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

Authors: Qiaomin Xiang; Pengxian Zhu
Journal: Journal of Function Spaces
 (2020)

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## 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.


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## Body

## 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:(1)D0cqLxt+Axt∈Ht,xt,t∈J:=0,T,xt=φt,t∈−τ,0,where D0cq (0 < q < 1) is the Caputo fractional order derivative, xt ∈ C([−τ, 0]; D(L)), xt(ζ) = x(t + ζ) (ζ ∈ [−τ, 0]), φ ∈ C([−τ, 0]; D(L)), and H : [0, T] × C([−τ, 0]; X) ⟶ 2Y 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 Rδ-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: Section2 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.LetC([α, β]; X) be the Banach space of all continuous functions from [α, β] to X, equipped with the sup-norm and Lp(J; X) (1≤p<∞) be the Banach space of all Bochner integrable functions f from [0, T] to X satisfying ∫0Tftdt<∞, endowed with its standard norm.Lemma 1 (see [20]).
Assume thatD ⊂ C([−τ, T]; X) is a bounded subset. If D is equicontinuous, then the convex closure of D, denoted by conv¯D, is bounded and equicontinuous.We introduce some facts about measures of noncompactness (MNC). Letχ(⋅) be the Hausdorff MNC in X; that is,(2)χD=infε>0:Dhas a finiteε−net,D⊂X.Then, the Hausdorff MNC satisfies the following properties (see [21] for details).(i)
LetE : X ⟶ X be a bounded linear operator and D ⊂ X, then the following inequality holds:(3)χED≤EχD.(ii)
Assume thatB ⊂ X is bounded; then, for every ϵ > 0, we can find {Bn} ⊂ B such that(4)χB≤2χBn+ε.We recall some additional properties of the Hausdorff MNC.Lemma 2 (see [22]).
Let {hn} ⊂ L1(J; X) be a sequence satisfying that, for a.e. t ∈ J,(5)hnt≤θtuniformly forhn,where θ∈L1J;ℝ+. Then, χhnt∈L1J;ℝ+ and(6)χ∫0thnsds≤2∫0tχhnsdsfor 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 by(7)Ltqft:=wq∗ft=1Γq∫0tt−sq−1fsds,t∈J,q>0,with Lt0ft=ft, where(8)wqs=1Γqsq−1,t>0,0,s≤0,w0s=0, and Γ(⋅) is the Gamma function.Definition 2.
(see [10]). If f ∈ Ck−1(J; Y), wk−q∗f∈Wk,1J;Y with k∈ℕ and q ∈ (k − 1, k). Then, the q-order Caputo derivative of the function f is defined by(9)D0cqft:=dkdtkLtk−qft−∑i=0k−1fi0wi+1t.See [2, 11] for more results of the fractional calculus.
Letp>1 and pq>1 and h ∈ Lp(J; Y). Consider the following system:(10)D0cqLxt+Axt=ht,t∈J,x0=x0∈DE,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 continuous
Combining (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 ℬ⋅ and ℛ⋅, which are given by(11)ℬt=∫0∞L−1ρqθTtqθdθ,ℛt=q∫0∞θL−1ρqθTtqθdθ,where(12)ρqt=1πq∑m=1∞−tm−1Γ1+qmm!sinmπq,t∈0,∞,with ρq(t) ≥ 0 and ∫0∞ρqtdt=1.
Obviously,ℬt and ℛt are equicontinuous for t > 0. According to (F1)–(F3), we also have(13)ℬty≤ML−1y,ℛty≤qML−1Γ1+qy,t≥0,y∈Y.Definition 3.
(see [13]). A mild solution of (10) is that a function x ∈ C(J; X), which satisfies(14)xt=ℬtLx0+∫0tt−sq−1ℛt−shsds,t∈J.
For eachx0 ∈ D(L), φ ∈ C([−τ, 0]; D(L)), and h ∈ Lp(J; Y), denote u(⋅, x0, h) by the mild solution of (10), which is also unique, and define the multivalued map Gφ : Lp(J; Y) ⟶ C([−τ, T]; X) by(15)Gφht=φt,t∈−τ,0,ut,φ0,h,t∈J.
Obviously,Gφh is the unique mild solution of the equation:(16)D0cqLxt+Axt=ht,t∈J,xt=φt,t∈−τ,0.
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 thatp>1,pq>1, and the conditions (F1) − (F3) hold.(i)
IfU ⊂ C([−τ, 0]; D(L)) is relatively compact and K ⊂ Lp(J; Y) satisfy that ‖h(t)‖ ≤ k(t) for a.e. t ∈ J and all h ∈ K, where k∈LpJ;ℝ+. Then GU(K) is equicontinuous in C([−τ, T]; X).(ii)
Let {hn} ⊂ Lp(J; Y) and {xn} ⊂ C([−τ, T); X) with xn = Gφfn be two sequences, and assume in addition that hn ⟶ h weakly in Lp(J; Y) and xn ⟶ x in C([−τ, T); X), we have x = Gφh.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).
Ifϕ 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).
LetU be a Banach space and D ⊂ U be a compact and convex subset. Assume that the multivalued map ϕ : D ⟶ 2D with closed contractible values is u.s.c., we have that ϕ has at least one fixed point.


## 3. Main Results

Suppose thatH : 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 H⋅,w has a Lp-integral selection for each w∈C−τ,0;X.(H2) H is uniformly Lp-integrable bounded, that is,(17)Ht,w:=suph;h∈Ht,w≤ηt,for a.et ∈ J and each w∈C−τ,0;X, where η∈LpJ;ℝ+.(H3) There exists μ∈CJ;ℝ+ such that χHt,B≤μtsups∈−τ,0χBs for a.e. t ∈ J and all bounded subsets B ⊂ C([−τ, 0]; X).Let us define two multivalued mapsSelH:C−τ,T;X⟶2LpJ;Y by(18)SelHx:=h∈LpJ;Yandht∈Ht,xtfor a.e.t∈J, S : C([−τ, T]; X) ⟶ 2C([−τ,T];X) by(19)Sx=Gφ∘SelHx.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 ∈ SelH(x).The following result gives the properties of SelH.Lemma 6.
LetY be reflexive. If the hypotheses (H1) and (H2) are satisfied, then SelH 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(20)a=ML−1L,b=qML−1Γ1+q,c=Tq−1/pp−1pq−11−1/p.Theorem 1.
LetY be reflexive and p>1andpq>1. If (F1)–(F3) and (H1)–(H3) are satisfied, then system (1) has at least a mild solution.Proof.
Letφ ∈ C([−τ, 0]; D(L)) be given and set(21)D0=x∈C−τ,T;X;xt=φtfort∈−τ,0andxt≤Cφ,ηfor allt∈J,Dn+1=conv¯SDn,n=0,1,…,D=∩n=0∞Dn,where Cφ,η=aφC−τ,0;DL+bcηLp0,T.
One sees thatD0 is bounded, convex, and closed. For each x ∈ D0 and y ∈ S(D0), there exists h ∈ SelH(x) such that y = Gφh. By (H2), it follows that(22)yt≤aφC−τ,0;DL+b∫0tt−sq−1ηsds=Cφ,η,t∈J.
This implies thatS(D0) ⊂ D0, which enable us to obtain that Dn+1 ⊂ Dn ⊂ ⋯ ⊂ D0. Hence, D is convex and closed.
It follows from (H2) that(23)ht≤ηt,for a.e.t∈Jandh∈SelHDn.
We also see that this inequality remains true uniformly forh ∈ SelH(Dn). Applying Lemma 3 (i), we obtain that S(Dn) = GφSelH(Dn) is equicontinuous. And thus, Dn+1 is also equicontinuous due to Lemma 1. Therefore, D is equicontinuous.
For givenϵ > 0, thanks to (4); it follows that there exists a sequence {hn} ⊂ SelH(Dn) such that(24)χDn+1t=χSDnt≤2χGφhnt+ε,t∈−τ,T.
From the definition ofGφ, we know that(25)Gφhnt=φt,t∈−τ,0,ℬtLφ0+∫0tt−sq−1ℛt−shnsds,t∈J.
Fort ∈ [−τ, 0], it is clear that χ(Dn+1(t)) = 0. Also, for t ∈ J and s < t, it follows from (H2) that(26)t−sq−1ℛt−shns≤bt−sq−1ηs.
Takingxn ∈ Dn such that hn ∈ SelH(xn) and xn = Gφhn, thanks to (H3), one has(27)χt−sq−1ℛt−shns≤bt−sq−1μssupσ∈0,sχxnσ,fort∈J,s<t.
Taking inequality (27) and Lemma 2 into account, we have that for t ∈ J,(28)χDn+1t≤2χGφhnt+ϵ≤4b∫0tt−sq−1μssupσ∈0,sχxnσds+ϵ≤4bμ0∫0tt−sq−1supσ∈0,sχDnσds+ϵ,where μ0 = supt∈Jμ(t). Since ϵ is arbitrary, then for t ∈ J,(29)χDn+1t≤4bμ0∫0tt−sq−1supσ∈0,sχDnσds.
Set(30)gt=4bμ0∫0tt−sq−1supσ∈0,sχDnσds.
For each 0 ≤t1 ≤ t,(31)gt−gt1=4bμ0∫0t1ξq−1supσ∈0,t−ξχDnσ−supσ∈0,t1−ξχDnσdξ+∫t1tt−sq−1supσ∈0,sχDnσds≥0.
This means thatgt is nondecreasing in [0, t]. Hence,(32)supσ∈0,tχDn+1σ≤4bμ0∫0tt−sq−1supσ∈0,sχDnσds,t∈J.
Takingn ⟶ ∞, we get(33)supσ∈0,tlimn⟶∞χDn+1σ≤4bμ0∫0tt−sq−1supσ∈0,slimn⟶∞χDnσds,t∈J.
Now, applying Gronwall’s inequality with singularity, one has(34)limn⟶∞χDnt=0,t∈J.
Therefore,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 thatS 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 {(xn, yn)} ⊂ Gra(S) with (xn, yn) ⟶ (x, y), we can find {hn} ⊂ Lp(J; Y) such that hn ∈ SelH(xn) and yn = Gφhn. Combining Lemma 2(ii) of [24] with Lemma 6, one can find h ∈ SelH(x) and a subsequence hnk of {hn}, such that hnk⟶h weakly in Lp(J; Y). By Lemma 3(ii), we get that y = Gφh 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 ofS, we can deduce that S has a closed value. Given x ∈ D and h0 ∈ SelH(x). For each y ∈ S(x), there exists h ∈ SelH(x) such that y = Gφh and put(35)xyt=ℬtLφ0+∫0λTt−sq−1ℛt−shsds+∫λTtt−sq−1ℛt−sh0sds,λ∈0,1,t∈λT,T.
Let us define a functionF : [0, 1] × S(x) ⟶ S(x) by(36)Fλ,yt=yt,t∈−τ,λT,xyt,t∈λT,T.
Denoting byh ∗ (t) := h(t)χ[0,λT](t) + h0(t)χ[λT,T](t), it follows that h ∗ ∈ SelH(x) and thus xy ∈ 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) = Gφh0 and F(1, y) = y for every y ∈ S(x). Accordingly, S has a contractible value.
Therefore, by applying the fixed point theorem of Lemma5 on S, the conclusion of the theorem holds, which completes the proof.
Denoted by(37)Φφ=x∈C−τ,T;X;xis a mild solution of1withxt=φt,t∈−τ,0.
We state the results of the compactness of Φ(φ) and the property of the map Φ.Theorem 2.
Let all conditions in Theorem1 hold, we have that the set Φ(φ) is compact and Φ is u.s.c.Proof.
Letφ be given and {xn} ⊂ Φ(φ), then there exists hn ∈ SelH(xn) such that xn = Gφhn. We verify that {xn} is relatively compact and closed. As argued in the proof of Theorem 1, the sequence {xn} is equicontinuous. Moreover, we know that(38)supσ∈0,tχxnσ≤2bμ0∫0tt−sq−1supσ∈0,sχxnσds,t∈J.
From this, we knowχ({xn(t)}) = 0 for each t ∈ J. Since {xn|[−τ,0]} = {φ}, one can deduce that χ({xn(t)}) = 0 for each t ∈ [−τ, 0]. Hence, {xn(t)} is relatively compact for each t ∈ [−τ, T]. Therefore, {xn} is relatively compact in C([−τ, T]; X). Then, there exists xnk of {xn} such that xnk⟶x. This together with Lemma 2(ii) of [24] and Lemma 6 implies that we can find a subsequence hnk of {hn} such that hnk⟶h weakly in Lp(J; Y) with h ∈ SelH(x). Again by Lemma 3(ii), we obtain that x = Gφh and thus x ∈ Φ(φ). This enable us to get that {xn} is closed in C([−τ, T]; X). Therefore, the compactness of Φ(φ) remains true.
LetK ⊂ C([−τ, 0]; D(L)) be an compact set and {xn} ⊂ Φ(K), take {φn} ⊂  K and hn ∈ SelH(xn) such that xn = Gφhn and xn ∈ Φ(φn). 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 {xn} is equicontinuous. On the one hand, one has χ({xn(t)}) = χ({φn(t)}) = 0 for each t ∈ [−τ, 0] since the compactness of K. On the other hand, it follows that for each t ∈ J,(39)χxnt≤aχφn0+2bμ0∫0tt−sq−1supσ∈0,sχxnσds=2bμ0∫0tt−sq−1supσ∈0,sχxnσds.
The last inequality implies thatχ({xn(t)}) = 0 for each t ∈ J due to Gronwall’s inequality with singularity. Therefore, χ({xn(t)}) = 0 for each [−τ, T], which yields that {xn(t)} is a relatively compact subset of X. Hence, {xn} 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:(40)D0cqut,x−uxxt,x−uxxt,x∈h1t,x,utx,h2t,x,utx,t,x∈0,d×0,π,ut,0=ut,π=0,t∈0,d,ut,x=φt,x,t,x∈−τ,0×0,π,where 1/2<q<1, d > 0, φ ∈ C([−τ, 0]; L2(0, π)), ut(ξ, x) = u(t + ξ, x), and hi:0,d×0,π×C−τ,0;L20,π⟶ℝi=1,2 are given functions such that h1 is l.s.c., h2 is u.s.c., and h1t,x,v≤h2t,x,v for each t,x,v∈0,d×0,π×C−τ,0;L20,π, and there exists l∈L∞0,T;ℝ+ such that(41)maxh1t,x,v,h2t,x,v≤lt,for each t,x,v∈0,d×0,π×C−τ,0;L20,π.TakingX = 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 = −uxx and Lu = u − uxx with D(A) = D(L) = {u ∈ X : u, ux are absolutely continuous, uxx ∈ X and u(t, 0) = u(t, π) = 0}, H : [0, d] × C([−τ, 0]; X) ⟶ 2Y is given by(42)Ht,v=h∈Y:hx∈h1t,x,v,h2t,x,va.e. in0,π,for each t,v∈0,d×C−τ,0;X. If we assume that H satisfies (H3), then assumptions (H1), (H2) (ηt=πlt), and (H3) hold.From the definition ofA, it follows that the spectrum σ(A) consists of the simple eigenvalues λk=k2,k∈ℕ+ with the corresponding eigenfunctions ekx=2/πsinkx, x ∈ [0, π]. Thus, we can see that(43)Aek=λkek=k2ek,ek,ej=δkj,fork,j=1,2,3,…,where δkj = 0 if k ≠ j and δkj = 1 if k = j. Define the projection operator Pk by(44)Pku=u,ekek,u∈X,then(45)I=∑k=1∞Pk.Therefore, operatorA can be explicitly expressed by(46)Au=AIu=A∑k=1∞Pku=∑k=1∞Au,ekek=∑k=1∞k2u,ekek,u∈DA,and operator L is given by(47)Lu=I+Au=∑k=1∞1+k2u,ekek,u∈DL.One can easily verify the following assertions:(i)
A and L are linear and closed(ii)
L−1 is continuous(iii)
For anyu∈X,(48)L−1u=∑k=1∞11+k2u,ekek,−AL−1u=∑k=1∞−k21+k2u,ekek.From the definition of −AL−1, we have(49)Ttu=∑k=1∞e−k2/1+k2tu,ekek,u∈X.Obviously,T(t) is equicontinuous for t > 0 and(50)Ttu2≤∑k=1∞e−1/2tu,ekek2=e−t∑k=1∞u,ekek2=e−tu2,which yields that Tt≤e−1/2t≤1.Accordingly, all hypotheses in Theorems1 and 2 are satisfied. Therefore, system (40) has at least a mild solution and the set of all mild solution of system (40) is compact.



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