R E S E A R C H
Open Access
Solutions to impulsive integrodifferential
evolution equations under a noncompact
evolution system
Shaochun Ji
1*and Min Wang
2*Correspondence:
[email protected] 1Faculty of Mathematics and
Physics, Huaiyin Institute of Technology, Huaian, 223003, P.R. China
Full list of author information is available at the end of the article
Abstract
In this paper, we study the existence of mild solutions to impulsive integrodifferential evolution equations in Banach spaces. Based on a measure of noncompactness and important properties of semicompact sets, new existence results are obtained. Here the evolution system is only supposed to be strongly continuous, without any compact or equicontinuous assumptions. Some applications are given to illustrate the effectiveness of our results.
MSC: 34A37; 34G20; 47D60
Keywords: impulsive conditions; nonlocal conditions; integrodifferential equation; mild solution; measure of noncompactness
1 Introduction
This paper is concerned with integrodifferential evolution equations with impulsive con-ditions and nonlocal concon-ditions:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
u(t) =A(t)u(t) +f(t,u(t),th(t,s,u(s)) ds), t∈J= [,b],t=ti,i= , , . . . ,p,
u(ti) =u(t+
i) –u(t–i) =Ii(u(ti)), i= , , . . . ,p, u() =g(u) +u,
(.)
whereA(t) is a family of linear operators which generates an evolution system{U(t,s) : ≤ s≤t≤b}. The state variableu(·) takes values in the real Banach spaceX. The operators h:T×X→X,f :J×X×X→Xare continuous,T={(t,s), ≤s≤t≤b}.Ii:X→X, i= , , . . . ,p, are impulsive functions, <t<t<· · ·<tp<tp+=b.u(ti) is the jump of
a functionuatti,u(t–i),u(t+i) denote the left and the right limit ofuatti, respectively. g:PC([,b];X)→Xis an appropriate continuous function to be specified later.
The impulsive differential systems can be used to model processes which are subject to short perturbations whose duration can be negligible in comparison with the duration of the process, such as the dynamics of populations subject to abrupt changes. For more details of this theory and its applications, we refer to the monographs of Lakshmikantham et al.[] and Benchohraet al.[], and to [–] and the references therein. In [], Fan and Li used the techniques of approximate solutions and fixed points to get the mild solutions
of nonlocal impulsive differential equation
u(t) =Au(t) +ft,u(t), t∈[,b],t=ti, u(ti) =ut+i–ut–i=Iiu(ti), i= , , . . . ,p,
u() =g(u),
whereAgenerates a compact semigroupT(t), the impulsive functionsIiare not compact. Abadaet al.[] studied the existence of integral solutions and extremal integral solutions for some nondensely defined impulsive semilinear functional differential inclusions in sep-arable Banach spaces. Jiet al.[, ] studied the existence and controllability of solutions to impulsive differential system when the semigroup is equicontinuous.
On the other hand, the abstract nonlocal initial problem was initiated by Byszewski and Lakshmikantham [, ], where the existence and uniqueness of solutions to semilinear nonlocal differential equations were discussed. The importance of the problem consists in the fact that it is more general and has a better effect than the classical initial conditions u() =u. Therefore the nonlocal Cauchy problem has been studied extensively under
var-ious conditions onA(t) andf,g, by several authors [–]. Ntouyas and Tsamatos [] studied the nonlocal semilinear differential equations with compact conditions. Xue [] discussed the semilinear nonlocal differential equations when the semigroupT(t) gener-ated by the coefficient operator is compact and the nonlocal functiong is not compact. Some classes of integrodifferential equations with nonlocal conditions have been investi-gated by Balachandranet al.[, ]. Wang and Wei [] and Machadoet al.[] discussed the existence of a class of impulsive integrodifferential evolution equations, where the evo-lution system is supposed to be equicontinuous.
In the above work, we find that the compactness of the evolution system plays a key role in this type of impulsive nonlocal Cauchy problem. However, sometimes it is difficult to satisfy. For example, letX=L(–∞, +∞). The ordinary differential operatorA= d/dxwith D(A) =H(–∞, +∞), generates a semigroupT(t) defined byT(t)u(s) =u(t+s), for every u∈X. TheC-semigroupT(t) is not compact onX.
Recently, by using a new two-component measure of noncompactness, Benchohra and Ziane [] proved the existence of mild solutions for a class of impulsive semilinear evo-lution differential inclusions with state-dependent delay whenA(t) generates a strongly continuous evolution operator. It is an interesting result. In this paper we explain that the existence results of differential systems under a noncompact evolution system can also be obtained via the classical Hausdorff measure of noncompactness. By applying the property of semicompact sets (see Lemma .), we discuss the existence of mild solutions to (.) without the compactness of evolution systemU(t,s), even its equicontinuity. This is one motivation of the present work. Note that the assumption on the evolution system here is weaker than that in [, , ], and no more conditions are added. The Banach space here is nonseparable. Another motivation of the present work is the exact controllability problem of the differential system. Our method can also be applied to an impulsive con-trol system and can deal with the technical error on the exact concon-trollability of differential system caused by the compactness of the evolution system (see Remark .).
2 Preliminaries
Let (X, · ) be a real Banach space. We denote byC([,b];X) the space ofX-valued con-tinuous functions on [,b] with the normx=sup{x(t),t∈[,b]}and byL([,b];X) the space of X-valued Bochner integrable functions on [,b] with the norm fL = b
f(t)dt.
For the sake of simplicity, we putJ= [,b];J= [,t];Ji= (ti,ti+],i= , . . . ,p. In order to
define the mild solution of problem (.), we introduce the setPC([,b];X) ={u: [,b]→ Xsuch thatu(·) is continuous except for a finite number of pointsti, at whichu(t+i),u(t–i) exist andu(ti) =u(t–
i),i= , . . . ,p}. It is easy to verify thatPC([,b];X) is a Banach space with the normuPC=sup{u(t),t∈[,b]}.
Now we recall the Hausdorff measure of noncompactness (in short MNC)β(·) defined by
β(B) =inf{ε> ;Bhas a finiteε-net inX}
for each bounded subsetBin Banach spaceX. We recall the following properties of the Hausdorff measure of noncompactnessβ.
Lemma .([]) Let X be a real Banach space and B,C⊆X be bounded.Then the fol-lowing properties are satisfied:
() Bis relatively compact if and only ifβ(B) = ;
() β(B) =β(B) =β(convB),whereBandconvBmean the closure and convex hull ofB,
respectively;
() β(B)≤β(C)whenB⊆C;
() β(B+C)≤β(B) +β(C),whereB+C={x+y:x∈B,y∈C}; () β(B∪C)≤max{β(B),β(C)};
() β(λB)≤ |λ|β(B)for anyλ∈R;
() if the mapQ:D(Q)⊆X→Zis Lipschitz continuous with constantk,then βZ(QB)≤kβ(B)for any bounded subsetB⊆D(Q),whereZis a Banach space.
A two parameter family of bounded linear operators{U(t,s), ≤s≤t≤b}onXis called anevolution systemif the following two conditions are satisfied:
(i) U(s,s) =I,U(t,r)U(r,s) =U(t,s)for≤s≤r≤t≤b;
(ii) (t,s)→U(t,s)is strongly continuous for≤s≤t≤b.
In a natural way, we can consider the respective evolution operatorU:J×J→L(X), where L(X) is the space of all bounded linear operators inX. Since the evolution sys-temU(t,s) is strongly continuous on the compact setJ×J, there existsM> such that
U(t,s) ≤Mfor any (t,s)∈J×J. More details as regards this evolution system can be found in Pazy [].
Definition . A functionu∈PC([,b];X) is said to be a mild solution of (.) if
u(t) =U(t, )u() + t
U(t,s)f
s,u(s), s
hs,τ,u(τ)dτ
ds+
<ti<t
U(t,ti)Iiu(ti)
Definition . A countable set{fn}+n∞=⊂L([,b];X) is said to be semicompact if
• the sequence{fn(t)}n+=∞is relatively compact inXfor a.a.t∈[,b];
• there is a functionμ∈L([,b];R+)satisfyingsup
n≥fn(t) ≤μ(t)for a.e.t∈[,b].
Lemma .([], Theorem ..) Let{fn}+∞
n= be a sequence of functions in L([,b];X).
Assume that there existμ,η∈L([,b];R+)satisfying
sup
n≥
fn(t)≤μ(t) and βfn(t)n+=∞≤η(t) a.e. t∈[,b].
Then for all t∈[,b],we have
β t
U(t,s)fn(s) ds:n≥
≤M t
η(s) ds.
The following lemma can be found in Theorem of [] and Theorem .. of [].
Lemma . Let(Gf)(t) =tU(t,s)f(s) ds.If{fn}+∞
n=⊂L([,b];X)is semicompact,then the
set{Gfn}+n=∞is relatively compact in C([,b];X)and,moreover,if fnf,then for all t∈
[,b], (Gfn)(t)→(Gf)(t)as n→+∞.
Lemma .([]) Let D be a bounded set of X.Then for anyε> ,there exists a sequence
{uk}∞k=⊂D such that
β(D)≤β{uk}∞k=
+ε.
Lemma .([]) If W⊆PC([,b];X)is bounded,thenβ(W(t))≤β(W)for all t∈[,b], where W(t) ={u(t);u∈W} ⊆X.Furthermore,suppose the following conditions are satis-fied:
() Wis equicontinuous onJ= [,t]and eachJi= (ti,ti+],i= , . . . ,p;
() Wis equicontinuous att=ti+,i= , . . . ,p.
Thensupt∈[,b]β(W(t)) =β(W).
Throughout this paper, we denoteM=sup{U(t,s): (t,s)∈J×J},Wr={u∈PC([,b]; X) :u(t) ≤r,∀t∈[,b]}. Without loss of generality, we letu= .
3 Main results
First we give the following hypotheses:
(H) A(t)is a family of linear (not necessarily bounded) operators andA(t) :D(A)→X
generates a strongly continuous evolution system{U(t,s) : ≤s≤t≤b},D(A)not
depending ontand a dense subset ofX(see []).
(H) g:PC([,b];X)→Xis continuous and compact.
(H) Ii:X→Xis continuous and compact for eachi= , , . . . ,p.
(H) The functionf : [,b]×X×X→Xsatisfies the following:
() For a.e.t∈[,b], the functionf(t,·,·) :X×X→Xis continuous and for all
() There exists a functionθ∈L(J;R+)such that
f(t,x,y)≤θ(t)x+y
for a.e.t∈[,b]and allx,y∈X.
() There exists a functionl∈L(J;R+)such that for every bounded setA,B⊂X,
βf(t,A,B)≤l(t)β(A) +β(B)
for a.e.t∈[,b].
(H) The functionh:T×X→Xsatisfies the following:
() For a.e.(t,s)∈T, the functionh(t,s,·) :X→Xis continuous and for allx∈X,
the functionh(·,·,x) :T→Xis strongly measurable.
() There exists a functionm∈L(T;R+)such that
h(t,s,x)≤m(t,s)x.
Let us takem∗=max(t,s)∈T
t
m(t,s) ds.
() There exist functionsζ,ζ∈L(J;R+)such that
βh(t,s,D)≤ζ(t)ζ(s)β(D)
for a.e.(t,s)∈T,D⊂Xa bounded set.
Now, we give the existence result under the above hypotheses.
Theorem . Assume that the hypotheses(H)-(H)are satisfied,then the nonlocal
im-pulsive problem(.)has at least one mild solution on[,b],provided that
M
sup
u∈Wr
g(u)+rθL+m∗rθL+ sup u∈Wr
p
i=
Iiu(ti)
≤r. (.)
Proof Put the mapK:PC([,b];X)→PC([,b];X) defined by
(Ku)(t) = (Ku)(t) + (Ku)(t),
with
(Ku)(t) =U(t, )g(u) +
t
U(t,s)f
s,u(s), s
hs,τ,u(τ)dτ
ds,
(Ku)(t) =
<ti<t
U(t,ti)Iiu(ti)
for allt∈[,b].
From our hypotheses, we see thatK is continuous onPC(J;X). For this purpose, we
Then from the continuity ofg,Iiand the dominated convergence theorem, we have
Kun–KuPC≤Mg(un) –g(u)
X) is bounded and convex.
DefineW=co{K(W),u}, wherecomeans the closure of the convex hull inPC([,b];
It is easy to show that{Wn}∞n=is a decreasing sequence of bounded, closed, convex subsets
inPC([,b];X). Then the setW=∞n=Wnis a nonempty bounded convex closed subset inPC([,b];X). Taking the limit in both sides of (.), we haveW=co{K(W),u}.
Now we shall prove thatWis relatively compact inPC([,b];X).
≤βU(t, )gu(kn)∞k=
is arbitrary, it follows from the above inequality that
nuity ofβ(Wn(t)) fort∈[,b] is based on the equicontinuity ofU(t,s), which is absent in our assumptions. Thus we look for a new way to consider the problem.
Setηn(t) =β(Wn(t)),ρn+(t) =
t
k(s)ηn(s) ds. As for eacht∈[,b],ηn(t) is a decreasing
sequence andη(t) =limn→∞ηn(t), we see thatlimn→∞ρn+(t) exists (the proof is similar
to the Levi lemma except for a trivial modification). Letρ(t) =limn→∞ρn+(t). Taking the
limit in both sides ofρn+(t) =
t
k(s)ηn(s) ds, we have
ρ(t)≤ t
k(s)η(s) ds, (.)
ρis a continuous function. From (.), we haveηn(t)≤tk(s)ηn–(s) ds=ρn(t). Taking the
limit in both sides, we get
η(t)≤ρ(t). (.)
From inequalities (.) and (.), we derive that
ρ(t)≤ t
k(s)ρ(s) ds,
hereρ(t) is continuous on [,b]. Using the Gronwall inequality, we conclude thatρ(t)≡ on [,b]. Asη(t)≤ρ(t), it follows thatη(t)≡ on [,b]. That is,
η(t) = lim
n→∞β
Wn(t)
= for eacht∈[,b]. (.)
AsW(t) =∞n=Wn(t) =limn→∞Wn(t), from (.), we have
βW(t)= (.)
for eacht∈[,b].
Subsequently, we shall prove thatβ(KW) = . To this end, from Lemma ., we see that, for anyε> , there exists a sequence{yn}∞n=⊂KWsuch that
β(KW)≤β{yn}∞n=+ε. (.)
Then there exists{un}∞n=⊂W, such that
yn=K(un) +K(un). (.)
Firstly, we show that {K(un)}∞n=⊂C([,b];X) is relatively compact, making full use of
Lemma .. From hypotheses (H)() and (H)(), we get
β
f
s,un(s), s
hs,τ,un(τ)dτ
∞
n=
≤l(s)
βun(s)∞n=+β s
hs,τ,un(τ)dτ ∞
n=
≤l(s)
βW(s)+ s
ζ(s)ζ(τ)β
for a.e.s∈[,b]. Then from (.),β(W(s)) = , we have
From the strong continuity ofU(t,s) and the compactness ofg, we see, for eacht∈[,b], that the set{U(t, )g(un) :n≥}is relatively compact inX. Now, for ≤t<t+h≤b, using the semigroup property,
U(t+h, )g(un) –U(t, )g(un)≤MU(t+h,t) –Ig(un).
Thus the functions in {U(·, )g(un) :n≥} are equicontinuous due to the compactness ofgand the strong continuity ofU(t,s). According to the Ascoli-Arzela theorem, we get
{U(·, )g(un) :n≥}is relatively compact inC([,b];X). So the set{Kun}∞n=is relatively
from which it follows that{Kun}∞n=is equicontinuous on eachJidue to the compactness
fort∈[,b]. Then, by Lemma ., we get
β{Kun}∞n=
= sup
t∈[,b]
β(Kun)(t)
∞
n=
= .
From (.), we have
β{yn}∞n=
≤β{Kun}∞n=
+β{Kun}∞n=
,
which impliesβ({yn}∞n=) = . Due to (.), we have
β(KW)≤β{yn}∞n=+ε.
Asεis arbitrary, it follows from the above inequality thatβ(KW) = . Thus, we get
β(W) =βco{KW,u}
=β(KW) = .
Therefore,Wis convex compact and nonempty inPC([,b];X), andK(W)⊂W. By the Schauder fixed point theorem, there exists at least one mild solution uof the problem (.), whereu∈Wis a fixed point of the continuous mapK. This completes the proof of
Theorem ..
Remark . If the functionf is compact or Lipschitz continuous (see,e.g., [, ]), then l(t) in hypothesis (H) becomes zero or a Lipschitz constant. In our proof, the measure
of noncompactness (MNC) is used to get rid of the compactness of the evolution system. Note that in [, , , ], MNC is adopted to discuss the differential and integrodifferential system in Banach spaces when the operation semigroup (evolution system) is compact or equicontinuous. Here the condition on the evolution systemU(t,s) is only supposed to be strongly continuous and they are weak in essence compared with the previous results. In our recent paper [], we get some existence results of fractional differential equations with nonlocal conditions when the semigroup is strongly continuous. There the work is based on a new regular measure of noncompactness defined by us (see Lemma . of []). We conjecture that the two different approaches in [] and in the present paper may be considered from a unified point of view in some way. It is an interesting problem and is worth discussing later.
4 Applications
Example . Consider the following integrodifferential evolution system with impulsive condition:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
ωt(t,θ) =ωθ(t,θ) +F(t,ω(t,θ),
t
h(t,s,ω(s,θ)) ds), ≤t≤b, ≤θ≤π,t=ti,
ω(t, ) =ω(t,π) = , ≤t≤b,
ω(ti+,θ) –ω(ti–,θ) =Ii(ω(ti,θ)), i= , , . . . ,p,
ω(,θ) =bg(s)log( +|ω(s,θ)|) ds+u(θ), ≤θ≤π,
whereF: [,b]×R×R→R,h: [,b]×[,b]×R→R, andu∈L[,π].
TakeX=L[,π]. DefineA(t)≡A:D(A)⊂X→XbyAz=zwith the domainD(A) =
semigroupT(t) defined byT(t)z(s) =z(t+s) for eachz∈X.T(t) is not a compact semi-group onXandβ(T(t)D)≤β(D) for a bounded subsetD, whereβis the Hausdorff MNC.
Now, we assume that:
() f : [,b]×X×X→Xis a continuous function defined by
f(t,x,h)(θ) =Ft,x(t,θ),h(t,θ), t∈[,b], ≤θ≤π,
h(t,θ) = t
h
t,s,x(s,θ)ds.
We take
F
t,x(t,θ), t
h
t,s,x(s,θ)ds
=csint,x(t,θ)+c t
x(ξ,θ) + dξ,
cis a constant. Thenf,hsatisfy hypotheses (H) and (H) of Section .
() Ii:X→Xis a continuous function for eachi= , , . . . ,p, defined by
Ii(x)(θ) =Iix(θ).
We take
Iix(θ)=
π
ρi(θ,y)cosx(y)dy,
x∈X,ρi∈C([,π]×[,π],R), for eachi= , , . . . ,p. ThenIiis compact and
satisfies hypothesis (H).
() g:PC([,b];X)→Xis a continuous function defined by
g(u)(θ) = b
g(s)log
+u(s)(θ)ds, u∈PC[,b];X,
withu(s)(θ) =ω(s,θ). Thengis a compact operator.
Under these assumptions, the above partial differential system can be reformulated as the abstract problem (.). Then due to Theorem ., the partial differential system has at least one mild solution on [,b].
Remark . We can extend our method to exact controllability for integrodifferential evolution system with impulsive conditions. That is,
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
u(t) =A(t)u(t) +f(t,u(t),th(t,s,u(s) ds)) +Bv(t), t∈J= [,b],t=ti,i= , , . . . ,p,
u(ti) =u(ti+) –u(t–
i) =Ii(u(ti)), i= , , . . . ,p, u() =g(u) +u,
hereBis a bounded linear operator from a Banach spaceVtoXand the control function v(·) is given inL([,b],V).
and other hypotheses are satisfied, that is, in this case the applications of controllability results are restricted to a finite-dimensional space. In order to fix this problem, Jiet al. [] and Machadoet al.[] used some measures of noncompactness to weaken the as-sumptions of compactness on the evolution systemU(t,s), where the evolution system is supposed to be equicontinuous. In this paper, the evolution system is only supposed to be strongly continuous, without any restrictions of compactness or equicontinuity. Since the method used in this paper is also available for controllability of the evolution equations with impulsive conditions, we can improve many controllability results under a noncom-pact semigroup, like in [, , ].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, 223003, P.R. China.2Library, Huaiyin
Institute of Technology, Huaian, 223003, P.R. China.
Acknowledgements
The research is supported by Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 14KJB110001), the Natural Science Foundation of Jiangsu Province (BK20150415), the Foundation of Huaiyin Institute of Technology (HGC1229).
Received: 22 April 2015 Accepted: 15 September 2015
References
1. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)
2. Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)
3. Ahmed, NU: Existence of optimal controls for a general class of impulsive systems on Banach spaces. SIAM J. Control Optim.42, 669-685 (2003)
4. Cardinali, T, Rubbioni, P: Impulsive semilinear differential inclusion: topological structure of the solution set and solutions on non-compact domains. Nonlinear Anal.14, 73-84 (2008)
5. Migorski, S, Ochal, A: Nonlinear impulsive evolution inclusions of second order. Dyn. Syst. Appl.16, 155-173 (2007) 6. Fan, Z, Li, G: Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct.
Anal.258, 1709-1727 (2010)
7. Abada, N, Benchohra, M, Hammouche, H: Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Differ. Equ.246, 3834-3863 (2009)
8. Ji, S, Li, G, Wang, M: Controllability of impulsive differential systems with nonlocal conditions. Appl. Math. Comput.
217, 6981-6989 (2011)
9. Ji, S, Li, G: A unified approach to nonlocal impulsive differential equations with the measure of noncompactness. Adv. Differ. Equ.2012, 182 (2012)
10. Byszewski, L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl.162, 494-505 (1991)
11. Byszewski, L, Lakshmikantham, V: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal.40, 11-19 (1991)
12. Aizicovici, S, Lee, H: Nonlinear nonlocal Cauchy problems in Banach spaces. Appl. Math. Lett.18, 401-407 (2005) 13. Zhu, L, Li, G: On a nonlocal problem for semilinear differential equations with upper semicontinuous nonlinearities in
general Banach spaces. J. Math. Anal. Appl.341, 660-675 (2008)
14. Ntouyas, SK, Tsamatos, PC: Global existence for semilinear evolution equations with nonlocal conditions. J. Math. Anal. Appl.210, 679-687 (1997)
15. Xue, X: Existence of semilinear differential equations with nonlocal initial conditions. Acta Math. Sin.23, 983-988 (2007)
16. Xue, X: Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces. Nonlinear Anal. TMA70, 2593-2601 (2009)
17. Balachandran, K, Trujillo, JJ: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Anal. TMA72, 4587-4593 (2010)
18. Arthi, G, Balachandran, K: Controllability of damped second-order impulsive neutral functional differential systems with infinite delay. J. Optim. Theory Appl.152, 799-813 (2012)
19. Wang, J, Wei, W: A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces. Results Math.58, 379-397 (2010)
21. Benchohra, M, Ziane, M: Impulsive evolution inclusions with state-dependent delay and multivalued jumps. Electron. J. Qual. Theory Differ. Equ.2013, 42 (2013)
22. Banas, J, Goebel, K: Measure of Noncompactness in Banach Spaces. Lect. Notes Pure Appl. Math., vol. 60. Dekker, New York (1980)
23. Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) 24. Kamenskii, M, Obukhovskii, V, Zecca, P: Condensing Multivalued Maps and Semilinear Differential Inclusions in
Banach Spaces. de Gruyter Series in Nonlinear Analysis and Applications, vol. 7. de Gruyter, Berlin (2001) 25. Cardinali, T, Rubbioni, P: On the existence of mild solutions of semilinear evolution differential inclusions. J. Math.
Anal. Appl.308, 620-635 (2005)
26. Bothe, D: Multivalued perturbation ofm-accretive differential inclusions. Isr. J. Math.108, 109-138 (1998) 27. Ji, S, Li, G: Solutions to nonlocal fractional differential equations using a noncompact semigroup. Electron. J. Differ.
Equ.2013, 240 (2013)
