Extremal solutions and strong relaxation for nonlinear multivalued systems with maximal monotone terms

Accepted Manuscript Extremal solutions and strong relaxation for nonlinear multivalued systems with maximal monotone terms

Nikolaos S. Papageorgiou, Calogero Vetro, Francesca Vetro

PII: DOI: Reference:

S0022-247X(18)30016-7 https://doi.org/10.1016/j.jmaa.2018.01.009 YJMAA 21942

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

25 October 2017

Please cite this article in press as: N.S. Papageorgiou et al., Extremal solutions and strong relaxation for nonlinear multivalued systems with maximal monotone terms, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2018.01.009

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Extremal solutions and strong relaxation for nonlinear multivalued systems with maximal monotone terms Nikolaos S. Papageorgioua , Calogero Vetrob,∗, Francesca Vetroc a Department

of Mathematics, National Technical University, Zografou campus, 15780, Athens, Greece b Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123, Palermo, Italy c Department of Energy, Information Engineering and Mathematical Models (DEIM), University of Palermo, Viale delle Scienze ed. 8, 90128, Palermo, Italy

Abstract We consider differential systems in RN driven by a nonlinear nonhomogeneous second order differential operator, a maximal monotone term and a multivalued perturbation F (t, u, u ). For periodic systems we prove the existence of extremal trajectories, that is solutions of the system in which F (t, u, u ) is replaced by extF (t, u, u ) (= the extreme points of F (t, u, u )). For Dirichlet systems we show that the extremal trajectories approximate the solutions of the “convex” problem in the C 1 (T, RN )-norm (strong relaxation). Keywords: Maximal monotone map, differential inclusion, extremal trajectories, strong relaxation, bang-bang controls 2010 MSC: 34B15, 34C25, 47H06

1. Introduction The starting point of our work in this paper is the following periodic system ⎧ ⎪ ⎨a(u (t)) ∈ A(u(t)) + F (t, u(t), u (t)) for a.a. t ∈ T = [0, b], (1) ⎪ ⎩u(0) = u(b), u (0) = u (b).

∗ Corresponding

author Email addresses: [email protected] (Nikolaos S. Papageorgiou), [email protected] (Calogero Vetro), [email protected] (Francesca Vetro)

Preprint submitted to Journal of Mathematical Analysis and Applications January 10, 2018

In this problem a : RN → RN and A : D(A) ⊆ RN → 2R

N

are two maximal

monotone maps with A(·) not necessarily defined everywhere. Also, F : T × N

RN × RN → 2R \ {∅} is a multivalued perturbation. The hypotheses on the 5

map a(·) involved in the differential operator, are general and incorporate as special cases differential operators of interest. Problem (1) was first studied by Kyritsi-Matzakos-Papageorgiou [15], who proved existence theorems for both the “convex” problem (that is, F is convex valued) and the “nonconvex” problem (that is, F is nonconvex valued). In this paper, we examine what happens if

10

in (1) we replace F (t, u, u ) by extF (t, u, u ) which is the set of extreme points of the set F (t, u, u ). This study is important in various applications such as in control systems in connection with the so-called “bang-bang principle”. So, the problem under consideration is the following: ⎧ ⎪ ⎨a(u (t)) ∈ A(u(t)) + extF (t, u(t), u (t)) ⎪ ⎩u(0) = u(b),

for a.a. t ∈ T = [0, b],

(2)

u (0) = u (b).

We know that even if F (t, ·, ·) has nice continuity properties (such as hcontinuity or even h-Lipschitzness), in general extF (t, ·, ·) need not have any regularity properties. So, the results of Kyritsi-Matzakos-Papageorgiou [15] are not applicable and a different approach is needed. We prove an existence theorem for problem (2) when F (t, ·, ·) is h-continuous. Under stronger conditions on a(·) and F (t, ·, ·) and with Dirichlet boundary conditions we show that every state u(·) of the system a(u ) ∈ A(u) + F (t, u, u ) can be approximated in the C 1 (T, RN )-norm by extremal trajectories, that is, by solutions of the system a(u ) ∈ A(u) + extF (t, u, u ) satisfying u(0) = u(b) = 0. Such an approximation result is known as “strong 15

relaxation theorem”. In the context of control systems, the result says that 2

any state of the system can be approximated at any desired level of accuracy, by extremal states which are generated by economizing in the use of control functions. It is an open problem whether the result is also valid for periodic systems. Versions of problem (1) were studied by Bader-Papageorgiou [1], 20

Erbe-Krawcewicz [4], Frigon [5], Frigon-Montoki [6], Gasi´ nski-Papageorgiou [7], Halidias-Papageorgiou [10], Kandilakis-Papageorgiou [14], Man´asevich-Mawhin [16], Pruszko [18], Zhang-Li [20, 21]. Of all the aforementioned works, only Gasi´ nski-Papageorgiou [7] examine the existence of extremal trajectories. However, they do it in the context of Dirichlet systems, driven by the vector p-

25

Laplacian (that is, a(y) = |y|p−2 y for all y ∈ RN with 2 ≤ p) and the multivalued perturbation is independent of u . They also prove a strong relaxation theorem but for the particular case of p = 2 (semilinear system).

2. Mathematical Background Our tools in the analysis of problem (2) come from multivalued analysis and 30

the theory of nonlinear operators of monotone type. Details on these issues can be found in the books of Gasi´ nski-Papageorgiou [8], Hu-Papageorgiou [12] and Zeidler [19]. Suppose that (Ω, Σ) is a measurable space and X a separable Banach space. We will use the following notation: Pf (c) (X) = {C ⊆ X : nonempty, closed (and convex)}, P(w)k(c) (X) = {C ⊆ X : nonempty, (weakly-) compact (and convex)}. Consider a multifunction (set-valued map) F : Ω → 2X \ {∅}. The “graph of F ” is the set GrF = {(ω, x) ∈ Ω × X : x ∈ F (ω)}. We say that F (·) is “graph measurable”, if GrF ∈ Σ ⊗ B(X), with B(X) being the Borel σ-field of X. Let μ(·) be a σ-finite measure defined on Σ. From the

35

Yankov-von Neumann-Aumann selection theorem (see Hu-Papageorgiou [12], p. 158), we know that there exists f : Ω → X a Σ-measurable function such that 3

f (ω) ∈ F (ω) μ-a.e.. We call f (·) a “measurable selection” of F (·). In fact one can show that there exists a whole sequence fn : Ω → X, n ∈ N, of measurable selections of F (·) such that F (ω) ⊆ {fn (ω)}n≥1 μ-a.e. (see Hu-Papageorgiou 40

[12], p. 159). Moreover, the result remains true if the separable Banach space X is replaced by a Souslin space (see Gasi´ nski-Papageorgiou [8], p. 905). A Souslin space is always separable but need not be metrizable. For example the dual of a separable Banach space furnished with the w∗ -topology, is Souslin. Now consider a multifunction F : Ω → Pf (X). We say that F (·) is “measur-

45

able”, if for all u ∈ X, the function ω → d(u, F (ω)) = inf[u − x : x ∈ F (ω)] is Σ-measurable. A measurable multifunction F (·) is graph measurable while the converse is true if (Ω, Σ) admits a σ-finite complete measure μ(·). Let (Ω, Σ, μ) be a σ-finite measure space, F : Ω → 2X \ {∅} a multifunction and p ∈ [1, +∞]. We introduce the following set SFp = {h ∈ Lp (Ω, X) : h(ω) ∈ F (ω)

μ-a.e.}.

This set is “decomposable” in the sense that, if (C, h1 , h2 ) ∈ Σ × SFp × SFp , then χC h1 + χΩ\C h2 ∈ SFp . Here for any D ∈ Σ, χD denotes the characteristic function of D, that is, ⎧ ⎪ ⎨1 if ω ∈ D, χD (ω) = ⎪ ⎩0 if ω ∈ Ω \ D. Since χΩ\C = 1 − χC the property of decomposability formally looks like that of convexity. It turns out that decomposable sets have several properties similar to 50

those of convex sets (see Hu-Papageorgiou [12], Section 2.3). A straightforward application of the Yankov-von Neumann-Aumann selection theorem, shows that for a graph measurable multifunction F : Ω → 2X \ {∅} we have “SFp = ∅ if and only if inf[u : u ∈ F (ω)] ∈ Lp (Ω).” Given any metric space (Y, d), we can define the Hausdorff generalized metric

4

on Pf (Y ) (= the collection of all nonempty, closed subsets of Y ), by h(C, E) = sup[|d(y, C) − d(y, E)| : y ∈ Y ] = max{sup d(c, E), sup d(e, C)] c∈C

e∈E

for all C, E ∈ Pf (Y ).

If Y is complete, then so is (Pf (Y ), h). Suppose that V is a Hausdorff topological space and G : V → Pf (Y ) a multifunction. We say that G(·) is Hausdorff continuous (h-continuous for short), if G(·) is continuous from V into (Pf (Y ), h). Also, a multifunction G : V → 2Y \ {∅} is said to be “lower semicontinuous” (“lsc” for short), if for all U ⊆ Y open the set G− (U ) = {v ∈ V : G(v) ∩ U = ∅} is open in V . Evidently G(·) is lsc if and only if for all C ⊆ Y closed G+ (C) = {v ∈ V : G(v) ⊆ C} 55

is closed in V . It is easy to check that x → G(x) is lsc if and only if x → G(x) is lsc. Next let us recall some basic definitions from the theory of nonlinear operators of monotone type. So, let X be a reflexive Banach space and X ∗ its topological dual. By ·, · we denote the duality brackets for the pair (X ∗ , X). ∗

A map A : D(A) ⊆ X → 2X is said to be “monotone”, if x∗ − u∗ , x − u ≥ 0

for all (x, x∗ ), (u, u∗ ) ∈ GrA.

Here D(A) is the domain of A(·) defined by D(A) = {x ∈ X : A(x) = ∅}. We say that A(·) is “strictly monotone”, if it is monotone and x∗ − u∗ , x − u = 0

=⇒

x = u.

A monotone map A(·) is “maximal monotone”, if “ x∗ − u∗ , x − u ≥ 0 for all (x, x∗ ) ∈ GrA

5

=⇒

(u, u∗ ) ∈ GrA”.

This implication says that GrA is maximal with respect to inclusion among the graphs of monotone maps. It is easy to see that if A(·) is maximal monotone, then GrA is sequentially closed in Xw × X ∗ and in X × Xw∗ . Here Xw (resp. 60

Xw∗ ) denotes the space X (resp. X ∗ ) furnished with the corresponding weak topology. Moreover, if A(·) is maximal monotone, then for all x ∈ D(A), A(x) ∈ Pf (c) (X ∗ ). Suppose that X = H is a Hilbert space identified with its dual (by the Riesz-Frechet representation theorem) and A : D(A) ⊆ H → 2H is a maximal monotone map. We introduce the following single valued approximations of the identity map I and of A: Jλ = (I + λA)−1 1 Aλ = [I − Jλ ] λ

(the “resolvent” of A), (the “Yosida approximation” of A).

We know that “A(·) is maximal monotone if and only if R(I + λA) = H”. 65

The result is known as “Minty’s Theorem” and can be found in Hu-Papageorgiou [12] (p. 321) and Zeidler [19] (p. 855). The single valued everywhere defined maps Jλ , Aλ exhibit some interesting properties listed in the next proposition. Proposition 1. If A : D(A) ⊆ H → 2H is maximal monotone and λ > 0, then (a) Jλ : H → H is nonexpansive (that is, Jλ (x) − Jλ (u) ≤ x − u for all

70

x, u ∈ H); (b) Aλ (x) ∈ A(Jλ (x)) for all x ∈ H; (c) Aλ : H → H is monotone and Lipschitz continuous with Lipschitz constant 1 λ

(therefore Aλ (·) is maximal monotone);

(d) for all x ∈ D(A), Aλ (x) ≤ A0 (x) = min[x∗  : x∗ ∈ A(x)] for all 75

λ > 0; (e) limλ→0+ Aλ (x) = A0 (x) for all x ∈ D(A); 6

(f ) D(A) is convex and limλ→0+ Jλ (x) = proj(x; D(A)) for all x ∈ H. Next we will introduce some more notions and notation which will be used in the sequel. So, let C ⊆ RN . We set |C| = sup[|c| : c ∈ C]. By (·, ·)RN we denote the inner product of RN , LT denotes the Lebesgue σ-field of T and B(RN ) the Borel σ-field of RN . Given 1 ≤ p < +∞, by 1 < p ≤ +∞ we denote the conjugate exponent of p, that is,

1 p

+ p1 = 1. By (·, ·)pp we denote



the duality brackets for the pair (Lp (T, RN ), Lp (T, RN )) (recall Lp (T, RN )∗ = 

Lp (T, RN )). So  (h, u)pp =

b 0



(h(t), u(t))RN dt for all h ∈ Lp (T, RN ), u ∈ Lp (T, RN ).

Given a Banach space Z, the “weak norm” on L1 (T, Z), denoted by  · w , is defined by

 hw = sup[

t s

h(τ )dτ Z : 0 ≤ s ≤ t ≤ b]

or equivalently by  hw = sup[

t 0

h(τ )dτ Z : 0 ≤ t ≤ b].

This norm is equivalent to the Pettis norm (see Egghe [3]). By L1w (T, Z) we denote the Lebesgue-Bochner space L1 (T, Z) furnished with the weak norm.

80

3. Extremal Periodic Solutions In this section we prove the existence of a solution for problem (2). To do this, we introduce the following hypotheses on the data of (2): H(a): a(y) = a0 (|y|)y for all y ∈ RN , with t → a0 (t)t continuous, nondecreasing, a0 (0) = 0, a0 (t) > 0 for all t > 0 and (a(y), y)RN ≥ c0 |y|p for all y ∈ RN ,

85

some c0 > 0 and 2 ≤ p < +∞.

7

Remark 1. If a(y) = |y|p−2 y with 2 ≤ p < +∞, then hypothesis H(a) is satisfied. This particular map corresponds to the vector p-Laplacian. Hypothesis H(a) also incorporates the vector (p, q)-Laplacian, that is a(y) = |y|p−2 y + |y|q−2 y

with 1 < q < p < +∞, p ≥ 2.

Note that a(·) need not have polynomial growth. So, we may also have the map a(y) = ∇θ(y) p

with θ(y) = 2e|y| − |y|p − 1 for all y ∈ RN . The conditions on the multivalued map A(·) are the following: H(A): A : D(A) ⊆ RN → 2R

N

is a maximal monotone map with 0 ∈ A(0).

Remark 2. Note that we do not assume that A is defined everywhere. This way 90

we include in our analysis also systems with unilateral constraints (differential variational inequalities). Finally the conditions on the multivalued perturbation F (t, x, y) are the following: H(F): F : T × RN × RN → Pkc (RN ) is a multifunction such that

95

(i) for all x, y ∈ RN , t → F (t, x, y) is graph measurable; (ii) for a.a. t ∈ T , (x, y) → F (t, x, y) is h-continuous; (iii) for a.a. t ∈ T and all x, y ∈ RN , we have |F (t, x, y)| ≤ γ1 (t, |x|) + γ2 (t, |x|)|y|p−1 where sup[γ1 (t, r) : 0 ≤ r ≤ ρ] ≤ η1,ρ (t) for a.a. t ∈ T , with η1,ρ ∈ L2 (T )+ sup[γ2 (t, r) : 0 ≤ r ≤ ρ] ≤ η2,ρ (t) for a.a. t ∈ T , with η2,ρ ∈ L∞ (T ); (iv) there exists M > 0 such that for a.a. t ∈ T , all x ∈ RN with |x| = M , all y ∈ RN and all h ∈ F (t, x, y) we have (h, x)RN ≥ 0. 8

Remark 3. Hypotheses H(F) (i), (ii) imply that (t, x, y) → F (t, x, y) is measurable. Hence F is superpositionally measurable (that is, for all (u, v) ∈ Lp (T, RN ) × Lp (T, RN ), t → F (t, u(t), v(t)) is measurable). Hypothesis H(F) 100

(iii) permits a general growth for x → |F (t, x, y)| not necessarily of polynomial type as is the case in most earlier works on the subject. Hypothesis H(F) (iv) is a multivalued version of the Hartman condition (see [11]). In what follows, let ϕ : RN → RN be the homeomorphism defined by ϕ(x) = |x|p−2 x

for all x ∈ RN .

Also let pM : RN → B M = {x ∈ RN : |x| ≤ M } be the M -radial retraction map defined by pM (x) =

⎧ ⎪ ⎨x

if |x| ≤ M,

⎪ ⎩ Mx

if M < |x|.

|x|

(3)

It is well-known that pM (·) is nonexpansive (that is, Lipschitz continuous with Lipschitz constant 1, see for example Gasi´ nski-Papageorgiou [8], p. 688). Using pM (·) we introduce the following “truncation-perturbation” of the multivalued perturbation F : F(t, x, y) = F (t, pM (x), y) − ϕ(pM (x)) for all (t, x, y) ∈ T × RN × RN . For every λ > 0 we consider the following auxiliary periodic system: ⎧ ⎪ ⎨a(u (t)) − ϕ(u(t)) ∈ Aλ (u(t)) + F (t, u(t), u (t)) for a.a. t ∈ T, ⎪ ⎩u(0) = u(b),

u (0) = u (b).

(Auλ )

Let S(λ) ⊆ C 1 (T, RN ) denote the solution set for problem (Auλ ). In the 105

next proposition, we establish an a priori bound for the solutions of (Auλ ). Proposition 2. If hypotheses H(a), H(A), H(F) hold, λ > 0 and u ∈ S(λ), then |u(t)| ≤ M for all t ∈ T (here M > 0 is as postulated by hypothesis H(F) (iv)).

9

Proof. We argue by contradiction. So, suppose that the conclusion of the propo110

sition is not true. We distinguish two cases: Case 1: There exist t1 , t2 ∈ T = [0, b] such that |u(t1 )| = M and |u(t2 )| = max |u(·)| > M. T

Case 2: |u(t)| > M for all t ∈ T . First we deal with Case 1. Without any loss of generality, we may assume that t1 < t2

and

|u(t)| > M for all t ∈ (t1 , t2 ].

Note that a(u ) ∈ L2 (T, RN ) and so a(u ) ∈ W 1,2 ((0, b), RN ). It follows that t → (a(u (t)), u(t))RN is absolutely continuous on T = [0, b]. By Lebesgue’s theorem t → (a(u (t)), u(t))RN is differentiable a.e. and we have

⇒

d (a(u ), u)RN = (a(u ) , u)RN + (a(u ), u )RN for a.a. t ∈ T, dt d − (a(u ) , u)RN = (a(u ), u )RN − (a(u ), u)RN for a.a. t ∈ T. dt

(4)

Since u ∈ S(λ), we can find h ∈ SF2 (·,pM (u(·)),u (·)) such that a(u (t)) − ϕ(u(t)) = Aλ (u(t)) + h(t) − ϕ(pM (u(t)))

for a.a. t ∈ T.

We take inner product with u(t) ∈ RN . Then −(a(u ) , u)RN + |u|p + (Aλ (u), u)RN + (h, u)RN −

|u| |pM (u)|p = 0 M

(5)

for a.a. t ∈ [t1 , t2 ] (see (3)). By hypothesis H(A) we have 0 ∈ A(0). Hence Aλ (0) = 0. Then the monotonicity of Aλ (·) (see Proposition 1), implies that (Aλ (u), u)RN ≥ 0

on T.

(6)

Returning to (5) and using (4) and (6), we obtain c0 |u |p −

d (a(u ), u)RN + |u|p − |u|M p−1 ≤ (−h, u)RN dt

(see hypothesis H(a) and (3)).

10

for a.a. t ∈ [t1 , t2 ] (7)

We set k(t) = |u(t)|2 for all t ∈ T . Then k(t2 ) = maxT k. If t2 ∈ (0, b), then we have k  (t2 ) = 0 and so (u(t2 ), u (t2 ))RN = 0.

(8)

Also from hypothesis H(F) (iv) we have (−h(t), u(t))RN =

|u(t)| (−h(t), pM (u(t)))RN ≤ 0 M

for a.a. t ∈ [t1 , t2 ].

Using (9) in (7) and since |u(t)| > M for all t ∈ (t1 , t2 ], we obtain 0<

d (a(u (t)), u(t))RN dt

for a.a. t ∈ [t1 , t2 ),

⇒ t → (a(u (t)), u(t))RN is strictly increasing on [t1 , t2 ]. It follows that (a(u (t)), u(t))RN < (a(u (t2 )), u(t2 ))RN

for all t ∈ [t1 , t2 ),

⇒

a0 (|u (t)|)(u (t), u(t))RN < a0 (|u (t2 )|)(u (t2 ), u(t2 ))RN = 0 (see (8)),

⇒

1  k (t) = (u(t), u (t))RN < 0 2

for all t ∈ [t1 , t2 ),

⇒

M 2 < k(t2 ) < k(t1 ) = M 2 ,

a contradiction.

If t2 = b, then we have |u(0)|2 = |u(b)|2 = max |u|2 , T

⇒





k (0) ≤ 0 ≤ k (b).

But the periodic boundary conditions imply that k  (0) = k  (b), hence k  (0) = k  (b) = 0 and so the previous argument applies. This takes care of Case 1. Next we deal with Case 2. So, we assume that |u(t)| > M

for all t ∈ T.

Then as above we show that 0< ⇒

d (a(u (t)), u(t))RN dt

for a.a. t ∈ T,

a0 (|u (0)|)(u (0), u(0))RN < a0 (|u (b)|)(u (b), u(b))RN . 11

(9)

But the periodic boundary conditions say that u (0) = u (b),

u(0) = u(b),

a contradiction. This takes care of Case 2 and we conclude that |u(t)| ≤ M

for all t ∈ T.

Using Proposition 2, we can produce a pointwise bound for |u (·)|.

115

>0 Proposition 3. If hypotheses H(a), H(A), H(F) hold, then there exists M  for all t ∈ T , all u ∈ S(λ), all λ > 0. such that |u (t)| ≤ M Proof. Since u ∈ S(λ), we can find h ∈ SF2 (·,pM (u(·)),u (·)) such that a(u (t)) − ϕ(u(t)) = Aλ (u(t)) + h(t) − ϕ(pM (u(t)))

for a.a. t ∈ T.

From Proposition 2 we have pM (u(t)) = u(t)

for all t ∈ T.

Therefore we have a(u (t)) = Aλ (u(t)) + h(t)

for a.a. t ∈ T and h ∈ SF2 (·,u(·),u (·)) .

(10)

Then (a(u (t)) , −u(t))RN = (Aλ (u(t)), −u(t))RN + (h(t), −u(t))RN for a.a. t ∈ T,  b  b  b (a(u ) , −u)RN dt = (Aλ (u), −u)RN dt + (h, −u)RN dt. (11) ⇒ 0

0

0

Using the integration by parts formula (see Gasi´ nski-Papageorgiou [8] (p. 148)) and the periodic boundary conditions, we obtain 

b 0

(a(u ) , −u)RN dt =



b 0

(a(u ), u )RN dt ≥ c0 u pp

(see hypothesis H(a)). (12)

12

Also, since 0 = Aλ (0) and Aλ (·) is monotone, we have  b (Aλ (u), −u)RN dt ≤ 0.

(13)

0

Returning to (11) and using (12), (13), we have  b |h|dt (see Proposition 2) c0 u pp ≤ M 0

⇒

≤ c1 [1 +

u p−1 ] p

⇒ u p ≤ c2

for some c1 > 0 (see hypothesis H(F) (iii))

for some c2 > 0 independent of λ > 0, u ∈ S(λ).

(14)

From (14) and Proposition 2 it follows that

S(λ) ⊆ W 1,p ((0, b), RN )

is bounded.

λ>0

From (10) and hypothesis H(F) (iii) we see that {a(u ) : u ∈ S(λ), λ > 0} ⊆ W 1,2 ((0, b), RN )

is bounded.

Exploiting the compact embedding of W 1,2 ((0, b), RN ) into C(T, RN ) we infer that {a(u ) : u ∈ S(λ), λ > 0} ⊆ C(T, RN )

is relatively compact.

So, there exists c3 > 0 such that |a(u (t))| ≤ c3 ⇒

for all t ∈ T , all u ∈ S(λ), all λ > 0,

 for some M  > 0, all t ∈ T , all u ∈ S(λ), all λ > 0 |u (t)| ≤ M

(since a(·) is a homeomorphism, see Man´ asevich-Mawhin [16]). On account of Propositions 2 and 3, we can replace F (t, x, y) by N N F0 (t, x, y) = F (t, pM (x), pM (y)) for all (t, x, y) ∈ T × R × R .

It is easy to see that F0 satisfies hypotheses H(F) (i), (ii), (iv) and we have

|F0 (t, x, y)| ≤ ϑ(t)

for a.a. t ∈ T , all x, y ∈ RN , with ϑ ∈ L2 (T ). 13

(15)

120

For g ∈ L2 (T, RN ), we consider the following single-valued periodic system ⎧ ⎪ ⎨a(u (t)) − ϕ(u(t)) = Aλ (u(t)) + g(t) ⎪ ⎩u(0) = u(b),

for a.a. t ∈ T,

(16)

u (0) = u (b).

Proposition 4. If hypotheses H(a), H(A) hold, then problem (16) admits a unique solution u = ξ(g) ∈ C 1 (T, RN ) and the solution map ξ : L2 (T, RN ) → C(T, RN ) is completely continuous. Proof. First we show the uniqueness of the solution of problem (16). To this end, suppose that u, v ∈ C 1 (T, RN ) are solutions of (16). Using the monotonicity of Aλ (·), we have −(a(u (t)) − a(v  (t)) , u(t) − v(t))RN + (ϕ(u(t)) − ϕ(v(t)), u(t) − v(t))RN ≤ 0 for a.a. t ∈ T . Integrating over T and using the integration by parts formula and the periodic boundary conditions, we obtain  b  b     (a(u ) − a(v ), u − v )RN dt + (ϕ(u) − ϕ(v), u − v)RN dt ≤ 0, 0

⇒

0

u=v

(follows from the monotonicity of a(·) and the strict monotonicity of ϕ(·)). 125

The existence of the unique solution is a special case of Proposition 4 of Kyritsi-Matzakos-Papageorgiou [15]. Finally we show the complete continuity of the solution map ξ(·). So, supw

→ g in L2 (T, RN ) and let un = ξ(gn ), n ∈ N. We have pose that gn − ⎧ ⎪ ⎨a(un (t)) − ϕ(un (t)) = Aλ (un (t)) + gn (t) ⎪ ⎩un (0) = un (b),

for a.a. t ∈ T,

(17)

un (0) = un (b).

As before taking inner product with un (t), using the integration by parts formula and the periodic boundary conditions and recalling that (Aλ (un (t), un (t))RN ≥ 0 for all t ∈ T,

14

we obtain



b 0

(a(un ), un )RN dt + un pp ≤ gn p un p

(recall p ≤ 2 ≤ p),

⇒ c0 un pp + un pp ≤ gn p un p , ⇒

{un }n≥1 ⊆ W 1,p ((0, b), RN )

is bounded.

By passing to a subsequence if necessary, we can have w

→ u in W 1,p ((0, b), RN ) un −

and

un → u in C(T, RN ).

(18)

Note that Aλ (un (·)) ∈ C(T, RN ) for all n ∈ N. From (17) we obtain  b  b (a(un ) , Aλ (un ))RN dt − (ϕ(un ), Aλ (un ))RN dt 0

= Aλ (un )22 +



0

b 0

(gn , Aλ (un ))RN dt

(19)

From Proposition 1 we have that Aλ : RN → RN is Lipschitz continuous and so by Rademacher’s theorem (see Gasi´ nski-Papageorgiou [8] (p. 561)), Aλ (·) is differentiable at almost all x ∈ RN . Also Aλ (·) is monotone (see Proposition 1). Hence if x ∈ RN is a point of differentiability of Aλ (·), then we have 

1 [Aλ (x + rh) − Aλ (x)], h ≥ 0 for all r ≥ 0, all h ∈ RN , r RN ⇒ (Aλ (x)h, h)RN ≥ 0 for all h ∈ RN . Also using the integration by parts formula  b (a(un ) , Aλ (un ))RN dt 0

= (a(un (b)), Aλ (un (b)))RN − (a(un (0)), Aλ (un (0)))RN  b d − (a(un ), Aλ (un ))RN dt dt 0  b d =− (a(un ), Aλ (un ))RN dt (using the periodic boundary conditions). dt 0 (20) From the chain rule of Marcus-Mizel [17] (see also Gasi´ nski-Papageorgiou [8] (p. 194)), we have d Aλ (un (t)) = Aλ (un (t))un (t) dt 15

for a.a. t ∈ T.

(21)

Using (21) in (20), we obtain  b (a(un ) , Aλ (un ))RN dt 0



=−  =− ≤0

b 0

0

b

(a(un ), Aλ (un )un )RN dt (a0 (|un |)(un , Aλ (un )un )RN dt

for all n ∈ N (see (21) and hypothesis H(a)).

(22)

Also we have 



b

0

(ϕ(un ), Aλ (un ))RN dt =

b 0

|un |p−2 (un , Aλ (un ))RN dt ≥ 0

for all n ∈ N. (23)

We return to (19) and use (20), (22), (23). Then Aλ (un )22 ≤ gn 2 Aλ (un )2 ⇒

{Aλ (un )}n≥1 ⊆ L2 (T, RN )

for all n ∈ N,

is bounded.

So, we have w

in L2 (T, RN ) (see (18)).

Aλ (un ) − → Aλ (u)

(24)

From (17) we have  b  b (a(un ) , un − u)RN dt − (ϕ(un ), un − u)RN dt 0



b

= 0

0

(Aλ (un ), un − u)RN dt +

Note that  b (ϕ(un ), un − u)RN dt → 0  

0

0

0

b

b



b 0

(gn , un − u)RN dt for all n ∈ N.

(25)

(see (18)),

(Aλ (un ), un − u)RN dt → 0 (see (18), (24)), (gn , un − u)RN dt → 0

w

(see (18) and recall that gn − → g in L2 (T, RN )).

So, if in (25) we pass to the limit as n → +∞, then  b (a(un ) , un − u)RN dt = 0. lim n→+∞

0

16

(26)



Let D = {u ∈ C 1 (T, RN ) : a(u ) ∈ W 1,p ((0, b), RN ), u(0) = u(b), u (0) = 

u (b)} and consider the operator K : D ⊆ Lp (T, RN ) → Lp (T, RN ) defined by K(u)(·) = −a(u (·))

for all u ∈ D.

From Proposition 3 of Kyritsi-Matzakos-Papageorgiou [15], we know that K(·) is maximal monotone. Then Proposition 3.2.47, p. 330, of Gasi´ nskiPapageorgiou [8] implies that K(·) is generalized pseudomonotone. From (26) it follows that u∈D

and

w



→ K(u) in Lp (T, RN ) K(un ) −

(27)

(see Definition 3.2.45, p. 330, of Gasi´ nski-Papageorgiou [8]). From (17) we have K(un ) − |un |p−2 un = Aλ (un ) + gn



in Lp (T, RN ) for all n ∈ N.

Passing to the limit as n → +∞ and using (18), (24) and (27), we obtain

K(u) − |u|p−2 u = Aλ (u) + g ⇒



in Lp (T, RN ), u ∈ D,

− a(u (t)) − ϕ(u(t)) = Aλ (u(t)) + g(t) for a.a. t ∈ T, u(0) = u(b),

u (0) = u (b),

⇒

u = ξ(g),

⇒

ξ(gn ) → ξ(g)

⇒

ξ(·) is completely continuous.

in C(T, RN ) (see (18)),

130

Now we consider the following multivalued periodic system: ⎧ ⎪ ⎪ a(u (t)) − ϕ(u(t)) ∈ Aλ (u(t)) + extF0 (t, u(t), u (t)) − ϕ(pM (u(t))) ⎪ ⎪ ⎨ for a.a. t ∈ T, ⎪ ⎪ ⎪ ⎪ ⎩u(0) = u(b), u (0) = u (b).

(28)λ

Proposition 5. If hypotheses H(a), H(A), H(F) hold and λ > 0, then problem  (28)λ has a solution uλ ∈ C 1 (T, RN ) such that |uλ (t)| ≤ M and |uλ (t)| ≤ M for all t ∈ T . 17

Proof. Let B = {h ∈ L2 (T, RN ) : |h(t)| ≤ ϑ(t) for a.a. t ∈ T } (see (15)). Proposition 4 and Theorem 5.86, p. 852, of Gasi´ nski-Papageorgiou [9], imply that E = conv ξ(B) ∈ Pkc (C(T, RN )). Invoking Theorem 8.31, p. 260 of Hu-Papageorgiou [12], we see that there exists a continuous map g : E → L1w (T, RN ) such that 2 g(u) ∈ extSF2 0 (·,u(·),u (·)) = SextF  0 (·,u(·),u (·))

for all u ∈ E

(see Hu-Papageorgiou [12], Theorem 4.6, p. 192). Then we consider the following single-valued periodic system ⎧ ⎪ ⎪ a(u (t)) − ϕ(u(t)) = Aλ (u(t)) + g(u)(t) − ϕ(pM (u(t))) ⎪ ⎪ ⎨ for a.a. t ∈ T, ⎪ ⎪ ⎪ ⎪ ⎩u(0) = u(b), u (0) = u (b).

(29)λ

Proposition 4 of Kyritsi-Matzakos-Papageorgiou [15], implies that problem (29)λ has a solution uλ ∈ C 1 (T, RN ). Recalling that F0 (t, x, y) still satisfies hypotheses H(F) (in fact H(F) (iii) takes the stronger form (15)), we can use Propositions 2 and 3 to conclude that |uλ (t)| ≤ M

and

 for all t ∈ T , all λ > 0. |uλ (t)| ≤ M

135

Proposition 5 and the definition of the multifunction F0 (t, x, y) imply that ⎧ ⎪ ⎨a(u (t)) ∈ Aλ (uλ (t)) + extF (t, uλ (t), u (t)) for a.a. t ∈ T, λ λ ⎪ ⎩uλ (0) = uλ (b),

uλ (0)

=

uλ (b).

(30)λ

 ⊆ Lp (T, RN ) → Lp (T, RN ) we denote the “lifting” In what follows by A : D 

of A on Lp (T, RN ) defined by 

A(u) = {h ∈ Lp (T, RN ) : h(t) ∈ A(u(t)) for a.a. t ∈ T } 18

for all  = {u ∈ Lp (T, RN ) : ∃ h ∈ Lp (T, RN ) such that h(t) ∈ A(u(t)) for a.a t ∈ T }. u∈D 

 From Bader-Papageorgiou [1] (p. 85) we know that Clearly 0 ∈ D. A(·) is maximal monotone. Now we are ready to produce a solution for problem (2). Theorem 1. If hypotheses H(a), H(A), H(F) hold, then problem (2) admits a solution u ∈ C 1 (T, RN ). Proof. Let λn → 0+ and let un = uλn ∈ C 1 (T, RN ), n ∈ N, be a solution of problem (28)λn . On account of Proposition 5, un ∈ C 1 (T, RN ) solves (30)λn . Moreover, {un }n≥1 ⊆ C 1 (T, RN )

is bounded.

We may assume that w

→ u in W 1,p ((0, b), RN ) un −

and

un → u in C(T, RN ).

(31)

We have a(un (t)) = Aλn (un (t)) + hn (t) for a.a. t ∈ T 140

with hn = g(un ) for all n ∈ N. From this equation, as in the proof of Proposition 4, we obtain

Aλn (un )22 ≤ hn 2 Aλn (un )2 ⇒ Aλn (un )2 ≤ hn 2

for all n ∈ N,

for all n ∈ N.

By hypothesis H(F) (iii) and Proposition 5, we have that {hn }n≥1 ⊆ L2 (T, RN ) ⇒

is bounded,

{Aλn (un )}n≥1 ⊆ L2 (T, RN )

19

is bounded.

So, we may assume that w



→ y in L2 (T, RN ) hence in Lp (T, RN ) too (recall that p ≤ 2 ≤ p). Aλn (un ) − (32) We have (un , −Aλn (un ) − hn ) ∈ GrK

for all n ∈ N (see the proof of Proposition 5). (33)

From (31) and Lemma 2.8, p. 24, of Hu-Papageorgiou [13], we have w

→ g(u) = h hn = g(un ) −



in Lp (T, RN ).

(34)

Since K is maximal monotone, GrK is sequentially closed in Lp (T, RN ) × 

Lp (T, RN )w . So, if in (33) we pass to the limit as n → +∞ and use (31), (32) and (34), then (u, −y − g(u)) ∈ GrK ⇒ a(u (t)) = y(t) + g(u)(t) for a.a. t ∈ T, u(0) = u(b),

(35)

u (0) = u (b).

From Proposition 1 we have Aλn (un (t)) ∈ A(Jλn (un (t))) for all t ∈ T , all n ∈ N, ⇒

(Jλn (un (·)), Aλn (un (·))) ∈ GrA for all n ∈ N.

(36)

Since u ∈ D (= the domain of K, see (35) and the proof of Proposition 4), from Proposition 1 we have Jλn (un (·)) → u

in Lp (T, RN ).

(37)

Therefore if in (36) we pass to the limit as n → +∞ and use (32), (37) and the fact that A(·) is maximal monotone (hence its graph is sequentially closed

20



in Lp (T, RN ) × Lp (T, RN )w ), then we obtain (u, y) ∈ GrA, ⇒

y(t) ∈ A(u(t))

for a.a. t ∈ T,

⇒ a(u (t)) ∈ A(u(t)) + extF (t, u(t), u (t)) u(0) = u(b),

u (0) = u (b)

for a.a. t ∈ T,

(see (35)).

4. Strong Relaxation 145

In this section we consider a parametric and Dirichlet version of problem (1) and we show that the trajectories of the system can be approximated in C 1 (T, RN ) by extremal ones. More precisely, let μ > 0 be a parameter and consider the following Dirichlet system

⎧ ⎪ ⎨a(u (t)) ∈ A(u(t)) + μF (t, u(t), u (t))

for a.a. t ∈ T,

⎪ ⎩u(0) = u(b) = 0.

(38)μ

By Sc (μ) ⊆ C 1 (T, RN ) denote the solution set of (38)μ . Assuming that F is Pkc (RN )-valued and that t → F (t, x, y) is graph measurable, (x, y) → 150

F (t, x, y) is h-continuous and hypotheses H(F) (iii), (iv) are satisfied, we have that Sc (μ) = ∅ (see Kyritsi-Matzakos-Papageorgiou [15]). Our aim is to approximate in the C 1 (T, RN )-norm the elements of Sc (μ) by extremal trajectories, namely solutions of the following Dirichlet system ⎧ ⎪ ⎨a(u (t)) ∈ A(u(t)) + μ extF (t, u(t), u (t)) for a.a. t ∈ T = [0, b], ⎪ ⎩u(0) = u(b) = 0.

(39)μ

We denote the solution set of (39)μ by Se (μ). Hence the goal is to provide conditions on F which guarantee that Sc (μ) = Se (μ)

21

C 1 (T,RN )

.

Such a result is known as “strong relaxation theorem” and has important implications in many applied problems. For example, in the context of control systems, the strong relaxation theorem implies that any state of the system 155

can be approximated by ones generated by “bang-bang controls”. This way we economize in the use of control functions. To have such a result we need to strengthen the conditions on the map a(·) and the multivalued perturbation F (t, x, y). The new conditions are the following:

160

H(a) : a(y) = a0 (|y|)y for all y ∈ RN , with t → a0 (t)t continuous, nondecreasing, a0 (0) = 0, a0 (t) > 0 for all t > 0, (a(y), y)RN ≥ c0 |y|p for all y ∈ RN , some c0 > 0 and 2 ≤ p < +∞, and for every r > 0, there exists cr > 0 such that (a(y) − a(u), y − u)RN ≥ cr |y − u|2 for all |y|, |u| ≤ r. Remark 4. Using the following elementary inequalities ⎧ ⎪ ⎨|y − u|p c (|y|p−2 y − |u|p−2 u, y − u)RN ≥  ⎪ |y−u|2 ⎩ 2−p (1+|y|+|u|)

if p > 2, if 1 < p ≤ 2

(see, for example, Gasi´ nski-Papageorgiou [8]), we see that the following maps satisfy the new hypotheses H(a) : a1 (y) = y

for all y ∈ RN ,

a2 (y) = |y|p−2 y + |y|q−2 y

for all y ∈ RN , with 1 < q ≤ 2 ≤ p,

a3 (y) = |y|q−2 y + (1 + |y|2 )

p−2 2

a4 (y) = |y|q−2 y +

|y|r−2 y 1 + |y|

y

for all y ∈ RN , with 1 < q ≤ 2 ≤ p,

for all y ∈ RN , with 1 < q ≤ 2 ≤ p = r − 1.

For the multivalued perturbation F (t, x, y) we introduce the following stronger 165

conditions: H(F) : F : T × RN × RN → Pkc (RN ) is a multifunction such that (i) for all x, y ∈ RN , t → F (t, x, y) is graph measurable;

22

(ii) for every r > 0, there exists kr ∈ L∞ (T )+ such that for a.a. t ∈ T , all |x|, |y|, |x |, |y  | ≤ r we have h(F (t, x, y), F (t, x , y  )) ≤ kr (t)[|x − x | + |y − y  |]. (iii) ⊕ (iv): these conditions are the same as the corresponding hypotheses H(F) (iii) and (iv). 170

We show that for all small parameter values μ > 0 strong relaxation holds. Theorem 2. If hypotheses H(a) , H(A), H(F) hold, then there exists μ∗ > 0 such that for all μ ∈ (0, μ∗ ) we have Sc (μ) = Se (μ)

C 1 (T,RN )

,

that is, for every u ∈ Sc (μ) we can find {un }n≥1 ⊆ Se (μ) such that un → u in C 1 (T, RN ). Proof. Let E = convξ(B) ∈ Pkc (C(T, RN )) be as in the proof of Proposition 5. Let u ∈ Sc (μ). We have a(u (t)) ∈ A(u(t)) + μh(t) for a.a. t ∈ T , u(0) = u(b) = 0 with h ∈ SF2 (·,u(·),u (·)) = SF2 0 (·,u(·),u (·)) (see Propositions 2 and 3 and observe that these bounds remain valid if the boundary conditions are changed from periodic to Dirichlet). We fix v ∈ E and ε > 0 and consider the multifunction N

Γv,ε : T → 2R \ {∅} defined by Γv,ε (t) = {y ∈ F0 (t, v(t), v  (t)) : |h(t) − y| < ε + d(h(t), F0 (t, v(t), v  (t)))}. Hypotheses H(F) (i), (ii) imply that GrΓv,ε ∈ LT ⊗ B(RN ). So, we can apply the Yankov-von Neumann-Aumann selection theorem (see Hu-Papageorgiou [12], Theorem 2.14, p. 158) and produce a measurable selec175

tion for the multifunction Γv,ε (·). 23

Now let Gε : E → 2L

2

(T,RN )

be the multifunction defined by

Gε (v) = {g ∈ SF2 0 (·,v(·),v (·)) : |h(t)−g(t)| < ε+d(h(t), F0 (t, v(t), v  (t))) a.e. on T } (that is, Gε (v) = SΓ2 ε,v for all v ∈ E). The previous argument guarantees that Gε (v) = ∅ for all v ∈ E. Also, using Proposition 4 of Bressan-Colombo [2] (see also Hu-Papageorgiou [12], Lemma 8.3, p. 239), we have that v → Gε (v) is lsc, ⇒

v → Gε (v) is lsc.

Also, it is clear that Gε (·) has decomposable values. So, we can apply Theorem 1 of Bressan-Colombo [2] and obtain a continuous map gε : E → L2 (T, RN ) such that gε (v) ∈ Gε (v)

for all v ∈ E.

Then using Theorem 8.31, p. 260, of Hu-Papageorgiou [12], we obtain a continuous map γε : E → L1w (T, RN ) such that

2 γε (v) ∈ SextF  0 (·,v(·),v (·))

and

γε (v) − gε (v)w ≤ ε

for all v ∈ E.

(40)

Now let εn → 0+ and set γn = γεn , gn = gεn for all n ∈ N. We consider the followig single-valued Dirichlet system ⎧ ⎪ ⎨a(v  (t)) − ϕ(v(t)) = A(v(t)) + μγn (v)(t) − ϕ(pM (v(t))) for a.a. t ∈ T, ⎪ ⎩v(0) = v(b) = 0. (41)nμ We know that for all n ∈ N, all μ > 0 problem (41)nμ admits a solution un ∈ C 1 (T, RN ) such that  for all t ∈ T , all n ∈ N and |un (t)| ≤ M, |un (t)| ≤ M {Aλ (un )}n≥1 ⊆ L2 (T, RN ) is bounded. 180

From (41)nμ and (42) it follows that 24

(42)



{a(un )}n≥1 ⊆ W 1,p ((0, b), RN ) is bounded, ⇒ {a(un )}n≥1 ⊆ C(T, RN ) is relatively compact, ⇒ {un }n≥1 ⊆ C(T, RN ) is relatively compact (since a(·) is a homeomorphism), ⇒ {un }n≥1 ⊆ C 1 (T, RN ) is relatively compact. So, we may assume that un → u  in C 1 (T, RN ).

(43)

Using the integration by parts formula, the Dirichlet boundary conditions and the monotonicity of A(·), we obtain  b (a(un ) − a(u ), u − un )pp ≤ μ(γn (un ) − h, u − un )RN dt for all n ∈ N, 0



where (·, ·)pp denote the duality brackets for the pair (Lp (T, RN ), Lp (T, RN )). So, we have (a(un ) − a(u ), u − un )pp  b    ≤ μ(γn (un ) − gn (un ), u − un )RN dt + 0

b 0

μ(gn (un ) − h, u − un )RN dt (44)

for all n ∈ N. From (40) and Lemma 2.8, p. 24 of Hu-Papageorgiou [13], we have w

in L2 (T, RN ) as n → +∞.

γn (un ) − gn (un ) − →0

From (44), hypotheses H(a) and H(F) (ii) and (42), we have  b 2  CM un − u2 ≤ εn + μh(F (t, un , un ), F (t, u, u ))|un − u|dt ≤ εn +



0

0

b

μkM0 (t)[|un − u| + |un − u |]|un − u|dt

} and εn → 0+ . Passing to the limit as n → +∞ and with M0 = max{M, M using (43), we obtain u− CM 

u22

 ≤ μkM0 ∞

b

0

[| u − u| + | u − u |]| u − u|dt.

25

(45)

Note that



| u(t) − u(t)| ≤ ⇒

 u−

u2∞

t 0

| u (s) − u (s)|ds for all t ∈ T,

2

≤ b  u − u 22

(using Jensen’s inequality).

(46)

Returning to (45) and using (46), and the Cauchy-Schwarz inequality we obtain CM  u − u22 ≤ μkM0 ∞ [b2  u − u 22 +  u − u 2  u − u2 ] ≤ μkM0 ∞ [b2  u − u 22 + b u − u 22 ]

(see (46))

u − u 22 . = μkM0 ∞ b(1 + b) If μ∗ =

CM kM0 ∞ b(1+b) ,

(47)

then for every μ ∈ (0, μ∗ ) from (47), we infer that

u  = u, ⇒

un → u

in C 1 (T, RN )

(see (43)).

Clearly un ∈ Se (μ) for all n ∈ N. Therefore we conclude that Se (μ)

C 1 (T,RN )

= Sc (μ)

for all μ ∈ (0, μ∗ )

(since Sc (μ) ⊆ C 1 (T, RN ) is closed, see [15]).

185

Remark 5. It is an open problem if strong relaxation also holds for periodic systems. Also, hypothesis H(a) excludes the p-Laplacian (p ≥ 2). So, it is another interesting open problem whether strong relaxation (for Dirichlet or periodic systems) holds for p-Laplacian equations (p ≥ 2). As an illustration of Theorem 2, we consider the following Dirichlet feedback

190

control system ⎧ ⎪ ⎪ (|u (t)|p−2 u (t)) + (|u (t)|q−2 u (t)) ∈ A(u(t)) + μ[f (t, u(t), u (t)) + v(t)] ⎪ ⎪ ⎨ for a.a. t ∈ T = [0, b], ⎪ ⎪ ⎪ ⎪ ⎩u(0) = u(b) = 0, v(t) ∈ K(t, u(t)) for a.a. t ∈ T. (48)μ 26

We assume that 1 < q ≤ 2 ≤ p and that • f (t, x, y) is a Carath´eodory function which is locally Lipschitz in (x, y) ∈ RN × RN ; • K : T × RN → Pkc (RN ) is a multifunction such that t → K(t, x) is graph measurable, x → K(t, x) is locally h-Lipschitz, |K(t, x)| ≤ γr (t) for a.a.

195

t ∈ T , all |x| ≤ ρ with γr ∈ L2 (T ); • there exists M > 0 such that for a.a. t ∈ T , all |x| = M , all y ∈ RN and all v ∈ K(t, x) we have (f (t, x, y) + v, x)RN ≥ 0. The control functions v(·) which satisfy v(t) ∈ extK(t, u(t)) for a.a. t ∈ T are 200

known as “bang-bang controls”. Let F (t, x, y) = f (t, x, y) + K(t, x) for all (t, x, y) ∈ T × RN × RN . Then problem (48)μ is equivalent to the following differential inclusion ⎧ ⎪ ⎪ (|u (t)|p−2 u (t)) + (|u (t)|q−2 u (t)) ∈ A(u(t)) + μF (t, u(t), u (t)) ⎪ ⎪ ⎨ for a.a. t ∈ T = [0, b], ⎪ ⎪ ⎪ ⎪ ⎩u(0) = u(b) = 0. (49)μ Let Sc (μ) be the solution set of (49)μ , while Se (μ) is the solution set when F (t, x, y) is replaced by extF (t, x, y) = f (t, x, y) + extK(t, x). Evidently Sc (μ) is the set of states of (48)μ , while Se (μ) are the states generated by bang-bang controls. On account of Theorem 2, for all μ > 0 small we have Sc (μ) = Se (μ)

C 1 (T,RN )

.

So, if u ∈ C 1 (T, RN ) is a state of (48)μ (μ > 0 small), then we can find states 2 {un }n≥1 ⊆ C 1 (T, RN ) generated by bang-bang controls vn ∈ SextK(·,u such n (·))

that un → u

in C 1 (T, RN ). 27

N Let RN and let + be the positive cone of R ⎧ ⎪ ⎨0 if x ∈ RN +, δR N (x) = + ⎪ ⎩+∞ otherwise.

is convex and lower semicontinuous. We set A = ∂δRN (= We know that δRN + + the convex subdifferential of δRN (·)). Evidently A(·) is maximal monotone and + 0 ∈ A(0). So, hypothesis H(A) is satisfied. In fact we have for all x = (xk )N k=1 ∈ RN A(x) = NRN (x) = +

⎧ ⎪ ⎨{0}

if xk > 0 for all k = 1, . . . , N,

⎪ ⎩−RN

if xk = 0 for some k ∈ {1, . . . , N }.

+

Then system (48)μ is the following differential variational inequality (|u (t)|p−2 u (t)) + (|u (t)|q−2 u (t)) = μ[f (t, u(t), u (t)) + v(t)] a.e. on I+ = {t ∈ T : uk (t) > 0 f or all k = 1, . . . , N } (|u (t)|p−2 u (t)) + (|u (t)|q−2 u (t)) ≤ μ[f (t, u(t), u (t)) + v(t)] a.e. on I0 = {t ∈ T : uk (t) = 0 f or some k ∈ {1, . . . , N }} u(0) = u(b) = 0, v(t) ∈ K(t, u(t)) for a.a. t ∈ T. For this system strong relaxation holds for μ > 0 small. Acknowledgement: The authors wish to thank a knowledgeable referee for his/her corrections and remarks that improved the paper.

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