R E S E A R C H
Open Access
A generalized Gronwall inequality and its
application to fractional neutral evolution
inclusions
Zufeng Zhang
1,2*and Zhangzhi Wei
1,2*Correspondence: [email protected] 1Laboratory of Intelligent
Information Processing, Suzhou University, Suzhou, Anhui 234000, P.R. China
2School of Mathematics and
Statistics, Suzhou University, Suzhou, Anhui 234000, P.R. China
Abstract
This paper deals with the fractional neutral evolution differential inclusions. The existence results are established by using the fractional power of operators and a fixed point theorem for multivalued map. Moreover, we present a new generalized Gronwall inequality with singularity, which is an important tool in the proof of solvability.
MSC: 26A33; 34A60; 34G25
Keywords: generalized Gronwall inequality; fractional neutral evolution inclusions; mild solution; analytic semigroup
1 Introduction
In recent years, the theory of fractional differential equations is an important area of in-vestigation for its wide applicability in the fields of physics, engineering, and economics. For more details, we refer to the monographs of Oldham and Spanier [], Miller and Ross [] and Podlubny [] and papers [–].
This paper deals with the fractional neutral evolution inclusions of the form ⎧
⎨ ⎩
Dq(x(t) –g(t,x
t))∈–Ax(t) +F(t,xt), t∈[,b],
x(t) =φ(t), t∈[–τ, ], (.)
whereDqrepresents the Caputo fractional derivative of order <q< ,b> , –Ais the infinitesimal generator of an analytic semigroup{T(t)}t≥on a Banach spaceX,F: [,b]×
C([–τ, ],X)→P(X) is a multivalued map with nonempty, bounded, convex, and closed values,P(X) is the family of all nonempty subsets ofX,g: [,b]×C([–τ, ],X)→Xis a given function to be introduced later, andφ∈C([–τ, ],X),τ> . For anyx∈C([–τ,b],X) and anyt∈[,b], definext(ς) byxt(ς) =x(t+ς) forς∈[–τ, ], wherext(·) represents the history of the state from timet–τ up to the present timet.
Neutral fractional differential systems arise in many areas of applied mathematics and have received much attention recently. For some applications and recent results, we re-fer to [–] and the rere-ferences therein. Very recently, Wang and Zhou [] studied the existence and controllability results for fractional evolution differential inclusions
ing the Caputo derivative in Banach spaces. Yan [] investigated the controllability of fractional-order functional integro-differential inclusions with infinite delay in Banach spaces. We remark that the existence or the controllability results in these mentioned pa-pers are obtained under strong assumptions on the nonlinear partsf andFin equations and inclusions, respectively. The purpose of this paper is to release the limitation on the nonlinear term. For this purpose, we first present a new generalized Gronwall inequality with singularity, which is effective in dealing with fractional neutral evolution systems.
The remainder of the paper is arranged as follows. In Section , we give some basic definitions and preliminary results. A new generalized Gronwall inequality is proved in Section . Section is devoted to the existence result of mild solutions for problem (.). An example is presented in Section to illustrate our main theorem.
2 Preliminaries
Let (X,| · |) be a Banach space, andJ⊂R. ThenC(J,X) is the Banach space consisting of all continuous functions fromJintoXequipped with the normyJ=sup{|y(t)|:t∈J},
andB(X) denotes the Banach space of all bounded linear operators fromXintoXwith the norm
T=supT(y):|y|= ,
whereT∈B(X) andy∈X.
For ≤p≤ ∞, define the norm of a measurable functionm:J→Rby
mLpJ= ⎧ ⎨ ⎩
(J|m(t)|pdt)p, ≤p<∞,
infμ(J)={supt∈J–J|m(t)|}, p=∞,
where μ(J) is the Lebesgue measure on J. Denote byLp(J, R) the Banach space of all Lebesgue-measurable functionsm:J→Rwith the norm satisfying · LpJ<∞.
Lemma .(Hölder inequality) Let r,p≥andr+p= .If l∈Lr(J, R)and m∈Lp(J, R),
then lm∈L(J, R)andlmLJ≤ lLrJmLpJ.
Lemma .(Bochner theorem) A measurable function x:J→X is Bochner-integrable if
|x|is Lebesgue-integrable.
Let –Abe the infinitesimal generator of an analytic semigroup{T(t)}t≥of uniformly
bounded linear operators onX. Let ∈ρ(A), whereρ(A) is the resolvent set ofA. Then we can define the fractional powerAα for <α≤ as a closed linear operator on the
domainD(Aα). We have the following known results:
(i) There isM≥ such that
M:= sup
t∈[,+∞)
T(t) ≤ ∞. (.)
(ii) For anyα∈(, ], there exists a positive constantCαsuch that
AαT(t) ≤Cα
tα, <t≤b. (.)
Definition .([]) The Riemann-Liouville fractional integral of orderα∈R+off : R+→
Xis defined by
Iαf(t) =
(α) t
(t–s)α–f(s)ds, t> ,
whereis the gamma function.
Definition .([]) The Caputo fractional derivative of order <α< off ∈C([,∞),
X) is defined by
Dαf(t) =
( –α) t
(t–s)–αf (s)ds, t> .
Next, we present some basic definitions and results on multivalued maps. See [] for more details.
Let (X,d) be a metric space, andP(X) be the family for all nonempty subsets ofX. We give the following notation:
Pcl(X) =Y∈P(X) :Yis closed, Pbd(X) =Y∈P(X) :Yis bounded, Pcv(X) =Y∈P(X) :Yis convex, Pcp(X) =Y∈P(X) :Yis compact.
A multivalued mapF:X→P(X) is convex (closed) valued ifF(x) is convex (closed) for allx∈X;F is bounded on bounded sets ifF(B) =Ux∈BF(x) is bounded inXfor all
B∈Pbd(X), that is,supx∈B{sup{|y|:y∈F(x)}}<∞;Fis said to be upper semi-continuous (u.s.c. for short) onXif for eachx∈X, the setF(x) is a nonempty, closed subset ofX
and if for each open setU ofXcontainingF(x), there is an open neighborhoodVofx
such thatF(V)⊂U; andFis completely continuous ifF(B) is relatively compact for every
B∈Pbd(X).
If a multivalued mapF is completely continuous and has nonempty compact values, thenFis u.s.c. if and only ifFhas a closed graph, that is,xn→x∗,yn→y∗,yn∈F(xn) implyy∗∈F(x∗).
ConsiderHd:P(X)×P(X)→R+∪ {∞}given by
Hd(A,B) =max
sup
x∈A
d(x,B),sup
y∈B
d(A,y)
,
whered(A,y) =infx∈Ad(x,y),d(x,B) =infy∈Bd(x,y). Then (Pbd,cl(X),Hd) is a metric space, and (Pcl(X),Hd) is a generalized metric space (see []).
Definition .([]) A multivalued operatorN :X→Pcl(X)
(i) is a contraction if and only if there exists <γ< such that, for eachx,y∈X,
Hd
N(x),N(y)≤γd(x,y);
Theorem .([]) Let J be a compact interval,and X be a Banach space.Let the map F:J×C(J,X)→Pbd,cl,cv(X), (t,u)→F(t,u),be measurable with respect to t for each u∈X
and upper semicontinuous with respect to u for each t∈J.Moreover,suppose that for each fixed u∈C(J,X),the set
NF,u=
f ∈L(J,X) :f(t)∈F(t,u)for a.e. t∈J
is nonempty.Also,letT be a linear continuous mapping from L(J,X)to C(J,X).Then the
operator
T ◦NF:C(J,X)→Pbd,cl,cv
C(J,X), u→(T ◦NF)(u) =T(NF,u),
is a closed graph operator in C(J,X)×C(J,X).
The main tool in our approach is the following fixed point theorem.
Theorem .([, ]) Let X be a Banach space,andA:X→Pcl,cv,bd(X)andB:X→
Pcp,cv(X)be two multivalued operators satisfying
(a) Ais a contraction,and
(b) Bis upper semicontinuous and completely continuous. Then,either
(i) the operator inclusionλx∈Ax+Bxhas a solution forλ= ,or
(ii) the setU={u∈X|u∈λAu+λBu, <λ< }is unbounded.
3 A generalized Gronwall inequality
In this section, we establish a generalized Gronwall inequality, which is important in prov-ing the existence result. The proof is based on an iteration argument.
Lemma .([]) For x≥,we have
x e
x√ πx
+
x
<(x+ ) <
x e
x√ πx
+ x– .
.
Theorem . Suppose thatα,β> ,a(t)is a nonnegative function locally integrable on
R+,g(t)and h(t)are nonnegative,nondecreasing continuous functions defined onR+,and
u(t)is a nonnegative and locally integrable function onR+such that
u(t)≤a(t) +g(t) t
(t–s)α–u(s)ds+h(t)
t
(t–s)β–u(s)ds, t∈R+.
Then,for each constant b≥,t∈[,b],we have
u(t)≤a(t) + t
∞
n=
n
k=
Ckn(g(t)(α))
k(h(t)(β))n–k
(kα+ (n–k)β) (t–s)
kα+(n–k)β–a(s)ds. (.)
Proof LetBψ(t) =g(t)t(t–s)α–ψ(s)ds+h(t)t
(t–s)
β–ψ(s)ds for locally integrable
functionsψ. Then
By iteration we have
u(t)≤ n–
k=
Bka(t) +Bnu(t).
Next, we shall prove the following two statements:
(i) Bnu(t)≤tnk=Cnk(g(t)(α(k))αk+((hn(–t)k)(ββ)))n–k(t–s)kα+(n–k)β–u(s)ds.
(ii) For eacht∈[,b],limn→∞Bnu(t) = , and∞n=Bna(t)is convergent.
Proof of (i). From the definition ofBwe have (i) is true forn= . Assume that (i) is true for somen=m. Then, forn=m+ ,
Bm+u(t) =BBmu(t) =g(t)
t
(t–s)α–Bmu(s)ds+h(t) t
(t–s)β–Bmu(s)ds
≤g(t) t
(t–s)α– s
m
k=
Cmk(g(t)(α))
k(h(t)(β))m–k
(kα+ (m–k)β)
×(s–s)kα+(m–k)β–u(s)dsds
+h(t) t
(t–s)β– s
m
k=
Ckm(g(t)(α))
k(h(t)(β))m–k
(kα+ (m–k)β)
×(s–s)kα+(m–k)β–u(s)dsds.
By interchanging the order of integration and by the equalityCn
m+Cnm+=Cnm++we have
Bm+u(t) ≤ t
g(t)u(s)
t
s m
k=
Cmk(g(t)(α))
k(h(t)(β))m–k
(kα+ (m–k)β) (t–s)
α–(s–s
)kα+(m–k)β–ds ds
+ t
h(t)u(s)
t
s m
k=
Cmk(g(t)(α))
k(h(t)(β))m–k
(kα+ (m–k)β)
×(t–s)β–(s–s)kα+(m–k)β–ds ds
= t
m
k=
Cmk(g(t)(α))
k+(h(t)(β))m–k
((k+ )α+ (m–k)β) u(s)(t–s)
(k+)α+(m–k)β–ds
+ t
m
k=
Cmk(g(t)(α))
k(h(t)(β))m+–k
(kα+ (m+ –k)β) u(s)(t–s)
kα+(m+–k)β–ds
= t
m+
k=
Cmk+(g(t)(α))
k(h(t)(β))m–k
(kα+ (m–k)β) u(s)(t–s)
kα+(m–k)β–ds .
Therefore, by induction we get that (i) is true.
Proof of (ii). First, there existsN> such that forn>N, we have
Take arbitraryb> . Then, forn>N,k= , , . . . ,n, we have
(t–s)kα+(n–k)β–≤bkα+(n–k)β–, t∈[,b],s∈[,t]. By Lemma ., forn>N, we have
n
k=
Ckn(g(t)(α))
k(h(t)(β))n–k
(kα+ (n–k)β) b
kα+(n–k)β–
≤ n
k=
Ckn (g(t)(α))
k(h(t)(β))n–k
(kα+(n–ek)β–)kα+(n–k)β–π(kα+ (n–k)β– )b
kα+(n–k)β–
≤ n
k=
Ckn
g(T)(α)bα
(kα+(n–ek)β–)α
k
h(T)(β)bβ
(kα+(n–ek)β–)β
n–k
× e
b(kα+ (n–k)β– )π(kα+ (n–k)β– ).
Choosing constantsc,c> satisfying <c+c< , we can easily get that there is an
integer constantN>Nsuch that, forn>N,k= , , , . . . ,n,
g(T)(α)bα
(kα+(n–ek)β–)α<c,
h(T)(β)bβ
(kα+(n–ek)β–)β <c,
e
b(kα+ (n–k)β– )π(kα+ (n–k)β– )< .
Therefore, forn>Nandt∈[,b], we have
lim
n→∞B
nu(t)≤ lim n→∞
t
n
k=
Cnk(g(t)(α))
k(h(t)(β))n–k
(kα+ (n–k)β) (t–s)
kα+(n–k)β–u(s)ds
≤ lim
n→∞ t
n
k=
Cnk(g(t)(α))
k(h(t)(β))n–k
(kα+ (n–k)β) b
kα+(n–k)β–u(s)ds
≤ lim
n→∞ n
k=
Cnkckcn–k
b
u(s)ds
= lim
n→∞(c+c) n
b
u(s)ds
= .
Similarly, we can prove that forn>Nandt∈[,b],
∞
n=
Bna(t) = N
n=
Bna(t) + ∞
n=N+
Bna(t)
≤ N
n=
Bna(t) + ∞
n=N+
(c+c)n
b
a(s)ds
≤ ∞.
As a consequence of (i) and (ii), we complete the proof of Theorem ..
Remark . Ifh(t)≡, then (.) becomes
u(t)≤a(t) + t
∞
n=
(g(t)(α))n
(nα) (t–s)
nα–a(s)ds,
which is Theorem in monograph [].
4 Existence result
According to [], we introduce the definition of the mild solution to problem (.).
Definition . A continuous functionx: [–τ,b]→Xis a mild solution of problem (.) ifx(t) =φ(t) on [–τ, ] and there existsf ∈L([,b],X) such thatf(t)∈F(t,x(t)) for a.e.
t∈[,b] and
x(t) =Sq(t)φ() –g(,φ)+g(t,xt) + t
(t–s)q–ATq(t–s)g(s,xs)ds
+ t
(t–s)q–Tq(t–s)f(s)ds,
where
Sq(t) = ∞
ξq(θ)Ttqθdθ, Tq(t) =q
∞
θ ξq(θ)Ttqθdθ,
ξq(θ) =
qθ
––q
q
θ–q,
q(θ) =
π
∞
n=
(–)n–θ–qn–(nq+ )
n! sin(nπq), θ∈R
+.
Remark .([]) The functionξq(θ) is a probability density function defined on R+, and
∞
θvξq(θ)dθ= ∞
θqvq(θ)dθ=
( +v)
( +qv).
Lemma .([]) The operatorsSqandTqhave the following properties:
(i) For any fixedt≥,SqandTqare linear and bounded operators,that is,for any
x∈X,
Sq(t)x≤M|x|, Tq(t)x≤
qM ( +q)|x|.
(ii) {Sq(t)}t≥and{Tq(t)}t≥are strongly continuous.
(iii) IfT(t)is compact fort> ,thenSq(t)andTq(t)are also compact operators fort> . (iv) For anyx∈X,β∈(, ),andη∈(, ],we have
and
AηTq(t) ≤ qCη( –η)
tqη( +q( –η)), <t≤b.
Let us list the following hypotheses:
(H)T(t) is compact operator for everyt> .
(H) The multivalued mapF: [,b]×C([–τ, ],X)→Pb,cl,cv(X) satisfies the following
conditions:
(i) For eacht∈[,b],F(t,·)is upper semicontinuous, for eachx∈C([–τ, ],X),F(·,x)
is measurable, and the setNF,x={f ∈L([,b],X) :f(t)∈F(t,x),for a.e.t∈[,b]}is
not empty.
(ii) For eachx∈C([–τ, ],X), there existm∈L/q([,b], R+)andr∈C([,b], R+)such
that
supf(t):f(t)∈F(t,x)≤m(t) +r(t)x[–τ,] for a.e.t∈[,b],
whereq∈[,q).
(H) There exists a constantβ∈(, ) such thatg∈D(Aβ),Aβgis continuous, and
(i) there exists a positive constantLsuch that
Aβg(t,x) –Aβg(t,x)≤L
|t–t|+x–x[–τ,]
for <t,t<b,x,x∈C([–τ, ],X), and
(ii) there exist positive constantsL,Lsuch that
Aβg(t,x)≤Lx[–τ,]+L
for anyx∈C([–τ, ],X).
Theorem . Assume that hypotheses(H)-(H)are satisfied.Then problem(.)has one mild solution,provided that
L=L A–β +
bqβC
–β( +β)
β( +qβ)
< , A–β L< . (.)
Proof Transform problem (.) into a fixed point problem. Consider the multivalued oper-ator:C([–τ,b],X)→P(C([–τ,b],X)) wherexis defined as the set ofρ∈C([–τ,b],X) such that
ρ(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
φ(t), t∈[–τ, ],
Sq(t)[φ() –g(,φ)] +g(t,xt) + t
(t–s)q–ATq(t–s)g(s,xs)ds
+t(t–s)q–Tq(t–s)f(s)ds, t∈[,b],
C([–τ,b],X) andB:C([–τ,b],X)→P(C([–τ,b],X)) defined by
(Ax)(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
φ(t), t∈[–τ, ],
Sq(t)[φ() –g(,φ)] +g(t,xt)
+t(t–s)q–ATq(t–s)g(s,xs)ds, t∈[,b],
and
Bx=
ρ∈C[–τ,b],X:ρ(t) =
, t∈[–τ, ], t
(t–s)q–Tq(t–s)f(s)ds, f∈NF,x,t∈[,b]
.
It is clear that=A+B. Mild solutions of problem (.) are converted to the fixed points ofx∈Ax+Bx. We shall show that the operatorsAandBsatisfy the conditions of Theo-rem . by the following steps.
Step.Ais a contraction.
Letx,y∈C([–τ,b],X). Then, for eacht∈[,b], from condition (H) we have that
(Ax)(t) – (Ay)(t)=g(t,xt) + t
(t–s)q–ATq(t–s)g(s,xs)ds
–g(t,yt) – t
(t–s)q–ATq(t–s)g(s,ys)ds
≤g(t,xt) –g(t,yt)+
t(t–s)q–ATq(t–s)g(s,xs) –g(t,ys)ds
≤L A–β x
t–yt[–τ,]
+L
t
(t–s)q– A–βTq(t–s) xs–ys[–τ,]ds
≤L A–β +qC–β( +β)
( +qβ) t
(t–s)qβ–ds
sup
≤s≤t
xs–ys[–τ,]
≤L A–β +b qβC
–β( +β)
β( +qβ)
x–y[–τ,b]
=Lx–y[–τ,b].
Therefore,
Ax–Ay[–τ,b]≤Lx–y[–τ,b],
so thatAis a contraction sinceL< .
Step.Bxis convex for eachx∈C([–τ,b],X).
Indeed, ifρandρbelong toBx, then there existf,f∈NF,xsuch that for eacht∈[,b], we have
ρi(t) = t
Let ≤λ≤. Then for eacht∈[,b], we have
λρ+ ( –λ)ρ
(t) =
t
(t–s)q–Tq(t–s)λf(s) + ( –λ)f(s)
ds.
SinceNF,xis convex, we haveλρ+ ( –λ)ρ∈Bx.
Step.Bsends bounded sets to bounded sets inC([–τ,b],X).
It suffices to prove that there exists a constantl> such that for eachρ∈Bx,x∈Bk= {x∈C([–τ,b],X),x[–τ,b]≤k}, we haveρ[–τ,b]≤l.
Letρ∈Bx. Then there existsf ∈NF,xsuch that fort∈[,b], we have
ρ(t) = t
(t–s)q–Tq(t–s)f(s)ds.
Then, fort∈[,b], we have
ρ(t)= t
(t–s)q–Tq(t–s)f(s)ds
≤ qM
( +q) t
(t–s)q–f(s)ds. (.)
From (H), fort∈[,b], we have t
(t–s)q–f(s)ds≤
t
(t–s) q– –qds
–q m
L
q[,t]+rk t
(t–s)q–ds
≤ M
( +a)–qb
(+a)(–q)+rkb q
q , (.)
wherea=–q–q
∈(–, ),M=mLq[,b], andr=sup{r(t),t∈[,b]}.
Then from (.) and (.) we get that
ρ[–τ,b]≤ ρ[–τ,]+ρ[,b]
≤ + qM
( +q)
M
( +a)–qb
(+a)(–q)+rkb q
q
:=l.
Step.Bmaps bounded sets to equicontinuous sets ofC([–τ,b],X).
Lett,t∈[,b],t<t, and letBk={x∈C([–τ,b],X),x[–τ,b]≤k}be a bounded set ofC([–τ,b],X). For eachx∈Bkandρ∈Bx, there existsf ∈NF,xsuch that
ρ(t) = t
(t–s)q–Tq(t–s)f(s)ds.
Then,
ρ(t) –ρ(t)=
t
(t–s)q–Tq(t–s)f(s)ds–
t
(t–s)q–Tq(t–s)f(s)ds
≤ t
t
(t–s)q–Tq(t–s)f(s)ds
+ t
(t–s)q–– (t–s)q–
Tq(t–s)f(s)ds
+ t
(t–s)q–
Tq(t–s) –Tq(t–s)
f(s)ds
=I+I+I,
where
I=
tt
(t–s)q–Tq(t–s)f(s)ds
,
I=
t(t–s)q–– (t–s)q–
Tq(t–s)f(s)ds
,
I=
t
(t–s)q–
Tq(t–s) –Tq(t–s)
f(s)ds.
By using a similar argument as that used in (.), we can conclude that
I≤
qM ( +q)
M
( +a)–q(t–t)
(+a)(–q)+rk(t–t) q
q
,
I≤
qM ( +q)
t
(t–s)q–– (t–s)q–
–qds
–q m
L q[,t] +rk
t
(t–s)q–– (t–s)q–ds
≤ qM
( +q)
M
t
(t–s)a– (t–s)a
ds
–q
+rk
(t–t)q
q –
tq q +
tq q
= qM
( +q)
M
( +a)–q
t+a–t+ a+ (t–t)+a
–q +rk
q
(t–t)q–tq+t
q
≤ MM
( +q)
M
( +a)–q(t–t)
(+a)(–q)+rk
q (t–t)
q
.
Hence,limt→tI= andlimt→tI= independently ofx∈Bk. On the other hand,
I≤
t
(t–s)q– Tq(t–s) –Tq(t–s) f(s)ds.
Hypothesis (H) and Lemma . imply thatTq(t) is continuous fort> . Then we get thatItends to independently ofx∈Bkast→t.
Consequently,|ρ(t) –ρ(t)| → independently ofx∈Bkast→t, which means that B(Bk) is equicontinuous.
Step. For eacht∈[,b],V(t) ={(Bx)(t),x∈Bk}is relatively compact inX.
Obviously,V() ={}is relatively compact inX. Let <t≤bbe fixed. Forx∈Bkand
ρ∈Bx, there existsf∈NF,xsuch that
ρ(t) = t
For arbitrary∈(,t) andδ> , define the operator,δonBkby
(,δx)(t) =q
t–
∞
δ
θ(t–s)q–ξq(θ)T
(t–s)qθf(s)dθds
=q
t–
∞
δ
θ(t–s)q–ξq(θ)TqδT(t–s)qθ–qδf(s)dθds
=Tqδq
t–
∞
δ
θ(t–s)q–ξ
q(θ)T
(t–s)qθ–qδf(s)dθds.
According to the compactness ofT(t),t> , we get that the setV,δ(t) ={(,δx)(t),x∈Bk} is relatively compact inXfor all∈(,t) andδ> . Moreover, for everyx∈Bk, we have
(Bx)(t) – (,δx)(t)
=q
t
δ
θ(t–s)q–ξq(θ)T
(t–s)qθf(s)dθds
+q
t
∞
δ
θ(t–s)q–ξq(θ)T(t–s)qθf(s)dθds
–q
t–
∞
δ
θ(t–s)q–ξq(θ)T
(t–s)qθf(s)dθds
≤q
t
δ
θ(t–s)q–ξq(θ)T
(t–s)qθf(s)dθds
+q
t
t–
∞
δ
θ(t–s)q–ξq(θ)T(t–s)qθf(s)dθds
≤qM
t
(t–s)q–f(s)ds
δ
θ ξq(θ)dθ
+qM
t
t–
(t–s)q–f(s)ds
∞
θ ξq(θ)dθ.
In view of (.), we have
(Bx)(t) – (,δx)(t)
≤qM
δ
θ ξq(θ)dθ
M
( +a)–qb
(+a)(–q)+rkbq
q
+ qM
( +q)
t
t–
(t–s) q– –qds
–q m
L
q[t–,t]+rk
t
t–
(t–s)q–ds
≤qM
δ
θ ξq(θ)dθ
M
( +a)–qb
(+a)(–q)+rkb q
q
+ qM
( +q)
M
( +a)–q
(+a)(–q)+rk q
q
.
Fromlimδ→
δ
Step.Bhas a closed graph. Letxn→x∗,ρ
n∈Bxn, andρn→ρ∗ asn→ ∞. We shall prove thatρ∗∈Bx∗. Since
ρn∈Bxn, there existsfn∈NF,xnsuch that
ρn(t) = t
(t–s)q–Tq(t–s)fn(s)ds, t∈[,b].
We need to prove that there existsf∗∈NF,x∗such that
ρ∗(t) = t
(t–s)q–Tq(t–s)f∗(s)ds, t∈[,b].
Consider the continuous operatorT :L([,b],X)→C([,b],X) defined by
(Tf)(t) = t
(t–s)q–Tq(t–s)f(s)ds.
We can easily obtain thatT is continuous. On the other hand,
ρn(t) –ρ∗(t)≤ ρn–ρ∗ → asn→ ∞.
From Theorem . it follows thatT ◦NF is a closed graph operator. Moreover, we have that
ρn∈T(NF,xn).
Sincexn→x∗, by Theorem . there existsf∗∈NF,x∗such that
ρ∗(t) = t
(t–s)q–Tq(t–s)f∗(s)ds, t∈[,b].
This implies thatρ∗∈Bx∗.
Therefore, the multivalued map B is completely continuous and u.s.c. with convex closed values.
Step. The setU={u∈C([–τ, ],X)|u∈λAu+λBu, <λ< }is bounded.
Letx∈U, thenx∈λxfor some <λ< . Thus, there existsf ∈NF,xsuch that for
t∈[,b],
x(t) =λ
Sq(t)φ() –g(,φ)+g(t,xt) + t
(t–s)q–ATq(t–s)g(s,xs)ds
+ t
(t–s)q–Tq(t–s)f(s)ds
.
By (H) and (H), for eacht∈[,b], we have
x(t)=λSq(t)φ() –g(,φ)+g(t,xt) + t
(t–s)q–ATq(t–s)g(s,xs)ds
+ t
≤ Sq(t) φ() –g(,φ)+ A–β Aβg(t,xt) +
t
(t–s)q– A–β
Tq(t–s) Aβg(s,xs)ds
+ t
(t–s)q– Tq(t–s) f(s)ds
≤Mφ()+g(,φ)+ A–β Lxt[–τ,]+L
+ t
(t–s)q– qC–β( +β) (t–s)q(–β)( +qβ)
Lxs[–τ,]+L
ds
+ qM
( +q) t
(t–s)q–f(s)ds
≤Mφ()+g(,φ)+ A–β Lxt[–τ,]+L
+qLC–β( +β)
( +qβ) t
(t–s)qβ–xs [–τ,]ds
+qLC–β( +β)
( +qβ) t
(t–s)qβ–ds
+ qM
( +q) t
(t–s)q–m(s)ds+ qM
( +q) t
(t–s)q–r(s)xs[–τ,]ds
≤Mφ()+g(,φ)+ A–β L
xt[–τ,]+L
+qLC–β( +β)
( +qβ) t
(t–s)qβ–xs
[–τ,]ds+
LC–βbqβ( +β)
β( +qβ)
+ qMM ( +a)–q( +q)b
(+a)(–q)+ qMr
( +q) t
(t–s)q–xs[–τ,]ds.
Consider the function
μ(t) =maxx(s):s∈[–τ,t], t∈[,b].
Lett∗∈[–τ,t] be such thatμ(t) =|x(t∗)|. Ift∗∈[,t], then fort∈[,b], we have
μ(t)≤Mφ()+g(,φ)+ A–β Lμ(t) +L
+qLC–β( +β)
( +qβ) t∗
t∗–sqβ–μ(s)ds+LC–βb
qβ( +β)
β( +qβ)
+ qMM ( +a)–q( +q)b
(+a)(–q)+ qMr
( +q) t∗
t∗–sq–μ(s)ds.
Sinceμ(t) is nondecreasing, fort∗∈[,t], we have
t∗
t∗–sq–μ(s)ds= t∗
sq–μt∗–sds
≤ t
sq–μ(t–s)ds= t
Therefore,
μ(t)≤Mφ()+g(,φ)+ A–β L
μ(t) +L
+qLC–β( +β)
( +qβ) t
(t–s)qβ–μ(s)ds+LC–βbqβ( +β)
β( +qβ)
+ qMM ( +a)–q( +q)b
(+a)(–q)+ qMr
( +q) t
(t–s)q–μ(s)ds.
Ift∗∈[–τ, ], thenμ(t) =φ[–τ,].
Therefore,
μ(t)≤ φ[–τ,]+Mφ()+g(,φ)+ A–β Lμ(t) +L
+qLC–β( +β)
( +qβ) t
(t–s)qβ–μ(s)ds+LC–βb
qβ( +β)
β( +qβ)
+ qMM ( +a)–q( +q)b
(+a)(–q)+ qMr
( +q) t
(t–s)q–μ(s)ds.
FromA–βL
< we get that
μ(t)≤
–A–βL
φ[–τ,]+Mφ()+g(,φ)+L A–β
+LC–βb
qβ( +β)
β( +qβ) +
qMM
( +a)–q( +q)b
(+a)(–q)
+ qLC–β( +β) ( –A–βL
)( +qβ)
t
(t–s)qβ–μ(s)ds
+ qMr
( –A–βL
)( +q)
t
(t–s)q–μ(s)ds
=C+G
t
(t–s)qβ–u(s)ds+K
t
(t–s)q–u(s)ds,
where
C=
–A–βL
φ[–τ,]+Mφ()+g(,φ)+L A–β
+LC–βb
qβ( +β)
β( +qβ) +
qMM
( +a)–q( +q)b
(+a)(–q)
,
G= qLC–β( +β) ( –A–βL
)( +qβ)
, K= qMr ( –A–βL
)( +q)
.
Then by Theorem . we have
μ(t)≤C+ t
∞
n=
n
k=
Cnk(G(qβ))
k(K(q))n–k
(kqβ+ (n–k)q) (t–s)
kqβ+(n–k)q–C ds
≤C+C
∞
n=
n
k=
Cnk (G(qβ))
Therefore, we obtain that
x[–τ,b]=μ(b)≤C+C
∞
n=
n
k=
Cnk (G(qβ))
k(K(q))n–kbkqβ+(n–k)q (kqβ+ (n–k)q)(kqβ+ (n–k)q).
This shows thatUis bounded.
As a result of Theorem ., we obtain thathas a fixed point, which is the mild solution of system (.). This completes the proof.
5 An example
As an application, we consider the following fractional differential inclusions:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂q
∂tq[u(t,z) – π
U(z,y)ut(ς,y)dy]
∈ ∂
∂zu(t,z) +G(t,ut(ς,z)), z∈[,π],t∈[,b],
u(t, ) =u(t,π) = , t∈[,b],
u(ς,z) =φ(ς,z), z∈[,π],ς∈[–τ, ],
(.)
where <q< , b> , the functionU(z,y), z,y∈[,π] satisfies some conditions, G: [,b]×C([–τ, ], R)→P(C([–τ, ], R)) satisfies assumptions (H), andut(ς,z) =u(t+
ς,z),t∈[,b],ς∈[–τ, ].
ConsiderX=L([,π]; R) and the operatorA:X→X defined byAw= –w with the domain
D(A) =w∈X,w,w are absolutely continuous,w ∈X,w() =w() = .
Then –Agenerates a strongly continuous semigroup{T(t)}t≥, which is compact, analytic,
and self-adjoint. Furthermore, –Ahas a discrete spectrum: the eigenvalues are –n,n∈N,
with the corresponding normalized eigenvectorsvn(z) = (π)
sin(nz). We have the next three properties.
(i) For eachw∈X,T(t)w=∞n=e–ntw,vnvn. In particular,T(·)is uniformly stable,
andT(t) ≤et.
(ii) For eachw∈X,A–w=∞
n=nw,vnvn, andA
–= .
(iii) The operatorA is given by
Aw= ∞
n=
nw,vnvn
on the spaceD(A) ={w(·)∈X,∞
n=nw,vnvn∈X}. Then the operatorAsatisfies (.), (.), and (H). Hence, system (.) can be reformulated as
⎧ ⎨ ⎩
Dq(x(t) –g(t,x
t))∈–Ax(t) +F(t,xt), t∈[,b],
x(t) =φ(t), t∈[–τ, ],
where x(t)(z) =u(t,z), F(t,xt)(z) = G(t,ut(θ,z)), g(t,xt)(z) = π
U(z,y)ut(θ,y)dy, and
The following conditions are also assumed to be true.
(i) The functionU(z,y),z,y∈[,π]is measurable, and
π
π
U(z,y)dy dz<∞.
(ii) The function∂∂zU(z,y)is measurable,U(,y) =U(π,y) = , and
π
π
∂ ∂zU(z,y)
<∞.
From [] we have that condition (H) of Theorem . holds withβ=.
Takeq= and assume thatFsatisfies (H). According to Theorem ., problem (.) admits a mild solution, provided that (.) holds.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by Natural Science Foundation of Anhui Province (1508085MA10) and Open Project of Laboratory of Intelligent Information Processing of Suzhou University (2014YKF38).
Received: 10 October 2015 Accepted: 22 January 2016 References
1. Oldham, KB, Spanier, J: The Fractional Calculus. Academic Press, New York (1974)
2. Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) 3. Podlubny, I: Fractional Differential Equation. Academic Press, San Diego (1999)
4. Delbosco, D, Rodino, L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609-625 (1996)
5. Aizicovici, S, Mckibben, M: Existence results for a class of abstract nonlocal Cauchy problems. Nonlinear Anal.39, 649-668 (2000)
6. Diethelm, K, Ford, NJ: Analysis of fractional differential equations. J. Math. Anal. Appl.265, 229-248 (2002) 7. Eidelman, SD, Kochubei, AN: Cauchy problem for fractional diffusion equations. J. Differ. Equ.199, 211-255 (2004) 8. Daftardar-Gejji, V, Jafari, H: Analysis of a system of nonautonomous fractional differential equations involving Caputo
derivatives. J. Math. Anal. Appl.328, 1026-1033 (2007)
9. Chang, YK, Nieto, JJ: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model.49, 605-609 (2009)
10. Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math.109, 973-1033 (2010)
11. Zhou, Y, Jiao, F, Li, J: Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal.71, 3249-3256 (2009)
12. Agarwal, RP, Zhou, Y, He, Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl.59, 1095-1100 (2010)
13. Dos Santos, JPC, Arjunan, MM, Cuevas, C: Existence results for fractional neutral integro-differential equations with state-dependent delay. Comput. Math. Appl.62, 1275-1283 (2011)
14. Zhou, Y, Feng, J: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl.59, 1063-1077 (2010)
15. Wang, J, Zhou, Y: Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal. 12, 3642-3653 (2011)
16. Yan, Z: Controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay. J. Franklin Inst.348, 2156-2178 (2011)
17. Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) 18. Deimling, K: Multivalued Differential Equations. de Gruyter, Berlin (1992)
19. Kisielewicz, M: Differential Inclusions and Optimal Control. Kluwer Academic, Dordrecht (1991)
20. Lasota, A, Opial, Z: An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys.13, 781-786 (1965)
21. Dhage, BC: Multi-valued mappings and fixed points I. Nonlinear Funct. Anal. Appl.10, 359-378 (2005) 22. Dhage, BC: Multi-valued mappings and fixed points II. Tamkang J. Math.37, 27-46 (2006)
24. Ye, H, Gao, J, Ding, Y: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl.328, 1075-1081 (2007)
