Existence and Controllability Result for the Nonlinear First Order Fuzzy Neutral Integrodifferential

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

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 mnagarajanphd@gmail.com

2

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 numbervalued 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.

DOI : 10.5121/ijfls.2013.3304

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

Fuzzy optimal control for the nonlinear fuzzy differential system with nonlocal initial condition in E N 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 ( u (t ) ≡ 0 ) with nonlocal conditions: t d ( x (t ) − h(t , x (t ))) = A(t )[ x (t ) + ∫ G (t − s ) x ( s ) ds ] + f (t , x (t )) + u (t ), t ∈ J = [0, b] 0 dt

x(0) + g ( x) = x0

(2) (3)

where A(t ) : J → E N is fuzzy coefficient, E N is the fuzzy set of all upper semicontinuous, convex, normal fuzzy numbers with bounded α − level intervals,

f : J × EN → EN ,

h : J × E N → E N , g : E N → E N are all nonlinear functions, G (t ) is n × n continuous matrix dG (t ) x such that is continuous for x ∈ E N and t ∈ J with G (t ) ≤ k , k > 0 , u : J → E N is dt 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 R n is defined in terms of a membership function which assigns to each point x ∈ R n a grade of membership in the fuzzy set. Such a membership function is denoted by

u : R n → [0,1].

Throughout this paper, we assume that u maps R n onto [0,1] , [u ]0 is a bounded subset of R n ,

u is upper semicontinuous, and u is fuzzy convex. We denote by E n the space of all fuzzy subsets u of R n which are normal, fuzzy convex, and upper semicontinuous fuzzy sets with bounded supports. In particular, E 1 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 χ a

χ a : R → [0,1].

A fuzzy number a is expressed as

a=∫

x∈R

χa x 40

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

with the understanding that χ a ( x) ∈ [0,1], represents the grade of membership of x in a and

∫ denotes the union of

χa x

.

Definition 2.1 A fuzzy number a ∈ R is said to be convex if, for any real numbers x, y, z in R with x ≤ y ≤ z ,

χ a ( y ) ≥ min{χ a ( x), χ a ( z )} Definition 2.2 If the height of a fuzzy set equals one, then the fuzzy set is called normal. Thus, a fuzzy number a ∈ R is called normal, if the followings holds:

max χ a ( x) = 1. x

Result 2.1 [10] Let E N be the set of all upper semicontinuous convex normal fuzzy numbers with bounded α - level intervals. This means that if a ∈ E N , then α − level set

[a ]α = {x ∈ R : a ( x) ≥ α ,0 ≤ α ≤ 1}, is a closed bounded interval, which we denote by

[a]α = [aqα , arα ] and there exists a t0 ∈ R such that a (t0 ) = 1.

Result 2.2 [10] Two fuzzy numbers a and b are called equal a = b , if χ a ( x) = χ b ( x), for all x ∈ R . It follows that a = b ⇔ [a ]α = [b]α , for all α ∈ (0,1].

Result 2.3 [10] A fuzzy number a may be decomposed into its level sets through the resolution identity 1

a = ∫ α [a ]α , 0

α

where α [a ] is the product of a scalar α with the set [a ]α and

∫ is the union of [a]

α

with α

ranging from 0 to 1. .

Definition 2.3 A fuzzy number a ∈ R is said to be positive if 0 < a1 < a2 holds for the support

Γa = [a1 , a2 ] of a, that is, Γa is in the positive real line. Similarly, a is called negative if

a1 ≤ a2 < 0 and zero if a1 ≤ 0 ≤ a2 .

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

Lemma: 2.1 [14] If a, b ∈ E N , then for α ∈ (0,1] ,

[a + b]α = [aqα + bqα , arα + brα ], [ab]α = [min{aiα biα }, max{aiα biα }], (i, j = q, r ),

[a − b]α = [aqα − bqα , arα − brα ].

Lemma: 2.2 [14] Let [aqα , arα ] , 0 < α ≤ 1 , be a given family of nonempty intervals. If

[aqβ , arβ ] ⊂ [aqα , arα ] for 0 < α ≤ β , α

α

[ lim aq k , lim ar k ] = [aqα , arα ], k →∞

k →∞

whenever (α k ) is nondecreasing sequence converting to α ∈ (0,1] , then the family [aqα , arα ] ,

0 < α ≤ 1 , are the α -level sets of a fuzzy number a ∈ E N .

Let x be a point in R n and A be a nonempty subsets of R n . We define the Hausdroff separation of B from A by

d ( x, A) = inf { x − a : a ∈ A}.

Now let A and B be nonempty subsets of R n . We define the Hausdroff separation of B from A by

d H* ( B, A) = sup{d (b, A) : b ∈ B}.

In general,

d H* ( A, B ) ≠ d H* ( B, A).

We define the Hausdroff distance between nonempty subsets of A and B of R n by

d H ( A, B ) = max{d H* ( A, B ), d H* ( B, A)}.

This is now symmetric in A and B . Consequently, 1. d H ( A, B ) ≥ 0 with d H ( A, B ) = 0 if and only if A = B; 2. d H ( A, B ) = d H ( B, A);

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

3. d H ( A, B ) ≤ d H ( A, C ) + d H (C , B ); for any nonempty subsets of A , B and C of R n . The Hausdroff distance is a metric, the Hausdroff metric.

The supremum metric d ∞ on E n is defined by

d ∞ (u , v) = sup{d H ([u ]α , [v]α ) : α ∈ (0,1]}, for all u , v ∈ E n , and is obviously metric on E n .

The supremum metric H 1 on C ( J , E n ) is defined by

H 1 ( x, y ) = sup{d ∞ ( x(t ), y (t ) : t ∈ J ), for all x, y ∈ C( J : E n )}.

We assume the following conditions to prove the existence of solution of the equation (1.2).

(H1). The nonlinear function g : J × E N → E N is a continuous function and satisfies the inequality

d H ([ g ( x(.))]α , [ g ( y (.))]α ) ≤ δ g d H ([ x(.)]α , [ y (.)]α ) (H2). The inhomomogeneous term f : J × E N → E N is continuous function and satisfies a global Libschitz

d H ([ f ( s, x( s ))]α , [ f ( s, y ( s ))]α ) ≤ δ f d H ([ x(.)]α , [ y (.)]α ) (H3.) The nonlinear function h : J × E N → E N is continuous function and satisfies the global lipschitz condition

d H ([h( s, x(.))]α , [h( s, y (.))]α ) ≤ δ h d H ([ x(.)]α , [ y (.)]α ) (H4). S(t) is the fuzzy number satisfies for y ∈ E N , S ′y ∈ C ( J , E N ) ∩ C ( J , E N ) the equation t d S (t ) y = A(t )( S (t ) y + ∫ G (t − s ) S ( s ) yds ) 0 dt t

= A(t ) S (t ) y + ∫ S (t − s ) A( s )G ( s )ds, t ∈ J 0

such that

[ S (t )]α = [ S qα , S rα ] and S iα (t ) , i = q, r are continuous. That is, a postive constant δ s such that S iα (t ) < δ s (H5) (∆1 + ∆ 2b) <1, where ∆1 = δ s (δ g + δ h + δ hδ g ) + δ h and ∆ 2 = δ s (δ f + M Aδ h ) 1

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

3.EXISTENCE AND UNIQUENESS OF FUZZY SOLUTION The solution of the equations (2)-(3) ( u ≡ 0 ) is of the form t

γx(t ) = S (t )[ x0 − g ( x) − h(0, x0 − g ( x))] + h(t , x(t )) + ∫ A( s) S (t − s)h( s, x( s ))ds 0

t

+ ∫ S (t − s ) f ( s, x( s ))ds

(4)

0

where γ is a continuous function from C ( J ; E N ) to itself.

Theorem: 3.1 Suppose that hypotheses (H1)-(H5) are satisfied. Then the equation (4) has unique fixed point in C ( J , E N ) .

Proof. For x, y ∈ C ( J ; E N ) , d H ([γx(t )]α , [γy (t )]α ) = d H ([ S (t )[ x0 − g ( x) − h(0, x0 − g ( x))] + h(t , x(t )) t

t

0

0

+ ∫ A( s ) S (t − s )h( s, x( s ))ds + ∫ S (t − s ) f ( s, x( s ))ds ]α t

[ S (t )[ x0 − g ( y ) − h(0, x0 − g ( y ))] + h(t , y (t )) + ∫ A( s ) S (t − s )h( s, y ( s ))ds 0

t

+ ∫ S (t − s ) f ( s, y ( s ))ds ]α ) 0

≤ d H ([ S (t )[ x 0 − g ( x) − h(0, x 0 − g ( x))]α , [ S (t ) x 0 − g ( y ) − h(0, x0 − g ( y ))]]α ) + d H ([h(t , x(t ))]α , [h(t , y (t ))]α ) t

t

+ d H ([ ∫ A( s ) S (t − s )h( s, x( s ))ds ]α , [ ∫ A( s ) S (t − s )h( s, y ( s ))ds ]α ) 0

0

t

t

+ d H ([ ∫ S (t − s ) f ( s, x( s ))ds ] , [ ∫ S (t − s ) f ( s, y ( s ))ds ]α ) α

0

0

α

≤ δ s d H ([ g ( x) + h(0, x0 − g ( x))] , [ g ( y ) + h(0, x0 − g ( y ))]α ) + δ h d H ([ x()]α , [ y ()]α ) t

t

0

0

+ δ s M A ∫ d H ([h( s, x( s ))]α , [h( s, y ( s ))]α )dt + δ s ∫ d H ([ f ( s, x( s ))]α , [ f ( s, y ( s ))]α )dt ≤ δ s (δ g + δ h + δ hδ g ) + δ h )d H ([ x(t )]α , [ y (t )]α ) 1

t

+ δ s (δ f + M Aδ h ) ∫ d H ([ x()]α , [ y ()]α )dt 0

t

= ∆1d H ([ x(t )] , [ y (t )]α ) + ∆ 2 ∫ d H ([ x(t )]α , [ y (t )]α )dt α

0

where, ∆1 = δ s (δ g + δ h + δ hδ g ) + δ h and ∆ 2 = δ s (δ f + M Aδ h ) .

1

Therefore,

d ∞ (γx(t ), γy (t )) = sup d H ([γx(t )]α , [γy (t )]α ) α ∈[0,1]

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

≤ ∆1 sup d H ([ x(t )]α , [ y (t )]α ) α ∈[0,1]

t

+ ∆ 2 sup ∫ d H ([ x(t )]α , [ y (t )]α )dt α ∈[0,1] 0

t

≤ ∆1d ∞ ( x(t ), y (t )) + ∆ 2 ∫ d ∞ ( x(t ), y (t ))dt 0

Hence,

H1 (γx(t ), γy (t )) = sup d ∞ (γx(t ), γy (t )) t∈J =[0,b ] t

≤ ∆1 sup d ∞ ( x(t ), y (t )) + ∆ 2 ∫ sup d ∞ ( x(), y ())dt 0 t∈[0,b ]

t∈J

≤ ∆1 H1 ( x(t ), y (t )) + ∆ 2bH1 ( x(t ), y (t )) = (∆1 + ∆ 2b) H1 ( x(t ), y (t )) Then by hypotheses, γ is a contraction mapping. By using Banach fixed point theorem, equations (2)-(3) have a unique fixed point, x ∈ C ( J , E N )

4. CONTROLLABILITY OF FUZZY SOLUTION In this secetion, we show for the controller term in equation(2)-(3), and the solution of the form t

x(t ) = S (t )[ x0 − g ( x) − h(0, x0 − g ( x))] + h(t , x(t )) + ∫ S (t − s ) A( s )h( s, x( s ))ds 0

t

t

0

0

+ ∫ S (t − s ) f ( s, x( s ))ds + ∫ S (t − s )u ( s )ds

(5)

Definition 4.1 [13] The equation (4.1) is controllable if there exists u (t ) such that the fuzzy solution x (t ) of (5) satisfies x (b) = x1 − g ( x ) , that is [ x (b)]α = [ x1 − g ( x )]α , where x1 is a target set.

The linear controll system is nonlocal controolable. That is, b

x(b) = S (b)[ x0 − g ( x)] + ∫ S (b − s )u ( s )ds 0

= x1 − g ( x) b

[ x(b)]α = [ S (b)[ x0 − g ( x)] + ∫ S (b − s )u ( s )ds ]α 0

b

b

= [ S qα (b)[ x0αq − g qα ( x)] + ∫ S qα (b − s )u qα ( s )ds, S rα (b) x0αr + ∫ S rα (b − s )u rα ( s )ds ] 0

1 α q

α

1 α r

0

α

= [( x ) − g q ( x), ( x ) − g 0 ( x)] = [ x1 − g ( x)]α .

We Define new fuzzy mapping ζ : P ( R ) → E N by 45

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

 tS (t − s )v( s )ds, v ⊂ Γ u , ζ (v) = ∫0 0, otherwise α Then there exists ζ i (i = q, r ) such that α

t

ζ qα (vq ) = ∫ S qα (t − s)vq ( s )ds, vq ∈ [uqα , u1 ] 0 t

ζ rα (vr ) = ∫ S rα (t − s)vr ( s)ds, vr ∈ [u1 , urα ] 0

We assume that ζ i 's are bijective mappings. Hence the α -set of u (s ) are

[u ( s )]α = [uqα ( s ), urα ( s )] α

= [(ζ q ) −1 (( x1 )αq − g qα ( x) − S qα (b)[( x0 )αq − g qα − hqα (0, x0 − g ( x))] t

− hqα (t , x(t )) − ∫ S qα (t − s ) Aqα ( s )hqα (t , x( s ))ds 0

t

− ∫ S qα (t − s ) f qα (t , x( s ))ds, 0 α

(ζ r ) −1 (( x1 )αr − g rα ( x) − S rα (b)[( x0 )αr − g rα ( x) − hrα (0, x0 − g ( x))] t

− hrα (t , x(t )) − ∫ S rα (t − s ) Arα ( s )hrα (t , x( s ))ds 0

t

− ∫ S rα (t − s ) f rα (t , x( s ))ds 0

Then substituting this expression into equation (5) yields α - level set of x (b)

[ x(b)]α

= [ S qα (b)[( x0 )αq − g qα ( x) − hqα (0, x0 − g ( x))] + hqα ( s, x( s )) b

b

+ ∫ S qα (b − s ) Aqα ( s )hqα (t , x( s ))ds + ∫ S qα (b − s ) f qα ( s, x( s ))ds 0

0

α b

+ ∫ S qα (b − s )(ζ q ) −1 (( x1 )αq − g qα ( x) − S qα (b)[( x0 )αq 0

− g qα ( x) − hqα (0, x0 )] − hqα ( s, x( s )) b

b

− ∫ S qα (b − s ) Aqα ( s )hqα ( s, x( s ))ds − ∫ S qα (b − s ) f qα ( s, x( s ))ds, 0

α

0

α

α

α

S r (b)[( x0 ) r − g r ( x) − hr (0, x0 − g ( x))] b

b

0

0

+ hrα (t , x(t )) + ∫ S rα (b − s ) Arα ( s )hrα ( s, x( s ))ds + ∫ S rα (b − s ) f rα ( s, x( s ))ds α b

+ ∫ S rα (b − s )(ζ r ) −1 (( x1 )αr − g rα ( x) − S rα (b)[( x0 )αr − g rα ( x) − hrα (0, x0 − g ( x))] 0

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 b

− hrα ( s, x( s )) − ∫ S rα (b − s ) Aqα ( s )hrα ( s, x( s ))ds 0

b

− ∫ S r (b − s ) f r ( s, x( s ))ds ] α

α

0

b

= [ S qα (b)[( x0 )αq − g qα ( x) − hqα (0, x0 − g ( x))] + h( s, x( s )) + ∫ S qα (b − s ) Aqα ( s )hqα ( s, x( s ))ds 0

α b

+ ∫ S qα (b − s ) f qα ( s, x( s ))ds + ζ qα (ζ q ) −1 (( x1 )αq − g qα ( x) − S qα (b)[( x0 )αq 0

b

− g qα ( x) − hqα (0, x0 − g ( x))] − hqα ( s, u ( s )) − ∫ S qα (b − s ) Aqα ( s )hqα ( s, x( s ))ds 0

b

− ∫ S qα (b − s ) f qα ( s, x( s ))ds, S rα (b)[( x0 )αr − g qα ( x) − hrα (0, x0 − g ( x))] 0

b

b

0

0

+ hr ( s, u ( s )) + ∫ S rα (b − s ) Arα ( s )hrα ( s, x( s ))ds + ∫ S rα (b − s ) f rα ( s, x( s ))ds α

α b

+ ζ rα (ζ r ) −1 (( x1 )αr − g rα ( x) − S rα (b)[( x0 )αr − g rα ( x) − hrα (0, x0 )] − ∫ S rα (b − s ) Arα ( s )hrα ( s, x( s ))ds 0

b

− ∫ S r (b − s ) f r ( s, x( s ))ds α

α

0

= [( x1 )αq − g qα ( x), ( x1 )αr − g rα ( x)] = [ x1 − g ( x)]α .

We now set t

Ωx(t ) = S (t )[ x0 − g ( x) − h(0, x0 − g ( x))] + h(t , u (t )) + ∫ S (t − s ) A( s )h(t , x( s ))ds 0

t

+ ∫ S (t − s ) f ( s, x( s ))ds 0

−1 t

+ ∫ S (t − s ) ζ ( x1 − g ( x) − S (b)[ x0 − g ( x) − h(0, x0 − g ( x))] 0

b

b

− h(t , x(t )) − ∫ S (b − s ) A( s )h( s, x( s ))ds − ∫ S (b − s ) f ( s, x( s ))ds 0

0

−1

where the fuzzy mapping ζ

satisfied above statement.

Now notice that Ωx (T ) = x1 − g ( x ), which means that the control u (t ) steers the equation(5) from the origin to x1 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 f ≡ 0 is nonlocal controllable.

(H7) (δ h + (δ s (δ f + δ k b) + δ h ) ≤ 1.

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

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 C ([0, b] : E N ) to itself. For

x, y ∈ C ([0, b] : E N ), d H ([Ωx(t )]α , [Ωy (t )]α ) = d H ([ S (t )[ x0 − g ( x) − h(0, x0 − g ( x))] + h(t , x(t )) t

t

+ ∫ S (t − s ) A( s )h( s, x( s ))ds + ∫ S (t − s ) f ( s, x( s ))ds 0

0

−1 b

+ ∫ S (b − s ) ζ ( x1 − g ( x) − S (b)[ x0 − g ( x) 0

b

− h(0, x0 − g ( x)] − h(t , x(t )) − ∫ S (b − s ) A( s )h( s, x( s ))ds 0

b

− ∫ S (b − s ) f ( s, x( s ))ds, 0

t

[ S (t )[ x0 − g ( y ) − h(0, x0 − g ( y ))] + h(t , y (t )) + ∫ S (t − s ) A( s )h( s, y ( s ))ds 0

t

+ ∫ S (t − s ) f ( s, y ( s ))ds 0

−1 b

+ ∫ S (b − s ) ζ ( x1 − S (b − s )[ x0 − g ( y ) − h(0, x0 − g ( y ))] − h(t , y (t )) 0 b

b

0

0

− ∫ S (b − s ) A( s )h( s, y ( s ))ds − ∫ S (b − s ) f ( s, y ( s ))ds )ds ]α ) t

≤ d H ([ S (t )[ g ( x) + h(0, x0 − g ( x))] + h(t , x(t )) + ∫ S (t − s ) A( s )h( s, x( s ))ds 0

t

+ ∫ S (t − s ) f ( s, x( s ))ds + 0

∑ S (t

k

− t ) I k ( x(t k− ))]α ,

0< t <t k t

[ S (t )[ g ( y ) + h(0, x0 − g ( y )) + h(t , y (t )) + ∫ S (t − s ) A( s )h( s, y ( s ))ds 0

t

+ ∫ S (t − s ) f ( s, y ( s ))ds 0

−1 t

+ d H ([ ∫ S (t − s ) ζ ( x1 − g ( x) − S (b)[ g ( x) + h(0, x0 − g ( x)) − h( s, x( s )) 0

b

b

− ∫ S (t − s ) A( s )h( s, x( s ))ds − ∫ S (b − s ) f ( s, x( s ))ds, 0

0

−1 t

[ ∫ S (t − s ) ζ ( x1 − g ( y ) − S (b)[ g ( y ) + h(0, x0 − g ( y ))] − h( s, y ( s )) 0

b

b

0

0

− ∫ S (b − s ) A( s )h( s, y ( s ))ds − ∫ S (b − s ) f ( s, y ( s ))ds ≤ (δ s (δ g + δ h + δ hδ g ) + δ h )d H ([ x( s )]α , [ y ]α ) 1

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 t

+ (δ s ( M Aδ h + δ f ) ∫ d H ([ x( s )]α , [ y ( s )]α ) 0

b

+ δ h d H ([ x( s )] , [ y ]α ) + (δ s (δ h + δ f ) ∫ d H ([ x( s )]α , [ y ( s )]α ) α

0

Let κ1 = δ s (δ g + δ h + δ hδ g ) + δ h , and κ 2 = δ s ( M Aδ h + δ f ) then we have 1

t

b

0

0

≤ 2κ1d H ([ x( s )]α , [ y ]α ) + κ 2 ( ∫ d H ([ x( s )]α , [ y ( s )]α ) + ∫ d H ([ x( s )]α , [ y ( s )]α )) Therefore

d ∞ (Ωx(t ), Ωy (t )) = sup d H ([Ωx(t )]α , [Ωy (t )]α ) α ∈[0,1]

α

≤ 2κ1d ∞ ([ x( s )] , [ y ]α ) t

b

0

0

+ κ 2 ( ∫ d ∞ ([ x( s )]α , [ y ( s )]α ) + ∫ d ∞ ([ x( s )]α , [ y ( s )]α )) Hence

H 1 (Ωx(t ), Ωy (t )) = sup d H ([Ωx(t )]α , [Ωy (t )]α ) t∈[0,b ]

≤ (2κ1 + 2κ 2b) H1 ( x, y ) = (2(κ1 + κ 2b)) H1 ( x, y ) By hypotheses ( H 6 ) , we take sufficiently small b , Ω is a contraction mapping. By Banach fixed point theorem, equation (5) has a unique fixed point x ∈ C ([0, b] : E N ).

5.EXAMPLE Consider the fuzzy solution of the nonlinear fuzzy neutral integrodifferential equation of the form: t d ( x(t ) − 2tx(t ) 2 )) = 2[ x(t ) − ∫ e −( t − s ) x( s )ds ] + 3tx(t ) 2 + u (t ), t ∈ J , 0 dt

(6)

n

x(0) + ∑ck x(t k ) = 0 ∈ E N ,

(7)

k =1

where x1 is target set, and the α - level set of fuzzy number 0,2 and 3 are

[0]α = [α − 1,1 − α ], for α ∈ [0,1] [2]α = [α + 1,3 − α ], for α ∈ [0,1] [3]α = [α + 2,4 − α ], for α ∈ [0,1].

Let

G (t − s ) = e − (t − s ) , f (t , u (t )) = 3tu (t ) 2 , h(t , u (t )) = 2tu (t ) 2 . Then α

- level set of

g ( x) = ∑k =1ck x(t k ) is n

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 n

[ g ( x)]α = [∑ck x(t k )]α k =1 n

n

k =1

k =1

= [∑ck xqα (t k ), ∑ck xrα (t k )] n

n

d H ([ g ( x)]α , [ g ( y )]α ) = d H ([∑ck x(t k )]α , [∑ck y (t k )]α ) k =1

k =1

n

n

n

n

k =1

k =1

k =1

k =1

= d H ([∑ck xqα (t k ), ∑ck xrα (t k )], [∑ck yqα (t k ), ∑ck yrα (t k )]) n

≤|| ∑c k || max {| x qα (t k ) − y qα (t k ) |, | x rα (t k ) − y rα (t k ) |} k =1

k

≤ δ g d H ([ x()]α , [ y ()]α ) where,

δ g =|| ∑k =1c k || ,where n

[ x(t )]α = [ xqα (t ), xrα (t )]

and

[3]α = [α + 2,4 − α ], for α ∈ [0,1] and the α - level set of f (t , x(t )) is [ f (t , x(t ))]α = [3tx(t ) 2 ]α = t[3]α [ x(t ) 2 ]α

= t[(α + 2)( xqα (t ) 2 ), (4 − α )( xrα (t )) 2 ], d H ([ f (t , x(t ))]α , [ f (t , y (t ))]α )

= d H (t[(α + 2)( xqα (t ) 2 ), (4 − α )( xrα (t )) 2 ], t[(α + 2)( yqα (t ) 2 ), (4 − α )( yrα (t )) 2 ]) = t max{(α + 2) | ( x qα (t ) 2 ) − ( y qα (t ) 2 ) |, (4 − α ) | ( x rα (t ) 2 ) − ( y rα (t ) 2 ) |}

≤ b(4 − α ) max{| ( xqα (t )) − ( y qα (t )) | | ( xqα (t )) + ( yqα (t )) |, | ( xrα (t )) − ( yrα (t )) || ( xrα (t )) + ( yrα (t )) |} ≤ 4b | ( x rα (t )) + ( y rα (t )) | max{| ( x qα (t )) − ( y qα (t )) |, | ( x rα (t )) − ( y rα (t )) |} = δ f d H ([ x(t )]α , [ y (t )]α ) where,

δ f = 4b | ( x rα (t )) + ( y rα (t )) |

[ x(t )]α = [ xqα (t ), xrα (t )]

and

[3]α = [α + 2,4 − α ], for α ∈ [0,1] and the α - level set of h(t , x(t )) is [h(t , x(t ))]α = [2tx(t ) 2 ]α = t[2]α [ x(t ) 2 ]α

= t[α + 1,3 − α ][( xqα (t )) 2 , ( xrα (t )) 2 ] = t[(α + 1)( xqα (t )) 2 , (3 − α )( xrα (t )) 2 ]. we introduced the α − set of Equation (5.1) − (5.2) Thus, d H ([h(t , x(t ))]α , [h(t , y (t ))]α ) 50

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

= d H (t[(α + 1)( xqα (t )) 2 , (3 − α )( xrα (t )) 2 ], t[(α + 1)( yqα (t )) 2 , (3 − α )( yrα (t )) 2 ]) = max{t (α + 1) | ( x qα (t )) 2 − ( y qα (t )) 2 |, t (3 − α ) | ( x rα (t )) 2 − ( y rα (t )) 2 |} ≤ b(3 − α ) max{| ( x qα (t )) − ( y qα (t )) || ( x qα (t )) + ( y qα (t )) |, | ( x rα (t )) − ( y rα (t )) || ( x rα (t )) + ( y rα (t )) |} ≤ 3b | ( x rα (t )) + ( y rα (t )) | max{| ( x qα (t )) − ( y qα (t )) |, | ( x rα (t )) − ( y rα (t )) |} = δ g d H ([ x(t )]α , [ y (t )]α ) where, δ g = 3b | ( x rα (t )) + ( y rα (t )) | Next we prove the nonlocal controllability parts, let us take target set, x1 = 2

[u ( s )]α

= [uqα , urα ] −1

n

= [ζ q ((1 + α ) − ∑ck xqα (t k ) − t (α + 1)( xqα ) 2 k =1 b

b

0

0 −1

− ∫ S qα (b − s )t (α + 1)( xqα ) 2 ( s )ds − ∫ S qα (b − s )t (α + 2)( xqα ) 2 ( s )ds n

b

− ∫ S qα (b − s )t (α + 1)( xqα ) 2 ( s )ds ), ζ r ((3 − α ) − ∑ck xrα (t k ) 0

k =1

b

b

0

0

− t (α + 1)( xrα ) 2 − ∫ S rα (b − s )t (α + 1)( xrα ) 2 ( s )ds − ∫ S rα (b − s )t (α + 2)( xrα ) 2 ( s )ds b

− ∫ S rα (b − s )t (α + 1)( xrα ) 2 ( s )ds )] 0

Then substituting this expression into the integral system with respect to (6)-(7) yields α − level set of x(b) .

[ x(b)]α n

= [ S qα (b)[(α − 1) − ∑ck x(t k )] + t (α + 1)( xqα ) 2 (t ) k =1 t

+ ∫ S qα (t − s )t (α + 1)t ( xqα ) 2 (t )ds q

t

t

+ ∫ S qα (t − s )t (α + 2)( xqα ) 2 )ds + ∫ S qα (t − s )t (α + 1)( xqα ) 2 (t )ds ) 0

0

α b

b

+ ∫ S qα (b − s )(ζ q ) −1 ((α + 1) − t (α + 1)( xqα ) 2 (t ) − ∫ S qα (b − s )t (α + 1)( xqα ) 2 (t )ds 0 t

0

t

− ∫ S qα (t − s )t (α + 2)( xqα ) 2 ( s )ds − ∫ S qα (t − s )t (α + 1)( xqα ( s )) 2 ds )ds ) 0

0 b

[ S r (b)(1 − α )]t (α + 1)t ( xr ) (t ) + ∫ S rα (b − s )t (3 − α )t ( xrα ) 2 (t )ds α

α 2

0

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 t

t

+ ∫ S rα (t − s )t (4 − α )( xrα ) 2 )ds + ∫ S qα (t − s )t (3 − α )( xrα ) 2 (t )ds ) 0

0

α b

b

+ ∫ S rα (b − s )(ζ r ) −1 ((3 − α ) − t (α + 1)( xrα ) 2 (t ) − ∫ S rα (b − s )t (3 − α )( xrα ) 2 (t )ds 0 b

0

b

− ∫ S r (b − s )t (4 − α )( xr ) ( s )ds − ∫ S r (b − s )t (3 − α )( xrα ( s )) 2 ds )ds )] α

α 2

0

α

0

n

n

k =1

k =1

= [(α + 1) − ∑ck xqα (t k ), (3 − α ) − ∑ck xrα (t k )] n

= [2 − ∑ck x(t k )]α k =1 1

= [ x − g ( x)] 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 E N , we study the existence and controllability for the nonlinear fuzzy neutral integrodifferential with nonlocal controll system in E N 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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[8]

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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

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