POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS WITH INFINITE DELAYS

Abstract. We prove the existence of decay global solutions to a class of fractional differential inclusions with infinite delays and estimate their decay rate. For this purpose, we have to construct a suitable regular measure of noncompactness on the space of solutions and then deploy the fixed point theory for condensing multivalued maps. An application to a class of differential variational inequalities is also given

POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS WITH INFINITE DELAYS CUNG THE ANH, TRAN DINH KE Abstract. We prove the existence of decay global solutions to a class of fractional differential inclusions with infinite delays and estimate their decay rate. For this purpose, we have to construct a suitable regular measure of noncompactness on the space of solutions and then deploy the fixed point theory for condensing multivalued maps. An application to a class of differential variational inequalities is also given. 1. Introduction Let X be a Banach space. We consider the Cauchy problem for fractional differential inclusions with infinite delays of the following form D0α x(t) ∈ Ax(t) + F (t, x(t), xt ), t > 0, (1.1) x(t) = ϕ(t), t ≤ 0, (1.2) D0α , where 0 < α ≤ 1, is the Caputo derivative of order α, A is the infinitesimal generator of a C0 -semigroup on X, ϕ ∈ B with B being an admissible phase space that will be specified later, F : [0, ∞) × X × B → P(X) is a multi-valued function, and xt : (−∞, 0] → X is the history of the state function defined by xt (θ) = x(t+θ) for θ ∈ (−∞, 0]. Differential inclusions (DIs) appeared as in (1.1) arise, for instance, from control theory in which control factor is taken in the form of feedbacks. In such control problems, the presence of delay terms is an inherent feature. Recently, the theory of differential variational inequalities (DVIs) has been an increasingly interested subject since DVIs come from various realistic problems (see [23]). In dealing with DVIs, an effective method is converting them to DIs. These brief mentions tell us that the study of DIs is able to range over many applications. Problem (1.1)-(1.2) in case α = 1 (with/without retarded terms) has been studied extensively. For a complete reference to DIs in infinite dimensional spaces, we refer the reader to monograph [15]. In addition, there are many contributions for semilinear DIs published in the last few years (see e.g. [1, 8, 9, 13, 20]). Concerning fractional DIs in infinite dimensional sapces, one can find a number of works devoted to the questions of solvability and controllability. Let us quote some investigations in [16, 21, 25, 28, 29] that are close to the problem under consideration. An important question raised for problem (1.1)-(1.2) is to study the existence of decay global solutions and estimate their decay rate. However, up to the best of 2010 Mathematics Subject Classification. 34A08, 35B35, 37C75, 47H08, 47H10. Key words and phrases. Decay solution; Fractional differential inclusion; Infinite delay; Condensing map; Fixed point; Measure of noncompactness; Differential variational inequality. Corresponding author: ketd@hnue.edu.vn. 1 2 CUNG THE ANH, TRAN DINH KE our knowledge, no such a result has been known. This is the motivation of the present paper. To study the stability of solutions to differential equations and functional differential equations, Burton and Furumochi [6, 7] introduced a new approach that deploys the fixed point theory to search for solutions lying in a stable subset of state spaces. We will exploit this idea to prove the existence of decay global solutions to problem (1.1)-(1.2) and determine the decay rate of solutions. To do this, we have to find a satisfactory space of solutions and construct on this space a regular measure of noncompactness (MNC). We also have to use some new asymptotic estimates on the family of operators {Sα (t), Pα (t), t ≥ 0} established in our previous work [3]. The designed solution space and MNC enable us to utilize the fixed point theory for condensing multivalued maps. Consequently, we obtain a compact set of decay solutions x with ||x(t)|| = O(t−γ ) as t → ∞ for some γ < α. Since the case α ∈ (0, 1) is more involved than the case α = 1, in this paper we only focus on the former one and make a note that our technique can be applied to the latter case by the same manner. The paper is organized as follows. In the next section, we recall some notions and facts related to fractional calculus, including some properties of fractional resolvent operators. We also recall concept of measure of noncompactness and the fixed point theory for condensing multivalued maps. For the sake of completeness, in Section 3 we prove the existence result of problem (1.1)-(1.2) on the interval (−∞, T ]. Section 4 is devoted to proving the existence of decay global solutions with a polynomial decay rate. In the last section, we apply the abstract results to a class of DVIs consisting of a functional partial differential equations in unbounded domain and a finite-dimensional variational inequality. 2. Preliminaries 2.1. Fractional calculus. Let L1 (0, T ; X) be the space of integrable functions on [0, T ] in the sense of Bochner. Definition 2.1. The fractional integral of order α > 0 of a function f ∈ L1 (0, T ; X) is defined by I0α f (t) = 1 Γ(α) t (t − s)α−1 f (s)ds, 0 where Γ is the Gamma function, provided the integral converges. Definition 2.2. For a function f ∈ C N ([0, T ]; X), the Caputo fractional derivative of order α ∈ (N − 1, N ] is defined by  t 1  (t − s)N −α−1 f (N ) (s)ds, if α ∈ (N − 1, N ), α D0 f (t) = Γ(N − α) 0  α D0 f (t) = f (N ) (t), if α = N. It should be noted that there are some notions of fractional derivatives, in which the Riemann-Liouville and Caputo definitions have been used widely. Many application problems, expressed by differential equations of fractional order, require initial conditions related to u(0), u (0), etc., and the Caputo fractional derivative POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 3 satisfies these demands. For u ∈ C N ([0, T ]; X), we have the following formulas D0α I0α u(t) = u(t), N −1 I0α D0α u(t) = u(t) − k=0 u(k) (0) k t . k! Consider the linear problem D0α x(t) = Ax(t) + f (t), x(0) = x0 . (2.1) (2.2) Let {Sα (t), Pα (t), t ≥ 0} be the family of operators such that t e−λt Sα (t)dt, (2.3) e−λt tα−1 Pα (t)dt. (2.4) λα−1 (λα − A)−1 = 0 t (λα − A)−1 = 0 Then we have the following representation of solution for the linear problem (2.1)(2.2): t (t − s)α−1 Pα (t − s)f (s)ds. x(t) = Sα (t)x0 + (2.5) 0 Let {S(t)} be the C0 -semigroup generated by A. We have the formulas for Sα and Pα as follows (see [30]) ∞ φα (θ)S(tα θ)zdθ, Sα (t)z = (2.6) 0 ∞ θφα (θ)S(tα θ)zdθ, z ∈ X, Pα (t)z = α (2.7) 0 where φα is a probability density function defined on (0, ∞), that is, φα (θ) ≥ 0 and ∞ φα (θ)dθ = 1. Moreover, φα has the expression 0 1 1 −1− 1 α ψ (θ − α ), θ α α ∞ Γ(nα + 1) 1 (−1)n−1 θ−αn−1 sin(nπα). ψα (θ) = π n=1 n! φα (θ) = Proposition 2.1. [27] We have the following properties (1) If the semigroup S(·) is norm continuous, that is t → S(t) is continuous for t > 0, then Sα (·) and Pα (·) are norm continuous as well; (2) If S(·) is a compact semigroup then Sα (t) and Pα (t) are compact for t > 0. Let p > 1 α. We define the operator Qα : Lp (0, T ; X) → C([0, T ]; X) as follows: t (t − s)α−1 Pα (t − s)f (s)ds. Qα (f )(t) = (2.8) 0 Using Proposition 2.1, we prove the following result. Proposition 2.2. If S(·) is a norm continuous semigroup, then the operator Qα defined by (2.8) maps any bounded set in Lp (0, T ; X) into an equicontinuous one in C([0, T ]; X). 4 CUNG THE ANH, TRAN DINH KE Proof. Since Pα is norm continuous, the operator Φ(t, s) = (t − s)α−1 Pα (t − s) satisfies the assumption of Lemma 1 in [22]. It follows that Qα (Ω) is equicontinuous for each bounded set Ω ⊂ Lp (0, T ; X). The proof is complete. To study the decay rate of solutions to problem (1.1)-(1.2), we need the following result. Proposition 2.3. [3] If the semigroup S(·) is exponentially stable, i.e. ||S(t)|| ≤ M e−at for some a, M > 0, then there exist two positive numbers CS and CP such that ||Sα (t)|| ≤ CS t−α , ||Pα (t)|| ≤ CP t−α , ∀t > 0. 2.2. Phase space. As is known, when we consider the differential equation with infinite delays, the way of choosing phase spaces plays an essential role. In what follows, we recall the axioms of phase spaces of Hale and Kato [17]. Let (B, | · |B ) be a semi-normed linear space, consisting of functions mapping (−∞, 0] into a Banach space X. The definition of a phase space B, introduced in [17], is based on the following axioms stating that if a function v : (−∞, T +σ] → X is such that v|[σ,T +σ] ∈ C([σ, T + σ]; X) and vσ ∈ B, then (B1) vt ∈ B for t ∈ [σ, T + σ]; (B2) the function t → vt is continuous on [σ, T + σ]; (B3) |vt |B ≤ K(t − σ) sup{ v(s) X : σ ≤ s ≤ t} + M (t − σ)|vσ |B , where K, M : [0, ∞) → [0, ∞), K is continuous, M is locally bounded, and they are independent of v. Let us give some examples of phase spaces. The first one is given by Cγ = {φ ∈ C((−∞, 0]; X) : lim eγθ ||φ(θ)|| exists in X}, θ→−∞ where γ is a positive number. This phase space satisfies (B1)-(B3) with K(t) = 1, M (t) = e−γt , and it is a Banach space with the norm |φ|γ = sup eγθ ||φ(θ)||. θ≤0 Regarding another typical example, suppose that 1 ≤ p < +∞, 0 ≤ r < +∞ and g : (−∞, −r] → R is nonnegative, Borel measurable function on (−∞, −r). Let CLpg is a class of functions ϕ : (−∞, 0] → X such that ϕ is continuous on [−r, 0] and g(θ) ϕ(θ) pX ∈ L1 (−∞, −r). A seminorm in CLpg is given by −r |ϕ|CLpg = sup { ϕ(θ) −r≤θ≤0 X} + g(θ) ϕ(θ) −∞ p X dθ 1 p . (2.9) Assume further that −r g(θ)dθ < +∞, for every s ∈ (−∞, −r), and (2.10) s g(s + θ) ≤ G(s)g(θ) for s ≤ 0, θ ∈ (−∞, −r), (2.11) POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 5 where G : (−∞, 0] → R+ is locally bounded. From [18], we know that if (2.10)(2.11) hold, then CLpg satisfies (B1)-(B3). Moreover, one can take  1 for 0 ≤ t ≤ r, 1/p K(t) = (2.12) −r 1 + g(θ)dθ for t > r; −t  1/p  max{1 + −r g(θ)dθ , G(−t)1/p } for 0 ≤ t ≤ r, −r−t M (t) = (2.13) 1/p  1/p max{ −t g(θ)dθ , G(−t) } for t > r. −r−t For more examples of phase spaces, see [18]. 2.3. Measure of noncompactness and condensing multivalued maps. Let E be a Banach space. Denote P(E) = {B ⊂ E : B = ∅}, Pb (E) = {B ∈ P(E) : B is bounded}, K(E) = {B ∈ P(E) : B is compact}, Kv(E) = {B ∈ K(E) : B is convex}. We will use the following definition of measure of noncompactness ([15]). Definition 2.3. A function β : Pb (E) → R+ is called a measure of noncompactness (MNC) on E if β(co Ω) = β(Ω) for every Ω ∈ Pb (E), where co Ω is the closure of the convex hull of Ω. An MNC β is called i) monotone if Ω0 , Ω1 ∈ Pb (E), Ω0 ⊂ Ω1 implies β(Ω0 ) ≤ β(Ω1 ); ii) nonsingular if β({a} ∪ Ω) = β(Ω) for any a ∈ E, Ω ∈ Pb (E); iii) invariant with respect to union with compact set if β(K ∪ Ω) = β(Ω) for every relatively compact set K ⊂ E and Ω ∈ Pb (E); iv) algebraically semi-additive if β(Ω0 + Ω1 ) ≤ β(Ω0 ) + β(Ω1 ) for any Ω0 , Ω1 ∈ Pb (E); v) regular if β(Ω) = 0 is equivalent to the relative compactness of Ω. An important example of MNC is the Hausdorff MNC χ(·), which is defined as follows: for Ω ∈ Pb (E) put χ(Ω) = inf{ε > 0 : Ω has a finite ε-net}. This MNC satisfies all properties given in Definition 2.3. We now give some basic estimates based on MNCs. We first recall the sequential MNC χ0 defined by χ0 (Ω) = sup{χ(D) : D ∈ ∆(Ω)}, (2.14) where ∆(Ω) is the collection of all at-most-countable subsets of Ω (see [2]). We know that 1 χ(Ω) ≤ χ0 (Ω) ≤ χ(Ω), (2.15) 2 for all bounded set Ω ⊂ E. Then the following property is evident. Proposition 2.4. Let χ be the Hausdorff MNC in E and Ω ⊂ E be a bounded set. Then for every > 0, there exists a sequence {xn } ⊂ Ω such that χ(Ω) ≤ 2χ({xn }) + . 6 CUNG THE ANH, TRAN DINH KE We need the following assertion, whose proof can be found in [15]. Proposition 2.5. If {wn } ⊂ L1 (0, T ; X) such that ||wn (t)||X ≤ ν(t), for a.e. t ∈ [0, T ], for some ν ∈ L1 (0, T ), then we have t t wn (s)ds}) ≤ 2 χ({ 0 χ({wn (s)})ds 0 for t ∈ [0, T ]. Using Propositions 2.4 and 2.5, we get Proposition 2.6. Let D ⊂ L1 (0, T ; X) such that (1) ||ξ(t)|| ≤ ν(t), for all ξ ∈ D and for a.e. t ∈ [0, T ], (2) χ(D(t)) ≤ q(t) for a.e. t ∈ [0, T ], where ν, q ∈ L1 (0, T ). Then t t D(s)ds ≤ 4 χ q(s)ds, 0 t t D(s)ds = { here 0 0 ξ(s)ds : ξ ∈ D}. 0 Proof. For > 0, there exists a sequence ξn ∈ D such that t t D(s)ds ≤ 2χ { χ 0 ξn (s)ds} + , 0 thanks to Proposition 2.4. Applying Proposition 2.5 for the last expression, we have t 0 Since t D(s)ds ≤ 4 χ t χ({ξn (s)})ds + ≤ 4 0 q(s)ds + . 0 is arbitrary, we get the desired conclusion. We make use of some notions and facts of set-valued analysis. Let Y be a metric space. Definition 2.4. A multivalued map (multimap) F : Y → P(E) is said to be: i) upper semicontinuous (u.s.c) if F −1 (V ) = {y ∈ Y : F(y) ∩ V = ∅} is a closed subset of Y for every closed set V ⊂ E; ii) weakly upper semicontinuous (weakly u.s.c) if F −1 (V ) is closed subset of Y for all weakly closed set V ⊂ E; iii) closed if its graph ΓF = {(y, z) : z ∈ F(y)} is a closed subset of Y × E; iv) compact if F(Y ) is relatively compact in E; v) quasicompact if its restriction to any compact subset A ⊂ Y is compact. The following lemmas give criteria for checking a given multimap is (weakly) u.s.c. Lemma 2.7 ([15, Theorem 1.1.12]). Let G : Y → P(E) be a closed quasicompact multimap with compact values. Then G is u.s.c. POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 7 Lemma 2.8 ([5, Proposition 2]). Let X be a Banach space and Ω be a nonempty subset of another Banach space. Assume that G : Ω → P(X) is a multimap with weakly compact, convex values. Then G is weakly u.s.c if and only if {xn } ⊂ Ω with xn → x0 ∈ Ω and yn ∈ G(xn ) implies yn y0 ∈ G(x0 ), up to a subsequence. We now introduce the concept of condensing multimaps. Definition 2.5. A multimap F : Z ⊆ E → P(E) is said to be condensing with respect to an MNC β (β-condensing) if for any bounded set Ω ⊂ Z, the relation β(Ω) ≤ β(F(Ω)) implies the relative compactness of Ω. Let β be a monotone nonsingular MNC in E. The application of the topological degree theory for condensing maps (see, e.g., [2, 15]) yields the following fixed point principle. Theorem 2.9. [15, Corollary 3.3.1] Let M be a bounded convex closed subset of E and let F : M → Kv(M) be a u.s.c and β-condensing multimap. Then Fix(F) := {x ∈ E : x ∈ F(x)} is nonempty and compact. Consequently, we prove the following result which will be used later for our existence theorem in Sect. 3. Theorem 2.10. Let M be a compact convex subset of E and let F : M → P(M) be a closed multimap with convex values. Then Fix(F) = ∅. Proof. Since F : M → P(M) is quasicompact and has closed, convex and compact values, Lemma 2.7 ensures that F is u.s.c. Obviously, F is χE -condensing with χE being the Hausdorff MNC on E. Then Fix(F) = ∅ thanks to Theorem 2.9. 3. Existence result To prove existence results for problem (1.1)-(1.2), we assume that (A) A is the infinitesimal generator of a C0 -semigroup {S(t)}t≥0 which is norm continuous. (B) The phase space B verifies (B1)-(B3). (F) F : [0, T ] × X × B → Kv(X) is a multimap satisfying that (1) t → F (t, v, w) admits a strongly measurable selection for each (v, w) ∈ X × B and (v, w) → F (t, v, w) is u.s.c for a.e. t ∈ (0, T ); (2) there exists a function m ∈ Lp (0, T ), p > α1 , such that ||F (t, v, w)|| ≤ m(t)(||v|| + |w|B ), ∀v ∈ X, w ∈ B, and for a.e. t ∈ (0, T ), here ||F (t, v, w)|| = sup{||ξ|| : ξ ∈ F (t, v, w)}; (3) if the semigroup S(·) is non-compact, then for any bounded sets B ⊂ X, C ⊂ B, we have χ(F (t, B, C)) ≤ k(t) χ(B) + sup χ(C(θ)) , θ≤0 p for a.e. t ∈ (0, T ), where k ∈ L (0, T ) is a nonnegative function. 8 CUNG THE ANH, TRAN DINH KE Put Cϕ = {y ∈ C([0, T ]; X) : y(0) = ϕ(0)}. Then Cϕ is a closed subspace of C([0, T ]; X) with the norm ||y||C = sup ||y(t)||. t∈[0,T ] For ϕ ∈ B and y ∈ C([0, T ]; X), we define the function y[ϕ] : (−∞, T ] → X as follows y(t) for t ∈ [0, T ], y[ϕ](t) = ϕ(t) for t < 0. For x ∈ Cϕ , we denote PFp (x) = {f ∈ Lp (0, T ; X) : f (t) ∈ F (t, x(t), x[ϕ]t )}. (3.1) Motivated by formula (2.5), we introduce the following definition. Definition 3.1. A function x : (−∞, T ] → X is said to be an integral solution of problem (1.1)-(1.2) if and only if x(t) = ϕ(t), t ≤ 0, and there exists a function f ∈ PFp (x) such that t (t − s)α−1 Pα (t − s)f (s)ds, t > 0. x(t) = Sα (t)ϕ(0) + 0 For ϕ ∈ B given, we define the solution operator Σ : Cϕ → P(Cϕ ) as follows t Σ(x)(t) = Sα (t)ϕ(0) + 0 (t − s)α−1 Pα (t − s)f (s)ds : f ∈ PFp (x) , (3.2) or equivalently, Σ(x) = Sα (·)ϕ(0) + Qα ◦ PFp (x), where Qα is defined by (2.8). Since F has convex values, so does PFp . This implies that Σ has convex values as well. It is obvious that if x is a fixed point of Σ, then x[ϕ] is an integral solution of (1.1)-(1.2) on (−∞, T ]. To establish the existence result, we need some properties of PFp . Arguing as in [14], PFp is well-defined. Moreover, we have the following lemma. Lemma 3.1. Under assumption (F), the multimap PFp is weakly u.s.c. Proof. We use Lemma 2.8. Let {xk } ⊂ Cϕ such that xk → x∗ , fk ∈ PFp (xk ). We see that {fk (t)} ⊂ C(t) := F (t, {xk (t), xk [ϕ]t }), and C(t) is a compact set for a.e t ∈ (0, T ). Furthermore, by (F)(2), {fk } is integrably bounded (bounded by an Lp integrable function). Therefore {fk } is weakly compact in Lp (0, T ; X) (see [10]). Let fk f ∗ . Then by Mazur’s lemma (see, e.g. [11]), there are f˜k ∈ co{fi : i ≥ k} such that f˜k → f ∗ in Lp (0, T ; X) and then f˜k (t) → f ∗ (t) for a.e. t ∈ (0, T ), up to a subsequence. Since F has compact values, the upper semicontinuity of F (t, ·, ·) means that F (t, xk (t), xk [ϕ]t ) ⊂ F (t, x∗ (t), x∗ [ϕ]t ) + B , for all large k, here radius . So > 0 is given and B is the ball in X centered at origin with fk (t) ∈ F (t, x∗ (t), x∗ [ϕ]t ) + B , for a.e. t ∈ (0, T ), POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 9 and the same inclusion holds for f˜k (t) thanks to the convexity of F (t, x∗ (t), x∗ [ϕ]t )+ B . Accordingly, f ∗ (t) ∈ F (t, x∗ (t), x∗ [ϕ]t ) + B for a.e. t ∈ (0, T ). Since is arbitrary, one gets f ∗ ∈ PFp (x∗ ). The lemma is proved. Using the last lemma, we prove the following property for the solution operator. Lemma 3.2. Under the assumptions (A), (B) and (F), the solution operator Σ is closed. Proof. Let {xn } ⊂ Cϕ , xn → x∗ , zn ∈ Σ(xn ) and zn → z ∗ . We show that z ∗ ∈ Σ(x∗ ). Take fn ∈ PFp (xn ) such that zn (t) = Sα (t)ϕ(0) + Qα (fn )(t), (3.3) where Qα is defined in (2.8). By Lemma 3.1, we get that fn f ∗ ∈ Lp (0, T ; X) and f ∗ ∈ PFp (x∗ ). In addition, C(t) = {fn (t) : n ≥ 1} is relatively compact and then t (t − s)α−1 Pα (t − s)fn (s)ds χ({Qα (fn )(t)}) ≤ χ 0 t (t − s)α−1 ||Pα (t − s)||χ({fn (s)})ds ≤2 0 = 0, according to Proposition 2.5. Due to Proposition 2.2, {Qα (fn )} is equicontinuous. Then by the Arzela-Ascoli theorem, we have the relative compactness of {Qα (fn )}. Since fn (t) → f ∗ (t) for a.e. t ∈ (0, T ), one has Qα (fn ) → Qα (f ∗ ). Therefore, it follows from (3.3) that z ∗ (t) = Sα (t)ϕ(0) + Qα (f ∗ )(t), ∀t ∈ [0, T ], where f ∗ ∈ PFp (x∗ ). Thus z ∗ ∈ Σ(x∗ ). The proof is complete. Now we prove the main result of this section. Theorem 3.3. Let (A), (B) and (F) hold. Then problem (1.1)-(1.2) has at least one integral solution on (−∞, T ]. Proof. We first look for a compact convex set M such that Σ(M) ⊂ M. For x ∈ Cϕ and z ∈ Σ(x), we have t (t − s)α−1 ||Pα (t − s)||m(s)[||x(s)|| + |x[ϕ]s |B ]ds ||z(t)|| ≤ ||Sα (t)|| · ||ϕ(0)|| + 0 t ≤ SαT ||ϕ(0)|| + PαT (t − s) (α−1)p p−1 p−1 p t ≤ + PαT p−1 αp − 1 0 p−1 p T t αp−1 p p p (m(s)) [||x(s)|| + |x[ϕ]s |B ] ds 0 (3.4) thanks to the H¨ older inequality, here SαT = sup ||S(t)||, PαT = sup ||Pα (t)||. t∈[0,T ] 1 p p (m(s)) [||x(s)|| + |x[ϕ]s |B ] ds ds 0 SαT ||ϕ(0)|| p t∈[0,T ] 1 p , 10 CUNG THE ANH, TRAN DINH KE Taking into account that |x[ϕ]s |B ≤ K(s) sup ||x(r)|| + M (s)|ϕ|B r∈[0,s] ≤ KT sup ||x(r)|| + MT |ϕ|B , r∈[0,s] with KT = sup K(s), MT = sup M (s), s∈[0,T ] s∈[0,T ] we deduce from (3.4) that 1 p t ||z(t)|| ≤ C1 + C2 p p (m(s)) [(1 + KT ) sup ||x(r)|| + MT |ϕ|B ] ds r∈[0,s] 0 ≤ C1 + 2 p−1 p t 1 p p p p (m(s)) [(1 + KT ) ( sup ||x(r)||) + C2 r∈[0,s] 0 MTp |ϕ|pB ]ds , (3.5) where C1 = SαT ||ϕ(0)||, C2 = PαT p−1 αp − 1 p−1 p T αp−1 p . Since the right-hand side of (3.5) is nondecreasing, we get t ( sup ||z(r)||)p ≤ C3 + C4 r∈[0,t] p (m(s)) ( sup ||x(r)||)p ds, (3.6) r∈[0,s] 0 where C3 = 2p−1 C1 + 22(p−1) C2p MTp |ϕ|pB ||m||pLp (0,T ) , C4 = 22(p−1) C2p (1 + KT )p . Let ψ be the unique solution of the integral equation t p (m(s)) ψ(s)ds, t ≥ 0, ψ(t) = C3 + C4 0 1 and M0 = {x ∈ Cϕ : supr∈[0,t] ||x(r)|| ≤ (ψ(t)) p }. It is easy to see that M0 is a closed convex subset of Cϕ . If x ∈ M0 , then z ∈ M0 according to (3.6). Thus Σ(M0 ) ⊂ M0 . Now for k ≥ 1, let Mk = co Σ(Mk−1 ). Then we see that Mk is closed convex and • Mk ⊂ Mk−1 , for all k ≥ 1; • Mk is equicontinuous due to Proposition 2.2. Mk . Then M is nonempty, convex and equicontinuous. In addition, Let M = k≥1 Σ(M) ⊂ M. We show that M is a compact set. This will be done if χ(M(t)) = 0 for all t ∈ [0, T ], by the Arzela-Ascoli theorem. To this end, we prove that µk (t) → 0 POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 11 as k → ∞, where µk (t) = χ(Mk (t)). Observing that µk (t) = χ(Σ(Mk−1 )(t)) t ≤χ 0 (t − s)α−1 Pα (t − s)PFp (Mk−1 )(s)ds t ≤4 0 (t − s)α−1 χ (Pα (t − s)PFp (Mk−1 )(s)) ds, thanks to Proposition 2.6, here for Λ ⊂ L1 (0, T ; X), t t Λ(s)ds := { f (s)ds : f ∈ Λ}. 0 0 If the semigroup S(·) is compact, so is Pα (·) and then µk (t) = 0. In the opposite case, using (F)(3) we have t (t − s)α−1 ||Pα (t − s)||k(s)[χ(Mk−1 (s)) + sup χ(Mk−1 [ϕ](s + θ))]ds µk (t) ≤ 4 θ≤0 0 t ≤ 4PαT (t − s)α−1 k(s)[χ(Mk−1 (s)) + sup χ(Mk−1 (r))]ds r∈[0,s] 0 t ≤ 8PαT (t − s)α−1 k(s) sup χ(Mk−1 (r))ds r∈[0,s] 0 t ≤ 8PαT (t − s) (α−1)p p−1 p−1 p p = r∈[0,s] 0 p−1 αp − 1 p−1 p T p (k(s)) ( sup µk−1 (r)) ds ds 0 8PαT 1 p t 1 p t αp−1 p p p (k(s)) ( sup µk−1 (r)) ds , r∈[0,s] 0 where we have used the H¨ older inequality. Since the last expression is nondecreasing in t, we have t ( sup µk (r))p ≤ C5 r∈[0,t] where C5 = (8PαT )p reads (k(s))p ( sup µk−1 (s))p ds, p−1 αp−1 p−1 (3.7) r∈[0,s] 0 T αp−1 . Let νk (t) = (supr∈[0,t] µk (r))p , then (3.7) t (k(s))p νk−1 (s)ds. νk (t) ≤ C5 0 Since {νk (t)} is decreasing, one can take the limit of the last inequality to get t (k(s))p ν∞ (s)ds, ν∞ (t) ≤ C5 0 where ν∞ (t) = lim νk (t). This implies that ν∞ (t) = 0 by using the Gronwall k→∞ 1 inequality. Taking into account that 0 ≤ µk (t) ≤ (νk (t)) p , we have lim µk (t) = k→∞ 0, ∀t ∈ [0, T ], as desired. We have proved that M ⊂ Cϕ is a compact convex set. Consider Σ : M → P(M). Applying Theorem 2.10, we have Fix(Σ) = ∅. The proof is complete. 12 CUNG THE ANH, TRAN DINH KE 4. Existence of decay global solutions In this section, we prove the existence of decay global solutions to problem (1.1)(1.2). To do this, we will consider the solution operator Σ on the following space: BCϕγ = {y ∈ C([0, +∞); X) : y(0) = ϕ(0) and sup tγ ||y(t)|| < ∞}, t≥0 where γ is a positive number chosen later. This space is endowed with the supremum norm ||y||BC = sup ||y(t)||, t≥0 and it becomes a closed subspace of the Banach space BC = {y ∈ C([0, +∞); X) : ||y||BC < ∞}. To apply fixed point theorems for condensing multimaps, one of our main task is to construct a regular MNC on the space BC([0, +∞); X). We first recall some usual MNCs on C([0, T ]; X). For given L > 0 and D ⊂ C([0, T ]; X), put ωT (D) = sup e−Lt χ(D(t)), (4.1) t∈[0,T ] modT (D) = lim sup max δ→0 x∈D t,s∈[0,T ],|t−s|0 (4.3) T >0 is an MNC on BC([0, ∞); X). One can check that this MNC satisfies all properties given in Definition 2.3, however it is not regular. Indeed, we will testify this claim by choosing the sequence {fk } ⊂ BC([0, ∞); R) as follows   t ∈ [k, k + 1], 0, fk (t) = 2t − 2k, t ∈ [k, k + 21 ],   −2t + 2k + 2, t ∈ [k + 12 , k + 1]. Then it is obvious that {πT (fk )} is compact (converging to 0 in C([0, T ]; R)) for any T > 0. However, sup ||fk (t) − fl (t)|| = 1 for k = l, t≥0 and thus {fk } is not a Cauchy sequence in BC([0, ∞); R). This implies that ωT (πT ({fk })) = 0 and modT (πT ({fk })) = 0 for any T > 0, and therefore χ∞ ({fk }) = 0, but {fk } is non-compact. Thus, the MNC χ∞ is not regular. POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 13 To overcome this, we will use the following MNCs on BC([0, ∞); X) (see [4] in the case X = RN ): dT (D) = sup sup ||x(t)||, (4.4) x∈D t≥T d∞ (D) = lim dT (D), (4.5) T →∞ ∗ χ (D) = χ∞ (D) + d∞ (D). (4.6) We now prove the following result. Lemma 4.1. The MNC χ∗ defined by (4.6) is regular. Proof. Let D ⊂ BC([0, ∞); X) be a bounded set such that χ∗ (D) = 0. It is obvious that πT (D) is relatively compact in C([0, T ]; X). We show that D is relatively compact in BC([0, ∞); X). Let P BC([0, ∞); X) be the space of piecewise continuous and bounded functions on R+ , taking values in X. This is a Banach space with the norm ||x||P BC = sup ||x(t)||, t≥0 and contains BC([0, ∞); X) as a closed subspace. For > 0, since d∞ (D) = 0 one can take T > 0 such that supt≥T ||x(t)|| < 2 , ∀x ∈ D. This means that ||x − πT (x)||P BC < , ∀x ∈ D, 2 here πT (x) agrees with a function in P BC([0, ∞); X) by the following manner πT (x) = x(t), t ∈ [0, T ], 0, t > T. Now since πT (D) is a compact set in C([0, T ]; X), we can write N πT (D) ⊂ i=1 BT (xi , ), 2 (4.7) where xi ∈ C([0, T ]; X), i = 1, ..., N , the notation BT (x, r) stands for the ball in C([0, T ]; X) centered at x with radius r. Defining x ˆi (t) = xi (t), t ∈ [0, T ], 0, t > T, then {ˆ xi } N i=1 belong to P BC([0, ∞); X). We assert that N D⊂ B∞ (ˆ xi , ), i=1 here B∞ (x, r) is the ball in P BC([0, ∞); X) with center x and radius r. Indeed, let x ∈ D then by (4.7), there is a number k ∈ {1, ..., N } such that ||πT (x) − xk ||C < where · C , 2 is the norm in C([0, T ]; X). This implies that ||πT (x) − x ˆk ||P BC < 2 . 14 CUNG THE ANH, TRAN DINH KE Then ||x − x ˆk ||P BC ≤ ||x − πT (x)||P BC + ||πT (x) − x ˆk ||P BC ≤ + 2 = . 2 N Thus x ∈ B∞ (ˆ xk , ). We have D ⊂ B∞ (ˆ xi , ), and hence D is relatively compact i=1 in P BC([0, ∞); X). Since BC([0, ∞); X) and P BC([0, ∞); X) have the same norm, we conclude that D is a relatively compact set in BC([0, ∞); X). The proof is complete. On BCϕγ we will use the regular MNC χ∗ given by (4.6). We now prove that Σ keeps BCϕγ invariant for some γ > 0, i.e. Σ(BCϕγ ) ⊂ BCϕγ , and Σ is χ∗ -condensing on BCϕγ . To this end, we have to replace (A), (B) and (F) by stronger ones. Specifically, we assume that for γ ≤ p1 : (A*) The operator A satisfies (A) such that the semigroup S(·) generated by A is exponentially stable, i.e. ||S(t)|| ≤ M e−at , for some a, M > 0. (B*) The phase space B satisfies (B) with K ∈ BC([0, +∞); R+ ) and M being such that tγ M (t) = O(1) as t → ∞. (F*) The nonlinearity F satisfies (F) for all T > 0. Put K∞ = sup K(t), M∞ = sup M (t). t≥0 t≥0 Lemma 4.2. Let (A*), (B*) and (F*) hold. Then Σ(BCϕγ ) ⊂ BCϕγ for all γ ≤ p1 . Proof. Let x ∈ BCϕγ with ||x||BC = r > 0. Then we have |x[ϕ]t |B ≤ K(t) sup ||x(s)|| + M (t)|ϕ|B s∈[0,t] ≤ K∞ r + M∞ |ϕ|B , ∀t ≥ 0. On the other hand |x[ϕ]t |B ≤ K t 2 t 2 sup ||x(s)|| + M s∈[ 2t ,t] t 2 ≤ K∞ sup ||x(s)|| + M s∈[ 2t ,t] |x[ϕ] 2t |B (K∞ r + M∞ |ϕ|B ). Thus tγ |x[ϕ]t |B ≤ K∞ tγ sup ||x(s)|| + tγ M s∈[ 2t ,t] ≤ K∞ 2γ sup sγ ||x(s)|| + 2γ s∈[ 2t ,t] t 2 t 2 (K∞ r + M∞ |ϕ|B ) γ M t 2 (K∞ r + M∞ |ϕ|B ) = O(1) as t → ∞, γ thanks to (B*). We show that t ||Σ(x)(t)|| = O(1) as t → ∞, here ||Σ(x)(t)|| = sup{||ξ|| : ξ ∈ Σ(x)(t)}. (4.8) POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 15 Take the decomposition of Σ as: Σ(x) = Σ1 (x) + Σ2 (x), where Σ1 (x)(t) = Sα (t)ϕ(0), t Σ2 (x)(t) = { 0 (t − s)α−1 Pα (t − s)f (s)ds : f ∈ PFp (x)}. We have ||Σ1 (x)(t)|| ≤ ||Sα (t)|| · ||ϕ(0)||. This implies tγ ||Σ1 (x)(t)|| ≤ tγ ||Sα (t)|| · ||ϕ(0)|| ≤ CS ||ϕ(0)||tγ−α = o(1) as t → ∞, (4.9) due to Proposition 2.3 and the fact that γ ≤ p1 < α. It remains to deal with Σ2 . Let z ∈ Σ2 (x), we have t (t − s)α−1 ||Pα (t − s)||m(s)[||x(s)|| + |x[ϕ]s |B ]ds ||z(t)|| ≤ 0 t 2 = t−1 + 0 t (t − s)α−1 ||Pα (t − s)||m(s)[||x(s)|| + |x[ϕ]s |B ]ds + t 2 t−1 = I1 (t) + I2 (t) + I3 (t), thanks to (F)(2). Using Proposition 2.3, we get t 2 I1 (t) ≤ C (t − s)−1 m(s)ds, 0 thanks to the boundedness of ||x(s)||+|x[ϕ]s |B . Here and hereafter, we use notation C for a generic constant which may be changed from line to line. Using the H¨older inequality, one has p−1 p t 2 I1 (t) ≤ C||m||Lp (R+ ) p − p−1 (t − s) 0 = C||m||Lp (R+ ) (p − 1) p−1 p 1 2 p−1 − 1 p−1 p 1 t− p . Thus 1 tγ I1 (t) ≤ Ctγ− p = O(1) as t → ∞. For I2 (t), using Proposition 2.3 once again, we see that t−1 (t − s)−1 m(s)[||x(s)|| + |x[ϕ]s |B ]ds tγ I2 (t) ≤ Ctγ t 2 t−1 s−γ (t − s)−1 m(s)[sγ ||x(s)|| + sγ |x[ϕ]s |B ]ds ≤ Ctγ t 2 t−1 (t − s)−1 m(s)ds, ≤C t 2 (4.10) 16 CUNG THE ANH, TRAN DINH KE where we have used the boundedness of sγ ||x(s)|| + sγ |x[ϕ]s |B and the fact that s ≥ 2t . Then the H¨ older inequality gives t 2 tγ I2 (t) ≤ C||m||Lp (R+ ) 1 − p−1 p 1 − p−1 = O(1) as t → ∞. (4.11) Now for I3 (t), we have t tγ I3 (t) ≤ Ctγ (t − s)α−1 m(s)[||x(s)|| + |x[ϕ]s |B ]ds t−1 t s−γ (t − s)α−1 m(s)[sγ ||x(s)|| + sγ |x[ϕ]s |B ]ds = Ctγ t−1 ≤C t t−1 γ t (t − s)α−1 m(s)ds. t−1 Then using the H¨ older inequality again, we get γ t I3 (t) ≤ C t t−1 γ t ||m||Lp (R+ ) (t − s) (α−1)p p−1 p−1 p ds t−1 γ p−1 αp − 1 t =C ||m||Lp (R+ ) t−1 = O(1) as t → ∞. p−1 p (4.12) Therefore, it follows from (4.10)-(4.12) that tγ ||Σ2 (x)(t)|| = O(1) as t → ∞. (4.13) Finally, combining (4.9) and (4.13) yields tγ ||Σ(x)(t)|| = O(1) as t → ∞. The proof is complete. Take L in the definition of ωT in (4.1) such that t (t − s)α−1 ||Pα (t − s)||e−L(t−s) k(s)ds < 1. := 8 sup t>0 (4.14) 0 This is possible since we have t t (t − s)α−1 ||Pα (t − s)||e−L(t−s) k(s)ds ≤ C 0 (t − s)α−1 e−L(t−s) k(s)ds 0 t ≤ C||k||Lp (R+ ) (t − s) (α−1)p p−1 Lp − p−1 (t−s) e p−1 p ds 0 ∞ ≤ C||k||Lp (R+ ) s (α−1)p p−1 e Lp − p−1 s p−1 p ds 0 p−1 = C||k||Lp (R+ ) Lp = o(1) as L → ∞, αp−1 p Γ αp − 1 p−1 p−1 p thanks to the H¨ older inequality and the fact that αp − 1 > 0. We will prove the χ∗ -condensing property for Σ. POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 17 Lemma 4.3. Let (A*), (B*) and (F*) hold. Then Σ is χ∗ -condensing on BCϕγ with 0 < γ ≤ p1 . Proof. Let D ⊂ BCϕγ be a bounded set. We first show that d∞ (Σ(D)) = 0. Indeed, taking r > 0 such that ||x||BC ≤ r for all x ∈ D, we have tγ ||Σ(x)(t)|| = O(1) as t → ∞, according to the proof of Lemma 4.2. This means that ||Σ(x)(t)|| ≤ Ct−γ , ∀x ∈ D, for all large t. Equivalently, for a large T , one has dT (Σ(D)) ≤ CT −γ . Then d∞ (Σ(D)) = lim dT (Σ(D)) = 0. (4.15) T →∞ On the other hand, since πT (D) is bounded in C([0, T ]; X) one sees that PFp (D) is bounded in Lp (0, T ; X). Due to Proposition 2.2, we have the set πT (Σ(D)) = Sα (·)ϕ(0) + Qα ◦ PFp (πT (D)) is equicontinuous in C([0, T ]; X). Then modT (πT (Σ(D))) = 0. (4.16) It remains to estimate ωT (πT (D)). For t ∈ [0, T ], we have t χ(Σ(D)(t)) ≤ χ 0 (t − s)α−1 Pα (t − s)PFp (D)(s)ds t ≤4 0 (t − s)α−1 χ(Pα (t − s)PFp (D)(s))ds (4.17) due to Proposition 2.6. If the semigroup S(·) is compact, then χ(Σ(D)(t)) = 0, thanks to the fact that χ(Pα (t − s)PFp (D)(s)) = 0 for s ∈ (0, t). In the opposite case, we have χ(Pα (t − s)PFp (D)(s)) ≤ ||Pα (t − s)||χ(PFp (D)(s)) ≤ ||Pα (t − s)||k(s)[χ(D(s)) + sup χ(D[ϕ](s + θ))] θ≤0 ≤ ||Pα (t − s)||k(s)[χ(D(s)) + sup χ(D(r))], r∈[0,s] here we employ (F)(3) and the fact that D[ϕ](r) = {ϕ(r)} (singleton) for r ≤ 0. Plugging this into (4.17), we get t (t − s)α−1 ||Pα (t − s)||k(s)[χ(D(s)) + sup χ(D(r))]ds χ(Σ(D)(t)) ≤ 4 r∈[0,s] 0 t (t − s)α−1 ||Pα (t − s)||k(s) sup χ(D(r))ds. ≤8 r∈[0,s] 0 This implies that t e−Lt χ(Σ(D)(t)) ≤ 8 (t − s)α−1 ||Pα (t − s)||e−L(t−s) k(s) sup e−Lr χ(D(r))ds r∈[0,s] 0 t ≤ (t − s)α−1 ||Pα (t − s)||e−L(t−s) k(s)ds ωT (πT (D)). 8 0 18 CUNG THE ANH, TRAN DINH KE Thus t ωT (πT (Σ(D))) ≤ (t − s)α−1 ||Pα (t − s)||e−L(t−s) k(s)ds ωT (πT (D)). 8 sup t>0 0 (4.18) Since the inequalities (4.16) and (4.18) take place for all T > 0, their combination give χ∞ (Σ(D)) ≤ · χ∞ (D), (4.19) with t (t − s)α−1 ||Pα (t − s)||e−L(t−s) k(s)ds < 1, = 8 sup t>0 0 where χ∞ is defined by (4.3). Now it follows from (4.15) and (4.19) that χ∗ (Σ(D)) ≤ · χ∗ (D), where χ∗ is defined by (4.6). Now if χ∗ (D) ≤ χ∗ (Σ(D)) then χ∗ (D) ≤ · χ∗ (D). Since < 1, one gets that χ∗ (D) = 0 and thus D is relatively compact due to Lemma 4.1. The proof is complete. The following theorem is our main result. Theorem 4.4. Let (A*), (B*) and (F*) hold. Then problem (1.1)-(1.2) has at least one integral solution on (−∞, +∞) satisfying ||x(t)|| = O(t−γ ) as t → +∞, provided that t (t − s)α−1 ||Pα (t − s)||m(s)[1 + K(s)]ds < 1. sup t≥0 (4.20) 0 Proof. By Lemma 4.3, Σ is χ∗ -condensing on BCϕγ . We prove that there is a number R > 0 such that Σ(BR ) ⊂ BR , where BR is the ball in BCϕγ centered at origin with radius R. Assume to the contrary that for each n ∈ N, there exist xn ∈ BCϕγ and zn ∈ Σ(xn ) satisfying that ||xn ||BC ≤ n but ||zn ||BC > n. From the formulation of Σ, one can find fn ∈ PFp (xn ) such that t (t − s)α−1 Pα (t − s)fn (s)ds, ∀t > 0. zn (t) = Sα (t)ϕ(0) + 0 Then t (t − s)α−1 ||Pα (t − s)||m(s)[||xn (s)|| + |xn [ϕ]s |B ]ds. ||zn (t)|| ≤ ||Sα (t)|| · ||ϕ(0)|| + 0 Noting that |xn [ϕ]s |B ≤ K(s) sup ||xn (r)|| + M (s)|ϕ|B r∈[0,s] ≤ nK(s) + M (s)|ϕ|B , we obtain t (t − s)α−1 ||Pα (t − s)||m(s)[(1 + K(s))n + M (s)|ϕ|B ]ds ||zn (t)|| ≤ ||Sα (t)|| · ||ϕ(0)|| + 0 t (t − s)α−1 ||Pα (t − s)||m(s)[1 + K(s)]ds + W (t)|ϕ|B , ≤ M ||ϕ(0)|| + n 0 (4.21) POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 19 where t (t − s)α−1 ||Pα (t − s)||m(s)M (s)ds. W (t) = 0 Since tγ M (t) = O(1) as t → ∞, using similar arguments as in the proof of Lemma 4.2, we see that W (t) = O(1) as t → ∞, that is, W (t) ≤ C for all t > 0. Hence (4.21) deduces 1< ||zn ||BC ≤ sup n t>0 t (t − s)α−1 ||Pα (t − s)||m(s)[1 + K(s)]ds + 0 M ||ϕ(0)|| + C|ϕ|B . n Passing limits through the last relation, we get a contradiction to (4.20). In the next step, we show that Σ is a quasicompact multimap. Let K ⊂ BCϕγ be a compact set and {zn } ⊂ Σ(K). We prove that {zn } has a convergent subsequence. Let {xn } ⊂ K such that zn ∈ Σ(xn ). Then one can assume that xn → x∗ in BCϕγ , up to a subsequence. For given T > 0, take fn ∈ PFp (πT (xn )) such that zn (t) = Sα (t)ϕ(0) + Qα (fn )(t), ∀t ∈ [0, T ]. Since {fn (s)} ⊂ F (s, {xn (s), xn [ϕ]s }), one sees that {fn (s)} is relatively compact for a.e. s ∈ (0, T ). Thus {Qα (fn )(t)} is a compact set for all t ∈ [0, T ]. In addition, {Qα (fn )} is equicontinuous due to Proposition 2.2. Then {zn } is relatively compact in C([0, T ]; X). Equivalently, ωT ({zn }) = modT ({zn }) = 0. Since T > 0 is arbitrary, we have χ∞ ({zn }) = 0. On the other hand, since {xn } ⊂ BCϕγ is bounded, we use the arguments in Lemma 4.2 to get that d∞ (Σ({xn })) = 0. In particular, d∞ ({zn }) = 0. Thus χ∗ ({zn }) = χ∞ ({zn }) + d∞ ({zn }) = 0 and therefore {zn } is relatively compact in BCϕγ . Now we have Σ is a quasicompact multimap. Moreover, following Lemma 3.2, Σ has a closed graph. This infers that Σ has closed and compact values. Hence Σ is u.s.c due to Lemma 2.7. Now we are able to state that Σ : BR → Kv(BR ) has a nonempty compact fixed point set according to Theorem 2.9. The proof is complete. 5. An application Let Ω ⊂ Rm be a closed convex set. In this section, we will apply our abstract results to the following differential variational inequality (DVI): m 1 ∂t2 u(t, x) = ψ(Dx )u(t, x) + f˜(t, x, u(t, x), ut ) + ρ(x) hi (t, u(t, ·) )ηi (t), (5.1) i=1 η(t) = (η1 (t), ..., ηm (t)) ∈ Ω, x ∈ Rn , t > 0, (5.2) w − η(t), G(t, u(t, ·) ) + Q(η(t)) ≥ 0, ∀w ∈ Ω, t > 0, (5.3) n u(s, x) = ϕ(s, x), x ∈ R , s ≤ 0, (5.4) where ∂tα (α > 0) stands for the Caputo derivative of order α with respect to t, the notation ψ(Dx ) stands for a partial differential operator defined by aα Dxα , ψ(Dx ) = |α|≤N where aα ∈ C and Dxα = ∂ ∂x1 α1 ∂ ∂x2 α2 ... ∂ ∂xn αn , 20 CUNG THE ANH, TRAN DINH KE for a multi-index α = (α1 , α2 , ..., αn ) ∈ Nn with length |α| = α1 + α2 + ... + αn ; the nonlinearity function f˜ is given by f˜(t, x, u(t, x), ut ) = f˜1 (t, x, u(t, x)) 0 ν(θ, y)f˜2 (y, u(t + θ, y))dydθ, + µ(t, x) −∞ Rn ρ ∈ L2 (Rn ), and the functions hi , G, Q will be described later. Here · denotes the norm in L2 (Rn ) and ·, · stands for the inner product in Rm . Problem (5.1)-(5.4) is a model of control systems subject to constraints, in which u is the state function and η = (η1 , ..., ηm ) is the control function with constraints imposed by the variational inequality (5.3). The symbol of ψ(Dx ) is given by aα (iξ)α = ψ(ξ) = |α|≤N aα (iξ1 )α1 (iξ2 )α2 ...(iξn )αn . |α|≤N The operator A = ψ(Dx ) is considered to be elliptic, i.e. the principal part ψN (ξ) = α |α|=N aα (iξ) of its symbol satisfies ψN (ξ) = 0 for all 0 = ξ ∈ Rn . Let X = L2 (Rn ), D(A) = H N (Rn ) and B = CL2g . In this example, we assume that (H1) The function ψ satisfies (a) ω = supξ∈Rn Re ψ(ξ) < 0; (b) there is δ ∈ (0, π2 ] such that ψ(Rn ) ⊂ C\Σδ+ π2 , where Σδ+ π2 = {λ ∈ C : |arg λ| < δ + π2 }\{0}. Under (H1), (A, D(A)) generates a bounded analytic semigroup {S(t)}t≥0 on X (see [12, Theorem 5.15]). Furthermore, adopting the result in [19, Theorem 2.2] (for the case p = 2, B = I), we have ||S(t)|| ≤ Ceωt , ∀t > 0, where C is a positive constant, ω is the constant in assumption (H1). Now we give the description for the nonlinearities f˜, hi , G and Q: (H2) f˜1 : R+ × Rn × R → R, µ : R+ × Rn → R, ν : (−∞, 0] × Rn → R and f˜2 : Rn × R → R such that (a) f˜1 is a continuous function such that f˜1 (t, x, 0) = 0 and |f˜1 (t, x, z1 ) − f˜1 (t, x, z2 )| ≤ m1 (t)|z1 − z2 | for all z1 , z2 ∈ R, here m1 ∈ Lp (R+ ) with p > 2; (b) µ ∈ Lp (R+ ; L2 (Rn )); (c) ν is continuous and satisfies |ν(t, x)| ≤ Cν eν0 t for all t ∈ (−∞, 0], x ∈ Rn , here 0 < ν0 < 1; (d) f˜2 is continuous and |f˜2 (y, z)| ≤ (y)|z| for ∈ L2 (Rn ). (H3) hi : R+ ×R+ → R, i = 1, ..., m, are continuous functions such that |hi (t, r)| ≤ ξ(t)r with ξ ∈ Lp (R+ ). (H4) G : R+ × R+ → Rm and Q : Ω → Rm are continuous functions such that (a) G is uniformly bounded, i.e. |G(t, r)| ≤ G0 for all t, r ∈ R+ , where G0 is a positive constant; (b) Q is monotone, i.e. w1 − w2 , Q(w1 ) − Q(w2 ) ≥ 0, for all w1 , w2 ∈ Ω; POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 21 w − v0 , Q(w) > 0. |w|2 w∈Ω,|w|→∞ For v ∈ X, w ∈ B, let f1 : R+ × X → X, f2 : R+ × B → X defined by f1 (t, v)(x) = f˜1 (t, x, v(x)) (c) there exists v0 ∈ Ω such that lim 0 ν(θ, y)f˜2 (y, w(θ, y))dydθ. f2 (t, w)(x) = µ(t, x) Rn −∞ Concerning f1 , we have ||f1 (t, v1 ) − f1 (t, v2 )|| ≤ m1 (t)||v1 − v2 ||. This implies that χ(f1 (t, V )) ≤ m1 (t)χ(V ), for all bounded set V ⊂ X, (5.5) where χ is the Hausdorff MNC on L2 (Rn ). Choose B = CL2g (described in (2.9)) with r = − ν10 ln(1 − ν0 ) and g(s) = eν0 s . Regarding f2 , using (H2)(b)-(H2)(d), we get that 2 0 ν(θ, y)f˜2 (y, w(θ, y))dydθ ||f2 (t, w)||2 = ||µ(t, ·)||2 Rn −∞ 2 0 ≤ ||µ(t, ·)||2 Cν2 eν0 s (y)|w(s, y)|dyds Rn −∞ 2 0 ≤ ||µ(t, ·)||2 Cν2 || ||2 ≤ 1 ||µ(t, ·)||2 Cν2 || ||2 ν0 1 = ||µ(t, ·)||2 Cν2 || ||2 ν0 ≤ e ν0 2 s e ν0 2 s ||w(s, ·)||ds −∞ 0 eν0 s ||w(s, ·)||2 ds −∞ 0 −r eν0 s ||w(s, ·)||2 ds + −r −∞ 1 ||µ(t, ·)||2 Cν2 || ||2 ||w||2C([−r,0];X) + ν0 −r eν0 s ||w(s, ·)||2 ds , −∞ thanks to the H¨ older inequality and the choosing of r. Then we have Cν ||f2 (t, w)|| ≤ √ ||µ(t, ·)|| · || || · |w|B . ν0 (5.6) On the other hand, for any bounded set W ⊂ B, we see that f2 (t, W ) ⊂ {λµ(t, ·) : λ ∈ R}, that is, f2 (t, W ) lies in an one-dimensional subspace of X. Hence χ(f2 (t, W )) = 0. (5.7) As far as the variational inequality (5.3) is concerned, we set Θ(z) = SOL(K, z + Q) := {v ∈ K : w − v, z + Q(v) ≥ 0, ∀w ∈ K}. It is known that (see [23, Proposition 6.2]), under (H4)(b)-(H4)(c), for each z ∈ Rm , Θ(z) is a nonempty, closed and convex set. Furthermore, Θ is a closed multimap and there exists Q0 > 0 such that |y| ≤ Q0 (1 + |z|), ∀y ∈ Θ(z). (5.8) 22 CUNG THE ANH, TRAN DINH KE So it follows from (H4)(a) that |y| ≤ Q0 (1 + G0 ), ∀y ∈ Θ(G(t, r)); ∀t, r ∈ R+ . (5.9) Note that Θ is a closed, locally bounded multimap with convex and compact values, then Θ is u.s.c. For v ∈ X, put m H(t, v)(x) = {ρ(x) hi (t, ||v||)ηi (t) : η(t) ∈ Θ(G(t, ||v||))}. i=1 Then H : R+ × X → P(X) has closed convex and compact values. Since Θ is u.s.c and hi , i = 1, ..., m, are continuous maps, we see that the composition multimap H is u.s.c. Obviously, for each bounded set V ⊂ X, one has H(t, V ) ⊂ {λρ(·) : λ ∈ R}, thanks to (5.8). Thus H(t, V ) is contained in an one dimensional subspace of X and then χ(H(t, V )) = 0. Moreover, we have the following estimate 1 2 m h2i (t, ||v||) ||H(t, v)|| ≤ ||ρ|| ≤ √ · sup{|η(t)| : η(t) ∈ Θ(G(t, ||v||))} i=1 mQ0 (1 + G0 )||ρ||ξ(t)||v||, thanks to (H3) and (5.9), here ||H(t, v)|| = sup{||w|| : w ∈ H(t, v)}. Let F : R+ × X × B → Kv(X) such that F (t, v, w)(x) = f1 (t, v)(x) + f2 (t, w)(x) + H(t, v)(x), It follows from (5.5) and (5.7) that χ(F (t, V, W )) ≤ χ(f1 (t, V )) + χ(f2 (t, W )) + χ(H(t, V )) ≤ m1 (t)χ(V ). Thus F fulfills (F) with k(t) = m1 (t) and m(t) = max{m1 (t) + √ Cν || || mQ0 (1 + G0 )||ρ||ξ(t), √ ||µ(t, ·)||}. ν0 As far as the phase space is concerned, for B = CL2g with r = − ν10 ln(1 − ν0 ) and g(s) = eν0 s , one sees that (2.10)-(2.11) are satisfied with G(s) = g(s). Then B verifies (B1)-(B3) with K(t) = 1, 1+ √1 ν0 √ 1 M (t) = e−ν0 r − max{e− 2 ν0 t , 1 + 1 e− 2 ν0 t , e−ν0 t , 0 ≤ t ≤ r, t > r. e−ν0 r ν0 (1 − e−ν0 t )}, 0 ≤ t ≤ r, t > r, thanks to the expressions of K and M in (2.12)-(2.13). This implies that B satisfies condition (B*). By the above description for (5.1)-(5.4), we can apply Theorem 4.4 to get the existence of decay global solutions with decay rate described by tγ ||u(t, ·)|| = O(1) as t → ∞, for 0 < γ ≤ p1 . POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 23 Acknowledgements. This work was done while the authors were visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM) as research fellows. The authors would like to thank the Institute for its hospitality and support. References [1] N. Abada, M. Benchohra and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations 246 (2009), 3834-863. [2] R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina and B.N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkh¨ auser, Boston-Basel-Berlin, 1992. [3] C.T. Anh and T.D. Ke, On nonlocal problems for retarded fractional differential equations in Banach spaces, Fixed Point Theory (2012), accepted for publication. [4] J. Banas and L. Olszowy, On a class of measures of noncompactness in Banach algebras and their application to nonlinear integral equations, J. Anal. Appl. 28 (2009), 475-498. [5] D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math. 108 (1998), 109-138. [6] T.A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006. [7] T.A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations, Dyn. Sys. Appl. 10 (2001), 89-116. [8] T. Cardinali and P. Rubbioni, Mild solutions for impulsive semilinear evolution differential inclusions, J. Appl. Funct. Anal. 1 (2006), 303-325. [9] N.M. Chuong and T.D. Ke, Generalized Cauchy problems involving nonlocal and impulsive conditions, J. Evol. Equ. 12 (2012), 367-392. [10] J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in Ll (µ, X), Proc. Amer. Math. Soc. 118 (1993), 447-453. [11] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. [12] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, in: Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. [13] E. P. Gatsori, L. Grniewicz, S. K. Ntouyas and G. Y. Sficas, Existence results for semilinear functional differential inclusions with infinite delay, Fixed Point Theory 6 (2005), 47-58. [14] C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay, Nonlinear Anal. 51 (2002), 765-782. [15] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Iclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter, Berlin, New York, 2001. [16] T.D. Ke, V. Obukhovskii, N.C. Wong and J.C. Yao, On a class of fractional order differential inclusions with infinite delays, Appl. Anal. 92 (2013), 115-137. [17] J.K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978), 11-41. [18] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay. Lecture Notes in Mathematics, Vol. 1473, Springer-Verlag, Berlin-Heidelberg-New York, 1991. [19] J. Liang and T.J. Xiao, Abstract degenerate Cauchy problems in locally convex spaces, J. Math. Anal. Appl. 259 (2001), 398-412. [20] V. Obukhovskii and J.-C. Yao, On impulsive functional differential inclusions with HilleYosida operators in Banach spaces, Nonlinear Anal. 73 (2010), 1715-728. [21] A. Ouahab, Fractional semilinear differential inclusions, Comp. Math. Appl. 64 (2012), 32353252. [22] T.I. Seidman, Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim. 25 (1987), 1173-1191. [23] J.-S. Pang and D.E. Steward, Differential variational inequalities, Math. Program. Ser. A 113 (2008), 345-424. [24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983. 24 CUNG THE ANH, TRAN DINH KE [25] P.D. Phung and L.X. Truong, On a fractional differential inclusion with integral boundary conditions in Banach space, Fract. Calc. Appl. Anal. 16 (2013), 538-558. [26] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, vol. 198, Sandiego, CA: Academic Press, 1999. [27] R.-N. Wang, D.-H. Chen and T.-J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations 252 (2012), 202-235. [28] J.R. Wang and Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal.: RWA 12 (2011) 3642-3653. [29] J.R. Wang, X.Z. Li and W. Wei, On controllability for fractional differential inclusions in Banach spaces, Opuscula Math. 32 (2012), 341-356. [30] Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comp. Math. Appl. 59 (2010), 1063-1077. Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam E-mail address: anhctmath@hnue.edu.vn (C.T.Anh), ketd@hnue.edu.vn (T.D.Ke) [...]... apply Theorem 4.4 to get the existence of decay global solutions with decay rate described by tγ ||u(t, ·)|| = O(1) as t → ∞, for 0 < γ ≤ p1 POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 23 Acknowledgements This work was done while the authors were visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM) as research fellows The authors would like to thank the Institute... D.-H Chen and T.-J Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J Differential Equations 252 (2012), 202-235 [28] J.R Wang and Y Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal.: RWA 12 (2011) 3642-3653 [29] J.R Wang, X.Z Li and W Wei, On controllability for fractional differential inclusions in Banach spaces, Opuscula... York, 1983 24 CUNG THE ANH, TRAN DINH KE [25] P.D Phung and L.X Truong, On a fractional differential inclusion with integral boundary conditions in Banach space, Fract Calc Appl Anal 16 (2013), 538-558 [26] I Podlubny, Fractional Differential Equations An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in... and Semilinear Differential Iclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications, vol 7, Walter de Gruyter, Berlin, New York, 2001 [16] T.D Ke, V Obukhovskii, N.C Wong and J.C Yao, On a class of fractional order differential inclusions with infinite delays, Appl Anal 92 (2013), 115-137 [17] J.K Hale and J Kato, Phase space for retarded equations with infinite delay,... ds 0 p−1 = C||k||Lp (R+ ) Lp = o(1) as L → ∞, αp−1 p Γ αp − 1 p−1 p−1 p thanks to the H¨ older inequality and the fact that αp − 1 > 0 We will prove the χ∗ -condensing property for Σ POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 17 Lemma 4.3 Let (A*), (B*) and (F*) hold Then Σ is χ∗ -condensing on BCϕγ with 0 < γ ≤ p1 Proof Let D ⊂ BCϕγ be a bounded set We first show that d∞ (Σ(D))... (converging to 0 in C([0, T ]; R)) for any T > 0 However, sup ||fk (t) − fl (t)|| = 1 for k = l, t≥0 and thus {fk } is not a Cauchy sequence in BC([0, ∞); R) This implies that ωT (πT ({fk })) = 0 and modT (πT ({fk })) = 0 for any T > 0, and therefore χ∞ ({fk }) = 0, but {fk } is non-compact Thus, the MNC χ∞ is not regular POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 13 To overcome... New York, 2000 [13] E P Gatsori, L Grniewicz, S K Ntouyas and G Y Sficas, Existence results for semilinear functional differential inclusions with infinite delay, Fixed Point Theory 6 (2005), 47-58 [14] C Gori, V Obukhovskii, M Ragni and P Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay, Nonlinear Anal 51 (2002), 765-782 [15]... ,t] ≤ K∞ 2γ sup sγ ||x(s)|| + 2γ s∈[ 2t ,t] t 2 t 2 (K∞ r + M∞ |ϕ|B ) γ M t 2 (K∞ r + M∞ |ϕ|B ) = O(1) as t → ∞, γ thanks to (B*) We show that t ||Σ(x)(t)|| = O(1) as t → ∞, here ||Σ(x)(t)|| = sup{||ξ|| : ξ ∈ Σ(x)(t)} (4.8) POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS 15 Take the decomposition of Σ as: Σ(x) = Σ1 (x) + Σ2 (x), where Σ1 (x)(t) = Sα (t)ϕ(0), t Σ2 (x)(t) = { 0 (t −... CUNG THE ANH, TRAN DINH KE 4 Existence of decay global solutions In this section, we prove the existence of decay global solutions to problem (1.1)(1.2) To do this, we will consider the solution operator Σ on the following space: BCϕγ = {y ∈ C([0, +∞); X) : y(0) = ϕ(0) and sup tγ ||y(t)|| < ∞}, t≥0 where γ is a positive number chosen later This space is endowed with the supremum norm ||y||BC = sup ||y(t)||,... Murakami and T Naito, Functional Differential Equations with Infinite Delay Lecture Notes in Mathematics, Vol 1473, Springer-Verlag, Berlin-Heidelberg-New York, 1991 [19] J Liang and T.J Xiao, Abstract degenerate Cauchy problems in locally convex spaces, J Math Anal Appl 259 (2001), 398-412 [20] V Obukhovskii and J.-C Yao, On impulsive functional differential inclusions with HilleYosida operators in Banach ... Theorem 4.4 to get the existence of decay global solutions with decay rate described by tγ ||u(t, ·)|| = O(1) as t → ∞, for < γ ≤ p1 POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS. .. order, require initial conditions related to u(0), u (0), etc., and the Caputo fractional derivative POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS satisfies these demands For... origin with fk (t) ∈ F (t, x∗ (t), x∗ [ϕ]t ) + B , for a.e t ∈ (0, T ), POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS and the same inclusion holds for f˜k (t) thanks to the