Some New Results for the Sobolev-Type Fractional Order Delay Systems with Noncompact Semigroup

/e topological structure of solution sets for the Sobolev-type fractional order delay systems with noncompact semigroup is studied. Based on a fixed point principle for multivalued maps, the existence result is obtained under certain mild conditions. With the help of multivalued analysis tools, the compactness of the solution set is also obtained. Finally, we apply the obtained abstract results to the partial differential inclusions.


Introduction
Let (X, ‖·‖) and (Y, ‖·‖) be Banach spaces. Given two operators L : D(L) ⊂ X ⟶ Y and A : D(A) ⊂ X ⟶ Y , consider the Sobolev-type fractional order delay system: 0 D q c (Lx(t)) + Ax(t) ∈ H t, x t , t ∈ J :� [0, T], where 0 D q c (0 < q < 1) is the Caputo fractional order derivative, Fractional calculus (an extension of ordinary calculus) is an important branch of mathematical analysis, and its theoretical and practical applications have been greatly developed. Many mathematicians have devoted themselves to the study of fractional differential equations for a long time and have made important contributions to theory and application of fractional differential equations. For more details about fractional calculus and fractional differential equations, we refer to the monographs [1][2][3] and the papers [4][5][6][7][8][9]. At present, much interest has been developing in the fractional differential inclusions, and we refer readers to [10,11] and the references therein.
Recently, the solvability and controllability of fractional order systems have been investigated by lots of authors. For a class of Sobolev-type fractional order evolution equations with nonlocal conditions, Debbouche and Nieto [12] obtained that the equation had at least one mild solution, which was proved to be unique. For the Sobolev-type fractional functional evolution equations, Feckan et al. [13] gave the controllability results by applying the Schauder fixed point theorem [14,15]. For the fractional evolution inclusions, by the multivalued analysis techniques, Wang et al. [10] proved the existence results. Wang and Zhou [16] investigated the existence and controllability results for the fractional semilinear differential inclusion with the Caputo fractional derivative by means of the Bohnenblust-Karlin's fixed point theorem. e readers can see [17][18][19] for more results of the fractional order system. Summarizing the above settings, one cause is that the nonlinearity with the compact value is upper semicontinuous, which has been proved to be too harsh and difficult to meet in practical application. e other cause is that the semigroup is compact. Our interest in system (1) is to analyze the topological properties (such as compactness, acyclicity, and R δ -structure) of its solution set. Unfortunately, the results on this direction are less known. Considering these, we aim to establish the topological properties of the set of all mild solutions for system (1). In this article, when the semigroup is noncompact and H is weakly upper semicontinuous to the solution variable, we firstly concern with the existence of mild solution of system (1), which can be obtained by using a fixed point theorem of multivalued maps (see Lemma 5). And then, by means of the theory of measure of noncompactness and a singular version of Gronwall inequality, we prove that the set of all mild solutions of system (1) is compact and the corresponding solution map is the upper semicontinuity. We should like to emphasize that system (1) involve the delay term, which makes it difficult for us to estimate the noncompactness measure of the solution set.
is paper is structured as follows: Section 2 presents some concepts and facts. Section 3 deals with the nonemptiness and compactness of the set of all mild solutions for system (1) and the upper semicontinuity of the corresponding solution map. We illustrate our abstract results by an example of partial differential inclusion in Section 4.

Preliminaries
In this section, we will introduce some notations and describe some results, which are used later in the paper.
Let C([α, β]; X) be the Banach space of all continuous functions from [α, β] to X, equipped with the sup-norm and L p (J; X) (1 ≤ p < ∞) be the Banach space of all Bochner integrable functions f from [0, T] to X satisfying T 0 ‖f(t)‖dt < ∞, endowed with its standard norm.
We introduce some facts about measures of noncompactness (MNC). Let χ(·) be the Hausdorff MNC in X; that is, en, the Hausdorff MNC satisfies the following properties (see [21] for details).
(i) Let E : X ⟶ X be a bounded linear operator and D ⊂ X, then the following inequality holds: (ii) Assume that B ⊂ X is bounded; then, for every ϵ > 0, we can find {B n } ⊂ B such that We recall some additional properties of the Hausdorff MNC.
See [2,11] for more results of the fractional calculus. Let p > 1 and pq > 1 and h ∈ L p (J; Y). Consider the following system: where A and L satisfy that (F 1 ) A is linear and closed (F 2 ) L is linear, closed, and bijective, and Combining (F 1 ) and (F 2 ) with the closed graph theorem, we know that the linear operator − AL − 1 : Y ⟶ Y is bounded. From this, we deduce that − AL − 1 generates a semigroup {T(t), t ≥ 0}. It is easy to know that T(t) is equicontinuous for t > 0 (see [23], eorem 1.2]). In this paper, we suppose sup Based on the semigroup {T(t), t ≥ 0} and operator L, we introduce two characteristic solution operators B(·) and R(·), which are given by where 2 Journal of Function Spaces Obviously, B(t) and R(t) are equicontinuous for t > 0. According to (F 1 )-(F 3 ), we also have Definition 3 (see [13]). A mild solution of (10) is that a function x ∈ C(J; X), which satisfies For each by the mild solution of (10), which is also unique, and define the multivalued map G φ : Obviously, G φ h is the unique mild solution of the equation e following assertion can be derived from the proof of Lemmas 2.4 and 3.1 of [10] (see also Lemma 2.6 of [24]). Lemma 3. Assume that p > 1, pq > 1, and the conditions ( x n � G φ f n be two sequences, and assume in addition that h n ⟶ h weakly in L p (J; Y) and In what follows, we introduce some notations and results about multivalued analysis. e definitions of a multivalued map to be upper semicontinuous (u.s.c.), lower semicontinuous (l.s.c.), and weakly upper semicontinuous (weakly u.s.c.), closed and quasi-compact, one can see [10] for details.
We recall the characterization about u.s.c. maps.
e following assertion provides us with a fixed point theorem.
Lemma 5 (see [25], Lemma 1). Let U be a Banach space and D ⊂ U be a compact and convex subset. Assume that the multivalued map ϕ : D ⟶ 2 D with closed contractible values is u.s.c., we have that ϕ has at least one fixed point.

Main Results
Suppose that H : J × C([− τ, 0]; X) ⟶ Y has convex closed values and (H 1 ) H(t, ·) is weakly u.s.c. for a.e. t ∈ J and H(·, w) has a L p -integral selection for each w ∈ C([− τ, 0]; X). (H 2 ) H is uniformly L p -integrable bounded, that is, for a.e t ∈ J and each w In this paper, x ∈ C([− τ, T]; X) is a mild solution of (1) which means that x is a mild solution of (16) with h ∈ Sel H (x). e following result gives the properties of Sel H .

Lemma 6. Let Y be reflexive. If the hypotheses (H 1 ) and (H 2 ) are satisfied, then Sel H is well defined, and it is weakly u.s.c. with weakly compact and convex values.
Proof. e proof can be obtained by the same argument as in Lemma 3.3 in [10].
For the sake of simplicity, write Journal of Function Spaces Theorem 1. Let Y be reflexive and p > 1 and pq > 1.
It follows from (H 2 ) that We also see that this inequality remains true uniformly for h ∈ Sel H (D n ). Applying Lemma 3 (i), we obtain that S(D n ) � G φ Sel H (D n ) is equicontinuous. And thus, D n+1 is also equicontinuous due to Lemma 1. erefore, D is equicontinuous.
For given ϵ > 0, thanks to (4); it follows that there exists a sequence {h n } ⊂ Sel H (D n ) such that From the definition of G φ , we know that For t ∈ [− τ, 0], it is clear that χ(D n+1 (t)) � 0. Also, for t ∈ J and s < t, it follows from (H 2 ) that Taking x n ∈ D n such that h n ∈ Sel H (x n ) and x n � G φ h n , thanks to (H 3 ), one has Taking inequality (27) and Lemma 2 into account, we have that for t ∈ J, where μ 0 � sup t∈J μ(t). Since ϵ is arbitrary, then for t ∈ J, (30) Journal of Function Spaces (32) Taking n ⟶ ∞, we get Now, applying Gronwall's inequality with singularity, one has lim n⟶∞ χ D n (t) � 0, t ∈ J. (34) us, D is compact due to the Arzela-Ascoli theorem. Now, we will verify that S is u.s.c. and has closed contractible values. It is easy to see that S(D) ⊂ D. Moreover, by the compactness of D, we see that G is quasi-compact. Taking {(x n , y n )} ⊂ Gra(S) with (x n , y n ) ⟶ (x, y), we can find {h n } ⊂ L p (J; Y) such that h n ∈ Sel H (x n ) and y n � G φ h n . Combining Lemma 2(ii) of [24] with Lemma 6, one can find h ∈ Sel H (x) and a subsequence h n k of {h n }, such that h n k ⟶ h weakly in L p (J; Y). By Lemma 3(ii), we get that y � G φ h which implies y ∈ S(x). From this, we verify that S is closed. us, thanks to Lemma 4, one concludes that S is u.s.c.
As a consequence of the closedness of S, we can deduce that S has a closed value. Given x ∈ D and h 0 ∈ Sel H (x). For each y ∈ S(x), there exists h ∈ Sel H (x) such that y � G φ h and put Let us define a function F : x y (t), t ∈ (λT, T].
Denoted by We state the results of the compactness of Φ(φ) and the property of the map Φ.
Theorem 2. Let all conditions in eorem 1 hold, we have that the set Φ(φ) is compact and Φ is u.s.c.
Proof. Let φ be given and {x n } ⊂ Φ(φ), then there exists h n ∈ Sel H (x n ) such that x n � G φ h n . We verify that {x n } is relatively compact and closed. As argued in the proof of eorem 1, the sequence {x n } is equicontinuous. Moreover, we know that en, there exists x n k of {x n } such that x n k ⟶ x. is together with Lemma 2(ii) of [24] and Lemma 6 implies that we can find a subsequence h n k of {h n } such that h n k ⟶ h weakly in L p (J; Y) with h ∈ Sel H (x). Again by Lemma 3(ii), we obtain that x � G φ h and thus x ∈ Φ(φ). is enable us to get that {x n } is closed in C([− τ, T]; X). erefore, the compactness of Φ(φ) remains true.
Let K ⊂ C([− τ, 0]; D(L)) be an compact set and {x n } ⊂ Φ(K), take {φ n } ⊂ K and h n ∈ Sel H (x n ) such that x n � G φ h n and x n ∈ Φ(φ n ). To apply Lemma 4, it is suffice to show that Φ is closed and quasi-compact. It is easy to Journal of Function Spaces know that Φ is closed. So, we only show that Φ is quasi-compact. Following the same method as above, we get {x n } is equicontinuous. On the one hand, one has χ({x n (t)}) � χ({φ n (t)}) � 0 for each t ∈ [− τ, 0] since the compactness of K. On the other hand, it follows that for each t ∈ J, e last inequality implies that χ({x n (t)}) � 0 for each t ∈ J due to Gronwall's inequality with singularity. erefore, χ({x n (t)}) � 0 for each [− τ, T], which yields that {x n (t)} is a relatively compact subset of X. Hence, {x n } is compact due to Arzela-Ascoli theorem. is completes the proof.
From the definition of A, it follows that the spectrum σ(A) consists of the simple eigenvalues λ k � k 2 , k ∈ N + with the corresponding eigenfunctions e k (x) � ��� 2/π √ sin(kx), x ∈ [0, π]. us, we can see that Ae k � λ k e k � k 2 e k , e k , e j � δ kj , for k, j � 1, 2, 3, . . . , where δ kj � 0 if k ≠ j and δ kj � 1 if k � j. Define the projection operator P k by P k u � u, e k e k , u ∈ X,