PERIODIC SOLUTIONS FOR NONLINEAR NONMONOTONE EVOLUTION INCLUSIONS

. We study periodic problems for nonlinear evolution inclusions deﬁned in the framework of an evolution triple ( X,H,X ∗ ) of spaces. The operator A ( t,x ) representing the spatial diﬀerential operator is not in general monotone. The reaction (source) term F ( t,x ) is deﬁned on [0 ,b ] × X with values in 2 X ∗ \{∅} . Using elliptic regularization, we approximate the problem, solve the approximation problem and pass to the limit. We also present some applications to periodic parabolic inclusions.

1. Introduction. In this paper we study periodic problems for a class of evolution inclusions defined in the framework of an evolution triple of spaces.
Let T = [0, b] and let (X, H, X * ) be an evolution triple (see Section 2). We assume that the embedding of X into H is compact (hence so is the embedding of H * = H into X * ). The problem under consideration is the following: u (t) + A(t, u(t)) ∈ F (t, u(t)) for a.a. t ∈ T, u(0) = u(b).
(1.1) and bounded. Browder [4] has a family {A(t)} t∈T of closed monotone operators defined on a Hilbert space and a single valued continuous reaction which is monotone too. Prüss [28] has a time-independent, linear, densely defined operator A(x) and a reaction term F (t, x) which is single-valued, continuous and satisfies a Nagumo-type tangential condition with respect to the tangent cone of a closed, convex, bounded set with nonempty interior. The first nonlinear result was proved by Vrabie [31], who assumed that A : X ⊇ D(A) −→ 2 X is a time-independent m-accretive operator defined on a Banach space X, which generates a compact nonlinear semigroup of contractions. Similar to Becker [3], Vrabie [31] assumes that for some a > 0, A − aI is m-accretive too and the reaction term F (t, x) is single-valued, continuous and satisfies the following asymptotic condition The approach of Vrabie [31] is based on the existence of a fixed point for an appropriately defined nonlinear map. The result of Vrabie [31] can be viewed as the nonlinear extension of that of Becker [3]. Soon thereafter, Hirano [17] partially improved the work of Vrabie [31]. Assuming that the underlying space is a Hilbert space and that A is of the subdifferential type, he was able to remove the requirements that A generates a compact semigroup and instead requires only that the resolvents of A are compact maps. The reaction F (t, x) is a single-valued Carathéodory map exhibiting a sublinear growth in x and A − F satisfies a coercivity condition. The approach of Hirano [17] uses a kind of elliptic regularization of the equation. Another extension of the nonlinear work of Vrabie [31] was produced by Caşcaval-Vrabie [5]. Additional nonlinear results can be found in the works of Shioji [30] and Sattayatham-Tangmanee-Wei [29]. Extensions to evolution inclusions were obtained by Bader-Papageorgiou [2], Hu-Papageorgiou [18], Kasyanov-Mel nik-Toscano [21], Lakshmikantham-Papageorgiou [22], Paicu [25], Xue-Cheng [32]. All imposed a monotonicity condition on the map A.
Here we go beyond all the aforementioned works. We consider evolution inclusions without any monotonicity condition on A(t, ·) and with a multivalued reaction term defined on T × X with values in 2 X * \ {∅}. It is an interesting open problem, if we can have A(t, x) to be set-valued too (as, for example, in Defranceschi [8]). Our approach is inspired by the elliptic regularization method developed by Lions [24] (see Section 3.1). Finally for other kinds of generalizations of the evolution inclusions we refer to Gasiński [9,10,11,12] and Gasiński-Smo lka [15,16].
In the next section for the convenience of the reader, we recall the main mathematical tools which we will use in this work.
2. Mathematical background. Let (Ω, Σ) be a measurable space and let X be a separable Banach space. We will use the following notation: For a multifunction F : Ω −→ 2 X \ {∅} the graph of F is the set We say that F is graph measurable, if Gr F ∈ Σ×B(X), with B(X) being the Borel σfield of X. According to the Yankov-von Neumann-Aumann selection theorem (see ), if µ is a σ-finite measure on Σ and F : Ω −→ 2 X \ {∅} is graph measurable, then F admits a measurable selection, that is there exists a Σ-measurable function f : Ω −→ X such that In fact the result is true if X is only Souslin and we can have a whole sequence {f n : Ω −→ X} n 1 of Σ-measurable functions such that in Ω.
We say that F : The converse is not in general true (see Hu-Papageorgiou [19]). Now assume that (Ω, Σ µ) is a σ-finite measure space and for 1 p +∞ consider the Lebesgue-Bochner space L p (Ω; X). Given a multifunction F : Let H be a separable Hilbert space with norm | · |. Also, let X be a separable reflexive Banach space with norm · which is embedded continuously and densely into H. Identifying H with its dual (pivot space), we have X → H → X * with all embeddings being continuous and dense. Such a triple of spaces is usually known in the literature as evolution triple (or Gelfand triple or spaces in normal position). Here we will also assume that the embedding X → H is compact, hence so is H → X * (Schauder theorem; see p. 275]). By · * we denote the norm of X * . Also by ·, · we denote the duality brackets for the pair (X * , X) and by (·, ·) the inner product of H. We know that ·, · H×X = (·, ·).
In this definition, the derivative of u is understood in the sense of vector valued distributions. Furnished with the norm the space W p (0, b) becomes a Banach space which is separable and reflexive. We know that 2) Moreover, we have the following integration by parts formula. If u, v ∈ W p (0, b), then t −→ (u(t), v(t)) is absolutely continuous and d dt For more details on evolution triples and related topics, we refer to the books of Gasiński-Papageorgiou [13] and Hu-Papageorgiou [20]. Let Y be a reflexive Banach space, Y * its topological dual and by ·, · Y the duality brackets for the pair (Y * , Y ). Let A : Y −→ 2 Y * be a multivalued map.
A is bounded (that is, maps bounded sets to bounded sets); We say that A is generalized pseudomonotone, if the following holds: In the next two propositions we state some useful results about these operators which we will need later. Details can be found in Gasiński-Papageorgiou [13, pp. 330-337].
is bounded (maps bounded sets to bounded sets) and generalized pseudomonotone, then A is pseudomonotone.
Pseudomonotone maps exhibit remarkable surjectivity properties and for this reason play an important role in the study of evolution equations.
then A is surjective.
u n k ∈ C n k , n k < n k+1 for all k 1 .
Here w denotes the weak topology in Y .
Returning to the setting of an evolution triple (X, H, X * ) described earlier by X w (respectively X * w ) we denote the space X (respectively X * ) equipped with the weak topology. In what follows using the Troyanski renorming theorem, we assume without any loss of generality that both X and X * are locally uniformly convex. Then the duality map F : X −→ X * is a homeomorphism and maximal monotone (see p. 316] and Zeidler [33, p. 860]). Recall that p. 129]) and both spaces are locally uniformly convex. Hence the duality map F : L p (T ; X) −→ L p (T ; X * ) defined by is a homeomorphism and maximal monotone. Let . Because of (2.1) the evaluations at t = 0 and t = b make sense. Of course W per Consider the operator L : [33, p. 855]) This is a linear densely defined operator which is maximal monotone (see Hu-Papageorgiou [19, Proposition 3.9, p. 419] and Zeidler [33, Proposition 32.10, p.855]). Note that D(L) furnished with the graph norm becomes the separable reflexive Banach space W per p (0, b). Since L is linear and densely defined, we can define its adjoint L * which is also linear, densely defined, maximal monotone and D(L * ) = D(L) = W per p (0, b). By ((·, ·)) we denote the duality brackets for the pair (L p (T ; X * ); L p (T ; X)). So, we have

Using integration by parts formula, for
So, we see that L * = −L and D(L) = D(L * ). Now we are ready to introduce the hypotheses on the data of (1.1).

LESZEK GASIŃSKI AND NIKOLAOS S. PAPAGEORGIOU
with β ∈ L 1 (T ), c > 0 and 1 < τ 2; if τ = 2 we assume additionally that cb Let a : L p (T ; X) −→ L p (T ; X * ) be the nonlinear map defined by 3. Existence of periodic solutions. First we consider the multivalued map Proof. Evidently the values of N are closed, convex and bounded subsets of L p (T ; X * ), hence weakly compact and convex. So, the only thing that we need to show, is that N has nonempty values. To this end, let {s n } n 1 ⊆ L p (T ; X) be simple functions such that and s n (t) u(t) for a.a. t ∈ T, all n 1.
(3.2) Hypothesis H(f )(i) implies that for every n 1, the multifunction t −→ F (t, s n (t)) is graph measurable. So, we can use the Yankov-von Neumann-Aumann selection theorem (see ) and find a measurable map f n : T −→ X * such that f n (t) ∈ F (t, s n (t)) for a.a. t ∈ T, all n 1, so the sequence {f n } n 1 ⊆ L p (T ; X * ) is bounded (see hypothesis H(F )(iii) and (3.1)-(3.2)). By passing to a suitable subsequence if necessary, we may assume that (3.2) and hypothesis H(F )(ii)), so f ∈ N (u) and thus This proves that N (u) ∈ P wkc (L p (T ; X * )) ∀u ∈ L p (T ; X).
The next proposition establishes the main property of this map.
Proposition 3.2. If hypotheses H(A) hold, then for every ε > 0, the operator Evidently K ε is bounded. So, according to Proposition 2.1, in order to prove the pseudomonotonicity of K ε , it suffices to show that it is generalized pseudomonotone. To this end, let {u n } n 1 ⊆ W per The monotonicity of F −1 implies that Then from (3.3) and (3.5), for all n ≥ 1 we have But recall that F −1 is maximal monotone, hence generalized pseudomonotone (see Gasiński (3.12) From (3.3), (3.4), (3.9) and (3.12) we conclude that so K ε is generalized pseudomonotone, hence pseudomonotone too. Proof. First we show the pseudomonotonicity of Q ε .
Since Q ε is bounded, it suffices to show that Q ε is generalized pseudomonotone (see Proposition 2.1). To this end, let {u n } n 1 ⊆ W per (3.14) We need to show that From (3.13) and (3.14) we have (see (3.14) and (3.16)). Let η n (t) = A(t, u n (t)), u n (t) − u(t) for n 1 and let E ⊆ T be the Lebesguenull set outside of which hypotheses H(A)(ii), (iii) and (iv) hold. Then for t ∈ T \E, we have Evidently D ⊆ T is Lebesgue measurable. Denoting by |·| 1 the Lebesgue measure on R, suppose that |D| 1 > 0. From (3.18) it is clear that the sequence {u n (t)} n 1 ⊆ X is bounded for all t ∈ D ∩ (T \ E) = ∅. Then from (3.14) and the Urysohn criterion for the convergence of sequences (see Gasiński-Papageorgiou [14, p. 331]), we have We have |η n (t)| = η + n (t) + η − n (t) = η n (t) + 2η − n (t) ∀n 1. Hence by passing to a suitable subsequence if necessary, we may assume that η n (t) −→ 0 for a.a. t ∈ T. Arguing by contradiction, suppose that the Claim is not true. Then we can find D * ⊆ T measurable with |D * | 1 > 0 such that sup n 1 u n (t) = +∞ ∀t ∈ D * .
So, we have proved that (3.15) holds and this implies the pseudomonotonicity of Q ε .
So, problem (4.1) incorporates equations with discontinuous reaction term. To deal with such problems, we pass to an inclusion by filling in the gaps at the discontinuity points (see Chang [6] and Hu-Papageorgiou [20]).
For problem (4.1) the evolution triple is: . From the Sobolev embedding theorem (recall that p 2), we know that X → H compactly. Hence, so does H * = H into X * .
Let A : T × X −→ X * be defined by From the Pettis measurability theorem (see Gasiński-Papageorgiou [13, Theorem 2.1.3, p. 109]), we have that for all x ∈ X, the map t −→ A(t, x) is measurable. Moreover, from Leray-Lions [23], we know that for almost all t ∈ T , the map x −→ A(t, x) is pseudomonotone. Finally hypotheses H(a)(iii) and (iv) imply that hypotheses H(A)(iii) and (iv) hold. Next consider the multifunction G : T × X −→ 2 X * defined by Evidently G has nonempty, weakly compact, convex values in L p (Ω) → X * . Moreover, hypotheses H(G)(i), (ii) and (iii) imply that for all x ∈ X, the map t −→ G(t, x) is graph measurable; (4.2) for a.a. t ∈ T , the map x −→ G(t, x) has a closed graph in X × L p (Ω) w . (4.3) Let V : X −→ X * be the nonlinear map defined by (4.5) From Zeidler [33, p. 593], we know that V is completely continuous, so (4.6) We have f n = g n + V (u n ) ∀n 1, with g n ∈ G(t, u n ). It is clear from hypothesis H(G)(iii) that the sequence { g n } n 1 ⊆ L p (Ω) is bounded. So, passing to a subsequence if necessary, we may assume that g n w −→ g in L p (Ω), thus g ∈ G(t, u) (4.7) (see (4.3)). From (4.4), (4.5), (4.6) and (4.7) it follows that Thus hypothesis H(F )(ii) is satisfied. Also from (4.2) we see that the map t −→ F (t, u) is graph measurable (see hypothesis H(F )(i)).
Hypothesis H(F )(iii) is a consequence of hypotheses H(G)(iii) and the definition of V .
From (4.8)-(4.9) we have u n −→ u in L p (T ; L 2 (Ω)) ⊆ L 2 (T × Ω) (4.11) (recall that 2 p). Note that from hypothesis H(G)(iii) we have that the sequence { g n } n 1 ⊆ L p (T × Ω) is bounded. So, passing to a subsequence if necessary, we may assume that Then we have (see (4.11), (4.12)). Also, from Zeidler [33, p. 593], we know that V is completely continuous, thus V (u n ) −→ V (u) in W −1,p (Ω) and so (4.14) From (4.13) and (4.14), we see that hypothesis H(F )(v) is satisfied. We rewrite problem (4.1) as the following equivalent abstract evolution inclusion: For this problem we can use Theorem 3.4 and state the following existence result for problem (4.1).
We can also consider the following parabolic inclusion: We assume the following on the coefficient ϑ(z): In this case V : X −→ X * is defined by We know that div (ϑ(z)Du) = ϑ(z)∆u + (Dϑ, Du) R N ∀u ∈ X, so for all u, h ∈ X = W 1,p 0 (Ω), we have Clearly Gr V is sequentially closed in X × X * w . Also, if u n w −→ u in X, then ϑ(z)∆u n , u n − u  ) admits a solution u ∈ L p (T ; W 1,p 0 (Ω)) ∩ C(T ; L 2 (Ω)) such that ∂u ∂t ∈ L p (T ; W −1,p (Ω)).
Also from the growth condition on G(t, z, ·, ·) and the condition on c (see hypothesis H(G)(iii)), we see that hypothesis H(F )(iv) holds.
Finally , assuming that u n w −→ u in X, we have that u n −→ u in L p (Ω), so f n , u n − u = Ω f n (u n − u) dz −→ 0, hence hypothesis H(F )(v) holds.