# New Controllability Results of Fractional Nonlocal Semilinear Evolution Systems with Finite Delay

Authors: Daliang Zhao; Juan Mao
Journal: Complexity
 (2020)

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## Abstract

In the present paper, sufficient conditions ensuring the complete controllability for a class of semilinear fractional nonlocal evolution systems with finite delay in Banach spaces are derived. The new results are obtained under a weaker definition of complete controllability we introduced, and then the Lipschitz continuity and other growth conditions for the nonlinearity and nonlocal item are not required in comparison with the existing literatures. In addition, an appropriate complete space and a corresponding time delay item are introduced to conquer the difficulties caused by time delay. Our main tools are properties of resolvent operators, theory of measure of noncompactness, and Mönch fixed point theorem.


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## Body

## 1. Introduction

In recent decades, various research studies on fractional differential systems are growing vigorously, which are chiefly due to the enormous scope and extensive applications of fractional calculus theory in mathematics, biology, physics, economics, and engineering science [1–9].As we all know, time delay effects exist widely in various fields such as communication security, weather predicting, and population dynamics. The difficulties in the study of these fields lie in the time lag effect for the systems caused by the delay. One can see [10–20], for further details. It should be noted that, in the case of research on fractional delay evolution systems, it is still in the initial stage. On the contrary, fractional calculus has also been applied to controllability issues in recent years. As one of the most important notions in control theory of mathematics, controllability has significant influence on the fields of control and engineering. We point out that complete controllability of infinite dimensional systems with noncompact semigroup is an important research direction, and there have been many outstanding achievements in this regard such as [21–26]. For more details of other research results on control theory, please refer to [27–40] and the reference therein.As we have seen, there is scarcely any results on complete controllability of fractional nonlocal evolution equations with delay in Banach spaces, except [21, 23, 24]. However, the Lipschitz and certain growth conditions on nonlocal item and nonlinearity are still necessary for [21]. In [23], the authors supposed the nonlinear function and nonlocal item to be Lipschitz continuous and the semigroup generated by the considered system to be compact. In [24], nonlinear function was imposed some growth conditions, and the resolvent operator used to define the mild solution was compact. However, it is a little pity that these suppositions are usually difficult to be made to work in numerous specific applications. In fact, many excellent achievements concerning complete controllability of various nonlinear differential systems, such as those in Debbouche and Baleanu, [11], Nirmala et al. [22], and Wang and Zhou [25], have been derived during recent years. However, the limitation also lies in that the nonlinear functions here are all provided with the Lipschitz continuity or other growth assumptions. In addition, the compactness or other assumptions of C0 semigroup actually prevents us from studying complete controllability in the infinite dimensional space.The current paper, inspired by the aforementioned analyses, is to address the complete controllability of the following fractional nonlocal semilinear evolution equations with finite delay in Banach spaces:(1)Dqut=Aut+ft,ut,ut+Bμt,a.e.t∈I≔0,T,ut+Mt,u=ϕt,t∈−r,0,where Dq denotes the Caputo derivative of order q∈0,1. u:I⟶X, μ∈L2I,U, and B:U⟶X is a bounded linear operator, where X and U are Banach spaces. A:D⊂X⟶X is a closed linear unbounded operator on X with dense domain D. ϕt∈C−r,0,X. f and M are given functions satisfying certain appropriate conditions that will be given later.The proposed fractional evolution system (1) here, which generalizes the case of integral (first) order differential equations studied in [41] about the complete controllability, has more extensive and valid applications in contrast to the abovementioned literatures [11, 21–25] as follows. First of all, under the new concept of complete controllability we drew into, that is, a weaker definition of complete controllability than the existing notion, the nonlinear function and nonlocal item here are only allowed to be continuous instead of other restrictions such as Lipschitz continuity and certain growth conditions. Secondly, compactness of the C0 semigroup and the resolvent operator used to define the mild solution for system (1) is no longer needed. Last but not the least, obstacles in the estimation of Kuratowski measures of noncompactness caused by time delay have been conquered in a new complete space with corresponding defined delay function.An outline of the current paper is as follows. Some essential preparations are made in Section2. In Section 3, we derive a sufficient condition on complete controllability for the addressed systems. In Section 4, we give an example to illustrate the obtained new results.


## 2. Preliminaries

LetCI,X be the space of continuous functions from I into X provided with the supreme norm ⋅CI,X. Similarly, C−r,T,X denotes the Banach space of continuous functions from −r,T to X with the usual supreme norm. If T=0, we denote the norm of this space simply by ⋅r. The domain D is endowed with the graph norm uD=u+Au, where ⋅ denotes the norm of Banach space X. For q∈0,1, CqI,X stands for the space of all the continuous functions from I into X equipped with the norm uCqI,X=uCI,X+uCqI,X, where uCqI,X=supt,s∈I,t≠sut−us/t−sq. Throughout this paper, ℒX,Y denotes the space of bounded linear operators from X into Banach space Y provided with the operator norm ⋅ℒX,Y.In order to conquer the inconveniences caused by time delay in the study of complete controllability, for eachu∈CI,X, t∈I, and the function ϕt in (1), we draw into the function ut defined by(2)utθ=ut+θ,t+θ≥0,ϕt+θ,t+θ≤0,for each θ∈−r,0. Simple check indicates xt∈L−r,0,X, where L−r,0,X stands for a complete integrable space consists of integrable functions from −r,0 into X.Remark 1.
Under the functionut defined in (2) in complete space L−r,0,X, we can get over the difficulties caused by time delay in the process of estimating Kuratowski measures of noncompactness (for details, please see Lemmas 4 and 5).
Next, we list the well-known definitions as follows.Definition 1 (see [5]).
The fractional integral with orderq>0 for a function u:0,+∞⟶R can be defined as(3)I0+qut=1Γq∫0tt−sq−1usds,provided that the right-side integral is pointwise defined on 0,+∞.Definition 2 (see [5]).
The Caputo fractional derivative with orderq>0 for a function u:0,+∞⟶R is written as(4)D0+qut=1Γn−q∫0tunst−sq−n+1ds,where n=q+1, provided the right-side integral is pointwise defined on 0,+∞.Definition 3 (see [42]).
The bounded linear operatorℑtt≥0⊂ℒX on X is defined as a resolvent operator of the following integral equation:(5)ut=∫0tσt−sAusds,t≥0,where scalar kernel σ∈Lloc1R+ and σ≡0, provided that it satisfies(i)
ℑt is strongly continuous, ℑ0=I(ii)
ℑt commutes with A, that is, ℑtD⊂D, and Aℑtu=ℑtAu for each u∈D and each t≥0(iii)
ℑtu=u+∫0tσt−sAℑsuds, for all u∈D, t≥0Definition 4 (see [42]).
Supposeℑ⋅u∈Wloc1,1R+,X for each u∈D, and there exists a function φ∈Lloc1R+, which satisfy(6)ℑ˙tu≤φtuD,a.e. onR+,∀u∈D.
Then, we callℑt a differentiable resolvent operator of (5).
Next, we focus on the following equation:(7)ut=gt+∫0tσt−sAusds,t∈I,where g∈L1I,X, and list the definition of mild solution of (7) as below on the basis of literature [42].Definition 5.
We callu∈CI,X a mild solution for (7) if ∫0tσt−susds∈D, which satisfies(8)ut=gt+A∫0tσt−susds,for each t∈I.
Taking advantage of properties of the differentiable resolvent operator, we present the equivalent definition of mild solution for (7).Lemma 1 (see [42]).
Assume thatℑt is a differentiable resolvent operator for (7) and g∈CI,D. Then,(9)ut=∫0tℑ˙t−sgsds+gt,t∈I,is a mild solution of (7).Before going further, we now recall below a few relevant properties of Kuratowski measures of noncompactness, which will play a critical role in our next proof of complete controllability. For further details, please see [43].Lemma 2.
LetX be a Banach space and γ⋅ be the Kuratowski measures of noncompactness which is given by γΩ=infδ>0:Ω=∪i=1kΩiwith diamΩi≤δ,i=1,2,…,k for a bounded subset Ω in X.(I)
LetD1,D2 be bounded sets of X and λ∈R. Then,(i)
γD1=0⟺D1 is relatively compact(ii)
γλD1=λγD1(iii)
γD1+D2≤γD1+γD2(II)
Assume thatD=un is a countable set of strongly measurable functions from I into Banach space X, and there exists a function ψ∈L1I such that unt≤ψt,a.e.t∈I,n=1,2,…, then γDt is integrable on I, which satisfies(10)γ∫Iuntdt:n∈N≤2∫IγDtdt.For the convenience of expression, the Kuratowski measures of noncompactness of a bounded subset in spacesX, CI,X and L−r,0,X are all written as γ⋅, on the premise of no confusion.Lemma 3 (Mönch).
If there is a closed and convex setD⊂X, where X is a Banach space, u0∈D, and suppose that A:D⟶D is continuous and satisfies C⊂D countable, C⊂co¯u0∪AC⟹C is relatively compact. Then, A has a fixed point in D.To end this section, we give the following useful lemmas.Lemma 4.
Assume thatun converges to u0 in CI,X as n⟶+∞. Then, unt converges to u0t in L−r,0,X for every t∈I as n⟶+∞.Proof.
Based on the definition in (2), we know(11)unt−u0tL−r,0=∫0tuns−u0sds,t≤r,∫t−rtuns−u0sds,t≥r,which implies that(12)unt−u0tL−r,0≤run−u0CI,X,∀t∈I.
This completes the proof.From the definition of Kuratowski measures of noncompactness in Lemma2, we can infer the following lemma.Lemma 5.
LetD=unn=1∞ be any bounded countable sequence in CI,X. Then, for each t∈I, one has(13)γDt≤rγD,where Dt=untn=1∞.


## 3. Main Results

In the sequel, we assume thatA admits a differentiable resolvent operator ℑtt≥0 on X. Based on Definition 5 and the Riemann–Liouville standard fractional integral, the mild solution of system (1) can be defined as follows.Definition 6.
For eachμ∈L2I,U, a function u∈CJ,X is said to be a mild solution of the considered system (1) on J, if ∫0tt−sq−1usds∈D for any t∈0,T∗ and(14)ut=ϕ0−M0,u+1ΓqA∫0tust−s1−qds+1Γq∫0tfs,us,us+Bμst−s1−qds,t∈0,T∗,ϕt−Mt,u,t∈−r,0,where J=−r,T∗,T∗∈0,T.Definition 7 (complete controllability).
We shall call system (1) is completely controllable on I=0,T, if for each initial function ϕt∈C−r,0,X and u1∈X, there exist a corresponding control μ∈L2I,U and a real number T∗∈0,T such that the mild solution u of (1) on J=−r,T∗ satisfies uT∗=u1.Remark 2.
In comparison with the existing notion in [11, 21, 22, 25, 41], in which T∗ is equal to T, our concept in which T∗∈0,T is weaker can be regarded as an extension of the present notion of complete controllability.
Before presenting and proving the main results, we firstly list the hypotheses as below:(H1)
f∈CI×X×L−r,0,X,D, ϕ0∈D and satisfies(i)
f maps bounded sets in I×X×L−r,0,X into bounded sets in D.(ii)
There exist a constantq1∈0,q and a function m∈L1/q1I,R+ such that, for any bounded subsets D1⊂X,D2⊂L−r,0,X(15)γft,D1,D2≤mtγD1+γD2,t∈I.(H2)
(i) The linear operatorB:L2I,U⟶L1I,D is bounded, and there exists a constant N1>0 satisfying BℒU,D≤N1.(ii)
Linear operatorsℜt,t∈I, denoted by ℜ⋅ from L2I,U to X are defined as(16)ℜtμ=ΠBt,μ+∫0tℑ˙t−sΠBs,μds,t∈I,where ΠBt,μ=1/Γq∫0tt−sq−1Bμsds has invertible operators ℜ−1⋅ taking values in L2I,U/kerℜ⋅, which satisfy, for some constant N2>0, supℜ−1⋅ℒX,L2I,U/kerℜ⋅≤N2, and there is a constant q2∈0,q and a function p∈L1/q2I,R+ satisfying(17)γℜ−1⋅Ds≤psγD,s∈I,for any bounded subset D⊂X.(H3)
(i) For everyt∈−r,0, Mt,⋅:C−r,T,X⟶X is a strict k-set contraction and for every u∈C−r,T,X, Mt,u is continuous with respect to each t∈−r,0.(ii)
M0,⋅:C−r,T,X⟶D and there exists a positive constant G such that M0,uD≤G for every u∈C−r,T,X.(H4)
The following estimation is valid:(18)kM1+2φL1I<1,where(19)M=1+2N1L1+2φL1IΓq,L=Tq−q21−q2q−q21−q2pL1/q2,and φ is the function mentioned in Definition 4.
In the following, letR0 be a fixed constant satisfying R0>ϕ0D+G1+φL1I≔N0. From (H1), take(20)M0=supft,u,vD:uCI,X≤R0,vL−r,0≤rϕr+R0,t∈I.
For simplicity, let(21)Πt,u,μ=1Γq∫0tt−sq−1fs,us,usds+1Γq∫0tt−sq−1Bμsds,and set(22)λ=2r+1Γq1−q1q−q11−q1mL1/q1.
By means of conditions (H2) and (H4), for anyu⋅∈CI,X and any u1∈X,t∈I, define a feedback control function(23)μut≔ℜ−1T∗u1−ϕ0+M0,u−ΠfT∗,u−∫0T∗ℑ˙T∗−sϕ0−M0,u+Πfs,udst,where(24)T∗=minT,R0−N0Γq+1M0+N1M11+φL1I1/q,1−kM1+2φL1IλM1+2φL1I1/q,(25)M1=N21+φL1Iϕ0D+G+M0TqΓq+1+u1,(26)Πft,u=1Γq∫0tt−sq−1fs,us,usds.
Denote(27)Ω=u∈CJ,X:uC0,T∗,X≤R0,supt∈0,T∗utL−r,0≤rϕr+R00;ut=ϕt−Mt,u,t∈−r,0,,then Ω is clearly a closed convex set in CJ,X. From Lemma 1, we thus define an operator Ψ:CJ,X⟶CJ,X which is given by(28)Ψut=ϕ0−M0,u+Πt,u,μu+∫0tℑ˙t−sϕ0−M0,u+Πs,u,μuds,t∈0,T∗,ϕt−Mt,u,t∈−r,0.
For the purpose of simplifying our next proof processes, we give the following conclusions.Lemma 6.
Assumef⋅,u⋅,u⋅∈CI×X×L−r,0,X,X and μ⋅∈L2I,U. Then, Π⋅,u,μ∈CqI,X, and(29)Π⋅,u,μCq≤2qf⋅,u⋅,u⋅CI,X+BℒU,Xμ⋅L2.Proof.
Fort∈0,T and h>0 such that t+h∈I, we have(30)Πt+h,u,μ−Πt,u,μ≤∫0tt−sq−1−t+h−sq−1fs,us,us+Bμsds+∫tt+ht+h−sq−1fs,us,us+Bμsds≤tq−t+hq+hqq+hqqf⋅,u⋅,u⋅CI,X+BℒU,Xμ⋅L2≤2hqqf⋅,u⋅,u⋅CI,X+BℒU,Xμ⋅L2,which implies that Π⋅,u,μCq≤2/qf⋅,u⋅,u⋅CI,X+BℒU,Xμ⋅L2 and Π⋅,u,μ∈CqI,X. The conclusion follows.Lemma 7.
Assume that conditions (H1) (i) and (H2)–(H4) hold. Then, the operatorΨ:Ω⟶Ω is equicontinuous on J.Proof.
Firstly, we prove thatΨΩ⊆Ω. In view of assumptions (H1) (i), (H2), and (H3) (ii), we can derive(31)μut≤N2u1+ϕ0D+G+∫0T∗φT∗−sϕ0D+Gds+N2Γq∫0T∗T∗−sq−1fs,us,usDds+∫0T∗φT∗−s∫0ss−ηq−1fη,uη,uηDdηds≤N2u1+ϕ0D+G+φL1Iϕ0D+G+N2M0TqΓq+1+φL1IM0TqΓq+1≤N2φL1I+1ϕ0D+G+M0TqΓq+1+u1=M1,t∈I.
Observing (H2), for anyu∈Ω and t∈0,T∗, we deduce(32)Πt,u,μuD≤1Γq∫0tt−sq−1fs,us,usDds+1Γq∫0tt−sq−1BμusDds≤M0T∗qΓq+1+N1M1T∗qΓq+1=M0+N1M1T∗qΓq+1.
Then, this together with (24) indicates(33)Ψut≤ϕ0D+M0,uD+Πt,u,μuD+∫0tφt−sϕ0D+M0,uD+Πs,u,μuDds≤‖ϕ0‖D+G+M0+N1M1T∗qΓq+1+φL1Iϕ0D+G+M0+N1M1T∗qΓq+1≤ϕ0D+GφL1I+1+M0+N1M1φL1I+1Γq+1T∗q≤R0.
On the contrary,(34)ΨutL−r,0=∫−r0Ψutθdθ=∫t−r0ϕsds+∫0tΨusds,t≤r,∫t−rtΨusds,t≥r,which infers that(35)ΨutL−r,0≤rϕr+rΨuC0,t,X.
Therefore, we can obtain(36)supt∈0,T∗ΨutL−r,0≤rϕr+R0.
It is obvious thatΨut=ϕt−Mt,u for any u∈Ω,t∈−r,0. Then, we conclude that ΨΩ⊆Ω.
Next, we shall demonstrate thatΨ:Ω⟶Ω is equicontinuous on J. For arbitrary u∈Ω and τ1,τ2∈J=−r,T∗ with τ1<τ2, we present the following discussions.(i)
If0≤τ1<τ2≤T∗, then(37)Ψuτ2−Ψuτ1=Πτ2,u,μu−Πτ1,u,μu+∫0τ2ℑ˙τ2−sϕ0−M0,uds−∫0τ1ℑ˙τ1−sϕ0−M0,uds+∫0τ2ℑ˙τ2−sΠs,u,μuds−∫0τ1ℑ˙τ1−sΠs,u,μuds.
Clearly,(38)Ψuτ2−Ψuτ1≤B1+B2+B3,where(39)B1=Πτ2,u,μu−Πτ1,u,μu,B2=∫0τ2ℑ˙τ2−sϕ0−M0,uds−∫0τ1ℑ˙τ1−sϕ0−M0,uds,B3=∫0τ2ℑ˙τ2−sΠs,u,μuds−∫0τ1ℑ˙τ1−sΠs,u,μuds.
Next, we shall show thatBi⟶0 is independent of u∈Ω as τ1−τ2⟶0,i=1,2,3. In fact, with regard to B1, we derive(40)B1≤∫0τ2τ2−sq−1fs,us,usds−∫0τ1τ1−sq−1fs,us,usds+∫0τ2τ2−sq−1Bμusds−∫0τ1τ1−sq−1Bμusds≤∫0τ1τ1−sq−1−τ2−sq−1fs,us,usDds+∫τ1τ2τ2−sq−1fs,us,usDds+∫0τ1τ1−sq−1−τ2−sq−1BμusDds+∫τ1τ2τ2−sq−1BμusDds≤M0qτ1q−τ2q+τ2−τ1q+M0qτ2−τ1q+N1M1qτ1q−τ2q+τ2−τ1q+N1M1qτ2−τ1q⟶0,asτ1−τ2⟶0.
Notice that(41)B2=∫0τ2ℑ˙τ2−sϕ0−M0,uds−∫0τ1ℑ˙τ1−sϕ0−M0,uds=∫0τ2−τ1ℑ˙τ2−sϕ0−M0,uds+∫τ2−τ1τ2ℑ˙τ2−sϕ0−M0,uds−∫0τ1ℑ˙τ1−sϕ0−M0,uds=∫0τ2−τ1ℑ˙τ2−sϕ0−M0,uds.
According to Definition4, we deduce(42)B2≤ϕ0D+M0,uD∫0τ2−τ1φτ2−sds≤ϕ0D+G∫τ1τ2φsds⟶0,asτ1−τ2⟶0.
From the proof process of Lemma6 and (32), it follows that(43)B3=∫0τ2ℑ˙τ2−sΠs,u,μuds−∫0τ1ℑ˙τ1−sΠs,u,μuds≤∫0τ2−τ1ℑ˙τ2−sΠs,u,μuds+∫τ2−τ1τ2ℑ˙τ2−sΠs,u,μuds−∫0τ1ℑ˙τ1−sΠs,u,μuds≤∫0τ2−τ1ℑ˙τ2−sΠs,u,μuds+∫0τ1ℑ˙sΠτ2−s,u,μuds−∫0τ1ℑ˙sΠτ1−s,u,μuds≤∫0τ2−τ1φτ2−sΠs,u,μuDds+∫0τ1φsΠτ2−s,u,μu−Πτ1−s,u,μuDds≤∫τ1τ2φsds⋅M0+N1M1T∗qΓq+1+∫0τ1φsds⋅2M0+N1M1qτ2−τ1q⟶0,asτ1−τ2⟶0.(ii)
If−r≤τ1<τ2≤0, then from the continuity of ϕ⋅ and (H3) (i), one has(44)Ψuτ2−Ψuτ1=ϕτ2−ϕτ1−Mτ2,u−Mτ1,u≤ϕτ2−ϕτ1+Mτ2,u−Mτ1,u⟶0,asτ1−τ2⟶0.(iii)
If−r≤τ1<0<τ2≤T∗, then(45)Ψuτ2−Ψuτ1≤Ψuτ2−Ψu0+Ψu0−Ψuτ1≤Πτ2,u,μu+∫0τ2ℑ˙τ2−sϕ0−M0,u+Πs,u,μuds+ϕ0−M0,u−ϕτ1−Mτ1,u≤M0+N1M1Γq+1τ2q+ϕ0D+M0,uD+Πs,u,μuD∫0τ2φτ2−sds+ϕ0−ϕτ1+M0,u−Mτ1,u≤M0+N1M1Γq+1τ2q+ϕ0D+G+M0+N1M1Γq+1Tq∫0τ2φsds+ϕ0−ϕτ1+M0,u−Mτ1,u⟶0,asτ1−τ2⟶0.
Thus, it implies thatΨuτ2−Ψuτ1⟶0, as τ1−τ2⟶0, for all u∈Ω. Consequently, Ψ:Ω⟶Ω is equicontinuous on J.Lemma 8.
Let conditions (H1) (i), (H2), (H3) (i), and (H4) be satisfied. Then, the operatorΨ:Ω⟶Ω is continuous.Proof.
It follows from Lemma7 that ΨΩ⊆Ω. Next, we show that Ψ is continuous. Let vn be a sequence such that vn⟶v in Ω as n⟶∞.
In view of condition (H1) (i) and Lebesgue dominated convergence theorem, we have(46)∫0tt−sq−1fs,vns,vns−fs,vs,vsDds⟶0,∀t∈0,T∗,asn⟶+∞.
Therefore, this together with the continuity ofM0,⋅ implies(47)Bμvns−BμvsD≤N1N2M0,vn−M0,vD+ΠfT∗,vn−ΠfT∗,vD+∫0T∗φT∗−sM0,vn−M0,vD+Πfs,vn−Πfs,vDds≤N1N2M0,vn−M0,vD+∫0T∗T∗−sq−1fs,vns,vns−fs,vs,vsDds+∫0T∗φT∗−sM0,vn−M0,vD+∫0ss−ηq−1fη,vnη,vnη−fη,vη,vηDdηds⟶0,asn⟶+∞.
Consequently, for eacht∈0,T∗, we have(48)Ψvnt−Ψvt≤Πt,vn,μvn−Πt,v,μvD+∫0tℑ˙t−sΠs,vn,μvn−Πs,v,μvds≤∫0t∫0tt−sq−1fs,vns,vns−fs,vs,vsDds+∫0tt−sq−1Bμvns−BμvsDds+∫0tφt−s∫0ss−ηq−1fη,vnη,vnη−fη,vη,vηDdηds+∫0tφt−s∫0ss−ηq−1Bμvnη−BμvηDdη⟶0,asn⟶+∞.
From the analogous proof of equicontinuous for operatorΨ in Lemma 7 and the well-known Ascoli–Arzelà theorem, it is not difficult to obtain Ψvn−ΨvCJ,X⟶0, as n⟶+∞, i.e., Ψ is continuous on Ω. This completes the proof.We present now our main results of this paper.Theorem 1.
If assumptions (H1)–(H4) hold, then the fractional evolution (1) is completely controllable on I.Proof.
Consider the operatorΨ defined as (28). In view of Lemma 1, it is enough to prove that when using the control μu, the operator Ψ has a fixed point u⋅ which is exactly a mild solution of (1) on J. From the verified fact uT∗=ΨuT∗=u1, it follows that the control μu steers system (1) from the initial function ϕ to u1 in finite time T∗. The complete controllability on I of system (1) is thus proved. To this end, we shall take advantage of Mönch fixed point theorem. The continuity of operator Ψ:Ω⟶Ω is given by Lemma 8. In the next step, we demonstrate that Mönch’s condition holds for operator Ψ:Ω⟶Ω.
LetB=co¯ΨΩ. It is not difficult to check that ΨB⊆B. Assume that bounded set D0⊂B is countable and D0⊂co¯u0∪ΨD0, and we shall prove that γD0=0. From Lemma 7, it is easy to derive that ΨD0 is equicontinuous on J. Notice that D0⊂co¯u0∪ΨD0, so D0 is also equicontinuous on J.
For anyu∈D0, let(49)Ψut=Ψ1ut+Ψ2ut,t∈0,T∗,ϕt−Mt,u,t∈−r,0,where(50)Ψ1ut=ϕ0−M0,u+Πt,u,μu,Ψ2ut=∫0tℑ˙t−sϕ0−M0,u+Πs,u,μuds.
No loss of generality, supposeD0=wnn=1∞. From hypothesis (H1) (ii) and Lemma 5, for any s∈I, it follows that(51)γfs,wns,wns≤msγwns+γwns≤msγwns+rγwn≤msγD0s+rγD0≤msr+1γD0.
Then, this indicates from Lemma2 and Hölder inequality that(52)γΠft,wn=γ1Γq∫0tt−sq−1fs,wns,wnsds≤2Γq∫0tt−sq−1γfs,wns,wnsds≤2r+1Γq∫0tt−sq−1msds⋅γwn≤2r+1Γq∫0tt−sq−11/1−q1ds1−q1∫0tms1/q1dsq1⋅γwn≤2r+1Γq1−q1q−q11−q1tq−q1mL1/q1⋅γwn≤λT∗qγD0,t∈0,T∗,which together with Lemma 2, (H2) (ii), and (H3) (i) implies(53)γμwns≤psγM0,wn+γΠfT∗,wn+2∫0T∗φT∗−sγM0,wn+γΠfs,wnds≤pskγwn+λT∗qγwn+2φL1Ikγwn+λT∗qγwn≤psk+λT∗q1+2φL1IγD0,s∈0,T∗.
Thus, by using Hölder inequality again, one has(54)γΠBt,μwn≤2N1Γq∫0tt−sq−1γμwnsds≤2N1k+λT∗q1+2φL1IΓq∫0tt−sq−1psdsγD0≤2N1Lk+λT∗q1+2φL1IΓqγD0,t∈0,T∗.
Consequently, from the abovementioned, we can derive(55)γΨ1D0t≤γM0,wn+γΠft,wn+γΠBt,μwn≤kγD0+λT∗qγD0+2N1Lk+λT∗q1+2φL1IΓqγD0≤k+λT∗q1+2N1L1+2φL1IΓqγD0≤λMT∗q+kMγD0.
Fort∈0,T∗, also in view of Lemma 2, one obtains(56)γΨ2D0t=γ∫0tℑ˙t−sϕ0−M0,wn+Πs,wn,μwnds≤2∫0tγℑ˙t−sϕ0−M0,wn+Πs,wn,μwnds≤2∫0tφt−sγΨ1D0sds≤2φL1IλMT∗q+kMγD0.
Then, by (55) and (56), we can infer(57)γΨD0t≤γΨ1D0t+γΨ2D0t≤λMT∗q+kMγD0+2φL1IλMT∗q+kMγD0≤λM1+2φL1IT∗q+kM1+2φL1IγD0,t∈0,T∗.
Besides, fort∈−r,0, we have from (H3) (i) that(58)γΨD0t≤γϕt−Mt,wn≤γMt,wn≤kγwn<γD0.
On the contrary, from the equicontinuity ofΨD0 on J, it follows that(59)γΨD0=maxt∈JγΨD0t.
Consequently, by (24) and (57)–(59), one can derive(60)γD0≤γco¯u0∪ΨD0≤γΨD0<γD0,which deduces γD0=0. Due to Lemma 2 (I) (i), we know that D0 is relatively compact. Then, from Lemma 3, Ψ has at least one fixed point u∈B. This shows that the complete controllability on I for system (1) is valid. The proof is now completed.Remark 3.
(i) Different from some papers [21, 25, 44] utilizing the definition of mild solution by probability density functions which are initially presented by El-Borai [3], the way of making use of differentiable resolvent operators to define mild solution in this paper can avoid the complexity of definitions and properties related to the probability density functions and its associated characteristic solution operators; sometimes, the limitation of fractional order q lying in (0, 1) due to the probability density functions can be extricated. (ii) In a comparative way, nonlocal item in this paper has better application effect in physics. In practical applications, it may be given by Mt,u=∑i=1kciuτi+t,t∈−r,0, where cii=1,2,…,k is a given constant and 0<τ1<τ2<⋅⋅⋅<τn≤T. At time t=0, we have M0,u=∑i=1kciuτi, which is exactly the case in [44].


## 4. An Example

As an application of our abstract results, we consider the following fractional partial differential nonlocal evolution equations with delay of the form:(61)∂1/3∂t1/3ut,τ=∂∂τut,τ+ϑtutτ2+utτ+∫−rtηt−sus,τds+δτμt,τ,t,τ∈0,T×0,π,ut,0=ut,π=0,t∈0,T,ut,τ+∑i=1lmisinuci+t,τ=ϕt,τ,t,τ∈−r,0×0,π,0<c1<c2<⋅⋅⋅<cl<T,where ϑ∈C0,T,R, η∈L−r,T+r,R, δ denotes the characteristic function of some subinterval D⊂0,π, μ∈C0,T×0,π,R and satisfies μt,0=μt,π=0 for t∈0,T. mi is a fixed constant for i=1,2,…,l satisfying ∑i=1lmi<1. ϕ∈C−r,0×0,π,X, which satisfies ϕt,0=ϕt,π=0 for t∈−r,0, X=C0,π,R, and D=u∈X:u′∈X,u0=uπ=0,Au=u′ for u∈D. Thus, A is an infinitesimal generator of a noncompact semigroup Tt:t≥0 which is given by Ttus=ut+s for u∈X. Let(62)D1/3utτ=∂1/3∂t1/3ut,τ,ft,ut,utτ=ϑtutτ2+utτ+∫−rtηt−sus,τds,utτ=ut,τ,Bμtτ=δτμt,τ,ϕtτ=ϕt,τ,Mt,uτ=∑i=1lmisinuci+t,τ,∑i=1lmi<1,0<c1<c2<⋅⋅⋅<cl<T.Under these assumptions, partial system (61) can be regarded as a control problem of the abstract form:(63)Dqut=Aut+ft,ut,ut+Bμt,a.e.t∈I≔0,T,ut+Mt,u=ϕt,t∈−r,0.For anyt∈−r,0 and for any bounded set D⊂C−r,T,X, we have that γMt,D≤∑i=1lmiγD<γD, which indicates that Mt,⋅:C−r,T,X⟶X is a strict k-set contraction with k=∑i=1lmi<1. In addition, it is not difficult to verify that f,B,andM satisfy other assumptions of Theorem 1. Thus, system (61) is completely controllable on 0,T.


## 5. Conclusions

In this work, complete controllability of a class of semilinear fractional nonlocal evolution systems with finite delay in Banach spaces is investigated by using properties of resolvent operators, theory of measure of noncompactness, and Mönch fixed point theorem. Under a weaker concept of complete controllability and a proper complete space we introduced, the controllability results of the addressed system are obtained without Lipschitz continuity and other growth conditions imposed on the nonlinearity and nonlocal item. In fact, the nonlinear function is only supposed to be continuous. Then, we improve and generalize some analogous results of fractional evolution systems such as [11, 21–25].



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