Extending Banach Space on Graph withConservation Laws and G -Flow Solutions of EquationsLinfan MAO(Chinese Academy of Mathematics and System Science, Beijing 100190, P.R.China)E-mail: maolinfan@163.com

Abstract: Let V be a Banach space over a field F . A G -flow is a graph

G embedded in a topological space S associated with an injective mappings L : uv L(uv ) V such that L(uv ) = L(v u ) for (u, v) X G holdingwith conservation laws

X

L (v u ) = 0 for v V

uNG (v)

G ,

where uv denotes the semi-arc of (u, v) X G , which is an abstract model,

also a mathematical object for things embedded in a topological space, or

matters happened in the world. The main purpose of this paper is to extend Banach spaces on topological graphs with operator actions and showall of these extensions are also Banach space with unique correspondence inelements on linear continuous functionals, which enables one to solve linearfunctional equations in such extended space, particularly, solve algebraic, differential or integral equations on a topological graph, i.e., find multi-spacesolutions for equations, for instance, the Einsteins gravitational equations. Ageneralization of some well-known results in classical mathematics, such asthose of the fundamental theorem in algebra, Hilbert and Schmidts result onintegral equations and the discussion on stability of such G -flow solutions withapplications to controlling of ecologically industrial systems can be also foundin this paper. All of these results in this paper establish the mathematicalfoundation for multi-spaces, i.e., mathematical combinatorics.

Key Words: Banach space, topological graph, conservation flow, topologicalgraph, differential flow, multi-space solution of equation, system control.

AMS(2010): 03A10,05C15,20A05, 34A26,35A01,51A05,51D20,53A35

1

1. IntroductionLet V be a Banach space over a field F . All graphs G , denoted by (V ( G ), X( G ))considered in this paper are strong-connected without loops. A topological graph Gis an embedding of an oriented graph G in a topological space C . All elements inV ( G ) or X( G ) are respectively called vertices or arcs of G .An arc e = (u, v) X( G ) can be divided into 2 semi-arcs, i.e., initial semi-arcuv and end semi-arc v u , such as those shown in Fig.1 following.

-

L(uv )

vu u

vu v

Fig.1 All these semi-arcs of a topological graph G are denoted by X 1 G .2LA vector labeling G on G is a 1 1 mapping L : G V such that L : uv LL(uv ) V for uv X 1 G , such as those shown in Fig.1. For all labelings G2on G , defineL1 L2 L1 +L2L LG +G =Gand G = G .Then, all these vector labelings on G naturally form a vector space.Particularly, a G -flow on G is such a labeling L : uv V for uv X 1 G hold with2

L (uv ) = L (v u ) and conservation lawsX

L (v u ) = 0

uNG (v)

for v V ( G ), where 0 is the zero-vector in V . For example, a conservation lawfor vertex v in Fig.2 is L(v u1 ) L(v u2 ) L(v u3 ) + L(v u4 ) + L(v u5 ) + L(v u6 ) = 0.

-

-L(vu ) u4

-

-L(vu ) u5

u1u1 L(v )u2u2 L(v )

4

5

v

-

-F (vu ) u6

u3u3 L(v )

6

Fig.2

2

LClearly, if V = Z and O = {1}, then the G -flow G is nothing else but the networkflow X( G ) Z on G .L L1 L2Let G , G , G be G -flows on a topological graph G and F a scalar.L1 L2LIt is clear that G + G and G are also G -flows, which implies that all0conservation G -flows on G also form a linear space over F with unit G underV0operations + and, denotedbyG,whereGissuchaG -flow with vector 0 on

uv for (u, v) X G , denoted by O if G is clear by the paragraph..The flow representation for graphs are first discussed in [5], and then applied to

differential operators in [6], which has shown its important role both in mathematicsand applied sciences. It should be noted that a conservation law naturally determines an autonomous systems in the world. We can also find G -flows by solvingconservation equationsX

L (v u ) = 0,

uNG (v)

vV

G .

Such a system of equations is non-solvable in general, only with G -flow solutionssuch as those discussions in references [10]-[19]. Thus we can also introduce G -flows

e Reby Smarandache multi-system ([21]-[22]). In fact, for any integer m 1 let ;

be a Smarandache multi-system consisting of m mathematical systemsh(1 ; Ri 1 ),L e e(2 ; R2 ), , (m ; Rm ), different two by two. A topological structure G ; R on

e Re is inherited by; hie Re = {1 , 2 , , m },V GL ; hiTe Re = {(i , j ) |i j 6= , 1 i 6= j m} with labelingE GL ;TL : i L (i ) = i and L : (i , j ) L (i , j ) = i jfor hintegersi 1 i 6= j m,Ti.e., a topological vertex-edge labeled graph. Clearly,e Re is a GL ;G -flow if i j = v V for integers 1 i, j m.The main purpose of this paper is to establish the theoretical foundation, i.e.,

extending Banach spaces, particularly, extended Hilbert spaces on topological graphswith operator actions and show all of these extensions are also Banach space withunique correspondence in elements on linear continuous functionals, which enablesone to solve linear functional equations in such extended space, particularly, solvealgebraic or differential equations on a topological graph, i.e., find multi-space solu3

tions for equations, such as those of algebraic equations, the Einstein gravitationalequations and integral equations with applications to controlling of ecologically industrial systems. All of these discussions provide new viewpoint for mathematicalelements, i.e., mathematical combinatorics.For terminologies and notations not mentioned in this section, we follow references [1] for functional analysis, [3] and [7] for topological graphs, [4] for linearspaces, [8]-[9], [21]-[22] for Smarandache multi-systems, [3], [20] and [23] for differential equations.2. G -Flow Spaces2.1 ExistenceDefinition 2.1 Let V be a Banach space. A family V of vectors v V is conservative if

X

v = 0,

vV

called a conservative family.

Let V be a Banach space over a field F with a basis {1 , 2 , , dimV }. Then,for v V there are scalars xv1 , xv2 , , xvdimV F such thatdimVX

v=

xvi i .

i=1

Consequently,X dimVX

xvi i =

vV i=1

implies that

dimVXi=1

X

X

vV

xvi

!

i = 0

xvi = 0

vV

for integers 1 i dimV .Conversely, if

X

xvi = 0, 1 i dimV ,

vV

define

i

v =

dimVXi=1

4

xvi i

and V = {vi , 1 i dimV }. Clearly,

P

v = 0, i.e., V is a family of conservation

vV

vectors. Whence, if denoted by xvi = (v, i ) for v V , we therefore get a conditionon families of conservation in V following.Theorem 2.2 Let V be a Banach space with a basis {1 , 2 , , dimV }. Then, avector family V V is conservation if and only ifX(v, i ) = 0vV

for integers 1 i dimV .For example, let V = {v1 , v2 , v3 , v4 } R3 withv1 = (1, 1, 1),

v2 = (1, 1, 1),

v3 = (1, 1, 1),

v4 = (1, 1, 1)

Then it is a conservation family of vectors in R3 .Clearly, a conservation flow consists of conservation families. The followingresult establishes its inverse.LTheorem 2.3 A G -flow G exists on G if and only if there are conservationfamilies L(v) in a Banach space V associated an index set V withL(v) = {L(v u ) V f or some u V }such that L(v u ) = L(uv ) andL(v)Proof Notice that

for v V

\

(L(u)) = L(v u ) or .

X

L(v u ) = 0

uNG (v)

G implies

L(v u ) =

X

L(v w ).

wNG (v)\{u}

Whence, if there is an index set V associated conservation families L(v) withL(v) = {L(v u ) V for some u V }5

Tfor v V such that L(v u ) = L(uv ) and L(v) (L(u)) = L(v u ) or , define atopological graph G by [V G = V and X G ={(v, u)|L(v u ) L(v)}vV

Lwith an orientation v u on its each arcs. Then, it is clear that G is a G -flowby definition.

LConversely, if G is a G -flow, let L(v) = {L(v u ) V for (v, u) X G }

G . Then, it is also clear that L(v), v V G are conservation families associated with an index set V = V G such that L(v, u) = L(u, v) andfor v V

L(v)

\

by definition.

L(v u ) if (v, u) X G (L(u)) = if (v, u) 6 X G

Theorems 2.2 and 2.3 enables one to get the following result.Corollary 2.4 There are always existing G -flows on a topological graph Gwith

weights v for v V , particularly, e i on e X G if X G V G +1. Proof Let e = (u, v) X G . By Theorems 2.2 and 2.3, for an integer1 i dimV , such a G -flow exists if and only if the system of linear equations X(v,u) = 0, v V G

uV G

is solvable. However, if X G V G + 1, such a system is indeed solvable

by theory of linear equations.

2.2 G -Flow SpacesDefine

L

G =

X

(u,v)X

6

kL(uv )k

G

L Vfor G G , where kL(uv )k is the norm of F (uv ) in V . Then

L

L L 0

(1) G 0 and G = 0 if and only if G = G = O.

L

L

(2) G = G for any scalar .

L

L L 1 L2 1 2

(3) G + G G + G because of

L

1 L2

G + G =X

(u,v)X

X

(u,v)Xv

G

kL1 (uv ) + L2 (uv )k

G

kL1 (u )k +

X

(u,v)X

L L 2 1

kL2 (u )k = G + G .v

G

VWhence, k k is a norm on linear space G .Furthermore, if V is an inner space with inner product h, i, defineD LEX 1 L2G ,G=hL1 (uv ), L2 (uv )i .

(u,v)X GThen we know thatD L LE (4) G , G=

D L LE vvhL(u),L(u)i0andG ,G= 0 if and only

(u,v)X GLif L(uv ) = 0 for (u, v) X( G ), i.e., G = O.D L1E D L2E L2 L1L1 L2 V(5) G , G= G ,Gfor G , G G because ofD

L1 L2G ,G

E

=

P

X

(u,v)X

=

X

(u,v)X

G

G

X

hL1 (uv ), L2 (uv )i =

(u,v)X

hL2 (uv ), L1 (uv )i

D

G

L2 L1= G ,G

L L1 L2 V(6) For G , G , G G , there isDEL1L2 LG+ G , GED L1ED L2 L L= G ,G + G ,G7

hL2 (uv ), L1 (uv )iE

because ofDL1G

+====

E D L1EL2 LL2 LG , G= G+G ,GXhL1 (uv ) + L2 (uv ), L(uv )i(u,v)X( G )XXhL1 (uv ), L(uv )i +hL2 (uv ), L(uv )i(u,v)X( G )(u,v)X( G )E D LED L 1 L 2 LG ,G + G ,GED L1ED L2 L L G ,G + G ,G .

VThus, G is an inner space also and as the usual, let

L rD L L E

G ,G

G =

LVfor G G . Then it is a normed space. Furthermore, we know the followingresult. VTheorem 2.5 For any topological graph G , G is a Banach space, and furthermore,Vif V is a Hilbert space, G is a Hilbert space also.VProof As shown in the previous, G is a linear normed space or inner space ifV is an inner space. We show nthat ito is also complete, i.e., any Cauchy sequence inVLnVG is converges. In fact, let Gbe a Cauchy sequence in G . Thus for anynumber > 0, there always exists an integer N() such that

L

n Lm

GG

0 such that

L

L L0 L0

T G T G

< if G G < ().16

The following result reveals the relation between conceptions of linear continuous with that of linear bounded.VVTheorem 3.4 An operator T : G G is linear continuous if and only if it isbounded.Proof If T is bounded, then

L

L L0

L

L0 L0

TGTG=TGGGG

L L0 Vfor an constant [0, ) and G , G G . Whence, if

L

L0

6 0,

G G < () with () = , =then there must be

L

L0

T G G

< ,

Vi.e., T is linear continuous on G . However, this is obvious for = 0.

n Ln oNow if T is linear continuous but unbounded, there exists a sequence GVin G such that

L

L n n

G n G .Let

1Ln

L n G .

n G

L 1

L n n

Then G = 0, i.e., T G

0 if n . However, by definitionn

Ln

L

G n

T

T G

=

Ln

n G

L

L n n

n G

T G

L L = 1,= n n

n G n G LnG =

a contradiction. Thus, T must be bounded.

The following result is a generalization of the representation theorem of FrechetVVand Riesz on linear continuous functionals, i.e., T : G C on G -flow space G ,where C is the complex field.17

VTheorem 3.5 Let T : G C be a linear continuous functional. Then there is aLb Vunique G G such that L L Lb T G= G ,GL Vfor G G .

VProof Define a closed subset of G by Ln LoV N (T) = G G T G=0

LVfor the linear continuous functional T. If N (T) = G , i.e., T G= 0 forL VLb G G , choose G = O. We then easily obtain the identity L L Lb T G= G ,G.VWhence, we assume that N (T) 6= G . In this case, there is an orthogonal decomposition

VG = N (T) N (T)

with N (T) 6= {O} and N (T) 6= {O}.L0L0Choose a G -flow G N (T) with G 6= O and define L L0 L0 LLG = T GG T GG

L Vfor G G . Calculation shows that L L L L L 0 0T G= T GT G T GT G= 0,Li.e., G N (T). We therefore get thatD LE L00 =G ,GD L L L L L E 0 0 0=T GG T GG ,G L D LE L D L L E 0 L0 0 0= T GG ,GT GG ,G.D LE L 2 0 L0 0

Notice that G , G= G 6= 0. We find that

L L*+ 0 0D L L0 E LTGTG LL0G ,G= G , .T G=

GL0 2L0 2

G

G 18

Let

L0 TGLbL0 L0G = G=G ,

L0 2

G

L 0 L L Lb T G = G ,G.where =

2 . We consequently get that T GL 0

G

LLb VL Lb Now if there is another G G such that T G= G ,Gfor

L VL Lb Lb G G , there must be G , G G= 0 by definition. Particularly, letL Lb Lb G = G G . We know that

b

LLb Lb Lb Lb Lb

= 0,

G G = G G , G G

Lb Lb Lb Lb which implies that G G = O, i.e., G = G .

3.2 Differential and Integral OperatorsLet V be Hilbert space consisting of measurable functions f (x1 , x2 , , xn ) on a set = {x = (x1 , x2 , , xn ) Rn |ai xi bi , 1 i n} ,i.e., the functional space L2 [], with inner productZhf (x) , g (x)i =f (x)g(x)dx for f (x), g(x) L2 []

Vand G its G -extension on a topological graph G . The differential operator andintegral operators

nX

D=

i=1

aixi

and

Von G are respectively defined by

Z

Z

,

L DL(uv )DG = Gand

Z

Z

LG =

L[y]K(x, y) G dy = G

R

K(x,y)L(uv )[y]dy

L[y]K(x, y) G dy = G

R

K(x,y)L(uv )[y]dy

LG =

Z

Z

19

,

ai C0 () for integers 1 i, j n and K(x, y) :for (u, v) X G , where ai ,xj C L2 ( , C) withZK(x, y)dxdy < .

Such integral operators are usually called adjoint for

Z

=

Z

by K (x, y) =

L1 L2 VK (x, y). Clearly, for G , G G and , F ,

L (uv )+L (uv ) 2L1 (uv )L2 (uv ) 1D(L1 (uv )+L2 (uv ))D G+ G=D G=GD(L1 (uv ))+D(L2 (uv )) D(L1 (uv )) D(L2 (uv ))=G=G+G

(L (uv ))

L (uv )

L2 (uv ) 1(L2 (uv )) 1=D G= D G+ D G+G

for (u, v) X G , i.e.,

L1L2L1L2D G + G= D G + D G .

Similarly, we know also thatZ ZZ

L1L2L1L2G + G=G +G ,Z ZZ

L1L2L1L2G + G=G +G .

Thus, operators D,

Z

and

Z

Vare al linear on G .

LFor example, let f (t) = t, g(t) = et , K(t, ) = t2 + 2 for = [0, 1] and let Gbe the G -flow shown on the left in Fig.6. Then, we know that Df = 1, Dg = et ,Z 1Z 1Z 1

t2 1K(t, )f ( )d =t2 + 2 d = + = a(t),K(t, )f ( )d =24000Z 1Z 1Z 1

K(t, )g( )d =K(t, )g( )d =t2 + 2 e d0

0

0

2

= (e 1)t + e 2 = b(t)

Land the actions D G ,

Z

LG and[0,1]

Z

LG are shown on the right in Fig.5.[0,1]

20

et-tet ?t=t e6 } t-t

et-1t= et ?11 e}66 -1

D

et

et

-et-tRR, [0,1]tt[0,1]t 6e}= e ?tt-

a(t)a(t)?a(t)6b(t) =} b(t)a(t)-b(t)

b(t)

et

Fig.5VFurthermore, we know that both of them are injections on G .ZVVVVTheorem 3.6 D : G G and: G G .

L VLProof For G G , we are needed to show that D G and

v

DL(u ) = 0 and

vNG (u)

hold with v V

L VG G ,

i.e., the conservation lawsX

Z

X Z

vNG (u)

L(uv ) = 0

L(uv ) VG . However, because of G G , there must beX

L(u ) = 0 for v V G ,v

vNG (u)

we immediately know that

0 = D

X

vNG (u)

and

0=

Z

for v V

G .

X

vNG (u)

L(uv ) =

L(u ) =v

21

X

DL(uv )

vNG (u)

X Z

vNG (u)

L(uv )

4. G -Flow Solutions of EquationsAs we mentioned, all G-solutions of non-solvable systems on algebraic, ordinary orpartial differential equations determined in [13]-[19] are in fact G -flows. We showthere are also G -flow solutions for solvable equations in this section.4.1 Linear EquationsLet V be a field (F ; +, ). We can further defineL1 L2 L1 L2G G =G with L1 L2 (u ) = L1 (u ) L2 (u ) for (u, v) X G . Then it can be verified F

Fc isomorphic to F if theeasily that G is also a field G ; +, with a subfield Fv

v

v

conservation laws is not emphasized, wheren L oFvcF = G G |L (u ) is constant in F for (u, v) X G.

FE G nE G F

if |F | = pn , where p is aClearly, G F. Thus G = pprime number. For this F -extension on G , the linear equation

LaX = G

a1 LFis uniquely solvable for X = Gin G if 0 6= a F . Particularly,if one views

Lan element b F as b = G if L(uv ) = b for (u, v) X G and 0 6= a F , thenan algebraic equation

ax = bFa1 Lin F also is an equation in G with a solution x = Gsuch as those shown in Fig.6 for G = C 4 , a = 3, b = 5 following.

-35

53

6

?53

53

Fig.622

Let [Lij ]mn be a matrix with entries Lij : uv V . Denoted by [Lij ]mn (uv )the matrix [Lij (uv )]mn . Then, a general result on G -flow solutions of linear systemsis known following.nTheorem 4.1 A linear system (LESm) of equationsL1 a11 X1 + a12 X2 + + a1n Xn = GL2 a X +a X ++a X = G21 122 22n n......................................Lm a X +a X ++a X = Gm1 1m2 2mn n

n(LESm)

VLi with aij C and G G for integers 1 i n and 1 j m is solvable forVXi G , 1 i m if and only ifvrank [aij ]mn = rank [aij ]+m(n+1) (u )

for (u, v) G , where

[aij ]+m(n+1)

a11

a12

a1n

L1

a21 a22 a2n L2= ... ... .. ... ...am1 am2 amn Lm

.

LxProof Let Xi = G i with Lxi (uv ) V on (u, v) X G for integers n1 i n. For (u, v) X G , the system (LESm) appears as a common linearsystem

a11 Lx1 (uv ) + a12 Lx2 (uv ) + + a1n Lxn (uv ) = L1 (uv ) a L (uv ) + a L (uv ) + + a L (uv ) = L (uv )21 x122 x22n xn2........................................................am1 Lx1 (uv ) + am2 Lx2 (uv ) + + amn Lxn (uv ) = Lm (uv )

By linear algebra, such a system is solvable if and only if ([4])vrank [aij ]mn = rank [aij ]+m(n+1) (u )

for (u, v) G .23

Labeling theuv respectively by solutions Lx1 (uv ), Lx2 (uv ), , Lxn (uv ) semi-arc

Lx LxLxnfor (u, v) X G , we get labeled graphs G 1 , G 2 , , G . We prove thatLx1 LxLxn VG , G 2, , GG .Let rank [aij ]mn = r. Similar to that of linear algebra, we are easily know thatmPLiX=c+ c1,r+1 Xjr+1 + c1n Xjnj11i Gi=1mLi Xj = P c2i G + c2,r+1 Xjr+1 + c2n Xjn2,i=1.........................................mPLicri G + cr,r+1 Xjr+1 + crn Xjn Xj r =i=1

LxLxVwhere {j1 , , jn } = {1, , n}. Whence, if G jr+1 , , G jn G , thenX

v

Lxk (u ) =

mX X

cki Li (uv )

vNG (u) i=1

vNG (u)

X

+

c2,r+1 Lxjr+1 (uv ) + +

vNG (u)

=

mXi=1

cki

+c2,r+1

X

vNG (u)

X

X

c2n Lxjn (uv )

vNG (u)

Li (uv )

Lxjr+1 (uv ) + + c2n

vNG (u)

X

Lxjn (uv ) = 0

vNG (u)

VnWhence, the system (LESm) is solvable in G .The following result is an immediate conclusion of Theorem 4.1.Corollary 4.2 A linear system of equationsa11 x1 + a12 x2 + + a1n xn = b1 a x +a x ++a x = b21 122 22n n2..................................am1 x1 + am2 x2 + + amn xn = bm

with aij , bj F for integers 1 i n, 1 j m holding withrank [aij ]mn = rank [aij ]+m(n+1)has G -flow solutions on infinitely many topological graphs G .24

Let the operator D and Rn be the same as in Subsection 3.2. We considerVdifferential equations in G following.VTheorem 4.3 For GL G , the Cauchy problem on differential equationDGX = GL0L0X|is uniquely solvable prescribed with G xn =xn = G . Proof For (u, v) X G , denoted by F (uv ) the flow on the semi-arc uv .

Then the differential equation DGX = GL transforms into a linear partial differential

equation

nX

ai

i=1

F (uv )= L (uv )xi

By assumption, ai C0 () and L (uv ) L2 [], which

on the semi-arc uv .

implies that there is a uniquely solution F (uv ) with initial value L0 (uv ) by thecharacteristic theory of partial differential equation of first order.

In fact, let

i (x1 , x2 , , xn , F ) , 1 i n be the n independent first integrals of its characteristic equations. Then

F (uv ) = F (uv ) L0 x1 , x2 , xn1 L2 [],

where, x1 , x2 , , xn1 and F are determined by system of equations1 (x1 , x2 , , xn1 , x0n , F ) = 1 (x , x , , x , x0 , F ) = 212n12n...............................n (x1 , x2 , , xn1 , x0n , F ) = n

Clearly,

D

X

vNG (u)

Notice that

F (uv ) =

X

DF (uv ) =

vNG (u)

X

v F (u )

vNG (u)

=

25

L(uv ) = 0.

vNG (u)

X

vNG (u)

xn =x0n

X

L0 (uv ) = 0.

We therefore know that

X

F (uv ) = 0.

vNG (u)

X F VThus, we get a uniquely solution G = G G for the equationDGX = GL0L0X|prescribed with initial data G xn =xn = G .

We know that the Cauchy problem on heat equationn

X 2uu= c2tx2ii=1is solvable in Rn R if u(x, t0 ) = (x) is continuous and bounded in Rn , c a non-zeroL Vconstant in R. For G G in Subsection 3.2, if we defineLG L= G tt

and

LG L= G xi , 1 i n,xi

Vthen we can also consider the Cauchy problem in G , i.e.,n

X 2XX= c2tx2ii=1with initial values X|t=t0 , and know the result following.L VTheorem 4.4 For G G and a non-zero constant c in R, the Cauchy problemson differential equations

n

X 2XX= c2tx2ii=1

L VVwith initial value X|t=t0 = G GissolvableinG if L (uv ) is continuous and

bounded in Rn for (u, v) X G . Proof For (u, v) X G , the Cauchy problem on the semi-arc uv appears asn

X 2uu= c2tx2ii=126

Fwith initial value u|t=0 = L (uv ) (x) if X = G . According to the theory of partialdifferential equations, we know thatZ +(x1 y1 )2 ++(xn yn )21v4tL (uv ) (y1 , , yn )dy1 dyn .eF (u ) (x, t) =n2(4t) Labeling the semi-arc uv by F (uv ) (x, t) for (u, v) X G , we get a labeledFF Vgraph G on G . We prove G G . L VBy assumption, G G , i.e., for u V G ,X

L (uv ) (x) = 0,

vNG (u)

we know thatX

F (uv ) (x, t)

vNG (u)

=

X

vNG

=

Z

1n(4t) 2(u)

1n(4t) 2

Z

+

+

e

(x1 y1 )2 ++(xn yn )24t

(x y )2 ++(xn yn )2 1 14t

e

Z

L (uv ) (y1 , , yn )dy1 dyn

X

vNG (u)

+

L (uv ) (y1 , , yn ) dy1 dyn

(x1 y1 )2 ++(xn yn )214te(0) dy1 dyn = 0n(4t) 2 F Vfor u V G . Therefore, G G and

=

n

X 2XX= c2tx2ii=1

L VVwith initial value X|t=t0 = G G is solvable in G .

Similarly, we can also get a result on Cauchy problem on 3-dimensional waveVequation in G following.L VTheorem 4.5 For G G and a non-zero constant c in R, the Cauchy problemson differential equations2X= c2t2

2X 2X 2X++x21x22x2327

L VVvwith initial value X|t=t0 = G G is solvable in G if L (u ) is continuous andbounded in Rn for (u, v) X G .For an integral kernel K(x, y), the two subspaces N , N

L2 [] are deter-

mined by=

2

(x) L []|

Z

K (x, y) (y)dy = (x) ,

Z2=(x) L []|K (x, y)(y)dy = (x) .

NN

Then we know the result following.VTheorem 4.6 For GL G , if dimN = 0, then the integral equationZXXG G = GL

Vis solvable in G with V = L2 [] if and only ifD L L E LG ,G= 0, G N . Proof For (u, v) X GZXXG G = GL and

v

D

EL LG ,G= 0,

LG N

on the semi-arc u respectively appear asZF (x) K (x, y) F (y) dy = L (uv ) [x]

v

if X (u ) = F (x) andZ LL (uv ) [x]L (uv ) [x]dx = 0 for G N .

Applying Hilbert and Schmidts theorem ([20]) on integral equation, we knowthe integral equationF (x)

Z

K (x, y) F (y) dy = L (uv ) [x]

2

is solvable in L [] if and only ifZL (uv ) [x]L (uv ) [x]dx = 0

28

Lfor G N . Thus, there are functions F (x) L2 [] hold for the integralequationF (x)

Z

K (x, y) F (y) dy = L (uv ) [x]

for (u, v) X G in this case. For u V G , it is clear thatX

v

F (u ) [x]

Z

vNG (u)

XL (uv ) [x] = 0,K(x, y)F (u ) [x] =v

vNG (u)

which implies that,Z

Thus,

K(x, y)

X

vNG (u)

F (uv ) [x] =

X

X

F (uv ) [x].

vNG (u)

F (uv ) [x] N .

vNG (u)

However, if dimN = 0, there must beX

F (uv ) [x] = 0

vNG (u)

for u V

F VG , i.e., G G . Whence, if dimN = 0, the integral equationXG

Z

XG = GL

Vis solvable in G with V = L2 [] if and only ifD L L E LG ,G= 0, G N .This completes the proof.

Theorem 4.7 Let the integral kernel K(x, y) : C L2 ( ) be givenwith

Z

|K(x, y)|2dxdy > 0,

dimN = 0 and K(x, y) = K(x, y)

29

for almostn allL (x,o y) . Then there is a finite or countably infinite system iG -flows G L2 (, C) with associate real numbers {i }i=1,2, R suchi=1,2,

that the integral equations

Z

Li [y]Li [x]K(x, y) Gdy = i G

hold with integers i = 1, 2, , and furthermore,|1 | |2 | 0

and

lim i = 0.

i

Proof Notice that the integral equationsZLi [y]Li [x]K(x, y) Gdy = i G

is appeared as

Z

K(x, y)Li (uv ) [y]dy = i Li (uv ) [x]

on (u, v) X G . By the spectral theorem of Hilbert and Schmidt ([20]), there

is indeed a finite or countably system of functions {Li (uv ) [x]}i=1,2, hold with this

integral equation, and furthermore,|1 | |2 | 0

with

lim i = 0.

i

Similar to the proof of Theorem 4.5, if dimN = 0, we know thatX

Li (uv ) [x] = 0

vNG (u)

for u V

Li VG , i.e., G G for integers i = 1, 2, .

4.2 Non-linear EquationsIf G is chosen with a special structure, we can get a general result on G -solutionsof equations, including non-linear equations following.Theorem 4.8 If the topological graph G can be decomposed into circuitsl

[G=Cii=1

30

such that L(u ) = Li (x) for (u, v) X C i , 1 i l and the Cauchy problemv

(

Fi (x, u, ux1 , , uxn , ux1 x2 , ) = 0u|x0 = Li (x)

is solvable in a Hilbert space V on domain Rn for integers 1 i l, then theCauchy problem

(

Fi (x, X, Xx1 , , Xxn , Xx1 x2 , ) = 0LX|x0 = G Vsuch that L (uv ) = Li (x) for (u, v) X C i is solvable for X G . Lu(x)vProof Let X = Gwith Lu(x) (u ) = u(x) for (u, v) X G . Notice thatthe Cauchy problem

(then appears as

Fi (x, X, Xx1 , , Xxn , Xx1 x2 , ) = 0X|x0 = GL

(

Fi (x, u, ux1 , , uxn , ux1 x2 , ) = 0

u|x0 = Li (x) on the semi-arc uv for (u, v) X G , which is solvable by assumption. Whence,there exists solution u (uv ) (x) holding with(Fi (x, u, ux1 , , uxn , ux1 x2 , ) = 0u|x0 = Li (x)

Lu(x)Let Gbe a labeling on G with u (uv ) (x) on uv for (u, v) X G . WeLu(x) Vshow that G G . Notice thatl

[G=Cii=1

and all flows on C i is the same, i.e., the solution u (uv ) (x). Clearly, it is holdenwith conservation on each vertex in C i for integers 1 i l. We therefore knowthat

X

Lx0 (uv ) = 0,

vNG (u)

31

uV

G .

Lu(x) VThus, G G . This completes the proof.

There are many interesting conclusions on G -flow solutions of equations byTheorem 4.8. For example, if Fi is nothing else but polynomials of degree n in onevariable x, we get a conclusion following, which generalizes the fundamental theoremin algebra.Corollary 4.9(Generalized Fundamental Theorem in Algebra) If G can be decomposed into circuitsl

[CiG=i=1

and Li (uv ) = ai C for (u, v) X C i and integers 1 i l, then thepolynomial

L1L2LnLn+1F (X) = G X n + G X n1 + + G X + GCL1always has roots, i.e., X0 G such that F (X0 ) = O if G 6= O and n 1.Particularly, an algebraic equationa1 xn + a2 xn1 + + an x + an+1 = 0Cwith a1 6= 0 has infinite many G -flow solutions in G on those topological graphsl [G with G =C i.i=1

Notice that Theorem 4.8 enables one to get G -flow solutions both on those

linear and non-linear equations in physics. For example, we know the sphericalsolution

rg 21ds2 = f (t) 1 dt dr 2 r 2 (d2 + sin2 d2 )r1 rrg

for the Einsteins gravitational equations ([9])

1R Rg = 8GT 2with R = R = g R , R = g R , G = 6.673108cm3 /gs2, = 8G/c4 =2.08 1048 cm1 g 1 s2. By Theorem 4.8, we get their G -flow solutions following.Corollary 4.10 The Einsteins gravitational equations1R Rg = 8GT ,232

Chas infinite many G -flow solutions in G , particularly on those topological graphsl [G=C i with spherical solutions of the equations on their arcs.i=1

For example, let G = C 4 . We are easily find C 4 -flow solution of Einsteins

gravitational equations,such as those shown in Fig.7.

-S1

v1

S4

v2

?S2

6y

v4

v3

S3

Fig.7where, each Si is a spherical solution

1rs 2dr 2 r 2 (d2 + sin2 d2 )ds2 = f (t) 1 dt r1 rrsof Einsteins gravitational equations for integers 1 i 4.As a by-product, Theorems 4.5-4.6 can be also generalized on those topologicalgraphs with circuit-decomposition following.Corollary 4.11 Let the integral kernel K(x, y) : C L2 ( ) be givenwith

Z

|K(x, y)|2 dxdy > 0,

K(x, y) = K(x, y)

for almost all (x, y) , andl

L [ G =Cii=1

such that L(uv ) = L[i] (x) for (u, v) X C i and integers 1 i l. Then, theintegral equation

V

is solvable in G

XG

Z

XG = GL

with V = L2 [] if and only ifD L L E LG ,G= 0, G N .33

Corollary 4.12 Let the integral kernel K(x, y) : C L2 ( ) be givenwith

Z

|K(x, y)|2 dxdy > 0,

K(x, y) = K(x, y)

for almost all (x, y) , andl

L [ G =Cii=1

such that L(u ) = L[i] (x) for (u, v) X C i and integers 1 i l. Then,n Lo ithere is a finite or countably infinite system G -flows G L2 (, C) withv

i=1,2,

associate real numbers {i }i=1,2, R such that the integral equationsZLi [y]Li [x]K(x, y) Gdy = i G

hold with integers i = 1, 2, , and furthermore,|1 | |2 | 0

and

lim i = 0.

i

5. Applications to System Control5.1 Stability of G -Flow SolutionsLu(x)Lu (x)Let X = Gand X2 = G 1 be respectively solutions ofF (x, Xx1 , , Xxn , Xx1 x2 , ) = 0LL1 Von the initial values X|x0 = G or X|x0 = G in G with V = L2 [], the Hilbertspace. The G -flow solution X is said to be stable if there exists a number () forany number > 0 such that

if

By definition,

L

u (x) Lu(x)

kX1 X2 k = G 1 G

0 there always is a number () (uv ) such thatku1 (uv ) (x) u (uv ) (x)k