Anti-periodic solutions for nonlinear evolution inclusions

We consider an anti-periodic evolution inclusion defined on an evolution triple of spaces, driven by an operator of monotone-type and with a multivalued reaction term F(t, x). We prove existence theorem for the “convex” problem (that is, F is convex-valued) and for the “nonconvex” problem (that is, F is nonconvex-valued) and we also show the existence of extremal trajectories (that is, when F is replaced by $$\mathrm {ext}\,F$$extF). Finally, we prove a “strong relaxation” theorem, showing that the extremal trajectories are dense in the set of solutions of the convex problems.

(1.1) (a) X is a separable reflexive Banach space and X * is its topological dual; (b) H is a separable Hilbert space which is identified with its dual (that is, H = H * ) by the Riesz-Fréchet representation theorem; (c) X → H .
As a consequence of (b) and (c) we also have that H → X * . In what follows by · (respectively |·|, · * ) we denote the norm of X (respectively of H , X * ). Also, by ·, · we denote the duality brackets for the pair (X * , X ) and by (·, ·) the inner product of H . We will also assume that the embedding X → H is compact.
Let 1 < p < +∞. The following space is important in the analysis of problem (1.1): Here by u we understand the distributional (weak) derivative of u. From the theory of Lebesgue-Bochner spaces (see p. 129]), we have L p (T ; X ) * = L p (T ; X * ) and the duality brackets for this pair of spaces are defined by If u ∈ W p (0, b), then u viewed as an X * -valued function, is absolutely continuous. So, since X * is reflexive, u : T −→ X * is almost everywhere differentiable in the classical sense (see p. 133]). This derivative coincides with the distributional one. Then we have W p (0, b) ⊆ AC 1, p (T ; X * ) = W 1, p (T ; X * ).
The space W p (0, b) becomes a separable reflexive Banach space, when given the norm An equivalent norm is given by |u| W p = u L p (T ;X ) + u L p (T ;X * ) .
The following properties of W p (0, b) will be important in our study of problem (1.1): • W p (0, b) → C(T ; H ); • W p (0, b) → L p (T ; H ) and the embedding is compact; ) for all t ∈ T , then η is absolutely continuous and ("integration by parts formula").
Suppose that V is a reflexive Banach space, V * its topological dual and by ·, · V we denote the duality brackets for the pair (V * , V ). Let L : V ⊇ D(L) −→ V * be a linear maximal monotone operator and A : V −→ 2 V * . We say that A is "Lpseudomonotone" if the following conditions hold: (a) for every v ∈ V , A(v) ⊆ V * is nonempty, convex and w-compact; (b) A is bounded (that is, maps bounded sets to bounded sets); The following surjectivity result is due to Papageorgiou et al. [20] and it extends an earlier single-valued result of Lions [15, p. 319].
Let ( , ) be a measurable space and Y a separable Banach space. We will use the following notation: A multifunction (set-valued function) G : −→ P f (Y ) is said to be "measurable" if for every y ∈ Y , the R + -valued function For P f (Y )-valued multifunctions measurability implies graph measurability. For the converse to be true, we need to have = (with being the universal σ -field). Recall that = , if there is a σ -finite measure μ on ( , ) with respect to which is complete. Now let μ be a σ -finite measure on . For 1 p +∞ we define For a graph measurable multifunction G : −→ 2 Y \ {∅}, the set S p G is nonempty if and only if the R + -valued function ω −→ inf{ u Y : u ∈ G(ω)} belongs in L p ( ) + . Note that by a corollary to the Yankov-von Neumann-Aumann selection theorem, we can find a sequence {g n } n 1 of -measurable selectors of G such that G(ω) ⊆ {g n (ω)} n 1 μ-a.e. Vol. 18 (2018) Anti-periodic solutions for nonlinear evolution 1029 The set S p G is "decomposable" in the sense that if (C, g 1 , g 2 ) ∈ × S p G × S p G , then with χ D being the characteristic function for the set D ∈ .
On P f (Y ) we can define a generalized metric h(·, ·) known as the "Hausdorff metric", by Then Suppose that Z , W are two Hausdorff topological spaces. A multifunction G : Z −→ 2 W \ {∅} is said to be • "upper semicontinuous", if for all closed sets C ⊆ W , the set • "lower semicontinuous", if for all closed sets C ⊆ W , the set Finally, if E is a Banach space and {C n } n 1 is a sequence of nonempty subsets of E, then we define wlim sup n→+∞ C n = u ∈ E : u = wlim k→+∞ u n k , u n k ∈ C n k , n 1 < n 2 < . . . .
For more details on the measurability and continuity properties of multifunctions, we refer to Hu-Papageorgiou [12].
Next we introduce the hypotheses on the map A: with 2 p < +∞; (iv) there exist a 2 ∈ L 1 (T ) + and c 2 > 0 such that Finally we mention that using the Troyanski renorming theorem (see, for example, Gasiński-Papageorgiou [7, p. 911]), without any loss of generality we may assume that both X and X * are locally uniformly convex and so L p (T ; X ) and L p (T ; X * ) are strictly convex. Moreover, by L 1 w (T ; H ) we will denote the Lebesgue-Bochner space L 1 (T ; H ) furnished with the "weak norm" defined by An equivalent definition of the weak norm is the following

Convex problem
In this section, we prove the existence of anti-periodic solutions for the case when the multivalued reaction term F is convex-valued. The hypotheses on F are the following: H w , we denote the spaces X and H endowed with the weak topologies); (iii) there exist a 3 ∈ L p (T ) + and c 3 > 0 such that This condition is known in the literature as "Hartman's condition" and it was first introduced by Hartman [11] in the context of second-order Dirichlet systems in R N .
Vol. 18 (2018) Anti-periodic solutions for nonlinear evolution 1031 From the Yankov-von Neumann-Aumann selection theorem (see p. 158]) and hypothesis H (A)(iii), we see that Based on the Hartman condition (see hypothesis H (F) 1 (v)), we introduce the following modification of the multivalued forcing term for all (t, x) ∈ T × X . On account of hypothesis H (F) 1 (iii) and of (3.1), we have with a ∈ L p (T ), c > 0. This modification of F(t, x) has the following properties.

Proof. (a) This follows from hypothesis
(d) This is immediate from (3.1) and hypothesis H (F) 1 (v).
Also we consider the linear operator L : From Proposition 1 of Liu [17], we have that L is maximal monotone (hence densely defined). Proof. We consider a sequence {u n } n 1 ⊆ D such that and u * n ∈ A(u n ) + N (u n ) for all n 1 such that We have Hypothesis H (A)(iii) and (3.4) imply that by passing to a subsequence if necessary, we may assume that We set ξ n (t) = h * n , u n (t) − u(t) , for all n 1. Then ξ n ∈ L 1 (T ) for all n 1 and from Hu-Papageorgiou [13] (proof of Theorem 2.35, p. 41), we have ξ n (t) −→ 0 for a.a. t ∈ T, as n → +∞.

PROPOSITION 3.5. If hypotheses H (A) and H (F) 1 hold, then A + N is strongly coercive.
Proof. Let h * ∈ A(u) and f ∈ N (u). We have for some c 3 > 0, so A + N is strongly coercive (recall that p > 1).
Let S c ⊆ W p (0, b) denote the solution set of problem (1.1) when the multivalued reaction term F is convex-valued. Then Proposition 3.7 and (3.1) lead to the following existence theorem.

Nonconvex problem
In this section, we look for solutions of problem (1.1) when F has nonconvex values. Now the hypotheses on F are the following. (iv) there exists ϑ ∈ L p (T ) + such that (v) there exists M > 0 such that By S ⊆ W p (0, b) we denote the solution set of problem (1.1) when the multivalued reaction term F is nonconvex-valued.
As in the proof of Proposition 3.3, we show that x) is lower semicontinuous; • there exist a 5 ∈ L p (T ) + and c 5 > 0 such that | v| a 5 (t) + c 5 |x| for a.a. t ∈ T, all x ∈ H.
These properties and Theorem 7.28 of Hu-Papageorgiou [12, p. 238] imply that N has nonempty, closed and decomposable values and it is lower semicontinuous. So, we can use the Bressan-Colombo [5] selection theorem and produce a continuous function e : We consider the following anti-periodic evolution inclusion: Reasoning as in the "convex" case and using (4.1), we show that (4.2) has a solution u ∈ W p (0, b) such that | u(t)| M for all t ∈ T . From (3.1) we conclude that u ∈ S.

Extremal solutions: strong relaxation
In this section we look for extremal solutions (that is, solutions of the inclusion in which the multivalued forcing term is ext F(t, x), the set of extremal points of  F(t, x)). Such solutions are important in control theory in connection with the bangbang principle.
The anti-periodic evolution inclusion under consideration is the following: To solve (5.1) we need to strengthen the hypotheses on A and on F. The new conditions on these items are the following.
x) is strictly monotone and maximal monotone; (iii) there exist a 6 ∈ L p (T ) and c 6 > 0 such that (iii) there exist a 8 ∈ L p (T ) + and c 8 > 0 such that (iv) there exists ϑ ∈ L p (T ) + such that with a 7 ∈ L 1 (T ) + as in hypothesis H (A) (iv).
Proof. Existence of a solution follows from Theorem 3.8. For the uniqueness, suppose that u, v ∈ W p (0, b) ⊆ C(T ; H ) are two solutions of (5.2). Then we have Suppose that u = v. Taking duality brackets with u(t) − v(t), using the integration by parts formula (see (2.1)), integrating first on [0, t] and then on [t, b], via the strict monotonicity of A(t, ·) and the anti-periodic boundary condition, we have a contradiction. So u ≡ v. Therefore, the solution of (5.2) is unique. Next we show that the solution map ξ : L p (T ; H ) −→ C(T ; H ) is completely continuous. So, suppose that g n w −→ g in L p (T ; H ) and let u n = ξ(g n ) for all n 1. We have − u n (t) = h * n (t) + g n (t) for a.a. t ∈ T, u n (0) = −u n (b), n 1.
We take duality brackets with u n (t) and obtain So, by passing to a suitable subsequence if necessary, we may assume that We set u * n = h * n + g n for n 1 and have Also, using the integration by parts formula and the anti-periodic boundary condition, we have (see (5.9)-(5.10)). Returning to (5.11) and using (5.12), (5.13), (5.14), we infer that so ξ is completely continuous.
As before we consider the modification F of F defined by (3.1) and consider the corresponding anti-periodic problem (3.18). From Proposition 3.7 we know that every solution u ∈ W p (0, b) of problem (3. Then we introduce the set V = g ∈ L p (T ; H ) : |g(t)| γ (t) for a.a. t ∈ T .