R E S E A R C H
Open Access
Approximate controllability of fractional
nonlocal evolution equations with multiple
delays
He Yang
*and Elyasa Ibrahim
*Correspondence:
[email protected] College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China
Abstract
This paper deals with the existence and approximate controllability for a class of fractional nonlocal control systems governed by abstract fractional evolution equations with multiple delays. Under some weaker assumptions, the existence as well as the approximate controllability is established by using fixed point theory. An example is given to illustrate the applicability of the abstract results.
MSC: 34A08; 34K35
Keywords: fractional evolution equations; nonlocal initial condition; multiple delays; compact analytic semigroup; approximate controllability
1 Introduction
Since the fractional differential equations have extensive physical background and realistic mathematical model, the theory has considerably developed in recent years; see [–] and the references therein. And the nonlocal initial conditions have better effects in applica-tions than the classical ones; see [, , , , –] and the references therein. Therefore, the theory of fractional nonlocal differential equations has been a research field of focus in recent years.
Controllability of deterministic and stochastic dynamical control systems in infinite-dimensional spaces is well-developed in which the details can be found in various papers; see [, –, , , ]. Several authors [, , ] investigated the exact controllability of control systems represented by nonlinear fractional evolution equations using fixed point approach. Debbouche and Baleanu [] established the exact null controllability result for a class of fractional integro-differential control systems governed by nonlinear fractional evolution equations with nonlocal initial conditions in Banach spaces. Liang and Yang [] investigated the exact controllability for a class of fractional integro-differential con-trol systems represented by nonlinear fractional evolution equations involving specific nonlocal functions. Sakthivelet al.[] studied the exact controllability for a class of frac-tional neutral control systems governed by abstract nonlinear fracfrac-tional neutral evolution equations.
However, in infinite-dimensional spaces, the concept of exact controllability is usually too strong []. Therefore, it is necessary to present a weaker concept of controllability,
namely approximate controllability for nonlinear control systems. In the recent literature, the approximate controllability of nonlinear fractional evolution systems has not yet been sufficiently studied. More precisely, there are limited papers regarding the approximate controllability of abstract nonlinear fractional evolution systems under different condi-tions [, , , ]. Kumar and Sukavanam [] obtained a new set of sufficient condicondi-tions of approximate controllability for a class of semilinear fractional control systems involving delay. Mahmudov and Zorlu [] established the sufficient conditions of approximate con-trollability for certain classes of abstract fractional evolution control systems. Sakthivalet al.[] investigated the approximate controllability for a class of nonlinear fractional dy-namical systems governed by abstract fractional evolution equations with nonlocal con-ditions.
However, to the best of our knowledge, the approximate controllability for nonlinear fractional nonlocal control systems governed by abstract fractional evolution equations involving multiple delays and compact analysis semigroup has not been investigated yet and it is also the motivation of this paper. In this paper, we consider the existence and approximate controllability for fractional nonlocal control system
⎧ ⎨ ⎩
Dqx(t) =Ax(t) +f(t,x(t),x(t–τ), . . . ,x(t–τ
n)) +Bu(t), t∈J:= [,T],
x(t) +g(x) =ϕ(t), t∈[–r, ], (.)
whereDq is Caputo fractional derivative of orderq∈(, ),T > is a constant,A
gen-erates a compact analytic semigroupS(t) (t≥) of uniformly bounded linear operator,
u∈L(J,Y) is a control,Y is a Banach space,B:Y →X
α is a linear bounded operator,
τ,τ, . . . ,τnare positive constants,r=max{τ,τ, . . . ,τn},ϕ: [–r, ]→Xαis continuous,f andgare given functions and will be specified later. The spaceXαwill be specified later.
The rest of this paper is organized as follows. In Section , some preliminaries are given on fractional calculus and fractional power of generator of compact analytic semigroup. In Section , we study the existence of mild solutions for fractional nonlocal control sys-tem (.). The approximate controllability of fractional nonlocal control syssys-tem (.) is discussed in Section . In Section , an example is given to illustrate the applicability of the abstract results.
2 Preliminaries
Let (X, · ) be a Banach space. Throughout this paper, we always assume thatA:D(A)⊂
X→Xgenerates a compact analytic semigroupS(t) (t≥) of uniformly bounded linear operator inX. Then there exists a constantM≥ such thatS(t) ≤Mfor allt≥. Let ∈ρ(A). Then
A–α:=
(α) ∞
tα–S(t)dt
for some <α< . Thus, we defineAαbyAα= (A–α)–withD(Aα) =A–α(X). LetX αbe the Banach space ofD(Aα) with normxα:=Aαxfor anyx∈D(Aα). Denote bySα(t) the restriction ofS(t) toXα for allt≥. ThenSα(t) (t≥) is aC-semigroup inXαand Sα(t)α≤ S(t)for allt≥. To prove our main results, the following lemmas are needed.
Lemma ([]) AαS(t)is bounded in X for any t> and there exists a constant M α>
such thatAαS(t) ≤M αt–α.
Lemma ([]) Sα(t) (t≥) is a compact semigroup in Xα, and hence it is
norm-continuous.
Denote byC([–r,T],Xα) the Banach space of all continuousXα-valued functions on the interval [–r,T] with normxC=maxt∈[–r,T]x(t)αfor anyx∈C([–r,T],Xα). Similarly, denote byC([–r, ],Xα) the Banach space of all continuousXα-valued functions on the interval [–r, ] with normxC[–r,]=maxt∈[–r,]x(t)αfor anyx∈C([–r, ],Xα). In this paper, we adopt the following definition of the mild solution of fractional nonlocal evolu-tion equaevolu-tion (.).
Definition By the mild solution of fractional nonlocal evolution equation (.), we mean a functionx∈C([–r,T],Xα) satisfying initial conditionx(t) +g(x) =ϕ(t),t∈[–r, ] and integral equation
x(t) =U(t)ϕ() –g(x)+ t
(t–s)q–V(t–s)F(x)(s) +Bu(s) ds, t∈J,
whereF(x)(s) =f(s,x(s),x(s–τ), . . . ,x(s–τn)), the operators{U(t)}t≥and{V(t)}t≥are defined by
U(t)x= ∞
ηq(θ)S
tqθx dθ, V(t)x=q
∞
θ ηq(θ)S
tqθx dθ, <q< ,
where
ηq(θ) =
qθ
––qρ
q
θ–q,
ρq(θ) =
π
∞
n=
(–)n–θ–qn–(nq+ )
n! sin(nπq), θ∈(,∞).
It is well known thatηq(θ)≥ for allθ∈(,∞) and
∞
ηq(θ)dθ= ,
∞
θ ηq(θ)dθ=
( +q).
Lemma ([, ]) The operators{U(t)}t≥and{V(t)}t≥satisfy the following properties: (i) For fixedt≥and anyx∈Xα,U(t)xα≤Mxα,V(t)xα≤(Mq)xα.
(ii) For fixedt> and anyx∈X,V(t)xα≤Cαt–qαx,whereCα=(+Mαq(–α)q(–α)). (iii) U(t)andV(t)are strongly continuous for allt≥.
(iv) U(t)andV(t)are norm-continuous inXfort> . (v) U(t)andV(t)are compact operators inXfort> .
(vi) For everyt> ,the restriction ofU(t)toXαand the restriction ofV(t)toXαare
norm-continuous.
(vii) For everyt> .the restriction ofU(t)toXαand the restriction ofV(t)toXαare
Letx(T;u) be the state value of fractional nonlocal evolution equation (.) at termi-nal timeT corresponding to controlu. Introduce the setR(T) byR(T) :={x(T;u) :u∈ L(J,Y)}.R(T) denotes its closure inXα.
Definition (Approximate controllability) The fractional nonlocal control system (.) is called approximately controllable on the interval [–r,T] ifR(T) =Xα.
Consider the following linear fractional differential system:
⎧ ⎨ ⎩
Dqx(t) =Ax(t) +Bu(t), t∈[,T],
x() =x∈Xα.
(.)
Let
T = T
(T–s)q–V(T–s)BB∗V∗(T–s)ds:Xα→Xα,
R,T=I+T–:Xα→Xα, > ,
whereB∗andV∗(t) denote the adjoint ofBandV(t), respectively. It is well known thatT
is a linear bounded operator andR(,T
) ≤.
Lemma ([]) The linear fractional differential system(.)is approximately control-lable on the interval[,T]if and only ifR(,T)→as→+in the strong operator topology.
Definition LetEbe a Banach space with norm · E. A mappingQ:E→Eis called a
nonlinear contraction if there exists a continuous and nondecreasing functionφ:R+→
R+satisfying
Qx–QyE≤φ
x–yE
for allx,y∈Ewithφ(τ) <τforτ> .
Remark It is clear ifφ(τ)≡kτ for somek∈(, ), the nonlinear contraction mapping degenerates into contraction mapping.
Lemma ([]) Let E be a Banach space and let Q,Q:E→E be two operators
satisfy-ing
(a) Qis a nonlinear contraction,and (b) Qis completely continuous.
Then either
(i) the operator equationx=Qx+Qxhas a solution,or (ii) the set:={x∈E:λ(Qx+Qx) =x, <λ< }is unbounded.
Lemma ([]) Letσ> and let a(t)be a nonnegative nondecreasing function locally
function defined on≤t<b,a(t)≤M(constant).Suppose that x(t)is nonnegative and local integrable on≤t<b with
x(t)≤a(t) +a(t) t
(t–s)σ–x(s)ds.
Then
x(t)≤a(t)
∞
k=
(a(t)(σ)tσ)k
(kσ+ ) , ≤t<b.
3 Existence of mild solutions We make the following assumptions:
(H) The functionf :J×Xαn+ →X is continuous and there exist positive constants
β,β, . . . ,βnandK≥ such that
f(t,ν,ν, . . . ,νn)≤ n
i=
βiνiα+K, t∈J, (ν,ν, . . . ,νn)∈Xαn+.
(H) The functiong:C([–r,T],Xα)→Xαis continuous and there exists a constantL≥
Msuch that
g(x) –g(y)α≤ x–yC
L+x–yC
, x,y∈C[–r,T],Xα
.
Remark The condition (H) can be replaced by the condition
(H) The function f : J× Xαn+ →X is continuous and there exist functions βi ∈
L(J,R+),i= , , . . . ,nand nondecreasing function: [, +∞)→(, +∞) such that
ft,x(t),x(t–τ), . . . ,x(t–τn)≤ n
i=
βi(t)x(t–τi)α
,
for anyt∈J,x∈C([–r,T],Xα), whereτ= . By (H), we have
ft,x(t),x(t–τ), . . . ,x(t–τn)≤ n
i=
βiLx(t–τi)α
.
Let(r) =rfor somer> . Then (H)⇒(H) withβi=βiL andK= . Since (H) would not be an essential generalization, we only consider (H) in the following.
Define an operatorQ:C([–r,T],Xα)→C([–r,T],Xα) by
(Qx)(t) = ⎧ ⎨ ⎩
ϕ(t) –g(x), t∈[–r, ],
U(t)(ϕ() –g(x)), t∈J. (.)
Proof Letx,y∈C([–r,T],Xα). Fort∈[–r, ], we have
(Qx)(t) – (Qy)(t)α=g(x) –g(y)α≤
x–yC
L+x–yC ≤
Mx–yC
L+x–yC
.
Fort∈[,T], we have
(Qx)(t) – (Qy)(t)α=U(t)
ϕ() –g(x)–U(t)ϕ() –g(y)α
≤Mg(x) –g(y)α≤ Mx–yC
L+x–yC
.
This implies that
Qx–QyC≤
Mx–yC
L+x–yC
.
Letφ(r) =LMr+r. Thenφ:R+→R+is continuous and nondecreasing andφ(r) <rforr> . HenceQis a nonlinear contraction inC([–r,T],Xα). This completes the proof.
For any> andh∈Xα, define a controlu(t) :=u(t;x) by
u(t;x) =B∗V∗(T–t)R,Th–U(T)
ϕ() –g(x)
– T
(T–s)q–V(T–s)F(x)(s)ds
.
Then, from assumptions (H) and (H), we have
Bu(t;x)α≤Lu, (.)
where
Lu=
LBA
–αhα+Mϕ
C[–r,]+M
+g()α
+LBMKT
q
(q+ )+
LBM
(q)
n
i= T
(T–s)q–βix(s–τi)αds,
LB=Bαsup t∈J
B∗V∗(T–t).
Define an operatorQ:C([–r,T],Xα)→C([–r,T],Xα) by
(Qx)(t) = ⎧ ⎨ ⎩
, t∈[–r, ],
t
(t–s)q–V(t–s)[F(x)(s) +Bu(s;x)]ds, t∈J.
(.)
Lemma If the assumptions(H)and(H)hold,Qis completely continuous.
Proof By the assumptions (H) and (H), it is easy to prove thatQ:C([–r,T],Xα)→
I= tt
(t–s)q–V(t–s)
F(x)(s) +Bu(s;x) ds
α .
By assumptions (H), (H) and (.), we can find
I≤Cα t
(t–s)q–– (t–s)q–(t–s)–qαF(x)(s)ds
+MLu[t
q
–t
q
– (t–t)q]
(q+ ) ,
I≤Cα t
t
(t–s)q(–α)–F(x)(s)ds
+MLu(t–t)
q
(q+ ) .
This implies thatIi→,i= , ast–t→. Fort> andη∈(,t) small enough, we have
I≤ t–η
(t–s)q–V(t–s) –V(t–s) F(x)(s) +Bu(s;x) αds
+ t
t–η
(t–s)q–V(t–s) –V(t–s) F(x)(s) +Bu(s;x) αds
≤ t–η
(t–s)q–F(x)(s)ds sup
s∈[,t–η]
V(t–s) –V(t–s)α
+Lu(t
q
–ηq)
q s∈[,supt–η]
V(t–s) –V(t–s)α
+Cα t
t–η
(t–s)q–(t–s)–qαF(x)(s)ds
+Cα t
t–η
(t–s)q(–α)–F(x)(s)ds
+MLuη
q
(q+ ).
Since compact semigroup is equicontinuous semigroup, it follows thatI→ ast–t→ andη→. Therefore, from the inequality
(Qx)(t) – (Qx)(t)α≤I+I+I,
we see that the operator Q is equicontinuous inC([–r,T],Xα). Hence, by the Ascoli-Arzela theorem,Qis a compact operator inC([–r,T],Xα). This completes the proof.
Theorem Assume that the conditions(H)and(H)hold.Then the fractional nonlocal control system(.)has at least one mild solution.
We next show that the conclusion (ii) is not possible. Equivalently, we prove that the set
Using the well-known singular version of Gronwall inequality [], we can deduce that there exists a constantM> such thatψ(t)≤M. Thus, for anyt≥, we have
Consequently,
xC= max
t∈[–r,T]
x(t)α≤M+MM.
This implies that the set is bounded. Therefore, by Lemma , the operator equation
x=Qx+Qx has at least one fixed point which is the mild solution of the fractional control system (.) onC([–r,T],Xα). This completes the proof.
Remark Even ifg(x)≡ and without controluin the fractional nonlocal control system (.), Theorem is still new.
The condition (H) can be replaced by the following condition:
(H)The functiong:C([–r,T],Xα)→Xα is Lipschitz continuous with constantL∈ (,
M), that is, for anyx,y∈C([–r,T],Xα), we have
g(x) –g(y)α≤Lx–yC.
Theorem Let the conditions(H)and(H)hold.Then the fractional nonlocal control
system(.)has at least one mild solution.
Proof By the condition (H), similar to the proof as in Lemma , we obtain
Qx–QyC≤MLx–yC, ∀x,y∈C
[–r,T],Xα
.
Letφ(r) =MLr. Thenφ:R+→R+is continuous and nondecreasing andφ(r) <rforr> . HenceQ is a nonlinear contraction onC([–r,T],Xα). The remaining proof is similar to the proof of Theorem , we omit it here. This completes the proof.
Remark In some existing literature, see [, , , ], the authors always assume that
f(t,x)≤m(t) with some functionsm∈L(J,R+) independent ofx, andgis either com-pletely continuous or Lipschitz continuous and the coefficients satisfy some inequality conditions. But in Theorems and , we only assume that the conditions (H) and (H) (or (H)) hold. Hence, Theorems and greatly extend the main results of [, , , , ].
4 Approximate controllability
To prove the approximate controllability of (.), the following assumption is required: (H) The functionf :J×Xn+
α →Xis bounded.
(H) The linear fractional control system (.) is approximately controllable.
Theorem In addition to the assumptions of Theorem,suppose that the conditions(H)
and(H)hold.Then the fractional nonlocal control system(.)is approximately control-lable.
Proof Letxbe a fixed point of the operatorQ+QonC([–r,T],Xα). By Theorem ,x is a mild solution of fractional nonlocal control system (.) with control
u(t;x) =B∗V∗(T–t)R
where
P(x) =h–U(T)
ϕ() –g(x)– T
(T–s)q–V(T–s)F(x)(s)ds,
F(x)(t) =f
t,x(t),x(t–τ), . . . ,x(t–τn)
, t∈J.
A direct calculation shows thatxsatisfies
x(T) =h–R
,TP(x). (.)
Moreover, by the assumption (H), there exists a constantN> such that T
F(x)(s)ds= T
fs,x(s),x(s–τ), . . . ,x(s–τn)
ds≤TN.
Consequently, there is a sequence still denoted by{F(x)(s)} weakly converging to, say, {F(s)}inL(J,X). Denote
ρ=h–U(T)
ϕ() –g(x)
– T
(T–s)q–V(T–s)F(s)ds.
Now, we have
P(x) –ρα≤
T
(T–s)q–V(T–s)F(x)(s) –F(s)αds
≤Cα T
(T–s)q(–α)–F(x)(s) –F(s)ds.
By using infinite-dimensional version of the Ascoli-Arzela theorem, we see that the oper-ator(·)→·(·–s)q(–α)–(s)ds:L(J,X)→C(J,X) is compact. Hence,P(x
) –ρα→ as→+. Moreover, by (.), we have
x(T) –hα≤R
,T(ρ)α+R,TP(x) –ρα ≤R,T(ρ)α+P(x) –ρα.
Hence, by Lemma and (H), we have
x(T) –hα→
as→+. This proves the approximate controllability of fractional nonlocal control sys-tem (.). This completes the proof.
Theorem In addition to the assumptions of Theorem,suppose that the conditions(H)
and(H)hold.Then the fractional nonlocal control system(.)is approximately control-lable.
Remark Compared with [], the present paper studies the fractional control system (.) with nonlocal condition and delays, and we suppose that the functionfmapsJ×Xαn+ toX. Hence, our results extend some existing results.
5 Existence and approximate controllability of delay parabolic equations Let∈RN be a bounded domain, whose boundary∂is sufficiently smooth. Let
A(z,D)x=
N
i,j=
∂ ∂zi
φij(z)
∂x ∂zj
–φ(z)x
be a uniformly elliptic differential operator of divergence form in. We assume that fol-lowing conditions are satisfied:
(A)φij∈C+μ() (i,j= , , . . . ,N) for someμ∈(, ), [φij(z)]N×N is a positive definite
symmetric matrix forz∈and there exists a constant> such that
N
i,j=
φij(z)ηiηj≥|η|, ∀η= (η,η, . . . ,ηN)∈RN,z∈.
(A)φ∈Cμ() for someμ∈(, ), andφ(z)≥ on. LetX=L(). Define an operatorA:D(A)⊂X→Xby
Ax=A(z,D)x, D(A) =H()∩H(). (.)
Then, from [],Agenerates a compact analysis semigroupS(t) (t≥) inXand there exists a constantM≥ such thatS(t) ≤M. It is well-known thatD(A) =H
(). Under above assumptions we discuss the approximate controllability of delay parabolic boundary value problem
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂q
∂tqx(z,t) =A(z,D)x(z,t) +F(z,t,x(z,t),x(z,t–τ), . . . ,x(z,t–τn))
+ωu(z,t), z∈,t∈[,T],
x|∂= ,
x(z,t) =ϕ(z,t) –g(x), z∈,t∈[–r, ],
(.)
where ∂∂tqq is the Caputo fractional partial derivative of orderq∈(, ),ω> is a constant,
τ, . . . ,τnare positive constants,r=max{τ, . . . ,τn},ϕ:×[–r, ]→R.
By Theorem , we have the following existence result.
Theorem Assume that the following conditions are satisfied:
(P)The function F:×[,T]×Rn+→Ris continuous and there exist positive
con-stantsβ,β, . . . ,βnand K such that
F(z,t,ξ,ξ, . . . ,ξn)≤ n
i=
βi|ξi|+K, z∈,t∈[,T].
(P)The function g is continuous and there exists a constant L≥M such that g(x) –g(y)≤ |x–y|
L+|x–y|,
Then the delay parabolic boundary value problem(.)has at least one mild solution.
By Theorem , we have the following approximate controllability result.
Theorem In addition to the assumptions of Theorem,suppose that the function F is bounded and the following condition holds:
(P)The linear fractional control system corresponding to(.)is approximately control-lable.
Then the delay parabolic boundary value problem(.)is approximately controllable.
By Theorem , we obtain the following theorem.
Theorem Assume that the conditions(P), (P)and
(P)The function g :C([–r,T],Xα)→Xα is Lipschitz continuous with constant L ∈ (,
M).
Then the delay parabolic boundary value problem(.)is approximately controllable.
Example Consider the approximate controllability of delay parabolic boundary value problem
6 Conclusion
In this paper, the approximate controllability of a nonlocal control system governed by fractional multi-delay evolution equation is investigated. By using fixed point theory and also the Gronwall-Bellman inequality of fractional order, sufficient conditions of approx-imate controllability are obtained. As an example, the approxapprox-imate controllability of frac-tional multi-delay parabolic boundary value problem is discussed and the abstract result is applied to a special fractional multi-delay parabolic function.
Acknowledgements
The research is supported by the NSF (No. 11661071).
Competing interests
None of the authors have any competing interests in the manuscript.
Authors’ contributions
All authors contributed equally in writing this paper. All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 24 April 2017 Accepted: 26 August 2017 References
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