# Complete Controllability for Fractional Evolution Equations

Authors: Xia Yang; Haibo Gu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The paper is concerned with the complete controllability of fractional
evolution equation with nonlocal condition by using a more general
concept for mild solution. By contraction fixed point theorem and
Krasnoselskii's fixed point theorem, we obtain some sufficient
conditions to ensure the complete controllability. Our obtained results are more general to known results.


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## Body

## 1. Introduction

Fractional differential equations have recently been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. It draws a great application in nonlinear oscillations of earthquakes and many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic model. There has been a significant development in fractional differential equations in recent years, see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3], Lakshmikantham et al. [4], and the papers [5–14] and the references therein.Some recent papers investigated the problem of the existence of a mild solution for abstract differential equation with fractional derivative [15–23]. However, the results in [15, 16, 18, 19] are incorrect since the considered variation of constant formulas is not appropriate [17]. Zhou and Jiao [22, 23] introduced two characteristic solution operators and gave a suitable concept on a mild solution by applying Laplace transform and probability density functions. But the condition that the analytic semigroup {T(t)}t≥0 was uniformly bounded was too strong. Shu et al. [20] researched the existence of mild solutions for impulsive fractional partial differential equation. But, Fec˘kan et al. [24] had pointed out that the definition of solution of impulsive fractional differential equation was not correct. By using Laplace transform, Shu and Wang [25] gave a definition of mild solution for fractional differential equation with order 1<α<2 and investigated its existence. Agarwal et al. [26] studied the existence and dimension of the set for mild solutions of semilinear fractional differential equations inclusions.In 1960, Kalman first introduced the concept of controllability which leads to some very important results regarding the behavior of linear and nonlinear dynamical systems. There are various works of complete controllability of systems represented by differential equations, integrodifferential equations, differential inclusions, neutral functional differential equations, and impulsive differential inclusions in Banach spaces (see [8, 27–29] and the references therein). Recently, more and more researchers also pay attention to study the controllability of fractional order evolution systems (see [21, 30, 31] and the references and therein). Unfortunately, the concept of mild solutions used in [30, 31] was not suitable for fractional evolution systems at all and the corresponding definition of mild solutions is only a simple extension of the mild solutions of integer order systems. Wang and Zhou [21] investigated the complete controllability of fractional evolution systems with two characteristic solution operators introduced by them.The nonlocal condition can be applied in physics with better effect than the classical initial conditionx(0)=x0. Nonlocal condition was initiated by Byszewski [32] when he proved the existence and uniqueness of mild and classical solutions of nonlocal Cauchy problems. As remarked by Byszewski and Lakshmikantham [33], the nonlocal condition can be more useful than the standard initial condition to describe some physical phenomena.Inspired by the above discussions, in this paper, we consider a class of fractional evolution equations. By using a more general definition of mild solution, we obtain some sufficient conditions to ensure the complete controllability.We consider the following fractional evolution equations:(1)CDtαx(t)=Ax(t)+f(t,x(t))+Bu(t),t∈J=[0,b],x(0)+g(x)=x0,
where CDtα is the Caputo fractional derivative of order α∈(0,1], the state x(·) takes value in a Banach space X, the control function u(·) is given in L2(J,U), with U as a Banach space, B is a bounded linear operator from U into X, A is a sectorial operator on X, f:J×X→X and g:X→X are given functions satisfying some assumptions, and x0∈X.The rest of this paper is organized as follows. In Section2, some notations and preparations are given. A suitable concept on a mild solution for our problem is introduced. In Section 3, the complete controllability results are obtained by using fixed point theorems. Some conclusions are given in Section 4.


## 2. Preliminaries

In this section, we will firstly introduce fractional integral and derivative, some notations about sectorial operators, solution operators, and analytic solution operators and then give the definition of a mild solution of system (1).Throughout this paper,ℝ, ℂ denote the sets of real and complex numbers, respectively, and ℝ+=[0,∞). By C(J,X), we denote the space of all continuous functions from J to X. ℒ(X) is the space of all bounded linear operators from X to X. D(A) denotes domain of A, while ρ(A) means resolvent set of A and R(λ,A)=(λI-A)-1 stands for the resolvent operator of A.Definition 1 (see [3]).
The fractional integral of orderp with the lower limit a for a function f:[a,∞)→ℝ is defined as
(2)aItpf(t)=1Γ(p)∫at‍f(s)(t-s)1-pds,t>a,p>0,
provided that the right side is point-wise defined on [a,∞), where Γ(·) is the gamma function.Definition 2 (see [3]).
The Riemann-Liouville derivative of orderp>0 for a function f:[a,∞)→ℝ is defined as
(3)aLDtpf(t)=1Γ(n-p)dndtn∫at‍f(s)(t-s)p+1-nds,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiit>a,n-1<p<n.Definition 3 (see [3]).
The Caputo derivative of orderp>0 for a function f:[a,∞)→ℝ is defined as
(4)aCDtpf(t)=aLDtp[f(t)-∑k=0n-1‍tkk!f(k)(0)],iiiiiiiiiiiiiiiiiiiiiiiiiit>a,n-1<p<n.
Let CDtpf(t)=D0Ctpf(t).Remark 4.
(i) Iff∈Cn[a,∞), then
(5)aCDtpf(t)=1Γ(n-p)∫at‍f(n)(s)(t-s)p+1-nds,iiiiiiiiiiiiiiiiiit>a,n-1<p<n.
The Caputo derivative of a constant is equal to zero.
Iff is an abstract function with values in X, then the integrals which appear in Definitions 1 and 2 are taken in Bochner’s sense.Definition 5.
An operatorA is said to be sectorial if there are constants ω∈ℝ, θ∈(0,π/2), and M>0 such that the resolvent of A exists outside the sector
(6)Σθ(ω)={λ∈ℂ:λ≠ω,|arg(λ-ω)|<θ}
with
(7)∥R(λ,A)∥ℒ(X)≤M|λ-ω|,λ∉Σθ(ω).Consider the following Cauchy problem for the Caputo derivative evolution equation of orderα(0<α≤1):
(8)DtαCu(t)=Au(t),t∈J=[0,b],u(0)=u0.Definition 6 (see [7]).
A family{Sα(t)}t≥0⊆ℒ(X) is called a solution operator for system (8), if the following conditions are satisfied:(a)
{Sα(t)}t≥0 is strongly continuous, for t≥0 and Sα(0)=I.(b)
Sα(t)D(A)⊆D(A) and ASα(t)x=Sα(t)Ax, for all x∈D(A) and t≥0.(c)
Sα(t)x is a solution of the following integral equation:
(9)u(t)=u0+1Γ(α)∫0t‍Au(t)(t-s)1-αds,for all x∈D(A) and t≥0.Remark 7 (see [7]).
If{Sα(t)}t≥0 is the solution operator of system (8), then
(10)Ax=Γ(α+1)lim⁡t→0Sα(t)x-xtα,
where D(A) consists of those x∈X for which this limit exists. We call A the infinitesimal generator of {Sα(t)}t≥0 or say that A generates {Sα(t)}t≥0.Remark 8 (see [7]).
The solution operator{Sα(t)}t≥0 of system (8) is defined as follows:
(11)Sα(t)=12πi∫γ‍eλtλα-1R(λα,A)dλ,
where γ is a suitable path such that λα∉Σθ(ω), for λ∈γ.A operatorA is said to belong to ℭα(X,M,ω) or ℭα(M,ω), if system (8) has solution operator {Sα(t)}t≥0 satisfying |Sα(t)|≤Meωt, t≥0. Denote ℭα(ω)=⋃‍{ℭα(M,ω),M≥1} and ℭα=⋃‍{ℭα(ω),ω≥0}.Definition 9 (see [7]).
A solution operator{Sα(t)}t≥0 of system (8) is called analytic, if {Sα(t)}t≥0 admits an analytic extension to a sectorial ∑θ0‍={λ∈ℂ∖{0}:|argλ|<θ0} for some θ0∈(0,π/2]. An analytic solution operator is said to be of analyticity type (θ0,ω0), if, for each θ<θ0 and ω>ω0, there is an M=M(θ,ω) such that |Sα(t)|≤eωRet, t∈∑θ‍={t∈ℂ∖{0}:|argt|<θ}. Denote
(12)𝒜α(θ0,ω0)={A∈ℭα:AgeneratesanalyticsolutioniiiiiiiiiiiiiiiiiiiSα(t)oftype(θ0,ω0)}.Lemma 10 (see [7]).
Letα∈(0,2); a linear closed densely defined operator A belongs to 𝒜α(θ0,ω0), if λα∈ρ(A), for each λ∈Σθ0+π/2 and, for any ω>ω0, θ<θ0, there is a constant C=C(θ,ω) such that
(13)∥λα-1R(λα,A)∥≤C|λ-ω|forλ∈Σθ+π/2(ω).
Next, we consider the definition of the mild solution of system (1).According to Definitions1 and 2, it is suitable to rewrite the nonlocal Cauchy problem (1) in the equivalent integral equation
(14)x(t)=x0-g(x)+1Γ(α)iiiiiiiii×∫0t‍(t-s)α-1(Ax(s)+f(s,x(s))+Bu(s))ds,
provided that the integral in (14) exists.The following Lemma11 is discussed in [20]; for the sake of completeness, we outline its proof here.Lemma 11.
If (14) holds and A is a sectorial operator, then we have
(15)x(t)=Sα(t)(x0-g(x))+∫0t‍Tα(t-s)f(s,x(s))dsiiiiiiiiiii+∫0t‍Tα(t-s)Bu(s)ds,
where(16)Sα(t)=12πi∫γ‍eλtλα-1R(λα,A)dλ,Tα(t)=12πi∫γ‍eλtR(λα,A)dλ,
and γ is a suitable path such that λα∉Σθ(ω), for λ∈γ.Proof.
By applying the Laplace transform to (14), we have
(17)ℒ(x)(λ)=x0-g(x)λ+Aℒ(x)(λ)+ℒ(f+Bu)(λ)λα.
Since (λαI-A)-1 exists, that is, λα∈ρ(A), from the above equation, we obtain
(18)ℒ(x)(λ)=λα-1(λαI-A)-1(x0-g(x))iiiiiiiiiiiiiiii+(λαI-A)-1ℒ(f+Bu)(λ).
Therefore, by the Laplace inverse transform, we have
(19)x(t)=Sα(t)(x0-g(x))+∫0t‍Tα(t-s)f(s,x(s))dsiiiiiiiiiii+∫0t‍Tα(t-s)Bu(s)ds.Lemma 12.
Ifα∈(0,1] and A∈𝒜α(θ0,ω0), then the operators Sα(t) and Tα(t) are continuous on t∈ℝ+.Proof.
For0≤t′<t′′, by Lemma 10, we have
(20)|Sα(t′′)-Sα(t′)|=|∫γ‍(eλt′′-eλt′)λα-1R(λα,A)dλ|iiiiiiiiiiiiiiiiiiiiiiiiii≤C∫γ‍|eλt′′-eλt′||dλ||λ-ω|;
choose the integration path γ as follows:
(21)γ={ω+re-i((π/2)+δ):ϱ≤r<∞}i⋃{ω+ϱeiφ:|φ|≤π2+δ}i⋃{ω+rei((π/2)+δ):ϱ≤r<∞},
such that γ is oriented counterclockwise, where δ∈(0,δ0), ω>ω0, and ϱ>0.
From (20), we have
(22)|Sα(t′′)-Sα(t′)|≤2C∫ϱ∞‍|eλt′′-eλt′|drr+C∫(-π/2)-δ(π/2)+δ‍|eλt′′-eλt′|dφ≤2Cϱ∫ϱ∞‍|e(ω+rcos⁡((π/2)+δ)-irsin((π/2)+δ))t′′iiiiiiiiiiiiiiiiiiii-e(ω+rcos⁡((π/2)+δ)-irsin((π/2)+δ))t′|dr+C∫(-π/2)-δ(π/2)+δ‍|e(ω+ϱcos⁡φ+iϱsinφ)t′′iiiiiiiiiiiiiiiiiiiiiiiiii-e(ω+ϱcos⁡φ+iϱsinφ)t′|dφ.
By noticing that cos⁡((π/2)+δ)<0, by the dominated convergence theorem, we have |Sα(t′′)-Sα(t′)|→0 as t′′→t′, which implies that Sα(t) is continuous on t. For the same reason, Tα(t) is too continuous on t. The proof is complete.Lemma 13 (see [20]).
Ifα∈(0,1] and A∈𝒜α(θ0,ω0), then, for any t>0, we have
(23)|Tα(t)|≤Ceωt(1+tα-1),ω>ω0.IfA∈𝒜α(θ0,ω0), then |Sα(t)|≤Meωt and |Tα(t)|≤Ceωt(1+tα-1), for all t∈(0,+∞). Let
(24)M~S=sup⁡0≤t≤b|Sα(t)|,M~T=sup⁡0≤t≤bCeωt(1+t1-α);
we have
(25)|Sα(t)|≤M~S,|Tα(t)|≤tα-1M~T∀t∈(0,+∞).Lemma 14 (Krasnoselskii’s fixed point theorem).
LetE be a Banach space, let B be a bounded closed and convex subset of E, and let F1 and F2 be maps of B into E such that F1x+F2y∈B for every pair x,y∈B. If F1 is a contraction and F2 is completely continuous, then the equation F1x+F2x=x has a solution on B.In [34], Reich gave a general fixed point theorem which contained Krasnoselskii’s fixed point theorem, for more details we can see the reference.Definition 15.
A functionx:J→X is called a mild solution of system (1), if x satisfies the following equation
(26)x(t)=Sα(t)(x0-g(x))+∫0t‍Tα(t-s)f(s,x(s))dsiiiiiiiiii+∫0t‍Tα(t-s)Bu(s)ds,
where
(27)Sα(t)=12πi∫γ‍eλtλα-1R(λα,A)dλ,Tα(t)=12πi∫γ‍eλtR(λα,A)dλ
and γ is a suitable path such that λα∉Σθ(ω), for λ∈γ.Remark 16.
Whenα=1, Sα(t)=Tα(t) is a C0-semigroup and system (1) degenerates into 1 order evolution equation. However, the limits of Sα(t) and Tα(t) in [21–23] did not exist as α→1-.Remark 17.
WhenA generates a C0-semigroup {T(t)}t≥0 in system (1), we have (28)Sα(t)=∫0∞‍ϕα(θ)T(tαθ)dθ,Tα(t)=αtα-1∫0∞‍θϕα(θ)T(tαθ)dθ,
where 0<α<1 and ϕ(θ) is a probability density function defined on (0,∞) in [21–23]. So, this definition is more general to that in [21–23].Remark 18.
It is easy to verify that a classical solution of system (1) is a mild solution of the same system.Definition 19.
The system (1) is said to be completely controllable on J, if, for every x0,x1∈X, there exists a control u∈L2(J,U), such that a mild solution x of system (1) satisfies x(b)=x1.
In this paper, we assume the following.(H1)
f:J×X→X is continuous and there exist constant q∈(0,α) and function m∈L1/q(J,ℝ+) such that
(29)|f(t,x(t))-f(t,y(t))|≤m(t)∥x-y∥,for allt∈J and x,y∈X;(H2)
g:X→X is continuous and there exists a constant Lg>0 such that
(30)∥g(x)-g(y)∥≤Lg∥x-y∥,for allx,y∈X.
It is easy to see that if(H2) holds, then the following assumption holds:(H2′)
g:X→X is continuous and there exist positive constants Kg and d such that
(31)∥g(x)∥≤Kg∥x∥+d,for allx∈X;(H3)
the operator family{Sα(t)}t≥0 is compact;(H4)
the linear operatorB:L2(J,U)→L(J,X) is bounded; W:L2(J,U)→X defined by
(32)Wu=∫0b‍Tα(b-s)Bu(s)dshas an inverse operatorW-1 which takes values in L2(J,U)/Ker⁡W and there exist two positive constants M2,M3>0 such that
(33)∥B∥≤M2,∥W-1∥≤M3.


## 3.  Complete Controllability Results

Theorem 20.
Suppose that(H1), (H2), and (H4) are satisfied; then system (1) is completely controllable on J, provided that A∈𝒜α(θ0,ω0) and
(34)Θ=(M~SLg+M~Tbα-q(1-qα-q)1-q∥m∥L1/q)IIIII×(1+M2M3M~Tbαα)<1.Proof.
Using hypothesis(H4) for an arbitrary function x∈C(J,X), we defined the control function ux(t) by
(35)ux(t)=W-1(∫0b‍x1-Sα(b)(x0-g(x))IIIIIIIII-∫0b‍Tα(b-s)f(s,x(s))ds).
We show that using this control, the operator F on C(J,X) by
(36)(Fx)(t)=Sα(t)(x0-g(x))+∫0t‍Tα(t-s)f(s,x(s))dsIIIIIIIIIIIII+∫0t‍Tα(t-s)Bux(s)ds
has a fixed point x, which is a mild solution of system (1).
It is obvious that(Fx)(b)=x1, which means that ux steers the mild x from x0 to x1 in finite time b. This implies that system (1) is completely controllable on J. Next, we will prove that F has a fixed point on C(J,X).
Takingt∈[0,b] and, for all x,y∈C(J,X), we have, from (H1), (H4), and (35),
(37)|ux(t)-uy(t)|=∥∫0b‍W-1(Sα∫0b‍(b)(g(x)-g(y))IIIIiiiiiiiiIIII+∫0b‍Tα(b-s)f((s,x(s))-f(s,y(s)))ds)∥≤M3M~T∫0b‍(b-s)α-1m(s)ds∥x-y∥+M3M~SLg∥x-y∥≤(M3M~Tbα-q(1-qα-q)1-q∥m∥L1/q+M3M~SLg)∥x-y∥;
by (25), we have
(38)|(Fx)(t)-(Fy)(t)|=|∫0t‍Sα(t)(g(y)-g(x))iiiiiiiiii+∫0t‍Tα(t-s)(f(s,x(s))-f(s,y(s)))dsiiiiiiiiii+∫0t‍Tα(t-s)B(ux(s)-uy(s))ds|≤|Sα(t)|∥g(y)-g(x)∥+|∫0t‍Tα(t-s)(f(s,x(s))-f(s,y(s)))ds|+|∫0t‍Tα(t-s)B(ux(s)-uy(s))ds|≤M~SLg∥y-x∥+M~T∫0t‍(t-s)α-1m(s)∥y-x∥ds+M~T∫0t‍(t-s)α-1|B(ux(s)-uy(s))|ds≤M~SLg∥y-x∥+M~T(∫0t‍(t-s)(α-1)/(1-q)ds)1-q×(∫0t‍(m(s))1/qds)q∥y-x∥+bααM2M3M~T×(M~Tb2α-qα(1-qα-q)1-q∥m∥L1/q+M~SLg)∥y-x∥≤[M~SLg+M~Tbα-q(1-qα-q)1-q∥m∥L1/qiiii×(1+M2M3M~Tbαα)]∥y-x∥=Θ∥y-x∥.
Hence, F is a contraction mapping and has a unique fixed point x*∈C(J,X). Therefore, this x* is a mild solution of system (1). The proof is complete.Theorem 21.
Suppose that(H1), (H2′), (H3), and (H4) are satisfied; then system (1) is completely controllable on J provided that A∈𝒜α(θ0,ω0) and
(39)M4=(M2M3M~Tbαα+1)d7×[M~SKg+M~T(1-qα-q)1-qbα-q∥m∥L1/q]<1.Proof.
Define(40)(F1x)(t)=∫0t‍Tα(t-s)f(s,x(s))dsiiiiiiiiiiiiiiii+∫0t‍Tα(t-s)Bux(s)ds,(F2x)(t)=Sα(t)(x0-g(x)),
for t∈[0,b] and any x∈C(J,X). Taking into account (35), by (H1), (H2′), and (H4), we have
(41)|ux(t)|=∥∫0b‍W-1(∫0b‍x1-Sα(b)(x0-g(x))iiiiiiiiiiiiiiiiiiiiiiii-∫0b‍Tα(b-s)f(s,x(s))ds)∥iiiiiiiii≤∥W-1∥(∫0b‍∥x1∥+|Sα(b)(x0-g(x))|iiiiiiiiiiiiiiiiiiiiiiiiiii+|∫0b‍Tα(b-s)f(s,x(s))ds|)iiiiiiiii≤M3(∫0b‍∥x1∥+M~S∥x0-g(x)∥iiiiiiiiiiiiiiiiiiiiii+M~T∫0b‍(b-s)α-1(m(s)∥x∥+|f(s,0)|)ds∫0b‍)iiiiiiiii≤M3[+bααsup⁡s∈J|f(s,0)|)∥x1∥+M~S(∥x0∥+Kg∥x∥+d)iiiiiiiiiiiiiiiiiiii+M~T((1-qα-q)1-qbα-q∥m∥L1/qiiiiiiiiiiiiiiiiiiiiiiiiiiii×∥x∥+bααsup⁡s∈J|f(s,0)|)].
In order to make the following process clear, we divide it into several steps.
S
t
e
p
I. For t∈[0,b] and any x,y∈C(J,X), we have
(42)|(F1x)(t)|=|∫0t‍Tα(t-s)f(s,x(s))dsiiiiiiiiii+∫0t‍Tα(t-s)Bux(s)ds|≤|∫0t‍Tα(t-s)(f(s,x(s))-f(s,0))ds|+|∫0t‍Tα(t-s)f(s,0)ds|+|∫0t‍Tα(t-s)Bux(s)ds|≤M~T∫0t‍(t-s)α-1m(s)∥x∥ds+M~Tsup⁡s∈J|f(s,0)|∫0t‍(t-s)α-1ds+M~T∫0t‍(t-s)α-1M2∥ux(s)∥ds≤M~T(∫0t‍(t-s)(α-1)/(1-q)ds)1-q∥m∥L1/q∥x∥+bααM~Tsup⁡s∈[0,b]|f(s,0)|+bαα×M~TM2M3[+bααsup⁡s∈J|f(s,0)|)∥x1∥+M~S(∥x0∥+Kg∥x∥+d)iiiiiiiiiiiiiiiiiiiiiiiiiii+M~T((1-qα-q)1-qbα-q∥m∥L1/q∥x∥iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii+bααsup⁡s∈J|f(s,0)|)]≤[(1+bααM~TM2M3)M~T(1-qα-q)1-qbα-q∥m∥L1/qiiiiiiiiii+M2M3M~TbααM~SKg]∥x∥+M2M3M~Tbαα[∥x1∥+M~S(∥x0∥+d)]+(M2M3M~Tbαα+1)bααM~Tsup⁡s∈J|f(s,0)|,|(F2y)(t)|=|Sα(t)(x0-g(y))|iiiiiiiiiiiiiii≤M~S(∥x0∥+Kg∥y∥+d).
By the condition M4<1, we can find k0>0 such that, for x,y∈Bk0={x∈C(J,X):∥x∥≤k0},
(43)∥F1x+F2y∥≤k0i.e.F1x+F2y∈Bk0.
S
t
e
p
II. F1 is a contraction mapping on Bk0.
For anyx,y∈Bk0 and t∈[0,b], we have
(44)|(F1x)(t)-(F1y)(t)|≤|∫0t‍Tα(t-s)(f(s,x(s))-f(s,y(s)))ds|+|∫0t‍Tα(t-s)B(ux(s)-uy(s))ds|≤M~T∫0t‍(t-s)α-1m(s)∥x-y∥ds+M~TM2∫0t‍(t-s)α-1|ux(s)-uy(s)|ds≤M~T(∫0t‍(t-s)(α-1)/(1-q)ds)1-q∥m∥L1/q∥x-y∥+M2M~Tbαα(M3M~Tbα-q(1-qα-q)1-q∥m∥L1/qiiiiiiiiiiiiiiiiiiiiiiiiiiii+M3M~SKg(1-qα-q)1-q)∥x-y∥≤(bααM~T(1-qα-q)1-qbα-q∥m∥L1/q(1+bααM~TM2M3)iiiiiiiiii+bααM~TM2M3M~SKg)∥x-y∥=:Υ∥x-y∥.
From the condition M4<1, we obtain Υ<1, which implies that F1 is a contraction mapping.
S
t
e
p
III. F2 is a completely continuous operator.
First, we will prove thatF2 is continuous on Bk0. Let {xn}⊆Bk0 with xn→x∈Bk0. By (H2′), we have
(45)g(xn)⟶g(x)asn⟶∞.
So, we have
(46)|(F2xn)(t)-(F2x)(t)|=|Sα(t)(x0-g(xn))-Sα(t)(x0-g(x))|=|Sα(t)(g(x)-g(xn))|≤M~S∥g(x)-g(xn)∥⟶0asn⟶∞,
which implies that F2 is continuous.
Next, we will show that{F2x,x∈Bk0} is relatively compact. It suffices to show that the family of function {F2x,x∈Bk0} is uniformly bounded and equicontinuous and, for any t∈[0,b], {(F2x)(t),x∈Bk0} is relatively compact.
For anyx∈Bk0, we have ∥F2x∥≤k0 which implies that {F2x,x∈Bk0} is uniformly bounded. In the following, we will show that {F2x,x∈Bk0} is a family of equicontinuous functions.
For anyx∈Bk0 and 0≤t′<t′′≤b, we have
(47)|(F2x)(t′′)-(F2x)(t′)|=|Sα(t′′)(x0-g(x))-Sα(t′)(x0-g(x))|≤|Sα(t′′)-Sα(t′)|∥x0-g(x)∥≤|Sα(t′′)-Sα(t′)|(∥x0∥+Kgk0+d).
From Lemma 12, we have |(F2x)(t′′)-(F2x)(t′)|→0 independently of x∈Bk0 as t′′-t′→0, which means that {F2x,x∈Bk0} is equicontinuous.
By the compactness of{Sα(t)}t≥0, we know that {(F2x)(t),x∈Bk0} is relatively compact. Therefore, {F2x,x∈Bk0} is relatively compact by Arzela-Ascoli theorem. The continuity of F2 and relative compactness of {F2x,x∈Bk0} imply that F2 is a completely continuous operator. By using Krasnoselskii’s fixed point theorem, we obtain that F1+F2 has a fixed point on Bk0. Therefore, the nonlocal Cauchy problem (1) has at least one mild solution. The proof is complete.When there is no control term, system (1) degenerates to the following system:
(48)DCtαx(t)=Ax(t)+f(t,x(t)),t∈J=[0,b],x(0)+g(x)=x0.
As a direct result of Theorems 20 and 21, we have the following corollaries.Corollary 22.
Suppose that(H1), (H2), and (H4) are satisfied; then system (48) has a unique mild solution, if A∈𝒜α(θ0,ω0) and
(49)Θ=M~SLg+M~Tbα-q(1-qα-q)1-q∥m∥L1/q<1.Corollary 23.
Suppose that(H1), (H2′), (H3), and (H4) are satisfied; then system (48) has at least one mild solution, if A∈𝒜α(θ0,ω0) and
(50)M4=M~SKg+M~T(1-qα-q)1-qbα-q∥m∥L1/q<1.


## 4. Conclusions

In this paper, we introduce a more general definition for mild solution of fractional evolution equation with nonlocal condition based on solution operator. By contraction fixed point theorem and Krasnoselskii’s fixed point theorem, we obtain some sufficient conditions to ensure the complete controllability for system (1). Here, we do not require the operator A to be the infinitesimal generator of an analytic semigroup {T(t)}t≥0 of uniform boundedness. So, the results we obtained are more general. For fractional evolution equation with Riemann-Liouville derivative, since it is equipped with a singular initial, it will be a difficult problem.



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*Source: 730695-2014-04-10.xml*