In this paper, we devoted study the existence and controllability for the nonlinear fuzzy neutral integrodifferential equations with control system in E_N. Moreover we study the fuzzy solution for the normal, convex, upper semicontinuous and compactly supported interval fuzzy number. The results are obtained by using the contraction principle theorem. An example to illustrate the theory.

1. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
DOI : 10.5121/ijfls.2013.3304 39
Existence and Controllability Result for the
Nonlinear First Order Fuzzy Neutral
Integrodifferential Equations with Nonlocal
Conditions
S.Narayanamoorthy1
and M.Nagarajan2
1,2
Department of Applied Mathematics, Bharathiar University, Coimbatore,
TamilNadu, India-46
1
snm_phd@yahoo.co.in
2
mnagarajanphd@gmail.com
ABSTRACT
In this paper, we devoted study the existence and controllability for the nonlinear fuzzy neutral
integrodifferential equations with control system in E_N. Moreover we study the fuzzy solution for the
normal, convex, upper semicontinuous and compactly supported interval fuzzy number. The results are
obtained by using the contraction principle theorem. An example to illustrate the theory.
KEYWORDS
Controllability, Fixed Point Theorem, Integrodifferential Equations, Nonlocal Condition.
1. INTRODUCTION
First introduced fuzzy set theory Zadeh[15] in 1965. The term "fuzzy differential equation" was
coined in 1978 by Kandel and Byatt[6] . This generalization was made by Puri and Ralescu [12]
and studied by Kaleva [5]. It soon appeared that the solution of fuzzy differential equation
interpreted by Hukuhara derivative has a drawback: it became fuzzier as time goes. Hence, the
fuzzy solution behaves quite differently from the crisp solution. To alleviate the situation,
Hullermeier[4] interpreted fuzzy differential equation as a family of differential inclusions. The
main short coming of using differential inclusions is that we do not have a derivative of a fuzzy
number valued function. There is another approach to solve fuzzy differential equations which is
known as Zadeh’s extension principle (Misukoshi, Chalco-Cano, Román-Flores, Bassanezi[9]),
the basic idea of the extension principle is: consider fuzzy differential equation as a deterministic
differential equation then solve the deterministic differential equation. After getting deterministic
solution, the fuzzy solution can be obtained by applying extension principle to deterministic
solution. But in Zadeh’s extension principle we do not have a derivative of a fuzzy number-
valued function either. In [2], Bede, Rudas, and Bencsik[4], strongly generalized derivative
concept was introduced. This concept allows us to solve the mentioned shortcomings and in
Khastan et al. [7] authors studied higher order fuzzy differential equations with strongly
generalized derivative concept. Recently, Gasilovet al. [3] proposed a new method to solve a
fuzzy initial value problem for the fuzzy linear system of differential equations based on
properties of linear transformations.

2. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
40
Fuzzy optimal control for the nonlinear fuzzy differential system with nonlocal initial condition
in NE was presented by Y.C.Kwun et al. [8] . Recently, the above concept has been extended to
the integrodifferential equations by J.H.Park et al. [12]. We will combine these earlier work and
extended the study to the following nonlinear first order fuzzy neutral integrodifferential
equations ( 0)( ≡tu ) with nonlocal conditions:
])()()()[(=)))(,()((
0
dssxstGtxtAtxthtx
dt
d t
−+− ∫ ][0,=),())(,( bJttutxtf ∈++ (2)
0=)((0) xxgx + (3)
where NEJtA →:)( is fuzzy coefficient, NE is the fuzzy set of all upper semicontinuous,
convex, normal fuzzy numbers with bounded −α level intervals, NN EEJf →×: ,
NN EEJh →×: , NN EEg →: are all nonlinear functions, )(tG is nn× continuous matrix
such that
dt
xtdG )(
is continuous for NEx∈ and Jt ∈ with ktG ≤)( , 0>k , NEJu →: is
control function.
The rest of this paper is organized as follows. In section 2, some preliminaries are presented. In
section 3, existence solution of fuzzy neutral integrodifferential equations. In section 4, we study
on nonlocal cotroll of solutions for the neutral system. In section 5, an example.
2.PRELIMINARIES
In section, we shall introduce some basic definitions, notations, lemmas and result which are used
throughout this paper. A fuzzy subset of n
R is defined in terms of a membership function which
assigns to each point n
x R∈ a grade of membership in the fuzzy set. Such a membership
function is denoted by
[0,1].: →n
u R
Throughout this paper, we assume that u maps n
R onto [0,1], 0
][u is a bounded subset of n
R ,
u is upper semicontinuous, and u is fuzzy convex. We denote by n
E the space of all fuzzy
subsets u of n
R which are normal, fuzzy convex, and upper semicontinuous fuzzy sets with
bounded supports. In particular, 1
E denotes the space of all fuzzy subsets u of .R
A fuzzy number a in real line R is a fuzzy set characterized by a membership function aχ
[0,1].: →Raχ
A fuzzy number a is expressed as
x
a a
x
χ
∫∈R
=

3. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
41
with the understanding that [0,1],)( ∈xaχ represents the grade of membership of x in a and
∫ denotes the union of .
x
aχ
Definition 2.1 A fuzzy number R∈a is said to be convex if, for any real numbers zyx ,, in R
with zyx ≤≤ ,
)}(),({min)( zxy aaa χχχ ≥
Definition 2.2 If the height of a fuzzy set equals one, then the fuzzy set is called normal. Thus, a
fuzzy number R∈a is called normal, if the followings holds:
1.=)(max xa
x
χ
Result 2.1 [10] Let NE be the set of all upper semicontinuous convex normal fuzzy numbers
with bounded α - level intervals. This means that if NEa∈ , then −α level set
1},,0)(:{=][ ≤≤≥∈ ααα
xaxa R
is a closed bounded interval, which we denote by
],[=][ ααα
rq aaa
and there exists a R∈0t such that 1.=)( 0ta
Result 2.2 [10] Two fuzzy numbers a and b are called equal ba = , if ),(=)( xx ba χχ for all
R∈x . It follows that
(0,1].,][=][= ∈⇔ ααα
allforbaba
Result 2.3 [10] A fuzzy number a may be decomposed into its level sets through the resolution
identity
,][=
1
0
α
α aa ∫
where α
α ][a is the product of a scalar α with the set α
][a and ∫ is the union of α
][a with α
ranging from 0 to 1.
.
Definition 2.3 A fuzzy number R∈a is said to be positive if 21 <<0 aa holds for the support
],[= 21 aaaΓ of a, that is, aΓ is in the positive real line. Similarly, a is called negative if
0<21 aa ≤ and zero if 21 0 aa ≤≤ .

4. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
42
Lemma: 2.1 [14] If NEba ∈, , then for (0,1]∈α ,
],,[=][ ααααα
rrqq bababa +++
),,=,(}],{max},{min[=][ rqjibabaab iiii
ααααα
].,[=][ ααααα
rrqq bababa −−−
Lemma: 2.2 [14] Let ],[ αα
rq aa , 1<0 ≤α , be a given family of nonempty intervals. If
,<0],[],[ βαααββ
≤⊂ foraaaa rqrq
],,[=]lim,lim[ αααα
rq
k
r
k
k
q
k
aaaa
∞→∞→
whenever )( kα is nondecreasing sequence converting to (0,1]∈α , then the family ],[ αα
rq aa ,
1<0 ≤α , are the α -level sets of a fuzzy number NEa∈ .
Let x be a point in n
R and A be a nonempty subsets of n
R . We define the Hausdroff separation
of B from A by
}.:{inf=),( AaaxAxd ∈−
Now let A and B be nonempty subsets of n
R . We define the Hausdroff separation of B from
A by
}.:),({sup=),(*
BbAbdABdH ∈
In general,
).,(),( **
ABdBAd HH ≠
We define the Hausdroff distance between nonempty subsets of A and B of n
R by
)}.,(),,({max=),( **
ABdBAdBAd HHH
This is now symmetric in A and B . Consequently,
1. 0=),(0),( BAdwithBAd HH ≥ if and only if ;= BA
2. );,(=),( ABdBAd HH

5. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
43
3. );,(),(),( BCdCAdBAd HHH +≤
for any nonempty subsets of A , B and C of n
R . The Hausdroff distance is a metric, the
Hausdroff metric.
The supremum metric ∞d on n
E is defined by
,,(0,1]},:)][,]([{sup=),( n
H Evuallforvudvud ∈∈∞ ααα
and is obviously metric on n
E .
The supremum metric 1H on ),( n
EJC is defined by
)}.:(,),:)(),(({sup=),(1
n
EJyxallforJttytxdyxH C∈∈∞
We assume the following conditions to prove the existence of solution of the equation (1.2).
(H1). The nonlinear function NN EEJg →×: is a continuous function and satisfies
the inequality
)(.)][,(.)]([)(.))]([,(.))](([ αααα
δ yxdygxgd HgH ≤
(H2). The inhomomogeneous term NN EEJf →×: is continuous function and
satisfies a global Libschitz
)(.)][,(.)]([)))](,([,))](,(([ αααα
δ yxdsysfsxsfd HfH ≤
(H3.) The nonlinear function NN EEJh →×: is continuous function and satisfies
the global lipschitz condition
)(.)][,(.)]([)(.))],([,(.))],(([ αααα
δ yxdyshxshd HhH ≤
(H4). S(t) is the fuzzy number satisfies for NEy∈ , ),(),( NN EJCEJCyS ∩∈′ the
equation
))()()()((=)(
0
ydssSstGytStAytS
dt
d t
−+ ∫
JtdssGsAstSytStA
t
∈−+ ∫ ,)()()()()(=
0
such that
],[=)]([ ααα
rq SStS
and )(tSi
α
, rqi ,= are continuous. That is, a postive constant sδ such that si tS δα
<)(
(H5) )( 21 b∆+∆ <1, where hghhgs δδδδδδ +++∆ )(=
11 and )(=2 hAfs M δδδ +∆

6. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
44
3.EXISTENCE AND UNIQUENESS OF FUZZY SOLUTION
The solution of the equations (2)-(3) ( 0≡u ) is of the form
dssxshstSsAtxthxgxhxgxtStx
t
))(,()()())(,())]((0,)()[(=)(
0
00 −++−−− ∫γ
dssxsfstS
t
))(,()(
0
−+ ∫ (4)
where γ is a continuous function from );( NEJC to itself.
Theorem: 3.1 Suppose that hypotheses (H1)-(H5) are satisfied. Then the equation (4) has
unique fixed point in ),( NEJC .
Proof. For );(, NEJCyx ∈ ,
))]([,)](([ αα
γγ tytxd H
))(,())]((0,)()[(([= 00 txthxgxhxgxtSdH +−−−
α
]))(,()())(,()()(
00
dssxsfstSdssxshstSsA
tt
−+−+ ∫∫
dssyshstSsAtythygxhygxtS
t
))(,()()())(,())]((0,)()[([
0
00 −++−−− ∫
)]))(,()(
0
α
dssysfstS
t
−+ ∫
)))]]((0,)()([,))]((0,)()[(([ 0000
αα
ygxhygxtSxgxhxgxtSdH −−−−−−≤
)))](,([,))](,(([ αα
tythtxthdH+
)]))(,()()([,]))(,()()(([
00
αα
dssyshstSsAdssxshstSsAd
tt
H −−+ ∫∫
)]))(,()([,]))(,()(([
00
αα
dssysfstSdssxsfstSd
tt
H −−+ ∫∫
)()][,()]([)))]((0,)([,))]((0,)(([ 00
αααα
δδ yxdygxhygxgxhxgd HhHs +−+−+≤
dtsysfsxsfddtsyshsxshdM H
t
sH
t
As )))](,([,))](,(([)))](,([,))](,(([
00
αααα
δδ ∫∫ ++
))]([,)](([))(
1
αα
δδδδδδ tytxdHhghhgs +++≤
dtyxdM H
t
hAfs )()][,()]([)(
0
αα
δδδ ∫++
dttytxdtytxd H
t
H ))]([,)](([))]([,)](([=
0
21
αααα
∫∆+∆
where, hghhgs δδδδδδ +++∆ )(=
11 and )(=2 hAfs M δδδ +∆ .
Therefore,
))]([,)](([sup=))(),((
[0,1]
αα
α
γγγγ tytxdtytxd H
∈
∞

7. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
45
))]([,)](([sup
[0,1]
1
αα
α
tytxdH
∈
∆≤
dttytxdH
t
))]([,)](([sup
0[0,1]
2
αα
α
∫∈
∆+
dttytxdtytxd
t
))(),(())(),((
0
21 ∞∞ ∫∆+∆≤
Hence,
))(),((sup=))(),((
][0,=
1 tytxdtytxH
bJt
γγγγ ∞
∈
dtyxdtytxd
bt
t
Jt
())(),(sup))(),((sup
][0,0
21 ∞
∈
∞
∈
∫∆+∆≤
))(),(())(),(( 1211 tytxbHtytxH ∆+∆≤
))(),(()(= 121 tytxHb∆+∆
Then by hypotheses, γ is a contraction mapping. By using Banach fixed point theorem,
equations (2)-(3) have a unique fixed point, ),( NEJCx∈
4. CONTROLLABILITY OF FUZZY SOLUTION
In this secetion, we show for the controller term in equation(2)-(3), and the solution of the form
dssxshsAstStxthxgxhxgxtStx
t
))(,()()())(,())]((0,)()[(=)(
0
00 −++−−− ∫
dssustSdssxsfstS
tt
)()())(,()(
00
−+−+ ∫∫ (5)
Definition 4.1 [13] The equation (4.1) is controllable if there exists )(tu such that the fuzzy
solution )(tx of (5) satisfies )(=)( 1
xgxbx − , that is αα
)]([=)]([ 1
xgxbx − , where 1
x is a
target set.
The linear controll system is nonlocal controolable. That is,
dssusbSxgxbSbx
b
)()()]()[(=)(
0
0 −+− ∫
)(= 1
xgx −
αα
])()()]()[([=)]([
0
0 dssusbSxgxbSbx
b
−+− ∫
])()()(,)()()]()[([=
0
0
0
0 dssusbSxbSdssusbSxgxbS rr
b
rrqq
b
qqq
ααααααααα
−+−+− ∫∫
)]()(),()[(= 0
11
xgxxgx rqq
αααα
−−
α
)]([= 1
xgx − .
We Define new fuzzy mapping NERP →)(:ζ by

8. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
46



 Γ⊂−∫
otherwise
vdssvstS
v u
t
0,
,,)()(
=)( 0
α
ζ
Then there exists ),=( rqii
α
ζ such that
],[,)()(=)( 1
0
uuvdssvstSv qqqq
t
qq
ααα
ζ ∈−∫
],[,)()(=)( 1
0
ααα
ζ rrrr
t
rr uuvdssvstSv ∈−∫
We assume that iζ 's are bijective mappings. Hence the α -set of )(su are
)](),([=)]([ sususu rq
ααα
))]((0,))[(()()(()[(= 00
11
xgxhgxbSxgx qqqqqqq −−−−−− αααααα
α
ζ
dssxthsAstStxth qqq
t
q ))(,()()())(,(
0
αααα
−−− ∫
,))(,()(
0
dssxtfstS qq
t
αα
−− ∫
))]((0,)())[(()()(()( 00
11
xgxhxgxbSxgx rrrrrrr −−−−−− αααααα
α
ζ
dssxthsAstStxth rrr
t
r ))(,()()())(,(
0
αααα
−−− ∫
dssxtfstS rr
t
))(,()(
0
αα
−− ∫
Then substituting this expression into equation (5) yields α - level set of )(bx
α
)]([ bx
))(,())]((0,)())[(([= 00 sxshxgxhxgxbS qqqqq
ααααα
+−−−
dssxsfsbSdssxthsAsbS qq
b
qqq
b
))(,()())(,()()(
00
ααααα
−+−+ ∫∫
αααα
α
α
ζ qqqqqq
b
xbSxgxsbS ))[(()()(())(( 0
11
0
−−−+ −
∫
))(,()](0,)( 0 sxshxhxg qqq
ααα
−−−
,))(,()())(,()()(
00
dssxsfsbSdssxshsAsbS qq
b
qqq
b
ααααα
−−−− ∫∫
))]((0,)())[(( 00 xgxhxgxbS rrrr −−− αααα
dssxsfsbSdssxshsAsbStxth rr
b
rrr
b
r ))(,()())(,()()())(,(
00
αααααα
−+−++ ∫∫
))]((0,)())[(()()(())(( 00
11
0
xgxhxgxbSxgxsbS rrrrrrrr
b
−−−−−−+ −
∫
αααααα
α
α
ζ

9. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
47
dssxshsAsbSsxsh rqr
b
r ))(,()()())(,(
0
αααα
−−− ∫
]))(,()(
0
dssxsfsbS rr
b
αα
−− ∫
dssxshsAsbSsxshxgxhxgxbS qqq
b
qqqq ))(,()()())(,())]((0,)())[(([=
0
00
ααααααα
−++−−− ∫
αααα
α
ααα
ζζ qqqqqqqq
b
xbSxgxdssxsfsbS ))[(()()(()())(,()( 0
11
0
−−+−+ −
∫
dssxshsAsbSsushxgxhxg qqq
b
qqq ))(,()()())(,())]((0,)(
0
0
αααααα
−−−−−− ∫
))]((0,)())[((,))(,()( 00
0
xgxhxgxbSdssxsfsbS rqrrqq
b
−−−−− ∫
αααααα
dssxsfsbSdssxshsAsbSsush rr
b
rrr
b
r ))(,()())(,()()())(,(
00
αααααα
−+−++ ∫∫
dssxshsAsbSxhxgxbSxgx rrr
b
rrrrrrrr ))(,()()()](0,)())[(()()(()(
0
00
11 ααααααααα
α
α
ζζ −−−−−−+ ∫
−
dssxsfsbS rr
b
))(,()(
0
αα
−− ∫
)]()(),()[(= 11
xgxxgx rrqq
αααα
−−
.)]([= 1 α
xgx −
We now set
dssxthsAstStuthxgxhxgxtStx
t
))(,()()())(,())]((0,)()[(=)(
0
00 −++−−−Ω ∫
dssxsfstS
t
))(,()(
0
−+ ∫
))]((0,)()[()(()( 00
1
1
0
xgxhxgxbSxgxstS
t
−−−−−−+
−
∫ ζ
dssxsfsbSdssxshsAsbStxth
bb
))(,()())(,()()())(,(
00
−−−−− ∫∫
where the fuzzy mapping
1−
ζ satisfied above statement.
Now notice that ),(=)( 1
xgxTx −Ω which means that the control )(tu steers the equation(5)
from the origin to 1
x in the time b provided we can obtain a fixed point of the nonlinear
operator Ω .
Assume that the hypotheses
(H6)The system (4.1) is linear 0≡f is nonlocal controllable.
(H7) 1.))((( ≤+++ hkfsh b δδδδδ

10. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
48
Theorem: 4.1 Suppose that the hypotheses (H1)-(H7) are satisfied. Then the equation (5) is a
nonlocal controllable.
Proof. We can easily check that Ω is continuous from ):]([0, NEbC to itself. For
),:]([0,, NEbCyx ∈
))]([,)](([ αα
tytxdH ΩΩ
))(,())]((0,)()[(([= 00 txthxgxhxgxtSdH +−−−
dssxsfstSdssxshsAstS
tt
))(,()())(,()()(
00
−+−+ ∫∫
)()[()(()( 0
1
1
0
xgxbSxgxsbS
b
−−−−+
−
∫ ζ
dssxshsAsbStxthxgxh
b
))(,()()())(,()]((0,
0
0 −−−−− ∫
,))(,()(
0
dssxsfsbS
b
−− ∫
dssyshsAstStythygxhygxtS
t
))(,()()())(,())]((0,)()[([
0
00 −++−−− ∫
dssysfstS
t
))(,()(
0
−+ ∫
))(,())]((0,)()[(()( 00
1
1
0
tythygxhygxsbSxsbS
b
−−−−−−−+
−
∫ ζ
)])))(,()())(,()()(
00
α
dsdssysfsbSdssyshsAsbS
bb
−−−− ∫∫
dssxshsAstStxthxgxhxgtSd
t
H ))(,()()())(,())]((0,)()[(([
0
0 −++−+≤ ∫
,))](()())(,()(
<<0
0
α−
−+−+ ∑∫ kkk
ktt
t
txIttSdssxsfstS
dssyshsAstStythygxhygtS
t
))(,()()())(,())((0,)()[([
0
0 −++−+ ∫
dssysfstS
t
))(,()(
0
−+ ∫
))(,())((0,)()[()(()(([ 0
1
1
0
sxshxgxhxgbSxgxstSd
t
H −−+−−−+
−
∫ ζ
,))(,()())(,()()(
00
dssxsfsbSdssxshsAstS
bb
−−−− ∫∫
))(,())]((0,)()[()(()([ 0
1
1
0
syshygxhygbSygxstS
t
−−+−−−
−
∫ ζ
dssysfsbSdssyshsAsbS
bb
))(,()())(,()()(
00
−−−− ∫∫
)][,)](([))((
1
αα
δδδδδδ ysxdHhghhgs +++≤

11. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
49
))]([,)](([)((
0
αα
δδδ sysxdM H
t
fhAs ∫++
))]([,)](([)(()][,)](([
0
αααα
δδδδ sysxdysxd H
b
fhsHh ∫+++
Let hghhgs δδδδδδκ +++ )(=
11 , and )(=2 fhAs M δδδκ + then we have
)))]([,)](([))]([,)](([()][,)](([2
00
21
αααααα
κκ sysxdsysxdysxd H
b
H
t
H ∫∫ ++≤
Therefore
))]([,)](([sup=))(),((
[0,1]
αα
α
tytxdtytxd H ΩΩΩΩ
∈
∞
)][,)](([2 1
αα
κ ysxd∞≤
)))]([,)](([))]([,)](([(
00
2
αααα
κ sysxdsysxd
bt
∞∞ ∫∫ ++
Hence
))]([,)](([sup=))(),((
][0,
1
αα
tytxdtytxH H
bt
ΩΩΩΩ
∈
),()2(2 121 yxHbκκ +≤
),())(2(= 121 yxHbκκ +
By hypotheses )( 6H , we take sufficiently small b , Ω is a contraction mapping. By Banach
fixed point theorem, equation (5) has a unique fixed point ).:]([0, NEbCx∈
5.EXAMPLE
Consider the fuzzy solution of the nonlinear fuzzy neutral integrodifferential equation of the
form:
,),()(3])()(2[=)))(2)(( 2)(
0
2
Jttuttxdssxetxttxtx
dt
d st
t
∈++−− −−
∫ (6)
,0=)((0)
1=
Nkk
n
k
Etxcx ∈+ ∑ (7)
where 1
x is target set, and the α - level set of fuzzy number 0,2 and 3 are
[0,1]],1,1[=[0] ∈−− αααα
for
[0,1]],1,3[=[2] ∈−+ αααα
for
[0,1].],2,4[=[3] ∈−+ αααα
for
Let .)(2=))(,(,)(3=))(,(,=)( 22)(
ttututhttututfestG st−−
− Then α - level set of
)(=)( 1= kk
n
k
txcxg ∑ is

12. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
50
αα
)]([=)]([
1=
kk
n
k
txcxg ∑
)](),([=
1=1=
krk
n
k
kqk
n
k
txctxc αα
∑∑
))]([,)](([=))]([,)](([
1=1=
αααα
kk
n
k
kk
n
k
HH tyctxcdygxgd ∑∑
)])(),([)],(),(([=
1=1=1=1=
krk
n
k
kqk
n
k
krk
n
k
kqk
n
k
H tyctyctxctxcd αααα
∑∑∑∑
})()(,)()({max
1=
|||||||| krkrkqkq
k
k
n
k
tytxtytxc αααα
−−≤ ∑
)()][,()]([ αα
δ yxdHg≤
where, |||| k
n
kg c∑ 1=
=δ ,where )](),([=)]([ txtxtx rq
ααα
and
[0,1]],2,4[=[3] ∈−+ αααα
for and the α - level set of ))(,( txtf is
αα
])([3=))](,([ 2
ttxtxtf
αα
])([[3]= 2
txt
],))()((4),)(2)([(= 22
txtxt rq
αα
αα −+
)))](,([,))](,(([ αα
tytftxtfdH
])))()((4),)(2)([(],))()((4),)(2)([((= 2222
tytyttxtxtd rqrqH
αααα
αααα −+−+
}|))(())(()(4,|))(())((2){(max= 2222
tytxtytxt rrqq
αααα
αα −−−+ ||
}|))(())((||))(())((|,))(())((||))(())(({|max)(4 tytxtytxtytxtytxb rrrrqqqq
αααααααα
α +−+−−≤ |
}))(())((|,))(())(({|max|))(())((|4 || tytxtytxtytxb rrqqrr
αααααα
−−+≤
))]([,)](([= αα
δ tytxdHf
where, |))(())((|4= tytxb rrf
αα
δ + )](),([=)]([ txtxtx rq
ααα
and
[0,1]],2,4[=[3] ∈−+ αααα
for and the α - level set of ))(,( txth is
αα
])([2=))](,([ 2
ttxtxth
αα
])([[2]= 2
txt
]))((,))(][(1,3[= 22
txtxt rq
αα
αα −+
].))()((3,))(1)([(= 22
txtxt rq
αα
αα −+
we introduced the −α set of Equation (5.2)(5.1) − Thus,
)))](,([,))](,(([ αα
tythtxthdH

13. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
51
])))()((3,))(1)([(],))()((3,))(1)([((= 2222
tytyttxtxtd rqrqH
αααα
αααα −+−+
}))(())(()(3,))(())((1)({max= 2222
|||| tytxttytxt rrqq
αααα
αα −−−+
}))(())((|))(())((|,))(())((|))(())(({|max)(3 |||| tytxtytxtytxtytxb rrrrqqqq
αααααααα
α +−+−−≤
}))(())((|,))(())(({|max))(())((|3 ||| tytxtytxtytxb rrqqrr
αααααα
−−+≤
))]([,)](([= αα
δ tytxdHg
where, |))(())((|3= tytxb rrg
αα
δ +
Next we prove the nonlocal controllability parts, let us take target set, 2=1
x
α
)]([ su
],[= αα
rq uu
2
1=
1
)1)(()()((1[= αα
ααζ qkqk
n
k
q xttxc +−−+ ∑
−
dssxtsbSdssxtsbS qq
b
qq
b
)()2)(()()()1)(()( 2
0
2
0
αααα
αα +−−+−− ∫∫
)()((3),)()1)(()(
1=
1
2
0
krk
n
k
rqq
b
txcdssxtsbS ααα
αζα ∑∫ −−+−−
−
dssxtsbSdssxtsbSxt rr
b
rr
b
r )()2)(()()()1)(()()1)(( 2
0
2
0
2 ααααα
ααα +−−+−−+− ∫∫
)])()1)(()( 2
0
dssxtsbS rr
b
αα
α +−− ∫
Then substituting this expression into the integral system with respect to (6)-(7) yields −α level
set of )(bx .
α
)]([ bx
)()1)(()](1))[(([= 2
1=
txttxcbS qkk
n
k
q
αα
αα ++−− ∑
dstxttstS qq
t
q
)()(1)()( 2αα
α +−+ ∫
))()1)(()())2)(()( 2
0
2
0
dstxtstSdsxtstS qq
t
qq
t
αααα
αα +−++−+ ∫∫
dstxtsbStxtsbS qq
b
qqq
b
)()1)(()()()1)((1)(())(( 2
0
21
0
ααα
α
α
αααζ +−−+−+−+ ∫∫
−
))))(1)(()()()2)(()( 2
0
2
0
dsdssxtstSdssxtstS qq
t
qq
t
αααα
αα +−−+−− ∫∫
dstxttsbStxttbS rr
b
rr )()()(3)()()(1)()])(1([ 2
0
2 αααα
ααα −−++− ∫

14. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013
52
))())((3)()))((4)( 2
0
2
0
dstxtstSdsxtstS rq
t
rr
t
αααα
αα −−+−−+ ∫∫
dstxtsbStxtsbS rr
b
rrr
b
)())((3)()()1)(()((3))(( 2
0
21
0
ααα
α
α
αααζ −−−+−−−+ ∫∫
−
)])))()((3)()())((4)( 2
0
2
0
dsdssxtsbSdssxtsbS rr
b
rr
b
αααα
αα −−−−−− ∫∫
)]()(3),(1)[(=
1=1=
krk
n
k
kqk
n
k
txctxc αα
αα ∑∑ −−−+
α
)]([2=
1=
kk
n
k
txc∑−
)]([= 1
xgx −
Then all condition stated in theorem4.1 are satisfied, so the system (6)-(7) is nonlocal
controllable on ].[0,b
6. CONCLUSION
In this paper, by using the concept of fuzzy number in NE , we study the existence and
controllability for the nonlinear fuzzy neutral integrodifferential with nonlocal controll system in
NE and find the sufficient conditions of controllability for the controll system (2)-(3).
ACKNOWLEDGEMENTS
The author’s are thankful to the anonymous referees for the improvement of this paper.
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AUTHORS
Dr.S.Narayanamoorthy received his doctoral degree from the Department of
Mathematics, Loyola College (Autonomous), Chennai-34, Affiliated to the University
of Madras and he obtained his graduate degree, post graduate degree and degree of
Master of Philosophy from the same institution. Currently he is working as Assistant
Professor of Applied Mathematics at Bharathiar University. His current research
interests are in the fields of Applications of Fuzzy Mathematics, Fuzzy Optimization,
Fuzzy Differential Equations. Email: snm_phd@yahoo.co.in
M. Nagarajan received the M.Sc. and M.Phil . degrees in Mathematics at Bharathiar
University, Coimbatore, India in 2007 and 2009, respectively, and is currently
pursuing the Ph.D, degree in Applied Mathematics at Bharathiar University,
Coimbatore, India. His current research interests are in the fields of Fuzzy Differential
equations. Email: mnagarajanphd@gmail.com