G -Flows on Differential Equations with Applications

Linfan 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 a mathematical

object for things embedded in a topological space. The main purpose of thispaper is to extend Banach spaces on topological graphs with operator actionsand show all of these extensions are also Banach space with a bijection witha bijection between linear continuous functionals and elements, which enablesone to solve linear functional equations in such extended space, particularly,solve algebraic, differential or integral equations on a topological graph, findmulti-space solutions on equations, for instance, the Einsteins gravitationalequations. A few well-known results in classical mathematics are generalizedsuch as those of the fundamental theorem in algebra, Hilbert and Schmidtsresult on integral equations, and the stability of such G -flow solutions withapplications to ecologically industrial systems are also discussed in this paper.

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

AMS(2010): 05C78,34A26,35A08,46B25,46E22,51D201. IntroductionLet V be a Banach space over a field F . All graphs G , denoted by (V ( G ), X( G ))

1

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.vu u

-

L(uv )

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, vva G -flow on G is such a labeling L : u V for u 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.2LClearly, 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 all2

0conservation 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 ; hie Re = {(i , j ) |i T 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, Hilbert spaces on topological graphs with

operator actions and show all of these extensions are also Banach space with abijection between linear continuous functionals and elements, which enables one tosolve linear functional equations in such extended space, particularly, solve algebraicor differential equations on a topological graph, i.e., find multi-space solutions forequations, such as those of algebraic equations, the Einstein gravitational equationsand integral equations with applications to controlling of ecologically industrial systems. All of these discussions provide new viewpoint for mathematical elements,i.e., mathematical combinatorics.3

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

dimVX

X

i=1

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 =

dimVX

xvi i

i=1

i

and V = {v , 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.4

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

G implies

\

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

X

L(v u ) = 0

uNG (v)

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 }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

5

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 }

for v V 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) andL(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 Gwith graph

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 GuV G

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

2.2 G -Flow SpacesDefine

L

G =

X

(u,v)X

kL(uv )kG

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 .6

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 (u )k +

G

kL1 (uv ) + L2 (uv )kX

(u,v)X

L L 1 2

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 LE D LE 1 L2 2 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 , GD LED LE 1 L 2 L= G ,G + G ,G

because ofDL1G

E D LEL2 L 1 L2 L+ G , G= G+G ,GX=hL1 (uv ) + L2 (uv ), L(uv )i(u,v)X( G )7

hL2 (uv ), L1 (uv )iE

=

X

hL1 (uv ), L(uv )i +

(u,v)X( G )

D

X

hL2 (uv ), L(uv )i

(u,v)X( G )

E D L2EL1 LL=G ,G + G ,GD LED LE 1 L 2 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

G G < if n, m N(). By definition,

L n Lm

kLn (uv ) Lm (uv )k G G <

i.e., {Ln (uv )} is also a Cauchy sequence for (u, v) X G , which is converges on

in V by definition.

Let L(uv ) = lim Ln (uv ) for (u, v) X G . Then it is clear thatn

Ln Llim G = G .

n

L VHowever, we are needed to show G G . By definition,X

Ln (uv ) = 0

vNG (u)

8

for u V G and integers n 1. Let n on its both sides. ThenXXlim Ln (uv ) =lim Ln (uv )n

vNG (u)

vNG (u)

X

=

n

L(uv ) = 0.

vNG (u)

L VThus, G G .

L1L2Similarly,two conservation G -flows G and G are said to be orthogonal ifEL1 L2G ,G= 0. The following result characterizes those of orthogonal pairs ofconservation G -flows.D

L1 L2 VL1L2Theorem 2.6 let G , G G . Then G is orthogonal to G if and only ifhL1 (uv ), L2 (uv )i = 0 for (u, v) X G . Proof Clearly, if hL1 (uv ), L2 (uv )i = 0 for (u, v) X G , then,D LEX 1 L2G ,G=hL1 (uv ), L2 (uv )i = 0,

(u,v)X GL1L2i.e., G is orthogonal to G .L1L2Conversely, if G is indeed orthogonal to G , thenD L1EX L2hL1 (uv ), L2 (uv )i = 0G ,G=

(u,v)X G

by definition. We therefore know that hL1 (u ), L2 (u )i = 0 for (u, v) X Gv

v

because of hL1 (uv ), L2 (uv )i 0.

Theorem 2.7 Let V be a Hilbert space with an orthogonal decomposition V =V V for a closed subspace V V . Then there is a decomposition

where,

Ve Ve ,G =V

n L o 1 V e = VG G L1 : X G V

n L2 oV eV = G G L2 : X G V ,9

L Vi.e., for G G , there is a uniquely decompositionL L1 L2G =G +G with L1 : X G V and L2 : X G V .

Proof By definition, L(uv ) V for (u, v) X G . Thus, there is a decom-

position

L(uv ) = L1 (uv ) + L2 (uv )vvwith uniquelyh L ideterminedh L iL1 (u ) V but L2 (u ) V . 1 2Let Gand Gbe two labeled graphs on G with L1 : X 1 G V2 h L1 i h L2 i DEand L2 : X 1 G V . We need to show that G, G G , V . In fact,2

the conservation laws show thatX

vNG (u)

for u V

X

L(uv ) = 0, i.e.,

(L1 (uv ) + L2 (uv )) = 0

vNG (u)

G . Consequently,X

X

L1 (uv ) +

vNG (u)

L2 (uv ) = 0.

vNG (u)

Whence,0 =

=

+

=+

*

*

*

*

X

L1 (uv ) +

vNG (u)

X

X

L1 (uv ),

vNG (u)

vNG (u)

X

X

L1 (uv ),

vNG (u)

vNG (u)

X

X

L1 (uv ),

vNG (u)

X

=

X

L1 (uv )L2 (uv )L1 (uv )

vNG (u)v

v

hL1 (u ), L2 (u )i +

vNG (u)

*

L2 (uv ),

vNG (u)

X

vNG (u)

X

+

+

+

X

vNG (u)

+

+

+

L1 (uv ),

vNG (u)

L1 (uv )

+

10

*

X

L2 (uv )

vNG (u)

L2 (uv ),

X

vNG (u)

vNG (u)

X

X

*

L2 (uv ),

vNG (u)

vNG (u)

X

X

*

L2 (uv ),

vNG (u)v

vNG (u)v

+

L2 (uv )L1 (uv )L2 (uv )

+

+

+

hL2 (u ), L1 (u )i

vNG (u)

X

L1 (uv ) +

X

+

*

X

vNG (u)

L2 (uv ),

X

vNG (u)

+

L2 (uv ) .

Notice that*+*+XXXXvvvvL1 (u ),L1 (u ) 0,L2 (u ),L2 (u ) 0.vNG (u)

vNG (u)

vNG (u)

vNG (u)

We therefore get that*+*+XXXXL1 (uv ),L1 (uv ) = 0,L2 (uv ),L2 (uv ) = 0,vNG (u)

i.e.,

vNG (u)

X

vNG (u)

X

L1 (uv ) = 0 and

vNG (u)

L2 (uv ) = 0.

vNG (u)

vNG (u)

h Li h Li 1 2VThus, G, G G . This completes the proof.

2.3 Solvable G -Flow Spaces

LLet G be a G -flow. If for v V G , all flows L (v u ) , u NG+ (v) \ u+are0determined by equations

L (v u ) ; L (w v ) , w NG (v) = 0

Fv

unless L(v u0 ), such a G -flow is called solvable, and L(v u0 ) the co-flow at vertex v.For example, a solvable G -flow is shown in Fig.3.v1

(0, 1)

-

(0, 1)(1, 0)

? 6(1, 0)v4

v2

(1, 0)

-

(1, 0)(0, 1)

? 6(0, 1)v3

Fig.3

L+A G -flow G is linear if each L v u , u+ NG+ (v) \ {u+0 } is determined by +L vu =

X

u NG(v)

11

au L u

v

,

with scalars au

+ F for v V G unless L v u0 , and is ordinary or partial

differential if L (v u ) is determined by ordinary differential equations

du+vLv;1 i nL(v ); L u, u NG (v) = 0dxi

or

+vLv; 1 i n L(v u ); L u , u NG (v) = 0xi

+ du0unless L vfor v V G , where, Lv; 1 i n , Lv;1 i ndxixidenote an ordinary or partial differential operators, respectively.LTheorem 2.8 For a strong-connected graph G , there exist linear G -flows G , notall flows being zero on G .Proof Notice that G is strong-connected. There must be a decomposition!!m1m2[ [ [G=CiTi ,i=1

i=1

where C i , T j are respectively directed circuit or path in G with1 1, m2 0. mkuFor an integer 1 k m1 , let C k = uk1 uk2 uksk and L uki i+1 = vk , wherei+Similarly, for integers 1 j m2 , if T j = w1j w2j wtj , let 1 t (mods).

wL wjt j+1 = 0. Clearly, the conservation law hold at v V G by definition, k uand each flow L uki i+1 , i + 1 (modsk ) is linear determined byL

k

uuki i+1

Xuk= L uki1 i + 0

X

j6=i vN (uk )iC

= vk + 0 +

m2X

X

j

m2 k XuiL v+

X

j=1 vN (uk )iTj

kL v ui

0 = vk .

j=1 vN (uk )iTj

LThus, G is a linear solvable G -flow and not all flows being zero on G .

All G -flows constructed in Theorem 2.8 can be also replaced by vectors dependent on the time t, i.e., v(t). Thus, if there is a vertex v V G with (v) 2 and + (v) 2, i.e., v is in at least 2 directed circuits C , C , let flows on C and Cbe respectively x and f(x, t). We then know the conservation laws hold for vertices12

in G , and there are indeed flows on G determined by ordinary differential equationssimilar to Theorem 2.8.Theorem2.9 Let G be a strong-connected graph. If there exists a vertex v V G with (v) 2 and + (v) 2 then there exist ordinary differential G -flowsLG , not all flows being zero on G .For example, the G -flow shown in Fig.4 is an ordinary differential G -flow in avector space if x = f(x, t) is solvable with x|t=t0 = x0 .

f(x, t)

-

v1

v2

x

f(x, t)

? 6x

x

x-

v4

? 6f(x, t)v3

f(x, t)Fig.4Similarly, let x = (x1 , x2 , , xn ). If xi = xi (t, s1 , s2 , , sn1)u = u(t, s1 , s2 , , sn1 ) p = p (t, s , s , , s ),i

i

1

2

n1

i = 1, 2, , n

is a solution of systemdx1dx2dxndu== == PnFp1Fp2Fpnpi Fpii=1

dp1dpn= = = = dtFx1 + p1 FuFxn + pn Fu

with initial values(xi0 = xi0 (s1 , s2 , , sn1), u0 = u0 (s1 , s2 , , sn1)pi0 = pi0 (s1 , s2 , , sn1),

such that

i = 1, 2, , n

F (x10 , x20 , , xn0 , u, p10 , p20 , , pn0 ) = 0nu0 X xi0pi0= 0, j = 1, 2, , n 1, s sjji=013

then it is the solution of Cauchy problem F (x1 , x2 , , xn , u, p1, p2 , , pn ) = 0

xi0 = xi0 (s1 , s2 , , sn1 ), u0 = u0 (s1 , s2 , , sn1 ) , p = p (s , s , , s ), i = 1, 2, , ni0i0 1 2n1

uFand Fpi =for integers 1 i n.xipiSimilarly, for partial differential equations of second order, the solutions of

where pi =

Cauchy problem on heat or wave equationsnXn2u2u2X2uu=a2=at2x2i2txi=1

,ii=1u = (x)u|t=0 = (x) u|t=0 = (x) ,t t=0are respectively known

1u (x, t) =n(4t) 2

for heat equation and

Z

+

e

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

(y1 , , yn )dy1 dyn

1u (x1 , x2 , x3 , t) =t 4a2 t

Z

MSat

dS +

14a2 t

Z

dS

MSat

Mfor wave equations in n = 3, where Satdenotes the sphere centered at M (x1 , x2 , x3 )with radius at. The following result on partial G -flows is known similar to that of

Theorem 2.8.Theorem2.10 Let G be a strong-connected graph. If there exists a vertex v V G with (v) 2 and + (v) 2 then there exist partial differential G -flowsLG , not all flows being zero on G .3. Operators on G -Flow Spaces3.1 Linear Continuous OperatorsDefinition 3.1 Let T O be an operator on Banach space V over a field F . AnVVoperator T : G G is bounded if

L

L

T G G 14

L Vfor G G with a constant [0, ) and furthermore, is a contractor if

L

L

L 1 2 1 L2

TGTGGG)

L1 L1 Vfor G , G G with [0, 1).

VVTheorem 3.2 Let T : G G be a contractor. Then there is a uniquelyL Vconservation G-flow G G such that LLT G=G .L0 VProof Let G G be a G-flow. Define a sequence LL1 0G =T G,

L0 L2L12 G =T G=T G,

n Ln oGby

................................ , L LLn n1 0n G =T G=T G,................................ .

We prove

n Ln oVGis a Cauchy sequence in G . Notice that T is a contractor.

For any integer m 1, we know that

L

L Lm1 m m+1 Lm

T G G = T G

G

L

Lm1Lm2 m Lm1

T G G G

= T G

LL1 L0 m1 Lm2

G 2 G

m G G .

Applying the triangle inequality, for integers m n we therefore get that

L

m Ln

G G

L

L

m Lm1 n1 Ln

G GG

+ + + G

L1 L0

mm1n1 +++ G G

n1 m L1 L0 = G G 1

L

n1 1 L0

G G for 0 < < 1.115

L

n Ln o m Ln

Consequently, G G 0 if m , n . So the sequence GLis a Cauchy sequence and converges to G . Similar to the proof of Theorem 2.5, weL Vknow it is a G-flow, i.e., G G . Notice that

L

L

L L L Lm m

G G + G T G

G T G

L

L

Lm m1 L

G G + G G .

L L

Let m . For 0 < < 1, we therefore get that G T G = 0, i.e., LLT G=G .LVFor the uniqueness,ifthereisananotherconservationG-flowG G L Lholding with T G= G , by

L

L

L

L L L

G G = T G T G G G ,

L Lit can be only happened in the case of G = G for 0 < < 1.

VVDefinition 3.3 An operator T : G G is linear if

L LL1L2 1 2T G + G= T G+ T G

L1 L2VL0for G , G G and , F , and is continuous at a G -flow G if therealways exist a number () for > 0 such that

L

L L L0 0

< if G G < ().

T G T G

The following result reveals the relation between conceptions of linear continu-

ous 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

T G T G

= T G G

G G

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

L

L0

6 0,

G G < () with () = , =16

then there must be

L

L0

T G G

< ,

Vi.e., T is linear continuous on G . However, this is obvious for = 0.Now if T is linear continuous but unbounded, there exists a sequenceVin G such that

L

L n n

G n G .Let

n Ln oG

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.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=017

LVfor the linear continuous functional T. If N (T) = G , i.e., T G= 0 forbL VL 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 L L LL 0 0 G = 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 L0 L0 L L0 E =T GG T GG ,G L D LE L D L L E 0 L0 0 0= T GG ,GT GG ,G.E L0 2 L0 L0

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

Let

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

GL0 2L0 2

G

G L0 TGLbL0 L0G = G=G ,

L0 2

G

L 0 L L Lb T G where = . We consequently get that T G= G ,G.

2L0

G 18

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

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

bb bb bb

LLLLLL

G G = G G , G G= 0,

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

Z

Z

,

Von G are respectively defined byL DL(uv )DG = GandZ

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

,

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

19

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 )

1(L2 (uv )) 1L2 (uv )=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 .

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

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

D

et

et

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

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

b(t)

Fig.5LFor 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 =K(t, )f ( )d =t2 + 2 d = + = a(t),2400020

Z

Z

1

K(t, )g( )d =

0

1

K(t, )g( )d =02

Z

10

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

Land the actions D G ,

Z

LG and[0,1]

Z

t2 + 2 e d

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

VFurthermore, 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(uv ) = 0 for v V

vNG (u)

G ,

we immediately know that

0 = D

X

vNG (u)

and

0=

Z

for v V G .

X

vNG (u)

L(uv ) =

L(u ) =v

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, which implies that the G -flowsolutions are fundamental for equations.21

4.1 Linear EquationsLet V be a field (F ; +, ). We can further defineL1 L2 L1 L2G G =G with L1 L2 (uv ) = L1 (uv ) L2 (uv ) 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 Fconservation laws is not emphasized, wheren L oFc= FG G |L (uv ) is constant in F for (u, v) X G.

FE G nE G F

Clearly, G F. Thus G = pif |F | = pn , where p is aprime 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.6Let [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.

22

nTheorem 4.1 A linear system (LESm) of equationsL1a11 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)

Li Vwith 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 .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 .23

Let rank [aij ]mn = r. Similar to that of linear algebra, we are easily know thatmPLiXc+ c1,r+1 Xjr+1 + c1n Xjnj1 =1i Gi=1mPLiXj 2 =c2i G + c2,r+1 Xjr+1 + c2n Xjn,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 .Let the operator D and Rn be the same as in Subsection 3.2. We considerVdifferential equations in G following.24

VTheorem 4.3 For GL G , the Cauchy problem on differential equationDGX = GL0X|L0is 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

vNG (u)

v F (u )

vNG (u)X

We therefore know that

DF (uv ) =

=

X

X

L0 (uv ) = 0.

F (uv ) = 0.

vNG (u)

25

L(uv ) = 0.

vNG (u)

vNG (u)

xn =x0n

X

X F VThus, we get a uniquely solution G = G G for the equationDGX = GL0X|L0prescribed 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=1

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 )21v4tF (u ) (x, t) =eL (uv ) (y1 , , yn )dy1 dyn .n(4t) 2 26

Labeling the semi-arc u by F (u ) (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 ,XL (uv ) (x) = 0,v

v

vNG (u)

we know thatXF (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

1e(0) dy1 dyn = 0n(4t) 2 F Vfor u V G . Therefore, G G and=

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

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++x21x22x23

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

mined byN

=

2

(x) L []|

Z

27

L2 [] are deter

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

N

=

2

(x) L []|

Z

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

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 GXG

Z

XG = GL

and

D

EL LG ,G= 0,

LG N

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

if X (uv ) = 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]

is solvable in L2 [] if and only ifZL (uv ) [x]L (uv ) [x]dx = 0

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]

28

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

v

F (u ) [x]

Z

vNG (u)

XK(x, y)F (u ) [x] =L (uv ) [x] = 0,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)

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

29

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)

Li Vfor u V G , i.e., G G for integers i = 1, 2, .

4.2 Non-linear EquationsIf G is strong-connected with a special structure, we can get a general result onG -solutions of equations, including non-linear equations following.Theorem 4.8 If the topological graph G is strong-connected with circuit decomposition

l

[G=Cii=1

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

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

30

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 Vvsuch that L (u ) = Li (x) for (u, v) X C i is solvable for X G . Lu(x)Proof Let X = Gwith Lu(x) (uv ) = 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,

uV

vNG (u)

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

G .

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 one31

variable x, we get a conclusion following, which generalizes the fundamental theoremin algebra.Corollary 4.9(Generalized Fundamental Theorem in Algebra) If G is strongconnected with circuit decompositionl

[G=Cii=1

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

polynomial

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