International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
EXISTENCE RESULT FOR THE NONLINEAR FIRST ORDER IMPULSIVE FUZZY INTEGRODIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITION S.Narayanamoorthy1and M.Nagarajan2 1,2
Department of Applied Mathematics, Bharathiar University, Coimbatore-641 046, INDIA Email:1snm phd@yahoo.co.in and 2mnagarajanphd@gmail.com
Abstract In this paper, the existence and uniqueness of a fuzzy solution for the impulsive nonlinear fuzzy integrodifferential equation with nonlocal condition is established via the Banach fixed-point theorem approach and using the fuzzy number whose values are normal, convex, upper semicontinuous, and compactly supported interval.
Keywords Fuzzy set, fuzzy number, impulsive fuzzy integrodifferential system,nonlocal conditions, fuzzy solution, fixed point theorem.
1 . INTRODUCTION In various fields of engineering and physics, many problems that are related to linear viscoelasticity, nonlinear elasticity have mathematical models and are described by the problems of differential or integral equations or integrodifferential equations. Integrodifferential equations are encountered in many areas of science and technology. It is well-known that the notion of aftereffect introduced in physics is very important. To model processes with aftereffect or delay it is not sufficient to employ ordinary or partial differential equations. An approach to resolve this problem is to use integrodifferential equations. Especially, one always describes a model which possesses hereditary properties by integrodifferential equations in practice. The word "fuzzy" means "vagueness". Fuzziness occurs when the boundary of a piece of information is not clearcut. Fuzzy set have been introduced by Zadeh [21] as an extension of the classical notion of set. Classical set theory allows the membership of the elements in the set in binary terms, a bivalent condition- an element either belongs or does not belong to the set. Fuzzy set theory permits the gradual assessment of the membership of elements in a set, described with the aid of a membership function valued in the real unit interval [0,1]. Fuzzy sets have been able provide solutions to many real world problems. Fuzzy set theory is an extension of classical set theory where elements have degrees of membership.
DOI : 10.5121/ijfls.2013.3108
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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
Generally, several systems are mostly related to uncertainty and inaccuracy. The problem of inaccuracy is considered in general an exact science and that of uncertainty is considered as vague or fuzzy and accidental. The definition given here generalizes that of Aumann [1] for setvalued mappings. A differential and integral calculus for fuzzy-set-valued, shortly fuzzy-valued, mappings was developed in recent papers of Dubois and Prade [5, 6, 7] and Puri and Ralescu [16]. For fuzzy concepts, recently Diamand and Kloeden [4] established the theory of metric space of fuzzy sets. In particular, Kaleva [9] researched the fuzzy differential equations, Cauchy problem for continuous fuzzy differential equations was studied by Nieto [14], and Song et al. [18] obtained the global solutions. Park and Han [15] studied the existence and uniqueness theorem for a solution of fuzzy Volterra integral equations by using the method of successive approximation. Seikkala [17] proved the existence and uniqueness of the fuzzy solution for the following systems:
u ′(t ) = f (t , u (t )), u (0) = u0 ,
t ∈ J = [0, b],
where f is a continuous mapping from R + × R into R and x0 is a fuzzy number. Recently, the above concept has been extended to the integrodifferential equations by Balasubramaniam and Muralisankar [2]. Ding and Kandel [8] analyzed a way to combine differential equations with fuzzy sets to form a fuzzy logic systems called a fuzzy dynamical system, which can be regarded to form a fuzzy neutral functional differential equations. During the past decades, the impulsive differential equations have attracted many authors since it is much richer than the corresponding theory of differential equations[2, 10, 20].Among the previous research, little is concerned with integrodifferential equations with impulsive conditions and nonlocal conditions. Here, motivated by[11, 12, 19], we will combine these earlier work and extend the study to the following impulsive nonlinear fuzzy integrodifferential equations with nonlocal conditions: t du (t ) = a(t )u (t ) + ∫ k (t , s, u ( s )) ds + f (t , u (t )), 0 dt u (0) + h(u ) = u0 ,
t ∈ [0, b],
∆u (t k ) = I k (u (t k− )) where a : J → E N is a fuzzy coefficient, E N is the set of all upper semicontinuous convex normal fuzzy numbers with bounded α - level intervals, f : J × E N → E N , h : E N → E N and
k : J × J × EN → EN
are nonlinear
continuous functions,
∆u (t k ) = u (t k+ ) − u (t k− )
and
I k ∈ C( E N , E N ) are continuous functions.
2. Preliminaries 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 80
International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
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
χa
a=∫
x∈R
x
∫
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 The height of a fuzzy set is the largest membership value attained by any point. Definition 2.3 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[13] Let E N be the set of all upper semicontinuous convex normal fuzzy numbers with bounded α - level intervals (see [13]). 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 t 0 ∈ R such that a (t0 ) = 1. Result 2.2[13] 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]. 81
International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
Result 2.3[13] A fuzzy number a may be decomposed into its level sets through the resolution identity 1
a = ∫ α [a ]α , 0
where α [a ]α 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 2.4 The support of a fuzzy set A in the universal set U is a crisp set that contains all the elements of U that have nonzero membership values in A , that is,
supp( A) = {x ∈ U : χ a ( x) > 0}, where supp( A) denotes the support of fuzzy set A. Hences the support Γa of a fuzzy number a is defined, as a special case of level set, by the following:
Γa = {x : χ a ( x) > 0}. Definition 2.5 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 . Lemma: 2.1 [17] 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 [17]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 82
International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
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); 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 H1 on C ( J , E n ) is defined by
H1 ( x, y ) = sup{d ∞ ( x(t ), y (t ) : t ∈ J ), for all x, y ∈ C( J : E n )}.
3 .Existence and Uniqueness of Fuzzy Solution In this section, we consider the existence and uniqueness of the fuzzy solution for impulsive nonlinear fuzzy integrodifferential equations with nonlocal conditions: t du (t ) = a (t )u (t ) + ∫ k (t , s, u ( s))ds + f (t , u (t )), 0 dt
u (0) + h(u ) = u0 ,
t ∈ [0, b],
(3.1) (3.2) 83
International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
∆u (t k ) = I k (u (t k− ))
(3.3)
where a : J → E N is a fuzzy coefficient, E N is the set of all upper semicontinuous convex normal fuzzy numbers with bounded α - level intervals, f : J × E N → E N , h : E N → E N and
k : J × J × EN → EN
are nonlinear
continuous functions,
∆u (t k ) = u (t k+ ) − u (t k− )
and
I k ∈ C( E N , E N ) are continuous functions. and satisfy a global Lipschitz condition, that is, there exist a finite constant l f , lk , lh and l I such that
d H ([ f ( s, u1 ( s ))]α , [ f ( s, u2 ( s ))]α ) ≤ l f d H ([u1 ( s ))]α , [u2 ( s ))]α ), d H ([ k (t , s, u1 ( s ))]α , [k (t , s, u 2 ( s ))]α ) ≤ lk d H ([u1 ( s ))]α , [u 2 ( s))]α ), d H ([ h(u1 ( s))]α , [ h(u 2 ( s ))]α ) ≤ lh d H ([u1 ( s))]α , [u 2 ( s ))]α ), d H ([ I k (u1 ( sk− ))]α , [ I k (u2 ( sk− ))]α ) ≤ li d H ([u1 ( s ))]α , [u2 ( s))]α ), k
∑l
= lI
i
i =1
for all u1 (t ), u 2 (t ) ∈ E N . Let I be an interval. A mapping u : I → E N of a fuzzy process u , then
[u′(t )]α = [(uqα )′, (urα )′],
0 < α ≤ 1.
The fuzzy integral b
∫ u(t )dt ,
a, b ∈ I ,
a
b
is defined by
∫ [u(t )]
α
a
b
b
a
a
= [ ∫ uqα , ∫ u rα ] provided Lebesgue integrals on the right exist.
Theorem 3.1 Let b > 0 , f , g and k satisfy a global Lipschitz condition, for every u0 ∈ E N , then the nonlinear impulsive fuzzy integrodifferential equation with nonlocal condition (3.1)-(3.3) has a unique solution u ∈ C( J , E N ). Proof. For each u (t ) ∈ E N , t ∈ J . t
s
0
0
(F0u (t )) = S (t )[u0 − h(u )] + ∫ S (t − s )( ∫ k ( s, r , u (r ))dr )ds t
+ ∫ S (t − s ) f ( s, u ( s )) ds + 0
∑ S (t − t
k
) I k (u (t k− )),
0<tk <t
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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
where S (t ) is a fuzzy number and
[ S (t )]α = [ S qα (t ), S rα (t )] t
t
0
0
= [exp{∫ aqα ( s )}, exp{∫ arα ( s )}], and Siα (t )(i = q, r ) is continuous. That is, there exist a constant l s > 0 such that Siα (t ) ≤ l s , for
t ∈ J . Thus F0u : J → E N u1 , u 2 ∈ C( J , E N ), we have
all
is
F0 : C( J , E N ) → C( J , E N ) .
continuous,
For
d H ([(F0 (u1 )(t ))]α , [(F0 (u 2 )(t ))]α ) t
s
t
0
0
0
= d H ([ S (t )(u0 − h(u1 )) + ∫ S (t − s)( ∫ k ( s, r , u1 (r ))dr )ds + ∫ S (t − s) f ( s, u1 ( s))ds +
∑ S (t − t
k
t
s
0
0
) I k (u1 (t k− ))]α , [ S (t )(u0 − h(u2 )) + ∫ S (t − s )( ∫ k ( s, r , u 2 ( r )) dr ) ds
0<tk <t
t
∑ S (t − t
+ ∫ S (t − s ) f ( s, u2 ( s )) ds + 0
k
) I k (u 2 (t k− ))]α )
0<tk <t
t
s
0
0
= d H ([ S (t )(u0 − h(u1 ))]α + [ ∫ S (t − s)( ∫ k ( s, r , u1 (r ))dr )ds ]α t
+ [ ∫ S (t − s ) f ( s, u1 ( s)) ds ]α + [ 0
∑ S (t − t
k
) I k (u1 (t k− ))]α ), [ S (t )(u0 − h(u2 ))]α
0 < tk <t
t
s
t
0
0
0
∑ S (t − t
k
+ [ ∫ S (t − s )( ∫ k ( s, r , u 2 (r ))dr )ds ]α + [ ∫ S (t − s ) f ( s, u 2 ( s ))ds ]α +[
) I k (u 2 (t k− ))]α )
0< tk <t
t
s
0
0
≤ d H ([ S (t )(u 0 − h(u1 ))]α , [ S (t )(u 0 − h(u 2 ))]α ) + d H ([ ∫ S (t − s)( ∫ k ( s, r , u1 (r ))dr )ds ]α , t
s
t
0
0
0
[ ∫ S (t − s)( ∫ k ( s, r , u 2 (r ))dr )ds ]α ) + d H ([ ∫ S (t − s ) f ( s, u1 ( s ))ds ]α , t
[ ∫ S (t − s ) f ( s, u 2 ( s ))ds ]α ) + d H ([ 0
[
∑ S (t − t
∑ S (t − t
k
) I k (u1 (t k− ))]α ,
0<tk <t
k
) I k (u 2 (t k− ))]α ),
0 < tk <t
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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
= d H ([ S (t )u0 ]α − [ S (t )h(u1 )]α , [ S (t )u0 ]α − [ S (t )h(u1 )]α ) t
s
0
0
+ d H ([ ∫ S (t − s )( ∫ k ( s, r , u1 (r ))dr )ds ]α , t
s
t
0
0
0
[ ∫ S (t − s)( ∫ k ( s, r , u 2 (r ))dr )ds ]α ) + d H ([ ∫ S (t − s ) f ( s, u1 ( s ))ds ]α , t
[ ∫ S (t − s) f ( s, u 2 ( s)) ds ]α ) + d H ([ 0
∑ S (t − t
k
) I k (u1 (t k− ))]α ,
0 <tk <t
[
∑ S (t − t
) I k (u 2 (t k− ))]α ),
k
0 < tk <t t
s
0
0
= d H ([ S (t )h(u1 )]α , [ S (t )h(u 2 )]α ) + d H ([ ∫ S (t − s)( ∫ k ( s, r , u1 (r ))dr )ds ]α , t
s
t
0
0
0
[ ∫ S (t − s)( ∫ k ( s, r , u 2 (r ))dr )ds ]α ) + d H ([ ∫ S (t − s ) f ( s, u1 ( s ))ds ]α , t
[ ∫ S (t − s ) f ( s, u 2 ( s ))ds ]α ) + d H ([ 0
[
∑ S (t − t
k
) I k (u1 (t k− ))]α ,
0<t k <t
∑ S (t − t
) I k (u 2 (t k− ))]α )
k
0 < tk <t
≤ d H ([ S qα (t )hqα (u1 ), S rα (t )hrα (u1 )], [ S qα (t )hqα (u2 ), S rα (t )hrα (u2 )]) t
s
s
0
0
0
+ ∫ d H ([ S qα (t − s)( ∫ k qα ( s, r , u1 (r ))dr ), S rα (t − s ) ∫ k rα ( s, r , u1 (r ))dr )], s
s
0
0
[ S qα (t − s)( ∫ k qα ( s, r , u 2 (r ))dr ), S rα (t − s ) ∫ k rα ( s, r , u 2 (r ))dr )])ds t
+ ∫ d H ([ S qα (t − s) f qα ( s, u1 ( s )), S rα (t − s ) f rα ( s, u1 ( s))], [ S qα (t − s) f qα ( s, u2 ( s)), 0
∑ S α (t − t
S rα (t − s ) f rα ( s, u 2 ( s ))])ds + d H ([
k
)( I k )αq (u1 (t k− )),
∑ S α (t − t
)( I k )αq (u 2 (t k− ))
q
0 <tk <t
∑ S α (t − t r
k
)( I k )αr (u1 (t k− ))], [
0<t k <t
q
k
0 < tk <t
∑ S α (t − t r
k
)( I k )αr (u 2 (t k− ))]),
0<t k < t
= max{ S qα (t ) hqα (u1 ) − S qα (t ) hqα (u 2 ) , S rα (t )hrα (u1 ) − S rα (t ) hrα (u 2 ) } 86
International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
t
s
s
0
0
0
+ ∫ max{ S qα (t − s )( ∫ k qα ( s, r , u1 ( r ))dr ) − S qα (t − s) ∫ k qα ( s, r , u 2 ( r ))dr ) , s
s
0
0
S rα (t − s )( ∫ k rα ( s, r , u1 ( r )) dr ) − S rα (t − s ) ∫ k rα ( s, r , u 2 ( r )) dr ) }ds t
+ ∫ max{| S qα (t − s ) f qα ( s, u1 ( s )) − S qα (t − s ) f qα ( s, u 2 ( s )) |, | S rα (t − s) f rα ( s, u1 ( s )) − 0
∑ S α (t − t
S rα (t − s ) f rα ( s, u2 ( s ) |}ds + max{|
q
k
)( I k )αq (u1 (t k− )) −
0<tk < t
∑ S α (t − t q
k
)( I k )αq (u 2 (t k− )) |, |
0<tk < t
r
k
)( I k )αr (u1 (t k− ))
0<tk < t
∑ S α (t − t
−
∑ S α (t − t
r
k
)( I k )αr (u2 (t k− )) |},
0< tk <t
= max{ S qα (t ) hqα (u1 ) − hqα (u2 ) , S rα (t ) hrα (u1 ) − hrα (u 2 ) } t
s
s
0
0
0
+ ∫ max{ S qα (t − s ) ( ∫ k qα ( s, r , u1 ( r ))dr − ∫ k qα ( s, r , u2 ( r ))dr ) , s
s
0
0
S rα (t − s ) ( ∫ k rα ( s, r , u1 (r )) dr − ∫ k rα ( s, r , u 2 ( r )) dr ) }ds t
+ ∫ max{| S qα (t − s ) || f qα ( s, u1 ( s )) − f qα ( s, u2 ( s )) |, | S rα (t − s ) || f rα ( s, u1 ( s)) − 0
f rα ( s, u 2 ( s)) |}ds + max{[
∑
| S qα (t − t k ) || ( I k )αq (u1 (t k− )) −
0<tk < t
( I k )αq (u 2 (t k− ))] |, [
∑
| S rα (t − t k ) || ( I k )αr (u1 (t k− )) − ( I k )αr (u2 (t k− ))] |},
0<tk < t
≤ l s max{ hqα (u1 ) − hqα (u 2 ) , hrα (u1 ) − hrα (u 2 ) } t
s
s
0
0
0
+ l s ∫ max{ ( ∫ k qα ( s, r , u1 ( r )) dr − ∫ k qα ( s, r , u 2 ( r )) dr ) , s
s
0
0
( ∫ k rα ( s, r , u2 ( r ))dr − ∫ k rα ( s, r , u 2 ( r )) dr ) }ds t
+ l s ∫ max{| f qα ( s, u1 ( s )) − f qα ( s, u 2 ( s )) |, | f rα ( s, u1 ( s)) − f rα ( s, u2 ( s ))) |}ds 0
87
International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
+ l s max{[
∑
| ( I k )αq (u1 (t k− )) − ( I k )αq (u 2 (t k− )) |,
0 < tk <t
[
∑
| ( I k )αr (u1 (t k− )) − ( I k )αr (u 2 (t k− )) |},
0 < tk <t t
t
0
0
≤ l s lh d H ([u1 ]α , [u 2 ]α ) + ls lk b ∫ d H ([u1 ]α , [u2 ]α )ds + l s l f ∫ d H ([u1 ]α , [u2 ]α )ds + l s l I d H ([u1 ]α , [u 2 ]α ) t
= l s (l h + l I )d H ([u1 ]α , [u 2 ]α ) + l s (l k b + l f ) ∫ d H ([u1 ]α , [u 2 ]α )ds 0
t
= L1d H ([u1 ]α , [u 2 ]α ) + L2 ∫ d H ([u1 ]α , [u 2 ]α )ds 0
where L1 = ls (lh + l I ), L2 = ls (lk b + l f ). Therefore
d ∞ ((F0u1 , F0u2 ) = sup d H ([(F0u1 )]α , [(F0u 2 )]α ) t∈J
t
≤ sup {L1d H ([u1 ]α , [u2 ]α ) + L2 ∫ d H ([u1 ]α , [u 2 ]α )ds} 0
t∈J
t
= L1 sup d H ([u1 ]α , [u 2 ]α ) + L2 sup ∫ d H ([u1 ]α , [u 2 ]α )ds t∈J
t∈J
0
t
≤ L1d ∞ ([u1 ]α , [u2 ]α ) + L2 ∫ d ∞ ([u1 ]α , [u2 ]α )ds. 0
Hence
H1 (F0u1 , F0u 2 ) = sup d ∞ (F0u1 , F0u 2 ) t∈J
t
≤ sup {L1d ∞ ([u1 ]α , [u 2 ]α ) + L2 ∫ d ∞ ([u1 ]α , [u 2 ]α )ds} 0
t∈J
t
= L1 sup d ∞ ([u1 ]α , [u2 ]α ) + L2 sup ∫ d ∞ ([u1 ]α , [u 2 ]α )ds t∈J
t∈J
0
≤ ( L1 + L2b) H1 (u1 , u 2 ).
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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
We take sufficiently small b , ( L1 + L2b) < 1 . Hence, F0 is a contraction mapping. By the Banach fixed point theorem,implusive fuzzy integrodifferential equation with nonlocal condition has a unique fixed point u ∈ C( J , E N ). 4 .Example Consider the fuzzy solution of the nonlinear impulsive fuzzy integrodifferential equation of the form:
du (t ) = 2u (t ) + 2tu (t ) 2 + 2tu (t ) 2 , dt
t ∈ J,
(4.1)
p
u (0) = ∑u (t k )
(4.2)
k =1
∆u (t k ) = I k (u (t k )) =
1 1 + u (t k )
(4.3)
The α - level set of fuzzy number 2 is
[2]α = [α + 1,3 − α ], for α ∈ [0,1] t
Let
∫ k (t , s, u( s))ds = 2tu (t ) , f (t , u(t )) = 2tu (t ) , h(u(t )) = ∑ 2
2
0
t
∫ k (t , s, u( s))ds = 2tu (t ) 0
2
p
u (t k ). Then α - level set of
k =1
is
t
[ ∫ k (t , s, u ( s ))ds]α = [2tu (t ) 2 ]α 0
= t[2]α [u (t ) 2 ]α
= t[α + 1,3 − α ][(u qα (t )) 2 , (urα (t )) 2 ] = t[(α + 1)(u qα (t )) 2 , (3 − α )(u rα (t )) 2 ], where [u (t )]α = [u qα (t ), u rα (t )] and [2]α = [α + 1,3 − α ], for α ∈ [0,1] and the α - level set of
f (t , x(t )) is [ f (t , u (t ))]α = [2tu (t ) 2 ]α = t[2]α [u (t ) 2 ]α
= t[α + 1,3 − α ][(u qα (t )) 2 , (urα (t )) 2 ] 89
International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
= t[(α + 1)(u qα (t )) 2 , (3 − α )(u rα (t )) 2 ], where [u (t )]α = [u qα (t ), u rα (t )] and [2]α = [α + 1,3 − α ], for α ∈ [0,1] and the α - level set of
h(u (t )) is p
[h(u (t ))]α = [∑ck u (t k )]α k =1 p
p
k =1
k =1
= [∑ck uqα (t k ), ∑ck u rα (t k )] and α - level set of I (u (t )) is
[ I (u (t k ))]α = [
=[
1 ]α 1 + u (t k )
1 1 , ] α 1 + u q (t k ) 1 + u rα (t k )
Thus
1 1 ]α , [ ]α ) d H ([ I (u (t k ))]α , [ I (v(t k ))]α ) = d H ([ 1 + u (t k ) 1 + v(t k )
= d H ([
1 1 1 1 , ],[ , ]) α α α 1 + u q (t k ) 1 + u r (t k ) 1 + vq (t k ) 1 + vrα (t k )
≤ max {| k
≤ max {| k
≤ max { k
1 1 1 1 − |, | − |} α α α 1 + u q (t ) 1 + vq (t ) 1 + u r (t ) 1 + vrα (t )
uqα (t ) − vqα (t k ) (1 + vqα (t k ))(1 + u qα (t k )) | u qα (t k ) − vqα (t k ) |
|, |
u rα (t ) − vrα (t ) |} (1 + urα (t ))(1 + vrα (t ))
| u rα (t k ) − vrα (t k ) | } (1+ | vqα (t k ) |)(1+ | u qα (t k ) |) (1+ | u rα (t k ) |)(1+ | vrα (t k ) |) ,
≤ l I max {| uqα − vqα |, | urα − vrα |} k
= l I d H ([u ]α , [v]α ) where l I = 1/(1+ | u rα (t k ) |)(1+ | vrα (t k ) |) 90
International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
d H ([ f (t , u (t ))]α , [ f (t , v(t ))]α ) = d H (t[(α + 1)(u qα (t )) 2 , (3 − α )(u αr (t )) 2 ], t[(α + 1)(vqα (t )) 2 , (3 − α )(vrα (t )) 2 ]) = t max{(α + 1) | (uqα (t )) 2 − (vqα (t )) 2 |, (3 − α ) | (u rα (t )) 2 − (vrα (t )) 2 |} = t max{(α + 1) | uqα (t ) + vqα (t ) || u qα (t ) − vqα (t ) |, (3 − α ) | u rα (t )) + (vrα (t )) || u rα (t ) − vrα (t ) |}
≤ (3 − α )t | urα (t )) + vrα (t ) | max{| uqα (t ) − vqα (t ) |, | u rα (t ) − vrα (t ) |} ≤ (3 − α )b | urα (t )) + vrα (t ) | max{| u qα (t ) − vqα (t ) |, | u rα (t ) − vrα (t ) |} ≤ 3b | urα (t )) + vrα (t ) | max{| uqα (t ) − vqα (t ) |, | urα (t ) − vrα (t ) |} = l f d H ([u (t )]α , [u (t )]α ) where l f = 3b | u rα (t )) + vrα (t ) | , t
t
0
0
d H ([ ∫ k (t , s, u ( s ))]α , [ ∫ k (t , s, v( s ))]α ) = d H (t[(α + 1)(uqα (t )) 2 , (3 − α )(u rα (t )) 2 ], t[(α + 1)(vqα (t )) 2 , (3 − α )(vrα (t )) 2 ])
= t max{(α + 1) | (u qα (t )) 2 − (vqα (t )) 2 |, (3 − α ) | (u rα (t )) 2 − (vrα (t )) 2 |}
= t max{(α + 1) | uqα (t ) + vqα (t ) || u qα (t ) − vqα (t ) |, (3 − α ) | u rα (t ) + vrα (t ) || u rα (t ) − vrα (t ) |}
≤ (3 − α )t | urα (t )) + vrα (t ) | max{| uqα (t ) − vqα (t ) |, | u rα (t ) − vrα (t ) |} ≤ (3 − α )b | urα (t ) + vrα (t ) | max{| uqα (t ) − vqα (t ) |, | urα (t ) − vrα (t ) |} ≤ 3b | urα (t ) + vrα (t ) | max{| u qα (t ) − vqα (t ) |, | u rα (t ) − vrα (t ) |} = lk bd H ([u (t )]α , [v(t )]α )
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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
Therefore, f , k , h, I k are satisfies the global lipschitz conditions and choose b is sufficiently small. Then all conditions stated in Theorem 3.1 are satisfied, so the problem (4.1)-(4.3) has a unique fuzzy solution.
5. CONCLUSIONS This paper contains some existence and uniqueness results for the nonlinear impulsive fuzzy integrodifferential equation with nonlocal condition in EN proved by using the Banach fixed point theorem approach and the fuzzy number. Under some hypotheses, the existence of firstorder nonlinear impulsive fuzzy integrodifferential equation is proved.
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
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