fixed and so there are constants Kz• such that

IEj(z.

k" .

�)-Ej

:

k" .

�)I �

Kz(E; )Ilz - z*lIlX.

}

1F;(

t.

z. �k '

Cr) - F;(t.

z: �k ' Cr)1 � Kz (F;)lIz - z*lIcx . (4.3.14) For the purposes of our existence proof. we firstly look at the monotone system

'�n . Bn

where we assume our coupling functionsJi and

Fj

are monotone nondecreasing in Cj and

Cj.

respectively for all }.

4.3.1 Upper and Lower Solutions and Monotone Iteration

We shall now introduce the concepts of upper and lower solutions relative to the monotone system

_{Sn. Bn .}

Definition 4.3.1 .

Assume that

(i) For components

i

E I. where Di > O.

fj

and

Cj

are continuous functions in

.Q

x A with continuous

first order Xj derivatives in

.Q

x A and continuous second order Xj derivatives in

.Q

x A ;

(ii) For components

i

E I. where

Dj

= II i = O. fi and C, arc continuous functions in £2 x A ;

(iii) For components

_i

E 1. where

�j

> O. C and

Cj

are continuous functions in if. with continuous first order Zj derivatives in if and continuous second order Zj derivatives in A;

(iv) For components

i

E 1. where

�i

= O.

u ·

'\lCi $0. �i and Ci are continuous functions in A . with continuous first order Zj derivatives in A;

4.3 _{EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM} 1 05

The ordered pair of functions

(fi' �i)

and

(ci' Ci)

with

fi

�

Ci

on n

x A

and

�i

�

Ci

on If arc said to be

lower

and

upper solutions

Sn' Bn

respectively, if Ihey satisfy:

and -D

iV;f

i � h(x, c;.;) in

,axA,

dC·

-=:!.. � 0 _on

d,alXA,

dn

d

·

D -=:!.. < H (C· - c· ) _on

d,a2XA

an

- ' -' -I '

-!llV2 c.+u· VC.+ H.

_I _{_ I} _ _I I

J

aDz

(

C

.- c-I -

)

� F( I z, (;:J') _ in

A,

-D;

V; Ci

� h (x, Cj) in

.(lxA,

de·

� 0 _on

d,alxA,

dc·

- _ D·_I-

dn

' _{- I}_>

I-f.(C.

-c·

·)

_on

d,a2XA

VC+ l-f.J

(E -c »

F(z

C )

A

I I I _aD2 I I - I ' J '

-

dE

VI C·+

9).

I __ , >

VIC.

I on

dAI

dnl

- I. '

dC·

__ , � o on

dAa,

a = 2, 3,

dna

respectively.

We nOLe from the counterexample at !he end of section 3 . 1 , !hat comparison !heorems analogous to Theorems 3 .2. 1 1 and 3.2. 1 2 do not hold in general in the case of the corresponding steady state or time independelll problem

Sn' Bn .

Hence, if there exist

(fi' �i)

and

(ci' C;)

which are

lower

and

upper

solutions of the steady stale problem

Sn' Bn

and

(Ci, Ci)

is a solution of

Sn' Bn ,

then in contrast to the unsteady state problem

Sn, Bn,

we cannot assert that

fi

�

ci

�

Ci

and

�i

�

Ci

�

Ci .

However, the method of monotone iteration is still applicable and shows !he existence of at least one solution

(Ci, Ci)

Sn' Bn

lying between

(fi' �i)

and

(ci' Ci) ·

Lower and upper solutions may not always exist for elliptic equations. Therefore, as a result of this, certain unstable solutions cannot be obtained by monotone iteration (PARTER [230] , KELLER and COHEN [ 1 39], AMANN [9]). However, it must be noted that as with parabolic equations (PAO [222]), there are geometric conditions which the nonlinear reaction functions f; and

Fi

may satisfy which guarantee the existence of either lower or upper solutions for elliptic equations (AMANN [9]). These lower and upper solutions may not necessarily exist simultaneously.

As for the system

Sn,

Bn, it is important to note that the lower and upper solutions provide lower and upper bounds for solutions of

,�'" Bn

which can be improved by monotone iteralive procedures.

4.3 _{EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM} ₁₀₆

In order to establish an existence theorem for

Sn' Bn

in terms of upper and lower solutions, we define a transformation !!i, by

and consider the sequences

{(c�k), Cfk» }

where

c�k)

is obtained from the linear system

and

Cj(k)

is obtained from the linear system

C(k)

- C

a

,

' + VI ' I on

'"

ac(k)

-a '

= 0 on

aAa,

a=2, 3, na 'th <

(k-I)

< - _d

C

C(k-I)

C-

_{f' k}_- ₁ WI £j _ Cj _{_j _ j} _ or - , .... (4.6.15) (4.3.16) (4.3.17) (4.3.18) (4.3.19) (4.3.20) (4.3.21)

For each k, the system (4.3.16) consists of

n(J)

linear, completely uncoupled boundary value problems with boundary conditions given by (4.3. 1 7)-(4.3. 1 8) and this system is uncoupled from the system (4.3. 1 9) which also consists of

n(J)

linear, completely uncoupled boundary valuc problems with boundary conditions given by (4.3.20)-(4.3.21).

Since c

fk)

(x, z) is not differentiated with respect to z in (4.3 . 1 6),

A

may be considered to be a parameter space in (4.3 . 1 6)-(4.3. 18). For functions

c[k)(x,

z) where

Dj

Hj =

0, n may also be considered to be a paramctcr spacc and for functions

C[k)(z),

where

9)j =

0, U ·

vcfk-1)

== 0, wc may similarly u'cat

A

as a parameter space. The cxistence and uniqucness of sequences

{(c?), Cj(k» }

may therefore fol low from solving standard scalar systems of linear elliptic equations (LADYSHENKA YA [ 154] or GILBARG and TRUDINGER [ 1 06]) which may or may not depend on parameters, systems of first order partial differential equations which may depend on parameters (LAKSIIMI KANTHAM et

al. I I GO]) and systcms of algebraic

equations in many variables.

The nonlinear algebraic equations h (x, Cj )

=

0 obtained when

Dj

Hj

°

may be expressed as C

j

= fi

(x,

C j)

+ Cj

in order to perfonn functional iterations to find

Cj

in terms of

Cj.

We shall develop a general theory for a broad class of monotone iterations involving such algebraic equations. This class of iterations includes Newton's method as wcll as a family of mCI,hods, which arc Ncwton-Gauss-Seidel processes

(ORTEGA and R I IElNBOLT 1 206, 207J, LADDE et

al.

1 1 53, p. 36 1). It is the Lipschitz propcrty that may be used instead of differentiability in many other iterative methods. Note that fi (x, Cj )

+ Cj

satisfy the Lipschitz and HOlder continuity properties that are assumed on our original fi (x,

Cj)

as well as the monotonicity in Cj (see Lemma 4. 1 . 1). We may therefore rewrite (4.3 . 1 6) as

Cfk)=

h (X,

C;k-l» +cfk-1)

when D, =

Hj

O .

The existence and uniqueness of sequences

( c?) }

follows from the uniquely dcfined solutions to algebraic equations.

4.3 _{EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM} _{1 07}

The rest of these theorems will require Holder continuity properties on the functions _!;(x,

_C)_k-I»

and F (z _I

_,

_J _I

C�k-I)

_I which are satisfied if either

_C(k-I)

_J E

Cl+a,a [{),

A Rn(l)] C(k-I)

_' _' _J E

Ca,a

[{),

A Rn(l)]

_, _{, J} E

CI+a[A Rn(J)]

' _or

C

J ₍

k-

)

Ca[A Rn(J)]

, .

These theorems will also require Holder continuity properLies in the boundary conditions and so we make the following assumptions on

Ci,l'

We will assume that

dQ

and

dA

belong to class

C2+a.

(H' ) ₃

_C.

_1,1E

CI+a[A Rn(J)]

_' •

The velocity distribution vector function u(z) is also required to satisfy the following Holder continuity property

where

_UI(Z)

is chosen without loss of generality to be the first component that is nonzero. For components i E J, where

9Ji

= 0,

u·VCi

¥'O, we shall also need the following additional assumptions

(Hs) Assume that

_A

A I

A

n-I , where

A I

E R.

(i) For each

(ZIO,

zo) E

A

I X

A n-l ,

there exists a unique solution

Z(ZI, ZIO,

zo) of

dz �z) -

- == -- ,

Z(ZIO)

= zo, on

A I,

dZI

ul

(z)

where

_ZI

correspondes to the nonzero component UI(Z);

(ii)

Z(ZIo ZIO, zo)

is continuously differentiable with respect to

(ZIO, zo);

(iii) The relationship

holds.

(H6) Assume that

A I

is the interval [a, b]

(i) For each

Zo

An-I

and

YiO

Rn(J),

there exists a unique solution

Yi(Z\,

YiO;

o

) of

F( (

) C(k-I» H

sYY. H

J

(k-I)(

( »

}

dJi

== I

Z), Z z), zlO, zo '

- I I + I

aaz ci z

lo

Z Zio zlO, Zo ,

dZ)

UI(z)

Yi(

a) =

YIO,

AI,

where

Z(ZI, ZIO,

o

) is the unique solution of (4.3.22);

(ii)

Yi(ZI,

YiO;

o

) is cont.inuously differentiable with respect to

(YiO,

zo).

(4.3.22)

(4.3.23)

4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 108

Note that assumptions (H3)-(H6) will hold in either our original system

5n, Bn

or the monotone system

S2,,' B2n

and (Hf;) can be shown to hold in the monotone system

52", B2"

if it holds in our original system

Sn' Bn .

In document A mathematical analysis of reaction diffusion systems in chemical and biological reactors with macro and micro structures :|ba thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy at Massey University (Page 112-116)