# Controllability of Second Order Impulsive Neutral Functional Differential Inclusions with Infinite Delay

**ABSTRACT** This paper is concerned with controllability of a partial neutral functional differential inclusion of second order with impulse effect and infinite delay. We introduce a new phase space to prove the controllability of an inclusion which consists of an impulse effect with infinite delay. We claim that the phase space considered by different authors is not correct. We establish the controllability of mild solutions using a fixed point theorem for contraction multi-valued maps and without assuming compactness of the family of cosine operators.

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**ABSTRACT:**We present some results on the existence of solutions for second-order impulsive differential equations with deviating argument subject to functional initial conditions. Our results are based on Schaefer's fixed point theorem for completely continuous operators.Journal of function spaces and applications 04/2013; 2013. · 0.58 Impact Factor - SourceAvailable from: Dimplekumar N. Chalishajar[Show abstract] [Hide abstract]

**ABSTRACT:**Abstract We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order � 2 (n−1, n) in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder’s inequality, a suitable singular Gronwall’s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.Acta Mathematica Scientia 01/2013; 33 B(3):1-16. · 0.49 Impact Factor - SourceAvailable from: Dr. Gunasekar T[Show abstract] [Hide abstract]

**ABSTRACT:**This paper is concerned with the controllability of a partial neutral func-tional integrodifferential inclusion of second order with impulse effect and infinite delay in Banach spaces. The controllability of mild solutions using a fixed point theorem for contraction multi-valued maps and without assuming compactness of the family of cosine operators.12/2013;

Page 1

J Optim Theory Appl (2012) 154:672–684

DOI 10.1007/s10957-012-0025-6

Controllability of Second Order Impulsive Neutral

Functional Differential Inclusions with Infinite Delay

Dimplekumar N. Chalishajar

Received: 3 November 2011 / Accepted: 7 March 2012 / Published online: 17 May 2012

© Springer Science+Business Media, LLC 2012

Abstract This paper is concerned with controllability of a partial neutral functional

differential inclusion of second order with impulse effect and infinite delay. We in-

troduce a new phase space to prove the controllability of an inclusion which consists

of an impulse effect with infinite delay. We claim that the phase space considered

by different authors is not correct. We establish the controllability of mild solutions

using a fixed point theorem for contraction multi-valued maps and without assuming

compactness of the family of cosine operators.

Keywords Controllability · Second order impulsive neutral differential inclusions ·

Fixed point theorem for multi-valued maps · Strongly continuous cosine family

1 Introduction

The problem of controllability for impulsive functional differential inclusions in Ba-

nach spaces has been studied extensively. Benchohra et al. [1] discussed the control-

labilityoffirstandsecondorderneutralfunctionaldifferentialandintegro-differential

inclusions in a Banach space with non-local conditions, without impulse effect.

Chang and Li [2] obtained the controllability result for functional integro-differential

inclusions on an unbounded domain without impulse term. Benchohra et al. [3] stud-

ied the existence result for damped differential inclusion with impulse effect. Hernan-

dez et al. [4] proved the existence of solutions for impulsive partial neutral functional

Communicated by Mark J. Balas.

D.N. Chalishajar

Department of Mathematics and Computer Science, Mallory Hall, Virginia Military Institute,

Lexington, VA 24450, USA

e-mail: chalishajardn@vmi.edu

D.N. Chalishajar (?)

e-mail: dipu17370@yahoo.com

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J Optim Theory Appl (2012) 154:672–684673

differentialequationsforfirstandsecondordersystemswithinfinitedelay.Ithasbeen

observed that the existence or the controllability results proved by different authors

are through an axiomatic definition of the phase space given by Hale and Kato [5].

However, as remarked by Hino, Murakami, and Naito [6], it has come to our attention

that these axioms for the phase space are not correct for the impulsive systems with

infinite delay.

On the other hand, researchers have been proving the controllability results using

compactness assumption of semigroups and the family of cosine operators. Bing Liu

[7] studied the controllability of first order impulsive neutral functional differential

inclusions with infinite delay in a Banach space with the assumption of compactness

of the semigroup. However, as remarked by Triggiani [8], in an infinite dimensional

Banach space, the linear control system is never exactly controllable on a given inter-

val of time, if either a bounded linear operator (from control space to state space) is

compact or a semigroup is compact. According to Triggiani [8], this is a typical case

for most control systems governed by parabolic partial differential equations, and

hence the concept of exact controllability is very limited for many parabolic partial

differential equations. Nowadays, researchers are engaged to overcome this problem,

refer to [4, 9]. Very recently, Chalishajar et al. [10–12] studied the controllability of

second order neutral functional differential inclusion, with infinite delay and impulse

effect on unbounded domain, without compactness of the family of cosine opera-

tors. Ntouyas and O’Regan [13] made some remarks on controllability of evolution

equations in Banach paces and proved a result without compactness assumption.

In this paper, we discuss the controllability for the second order impulsive neutral

functional differential inclusions, with infinite delay through the phase space defined

in [6], and without compactness of the family of cosine operators. To the best of our

knowledge, a controllability result has not been studied in this connection.

Section2containsthepreliminaries,whicharerequiredfor further investigationof

this paper. Section 3 deals with the main controllability result in a separable Banach

space X using a fixed point theorem for contraction multi-valued maps due to Covitz

and Nadler [14]. Concluding remarks are given in Sect. 4.

2 Preliminaries

The purpose of this paper is to study the controllability of impulsive partial neutral

functional differential inclusions of second order with infinite delay. Specifically, we

are concerned with the inclusions form

⎧

∈ Ay(t)+Bu(t)+F(t,yt,y?(t));

?y|t=tk= Ik(y(tk),y?(tk));

?y?|t=tk= Ik(y(tk),y?(tk));

y(0) = φ ∈ Bh,

where A : D(A) ⊂ X → X is the infinitesimal generator of a strongly continuous

cosine family {C(t) : t ∈ R} defined on X, and F : J × Bh× X ⇒ X is a bounded,

⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

d

dt[y?(t)−f(t,yt,y?(t))]

t ∈ J := [0,m]; t ?= tk;

k = 1,2,...,m;

k = 1,2,...,m;

y?(0) = x0,

(1)

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674J Optim Theory Appl (2012) 154:672–684

closed, and convex multi-valued map. Let J0= ]−∞,0], and non-local condition

φ ∈ Bh(defined below) and x0∈ X be the given initial values. f : J × Bh→ X is a

given function, the state function y(t) takes values in X, and the control function u ∈

L2(J,U), a Banach space of admissible control functions with U as a Banach space.

B is a bounded linear operator from U to X. X is a Banach space with norm |.|.

Also, 0 < t0< t1< ··· < tp< tp+1= m (→ ∞ as t → ∞); Ik,Ik∈ C(X ×

X,X), k = 1,2,...,p are bounded, ?y|t=tk= y(t+

y?(t−

y(t) and y?(t), respectively, at t = tk. Furthermore, for any continuous function y

defined on the interval J1= ]−∞,m[ with values in X and for any t ∈ J, we denote

by ytan element of C(J0,X) defined by yt(θ) = y(t +θ),θ ∈ J0.

We present the abstract phase space Bh. Assume that h : ]−∞,0] → ]0,∞[ be a

continuous function with l =?0

Bh:=

?0

Here, Bhis endowed with the norm

?0

Then it is easy to show that (Bh,?.?Bh) is a Banach space.

k) − y(t−

k), ?y?|t=tk= y?(t+

k) −

k), and y(t−

k) and y(t+

k), y?(t−

k) and y?(t+

k) represent the left and right limits of

−∞h(s)ds < +∞. Define

?

φ : ]−∞,0] → X such that, for any r > 0, φ(θ) is bounded and measurable

function on [−r,0] and

−∞

h(s) sup

s≤θ≤0

??φ(θ)??ds < +∞

?

.

?φ?Bh=

−∞

h(s) sup

s≤θ≤0

??φ(θ)??ds,

∀φ ∈ Bh.

Lemma 2.1 Suppose y ∈ Bh; then, for each t ∈ J,yt∈ Bh. Moreover,

l??y(t)??≤ ?yt?Bh≤ l sup

where l :=?0

Proof Forany t ∈ [0,a],itiseasytoseethat ytisboundedandmeasurableon [−a,0]

for a > 0, and

?0

?−t

?−t

?−t

?0

s≤θ≤0

???y(s)??+?y0?Bh

?,

−∞h(s)ds < +∞.

?yt?Bh=

−∞

h(s) sup

θ∈[s,0]

??yt(θ)??ds

??y(t +θ)??ds +

sup

θ1∈[t+s,t]

?

h(s) sup

θ1∈[0,t]

=

−∞

h(s) sup

θ∈[s,0]

?0

?0

−t

h(s) sup

θ∈[s,0]

??y(t +θ)??ds

??y(θ1)??ds

ds

=

−∞

h(s)

??y(θ1)??ds +

??y(θ1)??+ sup

??y(θ1)??ds

−t

h(s)

sup

θ1∈[t+s,t]

??y(θ1)???

≤

−∞

h(s)

sup

θ1∈[t+s,0]

θ1∈[0,t]

+

−t

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J Optim Theory Appl (2012) 154:672–684675

=

?−t

?−t

?0

?0

−∞

h(s)

sup

θ1∈[t+s,0]

??y(θ1)??ds +

??y(θ1)??ds +l sup

??y(θ1)??ds +l sup

??y0(θ1)??ds +l sup

?0

−∞

h(s)ds sup

s∈[0,t]

??y(s)??

≤

−∞

h(s) sup

θ1∈[s,0]

s∈[0,t]

??y(s)??

??y(s)??

??y(s)??

≤

−∞

h(s) sup

θ1∈[s,0]

s∈[0,t]

=

−∞

h(s) sup

θ1∈[s,0]

??y(s)??+?y0?Bh.

s∈[0,t]

= l sup

s∈[0,t]

Since φ ∈ Bh, then yt∈ Bh. Moreover,

?0

The proof is complete.

?yt?Bh=

−∞

h(s) sup

θ∈[s,0]

??yt(θ)??ds ≥??yt(θ)??

?0

−∞

h(s)ds = l??y(t)??.

?

Next, we introduce definitions, notation and preliminary facts from multi-valued

analysis, which are useful for the development of this paper. Let C(J,X) be the Ba-

nach space of continuous functions from J to X with the norm ?x?∞= supt∈J|x(t)|.

B(X) denotes the Banach space of bounded linear operators from X to X.

Let L1(J,X) denote the Banach space of continuous function y : J → X, which

are integrable and endowed with the norm ?y?L1 =?m

P(X) := {Y ∈ P(X) : Y ?= φ}, Pcl(X) := {Y ∈ P(X) : Y is closed}, Pb(X) :=

{Y ∈ P(X) : Y is bounded}, Pcp(X) := {Y ∈ P(X) : Y is compact}, Pb,cl(X) :=

{Y ∈ P(X) : Y is bounded and closed}. We define a Hausdorff space Hd: P(X) ×

P(X) → R+∪{∞} by

Hd(A,B) := max

a∈A

where d(A,b) := infa∈Ad(a,b),d(a,B) := infb∈Bd(a,b).

Then (Pb,cl(X),Hd) is a metric space, and (Pcl(X),Hd) is a generalized (com-

plete) metric space.

We now recall some preliminaries about multi-valued maps.

Let (X,?.?) be a Banach space. A multi-valued map G : X ⇒ X is convex (resp.

closed) iff G(x) is convex (resp. closed) in X, for all x ∈ X. The map G is bounded

on bounded sets iff G(B) = Ux∈BG(x) is bounded in X for any bounded set B of X

(i.e., supx∈B{sup{?y? : y ∈ G(x)}} < ∞). G is called upper semi-continuous (u.s.c.)

on X iff for each x0∈ X the set G(x0) is a nonempty, closed subset of X and if for

each open set B of X containing G(x0) there exists an open neighbourhood A of

x0such that G(A) ⊆ B. The map G is said to be completely continuous iff G(B) is

relatively compact for every bounded subset B ⊆ X.

If the multi-valued map G is completely continuous with nonempty compact val-

ues, then G is u.s.c. iff G has a closed graph. That is, if xn→ x0and yn→ y0, where

0|y(t)|dt, y ∈ L1(J,X). For a

metric space (X,d), we introduce the following notations:

?

sup

d(a,B),sup

b∈B

d(A,b)

?

,

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676J Optim Theory Appl (2012) 154:672–684

yn∈ G(xn), then y0∈ G(x0). We say that G has a fixed point iff there is x ∈ X such

that x ∈ G(x). In the following, BCC(X) denotes the set of all nonempty bounded,

closed and convex subsets of X.

A multi-valued map G : J → BCC(X) is said to be measurable iff for each x ∈ X,

the distance function Y : J → R, defined by

Y(t) := d?x,G(t)?= inf?|x −z| : z ∈ G(t)?,

is measurable. For more details on multi-valued maps see [16, 17].

An upper semi-continuous map G : X ⇒ X is said to be condensing iff for any

subset B ⊆ X, with α(B) ?= 0, we have α(G(B)) < α(B), where α denotes the Kura-

towski measure of non-compactness. For the properties of the Kuratowski measure,

we refer to Banas and Goebel [15].

We note that a completely continuous multi-valued map is the easiest example of

a condensing map. For more details on multi-valued maps, see the book of Deimling

[16] and the research article of Travis and Webb [18].

We say that the family {C(t) : t ∈ R} of operators in B(X) is a strongly continuous

cosine family iff

1. C(0) = I, I is the identity operator on X.

2. C(t +s)+C(t −s) = 2C(t)C(s) for all s,t ∈ R.

3. The map t ?→ C(t)x is strongly continuous for each x ∈ X.

The strongly continuous sine family {S(t) : t ∈ R}, associated with the strongly con-

tinuous cosine family {C(t) : t ∈ R}, is defined by

?t

The infinitesimal generator A : X → X of a cosine family C(t),t ∈ R is defined by

Ax :=d2

S(t)x :=

0

C(s)x ds,x ∈ X, t ∈ R.

dt2C(t)x

????t=0

,x ∈ D(A),

where D(A) = {x ∈ X : C(t)x is twice continuous differentiable}.

We refer to the book of Goldstein [19] for the detailed study of the family of cosine

and sine operators.

Definition 2.1 A multi-valued operator G : X ⇒ Pcl(X) is called

(a) γ-Lipschitz iff there exists γ > 0 such that Hd(G(x),G(y)) ≤ γd(x,y), for each

x,y ∈ X;

(b) a contraction iff it is γ-Lipschitz with γ < 1.

Definition 2.2 The integral formulation y(t) of system (1) is given by

y(t) := φ(t);

y(t) := C(t)φ(0)+S(t)?x0−f(0,φ,x0)?+

+

0

if t ∈ J0,

?t

0

C(t −s)f?s,ys,y?(s)?ds

S(t −s)v(s)ds

?t

S(t −s)Bu(s)ds +

?t

0

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J Optim Theory Appl (2012) 154:672–684677

+

?

F,z,y:= {v ∈ L1(J,X) : v(t) ∈ F(t,yt,y?(t)), for a.e. t ∈ J} ?= Φ and

x0= φ ∈ Bhis called a mild solution of the inclusion (1) provided?t

0<tk<t

C(t −tk)Ik

?ytk,y?(tk)?+

?

0<tk<t

S(t −tk)Ik

?ytk,y?(tk)?;

if t ∈ J,

where v ∈ S1

0C(t − s) ×

f(s,ys,y?(s))ds is integrable.

Definition 2.3 System (1) is said to be controllable on J?= J/{t1,t2,...,tp} iff

for every φ ∈ Bh with φ(0) = D(A),x0∈ X, and y1∈ X, there exists a con-

trol u ∈ L2(J,U) such that the mild solution y(.) of system (1) satisfies y(m) =

y1∈ D(A),y?(m) = x1∈ X, conditions ?y|t=tk= Ik(ytk,y?(tk)), and ?y?|t=tk=

Ik(ytk,y?(tk)); k = 1,2,...,p; and y?(0) = x0∈ Bh.

Lemma 2.2 (refer [18]) Let {C(t) : t ∈ R} be a strongly continuous cosine family in

X with infinitesimal generator A. If h1: R → X is continuously differentiable, with

x0∈ D(A),y0∈ X, and

?t

then w(t) ∈ D(A) for t ∈ R,w is twice continuously differentiable, and w satisfies

w??(t) = Aw(t)+h1(t),

Conversely, if h1: R → X is continuous, w(t) : R → X is twice continuously differ-

entiable, w(t) ∈ D(A) for t ∈ R, and w satisfies system (1). Then

?t

Lemma 2.3 (Covitz and Nadler [14]) Let (X,d) be a complete metric space. If G :

X ⇒ Pcl(X) is a contraction, then fix G = φ (where fix G denotes set of fixed points

of the multi-valued operator G).

w(t) = C(t)x0+S(t)y0+

0

S(t −s)h(s)ds, t ∈ R,

t ∈ R,w(0) = x1,w?(0) = y0.

w(t) = C(t)x0+S(t)y0+

0

S(t −s)h(s)ds, t ∈ R.

3 Controllability Result

In this section, we prove the controllability of system (1). We adopt the following

hypotheses:

(H1) An operator A is an infinitesimal generator of a strongly continuous and

bounded cosine family {C(t) : t ∈ R} with M = sup{|C(t)| : t ∈ J}, and sine

family {S(t) : t ∈ R} with?

to t for each ψ ∈ Bhand y ∈ X, and F is u.s.c. with respect to second and third

variables, for each t ∈ J.

Indeed, by Carathèodory condition of F, F has a measurable selection (see

Theorem III in [20]).

M = sup{|S(t)| : t ∈ J}.

(H2) F : J × Bh× X ⇒ Pcp(X) : (.,ψ,y) ⇒ F(.,ψ,y) is measurable with respect

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678J Optim Theory Appl (2012) 154:672–684

(H3) The linear operator W : L2(J,U) → X, defined by

Wu :=

?b

0

S(t −s)Bu(s)ds,

has an invertible operator W−1which takes the values in L2(J,U)/kerW

(since the restriction of W to the domain L2(J,U)/kerW is invertible) and

there exists positive constant M1such that and ?W−1? ≤ M1.

(H4) The linear operator W : L2(J,U) → X, defined by

Wu :=

0

has an invertible operator W−1which takes values in L2(J,U)/kerW and

there exists positive constant M2such that ?W−1? ≤ M2.

(H5) There exists a positive constant M3such that ?B? ≤ M3.

(H6) Hd(F(t,ψ,u),F(t,¯ψ, ¯ u)) ≤ l(t)(?ψ −¯ψ? + |u − ¯ u|), for each t ∈ J, and

ψ,¯ψ ∈ Bh, and u, ¯ u ∈ X; where l ∈ L1(J,R+) and d(0,F(t,0,0)) ≤ l(t), for

a.e. t ∈ J.

(H7) (i) Ik: Bh× X → X are completely continuous and there exist constants

cj

k,j = 1,2, such that

??Ik(ψ,y)??X≤ c1

(ii) Ik: Bh× X → X are completely continuous and there exist constants

dj

k,j = 1,2, such that

??Ik(ψ,y)??X≤ d1

Theorem 3.1 Assume that the hypotheses (H1)–(H7) be satisfied. Then system (1) is

controllable for x0and y1on J, provided m?

Proof For k = 1,2,...,p, consider

Bb(J1,X) =?y : ]−∞,m[ → X; y(t) is continuous at t ?= tk,y?t−

k

and

B1

and y??t+

with the norm ?y?b1 = ?y?Bh+ supt∈Jm{?y(s)?b,?y?(s)?b1};0 ≤ s ≤ b,y ∈ Bband

y?∈ B1

Here, ykisarestrictionof y to Jk= (tk,tk+1];suchthat ?yk?Jk= sups∈Jk?yk(s)?.

Let us define the spaces

Z1:=?Bb(J1,X)∩C2?J?

where J?

?b

C(t −s)Bu(s)ds,

k

??ψ?Bh+|y|?+c2

k;

k = 1,2,...,p;for every (ψ,y) ∈ Bh×X.

k

??ψ?Bh+|y|?+d2

k;

k = 1,2,...,p;for every (ψ,y) ∈ Bh×X.

ML[1+(1/2)(M1?

M +M2M)m] < 1.

k

?= y(tk),

and y?t+

?exists?,

b(J1,X) =?y ∈ Bb(J1,X) : y?(t) is continuous at t ?= tk,y??t−

k

k

?= y?(tk),

?exists?,

b; where ?y?b= supt∈Jm|y(t)|.

1,X??,Z2:=?B1

b(J1,X)∩C2?J?

1,X??;

1= J1\{tk,k = 1,2,...,p}.

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J Optim Theory Appl (2012) 154:672–684679

Using the hypotheses (H3) and (H4), we define the control formally as

?

−

0

?m

?

+W−1?

−

0

?m

?

where v ∈ S1

Using this control, we shall now show that the operators G1: Z1⇒ P(Z1) and

G2: Z2⇒ P(Z2), defined by

?

+

0

?t

+

0<tk<t

?

?x??:=

+

0

?t

ux(t) =1

2

W−1

?m

?

y1−C(m)φ(0)−S(m)?x(0)−f(0,φ,x0)?

C(m−s)f?s,ys,y?(s)?ds

S(m−s)v(s)ds −

0<tk<t

?ytk,y?(tk)??

x1−AS(m)φ(0)−C(m)?x(0)−f(0,φ,x0)?

AS(m−s)f?s,ys,y?(s)?ds

C(m−s)v(s)ds −

0<tk<t

?ytk,y?(tk)???

−

0

?

C(m−tk)Ik

?ytk,y?(tk)?

−

0<tk<t

S(m−tk)Ik

?m

−

0

?

AS(m−tk)Ik

?ytk,y?(tk)?

−

0<tk<t

C(m−tk)Ik

(t),

F,z,y.

G1(x) :=

y ∈ Z1: y(t) = C(t)φ(0)+S(t)?x0−f(0,φ,x0)?

?t

+

0

?

?ytk,y?(tk)?;v ∈ S1

?

?t

+

0

C(t −s)f?s,ys,y?(s)?ds

S(t −s)Bux(s)ds +

?t

0

S(t −s)v(s)ds

C(t −tk)Ik

?ytk,y?(tk)?

+

0<tk<t

y?∈ Z2: y(t) = AS(t)φ(0)+C(t)?x0−f(0,φ,x0)?

AS(t −s)f?s,ys,y?(s)?ds

C(t −s)Bux(s)ds +

S(t −tk)Ik

F,z,y

?

,

G2

?t

0

C(t −s)v(s)ds

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680J Optim Theory Appl (2012) 154:672–684

+

?

?

0<tk<t

AS(t −tk)Ik

?ytk,y?(tk)?

?ytk,y?(tk)?;t ∈ J

+

0<tk<t

C(t −tk)Ik

?

,

has a fixed point. This fixed point is then a mild solution of system (1).

Obviously, y1∈ (G1x)(m) and y2∈ (G2x?)(m). Next, we shall show that G1and

G2satisfy the assumptions of Lemma (2.3). The proof will be given in two steps.

Step 1: We show that G1(x) ∈ Pcl(Z1),G2(x?) ∈ Pcl(Z2).

Indeed, let {yn}n≥0∈ G1(x) such that yn→ y∗in Z1. Then y∗∈ Z1, and there

exists vn∈ S1

yn(t) = C(t)φ(0)+S(t)?x0−f(0,φ,x0)?+

+

0

+

0<tk<t

+1

2

0

?m

−

0<tk<t

+W−1?

−

0

−

0<tk<t

?

Using the fact that F has compact values and (H6) holds, we may pass a subsequence

which is necessary to find that vnconverges to v in L1(J,X); hence v ∈ S1

Then, for each t ∈ J,

yn(t) → y∗(t) = C(t)φ(0)+S(t)?x0−f(0,φ,x0)?+

+

0

0<tk<t

F,z,ysuch that, for each t ∈ J,

?t

0

C(t −s)f?s,ys,y?(s)?ds

?t

?

?t

S(t −s)vn(s)ds

C(t −tk)Ik

?ytk,y?(tk)?+

W−1

?

0<tk<t

S(t −tk)Ik

?ytk,y?(tk)?

S(t −s)B

C(m−τ)f?τ,yτ,y?(τ)?dτ −

?

x1−AS(m)φ(0)−C(m)?x(0)−f(0,φ,x0)?

?m

?

?ytk,y?(tk)???

?

?

y1−C(m)y0−S(m)?x(0)−f(0,φ,x0)?

?m

?ytk,y?(tk)?−

−

00

S(m−τ)vn(τ)dτ

C(m−tk)Ik

?

0<tk<t

S(m−tk)Ik

?ytk,y?(tk)??

AS(m−τ)f?τ,yτ,y?(τ)?dτ −

AS(m−tk)Ik

?m

0

C(m−τ)vn(τ)dτ

?ytk,y?(tk)?

+

0<tk<t

C(m−tk)Ik

(s)ds.

F,z,y.

?t

0

C(t −s)f?s,ys,y?(s)?ds

C(t −tk)Ik

?t

S(t −s)v(s)ds +

?

?y(tk,y?(tk)?

Page 10

J Optim Theory Appl (2012) 154:672–684681

+

?

?t

0<tk<t

+1

2

−S(m)?x(0)−f(0,φ,x0)?

+

0

?

+W−1?

−

0

?m

?

S(t −tk)Ik

?ytk,y?(tk)?

?

0

S(t −s)BW−1

?

y1−C(m)y0

?m

C(m−τ)f?τ,yτ,y?(τ)?dτ −

C(m−tk)Ik

?m

?

0

S(m−τ)v(τ)dτ

−

0<tk<t

?ytk,y?(tk)?−

0<tk<t

S(m−tk)Ik

?ytk,y?(tk)??

x1−AS(m)φ(0)−C(m)?x(0)−f(0,φ,x0)?

AS(m−τ)f?τ,yτ,y?(τ)?dτ

C(m−τ)v(τ)dτ −

0<tk<t

?ytk,y?(tk)???

?m

−

0

?

AS(m−tk)Ik

?ytk,y?(tk)?

+

0<tk<t

C(m−tk)Ik

(s)ds.

So, y∗∈ G1(x). In particular, G1(x) ∈ Pcl(Z1). Similarly, we can show that

G2(x?) ∈ Pcl(Z2).

Step 2: We shall show G1(x) and G2(x?) are contractive multi-valued maps for

each y ∈ Z1and y?∈ Z2.

Let yt,yt,y?,y?∈ Z1and x ∈ G1(x). Then there exists v ∈ S1

y(t) = C(t)φ(0)+S(t)?x0−f(0,φ,x0)?+

+

0

+

0<tk<t

+1

2

0

?m

−

0<tk<t

+W−1?

−

0

F,z,ysuch that

?t

0

C(t −s)f?s,ys,y?(s)?ds

?t

?

?t

S(t −s)v(s)ds

C(t −tk)Ik

?ytk,y?(tk)?+

W−1

?

0<tk<t

S(t −tk)Ik

?ytk,y?(tk)?

S(t −s)B

C(m−τ)f?τ,yτ,y?(τ)?dτ −

?

x1−AS(m)φ(0)−C(m)?x(0)−f(0,φ,x0)?

?m

?

?

y1−C(m)y0−S(m)?x(0)−f(0,φ,x0)?

?m

?ytk,y?(tk)?−

−

00

S(m−τ)v(τ)dτ

C(m−tk)Ik

?

0<tk<t

S(m−tk)Ik

?ytk,y?(tk)??

AS(m−τ)f?τ,yτ,y?(τ)?dτ −

?m

0

C(m−τ)vn(τ)dτ

Page 11

682J Optim Theory Appl (2012) 154:672–684

−

?

?

0<tk<t

AS(m−tk)Ik

?ytk,y?(tk)?

?ytk,y?(tk)???

+

0<tk<t

C(m−tk)Ik

(s)ds.

From (H6) it follows that, for each t ∈ J,

Hd

?F?t,yt,y?(t)?−F?t,yt,y?(t)??≤ l(t)??yt−yt?+??y?(t)−y?(t)???.

??v(t)−w(t)??≤ l(t)??yt−yt?+??y?(t)−y?(t)???,

V(t) :=?w(t) ∈ X :??v(t)−w(t)??≤ l(t)??yt−yt?+??y?(t)−y?(t)????.

Proposition III.4 in [20]), there exists a function v(t) which is a measurable selection

for W. So, v(t) ∈ F(t,yt,y?(t)), and

??v(t)−v(t)??≤ l(t)??yt−yt?+??y?(t)−y?(t)???,

y(t) = C(t)φ(0)+S(t)?x0−f(0,φ,x0)?+

+

0

?

+1

2

0

?m

−

0<tk<t

+W−1?

−

0

?

Hence, there exists w(t) ∈ F(t,yt,y?(t)) such that

t ∈ J.

Consider V : J ⇒ P(X), given by

Since the multi-valued operator W(t) = V(t) ∩ F(t,yt,y?(t)) is a measurable (see

for each t ∈ J.

Furthermore, for each t ∈ J, we define

?t

0

C(t −s)f?s,ys,y?(s)?ds

?t

S(t −s)v(s)ds

+

0<tk<t

?t

C(t −tk)Ik

?ytk,y?(tk)?+

W−1

?

0<tk<t

S(t −tk)Ik

?ytk,y?(tk)?

S(t −s)B

C(m−τ)f?τ,yτ,y?(τ)?dτ −

?

x1−AS(m)φ(0)−C(m)?x(0)−f(0,φ,x0)?

?m

AS(m−tk)Ik

?

?

y1−C(m)y0−S(m)?x(0)−f(0,φ,x0)?

?m

?ytk,y?(tk)?−

−

00

S(m−τ)v(τ)dτ

C(m−tk)Ik

?

0<tk<t

S(m−tk)Ik

?ytk,y?(tk)??

AS(m−τ)f?τ,yτ,y?(τ)?dτ −

?ytk,y?(tk)?+

?m

?

0

C(m−τ)v(τ)dτ

−

0<tk<t

0<tk<t

C(m−tk)Ik

?ytk,y?(tk)???

(s)ds.

Page 12

J Optim Theory Appl (2012) 154:672–684 683

Then, for each t ∈ J, we get

??y(t)−y(t)??≤

?????

?t

+1

0

S(t −s)?v(s)−v(s)?ds

?????

2

0

Mm??yt−yt?+??y?(t)−y?(t)????m

2

0

+??y?(t)−y?(t)????m

×

≤ m?

≤ m?

Then ?y −y? ≤ m?

Hd

?????

2

?t

?????

?t

0

S(t −s)B

?

?

W−1

?m

0

S(m−τ)?v(τ)−v(τ)?dτ

C(m−τ)?v(τ)−v(τ)?dτ

l(s)ds

???yt−yt?

l(τ)dτ

2

?

?

(s)ds

?????

+1

?t

S(t −s)BW−1?m

0

(s)ds

?????

≤?

0

+1

??S(t −s)??M3M1Mm

0

?

ds +1

?t

0

??S(t −s)??M3M2?

l(τ)dτ ds

Mm

???yt−yt?+??y?(t)−y?(t)????m

×(M1m?

0l(s)ds.

ML[1+(1/2)(M1?

?G1(x),G2(x)?≤ m?

Similarly, for any y?and y?, we obtain

?G1

Thus G1and G2are contractive. As a consequence of Lemma 2.3, we deduce that

G1and G2have a fixed point. Hence, system (1) is controllable over J.

0

?

ML??yt−yt?+??y?(t)−y?(t)???+(1/2)?

ML?1+(1/2)(M1m?

MmM3

M +M2mM)L??ytyt?+??y?(t)−y?(t)???

M +M2mM)???yt−yt?+??y?(t)−y?(t)???,

M +M2M)m](?yt−yt?+|y?(t)−y?(t)|).

where L =?m

By interchanging the roles of y and y, we get an analogous relation

ML?1+(1/2)(M1?

M +M2M)m?

×??yt−yt?+??y?(t)−y?(t)???.

?x???≤ mML?1+(1/2)(M1?

Hd

?x??,G2

M +M2M)m?

×??yt−yt?+??y?(t)−y?(t)???.

?

4 Conclusion

In this paper, we discussed controllability results for the second order impulsive neu-

tral functional differential inclusions, with infinite delay through phase space defined

in Sect. 2. We have proved the result without compactness of family of cosine op-

erators. The result is obtained using a Hausdorff space (defined in Sect. 2) and a

contraction mapping for multi-valued maps due to Covitz and Nadler [14].

Page 13

684J Optim Theory Appl (2012) 154:672–684

Acknowledgements

and suggestions which are helpful to improve the manuscript.

Author expresses his gratitude to the anonymous referee for valuable comments

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