Sensitivity of the Solution Set to Second Order Evolution Inclusions

In this note we study second order evolution inclusions in the framework of evolution triple of spaces. The existence of mild solutions (i.e. trajectory-selection pairs) to the inclusion, and the upper and lower semicontinuity properties of the solution set with respect to a parameter are established.

The goal is to establish the lower and upper semicontinuity properties of the solution set to Problem (P) with respect to the parameter n ∈ N. The main result concerns the Kuratowski convergence of the sequence of solution sets to Problem (P) n to that of Problem (P). Evolution inclusions of second order and their applications have been considered in several papers, see e.g. [6][7][8][9] and the references therein. We introduce below the notation and preliminary material needed in the next sections. For a Banach space X, we indicate by w-X, s-X the space X equipped with the weak and the strong (norm) topology, respectively. Let (Ω, Σ, µ) be a measure space. A multifunction F defined on Ω with values in the space is the family of all Borel subsets of X. We denote by S r F , 1 ≤ r ≤ ∞, the set of all selectors of F that belong to L r (Ω; X), i.e., S r The symbol P f (c) (X) stands for the family of all closed, (convex) subsets of 2 X . On P f (X) we define the Hausdorff metric, by setting h(A, B) = max {sup a∈A d(a, B), sup b∈B d(b, A)}. We also write |A| = sup{|a| | a ∈ A}.
Given {S n , S} n∈N ⊂ 2 Z , we define (see e.g. [3]) the sequential Kuratowski lower and upper limits respectively by τ Z -lim inf S n = {z ∈ Z | ∃ z n ∈ S n , z n → z in τ Z -Z, as n → +∞} and τ Z -lim sup S n = {z ∈ Z | ∃ {n ν }, z nν ∈ S nν , z nν → z in τ Z -Z, as ν → +∞}. We say that S n converge to S in the Kuratowski sense (denoted by S n The definition of lsc is equivalent to saying that if y n → y in τ Y -Y , then G(y) ⊂ τ Z -lim inf G(y n ). For a sequence of multifunctions G, G n : Y → 2 Z , we write if τ Z -lim sup G n (y n ) ⊂ G(y) for every y n → y in τ Y -Y . Similar notation is used for τ Z -lim inf.
Let H be a separable Hilbert space and V be a reflexive Banach space which is densely, continuously and compactly embedded in H. Identifying H with its dual H * , we have the Gelfand triple V ⊂ H ⊂ V * , where V * is the dual of V . Let ·, · be the duality of V and V * as well as the inner product on H, let · , | · | and · V * denote the norms in V , H and V * , respectively. For T > 0 and 2 ≤ p < +∞, we introduce the following spaces V = L p (0, T ; V ), The pairing of V and V * and the duality between H and H * are denoted by It is well known that the embedding W ⊂ C(0, T ; H) is continuous. Since V ⊂ H compactly we know that the embedding W ⊂ H is also compact. Finally, the class of linear bounded operators from V into V * is denoted by L(V, V * ). For additional details on the material, we refer to [3,11].

Results on evolution equations
In this section we investigate the existence, uniqueness and continuous dependence of solutions on the data for an evolution equation of second order. We consider the following problem A function x ∈ C(0, T ; V ) is called a solution to the problem (E) if and only iḟ x ∈ W and (E) is satisfied.
We will need the following hypotheses.
The proof of the following result follows from the standard application of the Galerkin method and can be found in [1,5,6].

Proposition 1. Under hypotheses H(A), H(B)
, (H 0 ) and f ∈ H * , the problem (E) admits a unique solution which satisfies x ∈ C(0, T ; V ),ẋ ∈ W, and the following estimate We present now a result on the continuous dependence of solutions to the problem on the data. We will need the following assumptions.
uniformly with respect to n ∈ N and the condition For every n ∈ N, let x n be a solution of the problem (E) n and let x be a solution of the problem (E). We have Proof. By Proposition 1 we know that, for every n ∈ N, the problem (E) n has the unique solution x n ∈ C(0, T ; V ) such thatẋ n ∈ W. From (E) n and (E), we have for every n ∈ N. Integrating this equality and using the monotonicity of A n (s, ·), we get for all t ∈ [0, T ]. Hence using H(B) 1 and applying the Hölder inequality, we obtain where A n and A are the Nemitsky operators corresponding to A n and A, respectively, and C is a positive constant independent of n. On the other hand, due to H(A) 1 , H(B) 1 and (H 0 ) 1 , from Proposition 1, we have Hence, it follows that {ẋ n } lies in a bounded subset of W. Thus, up to a subsequence,ẋ n converges weakly in W and (since W ⊂ H compactly) strongly in H. So we have lim Using the assumptions, (2) and (3), from (1), we get (x n (t),ẋ n (t)) → (x(t),ẋ(t)) in s-(V ×H) for all t ∈ [0, T ], as n → +∞. Since the solution to (E) is unique, we deduce that the whole sequence {(x n ,ẋ n )} converges to (x,ẋ) in C(0, T ; V × H).
The proof is completed.
In the sequel, we make use of the solution map r : H * → C(0, T ; V ) × W for (E) defined by r(f ) = (x,ẋ), where x (respectivelyẋ) is the solution (and its derivative, respectively) to (E). By Proposition 1 this map is well defined and Proposition 2 implies the following result.

Existence result for inclusions
In this section we study the existence of solutions to Problem (P). We start with the following Prior to the existence theorem, we state the a priori bound on the solution to the evolution inclusion. We need the following hypotheses.   In the proof of the next result, we follow methods used in [4,10]. Proof. From Lemma 1, it is clear that every solution to Problem (P) satisfies Since the map p is Lipschitz continuous, from the properties of F , we deduce that F satisfies H(F )(1)(2). Furthermore, we note that | F (t, x, y)| ≤ a 1 (t) a.e. t ∈ (0, T ), where a 1 ∈ L q + (0, T ) is given by a 1 (t) = a 1 (t) is the solution map for the equation (E)). Since F is graph measurable and L 1 integrably bounded, using the Aumann selection theorem (see Theorem 4.3.7 of [3]), we have R(f ) = ∅ for f ∈ Z. Moreover, because F is P f c (H)-valued and | F (t, r(f )(t))| ≤ a 1 (t) a.e. t ∈ (0, T ), we obtain that R : Z → P f c (Z). We will show that R is (w-H * )×(w-H * ) usc on Z. Since Z is compact in w-H * , it suffices to prove (see Chapter I of [2], Section 4.1 of [3]) that GrR is weakly-weakly closed in Z × Z. Let (f n , z n ) ∈ GrR, f n → f and z n → z both in w-H * . By Corollary 1, we know that r(f n )(t) → r(f )(t) in (s-H) × (s-H) for all t ∈ [0, T ]. Since F satisfies H(F )(1)(2), we deduce that w-lim sup F (t, r(f n )(t)) ⊂ F (t, r(f )(t)) a.e. t ∈ (0, T ). Using Theorem 4.7.51 of [3], we obtain w-lim sup R(f n ) = w-lim sup S 1 F (·,r(fn)(·)) ⊂ ⊂ S 1 w−lim sup F (·,r(fn)(·)) ⊂ S 1 F (·,r(f )(·)) = R(f ). From these inclusions we have (f, z) ∈ GrR. This means that GrR is closed in (w-Z) × (w-Z) and proves that R is weakly-weakly usc on Z.
for all u, v ∈ H, then Theorem 1 ensures that the Cauchy problem for the nonlinear equationẍ(t) + A(t,ẋ(t)) + Bx(t) = f (t, x(t),ẋ(t)) has at least one solution.
Let S be the set of mild solutions to Problem (P) and let Proof. The nonemptiness of M follows from Theorem 1. Let {(x k ,ẋ k , f k )} k∈N ⊂ M. We will show that this sequence has a subsequence which converges in an appropriate topology to an element of M. By the definition, x k satisfies the evolution equation (E) with the right-hand side f k and f k (·) ∈ S q F (·,x k (·),ẋ k (·)) . From Lemma 1, we obtain in particular that f k remains in a bounded subset of H * . So after a possible passing to subsequence, we have f k → f weakly in H * , as k → +∞, with f ∈ H * . Corollary 1 says that where r(f ) = (x,ẋ) is a solution to (E). In order to conclude the proof, it suffices to show that f is a selection for F (·, x(·),ẋ(·)). From Theorem 4.7.44 of [3], we have a.e. t ∈ (0, T ). Since (x k (t),ẋ k (t)) → (x(t),ẋ(t)) in s-(H × H) for all t ∈ [0, T ], from H(F )(1) (2), we easily deduce that w-lim sup F (t, x k (t),ẋ k (t)) ⊂ F (t, x(t),ẋ(t)) a.e. t ∈ (0, T ). Hence, we get f (t) ∈ F (t, x(t),ẋ(t)) a.e. t ∈ (0, T ). So we have obtained (x,ẋ, f ) ∈ M which completes the proof.

Upper semicontinuity property of the solution set
Consider now a sequence of evolution inclusions Problem (P) n . Let us denote by S n the set of mild solutions to Problem (P) n , i.e., S n = {(x, f ) ∈ C(0, T ; V )×H * | (x, f ) is a mild solution to Problem (P) n }.
If (x n , f n ) ∈ S n , n ∈ N and f n → f in w-H * , then (x, f ) ∈ S.
Proof. From Theorem 1 we know that S n , S = ∅. Let (x n , f n ) ∈ S n for n ∈ N and f n → f weakly in H * . By Proposition 2, we infer that (x n ,ẋ n ) converges in C(0, T ; V × H) to (x,ẋ), as n → +∞, where x is a solution to the equation (E) (corresponding to the right hand side f ). It remains to prove that f (·) ∈ S q F (·,x(·),ẋ(·)) . From Theorem 4.7.44 of [3], we have a.e. t ∈ (0, T ) and by (6) we obtain This facts imply f (t) ∈ F (t, x(t),ẋ(t)) a.e. t ∈ (0, T ). Hence (x, f ) is a mild solution to Problem (P) which concludes the proof.
We introduce the sets M n = {(x,ẋ, f ) ∈ C(0, T ; V × H) × H * | (x, f ) ∈ S n } for every n ∈ N. We have the following upper semicontinuity property.

Lower semicontinuity property of the solution set
In order to state a result on lower semicontinuity of the set of mild solutions, we admit the following stronger assumption on the multivalued term.
In order to prove (9), by using (10), we write In view of (8) and the convergence β n → 0 in L 2 (0, T ), we easily get (9). This completes the proof.