Controllability for the Impulsive Semilinear Nonlocal Fuzzy Integrodifferential Equations in n-Dimensional Fuzzy Vector Space

Advances in Difference Equations Volume 2010, Article ID 983483,22pages 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,

Jeong Soon Kim,

Min Ji Park,

and Jin Han Park

1Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

2Department 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

Copyrightq2010 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 inn-dimensional fuzzy vector spaceENⁿ 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., 2009to 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.

Rogovchenko4followed 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 McRae5studied basic results for fuzzy impulsive differential equations. Park et al.6studied the existence and uniqueness of fuzzy solutions and controllability for the impulsive semilinear fuzzy integrodifferential equations in one- dimensional fuzzy vector spaceE¹_N. Rodr´ıguez-L ´opez7studied periodic boundary value

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 Muralisankar8proved 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 domain0,1for material with memory. In one-dimensional fuzzy vector space E¹_N, Park et al. 9 proved the existence and uniqueness of fuzzy solutions and 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-number- valued functions. Gal and N’Guerekata 13 studied almost automorphic fuzzy-number- valued functions. More recently, Kwun et al. 14studied the existence and uniqueness of solutions and nonlocal controllability for the semilinear fuzzy integrodifferential equations inn-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 inn-dimensional fuzzy vector space by using short-term perturbations techniques and Banach fixed point theorem:

dx_it dt A_i

x_it

_t

0

Gt−sxisds

f_i

t, x_it, _t

0

q_it, s, xisds

u_it onE_Nⁱ ,

x_i0 g_ixi x_0i ∈Eⁱ_N,

Δxitk I_kxitk, t /t_k, k1,2, . . . , m, i1,2, . . . , n,

1.1

where Ai : 0, T → Eⁱ_N is fuzzy coefficient, Eⁱ_N is the set of all upper semicontinuously convex fuzzy numbers onRwithEⁱ_N/E^j_N i /j,f_i:0, T×Eⁱ_N×Eⁱ_N → Eⁱ_Nandq_i:0, T× 0, T×Eⁱ_N are nonlinear regular fuzzy functions,g_i : Eⁱ_N → E_Nⁱ is a nonlinear continuous function,Gtis ann×ncontinuous matrix such thatdGtxi/dtis continuous forxi ∈Eⁱ_N andt∈ 0, TwithGt ≤ k,k > 0,u_i :0, T → Eⁱ_Nis a control function,x_0i ∈ Eⁱ_Nis an initial value andI_k∈CEⁱ_N, Eⁱ_Nare bounded functions,Δxitk x_it_k−x_it⁻_k, wherex_it⁻_k andxit_krepresent the left and right limits ofxitatttk, respectively.

2. Preliminaries

A fuzzy setuofRⁿ is a functionu :Rⁿ → 0,1. For each fuzzy setu, we denote byu^α {x∈Rⁿ:ux≥α}for anyα∈0,1itsα-level set.

Letu, vbe fuzzy sets ofRⁿ. It is well known thatu^α v^αfor eachα∈0,1implies uv.

Let Eⁿ denote the collection of all fuzzy sets of Rⁿ that satisfies the following conditions:

1uis normal, that is, there exists anx0∈Rⁿsuch thatuxo 1;

2uis fuzzy convex, that is, uλx 1−λy ≥ min{ux, uy}for anyx, y ∈ Rⁿ, 0≤λ≤1;

3uxis upper semicontinuous, that is,ux0 ≥ lim_{k→ ∞}uxkfor anyx_k ∈ Rⁿk 0,1,2, . . .,x_k → x₀;

4 u⁰is compact.

We callu∈Eⁿann-dimension fuzzy number.

Wang et al. 15 defined n-dimensional fuzzy vector space and investigated its properties.

For anyu_i ∈ E,i1,2, . . . , n, we call the ordered one-dimension fuzzy number class u1, u2, . . . , uni.e., the Cartesian product of one-dimension fuzzy numberu1, u2, . . . , unann- dimension fuzzy vector, denote it asu1, u₂, . . . , u_n, and call the collection of alln-dimension fuzzy vectorsi.e., the Cartesian product

E×E× · · · ×En-dimensional fuzzy vector space, and denote it asEⁿ.

Definition 2.1see15. Ifu∈Eⁿ, andu^αis a hyperrectangle, that is,u^αcan be represented by_n

i1u^α_il, u^α_ir, that is,u^α_1l, u^α_1r×u^α_2l, u^α_2r×· · ·×u^α_nl, u^α_nrfor everyα∈0,1, whereu^α_il, u^α_ir∈R withu^α_il ≤u^α_irwhenα∈0,1,i1,2, . . . , n, then we callua fuzzyn-cell number. We denote the collection of all fuzzyn-cell numbers byLEⁿ.

Theorem 2.2see15. For anyu ∈LEⁿwithu^α _n

i1u^α_il, u^α_irα∈0,1, there exists a uniqueu1, u2, . . . , un∈Eⁿsuch thatui^α u^α_il, u^α_ir(i1,2, . . . , nandα∈0,1). Conversely, for anyu1, u₂, . . . , u_n ∈Eⁿwithui^α u^α_il, u^α_ir(i 1,2, . . . , nandα∈0,1), there exists a uniqueu∈LEⁿsuch thatu^α_n

i1u^α_il, u^α_irα∈0,1.

Note 1see15. Theorem2.2indicates that fuzzy n-cell numbers andn-dimension fuzzy vectors can represent each other, so LEⁿ and Eⁿ may be regarded as identity. If u1, u2, . . . , un ∈ Eⁿ is the uniquen-dimension fuzzy vector determined by u ∈ LEⁿ, then we denoteu u1, u2, . . . , un.

LetE_N^{i n}E¹_N×E²_N× · · · ×Eⁿ_N, whereEⁱ_N i1,2, . . . , nis a fuzzy subset ofR. Then Eⁱ_Nⁿ⊆Eⁿ.

Definition 2.3see15. The complete metricDLonEⁱ_Nⁿis defined by D_Lu, v sup

0<α≤1

d_L

u^α,v^α sup

0<α≤1

max1≤i≤nu^α_il−v^α_il,u^α_ir−v_ir^α 2.1

for anyu, v∈Eⁱ_Nⁿ, which satisfiesd_Lu w, v w d_Lu, v.

Definition 2.4. Letu, v∈C0, T:Eⁱ_Nⁿ, H1u, v sup

0≤t≤TDLut, vt. 2.2

Definition 2.5see15. The derivativextof a fuzzy processx∈E_N^{i n}is defined by xtαⁿ

i1

x^α_il t,

x^α_ir t

2.3

provided that equation defines a fuzzyxt∈Eⁱ_Nⁿ. Definition 2.6see15. The fuzzy integral_a

bxtdt,a, b∈0, Tis defined by _a

b

xtdt α

ⁿ

i1

_a

b

x^α_iltdt, _a

b

x^α_irtdt

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 for1.1 u≡0.

We define

A A1, A2, . . . , An, x x₁, x₂, . . . , x_n,

f

f1, f2, . . . , fn

,

q

q₁, q₂, . . . , q_n , u u₁, u₂, . . . , u_n,

g

g1, g2, . . . , gn

,

3.1

x0 x01, x02, . . . , x0n. 3.2 Then

A, x, f, q, x0, u, g∈ Eⁱ_Nn

. 3.3

Instead of1.1, we consider the following fuzzy integrodifferential equations inEⁱ_Nⁿ dxt

dt A

xt

_t

0

Gt−sxsds

f

t, xt, _t

0

qt, s, xsds

ut on

Eⁱ_Nn , 3.4 x0 gx x0∈

E_Nⁱ _n

, 3.5

Δxtk Ikxtk, t /tk, k1,2, . . . , m, i1,2, . . . , n, 3.6

with fuzzy coefficientA:0, T → Eⁱ_Nⁿ, initial valuex0 ∈E_N^{i n}, andu:0, T → Eⁱ_Nⁿ being a control function. Given nonlinear regular fuzzy functionsf :0, T×E_N^{i n}×Eⁱ_Nⁿ → Eⁱ_Nⁿandq:0, T×0, T×Eⁱ_Nⁿ → Eⁱ_Nⁿsatisfy global Lipschitz conditions, that is, there exist finite constantsk1, k2, M >0 such that

dL

f

s, ξ1s, η1s_α ,

f

s, ξ2s, η2s_α

≤k1dL

ξ1s^α,ξ2s^α k2dL

η1s_α ,

η2s_α

, 3.7

d_L q

t, s, ϕ₁sα

, q

t, s, ϕ₂sα

≤Md_L ϕ₁sα

,

ϕ₂sα

3.8

for all ξ_js, ηjs, ϕjs ∈ Eⁱ_Nⁿj 1,2, the nonlinear functiong : Eⁱ_Nⁿ → E_N^{i n} is continuous and satisfies the Lipschitz condition

dL

gx·_α ,

g

y·_α

≤hdL

x·^α, y·_α

3.9

for allx·, y·∈E_N^{i n},his a finite positive constant.

Definition 3.1. The fuzzy processx:I 0, T → Eⁱ_Nⁿwithα-level setxt^α Πⁿ_i1xi^α Πⁿ_i1x^α_il, x^α_iris a fuzzy solution of3.4and3.5without nonhomogeneous term if and only if

x^α_il

t min

A^α_ijt

x^α_ikt _t

0

Gt−sx^α_iksds

:j, kl, r

, x^α_ir

t max

A^α_ijt

x^α_ikt _t

0

Gt−sx^α_iksds

:j, kl, r

,

x^α_il0 g_il^α x^α_il

x^α₀

il, x^α_ir0 g^α_ir x_ir^α

x^α₀

ir, i1,2, . . . , n.

3.10

For the sequel, we need the following assumption:

H1St is a fuzzy number satisfying, for y ∈ E_N^{i n}, d/dtSty ∈ C¹I : Eⁱ_Nⁿ

CI:Eⁱ_Nⁿ, the equation

d

dtStyA

Sty _t

0

Gt−sSsy ds

, t∈I, 3.11

where

St^αn

i1

Sit^αn

i1

S^α_ilt, S^α_irt

, S0 I 3.12

andS^α_ijtjl, ris continuous with|S^α_ijt| ≤c,c >0, for allt∈I 0, T.

In order to define the solution of3.4–3.6, we will consider the spaceΩi{xi:J → Eⁱ_N:xi_k ∈CJk, Eⁱ_N, Jk tk, t_{k 1}, k0,1, . . . , m,and there existxit⁻_kandxit_k k 1,2, . . . , m,withxit⁻_k xitk}, i1,2, . . . , n.

LetΩ Πⁿ_i1Ω_i, Ω_i Ωi

C0, T:Eⁱ_N, i1,2, . . . , n.

Lemma 3.2. Ifxis an integral solution of 3.4–3.6 u≡0, thenxis given by

xt St

x0−gx _t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds

0<tk<t

St−tkIk

x t⁻_k

, fort∈J.

3.13

Proof. Letxbe a solution of3.4–3.6. Defineωs St−sxs. Then we have that

dωs

ds −dSt−s

ds xs St−sdxs ds −A

St−sxs _t

0

Gt−sSsxsds

St−sdxs ds St−sf

s, xs,

_s

0

qs, τ, xτdτ

.

3.14

Considert_k< t, k1,2, . . . , m. Then integrating the previous equation, we have _t

0

dωs ds ds

_t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds. 3.15

Fork1,

ωt−ω0

_t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds 3.16

or

xt St

x₀−gx _t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds. 3.17

Now fork2, . . . , m,we have that _t1

0

dωs ds ds

_t2

t1

dωs ds ds · · ·

_t

tk

dωs ds ds

_t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds.

3.18

Then

ω t⁻₁

−ω0 ω t⁻₂

−ω t₁

· · · −ω t_k

ωt

_t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds

3.19

if and only if

ωt ω0

_t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds

0<tk<t

ω t_k

−ω t⁻_k

. 3.20

Hence

xt St

x₀−gx _t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds

0<tk<t

St−t_kIk

x t⁻_k

, 3.21

which proves the lemma.

Assume the following:

H2there existsd >0 such that

dL

Ik

x t⁻_kα

, Ik

y t⁻_kα

≤ddL

xt^α, yt_α

, 3.22

wherext, yt∈Ω; H3

c

T

!

h1 c cd k₁

1 cT 2

k₂MT

12 cT 3

"

h d

<1. 3.23

Theorem 3.3. LetT >0. If hypotheses (H1)–(H3) are hold, then, for everyx0 ∈Eⁱ_Nⁿ,3.13has a unique fuzzy solutionx∈Ω.

Proof. For eachxt∈Ω andt∈0, T, defineG0xt∈Ω by

G0xt St

x₀−gx _t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds

0<tk<t

St−t_kIk

x t⁻_k

.

3.24

Thus,G₀x:0, T → Ω is continuous, soG₀is a mapping fromΩ into itself. By Definitions 2.3and2.4, some properties ofd_Land inequalities3.7,3.8, and3.9, we have the following inequalities. Forx, y∈Ω,

dL

G0xt^α, G0y

t_α

≤dL

St

x0−gx _t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds _α

,

St

x₀−g

y _t

0

St−sf

s, ys, _s

0

q

s, τ, yτ dτ

ds

_α

d_L

0<tk<t

St−t_kIk

x t⁻_kα

,

0<tk<t

St−t_kIk

y

t⁻_kα

≤d_L

Stgx_α ,

Stg y_{α t}

0

dL

St−sf

s, xs, _s

0

qs, τ, xτdτ α

,

St−sf

s, ys, _s

0

q

s, τ, yτ dτ

α ds

d_L

0<tk<t

St−t_kIk

x t⁻_kα

,

0<tk<t

St−t_kIk

y

t⁻_kα

≤chdL

x·^α, y·_α

c _t

0

k₁d_L

xs^α, ysα

k₂M _s

0

d_L

xτ^α,

yτα dτ

ds

cddL

xt^α, yt_α

.

3.25

Therefore

DL

G0xt, G0y

t sup

0<α≤1dL

G0xt^α, G0y

t_α

≤chsup

0<α≤1dL

x·^α, y·_α

c _t

0

k₁sup

0<α≤1d_L

xs^α, ysα

k₂M _s

0

sup

0<α≤1d_L

xτ^α, yτα

dτ

ds

cdsup

0<α≤1dL

xt^α, yt_α

≤chD_L

x·, y·

c _t

0

k₁D_L

xs, ys k₂M

_s

0

D_L

xτ, yτ dτ

ds cdD_L

xt, yt .

3.26

Hence

H₁

G₀x, G₀y sup

0≤t≤TD_L

G0xt, G₀y

t

≤chsup

0≤t≤TDL

x·, y·

csup

0≤t≤T

_t

0

k1DL

xs, ys k2M

_s

0

DL

xτ, yτ dτ

ds

cdsup

0≤t≤TDL

xt, yt

≤c

h d

k₁ k₂MT 2

T

H₁

x, y .

3.27

By hypothesis H3, G₀ is a contraction mapping. Using the Banach fixed point theorem, 3.13has a unique fixed pointx∈Ω.

4. Nonlocal Controllability

In this section, we show the nonlocal controllability for the control system1.1.

The control system1.1is related to the following fuzzy integral system:

xt St

x₀−gx _t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds _t

0

St−susds

0<tk<t

St−t_kIk

x t⁻_k

.

4.1

Definition 4.1. Equations1.1–3are nonlocal controllable. Then there existsutsuch that the fuzzy solutionxtfor4.1asxT x¹−gxi.e.,xT^α x¹−gx^α, wherex¹ ∈ Eⁱ_Nⁿ, is target set.

Define the fuzzy mappingβ#:PR# ⁿ → E_N^{i n}by

β#^αv

⎧⎪

⎨

⎪⎩ _T

0

S^αT−svsds, v⊂Γu,

0, otherwise,

4.2

whereΓuis closed support ofu. Then there exists

β#i:PR# −→Eⁱ_N i1,2, . . . , n 4.3

such that

β#_i^αvi

⎧⎪

⎨

⎪⎩ _T

0

S^α_iT−svisds, vis⊂Γui

0, otherwise.

4.4

Then there existsβ#^α_ijj l, rsuch that

β#_il^αvil

_T

0

S^α_ilT−svilsds, v_ils∈

u^α_ils, u¹_i ,

β#^α_irvir

_T

0

S^α_irT−svirsds, virs∈

u¹_i, u^α_irs .

4.5

We assume thatβ#_il^α,β#^α_irare bijective mappings.

We can introduceα-level set ofusof4.1:

us^αn

i1

uis^αn

i1

u^α_ils, u^α_irs

ⁿ

i1

β#_il^α₋₁ x¹_α

il−g_il^α x^α_il

−S^α_ilT x^α₀

il−g_il^α x_il^α

− _T

0

S^α_ilT−sf_il^α

s, x_il^αs, _s

0

q^α_il

s, τ, x^α_ilτ dτ

ds

−

0<tk<T

S^α_ilT−t_kI_k^α_il x_il^α

t⁻_k , β#^α_ir−1

x¹_α

ir−g_ir^α x^α_ir

−S^α_irT x^α₀

ir −g_ir^α x^α_ir

− _T

0

S^α_irT−sf_ir^α

s, x^α_irs, _s

0

q^α_ir

s, τ, x^α_irτ dτ

ds

−

0<tk<T

S^α_irT−tkI_k^α_ir x_ir^α

t⁻_k .

4.6

Then substituting this expression into4.1yieldsα-level ofxT. For eachi1,2, . . . , n,

xiT^α

S^α_ilT

x^α_0il−g_il^α

x^α_{il T}

0

S^α_ilT−s

×f_il^α

s, x^α_ils, _s

0

q^α_il

s, τ, x^α_ilτ dτ

ds

0<tk<T

S^α_ilT−t_kI_k^α_il x^α_il

t⁻_{k T}

0

S^α_ilT−s

β#^α_il−1 x¹_α

il−g_il^α x_il^α

−S^α_ilT

x^α_0il−g_il^α x^α_il

− _T

0

S^α_ilT−sf_il^α

s, x^α_ils, _s

0

q^α_il

s, τ, x_il^ατ dτ

ds

−

0<tk<T

S^α_ilT−t_kI_k^α_il x^α_il

t⁻_k ds,

S^α_irT

x^α_0ir −g_ir^α

x^α_{ir T}

0

S^α_irT−s×f_ir^α

s, x^α_irs, _s

0

q^α_il

s, τ, x^α_irτ dτ

ds

0<tk<T

S^α_irT−t_kI_k^α_ir x^α_ir

t⁻_{k T}

0

S^α_irT−s β#^α_ir−1

×

x¹α ir−g_ir^α

x^α_ir

−S^α_irT x^α₀

ir −g_ir^α x^α_ir

− _T

0

S^α_irT−sf_ir^α

s, x^α_irs, _s

0

q^α_il

s, τ, x^α_irτ dτ

ds

−

0<tk<T

S^α_irT−tkI_k^α_ir x_ir^α

t⁻_k ds

x¹−gx_α

il,

x¹−gx_α

ir

x¹−gx

i

_α .

4.7 Therefore

xT^αn

i1

xiT^αn

i1

x¹−gx

i

α

x¹−gxα

. 4.8

We now set

Φxt St

x₀−gx _t

0

St−sf

s, xs, _t

0

qs, τ, xτdτ

ds

0<tk<t

St−t_kIk

x t⁻_{k t}

0

St−sβ#⁻¹

x¹−gx−ST

x0−gx

− _T

0

ST−sf

s, xs, _s

0

qs, τ, xτdτ

ds

−

0<tk<T

ST−t_kIk

x t⁻_k

ds,

4.9

where the fuzzy mappingβ#⁻¹satisfies the previous statements.

Notice thatΦxT x¹−gx, which means that the controlutsteers4.9from the origin tox¹−gxin timeTprovided we can obtain a fixed point of the operatorΦ.

H4Assume that the linear system of4.9 f≡0is controllable.

Theorem 4.2. Suppose that hypotheses (H1)–(H4) are satisfied. Then4.9is nonlocal controllable.

Proof. We can easily check thatΦis continuous function fromΩ to itself. By Definitions2.3 and 2.4, some properties ofd_L, and inequalities 3.7,3.8, and 3.9, we have following inequalities. For anyx, y∈Ω,

d_L

Φxt^α,

Φyt_α dL

St

x0−gx _t

0

St−sf

s, xs, _s

0

qs, τ, xτdτ

ds

0<tk<t

St−tkIk

x t⁻_{k t}

0

St−sβ#⁻¹

x¹−gx−ST

x0−gx

− _T

0

ST−sf

s, xs, _s

0

qs, τ, xτdτ

ds

−

0<tk<T

ST−t_kIk

x t⁻_k

ds α

,

St

x0−g

y _t

0

St−sf

s, ys, _s

0

q

s, τ, yτ dτ

ds

0<tk<t

St−t_kIk

y t⁻_{k t}

0

St−sβ#⁻¹

x¹−g y

−ST x₀−g

y

− _T

0

ST−sf

s, ys, _s

0

q

s, τ, yτ dτ

ds

−

0<tk<T

ST−t_kIk

y t⁻_k

ds _α

≤chd_L

x·^α, y·α

c _t

0

k1dL

xs^α, ys_α

k2M _s

0

dL

xτ^α,

yτ_α dτ

ds cdd_L

xt^α, yt_α

c _t

0

hdL

x·^α, y·_α

chdL

x·^α, y·_α

c _T

0

k₁d_L

xs^α, ysα

k₂M _s

0

d_L

xτ^α, yτα

dτ

ds cddL

xt^α,

ytα ds.

4.10

Therefore

DL

Φxt,Φyt sup

0<α≤1dL

Φxt^α,

Φyt_α

≤chsup

0<α≤1d_L

x·^α, y·_α

cdsup

0<α≤1d_L

xt^α, yt_α

c _t

0

k₁sup

0<α≤1d_L

xs^α, ysα

k2M _s

0

sup

0<α≤1d_L

xτ^α, yτα

dτ

ds

c _t

0

h1 csup

0<α≤1d_L

x·^α, y·α

c _T

0

k₁sup

0<α≤1d_L

xs^α, ysα

k2M _s

0

sup

0<α≤1d_L

xτ^α, yτα

dτ

ds

cdsup

0<α≤1d_L

xs^α,

ysα ds

≤chD_L

x·, y·

cdD_L

xt, yt

c _t

0

k₁D_L

xs, ys k₂M

_s

0

D_L

xτ, yτ dτ

ds

c _t

0

h1 cDL

x·, y·

c _T

0

k1DL

xs, ys k2M

_s

0

DL

xτ, yτ dτ

ds cdDL

xs, ys

ds. 4.11

Hence

H1

Φx,Φy sup

0≤t≤TDL

Φxt,Φyt

≤chsup

0≤t≤TD_L

x·, y·

cdsup

0≤t≤TD_L

xt, yt

csup

0≤t≤T

_t

0

k1DL

x·, y·

k2M _s

0

DL

xτ, yτ dτ

ds

csup

0≤t≤T

_t

0

h1 cDL

x·, y·

c _T

0

k₁D_L

xs, ys k₂M

_s

0

D_L

xτ, yτ dτ

ds

cdDL

xs, ys ds

≤chH1

x, y

cdH1

x, y c

_T

0

k1H1

x, y k2M

_s

0

H1

x, y dτ

ds

c _T

0

h1 cH1

x, y

c _T

0

k₁H₁

x, y k₂M

_s

0

H₁ x, y

dτ

ds cdH₁ x, y

ds c

T

!

h1 c cd k1

1 cT

2

k2MT 1

2 cT

3

"

h d

H1

x, y . 4.12 By hypothesisH3,Φis a contraction mapping. Using the Banach fixed point theorem,4.9 has a unique fixed pointx∈Ω.

5. Example

Consider the two semilinear one-dimensional heat equations on a connected domain0,1 for material with memory onE_Nⁱ , i1,2,boundary condition

x_it,0 x_it,1 0, i1,2 5.1

and with initial conditions

xi0, zi

p

k1

ck_ixitk, zi x0izi, 5.2

wherex0izi∈E_Nⁱ ,

p

k1

ck_ix_itk, z_i g_ixi, i1,2. 5.3

Letx_it, zi, i1,2 be the internal energy and

f_it, xit, zi, _t

0

q_it,s, xis, zids#2tx_it, zi² _t

0

t−sxis, zids, i1,2 5.4