Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 983483, 22 pages
doi:10.1155/2010/983483
Research Article
Controllability for the Impulsive Semilinear
Nonlocal Fuzzy Integrodifferential Equations in
n-Dimensional Fuzzy Vector Space
Young Chel Kwun,1 Jeong Soon Kim,1 Min Ji Park,1
and Jin Han Park2
1
2
Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea
Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea
Correspondence should be addressed to Jin Han Park, jihpark@pknu.ac.kr
Received 14 March 2010; Accepted 21 June 2010
Academic Editor: T. Bhaskar
Copyright q 2010 Young Chel Kwun et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We study the existence and uniqueness of solutions and nonlocal controllability for the impulsive
semilinear nonlocal fuzzy integrodifferential equations in n-dimensional fuzzy vector space EN n
by using short-term perturbations techniques and Banach fixed point theorem. This is an extension
of the result of Kwun et al. Kwun et al., 2009 to impulsive system.
1. Introduction
The theory of differential equations with discontinuous trajectories during the last twenty
years has been to a great extent stimulated by their numerous applications to problem arising
in mechanics, electrical engineering, the theory of automatic control, medicine and biology.
For the monographs of the theory of impulsive differential equations, see the papers of Bainov
and Simenov 1, Lakshmikantham et al. 2 and Samoileuko and Perestyuk 3, where
numerous properties of their solutions are studied and detailed bibliographies are given.
Rogovchenko 4 followed the ideas of the theory of impulsive differential equations which
treats the changes of the state of the evolution process due to a short-term perturbations
whose duration can be negligible in comparison with the duration of the process as an instant
impulses. In 2001, Lakshmikantham and McRae 5 studied basic results for fuzzy impulsive
differential equations. Park et al. 6 studied the existence and uniqueness of fuzzy solutions
and controllability for the impulsive semilinear fuzzy integrodifferential equations in one1
. Rodrı́guez-López 7 studied periodic boundary value
dimensional fuzzy vector space EN
2
Advances in Difference Equations
problems for impulsive fuzzy differential equations. Fuzzy integrodifferential equations are
a field of interest, due to their applicability to the analysis of phenomena with memory
where imprecision is inherent. Balasubramaniam and Muralisankar 8 proved the existence
and uniqueness of fuzzy solutions for the semilinear fuzzy integrodifferential equation with
nonlocal initial condition. They considered the semilinear one-dimensional heat equation
on a connected domain 0, 1 for material with memory. In one-dimensional fuzzy vector
1
, Park et al. 9 proved the existence and uniqueness of fuzzy solutions and
space EN
presented the sufficient condition of nonlocal controllability for the following semilinear
fuzzy integrodifferential equation with nonlocal initial condition.
In 10, Kwun et al. proved the existence and uniqueness of fuzzy solutions for
the semilinear fuzzy integrodifferential equations by using successive iteration. In 11,
Kwun et al. investigated the continuously initial observability for the semilinear fuzzy
integrodifferential equations. Bede and Gal 12 studied almost periodic fuzzy-numbervalued functions. Gal and N’Guerekata 13 studied almost automorphic fuzzy-numbervalued functions. More recently, Kwun et al. 14 studied the existence and uniqueness of
solutions and nonlocal controllability for the semilinear fuzzy integrodifferential equations
in n-dimensional fuzzy vector space.
In this paper, we study the existence and uniqueness of solutions and nonlocal
controllability for the following impulsive semilinear nonlocal fuzzy integrodifferential
equations in n-dimensional fuzzy vector space by using short-term perturbations techniques
and Banach fixed point theorem:
t
t
dxi t
Ai xi t Gt − sxi sds fi t, xi t, qi t, s, xi sds ui t
dt
0
0
i
,
on EN
i
xi 0 gi xi x0i ∈ EN
,
Δxi tk Ik xi tk ,
t/
tk , k 1, 2, . . . , m, i 1, 2, . . . , n,
1.1
i
i
where Ai : 0, T → EN
is fuzzy coefficient, EN
is the set of all upper semicontinuously
j
i
i
i
i
convex fuzzy numbers on R with EN /
j, fi : 0, T × EN
× EN
→ EN
and qi : 0, T ×
EN i /
i
i
i
0, T × EN are nonlinear regular fuzzy functions, gi : EN → EN is a nonlinear continuous
i
function, Gt is an n × n continuous matrix such that dGtxi /dt is continuous for xi ∈ EN
i
i
and t ∈ 0, T with Gt ≤ k, k > 0, ui : 0, T → EN is a control function, x0i ∈ EN is an
i
i
, EN
are bounded functions, Δxi tk xi tk − xi t−k , where xi t−k initial value and Ik ∈ CEN
and xi tk represent the left and right limits of xi t at t tk , respectively.
2. Preliminaries
A fuzzy set u of Rn is a function u : Rn → 0, 1. For each fuzzy set u, we denote by uα {x ∈ Rn : ux ≥ α} for any α ∈ 0, 1 its α-level set.
Let u, v be fuzzy sets of Rn . It is well known that uα vα for each α ∈ 0, 1 implies
u v.
Advances in Difference Equations
3
Let En denote the collection of all fuzzy sets of Rn that satisfies the following
conditions:
1 u is normal, that is, there exists an x0 ∈ Rn such that uxo 1;
2 u is fuzzy convex, that is, uλx 1 − λy ≥ min{ux, uy} for any x, y ∈ Rn ,
0 ≤ λ ≤ 1;
3 ux is upper semicontinuous, that is, ux0 ≥ limk → ∞ uxk for any xk ∈ Rn k 0, 1, 2, . . ., xk → x0 ;
4 u0 is compact.
We call u ∈ En an n-dimension fuzzy number.
Wang et al. 15 defined n-dimensional fuzzy vector space and investigated its
properties.
For any ui ∈ E, i 1, 2, . . . , n, we call the ordered one-dimension fuzzy number class
u1 , u2 , . . . , un i.e., the Cartesian product of one-dimension fuzzy number u1 , u2 , . . . , un an ndimension fuzzy vector, denote it as u1 , u2 , . . . , un , and call the collection of all n-dimension
fuzzy vectors i.e., the Cartesian product E × E × · · · × E n-dimensional fuzzy vector space,
and denote it as En .
Definition 2.1 see 15. If u ∈ En , and uα is a hyperrectangle, that is, uα can be represented
by ni1 uαil , uαir , that is, uα1l , uα1r ×uα2l , uα2r ×· · ·×uαnl , uαnr for every α ∈ 0, 1, where uαil , uαir ∈ R
with uαil ≤ uαir when α ∈ 0, 1, i 1, 2, . . . , n, then we call u a fuzzy n-cell number. We denote
the collection of all fuzzy n-cell numbers by LEn .
Theorem 2.2 see 15. For any u ∈ LEn with uα ni1 uαil , uαir α ∈ 0, 1, there exists a
unique u1 , u2 , . . . , un ∈ En such that ui α uαil , uαir (i 1, 2, . . . , n and α ∈ 0, 1). Conversely,
for any u1 , u2 , . . . , un ∈ En with ui α uαil , uαir (i 1, 2, . . . , n and α ∈ 0, 1), there exists a
unique u ∈ LEn such that uα ni1 uαil , uαir α ∈ 0, 1.
Note 1 see 15. Theorem 2.2 indicates that fuzzy n-cell numbers and n-dimension fuzzy
vectors can represent each other, so LEn and En may be regarded as identity. If
u1 , u2 , . . . , un ∈ En is the unique n-dimension fuzzy vector determined by u ∈ LEn ,
then we denote u u1 , u2 , . . . , un .
n
i n
i
1
2
EN
× EN
× · · · × EN
, where EN
i 1, 2, . . . , n is a fuzzy subset of R. Then
Let EN
n
i n
EN ⊆ E .
i n
Definition 2.3 see 15. The complete metric DL on EN
is defined by
DL u, v sup dL uα , vα sup max uαil − vilα , uαir − virα 0<α≤1 1≤i≤n
0<α≤1
2.1
i n
for any u, v ∈ EN
, which satisfies dL u w, v w dL u, v.
i n
Definition 2.4. Let u, v ∈ C0, T : EN
,
H1 u, v sup DL ut, vt.
0≤t≤T
2.2
4
Advances in Difference Equations
i n
is defined by
Definition 2.5 see 15. The derivative x t of a fuzzy process x ∈ EN
x t
α
xilα t, xirα t
n
2.3
i1
i n
provided that equation defines a fuzzy x t ∈ EN
.
a
Definition 2.6 see 15. The fuzzy integral b xtdt, a, b ∈ 0, T is defined by
a
α
xtdt
b
n a
i1
b
xilα tdt,
a
b
xirα tdt
2.4
provided that the Lebesgue integrals on the right-hand side exist.
3. Existence and Uniqueness
In this section we consider the existence and uniqueness of the fuzzy solution for 1.1 u ≡ 0.
We define
A A1 , A2 , . . . , An ,
x x1 , x2 , . . . , xn ,
f f1 , f2 , . . . , fn ,
q q1 , q2 , . . . , qn ,
3.1
u u1 , u2 , . . . , un ,
g g1 , g2 , . . . , gn ,
x0 x01 , x02 , . . . , x0n .
3.2
n
i
A, x, f, q, x0 , u, g ∈ EN
.
3.3
Then
i n
Instead of 1.1, we consider the following fuzzy integrodifferential equations in EN
t
t
dxt
A xt Gt − sxsds f t, xt, qt, s, xsds ut
dt
0
0
n
i
on EN
,
3.4
n
i
x0 gx x0 ∈ EN
,
Δxtk Ik xtk ,
t/
tk , k 1, 2, . . . , m, i 1, 2, . . . , n,
3.5
3.6
Advances in Difference Equations
5
i n
i n
i n
, initial value x0 ∈ EN
, and u : 0, T → EN
with fuzzy coefficient A : 0, T → EN
i n
i n
being a control function. Given nonlinear regular fuzzy functions f : 0, T ×EN ×EN →
i n
i n
i n
and q : 0, T × 0, T × EN
→ EN
satisfy global Lipschitz conditions, that is, there
EN
exist finite constants k1 , k2 , M > 0 such that
α α dL f s, ξ1 s, η1 s , f s, ξ2 s, η2 s
α α ≤ k1 dL ξ1 sα , ξ2 sα k2 dL η1 s , η2 s ,
α α α α ≤ MdL ϕ1 s , ϕ2 s
dL q t, s, ϕ1 s , q t, s, ϕ2 s
3.7
3.8
i n
i n
i n
j 1, 2, the nonlinear function g : EN
→ EN
is
for all ξj s, ηj s, ϕj s ∈ EN
continuous and satisfies the Lipschitz condition
α α α ≤ hdL x·α , y·
dL gx· , g y·
3.9
i n
for all x·, y· ∈ EN
, h is a finite positive constant.
i n
Definition 3.1. The fuzzy process x : I 0, T → EN
with α-level set xtα Πni1 xi α n
α
α
Πi1 xil , xir is a fuzzy solution of 3.4 and 3.5 without nonhomogeneous term if and only
if
xilα t
xirα t
min
Aαij t
max
α
α
xik
t
α
xilα 0 gil xil x0αil ,
α
xik
t
t
0
Gt −
α
sxik
sds
Gt −
α
sxik
sds
0
Aαij t
t
: j, k l, r ,
xirα 0 girα xirα x0αir ,
: j, k l, r ,
3.10
i 1, 2, . . . , n.
For the sequel, we need the following assumption:
i n
H1 St is a fuzzy number satisfying, for y ∈ EN
, d/dtSty ∈ C1 I :
i n i n
EN CI : EN , the equation
t
d
Sty A Sty Gt − sSsy ds ,
dt
0
t ∈ I,
3.11
S0 I
3.12
where
Stα n
n
α
Sil t, Sαir t ,
Si tα i1
i1
and Sαij tj l, r is continuous with |Sαij t| ≤ c, c > 0, for all t ∈ I 0, T .
6
Advances in Difference Equations
In order to define the solution of 3.4–3.6, we will consider the space Ωi {xi : J →
i
i
: xi k ∈ CJk , EN
, Jk tk , tk1 , k 0, 1, . . . , m, and there exist xi t−k and xi tk k EN
1, 2, . . . , m, with xi t−k xi tk }, i 1, 2, . . . , n.
i
, i 1, 2, . . . , n.
Let Ω Πni1 Ωi , Ωi Ωi C0, T : EN
Lemma 3.2. If x is an integral solution of 3.4–3.6 u ≡ 0, then x is given by
xt St x0 − gx t
St − sf s, xs,
0
St − tk Ik x t−k ,
s
qs, τ, xτdτ ds
0
3.13
for t ∈ J.
0<tk <t
Proof. Let x be a solution of 3.4–3.6. Define ωs St − sxs. Then we have that
dωs
dxs
dSt − s
−
xs St − s
ds
ds
ds
t
dxs
−A St − sxs Gt − sSsxsds St − s
ds
0
s
St − sf s, xs, qs, τ, xτdτ .
3.14
0
Consider tk < t, k 1, 2, . . . , m. Then integrating the previous equation, we have
t
0
dωs
ds ds
t
s
St − sf s, xs, qs, τ, xτdτ ds.
0
3.15
0
For k 1,
ωt − ω0 t
St − sf s, xs,
0
s
qs, τ, xτdτ ds
3.16
0
or
xt St x0 − gx t
0
St − sf s, xs,
s
0
qs, τ, xτdτ ds.
3.17
Advances in Difference Equations
7
Now for k 2, . . . , m, we have that
t2
t
dωs
dωs
dωs
ds ds · · · ds
ds
ds
ds
tk
t1
t
s
St − sf s, xs, qs, τ, xτdτ ds.
t1
0
0
3.18
0
Then
ω t−1 − ω0 ω t−2 − ω t1 · · · − ω tk ωt
t
s
St − sf s, xs, qs, τ, xτdτ ds
0
3.19
0
if and only if
ωt ω0 t
St − sf s, xs,
0
s
qs, τ, xτdτ ds 0
ω tk − ω t−k .
3.20
0<tk <t
Hence
xt St x0 − gx t
s
St − sf s, xs, qs, τ, xτdτ ds 0
0
St − tk Ik x t−k ,
0<tk <t
3.21
which proves the lemma.
Assume the following:
H2 there exists d > 0 such that
α α − α dL Ik x t−k
, I k y tk
≤ ddL xtα , yt ,
3.22
where xt, yt ∈ Ω ;
H3
!
"
cT
cT
c T h1 c cd k1 1 k2 MT 12 h d < 1.
2
3
3.23
8
Advances in Difference Equations
i n
, 3.13 has a
Theorem 3.3. Let T > 0. If hypotheses (H1)–(H3) are hold, then, for every x0 ∈ EN
unique fuzzy solution x ∈ Ω .
Proof. For each xt ∈ Ω and t ∈ 0, T , define G0 xt ∈ Ω by
G0 xt St x0 − gx t
s
St − sf s, xs, qs, τ, xτdτ ds
0
0
3.24
St − tk Ik x t−k .
0<tk <t
Thus, G0 x : 0, T → Ω is continuous, so G0 is a mapping from Ω into itself. By Definitions
2.3 and 2.4, some properties of dL and inequalities 3.7, 3.8, and 3.9, we have the following
inequalities. For x, y ∈ Ω ,
α dL G0 xtα , G0 y t
≤ dL
St x0 − gx t
St − sf s, xs,
s
0
St x0 − g y t
0
St − sf s, ys,
s
0
α q s, τ, yτ dτ ds
0
α α St − tk Ik x t−k
dL
α
qs, τ, xτdτ ds ,
St − tk Ik y t−k
,
0<tk <t
0<tk <t
α α ≤ dL Stgx , Stg y
t
St − sf s, xs,
dL
0
s
α
qs, τ, xτdτ
3.25
,
0
s
α St − sf s, ys, q s, τ, yτ dτ
ds
0
dL
α α St − tk Ik x t−k
0<tk <t
St − tk Ik y t−k
,
0<tk <t
α ≤ chdL x·α , y·
c
t
α
k1 dL xs , ys
0
α cddL xtα , yt .
α k2 M
s
0
α
dL xτ , yτ
dτ ds
α Advances in Difference Equations
9
Therefore
DL G0 xt, G0 y t
α sup dL G0 xtα , G0 y t
0<α≤1
α ≤ ch sup dL x·α , y·
0<α≤1
c
t
α k1 sup dL xs , ys
k2 M
0
α
s
α
sup dL xτ , yτ
α dτ ds
0 0<α≤1
0<α≤1
α cd sup dL xtα , yt
0<α≤1
≤ chDL x·, y·
c
t
k1 DL xs, ys k2 M
0
s
DL xτ, yτ dτ ds cdDL xt, yt .
0
3.26
Hence
H1 G0 x, G0 y sup DL G0 xt, G0 y t
0≤t≤T
≤ ch sup DL x·, y·
0≤t≤T
c sup
t
s
k1 DL xs, ys k2 M DL xτ, yτ dτ ds
0≤t≤T
0
0
3.27
cd sup DL xt, yt
0≤t≤T
T
≤ c h d k1 k2 M
T H1 x, y .
2
By hypothesis H3, G0 is a contraction mapping. Using the Banach fixed point theorem,
3.13 has a unique fixed point x ∈ Ω .
4. Nonlocal Controllability
In this section, we show the nonlocal controllability for the control system 1.1.
10
Advances in Difference Equations
The control system 1.1 is related to the following fuzzy integral system:
xt St x0 − gx t
t
St − sf s, xs,
0
qs, τ, xτdτ ds
0
4.1
St − tk Ik x t−k .
St − susds 0
s
0<tk <t
Definition 4.1. Equations 1.1–3 are nonlocal controllable. Then there exists ut such that
the fuzzy solution xt for 4.1 as xT x1 − gxi.e., xT α x1 − gxα , where x1 ∈
i n
, is target set.
EN
i n
Define the fuzzy mapping β# : P# Rn → EN
by
⎧ T
⎪
⎨ Sα T − svsds, v ⊂ Γu ,
β#α v 0
⎪
⎩
0,
otherwise,
4.2
where Γu is closed support of u. Then there exists
i
β#i : P# R −→ EN
4.3
i 1, 2, . . . , n
such that
⎧ T
⎪
⎨ Sα T − svi sds, vi s ⊂ Γu
i
i
β#iα vi ⎪ 0
⎩
0,
otherwise.
4.4
Then there exists β#ijα j l, r such that
β#ilα vil β#irα vir T
0
T
0
Sαil T − svil sds,
Sαir T
− svir sds,
We assume that β#ilα , β#irα are bijective mappings.
vil s ∈ uαil s, u1i ,
vir s ∈
u1i , uαir s
.
4.5
Advances in Difference Equations
11
We can introduce α-level set of us of 4.1:
usα n
n
α
uil s, uαir s
ui sα i1
n
β#ilα
i1
−1 α
x1 − gilα xilα − Sαil T x0αil − gilα xilα
il
i1
−
T
0
Sαil T
−
−
Sαil T
−
0<tk <T
β#irα
s, xilα s,
sfilα
tk Ikαil
xilα
s
0
qilα
s, τ, xilα τ
dτ ds
− tk
,
4.6
−1 α
x1 − girα xirα − Sαir T x0αir − girα xirα
ir
−
T
0
Sαir T
−
−
sfirα
−
Sαir T
0<tk <T
s, xirα s,
tk Ikαir
s
0
qirα
s, τ, xirα τ
dτ
ds
xirα
− tk
.
Then substituting this expression into 4.1 yields α-level of xT .
For each i 1, 2, . . . , n,
T
Sαil T − s
xi T Sαil T x0αil − gilα xilα α
0
× filα s, xilα s,
T
0
s
0
qilα s, τ, xilα τ dτ ds 0<tk <T
Sαil T − tk Ikαil xilα t−k
−1 α
x1 − gilα xilα − Sαil T x0αil − gilα xilα
Sαil T − s β#ilα
il
−
T
0
Sαil T
−
0<tk <T
−
sfilα
s
α
s, xil s, qilα s, τ, xilα τ dτ ds
0
ds,
Sαil T − tk Ikαil xilα t−k
12
Advances in Difference Equations
T
s
Sαir T − s × firα s, xirα s, qilα s, τ, xirα τ dτ ds
Sαir T x0αir − girα xirα Sαir T
−
tk Ikαir
0<tk <T
T
0
0
xirα
−1
Sαir T − s β#irα
α
− Sαir T x0αir − girα xirα
x1 − girα xirα
×
ir
−
T
0
s
Sαir T − sfirα s, xirα s, qilα s, τ, xirα τ dτ ds
0
−
Sαir T
−
tk Ikαir
0<tk <T
0
− tk
x1 − gx
xirα
− ds
tk
α α α
x1 − gx
, x1 − gx
.
il
i
ir
4.7
Therefore
xT α n
xi T α i1
n x1 − gx
α
i1
i
α
x1 − gx .
4.8
We now set
Φxt St x0 − gx t
St − sf
0
s, xs,
qs, τ, xτdτ ds
0
St − tk Ik x t−k
0<tk <t
t
t
#−1
St − sβ
x1 − gx − ST x0 − gx
0
−
T
4.9
s
ST − sf s, xs, qs, τ, xτdτ ds
0
−
0
− ds,
ST − tk Ik x tk
0<tk <T
where the fuzzy mapping β#−1 satisfies the previous statements.
Notice that ΦxT x1 − gx, which means that the control ut steers 4.9 from the
origin to x1 − gx in time T provided we can obtain a fixed point of the operator Φ.
H4 Assume that the linear system of 4.9 f ≡ 0 is controllable.
Advances in Difference Equations
13
Theorem 4.2. Suppose that hypotheses (H1)–(H4) are satisfied. Then 4.9 is nonlocal controllable.
Proof. We can easily check that Φ is continuous function from Ω to itself. By Definitions 2.3
and 2.4, some properties of dL , and inequalities 3.7, 3.8, and 3.9, we have following
inequalities. For any x, y ∈ Ω ,
α dL Φxtα , Φyt
t
s
St x0 − gx St − sf s, xs, qs, τ, xτdτ ds
dL
0
0
St − tk Ik x t−k
0<tk <t
t
−1
#
x1 − gx − ST x0 − gx
St − sβ
0
T
−
s
qs, τ, xτdτ ds
0
α
− ds ,
ST − tk Ik x tk
ST − sf s, xs,
0
−
0<tk <T
St x0 − g y t
s
St − sf s, ys, q s, τ, yτ dτ ds
0
0
St − tk Ik y t−k
0<tk <t
t
−1
#
x1 − g y − ST x0 − g y
St − sβ
0
−
T
q s, τ, yτ dτ ds
0
α − ds
ST − tk Ik y tk
ST − sf s, ys,
0
−
s
0<tk <T
α ≤ chdL x·α , y·
t
s
α α c
k1 dL xsα , ys
k2 M dL xτα , yτ dτ ds
0
0
α cddL xt , yt
t
α α hdL x·α , y·
c
chdL x·α , y·
0
α
T
α k1 dL xsα , ys
k2 M
0
α α ds.
cddL xt , yt
c
s
α dL xτα , yτ dτ ds
0
4.10
14
Advances in Difference Equations
Therefore
α DL Φxt, Φyt sup dL Φxtα , Φyt
0<α≤1
α α ≤ ch sup dL x·α , y·
cd sup dL xtα , yt
0<α≤1
c
0<α≤1
t
α k1 sup dL xsα , ys
0
0<α≤1
k2 M
s
α
sup dL xτ , yτ
α dτ ds
0 0<α≤1
t
α h1 c sup dL x·α , y·
c
0
0<α≤1
c
T
α k1 sup dL xsα , ys
0
0<α≤1
k2 M
s
α
sup dL xτ , yτ
α dτ ds
0 0<α≤1
α
cd sup dL xs , ys
α ds
0<α≤1
≤ chDL x·, y· cdDL xt, yt
t
s
c
k1 DL xs, ys k2 M DL xτ, yτ dτ ds
0
0
t
h1 cDL x·, y·
c
0
T
k1 DL xs, ys k2 M
0
cdDL xs, ys ds.
c
s
DL xτ, yτ dτ ds
0
Hence
H1 Φx, Φy sup DL Φxt, Φyt
0≤t≤T
≤ ch sup DL x·, y· cd sup DL xt, yt
0≤t≤T
c sup
0≤t≤T
0≤t≤T
t
0
k1 DL x·, y· k2 M
s
0
DL xτ, yτ dτ ds
4.11
Advances in Difference Equations
t
h1 cDL x·, y·
c sup
0≤t≤T
15
0
T
c
k1 DL xs, ys k2 M
0
s
DL xτ, yτ dτ ds
0
cdDL xs, ys ds
s
T
≤ chH1 x, y cdH1 x, y c
k1 H1 x, y k2 M H1 x, y dτ ds
c
T
0
0
h1 cH1 x, y
0
c
T
k1 H1 x, y k2 M
s
0
H1 x, y dτ ds cdH1 x, y ds
0
!
"
1 cT
cT
c T h1 c cd k1 1 k2 MT
h d H1 x, y .
2
2
3
4.12
By hypothesis H3, Φ is a contraction mapping. Using the Banach fixed point theorem, 4.9
has a unique fixed point x ∈ Ω .
5. Example
Consider the two semilinear one-dimensional heat equations on a connected domain 0, 1
i
, i 1, 2, boundary condition
for material with memory on EN
xi t, 0 xi t, 1 0,
i 1, 2
5.1
and with initial conditions
p
xi 0, zi ck i xi tk , zi x0i zi ,
5.2
k1
i
,
where x0i zi ∈ EN
p
ck i xi tk , zi gi xi ,
i 1, 2.
5.3
k1
Let xi t, zi , i 1, 2 be the internal energy and
fi t, xi t, zi ,
t
0
# i t, zi 2 qi t,s, xi s, zi ds 2tx
t
0
t − sxi s, zi ds,
i 1, 2
5.4
16
Advances in Difference Equations
be the external heat with memory.
Δxi tk , zi xi tk , zi − xi t−k , zi
5.5
is impulsive effect at t tk k 1, 2, . . . , m.
Let
2
2
#2 ∂ , 2# ∂
,
∂z21 ∂z22
A A1 , A2 f
t
t, xt,
qt, s, xsds
0
f1 t, x1 t,
t
q1 t, s, x1 sds , f2 t, x2 t,
0
# 1 t, z1 2 2tx
t
q2 t, s, x2 sds
0
t
# 2 t, z2 2 t − sx1 s, z1 ds, 2tx
t
0
t − sx2 s, z2 ds ,
0
p
gx g1 x1 , g2 x2 p
ck 1 x1 tk , z1 ,
k1
x0 gx x1 0 g1 x, x2 0 g2 x ,
ck 2 x2 tk , z2 ,
k1
5.6
# 0# ,
x0 x01 , x02 0,
t/
tk , k 1, 2, . . . , m,
Δxtk Δx1 tk , Δx2 tk ,
Ik xtk Ik x1 tk , Ik x2 tk x1 tk − x1 t−k , x2 tk − x2 t−k
x1 tk , z1 − x1 t−k , z1 , x2 tk , z2 − x2 t−k , z2 ,
t/
tk , k 1, 2, . . . , m,
Gt − s e−t−s , e−t−s
then the balance equations become
t
dxt
A xt Gt − sxsds
dt
0
t
f
qt, s, xsds
t, xt,
0
2
i
ut on EN
,
2
i
,
x0 gx x0 ∈ EN
Δxtk Ik xtk ,
t/
tk , k 1, 2, . . . , m, i 1, 2.
5.7
Advances in Difference Equations
17
The α-level sets of fuzzy numbers are the following
# α α 1, 3 − α for all α ∈ 0, 1. Then α-level set of
# α α − 1, 1 − α, 2
0
t
ft, xt, 0 qt, s, xsds is
f
α
t
t, xt,
qt, s, xsds
0
# 1 t 2tx
2
α
t
# 2 t × 2tx
2
t − sx1 sds
t
0
α
t − sx2 sds
0
α t
α 2
α
2# · t x1 t
t − sx1 s ds
0
α t
α #2 · t x2 t2 t − sx2 sα ds
×
0
2 α 2 α
t , x1r
t
α 1, 3 − α · t x1l
×
α 1, 3 − α · t
α 1t
2
α
x1l
t
× α 1t
2
α
x2l
t
2 α 2
α
x2l
t , x2r
t
t
t −
0
t
α
sx1l
sds, 3
t −
0
t
0
t
t − s
0
− αt
α
sx2l
sds, 3
5.8
α
α
t − s x1l s , x1r s ds
α
x2l
s
2
α
x1r
t
− αt
2
α
x2r
t
α
, x2r s ds
t
t −
0
t
α
sx1r
sds
t −
0
α
sx2r
sds
.
Further, we have
dL
f
α t
t, xt,
qt, s, xsds
, f
t, yt,
0
dL
α 1t
α 1t
α q t, s, ys ds
0
2
xilα t
t
2
yilα t
t
t −
0
t
0
t −
sxilα sds, 3
syilα sds, 3
− αt
− αt
2
xirα t
2
yirα t
t
t −
0
t
1≤i≤2
,
t −
0
(
2 2 2 2 )
t max α 1 xilα t − yilα t , 3 − α xirα t − yirα t 1≤i≤2
t
t − smax xilα s − yilα s, xirα s − yirα s ds
0
sxirα sds
syirα sds
18
Advances in Difference Equations
≤ T 3 − αmax xilα t − yilα txilα t yilα t, xirα t − yirα txirα t yirα t
1≤i≤2
T2
max xilα t − yilα t, xirα t − yirα t
2 1≤i≤2
α T 2 α ≤ 3T xirα t yirα tdL xtα , yt
dL xtα , yt
2
α α k1 dL xtα , yt
k2 dL xtα , yt ,
p
α p
α α α ck xtk ,
ck ytk dL gx· , g y·
dL
k1
k1
p
p
p
p
α
α
α
α
max ck i xil tk − ck i yil tk , ck i xir tk − ck i yir tk k1
1≤i≤2 k1
k1
k1
p α
p α α
α
α
≤ ck max xil tk − yil tk , xir tk − yir tk ck dL xtk α , ytk k1 1≤i≤2
k1 p α α ≤ ck max dL xtk α , ytk hdL x·α , y· ,
k1 k
5.9
where k1 , k2 , and h satisfy inequalities 3.7, 3.8, and 3.9, respectively. Choose T such that
T < 1 − ch d/c. Then all conditions stated in Theorem 3.3 are satisfied, so the problem
5.7 has a unique fuzzy solution.
# 3.
# The α-level set of fuzzy numbersis 3
# 3
# α Let target set be x1 x11 , x21 2,
α 2, 4 − α.
From the definition of fuzzy solution,
xilα t
Sαil t
t
0
t
0
xirα t
0
t
0
where i 1, 2.
−
ck i xilα tk k1
s
2
Sαil t − s α 1s xilα s s − τxilα dτ ds
0
− α
Sαil t − tk Ikil
x tk ,
Sαil t − suαil sds 0<tk <t
Sαir t
t
p
x0 αil
p
x0 αir
−
ck i xirα tk k1
2
Sαir t − s 3 − αs xirα s Sαir t − suαir sds s
0
xirα dτ ds
− α
Sαir t − tk Ikir
x tk ,
0<tk <t
5.10
Advances in Difference Equations
19
Thus the α-levels of us
uα1l s
α
β#1l
−1
p
ck 1 xilα tk α 1 −
k1
−
T
−
p
α 1Sα1l T
s
α 2
α
− s s x1l s s − τx1l τdτ ds
0
0<tk <T
−1
α
uα1r s β#1r
3 − α −
−
p
ck 1 xirα tk k1
T
p
1 − α −
Sα1r T −
α
ck 1 x1r
tk k1
3 −
0
αSα1r T
s
α
2
α
− s s x1r s s − τx1r τdτ ds
0
α − α
,
Sα1r T − tk Ik1r
x1r tk
−
0<tk <T
−1
α
uα2l s β#2l
α 2 −
−
p
ck 2 xilα tk k1
T
p
α − 1 −
Sα2l T −
α
ck 2 x2l
tk k1
α 0
1Sα2l T
s
α 2
α
− s s x2l s s − τx2l τdτ ds
0
α − α
,
Sα2l T − tk Ik2l
x2l tk
−
α
uα2r s β#2r
α − α
,
Sα1l T − tk Ik1l
x1l tk
−
−1
α
ck 1 x1l
tk k1
0
α − 1 −
Sα1l T 0<tk <T
p
4 − α −
k1
−
Sα2r T −
ck 2 xirα tk T
0
p
1 − α −
α
ck 2 x2r
tk k1
3 −
αSα2r T
s
α
2
α
− s s x2r s s − τx2r τdτ ds
0
α − α
.
Sα2r T − tk Ik2r
x2r tk
−
0<tk <T
5.11
20
Advances in Difference Equations
Then α-level of xT x1 T , x2 T is
α
α
x1 T α x1l
T , x1r
T T
p
α − 1 −
Sα1l T α
ck 1 x1l
tk k1
α 0
2
α
x1l
s
− s s
1Sα1l T
−1
α #α
β#1l
β1l
α 1 −
p
0
α
s − τx1l
τdτ ds
ck 1 xilα tk k1
−
s
p
α − 1 −
Sα1l T α
ck 1 x1l
tk k1
T
0
s
α 2
α
α 1Sα1l T − s s x1l
s s − τx1l
τdτ ds
0
α − α
Sα1l T − tk Ik1l
x1l tk
0<tk <T
Sα1l T
−
α
tk Ik1l
α
x1l
− α
tk , S1r T 1 − α −
p
α
ck 1 x1r
tk 0<tk <T
α
2
3 − αSα1r T − s s x1r
s α #α
β#1r
β1r
−1
p
−
p
α
ck 1 x1r
tk k1
T
3 −
αSα1r T
− s s
2
α
x1r
s
α
tk Ik1r
α
x1r
α 1 −
α
ck 1 x1l
tk k1
p
ck 1 x1 tk 0
α
s − τx1r
τdτ ds
− tk
0<tk <T
p
s
α − α
Sα1r T − tk Ik1r
x1r tk
0<tk <T
−
1 − α −
Sα1r T 0
0
α
s − τx1r
τdτ ds
k1
Sα1r T
s
α
ck 1 x1r
tk 3 − α −
2# −
k1
T
0
p
, 3 − α −
α
ck 1 x1r
tk k1
α
.
k1
5.12
Advances in Difference Equations
21
Similarly,
α
α
x2 T x2l T , x2r T 3# −
α
p
α
ck 2 x2 tk .
5.13
x1 − gx.
5.14
k1
Hence
xT x1 T , x2 T 2# −
p
k1
ck 1 x1 tk , 3# −
p
ck 2 x2 tk k1
Then all the conditions stated in Theorem 4.2 are satisfied, so the system 5.7 is nonlocal
controllable on 0, T .
Acknowledgment
This study was supported by research funds from Dong-A University.
References
1 D. D. Bainov and P. S. Simenov, Impulsive Differential Equations: Periodic Solutions and Applications,
Longman, Harlow, UK, 1993.
2 V. Lakshmikantham, D. D. Baı̆nov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6
of Series in Modern Applied Mathematics, World Scientific, Singapore, 1989.
3 A. M. Samoileuko and N. A. Perestyuk, Impulsive Differential Equations, vol. 28 of Series on Advances in
Mathematics for Applied Sciences, World Scientific, Singapore, 1995.
4 Y. V. Rogovchenko, “Impulsive evolution systems: main results and new trends,” Dynamics of
Continuous, Discrete and Impulsive Systems, vol. 3, no. 1, pp. 57–88, 1997.
5 V. Lakshmikantham and F. A. McRae, “Basic results for fuzzy impulsive differential equations,”
Mathematical Inequalities & Applications, vol. 4, no. 2, pp. 239–246, 2001.
6 J. H. Park, J. S. Park, Y. C. Ahn, and Y. C. Kwun, “Controllability for the impulsive semilinear fuzzy
integrodifferential equations,” Advances in Soft Computing, vol. 40, pp. 704–713, 2007.
7 R. Rodrı́guez-López, “Periodic boundary value problems for impulsive fuzzy differential equations,”
Fuzzy Sets and Systems, vol. 159, no. 11, pp. 1384–1409, 2008.
8 P. Balasubramaniam and S. Muralisankar, “Existence and uniqueness of fuzzy solution for semilinear
fuzzy integrodifferential equations with nonlocal conditions,” Computers & Mathematics with
Applications, vol. 47, no. 6-7, pp. 1115–1122, 2004.
9 J. H. Park, J. S. Park, and Y. C. Kwun, “Controllability for the semilinear fuzzy integrodifferential
equations with nonlocal conditions,” in Proceedings of the 3rd International Conference on Fuzzy Systems
and Knowledge Discovery (FSKD ’06), vol. 4223 of Lecture Notes in Artificial Intelligence, pp. 221–230,
2006.
10 Y. C. Kwun, M. J. Kim, B. Y. Lee, and J. H. Park, “Existence of solutions for the semilinear fuzzy
integrodifferential equations using by succesive iteration,” Journal of Korean Institute of Intelligent
Systems, vol. 18, pp. 543–548, 2008.
11 Y. C. Kwun, M. J. Kim, J. S. Park, and J. H. Park, “Continuously initial observability for the semilinear
fuzzy integrodifferential equations,” in Proceedings of the 5th International Conference on Fuzzy Systems
and Knowledge Discovery (FSKD ’08), vol. 1, pp. 225–229, 2008.
12 B. Bede and S. G. Gal, “Almost periodic fuzzy-number-valued functions,” Fuzzy Sets and Systems, vol.
147, no. 3, pp. 385–403, 2004.
22
Advances in Difference Equations
13 S. G. Gal and G. M. N’Guerekata, “Almost automorphic fuzzy-number-valued functions,” Journal of
Fuzzy Mathematics, vol. 13, no. 1, pp. 185–208, 2005.
14 Y. C. Kwun, J. S. Kim, M. J. Park, and J. H. Park, “Nonlocal controllability for the semilinear fuzzy
integrodifferential equations in n-dimensional fuzzy vector space,” Advances in Difference Equations,
vol. 2009, Article ID 734090, 16 pages, 2009.
15 G. Wang, Y. Li, and C. Wen, “On fuzzy n-cell numbers and n-dimension fuzzy vectors,” Fuzzy Sets
and Systems, vol. 158, no. 1, pp. 71–84, 2007.