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Abstract and Applied Analysis Attractors of multivalued semiflows generated by differential inclusions and their approximations
Attractors of multivalued semiflows generated by differential inclusions and their approximations
Kapustian, Alexei V., Valero, José이 책이 얼마나 마음에 드셨습니까?
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5
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2000
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english
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Abstract and Applied Analysis
DOI:
10.1155/s1085337500000191
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ATTRACTORS OF MULTIVALUED SEMIFLOWS GENERATED BY DIFFERENTIAL INCLUSIONS AND THEIR APPROXIMATIONS ALEXEI V. KAPUSTIAN AND JOSÉ VALERO Received 20 October 1999 We prove the existence of global compact attractors for differential inclusions and obtain some results concerning the continuity and upper semicontinuity of the attractors for approximating and perturbed inclusions. Applications are given to a model of regional economic growth. 1. Introduction The theory of multivalued dynamical systems is motivated by differential equations for which it is not known whether the solution corresponding to each initial data is unique or not. In such a case it is not possible to define a semigroup of operators. However, by taking the union of all solutions belonging to a certain class we can define a multivalued semiflow and study in this way the asymptotic behavior of the trajectories. We will recall some results of the abstract theory of attractors for multivalued semiflows developed in [11, 13, 14] (see also [3, 5]). Denote by X a complete metric space with the metric ρ and by 2X (β(X); Cv (X); comp(X)) the family of all (nonempty bounded; nonempty, bounded, closed, convex; nonempty compact) subsets of X. As usual, dist(A, B) = supy∈A inf x∈B ρ(y, x) and dist H (A, B) = max{dist(A, B), dist(B, A)}, A, B ∈ β(X), is the Hausdorff metric. Let B (A) = {y ∈ X  dist(y, A) ≤ } be an neighborhood of the set A ⊂ X. A multivalued map F : X → 2X is said to be wupper semicontinuous if ∀x0 ∈ D(F ), ∀ > 0, ∃δ > 0 such that F (x) ⊂ B (F (x0 )), ∀x ∈ Bδ (x0 ), where D(F ) = {x  F (x) ∈ P (X)}. It is said to be upper semicontinuous if ∀x0 ∈ D(F ) and any neighborhood O(F (x0 )) there exists δ > 0 such that F (x) ⊂ O(F (x0 )), ∀x ∈ Bδ (x0 ). Obviously, any upper semicontinuous map is wupper semicontinuous, the converse being valid if F has compact values [1, page 45]. A multivalued map G : R+ × X → P (X) is said to be a multivalued semiflow (msemiflow for short) if G(0, ·) = I d and G(t1 + t2 , x) ⊂ G(t1 , G(t2 , ; x)), ∀t1 , t2 ∈ R+ , ∀x ∈ X. The set is called a global attractor of G if ⊂ G(t, ), ∀t ∈ R+ , and Copyright © 2000 Hindawi Publishing Corporation Abstract and Applied Analysis 5:1 (2000) 33–46 2000 Mathematics Subject Classification: 35B40, 35B41, 35K55, 35K57, 35K90 URL: http://aaa.hindawi.com/volume5/S1085337500000191.html 34 Attractors of multivalued semiflows generated by differential inclusions . . . t→∞ dist(G(t, B), ) −−−→, ∀B ∈ β(X). It is said to be invariant if = G(t, ), ∀t ∈ R+ . If is compact then it is the minimal closed set attracting all bounded sets. The msemiflow G is called point dissipative if there exists B0 ∈ β(X) such that t→∞ dist(G(t, x), B0 ) −−−→ 0, ∀x ∈ X. Theorem 1.1 (see [14, Theorem 3 and Proposition 1]). Let for any t ∈ R+ , G(t, ·) : X → C(X) be upper semicontinuous. Suppose that G is point dissipative and that for some t0 > 0 the operator G(t0 , ·) is compact. Then G has the global compact attractor . Concerning the dependence of attractors on a parameter from the proof of [11, Theorem 4] it follows the following theorem. Theorem 1.2. Let be a metric space, λ0 be a nonisolated point and Gλ : R+ ×X → P (X), λ ∈ , be a family of msemiflows satisfying: (1) for each λ ∈ , Gλ has a global attractor λ and ∪λ∈ λ ∈ β(X); (2) the map λ → Gλ (t, ), = ∪λ∈ λ , is wupper semicontinuous at λ0 for large t. Then dist(λ , λ0 ) → 0, as λ → λ0 . Other approaches to the problem of nonuniqueness is the construction of the socalled trajectory attractors (see [8, 15, 18]) or multivalued semiflows via the nonstandard framework [7]. Whereas in [4, 21] are considered differential inclusions generating a semigroup of operators in this paper we study, as in [14], inclusions generating a multivalued semiflow. This paper is organized as follows. In Section 2, we extend the results of [14] on existence of a global compact attractor for the differential inclusion dy ∈ −∂φ(y) + F (y), t ∈ [0; T ], dt y(0) = y0 , (1.1) where F : H → 2H is a multivalued map in a Hilbert space H . In Sections 3 and 4, we prove that for a certain class of approximating maps Fn of the multivalued righthand side F the corresponding attractors n converge in the Hausdorff metric to . Finally, in Section 5 we prove the upper semicontinuity of the global attractor under a small perturbation of the map F , F = F + S, > 0. All these results are applied to boundary value problems and in particular to a model of regional economic growth. 2. Existence of the global attractor Let H be a real separable Hilbert space, (·, ·), · be the scalar product and norm in H , respectively, φ : H → (−∞, +∞] be a proper, convex, lower semicontinuous function and let ∂φ : D(∂φ) ⊂ H → 2H be its subdifferential. A. V. Kapustian and J. Valero 35 Consider the problem dy ∈ −∂φ(y) + F (y), t ∈ [0; T ], dt y(0) = y0 ∈ H, (2.1) where F : H → 2H and satisfy the properties: (G1) F : H → Cv (H ); (G2) ∃D1 , D2 ≥ 0 such that supu∈F (v) u ≤ D1 + D2 v, ∀v ∈ H ; (G3) F is wupper semicontinuous; (G4) ∃δ > 0, M > 0 such that ∀u ∈ D(∂φ), u > M, ∀y ∈ −∂φ(u) + F (u), (y, u) ≤ −δ; (2.2) (G5) ∀R > 0 the set MR = {u ∈ H  u ≤ R, φ(u) ≤ R} is compact in H . Further we denote X = D(φ). Definition 2.1. The continuous function y : [0, T ] → X is called an integral solution of problem (2.1) if y(0) = y0 and there exists f ∈ L1 ([0, T ], X), f (τ ) ∈ F (y(τ )), a.e. τ ∈ (0, T ), such that ∀u ∈ D(∂ϕ), ∀v ∈ −∂ϕ(u), y(t) − u ≤ y(s) − u + 2 2 t 2 s f (τ ) + v, y(τ ) − u dτ, t ≥ s. (2.3) Further we shall denote each integral solution by y(·) = I (y0 )f (·). The integral solution y(·) is called a strong one if it is absolutely continuous on (0, T ) and dy/dt ∈ −∂φ(y(τ )) + f (τ ), a.e. on (0, T ). According to [20, Theorem 2.1] ∀x0 ∈ X, ∀T > 0, there exists an integral solution of (2.1), x(·) = I (x0 )f (·), x(0) = x0 . Moreover, the set of all integral solutions on [0, T ] starting from the point x0 (denoted by TF (x0 )) is a connected compact set in the space C(0, T ; X) and the map x → TF (x) is wupper semicontinuous [20, Theorems 2.1 and 4.3]. Lemma 2.2. Under condition (G2) each integral solution of (2.1) is a strong solution. Proof. According to [6, page 189] it is sufficient to prove that any selection f (·) ∈ F (y(·)), where y(·) = I (u0 )f (·), belongs to L2 (0, T ; X). It follows from (G2) that f (t) ≤ D1 + D2 y(t), but y(·) ∈ C(0, T ; X), so that f (·) ∈ L2 (0, T ; X). Now in the same way as in [14] we define the msemiflow G : R+ × X → P (X), G(t, y0 ) = {y(t)  y(·) is a strong solution of (2.1), y(0) = y0 }. Following [14, Lemma 6] we can prove that G(t1 + t2 , x) = G(t1 , G(t2 , x)), ∀x ∈ X, ∀t1 , t2 ∈ R+ . Theorem 2.3. Let (G1)–(G5) hold. Then G has the global compact invariant attractor , which is the minimal closed set attracting all bounded sets. 36 Attractors of multivalued semiflows generated by differential inclusions . . . Proof. We obtain some properties of G. First we prove that ∀t ≥ 0, ∀x ∈ X, G(t, x) is compact in X. Indeed, from the fact that TF (x) is compact in C(0, T ; X) we have that ∀{yn (·)} ⊂ TF (x) there exist a subsequence and y(·) ∈ TF (x) such as yn → y in C(0, T ; X). Hence, yn (t) → y(t), ∀t ∈ [0, T ], in X. It follows that G(t, x) is compact ∀t ∈ [0, T ]. On the other hand, we obtain that G(t, ·) : X → P (X) is upper semicontinuous. Indeed, from the fact that x → TF (x) is wupper semicontinuous, we have that ∀ > 0, ∀x ∈ X, ∃δ > 0 such that x −x0 < δ implies TF (x) ⊂ B (TF (x0 )), that is, for an arbitrary y(·) ∈ TF (x), ∃y0 (·) ∈ TF (x0 ) such that maxt∈[0,T ] y(t) − y0 (t) ≤ and then ∀t ∈ [0, T ], y(t) − y0 (t) ≤ . Thus G(t, x) ⊂ B (G(t, x0 )) and by virtue of the compactness of G(t, x) the upper semicontinuity is proved. Let B0 = {u ∈ X  u ≤ M + }, > 0. We show that G(t, B0 ) ⊂ B0 , ∀t ≥ 0. Let x0 ∈ B0 , x(·) ∈ TF (x0 ) be such that ∃t > 0 for which x(t) ∈ / B0 , that is, x(t) > M +. As x(·) is continuous, then there exists t0 such that x(t0 ) = M + , x(τ ) ≥ M + , ∀τ ∈ [t0 , t]. Therefore, using (G4) and the fact that x(·) is a strong solution of (2.1), in a standard way we obtain that (1/2)(d/dτ )x(τ )2 ≤ −δ, ∀τ ∈ [t0 , t], so that x(t)2 ≤ x(t0 )2 − 2δ(t − t0 ), which is a contradiction. Hence, G(t, B0 ) ⊂ B0 , ∀t ≥ 0. Thus, repeating the proof of [14, Theorem 7], we obtain that ∀x ∈ X, ∃tx > 0 such that G(t, x) ⊂ B0 , ∀t ≥ tx . In the same way we also prove that G(t, BN ) ⊂ BN , ∀N > M, ∀t ≥ 0, where BN = {u ∈ X  u ≤ N }. Therefore, G is pointwise dissipative and τ ≥0 G(τ, B) ∈ β(X), ∀B ∈ β(X). Now we prove that G(t, B) is precompact in X for any t > 0 and B ∈ β(X). According to (G5) it is sufficient to prove that ∃R = R(t, B) such that G(t, B) ⊂ MR . First we shall show that the set M(B, T ) = {f (·)  y(·) = I (y0 )f (·), y ∈ TF (y0 ), y0 ∈ B} is bounded in L2 (0, T ; X). Indeed, there exists N for which G(t, B) ⊂ BN , ∀t ≥ 0, and then maxt∈[0,T ] y(t) ≤ N, ∀y(·) ∈ TF (B). By virtue of (G2), f (t) ≤ D1 + D2 y(t) ≤ D1 + D2 N, ∀f (·) ∈ M(B, T ). Thus M(B, T ) is bounded in L2 (0, T ; X). So, repeating the proof of [14, Theorem 8] we obtain that ∀t > 0, ∃R = R(t, B) such that G(t, B) ⊂ MR . Therefore G(t, B) is precompact in X. Hence, it follows from Theorem 1.1 that there exists the global compact attractor . Moreover, by [14, Remarks 5 and 8], = G(t, ), ∀t ≥ 0, and the minimality property holds. Remark 2.4. Theorem 2.3 generalizes Theorem 9 from [14], in which F is supposed to be Lipschitz in the multivalued sense. Consider the application of the previous result to the problem ∂y ∈ y + f (y) + h, on 1 × (0, T ), ∂t y∂1 = 0, y(x, 0) = y0 (x), (2.4) x ∈ 1, where h ∈ L2 (1), 1 ⊂ Rn is a bounded open domain with smooth boundary ∂1 and f : R → 2R satisfies: (H1) f : R → Cv (R); A. V. Kapustian and J. Valero 37 (H2) ∃D1 , D2 ≥ 0 such that supy∈f (s) y ≤ D1 + D2 s, ∀s ∈ R; (H3) f is wupper semicontinuous; (H4) ∃M ≥ 0, α > 0 such that ∀s ∈ R, ∀y ∈ f (s), ys ≤ (λ1 − α)s2 + M, where λ1 is the first eigenvalue of −3 in H01 (1). To come to problem (2.1), we define F : H → 2H , H = L2 (1), F (y) = ξ + h  ξ ∈ H, ξ(x) ∈ f y(x) a.e. x ∈ 1 . (2.5) It is well known that −3 is the subdifferential of the proper convex lower semicontinuous function φ(u) = 1 (1/2)∇u2 dx with D(φ) = H01 (1) and (G5) holds [6]. Proposition 2.5. The map F satisfies (G1)–(G4). Proof. Condition (H4) in a standard way [14, Theorem 10] provides that (G4) holds. The map f has compact values and then it is upper semicontinuous, so that it is measurable [2, Proposition 8.2.1]. Hence, there exists a measurable selection g(s) ∈ f (s), s ∈ R [2, Theorem 8.1.3]. Then for any y ∈ H , g(y(x)) is a measurable selection of we have that ∀y ∈ H , ∀(ξ + h) ∈ F (y), ξ + h ≤ f (y(x)). In view of(H2), 2 2 1 ξ(x) dx +h ≤ 1 (D1 + D2 y(x)) dx +h ≤ D̃1 + D̃2 y, so that F (y) = ∅, ∀y ∈ H , and (G2) holds. Following [14, Lemma 11] we obtain that F : H → Cv (H ). Now we prove that if f : R → Cv (R) is upper semicontinuous and satisfies (H2) then F is upper semicontinuous on H . Since the map f is upper semicontinuous, is upper hemicontinuous [1, page 60]. We prove that F is also hemicontinuous, that is, from un → u in H and σn (p) := σ (F (un ), p) = supv∈F (un ) (p, v) → σ0 (p), ∀p ∈ H , it follows that σ (F (u), p) ≥ σ0 (p). Indeed, ∀p ∈ H , ∀n ≥ 1 ∃vn ∈ F (un ) such that (p, vn ) > σn (p) − 1/n. Moreover, by virtue of (G2) with accuracy to a subsequence vn → v weakly in H . Now we can use [16, Chapter 3, Theorem 6], taking X = Y = R, p = q = 2. Since (un (x), vn (x)) ∈ graph (f ) for a.e. x ∈ 1, un → u in H , vn → v weakly in H , all the conditions of the mentioned theorem hold and we have v(x) ∈ f (u(x)) for a.e. x ∈ 1. Then passing to the limit in the last inequality we have (p, v) ≥ σ0 (p), v ∈ F (u). Thus, supv∈F (u) (p, v) = σ (F (u), p) ≥ σ0 (p) and hence F : H → Cv (H ) is hemicontinuous. For arbitrary u0 ∈ H conditions (G1)–(G2) hold, so that F (u0 ) is weakly compact and convex in H and hence according to [16, Chapter 3, Theorem 10] F is upper semicontinuous at u0 . Therefore, G3 is satisfied. Now, Theorem 2.3 implies the following theorem. Theorem 2.6. Let (H1)–(H4) hold. The semiflow generated by (2.4) has the global compact invariant attractor , which is the minimal closed set attracting all bounded sets. Example 2.7. A model of regional economic growth. Consider a closed economy on a bounded domain 1 ⊂ Rn and the following variables: y(x, t) is the stock of available capital; u(x, t) is the rate of investment. From the local conservation of capital it follows, as a particular case, that the equation (see 38 Attractors of multivalued semiflows generated by differential inclusions . . . [17, page 603]): ∂y = y + ω(y) + g(y) + u, ∂t y∂1 = 0, on 1 × (0, T ), y(x, 0) = y0 (x), x ∈ 1, 0 ≤ u(x, t) ≤ θ y(x, t) , on 1 × (0, T ), (2.6) where −ω(y), ω being nondecreasing, represents a recursive depreciation of capital and −g(y) is the nonlinear rate of demand. The Dirichlet boundary conditions imply the fact that the economy is closed. We assume that the functions ω, g : R → R, θ : R → R+ are continuous and have at most linear growth. Define the multivalued map f : R → 2R , f (s) = ω(s) + g(s) + ξ  0 ≤ ξ ≤ θ(s) . (2.7) It is straightforward to check that (H1)–(H3) hold. If we assume that ω(s) + g(s) + θ (s) s ≤ λ1 − α s 2 + M, ∀s ≥ 0, ω(s) + g(s) s ≤ λ1 − α s 2 + M, ∀s ≤ 0, (2.8) then (H4) is also satisfied. Therefore, equation (2.6) is a particular case of (2.4) and Theorem 2.6 holds. 3. Approximation of the attractor Now we are interested in the possibility of the approximation of the attractor . For this we assume that the following stronger conditions hold instead of (G2) and (G4): (G2*) ∃C > 0 such that supu∈F (v) u ≤ C, ∀v ∈ H ; (G4*) ∃γ > 0 such that (∂ϕ(y), y) ≥ γ y2 , ∀y ∈ D(∂ϕ). Conditions (G2∗ ), (G4∗ ) imply (G2), (G4). Indeed, for any ξ ∈ −∂ϕ(y) + F (y) we have (ξ, y) ≤ −γ y2 + supu∈F (y) uy ≤ −γ y2 + Cy. Hence (ξ, y) ≤ y(−γ y + C) and condition (G4) holds for δ = M = (1/γ )(C + 1). Due to condition (G2∗ ) we can use [20, Theorem 1.1] and construct the sequence {Fn : H → Cv (H )} such that ∀u ∈ H , F (u) = ∞ n=1 Fn (u), Fn+1 (u) ⊂ Fn (u), Fn are locally Lipschitz (in the multivalued sense) and have locally Lipschitz selections and for each Fn condition (G2∗ ) holds with the same constant C. Moreover, dist(Fn (u), F (u)) → 0, ∀u ∈ H . By Fn we construct in the same way as before the msemiflows Gn , since (G1)–(G4) are satisfied for the maps Fn . From Theorem 2.3 it follows the existence of the compact global invariant attractor n for each Gn , n ≥ 1. The maps Fn are more regular than F , so it is interesting to consider whether the attractors n converge to in the Hausdorff metric. Theorem 3.1. Let (G1), (G2∗ ), (G3), (G4∗ ) hold. Then dist H (, n ) → 0, as n → ∞. Proof. We note that = G(t, ) ⊂ Gn (t, ) ⊂ B (n ), ∀ > 0, t ≥ T (), and since the sets n are compact, we have ⊂ n , ∀n ≥ 1. Analogously, n+1 ⊂ n . Hence, A. V. Kapustian and J. Valero ∞ n=1 n ∪ = 39 ∞ n=1 n = 1 . We must show that ∀ > 0, ∃N such that n ⊂ B (), ∀n ≥ N. In view of Theorem 1.2 we have to prove that ∞ n=1 n ∈ β(H ) (but we have already shown that such a set is compact) and for large t the next property holds: ∀ > 0, ∃N such that Gn (t, 1 ) ⊂ B (G(t, 1 )), ∀n ≥ N. Now we prove it. On the set = {n, n ≥ 1, +∞} we introduce the metric ρ(m, n) = 1/m − 1/n (1/∞ = 0). Hence (, ρ) is a metric compact space. Let λ0 := +∞. Now it is sufficient to verify that the map λ → Gλ (t, 1 ) is upper semicontinuous at λ0 . Since Gλ (t, 1 ) is compact for any λ ∈ (this follows from the fact that the map Gλ (t, ·) is upper semicontinuous and have compact values [1, page 42]), (, ρ) is a compact metric space and Gλ ⊂ G1 , ∀λ ∈ , it is sufficient to prove that its graph on is compact in ×H [2, Proposition 1.4.8], that is, the set D = {(λ, u)  λ ∈ , u ∈ Gλ (t, 1 )} is compact in × H . Let {(λn , un )} ⊂ D. Hence λn → λ0 and we have to prove that there exists u1 ∈ Gλ0 (t, 1 ) such that un → u1 in H (with accuracy to a subsequence). We have un = un (t), un (·) = I (ηn )fn (·), un (0) = ηn ∈ 1 . Hence, there exists η0 ∈ 1 and a subsequence such that ηn → η0 . We consider zn (·) = I (η0 )fn (·). Let σ − L1 (0, T ; H ) be the space L1 (0, T ; H ) endowed with the weak topology. In view of the inequality fn (τ ) ≤ C, a.e. τ ∈ (0, T ), for a subsequence fn → f in σ − L1 (0, T ; H ). Since {fn } are uniformly integrable and the semigroup S(t, ·) generated by −∂φ is compact (this follows from (G5) [10, page 1398]), there exist a subsequence {zn (·)} such that zn → z in C(0, T ; H ) [9, Theorem 2.3]. Hence, zn → z in C(0, T ; H ), fn → f in σ −L1 (0, T ; H ) and z(·) = I (η0 )f (·) [19, Lemma 1.3]. Therefore maxt∈[0,T ] un (t)− z(t) ≤ maxt∈[0,T ] I (ηn )fn (t)−I (η0 )fn (t)+maxt∈[0,T ] I (η0 )fn (t)−I (η0 )f (t) ≤ ηn − η0 + maxt∈[0,T ] zn (t) − z(t) → 0, n → ∞. Thus un (t) → z(t), ∀t ∈ [0, T ], z(0) = η0 ∈ 1 . We prove the fact that f (t) ∈ F (z(t)) for a.e. t ∈ [0, T ]. First we note that fn (t) ∈ Fn (zn (t)), a.e. t ∈ [0, T ]. We prove that ∃N such that ∀n ≥ N, f (t) ∈ B1/n (Fn (z(t))), a.e. on (0, T ). Indeed, let it not be so. Then ∀N ≥ 1, ∃ n ≥ N such that f (t) ∈ / B1/n (Fn (z(t))). On the other hand, from the wsemicontinuity and the facts proved above ∀n ≥ 1, ∃m(n) ≥ n such that Fn (zk (t)) ⊂ B1/2n (F n (z(t))), ∀k ≥ m(n). So k≥m(n) Fn (zk (t)) ⊂ B1/2n (Fn (z(t))). As k ≥ m(n) ≥ n, so k≥m(n) Fk (zk (t)) ⊂ B1/2n (Fn (z(t))). Hence, by virtue of the convexity of Fn (z) we have co k≥m(n) fk (t) ⊂ B1/n (Fn (z(t))) and therefore f (t) ∈ / co k≥m(n) fk (t). From [19, Proposition 1.1] we obtain a contradiction. Thus ∀n ≥ N, ∃gn ∈ Fn (z(t)) such that gn − f (t) ≤ 1/n. Hence gn → f (t) in H and from Fn+1 (z(t)) ⊂ Fn (z(t)) it follows that f (t) ∈ Fn (z(t)), ∀n ≥ N. Thus f (t) ∈ F (z(t)), a.e. on (0, T ), and un = un (t) → z(t) = u1 ∈ Gλ0 (t, 1 ). Remark 3.2. Theorem 3.1 holds for inclusion (2.4) if we assume that D2 = 0 in condition (H2). (G2∗ ) and (G4∗ ) will be satisfied with C = D1 (µ(1))1/2 and γ = λ1 . 4. Dependence on a parameter Now we are interested in the continuous dependence on a parameter. Consider the sequence of problems (2.1) with righthand sides Fn satisfying: (R1) Fn : H → Cv (H ); ∀u ∈ H, ∀n ≥ 1; (R2) Fn+1 (u) ⊂ Fn (u), ∀n ≥ 1; 40 Attractors of multivalued semiflows generated by differential inclusions . . . (R3) ∃D1 , D2 ≥ 0 such that supv∈F1 (u) v ≤ D1 + D2 u, ∀u ∈ H ; (R4) Fn are wupper semicontinuous ∀n ≥ 1; ∞ (R5) ∀u ∈ H, ∞ n=1 Fn (u) = ∅ and F (u) = n=1 Fn (u) is wupper semicontinuous; (R6) ∃δ > 0, M > 0 such that ∀u ∈ D(∂φ), u > M, ∀n ≥ 1, ∀y ∈ −∂φ(u)+Fn (u), (y, u) ≤ −δ. (4.1) As before we assume that (G5) holds. Since F (u) ⊂ Fn (u), Fn+1 (u) ⊂ Fn (u), ∀u ∈ H , ∀n ≥ 1, conditions (G1)–(G4) hold for all Fn , F (with the same constants D1 , D2 ). Let Gn , G be the semiflows corresponding to Fn , F . Then in view of Theorem 2.3 there exist the global compact attractors n , corresponding to Gn , G, respectively. Theorem 4.1. Let (R1)–(R6) and (G5) hold. Then dist H (n , ) → 0, as n → ∞. Proof. As in Theorem 1.2, ⊂ · · · ⊂ n+1 ⊂ n ⊂ · · · ⊂ 1 , ∀n ≥ 1, and the desired result will be obtained if we show that for any sequence un ∈ Gn (t, 1 ) there exists u1 ∈ G(t, 1 ) such that un → u1 in H (with accuracy to a subsequence). From the proof of Theorem 2.3 it follows that G(t, BN ) ⊂ BN , Gn (t, BN ) ⊂ BN , ∀n ≥ 1, ∀N > M. Let un = un (t), un (·) = I (ηn )fn (·), un (0) = ηn ∈ 1 , ηn ≤ N, ∀n ≥ 1, where N > M. Then maxt∈[0,T ] un (t) ≤ N . Hence, fn (t) ≤ D1 +D2 N, a.e. on (0, T ), and we can use the same arguments as in the final part of the proof of Theorem 1.2. Remark 4.2. We note that conditions (R1)–(R5) do not imply that dist(Fn (u), F (u)) → 0, as n → ∞. 2 Proof. Consider the space H = l2 = {y = (y1 , y2 , . . .)  ∞ i=1 yi  < ∞} and the sequence of constant maps Fn (u) ≡ Yn = {y ∈ l2  y1 = · · · = yn = 0, y ≤ 1}, n ≥ 1. The sets Yn are nonempty, bounded, closed and convex and F (u) = ∩∞ n=1 Fn (u) = {0}. It is obvious that the maps Fn , F are wupper semicontinuous and satisfy (R2)–(R3) n times (with D1 = 1, D2 = 0). We take ξn = (0, . . . , 0, 1, 0, . . .) ∈ Fn . Since ξn − 0 = 1, we have dist(Fn (u), F (u)) ≥ 1, ∀n ≥ 1. Consider the sequence of problems ∂y ∈ y + fn (y) + h, 1 × (0, T ), ∂t y∂1 = 0, y(x, 0) = y0 (x), (4.2) x ∈ 1, where h ∈ L2 (1), 1 ⊂ Rn is a bounded open domain with smooth boundary ∂1 and fn : R → 2R satisfy: (L1) fn : R → Cv (R), fn+1 (t) ⊂ fn (t), ∀t ∈ R, ∀n ≥ 1; (L2) ∃D1 , D2 ≥ 0 such that supy∈f1 (s) y ≤ D1 + D2 s, ∀s ∈ R; A. V. Kapustian and J. Valero 41 (L3) fn are wupper semicontinuous ∀n ≥ 1; (L4) ∃M ≥ 0, α > 0 such that ∀s ∈ R, ∀n ≥ 1, ∀y ∈ fn (s), ys ≤ (λ1 − α)s2 + M. Define Fn , F : H → 2H , H = L2 (1), Fn (y) = ξ + h  ξ ∈ H, ξ(x) ∈ fn y(x) a.e. x ∈ 1 , F (y) = ξ + h  ξ ∈ H, ξ(x) ∈ ∩∞ n=1 fn y(x) a.e. x ∈ 1 . (4.3) Proposition 4.3. The maps F, Fn satisfy (R1)–(R6). Proof. Condition (L4) in a standard way [14, Theorem 10] provides that (R6) holds. It follows from (L1)–(L4) and Proposition 2.5 that the maps Fn satisfy (R1)–(R4). (L1)–(L3) imply that all fn are upper semicontinuous (because they are compactvalued) and for any t ∈ R, a > 0, map the ball Ba (t) into subsets of some compact set in R. As {fn (t)} is a centered family of compacts, so ∞ n=1 fn (t) = ∅ and in view of [12, page 60] f (·) = ∞ f (·) is upper semicontinuous at t. It follows now from n=1 n (L1)–(L4) that f satisfies (H1)–(H4). Then using again Proposition 2.5 we obtain that (R5) holds. Let n , be the global attractors corresponding to fn , f , respectively. As a consequence of Theorem 4.1 we have the following theorem. Theorem 4.4. Let (L1)–(L4) hold. Then dist H (n , ) → 0, as n → ∞. Example 4.5. A model of regional economic growth. Consider in (2.6) a sequence of functions θn such that θn+1 (s) ≤ θn (s), ∀n ≥ 1, ∀s ∈ R, and θ1 satisfies (2.8). Then (L1)–(L4) hold and Theorem 4.4 takes place. 5. Perturbed differential inclusions We are now interested in the upper semicontinuity of the global attractor for inclusion (2.1) under small perturbations. Consider the family of differential inclusions du ∈ −∂ϕ(u) + F (u) + S(u), dt u(0) = u0 , (5.1) where ≥ 0 is a small parameter and S, F : H → 2H are multivalued maps satisfying (G1)–(G3) and (G4**) there exist 0 > 0, δ > 0, M > 0 such that ∀ ≤ 0 , ∀u ∈ D(∂ϕ), u > M, ∀y ∈ −∂ϕ(u) + F (u) + S(u), (y, u) ≤ −δ. Lemma 5.1. The maps S (u) = F (u) + S(u) are wupper semicontinuous. (5.2) 42 Attractors of multivalued semiflows generated by differential inclusions . . . Proof. Let η > 0 be arbitrary and γ > 0 be such that γ + γ ≤ η. In view of the wupper semicontinuous of F, S there exists δ > 0 such that if u − u0 ≤ δ then (5.3) dist S (u), S u0 ≤ dist F (u), F u0 + dist S(u), S u0 ≤ η. On the other hand, it is evident that S satisfy (G1) and (G2) with D1 = D1S +D1F , D2 = D2S +D2F , where DiS , DiF are the constants in condition (G2) corresponding to S and F , respectively. If condition (G5) is also satisfied then in view of Theorem 2.3 for each ≤ 0 inclusion (5.1) generates the multivalued semiflow G : R+ × D(ϕ) → Comp(D(ϕ)) which has the global compact invariant attractor . Define the setvalued map R(u) = ∪0≤≤0 S(u). Lemma 5.2. The map R satisfies (G1)–(G3) and (G4∗∗ ) replacing S by R. n→∞ Proof. It is clear that the set R(u) is nonempty and bounded. Let yn ∈ R(u), yn −−−→ y. Then yn = n zn , zn ∈ S(u). If there exists a subsequence n! → 0 then y = 0 ∈ R(u). In another case there exists n0 such that n ∈ [δ, 0 ], ∀n ≥ n0 , for some δ > 0. Take a converging subsequence n! → 1 ∈ [δ, 0 ]. It follows that zn! = yn! /n! → y/1 = z ∈ S(u), since S(u) is closed. Hence, y = 1 z ∈ R(u), so that R(u) is closed. Further, let y, 1 z ∈ R(u) be arbitrary. Suppose that ≤ 1 . Then for any α ∈ [0, 1], (5.4) αy + (1 − α)1 z = 2 α ! y + (1 − α ! )z = 2 v, where 2 = α + (1 − α)1 , α ! = α(/2 ) ∈ [0, 1]. Since S(u) is convex, v ∈ S(u) and then R(u) is convex. Therefore, R(u) ∈ Cv (H ) and (G1) holds. Let us check (G3). Let u be arbitrary. Since S is wupper semicontinuous, for any γ > 0 there exists δ > 0 such that if u−v ≤ δ, then S(v) ⊂ Oγ (S(u)). Let y ∈ R(v) be arbitrary. We take h ∈ S(u) such that dist(y, R(u)) = y − h. Then (5.5) dist y, R(u) ≤ y − h ≤ 0 γ . It follows that dist(R(v), R(u)) ≤ 0 γ , if u − v ≤ δ, so that R is wupper semicontinuous. Finally, it is evident that R satisfies (G2) with D1R = 0 D1S , D2R = 0 D2S , and also that (G4∗∗ ) holds. Theorem 5.3. Let the maps F, S satisfy (G1)–(G3), (G4∗∗ ) and (G5) hold. Then dist( , 0 ) → 0, as → 0+ . Proof. From Theorem 1.2 it follows that it is sufficient to check that ∪≤0 ∈ β(D(ϕ)) and that the map → G (t, ∪≤0 ) is wupper semicontinuous at = 0 for any t ≥ 0. First, we note that for any ≤ 0 , belongs to the ball B α = {u ∈ H  u ≤ M + α}, where α > 0. To prove this fact we shall use that for any γ > 0 and u ∈ D(ϕ) there exists T (u, ) such that G (T , u) ∈ B γ and also that G (t, B γ ) ⊂ B γ , A. V. Kapustian and J. Valero 43 ∀t ≥ 0, ∀ ≤ 0 (see the proof of Theorem 2.3). Let γ < α. Since G (T , ·) is upper semicontinuous (see again Theorem 2.3), for any u ∈ we can find a neighborhood O(u) such that G (T , O(u)) ⊂ B α . Since is compact, from the covering ∪u∈ O(u) we can obtain a finite subcovering ∪ni=1 O(ui ). Hence, ⊂ G (t, ) ⊂ B α (we take t ≥ maxi {T (ui , )}), as required. Hence, ∪≤0 ∈ β(D(ϕ)). In order to check the second property we shall prove first that the set K0 = ∪≤0 is compact. Let GR be the semiflow generated by inclusion (5.1) if we replace the map S by R. Since S(u) ⊂ R(u), ∀ ≤ 0 , it is clear that G (u) ⊂ GR (u), ∀u ∈ D(ϕ), ∀ ≤ 0 . From Theorem 2.3 and Lemma 5.2 it follows that GR has a compact global attractor R . Obviously, R is a globally attracting set for each G , ≤ 0 . Hence, since is the minimal closed set that attracts any bounded set for G , it follows that ⊂ R , ∀ ≤ 0 . Therefore, K0 is compact. Suppose that the map → G (t, ∪≤0 ) is not wupper semicontinuous at = 0 for some t > 0. Then there exists a γ neighborhood Oγ of G0 (t, K0 ) and a sequence un ∈ Gn (t, K0 ), n → 0+ , such that un ∈ / Oγ . Then un = un (t), where un (·) = 0 0 I (un )fn (·), un ∈ K0 , and fn (τ ) ∈ F (un (τ )) + n S(un (τ )), a.e. τ ∈ (0, t). Arguing as in Theorems 3.1, 4.1 we obtain the existence of a subsequence (denoted again by n ) and functions f, u such that fn → f in σ − L1 ([0, t], H ), u0n → u0 ∈ K0 , un → u in C([0, t], H ) and u(·) = I (u0 )f (·). We have to prove that f (τ ) ∈ F (u(τ )), a.e. τ ∈ (0, t). In view of [19, Proposition 1.1] for a.a. τ ∈ (0, t), f (τ ) ∈ ∩∞ m=1 co ∪n≥m fεn (τ ). Fix τ ∈ (0, t). Since F is wupper semicontinuous and using condition (G2) for the map S, we obtain that for any δ > 0 there exists n > 0 such that ∀k ≥ n, dist F uk (τ ) + k S uk (τ ) , F u(τ ) ≤ dist F uk (τ ) + k S uk (τ ) , F uk (τ ) + dist F uk (τ ) , F u(τ ) (5.6) δ ≤ k D1S + D2S uk (τ ) + ≤ δ. 2 Since F (u(τ )) is convex, this implies that co ∪k≥n fεk (τ ) ⊂ Oδ (F (u(τ ))). Hence, since F (u(τ )) is closed, f (τ ) ∈ F (u(τ )), a.e. t ∈ (0, t). Then uεn → u(t) ∈ G0 (t, K0 ), which is a contradiction. Consider now the family of boundary value problems ∂u ∈ 3u + f (u) + j (u) + h, ∂t u ∂1 = 0, on 1 × (0, T ), (5.7) u(0) = u0 , where h ∈ L2 (1), ≥ 0 is small, f, j : R → 2R satisfy (H1)–(H3) and f satisfies (H4). Define the maps F, S : H → 2H , H = L2 (1), by F (u) = y ∈ H  y(x) ∈ f u(x) + h(x), a.e. on 1 , (5.8) S(u) = y ∈ H  y(x) ∈ j u(x) , a.e. on 1 . 44 Attractors of multivalued semiflows generated by differential inclusions . . . It follows from Proposition 2.5 that the maps F, S satisfy (G1)–(G3). Lemma 5.4. Condition (G4∗∗ ) holds. Proof. Since f satisfies (H4) and (G2) holds for S, we have that ∀u ∈ D(∂ϕ), ∀y ∈ −∂ϕ(u) + F (u) + S(u), (y, u) ≤ −λ1 u2 + λ1 − α u2 + Mµ(1) + D1 + D2 u u + uh (5.9) 2 D12 1 α ≤ − + D2 u2 + Mµ(1) + + h2 . 2 α α Taking 0 = α/4D2 the last inequality implies that condition (G4∗∗ ) holds. Since (G5) is also satisfied, we have obtained a particular case of inclusion (5.1), so that Theorem 5.3 implies the following result. Theorem 5.5. Let f, j satisfy (H1)–(H3) and f satisfy (H4). Then dist( , 0 ) → 0, as → 0+ . Example 5.6. A model of regional economic growth. Consider in (2.6) the family of functions g = g1 + g2 , θ = θ1 + θ2 , where g1 , θ1 satisfy the same conditions as g, θ and g2 , θ2 are continuous and have at most linear growth. Define the multivalued maps f, j : R → 2R , f (s) = ω(s) + g1 (s) + ξ  0 ≤ ξ ≤ θ1 (s) , (5.10) j (s) = g2 (s) + ξ  0 ≤ ξ ≤ θ2 (s) . Then we obtain a particular case of inclusion (5.7), so that Theorem 5.5 holds. Finally, we remark that if in problems (2.4), (4.2), and (5.7) we replace the operator −3 by A(u) = − ni=1 (∂/∂xi )(∂y/∂xi p−2 (∂y/∂xi )), p > 2, then all the results remain valid. In this case, conditions (H4), (L4) are not necessary. Indeed, we prove that (G4∗∗ ) holds ((G4) and (R6) can be proved in a similar way). It follows from p p Poincaré inequality that "Au, u# = ∇uLp ≥ DuLp for some D > 0. Let 0 > 0 be arbitrary but fixed. 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