# Controllability of Impulsive Neutral Functional Differential Inclusions in Banach Spaces

Authors: X. J. Wan; Y. P. Zhang; J. T. Sun
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the controllability of impulsive neutral functional differential inclusions in Banach spaces. Our main aim is to find an effective method to solve the controllability problem of impulsive neutral functional differential inclusions with multivalued jump sizes in Banach spaces. Based on a fixed point theorem with regard to condensing map, sufficient conditions for the controllability of the impulsive neutral functional differential inclusions in Banach spaces are derived. Moreover, a remark is given to explain less conservative criteria for special cases, and work is improved in the previous literature.


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## Body

## 1. Introduction

During the last decade, differential inclusions [1–3] were well known for applications to mechanics, engineering, and so on. Impulsive differential equations [4–9] were important in the study of physical fields. Ahmed [10] first introduced three different models of impulsive differential inclusions and studied the existence of them, respectively. From then on, there have been many focuses on various properties of impulsive differential inclusions, see [11–17] and references therein.Controllability is one of the primary problems in control theory [11, 13, 14, 17–24]. Study on controllability has always been considered as a hot topic given its numerous applications to mechanics, electrical engineering, medicine, biology, and so forth. Because of their various application backgrounds, there were a number of researches on controllability of differential inclusions, see [11, 13, 14, 17]. Controllability of impulsive functional differential inclusions is an attractive subject, thanks to their outstanding performance in applications. But as far as we are concerned, there were very few results on controllability of the model with multivalued jump sizes [13]. As for the third model initiated by Ahmed [10], we were impressed by the statement that the model of differential inclusions with multi-valued jump sizes may arise under many different situations, for example, in case of a control problem where one wishes to control the jump sizes in order to achieve certain objectives. In this paper, we aim to find an effective method to solve the controllability problem of impulsive neutral functional differential inclusions with multi-valued jump sizes in Banach spaces.Liu [11] studied impulsive neutral functional differential inclusions in Banach spaces. However, to the best of our knowledge, there has not any result considering the controllability of the impulsive neutral functional differential inclusions with multi-valued jump sizes in Banach spaces. This work is both challenging and interesting, since our systems are more general than those studied ever before. Based on a fixed point theorem with regard to condensing map, we work out the sufficient conditions for the controllability of impulsive neutral functional differential inclusions in Banach spaces. In [11], Liu considered the controllability basing on Martelli’s fixed point theorem [25]. He took advantage of the statement that a completely continuous map is a condensing map. However, condensing map may not be completely continuous. We notice this inequality and consider the controllability on the strength of a special property of Kuratowski measure of noncompactness in Banach spaces. Due to the property, we are allowed to prove that a map is condensing according to its definition. When jumps are single-valued maps in our system, the system degenerates into the system (1.1) in [11]. At this time, less conservative criteria can be given for controllability of system (1.1) [11] after appropriate degeneration. Work in [11] is improved.The content of this paper is organized as follows. In Section2, some preliminaries are recalled; the impulsive neutral functional differential inclusions is proposed. In Section 3, the results on controllability of impulsive neutral functional differential inclusions in Banach spaces are derived, as well as strictly proof; a remark is given to show our criteria are less conservative. In Section 4, conclusions are given to explain our work in this paper.


## 2. Preliminaries

Definition 1.
LetX be a Banach space, a multi-valued map 𝔉:X→2X is called convex valued, if 𝔉(x) is convex for all x∈X.
𝔉 is called closed valued, if 𝔉(x) is closed for all x∈X.
𝔉 is called bounded on bounded set, if 𝔉(E)=∪x∈E𝔉(x) is bounded in X for any bounded subset E⊂X.
𝔉 is called upper semicontinuous on X, if for every x^∈X, the set 𝔉(x^) is a nonempty and closed subset of X, and for every open set E of X containing 𝔉(x^), there is an open neighborhood B of x^, such that 𝔉(B)⊆E.We make the following notations:𝔅h={χ:(-∞,0]→X; for any ε>0, χ is a bounded and measurable function on [-ε,0], and ∫-∞0‍h(s)sup[s,0]|χ|ds<+∞}, where h:(-∞,0]→(0,+∞) is a continuous function. Define norm on 𝔅h, as ∥χ∥𝔅h=∫-∞0‍h(s)sup[s,0]|χ|ds. (𝔅h,∥·∥𝔅h) is a Banach space [11].H
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E is bounded, convex, and closed in X}.In this paper, we consider the neutral functional differential inclusions in Banach spaceX as follows:
(1){ddt[x(t)-g(t,xt)]∈Ax(t)+F(t,xt)+Bu(t),t∈J∖{tk}k=1m,Δx∣t=tk=x(tk+)-x(tk-)∈Ik(x(tk-)),k=1,2,…,m,x0=ϕ∈𝔅h,
where x∈X. For t∈J,xt represents the xt:(-∞,0]→𝔅h defined by xt(θ)=x(t+θ),θ∈(-∞,0] which belongs to some abstract phase space 𝔅h;g:J×𝔅h→X;J=[0,b], where b is a positive constant; A is the infinitesimal generator of a strongly continuous operator semigroup (T(t))t>0 [26]; F:J×𝔅h→2X is a closed, bounded, and convex valued multivalued map; B:U→X is a continuous linear operator, where U is a Banach space with u(·)∈L2(J,U), here u(·) is the control function; {Ik:X→2X}k=1m are closed, bounded, and convex valued multi-valued maps, x(tk+), and x(tk-) represent the left and right limits of x(·) at t=tk, respectively. The histories xt:(-∞,0]→X,xt(θ)=x(t+θ).We introduce definitions the following.Definition 2.
A functionx:(-∞,b]→X is called a mild solution of system (1) if the following holds: x0=ϕ∈𝔅h on (-∞,0] and for each s∈[0,t), the function AT(t-s)g(s,xs) is integrable, and there exists ℐk(x(tk-))∈Ik(x(tk-)), such that the integral equation
(2)x(t)=T(t)[ϕ(0)-g(0,ϕ)]+g(t,xt)+∫0t‍AT(t-s)g(s,xs)ds+∫0t‍T(t-s)f(s)ds+∫0t‍T(t-s)(Bu)(s)ds+∑0<tk<t‍T(t-tk)ℐk(x(tk-)),t∈J,
is satisfied, where f∈SF,x={f∈L1(J,X):f(t)∈F(t,xt), for a.e. t∈J}.Definition 3.
The system (1) is said to be controllable on the interval J, if for every initial function ϕ∈𝔅h and every x1∈X, there exists a control function u∈L2(J,U), such that the mild solution x of (1) satisfies x(b)=x1.Definition 4 (see [27]).
A mapN:X→X is called α-condensing, if for any bounded subset S of X, N(S) is bounded and α(N(S))<α(S), α(S)>0.Remark 5.
Theα(·) in the Definition 4 is called the Kuratowski measure of noncompactness, which is defined as α(S)=inf{d>0 : there exist finitely many sets of diameter at most d which cover S}. Measures of noncompactness are useful in the study of infinite-dimensional Banach spaces, where any ball ℬ of diameter d has α(ℬ)=d.Lemma 6 (see [25]).
LetE be a Banach space, and N:E→Pbd,cv,cl(E) is a condensing map. If the set
(3)Ω={x∈E:λx∈Nx,forsomeλ>1}
is bounded, then N has a fixed point.


## 3. Main Results

In order to study system (1), we introduce hypotheses hereinafter: (A0)
T(t) is bounded, that is to say there are constants M1, such that ∥T∥≤M1.(A1)
The linear operatorW:L2(J,U)→X defined by
(4)Wu=∫0b‍T(b-s)Bu(s)ds
has an inverse operator W-1, which takes value in L2(J,U)/kerW. And W-1 is bounded. There exist positive constants M2 and M3 satisfying ∥B∥≤M2 and ∥W-1∥≤M3.(A2)
For eachk∈{1,2,…,m}, there is a positive constant αk, such that ∥I(x(tk-))∥=sup{|ℐk(x(tk-))|:ℐ(x(tk-))∈I(x(tk-))}≤αk for all x∈X.(A3)
There exist constantsa1, a2, b1, and b2, satisfying ∥Ag(t,χ)∥≤a1∥χ∥𝔅h+a2,|g(t,χ)|≤b1∥χ∥𝔅h+b2, and  t∈J,χ∈𝔅h.(A4)
F:J×𝔅h→Pbd,cv,cl(X);(t,χ)↦F(t,χ) is measurable with respect to t for every χ∈𝔅h, upper semicontinuous with respect to χ for every t∈J, and for every fixed χ∈𝔅h(5)SF,χ={f∈L1(J,X):f(t)∈F(t,χ),a.e.t∈J}
is nonempty, or equivalently, inf{|f|:f(t)∈F(t,χ)}∈L1{J,X}.(A5)
There is an integrable functionφ:J→[0,∞) and a continuous and nondecreasing function ψ:[0,∞)→(0,∞), such that
(6)∥F(t,χ)∥=sup{|f|:f∈F(t,χ)}≤φ(t)ψ(∥χ∥𝔅h),t∈J,χ∈𝔅h.Lemma 7 (see [28]).
LetI be a compact real interval, and let, E be a Banach space. Let F be a multivalued map satisfying (A4), and let Γ be a linear continuous mapping from L1(I,E)→C(I,E). Then the operator
(7)Γ∘SF:C(I,E)⟶Pbd,cv,cl(C(I,E)),x↦(Γ∘SF)(x)=Γ(SF,x)
is a closed graph operator in C(I,E)×C(I,E).We denote the Banach space𝔅b={x(t):(-∞,b]→X:xk∈C(Jk,X),k=1,2,…,m, such thatxis left continuous and right limits}, with seminorm defined by ∥x∥𝔅b:=∥x0∥𝔅h+sup0≤s≤b|x(s)|, where xk is the restriction of x to Jk=(tk,tk+1].Lemma 8.
Ifx∈𝔅b and η=∫-∞0‍h(s)ds<+∞, then for xt∈𝔅h, η|x(t)| ≤ ∥xt∥𝔅h ≤ ∥x0∥𝔅h+ηsup0≤s≤t|x(s)| holds.Proof.
On the one hand, we have(8)∥xt∥𝔅h=∫-∞0‍h(s)sups≤θ≤0|xt(θ)|ds=∫-∞0‍h(s)supt+s≤θ≤t|x(θ)|ds=∫-∞-t‍h(s)supt+s≤θ≤t|x(θ)|ds+∫-t0‍h(s)supt+s≤θ≤t|x(θ)|ds≤∫-∞-t‍h(s)[supt+s≤θ≤0|x(θ)|+sup0≤θ≤t|x(θ)|]ds+∫-t0‍h(s)sup0≤θ≤t|x(θ)|ds≤∫-∞-t‍h(s)supt+s≤θ≤0|x(θ)|ds+∫-∞0‍h(s)sup0≤θ≤t|x(θ)|ds≤∫-∞-t‍h(s)sups≤θ≤0|x(θ)|ds+ηsup0≤s≤t|x(s)|≤∫-∞0‍h(s)sups≤θ≤0|x(θ)|ds+ηsup0≤s≤t|x(s)|=∥x0∥𝔅h+ηsup0≤s≤t|x(s)|.
On the other hand, ∥xt∥𝔅h=∫-∞0‍h(s)sups≤θ≤0|xt(θ)|ds≥|xt(0)|∫-∞0‍h(s)ds=η|x(t)|.
The proof is thus completed.For anyx1∈X, we design the control function u(t) in system (1) as
(9)u(t)=W-1{∑k=1m‍x1-T(b)[ϕ(0)-g(0,ϕ)]-g(b,xb)-∫0b‍AT(b-ξ)g(ξ,xξ)dξ-∫0b‍T(b-ξ)f(ξ)dξ-∑k=1m‍T(b-tk)ℐk(x(tk-))}(t).
Then we consider the multi-valued function ℱ:𝔅b→2𝔅b, for any x∈𝔅b,
(10)ℱx(t)={ϕ(t),t∈(-∞,0]T(t)[ϕ(0)-g(0,ϕ)]+g(t,xt)+∫0t‍AT(t-s)g(s,xs)ds+∫0t‍T(t-s)f(s)ds+∫0t‍T(t-s)(Bu)(s)ds+∑0<tk<t‍T(t-tk)ℐk(x(tk-)),t∈J,
where u(t) is described by (9), f(t)∈SF,x, ℐk(x(tk-))∈Ik(x(tk-)).If we define functionϕ~:(-∞,b]→𝔅b as
(11)ϕ~(t)={ϕ(t),t∈(-∞,0],T(t)ϕ(0),t∈[0,b],
where ϕ(t)∈𝔅h, and denote y(t)=x(t)-ϕ~(t), then x(t) is a mild solution of system (1) if and only if y0=0 and
(12)y(t)=-T(t)g(0,ϕ)+g(t,xt)+∫0t‍AT(t-s)g(s,xs)ds+∫0t‍T(t-s)f(s)ds+∫0t‍T(t-s)(Bu)(s)ds+∑0<tk<t‍T(t-tk)ℐk(x(tk-)),t∈J,
where xt=yt+ϕ~t, x(tk-)=y(tk-)+ϕ~(tk-).Now we denote𝔅b0={y(t)∈𝔅b:y0=0∈𝔅h}, and define norm ∥y(t)∥𝔅b0=sup0≤t≤b|y(t)|, thus 𝔅b0 is a Banach space. Then we can define another multi-valued function ℱ1:𝔅b0→2𝔅b0;ℱ1y(t)=ℱx(t)-ϕ~(t), so
(13)ℱ1y(t)={0,t∈(-∞,0]-T(t)g(0,ϕ)+g(t,xt)+∫0t‍AT(t-s)g(s,xs)ds+∫0t‍T(t-s)f(s)ds+∫0t‍T(t-s)(Bu)(s)ds+∑0<tk<t‍T(t-tk)ℐk(x(tk-)),t∈J,
where u(t) is the same as that in (10). Thus ℱ has a fixed point in 𝔅b if and only if ℱ1 has a fixed point in 𝔅b0, y(t)=x(t)-ϕ~(t).Let𝔅r={y(t)∈𝔅b0:∥y(t)∥𝔅b0=sup0≤s≤b|y(s)|≤r}, and r is a positive constant. It is true that 𝔅r is a closed subspace of 𝔅b0, so 𝔅r is also a Banach space. Next we show that ℱ1 has a fixed point in 𝔅r.Lemma 9.
ℱ
1 is bounded, convex, and closed on 𝔅r.Proof.
(I)ℱ1 is bounded on 𝔅r. Let y∈𝔅r, thanks to Lemma 8,
(14)∥xt∥𝔅h≤∥yt∥𝔅h+∥ϕ~t∥𝔅h≤ηsup0≤s≤t|y(s)|+∥y0∥𝔅h+ηsup0≤s≤t|ϕ~(s)|+∥ϕ~0∥𝔅h≤η(r+M1|ϕ(0)|)+∥ϕ∥𝔅h≜r1,(15)|u(t)|=|W-1{∑k=1m‍x1-T(b)[ϕ(0)-g(0,ϕ)]-g(b,xb)-∫0b‍AT(b-ξ)g(ξ,xξ)dξ-∫0b‍T(b-ξ)f(ξ)dξ-∑k=1m‍T(b-tk)ℐk(x(tk-))}(t)|≤M3[∑k=1m‍|x1|+M1(|ϕ(0)|+|g(0,ϕ)|)+|g(b,xb)|+M1b(a1r1+a2)+M1∫0b‍|f(ξ)|dξ+M1∑k=1m‍|ℐk(x(tk-))|]≤M3[∑k=1m‍|x1|+M1(|ϕ(0)|+|g(0,ϕ)|)+b1r1+b2+M1b(a1r1+a2)+M1∫0b‍φ(ξ)ψ(∥xξ∥𝔅h)ds+M1∑k=1m‍αk].
Then we have
(16)|ℱ1y(t)|≤|T(t)g(0,ϕ)|+|g(t,xt)|+∫0t‍|AT(t-s)g(s,xs)|ds+∫0t‍|T(t-s)f(s)|ds+∫0t‍|T(t-s)(Bu)(s)|ds+∑0<tk<t‍|T(t-tk)ℐk(x(tk-))|≤M1|g(0,ϕ)|+M1b(a1r1+a2)+M1∫0b‍φ(s)ψ(∥xs∥𝔅h)ds+bM1M2M3×[∑k=1m‍|x1|+M1(|ϕ(0)|+|g(0,ϕ)|)+(1+bM2)(a1∥xb∥𝔅h+a2)+M1∫0b‍φ(s)ψ(∥xs∥𝔅h)ds+M1∑k=1m‍αk]+M1∑k=1m‍αk≤M1|g(0,ϕ)|+(1+bM2)(a1r1+a2)+M1ψ(r1)∫0b‍φ(s)ds+bM1M2M4×[∑k=1m‍|x1|+M1(|ϕ(0)|+|g(0,ϕ)|)+b1r1+b2+M1b(a1r1+a2)+M1ψ(r1)∫0b‍φ(s)ds+M1∑k=1m‍αk]+M1∑k=1m‍αk≜M4.
Consequently, ∥ℱ1y(t)∥𝔅b≤M4.
(II)ℱ1 is convex on 𝔅r. Let y∈{𝔅r} and z1,z2∈ℱ1y, there must be f1,f2∈SF,x and ℐk1,ℐk2∈Ik(x(tk-)), such that
(17)zγ=-T(t)g(0,ϕ)+g(t,xt)+∫0t‍AT(t-s)g(s,xs)ds+∫0t‍T(t-s)fγ(s)ds+∫0t‍T(t-s)BW-1×{∑k=1m‍x1-T(b)[ϕ(0)-g(0,ϕ)]-g(b,xb)-∫0b‍AT(b-ξ)g(ξ,xξ)dξ-∫0b‍T(b-ξ)fγ(ξ)dξ-∑k=1m‍T(b-tk)ℐkγ(x(tk-))}sds+∑0<tk<t‍T(t-tk)ℐkγ(x(tk-)),
for γ=1,2. Then for any τ∈[0,1], we have
(18)[τz1+(1-τ)z2](t)=-T(t)g(0,ϕ)+g(t,xt)+∫0t‍AT(t-s)g(s,xs)ds+∫0t‍T(t-s)[τf1+(1-τ)f2](s)ds+∫0t‍T(t-s)BW-1×{×[τℐk1+(1-τ)ℐk2](x(tk-))x1-T(b)[ϕ(0)-g(0,ϕ)]-g(b,xb)-∫0b‍AT(b-ξ)g(ξ,xξ)dξ-∫0b‍T(b-ξ)[τf1+(1-τ)f2](ξ)dξ-∑k=1m‍T(b-tk)×[τℐk1+(1-τ)ℐk2](x(tk-))}(s)ds+∑0<tk<t‍T(t-tk)[τℐk1+(1-τ)ℐk2](x(tk-))(s)ds.
Combining SF,x and Ik(x(tk-)) is convex, and ℱ1 is convex.
(III)ℱ1 is closed on 𝔅r. Let y∈{𝔅r}. Here we should proof that if there are sequences {zn} satisfying zn→z* and zn∈ℱ1y, then z*∈ℱ1y holds.
For everyzn∈ℱ1y, there is a fn∈SF,x and a ℐkn∈Ik(x(tk-)), such that
(19)zn=-T(t)g(0,ϕ)+g(t,xt)+∫0t‍AT(t-s)g(s,xs)ds+∫0t‍T(t-s)fn(s)ds+∫0t‍T(t-s)BW-1×{∑k=1m‍x1-T(b)[ϕ(0)-g(0,ϕ)]-g(b,xb)-∫0b‍AT(b-ξ)g(ξ,xξ)dξ-∫0b‍T(b-ξ)fn(ξ)dξ-∑k=1m‍T(b-tk)ℐkn(x(tk-))}(s)ds+∑0<tk<t‍T(t-tk)ℐkn(x(tk-)).
And forz*, we should prove that there must be some f*∈SF,x and some ℐk*∈Ik(x(tk-)), such that
(20)z*=-T(t)g(0,ϕ)+g(t,xt)+∫0t‍AT(t-s)g(s,xs)ds+∫0t‍T(t-s)f*(s)ds+∫0t‍T(t-s)BW-1×{∑k=1m‍x1-T(b)[ϕ(0)-g(0,ϕ)]-g(b,xb)-∫0b‍AT(b-ξ)g(ξ,xξ)dξ-∫0b‍T(b-ξ)f*(ξ)dξ-∑k=1m‍T(b-tk)ℐk*(x(tk-))}(s)ds+∑0<tk<t‍T(t-tk)ℐk*(x(tk-)).
Considering calculation of (20) subtracting (19), we get
(21)z*-zn=∫0t‍T(t-s)×[BW-1(∫0b‍T(b-ξ)(fn(ξ)-f*(ξ))dξ)+∑k=1m‍T(t-tk)(ℐkn(x(tk-))-ℐk*(x(tk-)))+(fn-f*)∫0b‍](s)ds+∑0<tk<t‍T(t-tk)(ℐkn(x(tk-))-ℐk*(x(tk-)))⟶0,n⟶+∞.
Meanwhile Ik is a closed multi-valued map, there truly exists some ℐk*∈Ik(x(tk-)) for (20).
Accordingly, (21) can be transformed to
(22)∫0t‍T(t-s)[BW-1(∫0b‍T(b-ξ)(fn-f*)(ξ)dξ)+(fn-f*)∫0b‍](s)ds⟶0,n⟶+∞.
We construct a linear and continuous operator like
(23)Γ:L1(J,X)⟶C(J,X);Γ∘SF(x)=Γ(f)(t)=∫0t‍T(t-s)[BW-1(∫0b‍T(b-ξ)f(ξ)dξ)×(s)+f(s)(∫0b‍T(b-ξ)f(ξ)dξ)]ds,
moreover,
(24)∥Γ(f)(t)∥L1≤(bM1M2M3+1)M1∥f∥L1.
From Lemma 7, Γ∘SF is a closed graph operator. Then (22) implies that ∫0t‍T(t-s)[BW-1(∫0b‍T(b-ξ)f*(ξ)dξ)(s)+f*(s)]ds∈Γ(SF,x), with f*∈SF,x.
From the foregoing, (20) holds, which means z*∈ℱ1y. Thus ℱ1 is closed.
The proof is thus completed.Theorem 10.
Assume that hypotheses(A0)–(A5) hold, and
(25)M1|g(0,ϕ)|+b1r1+b2+M1b(a1r1+a2)+M1ψ(r1)∫0b‍φ(s)ds+bM1M2M3×[∑k=1m‍|x1|+M1(|ϕ(0)|+|g(0,ϕ)|)+b1r1+b2+M1b(a1r1+a2)+M1ψ(r1)∫0b‍φ(s)ds+M1∑k=1m‍αk]+M1∑k=1m‍αk<r,
then system (1) is controllable on J under control function (9).Proof.
First,ℱ1 is a condensing map on 𝔅b0. Considering Remark 5, we just have to prove that for any bounded subset E of 𝔅b0, diam(ℱ1(E))<diam(E). It is obvious, because ∥ℱ1x(t)∥≤M4<r= diam(𝔅r) for any x(t)∈𝔅r by combining inequalities (19) and (25).
Second, here we show that the setΩ={y∈𝔅b0:λy∈ℱ1y for some λ>1} is bounded. Let y∈Ω, then there are f∈SF,x and ℐk∈Ik(x(tk-)), such that
(26)y(t)=1λ[∑k=1m‍-T(t)g(0,ϕ)+g(t,xt)+∫0t‍AT(t-s)g(s,xs)ds+∫0t‍T(t-s)f(s)ds+∫0t‍T(t-s)(Bu)(s)ds+∑0<tk<t‍T(t-tk)ℐk(x(tk-))].
So |y(t)|≤(1/λ)M4<M4. Due to Lemma 6, ℱ1 has a fixed point in 𝔅r. Consequently, ℱ has a fixed point, which means system (1) is controllable.
The proof is thus completed.Remark 11.
In case{Ik} in system (1) are single-valued maps, then the system (1) degenerates into the system (1.1) in [11]. Accordingly, our degenerated assumptions for the controllability Theorem 3.1 [11] are less conservative, which means the following: firstly, the hypothesis (H4)(i) [11] is unnecessary; secondly, the very complex hypothesis (H6) [11] can be replaced by inequality (25); finally, the assumption |AT|≤M2 [11] is replaced by ∥Ag(t,χ)∥≤a1∥χ∥𝔅h+a2, so A is not necessary to be bounded; otherwise, our results can only be applied to finite Banach spaces [29].


## 4. Conclusion

In this paper, we have investigated the controllability of impulsive neutral functional differential inclusions in Banach spaces. Based on a fixed point theorem with regard to condensing map, sufficient conditions for the controllability of the impulsive neutral functional differential inclusions in Banach spaces have been derived. Moreover, a remark has been given to explain less conservative criteria for special cases. We have found an effective method to solve the controllability problem of impulsive neutral functional differential inclusions with multi-valued jump sizes in Banach spaces. Work has been improved in the previous literature.



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*Source: 861568-2013-03-06.xml*