Research ArticleExact Controllability for Hilfer Fractional Differential InclusionsInvolving Nonlocal Initial Conditions

Jun Du ,1,2 Wei Jiang ,1 Denghao Pang,1 and Azmat Ullah Khan Niazi 1

1School of Mathematical Sciences, Anhui University, Hefei 230601, China2Department of Applied Mathematics, Huainan Normal University, Huainan 232038, China

Correspondence should be addressed to Jun Du; djwlm@163.com and Wei Jiang; jiangwei@ahu.edu.cn

Received 27 March 2018; Accepted 11 July 2018; Published 19 August 2018

Academic Editor: Dimitri Volchenkov

Copyright © 2018 Jun Du et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The exact controllability results for Hilfer fractional differential inclusions involving nonlocal initial conditions are presented andproved. By means of the multivalued analysis, measure of noncompactness method, fractional calculus combined with thegeneralized Monch fixed point theorem, we derive some sufficient conditions to ensure the controllability for the nonlocal Hilferfractional differential system. The results are new and generalize the existing results. Finally, we talk about an example tointerpret the applications of our abstract results.

1. Introduction

Fractional calculus generalizes the standard integer calculusto arbitrary order. It provides a valuable tool for the descrip-tion of memory and hereditary properties of diversifiedmaterials and processes. In the past twenty years, the subjectof the fractional calculus is picking up considerable popular-ity and importance. We can refer to the monographs ofDiethelm and Freed [1], Kilbas et al. [2], Miller and Ross[3], Podlubny [4], and Zhou [5]. Fractional differentialequations and inclusions involving Caputo derivative orRiemann-Liouville derivative have obtained more and moreresults (see [6–15]). Recently, Hilfer [16] initiated anextended Riemann-Liouville fractional derivative, namedHilfer fractional derivative, which interpolates Caputo frac-tional derivative and Riemann-Liouville fractional derivative.This operator appeared in the theoretical simulation ofdielectric relaxation in glass forming materials. Hilfer et al.[17] initially presented linear differential equations with thenew Hilfer fractional derivative and applied operationalcalculus to solve such generalized fractional differentialequations. Subsequently, Furati et al. [18] and Gu andTrujillo [19] generalized to consider nonlinear problemsand proved the existence, nonexistence, and stability resultsfor initial value problems of nonlinear fractional differential

equations with Hilfer fractional derivative in a suitableweighted space of continuous functions.

Control theory is an interdisciplinary branch of engineer-ing and mathematics that deals with influence behavior ofdynamical systems. Controllability is one of the fundamentalconcepts in mathematical control theory, it means that it ispossible to steer a dynamical system from an arbitrary initialstate to arbitrary final state using the set of admissiblecontrols. Recently, the controllability conditions for variouslinear and nonlinear integer or fractional order systems havebeen considered in many papers by using different methods[20–33] and the references. There have also been someresults [20–24, 32, 33] about the investigations of the exactcontrollability of systems represented by nonlinear evolutionequations in infinite dimensional space. But when thesemigroup or the control action operator B is compact, thenthe controllability operator is also compact and the applica-tions of exact controllability results is just restricted to thefinite dimensional space [20]. Therefore, we investigate theexact controllability of the fractional evolution systems onlyinvolving noncompact semigroups.

The nonlocal initial problems have been initially pro-posed by Byszewski et al. [34, 35] to generalize the study ofthe canonical initial problem, comes from physical science.For instance, it used to determine the unknown physical

HindawiComplexityVolume 2018, Article ID 9472847, 13 pageshttps://doi.org/10.1155/2018/9472847

parameters in some inverse heat condition problems. Ithas been found that the nonlocal initial condition is moreexact to describe the nature phenomena than the classicalinitial condition, since more data is taken into account,therefore abating the negative influences induced by a pos-sible inaccurate single estimation taken at the start time.For more discussion on this type of differential equationsand inclusions, we can see papers [36–42] and referencesgiven therein.

Boucherif and Precup [36] proved the existence formild solutions to the following nonlocal initial problemfor first-order evolution equations using Schaefer fixedpoint theorem:

x′ t + Ax t = f t, x t , t ∈ J ,

x 0 + 〠m

k=1akx tk = 0, 0 < t1 < t2 <⋯ < tm < b,

1

where A D A ⊆ X→ X is the infinitesimal generatorof a C0-semigroup T t t≥0 on a Banach space X andf J × X→ X is a known function.

Liang and Yang [33] concerned the controllability forthe following fractional integrodifferential evolution equa-tions involving nonlocal conditions using the Monch fixedpoint theorem:

Dqx t + Ax t = f t, x t ,Gx t + Bu t , t ∈ J ,

x 0 = 〠m

k=1ckx tk , 0 < t1 < t2 <⋯ < tm < b,

2

where Dq is the Caputo derivative of order q ∈ 0, 1 ,−A D A ⊆ X → X is the infinitesimal generator of C0-semi-group T t t≥0 of uniformly bounded linear operator, thecontrol function u is known in L2 J ,U ; U is a Banachspace, B is a linear bounded operator from U to X; f is aknown function and Gx t = t

0K t, s x s ds is a Volterraintegral operator.

Du et al. [43] generalized the results of [33] and gavethe controllability for a new class of fractional neutral inte-grodifferential evolution equations with infinite delay andnonlocal conditions using Mönch fixed point theorem.However, it should be emphasized that to the best of ourknowledge, the exact controllability of Hilfer fractional dif-ferential system has not been investigated yet. Motivated by[19, 30, 33, 36, 43], in this paper, we concern the controlla-bility of the following fractional differential inclusionsinvolving a more general fractional derivative with nonlocalinitial conditions:

Dp,q0+ x t ∈ Ax t + F t, x t + Bu t , t ∈ 0, a ,

I 1−q 1−p0+ x 0+ = 〠

n

i=1aix ti , 0 < t1 < t2 <⋯ < tn < a,

3

where Dp,q0+ is the Hilfer fractional derivative of order p

(p obeys 1/2 < p ≤ 1) and type q (q obeys 0 ≤ q ≤ 1) which willbe given in Section 2; E is separable and A is bounded, so S ·is a uniformly continuous semigroup and S t = eAt . Thenonlinear term F J × E→ 2E \ ∅ is multivalued function.Let J = 0, a , J′ = 0, a , a > 0 are two finite intervals of ℝ;ai ∈ℝ, ai ≠ 0 i = 1, 2,… , n , n ∈N The control function utakes values in L2 J ,U , with U as a Banach space; B isa linear bounded operator from U to E.

In this paper, by means of a concrete nonlocal function,we do not have to suppose the compactness and Lipschitzconditions on the nonlocal function but only assume that aii = 1, 2,… , n satisfy the hypothesis (H0) (see Section 3).Furthermore, the proofs of our main results are based onfractional calculus theory, the multivalued analysis, measureof noncompactness method, in addition to the O’Regan-Precup fixed point theorem, which is an extension of theMönch fixed point theorem.

2. Preliminaries and Notations

Let C J′, E and C J , E denote the space of E-valuedcontinuous functions from J′ to E and from J to E,respectively; Let r = p + q − pq.

Define Y = x ∈ C J′, E : limt→0+ t1−rx t exists and is

finite}, involving the norm · Y defined by x Y =supt∈J′ t1−r x t . Then, Y is a Banach space. We alsonote that

(1) When r = 1, then Y = C J , E and · Y = · ;

(2) Let x t = tr−1y t for t ∈ J′, x ∈ Y if and only ify ∈ C J , E , and x Y = y .

For γ > 0, define BYγ J′ = x ∈ Y x Y ≤ γ Thus BY

γ isa bounded closed and convex subset of Y .

Let Bγ J = y ∈ C J , E : y ≤ γ , Then Bγ is a closedball of the space C J , E with the radius γ and centerat 0. And Bγ is also a bounded closed and convex subsetof C J , E .

Next, we list some definitions and properties in fractionalcalculus, multivalued analysis, semigroup theory, and mea-sure of noncompactness.

The following definitions concerning fractional calculuscan be found in the books [2–4, 16].

Definition 1. The fractional integral for function f from lowerlimit 0 and order α can be expressed by

Iα0+ f t = 1Γ α

t

0

f s

t − s 1−α ds, α > 0, t > 0, 4

where Γ is the gamma function, and right side of upperequality is point-wise defined on 0, +∞ .

2 Complexity

Definition 2. The Riemann-Liouville derivative of order αwith the lower limit 0 for function f 0, +∞ →ℝ can beexpressed by

R−LDα0+ f t = 1

Γ n − α

dn

dtn

t

0

f s

t − s α+1−n ds, t > 0, 0 ≤ n − 1 < α < n

5

Definition 3. The Caputo derivative of order α for functionf 0, +∞ →ℝ can be denoted by

CDα0+ f t = R−LDα f t − 〠

n−1

k=0

tk

kf k 0 , t > 0, 0 ≤ n − 1 < α < n

6

Definition 4. The left Hilfer derivative of order 0 < p ≤ 1 andtype 0 ≤ q ≤ 1 of function f is defined by

Dp,q0+ f t = Iq 1−p

0+ D I 1−q 1−p0+ f t , 7

where D≔ d/dt

Remark 1.

(i) The operator Dp,q0+ can be written as

Dp,q0+ f t = Iq 1−p

0+ D I1−r0+ f t

= Iq 1−p0+ Dr f t , r = p + q − pq

8

(ii) When q = 0 and 0 < p ≤ 1, the Hilfer fractionalderivative coincides with the Riemann-Liouvillederivative:

Dp,00+ f t = d

dtI1−p0+ f t = R−LDp

0+ f t 9

(iii) When q = 1 and 0 < p ≤ 1, the Hilfer fractionalderivative coincides with the Caputo derivative:

Dp,10+ f t = I1−p0+

ddt

f t = CDp0+ f t 10

Let P E be the set of all nonempty subsets of E. Wewill use the following notations:

P cl E ≔ Y ∈P E Y is closed ,P b E ≔ Y ∈P E Y is bounded ,P cv E ≔ Y ∈P E Y is convex ,P cp E ≔ Y ∈P E Y is compact

11

Remark 2.

(i) A measurable function u J → E is Bochner integra-ble if and only if u is Lebesgue integrable.

(ii) A multivalued map F E→ 2E is said to be convexvalued (closed valued) if F u is convex (closed)for all u ∈ E is said to be bounded on bounded setsif F B = ∪u∈B is bounded in E for all B ∈P b E

(iii) A multivalued map F is said to be upper semicontin-uous (u.s.c.) on E if for each u0 ∈ E, the set F u0 is anonempty closed subset of E, and if for each opensubset Ω of E containing F u0 , there exists an openneighborhood ∇ of u0 such that F ∇ ⊆Ω

(iv) A multivalued map F is said to be completelycontinuous if F B is relatively compact for everyB ∈P b E If the multivalued map F is completelycontinuous with nonempty compact values, thenF is u.s.c. if and only if F has a closed graph, thatis, un → u, yn → y, yn ∈ F u0 imply y ∈ F u Wesay that F has a fixed point if there is u ∈ E, suchthat u ∈ F u

(v) A multivalued map F J →P cl E is said to be mea-surable if for each u ∈ E, the function y J → Rdefined by y t = d u, F t = inf u − z , z ∈ F tis measurable.

Lemma 1 (see [44]). Let E be a Banach space. The multiva-lued map satisfies the following: for each t ∈ J , F t, · : E→P b,cl,cv E is u.s.c.; for each x ∈ E, the function F ·, x : E→P b,cl,cv E is strongly measurable and the set SF,x = f ∈ L1J , E : f t ∈ F t, x t , for a e t ∈ J is nonempty. Let Γbe a linear continuous mapping from L1 J , E to C J , E ,then the operator

Γ ∘ SF C J , E →P b,cl,cv C J , E , x↦ Γ ∘ SF x = Γ SF,x ,12

is a closed graph operator in C J , E × C J , EConsider

Kp t = p∞

0θξp θ S tpθ dθ, t ≥ 0, 13

where ξp θ = 1/p θ−1−1/2ωp θ−1/2 , and

ωp θ = 1π〠∞

n=1−1 n−1θ−pn−1

Γ np + 1n

sin nπp , θ ∈ 0,∞ ,

14

and ξp is a probability density function defined on 0,∞ , thatis ∞

0 ξp θ dθ = 1

3Complexity

Remark 3 (see [2]). As we all know from [2] that Kp can bedenoted by the Mittag-Leffler functions:

Kp t = 〠∞

i=0

Aitpi

Γ pi + p15

The following essential propositions can be found in thepapers [19, 38].

Lemma 2. If S t ≤N , t ∈ J , then for each x ∈ E,

(i) Kp t x ≤ N/Γ p x ;

(ii) tp−1Kp t x ≤ Ntp−1/Γ p x and Iq 1−p0+ tp−1

Kp t x ≤ Ntr−1/Γ r x .

Lemma 3 (Example 2.1.3 [45]).For each G ⊂ C J , E and t ∈ J , define G t = g t :

g ∈G If G is equicontinuous and bounded, then β G tis continuous on J and

β G =maxt∈J

β G t 16

Here, β is the Hausdorff noncompact measure on Edefined on every bounded subset U of Banach space E by

β U = inf ε > 0,U has a f inite ε − net in E 17

Lemma 4 (see Lemma 5 [46]). Let G ⊆ L1 J , E be a countablesubset with g t ≤ φ t , for almost everywhere t ∈ J and anyg ∈G, where φ ∈ L1 J ,ℝ+ . Then

βJg t dt g ∈G ≤

Jβ G t dt 18

To end this section, we reintroduce the O’Regan-Precupfixed point theorem.

Lemma 5 (see Theorem 3.2 [47]). LetD be a subset of Banachspace E which is closed and convex. Ω is a relatively opensubset of D, and T Ω→P cv D Suppose graph (T ) isclosed, T maps compact sets into relatively compact sets,and that for some x0 ∈Ω, the following two conditionsare satisfied:

Z ⊆D, Z ⊂ conv x0 ∪T Z

Z = G with G ⊆ Z countable⇒Z compact,

x ∉ 1 − λ x0 + λT x , for all x ∈Ω\Ω, λ ∈ 0, 1

19

Then T has a fixed point.

3. Controllability Results

We first consider linear Hilfer fractional differential equa-tions of the form

Dp,q0+ x t = Ax t + h t , t ∈ J′,

I1−r0+ x 0+ = 〠n

i=1aix ti ,

20

where h ∈ C J , E .Assume that there exists the bounded operator K E→ E

given by

K ≔ I − 〠n

i=1aiΦp,r A, ti

−1

, 21

where Φp,r A, t = Iq 1−p0+ tp−1Kp t .

By means of [48], we can present the sufficient conditionsfor the existence and boundedness of the operator K .

Lemma 6. If the hypothesis (H0) ∑ni=1 ai < Γ r /N t1

1−r

holds, the operator K defined in (21) exists and is bounded.

Proof 1. From the hypothesis H0 , we have

〠n

i=1aiΦp,r A, ti ≤

NΓ r

〠n

i=1ai · tr−1i < Ntr−11

Γ r〠n

i=1ai < 1

22

By operator spectrum theorem, the operator K ≔I − ∑n

i=1aiΦp,r A, ti−1 exists and is bounded. Furthermore,

by Neumann expression, we get

K ≤ 〠∞

n=0〠n

i=1aiΦp,r A, ti

n

= 11 − ∑n

i=1aiΦp,r A, ti

≤1

1 − Ntr−11 /Γ r ∑ni=1 ai

23

Using Lemma 6 and [19], we give the following definitionof mild solution for the Hilfer fractional system (20)involving nonlocal initial conditions.

Definition 5. A function x ∈ C J′, X is called a mild solu-tion for the Hilfer fractional system (20) if it satisfiesthe following equation:

x t =Φp,r A, t 〠n

i=1aiK h ti + h t , t ∈ J′, 24

where h t = t0Φp,p A, t − s h s ds.

4 Complexity

Remark 4. By virtue of [19], a mild solution for Hilferfractional evolution (20) with the initial condition is

x t =Φp,r A, t I1−r0+ x 0+ +t

0Φp,p A, t − s h s ds 25

Specially,

x ti =Φp,r A, ti I1−r0+ x 0+ +ti

0Φp,p A, ti − s h s ds

26Using (20) and (26), we get

I − 〠n

i=1aiΦp,r A, ti · I1−r0+ x 0+ = 〠

n

i=1ai

ti

0Φp,p A, ti − s h s ds

27Since I −∑n

i=1aiΦp,r A, ti exists a bounded inverse operatorwhich is denoted by K , so

I1−r0+ x 0+ = 〠n

i=1aiK

ti

0Φp,p A, ti − s h s ds 28

And hence,

x t =Φp,r A, t 〠n

i=1aiK h ti + h t , t ∈ J′, 29

it is indeed (24).

Using Definition 5, we give a new definition of themild solution for the Hilfer fractional nonlocal differentialinclusions (3) as follows:

Definition 6. A function x ∈ C J′, E is called a mild solu-tion of the Hilfer fractional nonlocal differential inclusions(3) if for any u ∈ L2 J ,U , the following integral equationis satisfied:

x t =Φp,r A, t 〠n

i=1aiK f ti + f t , t ∈ J′, 30

where f t = t0Φp,p A, t − s Bu s + f s ds, Φp,p A, t =

tp−1Kp t and f ∈ SF,x.

To present and prove the main results of this paper,we enumerate the following hypotheses:

(H1) E is a separable Banach space, A is bounded, henceS · = eA· is a uniformly continuous semigroup.

(H2) The multivalued map F J × E→P b,cl,cv E sat-isfies the following:

(2a) For every t ∈ J , F t, · : E→P b,cl,cv is u.s.c., foreach x ∈ E, the function F ·, x : J →P b,cl,cv isstrongly measurable. The set SF,x = f ∈ L1J , E : f t ∈ F t, x t , for almost everywheret ∈ J is nonempty;

(2b) There exists a function ξ1 ∈ L1/p1 J ,ℝ+ ,p1 ∈ 0, p and a continuous nondecreasingfunction ψ 0,∞ → 0,∞ , such that forany t, x ∈ J × E, we have F t, x t = sup

f t : f t ∈ F t, x t ≤ ξ1 t ψ x Y ,limm→∞ inf ψ m /m =Λ <∞;

(2c) There exists a constant p2 ∈ 0, p and afunction ξ2 ∈ L1/p2 J′,ℝ+ s.t. β F t, Z ≤ξ2 t β t1−rZ for any countable subset Z ⊂ E.

(H3) (3a) Linear operator V L2 J ,U → E defined by

Vu = a1−r Φp,r A, a 〠n

i=1aiK

ti

0Φp,p A, ti − s Bu s ds

+a

0Φp,p A, a − s Bu s ds

31

is reversible, the inverse operator denoted by V−1

and takes values in L2 J ,U ker V , and there existtwo constants Nu > 0, Nv > 0 such that

B ≤Nu, V−1 ≤Nv ; 32

(3b) There exists a constant p3 ∈ 0, p and ξ3 ∈L1/p3 J′,ℝ+ such that

β V−1 Z t ≤ ξ3 t β Z , t ∈ J , 33

for any countable subset Z ⊂ E

(H4) limη→∞ sup ηar−1 Np x1 +N1Nr Np + 1 ψ η −1

> 1, where Nr = Γ r N/Γ p Γ r −Ntr−11 ∑ni=1 ai ,

Np =NuNvNr a1−q 1−p /p a1−q 1−p /p and di = biap−pi , bi = 1 − pi/p − pi

1−pi , Ni = di ξi s L1/pi , i =1, 2, 3

For any x ∈ BYγ J′ , define an operator T as follows:

Tx t =Φp,r A, t 〠n

i=1aiK

ti

0Φp,p A, ti − s f s + Bu s ds

+t

0Φp,p A, t − s f s + Bu s ds, t ∈ J′,

34

where f ∈ SF,x.It is evident to see that limt→0t

1−r Tx t = 1/Γ r∑n

i=1aix ti .For any y ∈ Bγ J , let x t = tr−1y t for t ∈ J′, then

x ∈ BYγ J′ . Define T as follows

5Complexity

T y t =t1−r Tx t , t ∈ J′,1

Γ r〠n

i=1aix ti , t = 0

35

Clearly, x is a mild solution of (3) in Y if and only ify =T y has a solution y ∈ C J , E

u t ; x = a1−rV−1 x1 −Φp,r A, a 〠n

i=1aiK

ti

0Φp,p A, ti − s f s ds

−a

0Φp,p A, a − s f s ds t , t ∈ J , f ∈ SF,x

36

For brevity, let us take the following notations

P t ; x = Bu t ; x + f t , f ∈ SF,x ;

P x = 〠n

i=1aiK

ti

0Φp,p A, ti − s P s ; x ds

37

In view of Lemma 2, we obtain the following lemma thatwill be useful in the proof of the main results.

Lemma 7. Under the hypothesis (H2) (2b), (H3) (3a), for eachy ∈ Bγ J , set x t = tr−1y t , t ∈ J′, we have

P t ; x ≤NuNva1−r x1 +N1Nrψ y

+ ξ1 t ψ y , P x ≤Np 〠n

i=1ai x1

+N1Nr Np + 1 〠n

i=1ai ψ y

38

Proof 2. By Lemma 2, for each y ∈ Bγ J , set t = tr−1y t ,t ∈ J′, it is easy to get

Bu t ; x ≤NuNv a1−r x1 + N2 K ψ x Y ∑ni=1 ai

Γ r Γ pti

0ti − s p−1ξ1 s ds

+ a1−rNψ x Y

Γ p

a

0a − s p−1ξ1 s ds

≤NuNv a1−r x1 + N2ψ y ∑ni=1 ai d1 ξ1 L1/p1

Γ p Γ r −Ntr−11 ∑ni=1 ai

+ Nd1a1−rψ yΓ p

ξ1 L1/p1

≤NuNva1−r x1 + Γ r NN1

Γ p Γ r −Ntr−11 ∑ni=1 ai

ψ y

≜NuNva1−r x1 +N1Nrψ y

39

Hence, we easily see that

P t ; x ≤NuNva1−r x1 +N1Nrψ y

+ ξ1 t ψ y ;

P x ≤NΓ r ∑n

i=1 aiΓ p Γ r −Ntr−11 ∑n

i=1 aiti

0ti − s p−1 P s ; x ds

≤NΓ r ∑n

i=1 aiΓ p Γ r −Ntr−11 ∑n

i=1 aiti

0ti − s p−1 NuNva

1−r x1 +N1Nrψ y

+ ξ1 s ψ y ds

≤Γ r NNuNva

1−q 1−p ∑ni=1 ai

Γ p + 1 Γ r −Ntr−11 ∑ni=1 ai

x1 +N1Nrψ y +N1Nr 〠n

i=1ai ψ y

≜Np 〠n

i=1ai x1 +N1Nr Np + 1 〠

n

i=1ai ψ y

40This completes the proof.Next, we derive the controllability results for the Hilfer

fractional nonlocal differential inclusions (3).

Theorem 1. Assume the hypotheses (H0)–(H4) hold, then theHilfer fractional nonlocal differential inclusions (3) are exactcontrollable on J provided that

Λ ≜NN2 1 +N3NrNua1−r Nr

Γ r〠n

i=1∣ ai ∣ +

1Γ p

< 1

41

Proof 3. According to (H2) (2a) and [22], for each x ∈ CJ′, E , the multivalued function t→ F t, x t has a mea-surable selection, and in view of (H2) (2b), this selectionbelongs to SF,x. Thus, we can define a multivalued functionT ′ C J , E → 2C J ,E as follows. For every y ∈ C J , E , letx t = tr−1y t ∈ Y , t ∈ J′, and a function z ∈T ′y if and only if

z t =Kr t 〠

n

i=1aiK f ti + t1−r f t , t ∈ J′,

1Γ r

〠n

i=1aix ti , t = 0

42

where Kr t = t1−rΦp,r A, t , f t = t0 t − s p−1Kp t − s Bu

s + f s ds and f ∈ SF,x.Note that according to (H4), there is R∗ such that for

all R > R∗,

a1−r Np x1 +N1Nr Np + 1 ψ R <R 43

6 Complexity

Denote Ω = y ∈ C J , E : y ∞ < R0 and D = y ∈ CJ , E : y ∞ ≤ R0 , where R0 ≥ γ and R0 = R∗ + 1 We justprove that the multivalued operator mathcalT′ D→ 2C J ,E

meets the conditions of Lemma 5. Obviously, since the valuesof F are convex, the values of T ′ are also convex.

Claim 1. Each solution to the inclusion

y ∈ λT ′ y , y ∈D, λ ∈ 0, 1 44

satisfies y ∉D −Ω.Let y be a solution of (44). Then, by reminding the

definition of T ′, the hypothesis (H2) (2b) and recalling alsoLemma 2, for each t ∈ J , we derive

This inequality with (43) deduces y ∞ < R0, So y ∉D −Ω.

Claim 2. Every function in T ′y y ∈ Bγ J isequicontinuous.

For each y ∈ Bγ J , set x t = tr−1y t , t ∈ J′ such that

z ∈T ′ y By (42), there is f ∈ SF,x, for τ1, τ2 ∈ J , τ1 < τ2,we get

y t ≤N2∑n

i=1 aiΓ p Γ r −Ntr−11 ∑n

i=1 ai

ti

0ti − s p−1 P s ; x ds + Nt1−r

Γ p·

t

0t − s p−1 P s ; x ds

≤N2NuNva

p+1−r∑ni=1 ai

pΓ p Γ r −Ntr−11 ∑ni=1 ai

x1 +N1Nrψ y + NNuNvap+2−2r

pΓ p· x1 +N1Nrψ y

+ Na1−rψ yΓ p

t

0t − s p−1ξ1 s ds + N2ψ y ∑n

i=1 aiΓ p Γ r −Ntr−11 ∑n

i=1 ai

ti

0ti − s p−1ξ1 s ds

≤N2NuNva

p+1−r∑ni=1 ai

pΓ p Γ r −Ntr−11 ∑ni=1 ai

+ NNuNvΓ r ap+2−2r −N2NuNvap+2−2rtr−11 ∑n

i=1 aipΓ p Γ r −Ntr−11 ∑n

i=1 ai· x1 +N1Nrψ y

+ Na1−rψ y Γ r −N2a1−rtr−11 ψ y ∑ni=1 ai

Γ p Γ r −Ntr−11 ∑ni=1 ai

+ N2ψ y ∑ni=1 ai

Γ p Γ r −Ntr−11 ∑ni=1 ai

· d1 ξ1 L1/p1

≤Γ r NNuNva

p+2−2r

Γ p + 1 Γ r −Ntr−11 ∑ni=1 ai

x1 +N1Nrψ y + Γ r a1−rNd1 ξ1 L1/p1ψ y

Γ p Γ r −Ntr−11 ∑ni=1 ai

≤Γ r NNuNva

p+2−2r x1Γ p + 1 Γ r −Ntr−11 ∑n

i=1 ai+ a1−rN1Nrψ R0

Γ r NNuNvap+1−r

Γ p + 1 Γ r −Ntr−11 ∑ni=1 ai

+ 1

≜ a1−r Np x1 +N1Nr Np + 1 ψ R0

45

z τ2 − z τ1 ≤ Kr τ2 P x − Kr τ1 P x

+ τ1−r2

τ2

0τ2 − s p−1Kp τ2 − s P s ; x ds − τ1−r1

τ1

0τ1 − s p−1Kp τ1 − s P s ; x ds

≤ Kr τ2 − Kr τ1 P x +τ2

τ1

τ1−r2 τ2 − s p−1Kp τ2 − s P s ; x ds

+τ1

0τ1−r1 τ1 − s p−1 Kp τ2 − s − Kp τ1 − s P s ; x ds

+τ1

0τ1−r1 τ1 − s p−1 − τ1−r2 τ2 − s p−1 Kp τ2 − s P s ; x ds

≤ I1 + I2 + I3 + I4,

46

7Complexity

where

I1 = Kr τ2 − Kr τ1 P x ,

I2 =NΓ p

τ2

τ1

τ1−r2 τ2 − s p−1P s ; x ds ,

I3 =NΓ p

τ1

0τ1−r1 τ1 − s p−1 − τ1−r2 τ2 − s p−1 P s ; x ds ,

I4 =τ1

0τ1−r1 τ1 − s p−1 Kp τ2 − s − Kp τ1 − s P s ; x ds

47From (H1), one can deduce that I1 → 0 as τ2 − τ1 → 0 FromLemma 7, we have

I2 ≤NNuNva

2−2r

Γ p + 1 x1 +N1Nrψ γ τ2 − τ1p

+ Nb1a1−r

Γ pτ2 − τ1

p−p1 ξ1 L1/p1ψ γ ,48

which implies that I2 → 0 as τ2 − τ1 → 0

I3 ≤NNuNva

1−r

Γ px1 +N1Nrψ γ

τ1

0τ1−r1 τ1 − s p−1 − τ1−r2 τ2 − s p−1 ds

+ NΓ p

ψ γτ1

0τ1−r1 τ1 − s p−1 − τ1−r2 τ2 − s p−1 ξ1 s ds

49

Noting that

τ1−r1 τ1 − s p−1 − τ1−r2 τ2 − s p−1 ξ1 s ≤ τ1−r1 τ1 − s p−1ξ1 s ,50

and τ10 τ

1−r1 τ1 − s p−1ξ1 s ds, s ∈ 0, τ1 exists, then by Lebes-

gue’s dominated convergence theorem, we obtain I3 → 0 asτ2 − τ1 → 0.For ε > 0, we get

The assumption (H1) guarantees that I4 → 0 as τ2 −τ1 → 0 and ε→ 0 We prove that Kr t is uniformly contin-uous on J Consequently, T ′ is equicontinuous on Bγ J .

Claim 3. The inference (13) holds with y0 = 0.Let Z = conv y0 ∪T ′ Z ⊆D, Z =G with G ⊆ Z

countable. We assert that Z is relatively compact. In fact,since G is countable and G ⊆ Z = conv y0 ∪T ′ Z , wecan chase down a countable set H = zn n ≥ 1 ⊆T ′ Zwith G ⊆ conv y0 ∪H . Then, there exists yn ∈ Z withzn ∈T ′ yn . This means that there is f n ∈ SF,xn such thatfor t ∈ J

zn t = Kr t 〠n

i=1aiK f n ti + f n t , 52

where

f n t =t

0t − s p−1Kp t − s Bun s + f n s ds 53

From Z ⊆ G ⊆ conv y0 ∪H , for t ∈ J , s ∈ 0, t . Byutilizing Lemmas 2, 4, (H2) (2c), (H3) (3b), and the proper-ties of the noncompact measure, we can derive

I4 ≤τ1−ε

0τ1−r1 τ1 − s p−1 Kp τ2 − s − Kp τ1 − s P s ; x ds

+τ1

τ1−ετ1−r1 τ1 − s p−1 Kp τ2 − s − Kp τ1 − s P s ; x ds

≤NuNva2−2r x1 +N1Nrψ γ

τp1 − εp

psup

s∈ 0,τ1−εKp τ2 − s − Kp τ1 − s

+ τ1−r1

τ1−ε

0τ1 − s p−1ξ1 s ds sup

s∈ 0,τ1−εKp τ2 − s − Kp τ1 − s ψ γ

+ 2NNuNva2−2rεp

Γ p + 1 x1 +N1Nrψ γ + 2Na1−r

Γ pb1ε

p−p1ψ γ ξ1 L1/p1

51

8 Complexity

and

β Z t ≤ β G t ≤ β H t

≤ β Kr t 〠n

i=1aiK

ti

0ti − s p−1Kp ti − s

Bun s + f n s ds n ≥ 1

+ βt

0t − s p−1Kp t − s Bun s + f n s ds n ≥ 1

≤N2∑n

i=1 aiΓ p Γ r −Ntr−11 ∑n

i=1 aiti

0ti − s p−1 N2NuNra

1−rξ3 s + ξ2 s ds · β Z

+ NΓ p

t

0t − s p−1 N2NuNra

1−rξ3 s + ξ2 s ds · β Z

≤NN2 1 +N3NrNua1−r Nr

Γ r〠n

i=1ai +

1Γ p

· β Z

≜Λ · β Z

55

Reminding Z = conv y0 ∪T ′ Z , by Claim 2, Z isequicontinuous. So we find from Lemma 3 that

β Z =maxt∈J

β Z t ≤Λβ Z 56

Since Λ < 1, we obtain β Z = 0 That is, Z is compact.

Claim 4. T ′ maps compact sets into relatively compact sets.LetQ be a compact subset of Z. From Claim 2,T ′ Q is equi-continuous. Let t ∈ J , by the definition of T ′, for any y ∈Qand z ∈T ′ y , there is f y ∈ SF,x such that

z t = Kr t 〠n

i=1aiK f y ti + f y t , t ∈ J , 57

where f y t = t0 t − s p−1Kp t − s Buy s + f y s ds

Therefore

β T ′ Q t ≤ β z t : z ∈T ′ y , y ∈Q

≤ β Kr t 〠n

i=1aiK

ti

0ti − s p−1Kp ti − s

Buy s + f y s ds y ∈ G

+ βt

0t − s p−1Kp t − s

Buy s + f y s ds y ∈ G

≤NN2 1 +N3NrNua1−r

Nr

Γ r〠n

i=1ai +

1Γ p

β Q = 0

58

Then, Lemma 3 indicates β T ′ Q =maxt∈Jβ T ′ Qt = 0, that is, the set T ′ Q is relatively compact.

Claim 5. The graph T ′ is closed.For any y n ∈ Bγ J , let x n t = tr−1y n t and x∗

t = tr−1y∗ t , t ∈ J′. Let y n → y∗ n→∞ , μ n ∈T ′ y n ,μ n → μ∗ n→∞ We will show that μ∗ ∈T ′ y∗ . Sinceμ n ∈T ′ y n , there is f n ∈ SF,x n such that for any t ∈ J ,

β Bun s + f n s ≤Nuξ3 sN2∑n

i=1 aiΓ p Γ r −Ntr−11 ∑n

i=1 ai

ti

0ti − s p−1β f n s : n ≥ 1 ds

+ Na1−r

Γ p

a

0a − s p−1β f n s : n ≥ 1 ds + β f n s : n ≥ 1

≤Nuξ3 sN2∑n

i=1 aiΓ p Γ r −Ntr−11 ∑n

i=1 ai

ti

0ti − s p−1ξ2 s β yn s : n ≥ 1 ds

+ Na1−r

Γ p

a

0a − s p−1ξ2 s β yn s : n ≥ 1 ds + ξ2 s β yn s : n ≥ 1

≤ N2NuNra1−rξ3 s + ξ2 s · β Z ,

54

9Complexity

μ n t = Kr t 〠n

i=1aiK

ti

0ti − s p−1Kp ti − s f n s ds

+ Kr t 〠n

i=1aiK

ti

0ti − s p−1 · Kp ti − s BV−1

· a1−rx1 − Kr a 〠n

i=1aiK

ti

0ti − τ p−1Kp ti − τ f n τ dτ

− a1−ra

0a − τ p−1Kp a − τ f n τ dτ ds

+ t1−rt

0t − s p−1Kp t − s f n s ds

+ t1−rt

0t − s p−1Kp t − s BV−1

a1−rx1 − Kr s 〠n

i=1aiK

·ti

0ti − τ p−1 · Kp ti − τ f n τ dτ − a1−r

a

0a − τ p−1Kp a − τ f n τ dτ ds

59

So we just need to demonstrate the existence of f ∗ ∈ SF,x∗such that for any t ∈ J ,

μ∗ t = Kr t 〠n

i=1aiK

ti

0ti − s p−1Kp ti − s f ∗ s ds

+ Kr t 〠n

i=1aiK

ti

0ti − s p−1 · Kp ti − s BV−1

a1−rx1 − Kr a 〠n

i=1aiK

ti

0ti − τ p−1Kp ti − τ f ∗ τ dτ

− a1−ra

0a − τ p−1Kp a − τ f ∗ τ dτ ds

+ t1−rt

0t − s p−1Kp t − s · f ∗ s ds

+ t1−rt

0t − s p−1Kp t − s BV−1

a1−rx1 − Kr s 〠n

i=1aiK ·

ti

0ti − τ p−1Kp ti − τ f ∗ τ dτ

− a1−ra

0a − τ p−1Kp a − τ f∗ τ dτ ds

60

Take into account the linear continuous operator

Γ L1/p J , E → Bγ J , 61

where

Γf t = Kr t 〠n

i=1aiK

ti

0ti − s p−1Kp ti − s

f s − BV−1 Kr s 〠n

i=1aiK

·ti

0ti − τ p−1Kp ti − τ f τ dτ + a1−r

a

0a − τ p−1Kp a − τ f τ dτ ds

+ t1−rt

0t − s p−1Kp t − s

f s − BV−1 · Kr s 〠n

i=1aiK

ti

0ti − τ p−1 · Kp ti − τ f τ dτ

+ a1−ra

0a − τ p−1Kp a − τ f τ dτ ds

62

Clearly, we can get from Lemma 1 that the operator Γ ∘ SFis a closed graph.

Since μ n → μ∗, n→∞, we can obtain that as n→∞,

μ n t − Kr t 〠n

i=1aiK

ti

0ti − s p−1Kp ti − s BV−1a1−rx1ds

− t1−rt

0t − s p−1 · Kp t − s BV−1a1−rx1ds

− μ∗ t − Kr t 〠n

i=1aiK

ti

0ti − s p−1Kp ti − s

· BV−1a1−rx1ds − t1−r

⋅t

0t − s p−1Kp t − s BV−1a1−rx1ds → 0

63

In addition, we get

μ n t − Kr t 〠n

i=1aiK

ti

0ti − s p−1Kp ti − s BV−1a1−rx1ds

− t1−rt

0t − s p−1Kp t − s BV−1a1−rx1ds ∈ Γ SF,x n

64

Since y n → y∗ n→∞ , we can obtain from Lemma 1that

10 Complexity

μ∗ t − Kr t 〠n

i=1aiK

ti

0ti − s p−1Kp ti − s BV−1a1−rx1ds

− t1−rt

0t − s p−1Kp t − s BV−1a1−rx1ds

= Kr t 〠n

i=1aiK

ti

0ti − s p−1Kp ti − s f ∗ s ds

− Kr t 〠n

i=1aiK

ti

0ti − s p−1 · Kp ti − s BV−1

Kr s 〠n

i=1aiK

ti

0ti − τ p−1Kp ti − τ f ∗ τ dτ + a1−r

·a

0a − τ p−1Kp a − τ f ∗ τ dτ ds

+ t1−rt

0t − s p−1Kp t − s f ∗ s ds

− t1−rt

0t − s p−1Kp t − s BV−1

Kr s 〠n

i=1aiK

ti

0ti − τ p−1Kp ti − τ · f ∗ τ dτ

+ a1−ra

0a − τ p−1Kp a − τ f ∗ τ dτ ds

65

For some f ∗ ∈ SF,x∗ , this infers that μ∗ ∈T ′ y∗ Hence,T ′ has a closed graph. Thus, Claims 1–5 are completed. Byuse of Lemma 5, we know operator T ′ has a fixed point inBγ J . Let x t = tr−1y t , then x is a mild solution of (3)and it satisfies x a = x1. Therefore, the Hilfer fractional non-local differential inclusions (3) are exact controllable on J

4. Applications

Consider the following partial differential system

D3/4,2/30+ x t, s ∈ −

∂∂s

x t, s

+ e−2t

1 + etx t, s + η t, s , t ∈ 0, a , s ∈ 0, 1 ,

x t, 0 = x t, 1 = 0, t ∈ 0, a ,

I1/3 1−p0+ x t, s

t=0= 〠

n

i=1Γ 11

12 arctan 12i2

x ti, s , s ∈ 0, 1 ,

66

where n ∈ 0, a is constants, η J × 0, 1 → 0, 1 iscontinuous.

Let E =Ω≕ C 0, 1 and A is defined by

D A = w ∈ E w′ ∈ E,w 0 =w 1 = 0 ,

Aw = −w′,w ∈D A67

As we all know that −A generates an equicontinuoussemigroup S t t ≥ 0 in E and it satisfies

T t w υ =w t + υ , 68

for w ∈ E Thus, S t t ≥ 0 is not compact in E andsup0≤t≤a T t ≤ 1 Take

x t s = x t, s ,

D3/4,2/30+ x t s = I1/60+ D

11/120+ x t, s

= 1Γ 1/6

t

0t − τ −5/6 ∂11/12

∂τ11/12x τ, s dτ,

f t, x t s = e−2t

1 + etx t, s ,

u t s = μ t, s ,

ai = Γ 1112 arctan 1

2i2 , ti = i, i = 1, 2,… , n

69

Then, for any y ∈ Bγ ≔ y ∈ C J , E : y ≤ γ , where J =0, a , let x t = t− 1/12 y t , t ∈ J′≔ 0, a , then x ∈ BY

γ J′ ,and we obtain

f t, x t s ≤e−2t

1 + etx t, s ≤

e−2tγ1 + et

≤γ

270

Thus, the hypothesis H2 holds for β = 1/2 and ξ2 t =1/2 for all t ∈ J′ By

〠n

i=1ai ≤ Γ 11

12 · 〠∞

i=1arctan 1

2i2= Γ 11

12 · π4 < Γ 1112 ,

71

we verify that the hypothesis (H0) holds.For s ∈ 0, 1 , the operator V is defined as

V t s = Kr a I − Γ 1112 〠

n

i=1i− 1/12 Kr i arctan 1

2i2−1

Γ 1112 〠

n

i=1arctan 1

2i2

·i

0i − τ − 1/4 Kp i − τ η τ, s dτ

+a

0a − τ − 1/4 Kp a − τ η τ, s dτ,

72

where Kr t t≥0 and Kp tt≥0 satisfy

Kr t = I1/60+ t− 1/4 Kp t ,

Kp t = 〠∞

i=0

Ait 3/4 i

Γ 3/4 i + 3/4

73

11Complexity

IfV satisfies the hypothesis (H3), from Theorem 1, we getthat the Hilfer fractional differential inclusion (66) involvingnonlocal initial conditions is exact controllable on 0, aprovided that (H4) and (41) are satisfied.

Data Availability

The data used to support the findings of this study areincluded within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (nos. 1371027, 11471015, and1601003), Natural Science Foundation of Anhui Province(no. 1708085MA15), and Natural Science Fund of Collegesand Universities in Anhui Province (no. KJ2018A0470).

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