Abstract

The 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.

1. 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:where (0 < q < 1) is the Caputo fractional order derivative, x_t ∈ C([−τ, 0]; D(L)), x_t(ζ) = x(t + ζ) (ζ ∈ [−τ, 0]), φ ∈ C([−τ, 0]; D(L)), and H : [0, T] × C([−τ, 0]; X) ⟶ 2^Y is a multivalued map.

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 [1–3] and the papers [4–9]. 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, Fec̆kan 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. The readers can see [17–19] 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. The 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.

This 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.

2. 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) () be the Banach space of all Bochner integrable functions f from [0, T] to X satisfying , endowed with its standard norm.

Lemma 1 (see [20]). Assume that D ⊂ C([−τ, T]; X) is a bounded subset. If D is equicontinuous, then the convex closure of D, denoted by , is bounded and equicontinuous.

We introduce some facts about measures of noncompactness (MNC). Let χ(⋅) be the Hausdorff MNC in X; that is,

Then, 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.

Lemma 2 (see [22]). Let {h_n} ⊂ L¹(J; X) be a sequence satisfying that, for a.e. t ∈ J,where . Then, andfor each t ∈ J.

Definition 1. (see [10]). If f ∈ L¹(J; Y), q ≥ 0, then the Riemann–Liouville fractional integral of order q of f is defined bywith , where, and Γ(⋅) is the Gamma function.

Definition 2. (see [10]). If f ∈ C^k−1(J; Y), with and q ∈ (k − 1, k). Then, the q-order Caputo derivative of the function f is defined bySee [2, 11] for more results of the fractional calculus.
Let and and h ∈ L^p(J; Y). Consider the following system:where A and L satisfy that (F₁) A is linear and closed (F₂) L is linear, closed, and bijective, and D(L) ⊂ D(A) (F₃) L⁻¹ : Y ⟶ D(L) is continuousCombining (F₁) and (F₂) 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], Theorem 1.2]). In this paper, we suppose sup{‖T(t)‖, t ≥ 0} ≤ M with some M > 0.
Based on the semigroup {T(t), t ≥ 0} and operator L, we introduce two characteristic solution operators and , which are given bywherewith ρ_q(t) ≥ 0 and .
Obviously, and are equicontinuous for t > 0. According to (F₁)–(F₃), we also have

Definition 3. (see [13]). A mild solution of (10) is that a function x ∈ C(J; X), which satisfiesFor each x₀ ∈ D(L), φ ∈ C([−τ, 0]; D(L)), and h ∈ L^p(J; Y), denote u(⋅, x₀, h) by the mild solution of (10), which is also unique, and define the multivalued map G_φ : L^p(J; Y) ⟶ C([−τ, T]; X) byObviously, G_φh is the unique mild solution of the equation:The 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 , and the conditions (F₁) − (F₃) hold.(i)If U ⊂ C([−τ, 0]; D(L)) is relatively compact and K ⊂ L^p(J; Y) satisfy that ‖h(t)‖ ≤ k(t) for a.e. t ∈ J and all h ∈ K, where . Then G_U(K) is equicontinuous in C([−τ, T]; X).(ii)Let {h_n} ⊂ L^p(J; Y) and {x_n} ⊂ C([−τ, T); X) with x_n = G_φf_n be two sequences, and assume in addition that h_n ⟶ h weakly in L^p(J; Y) and x_n ⟶ x in C([−τ, T); X), we have x = G_φh.

In what follows, we introduce some notations and results about multivalued analysis. The 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.

Lemma 4 (see [21], Theorem 1.1.12). If ϕ with convex, closed, and compact values is closed and quasi-compact, we have that ϕ is u.s.c.

The 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.

3. Main Results

Suppose that H : J × C([−τ, 0]; X) ⟶ Y has convex closed values and (H₁) H(t, ⋅) is weakly u.s.c. for a.e. t ∈ J and has a L^p-integral selection for each . (H₂) H is uniformly L^p-integrable bounded, that is, for a.e t ∈ J and each , where . (H₃) There exists such that for a.e. t ∈ J and all bounded subsets B ⊂ C([−τ, 0]; X).

Let us define two multivalued maps byS : C([−τ, T]; X) ⟶ 2^{C([−τ,T];X)} by

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).

The following result gives the properties of Sel_H.

Lemma 6. Let Y be reflexive. If the hypotheses (H₁) and (H₂) are satisfied, then Sel_H is well defined, and it is weakly u.s.c. with weakly compact and convex values.

Proof. The proof can be obtained by the same argument as in Lemma 3.3 in [10].
For the sake of simplicity, write

Theorem 1. Let Y be reflexive and . If (F₁)–(F₃) and (H₁)–(H₃) are satisfied, then system (1) has at least a mild solution.

Proof. Let φ ∈ C([−τ, 0]; D(L)) be given and setwhere .
One sees that D₀ is bounded, convex, and closed. For each x ∈ D₀ and y ∈ S(D₀), there exists h ∈ Sel_H(x) such that y = G_φh. By (H₂), it follows thatThis implies that S(D₀) ⊂ D₀, which enable us to obtain that D_n+1 ⊂ D_n ⊂ ⋯ ⊂ D₀. Hence, D is convex and closed.
It follows from (H₂) thatWe 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_φSel_H(D_n) is equicontinuous. And thus, D_n+1 is also equicontinuous due to Lemma 1. Therefore, D is equicontinuous.
For given ϵ > 0, thanks to (4); it follows that there exists a sequence {h_n} ⊂ Sel_H(D_n) such thatFrom the definition of G_φ, we know thatFor t ∈ [−τ, 0], it is clear that χ(D_n+1(t)) = 0. Also, for t ∈ J and s < t, it follows from (H₂) thatTaking x_n ∈ D_n such that h_n ∈ Sel_H(x_n) and x_n = G_φh_n, thanks to (H₃), one hasTaking inequality (27) and Lemma 2 into account, we have that for t ∈ J,where μ₀ = sup_t∈Jμ(t). Since ϵ is arbitrary, then for t ∈ J,SetFor each 0 ≤ t₁ ≤ t,This means that is nondecreasing in [0, t]. Hence,Taking n ⟶ ∞, we getNow, applying Gronwall’s inequality with singularity, one hasTherefore, χ(D(t)) = 0 for each t ∈ [−τ, T], which yields that D(t) is relatively compact in X for each t ∈ [−τ, T].
Thus, 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_φh_n. Combining Lemma 2(ii) of [24] with Lemma 6, one can find h ∈ Sel_H(x) and a subsequence of {h_n}, such that weakly in L^p(J; Y). By Lemma 3(ii), we get that y = G_φh which implies y ∈ S(x). From this, we verify that S is closed. Thus, 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 h₀ ∈ Sel_H(x). For each y ∈ S(x), there exists h ∈ Sel_H(x) such that y = G_φh and putLet us define a function F : [0, 1] × S(x) ⟶ S(x) byDenoting by h ∗ (t) := h(t)χ_[0,λT](t) + h₀(t)χ_[λT,T](t), it follows that h ∗ ∈ Sel_H(x) and thus x_y ∈ S(x). From this, we conclude that F is well defined, i.e., F(λ, y) ∈ S(x) for each (λ, y) ∈ [0, 1] × S(x). One can see that F is continuous, and F(0, y) = G_φh₀ and F(1, y) = y for every y ∈ S(x). Accordingly, S has a contractible value.
Therefore, by applying the fixed point theorem of Lemma 5 on S, the conclusion of the theorem holds, which completes the proof.
Denoted byWe state the results of the compactness of Φ(φ) and the property of the map Φ.

Theorem 2. Let all conditions in Theorem 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_φh_n. We verify that {x_n} is relatively compact and closed. As argued in the proof of Theorem 1, the sequence {x_n} is equicontinuous. Moreover, we know thatFrom this, we know χ({x_n(t)}) = 0 for each t ∈ J. Since {x_n|_[−τ,0]} = {φ}, one can deduce that χ({x_n(t)}) = 0 for each t ∈ [−τ, 0]. Hence, {x_n(t)} is relatively compact for each t ∈ [−τ, T]. Therefore, {x_n} is relatively compact in C([−τ, T]; X). Then, there exists of {x_n} such that . This together with Lemma 2(ii) of [24] and Lemma 6 implies that we can find a subsequence of {h_n} such that weakly in L^p(J; Y) with h ∈ Sel_H(x). Again by Lemma 3(ii), we obtain that x = G_φh and thus x ∈ Φ(φ). This enable us to get that {x_n} is closed in C([−τ, T]; X). Therefore, 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_φh_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 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,The last inequality implies that χ({x_n(t)}) = 0 for each t ∈ J due to Gronwall’s inequality with singularity. Therefore, 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. This completes the proof.

4. An Example

We apply the abstract results to the system of partial differential inclusion as follows:where , d > 0, φ ∈ C([−τ, 0]; L²(0, π)), u_t(ξ, x) = u(t + ξ, x), and are given functions such that h₁ is l.s.c., h₂ is u.s.c., and for each , and there exists such thatfor each .

Taking X = Y = L²(0, π), we can rewrite system (40) abstractly in the form of system (1), where the operators A and L are defined by Au = −u_xx and Lu = u − u_xx with D(A) = D(L) = {u ∈ X : u, u_x are absolutely continuous, u_xx ∈ X and u(t, 0) = u(t, π) = 0}, H : [0, d] × C([−τ, 0]; X) ⟶ 2^Y is given byfor each . If we assume that H satisfies (H₃), then assumptions (H₁), (H₂) (), and (H₃) hold.