selection theorem implies that we can find h n ∈ L p such that h n (t) ∈ L n (t) for a.a. t ∈ T , all n ∈ N.
In addition this multifunction has decomposable values. So, we can apply Theorem 2 and obtain a continuous map γ n : V → L p , n ∈ N, such that γ n (y) ∈ n (y) · p for all y ∈ V , all n ∈ N.
Then we consider the following boundary value problem This problem has a solution v n ∈ C 1 , n ∈ N. We return to (73) and use (74), (75) and hypotheses H 10 (ii) and H 13 (ii). We obtain for some c 23 > 0, k 0 ∈ L 1 (T ) and all n ∈ N.
Passing to the limit as n → +∞ and using (72) and (75), we obtain for all t ∈ T . Then Proposition 1.7.87, p. 128, of Denkowski-Migorski-Papageorgiou [3], implies that The Dirichlet boundary condition implies η = 0. We have v n → u in C 1 and v n ∈ S for all n ∈ N,

Applications
First we present an application on differential variational inequalities. So, let R N + be the positive cone of R N . We consider the indicator function of this cone, namely the function This function is convex and lower semicontinuous. We consider the subdifferential in the sense of convex analysis ∂i R N + (x). We know that for all x = (x k ) N k=1 ∈ R N + we have and ∂i R N + (x) = ∅ if x / ∈ R N + (see , p. 526). The set N R N + (x) is known as the normal cone to R N + at x. We set Evidently A(·) is maximal monotone, 0 = A(0) and D(A) = R N . Given u = (u k ) N k=1 ∈ W 1,p , we introduce the following sets (iv) for every r > 0, there exists c r > 0 such that f (t, x) L ≤ c r for a.a. t ∈ T , all |x| ≤ r; H 15 : B ∈ L p (T , L(R m , R N )). H 16 : K : T × R N → P kc (R m ) is a multifunction such that (i) (t, x) → K(t, x) is measurable; (ii) for a.a. t ∈ T , x → K(t, x) is closed; (iii) |K(t, x)| ≤ M for a.a. t ∈ T , all x ∈ R N with M > 0.  L(t, x, y, v) for a.a. t ∈ T , all x, y ∈ R N , v ∈ R m and with η ∈ L 1 (T ).
On account of hypothesis H 16 (i) we can find k n : T × R N → R m , n ∈ N, measurable functions such that

F (t, x, y) = f (t, x)y + G(t, x)
is a measurable multifunction. A straightforward application of the Yankov-von Neumann-Aumann selection shows that the dynamic constraint of the optimal control problem is equivalent to the following multivalued boundary value problem a(u (t)) ∈ A(u(t)) + F (t, u(t), u (t)) for a.a. t ∈ T , u ∈ BC.
On account of hypotheses H 14 , H 15 , H 16 , the multifunction F (t, x, y) satisfies hypotheses H 2 .
Finally we assume that a(·) and A(·) satisfy hypotheses H 0 and H 1 respectively. Let {(u n , v n )} n∈N ⊆ C 1 × L 1 be a minimizing sequence for problem (78). Then we can show that {u n } n∈N ⊆ W 1,p is bounded. So, we may assume that u n w − → u in W 1,p and u n → u in C(T , R N ).
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