G -Flows on Differential Equations with Applications

Page 1

− → 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 holding with conservation laws

X

L (v u ) = 0 for ∀v ∈ V

u∈NG (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 this paper is to extend Banach spaces on topological graphs with operator actions and show all of these extensions are also Banach space with a bijection with a bijection between linear continuous functionals and elements, which enables one to solve linear functional equations in such extended space, particularly, solve algebraic, differential or integral equations on a topological graph, find multi-space solutions on equations, for instance, the Einstein’s gravitational equations. A few well-known results in classical mathematics are generalized such as those of the fundamental theorem in algebra, Hilbert and Schmidt’s − → result on integral equations, and the stability of such G -flow solutions with applications to ecologically industrial systems are also discussed in this paper.

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

AMS(2010): 05C78,34A26,35A08,46B25,46E22,51D20 §1. Introduction − → − → − → Let 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 G − → is an embedding of an oriented graph G in a topological space C . All elements in − → − → − → V ( 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-arc uv and end semi-arc v u , such as those shown in Fig.1 following. v u u

-

L(uv )

vu v

Fig.1 − → − → All these semi-arcs of a topological graph G are denoted by X 1 G . 2 − →L − → − → A vector labeling G on G is a 1 − 1 mapping L : G → V such that L : uv → − → − →L L(uv ) ∈ V for ∀uv ∈ X 1 G , such as those shown in Fig.1. For all labelings G 2 − → on G , define − →L1 − →L2 − →L1 +L2 − →L − →λL G +G =G and λ G = G . − → Then, all these vector labelings on G naturally form a vector space. Particularly, − → − → − → v v a G -flow on G is such a labeling L : u → V for ∀u ∈ X 1 G hold with 2

L (uv ) = −L (v u ) and conservation laws X

L (v u ) = 0

u∈NG (v)

− → for ∀v ∈ V ( G ), where 0 is the zero-vector in V . For example, a conservation law for 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

u1 u1 L(v ) u2 u2 L(v )

4

5

v

-

-F (vu ) u6

u3 u3 L(v )

6

Fig.2 − → − →L Clearly, if V = Z and O = {1}, then the G -flow G is nothing else but the network − → − → flow X( G ) → Z on G . − →L − →L1 − →L2 − → − → Let G , G , G be G -flows on a topological graph G and ξ ∈ F a scalar. − →L1 − →L2 − →L − → It is clear that G + G and ξ · G are also G -flows, which implies that all 2

− → − → − →0 conservation G -flows on G also form a linear space over F with unit G under − →V − →0 − → operations + and ·, denoted by G , where G is such a G -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 mathematics and applied sciences. It should be noted that a conservation law naturally deter− → mines an autonomous systems in the world. We can also find G -flows by solving conservation equations X

L (v u ) = 0,

u∈NG (v)

v∈V

− → G .

− → Such a system of equations is non-solvable in general, only with G -flow solutions − → such as those discussions in references [10]-[19]. Thus we can also introduce G -flows e R e by Smarandache multi-system ([21]-[22]). In fact, for any integer m ≥ 1 let Σ;

be a Smarandache multi-system consisting of m mathematical systemsh(Σ1 ; R i 1 ), L e e (Σ2 ; R2 ), · · ·, (Σm ; Rm ), different two by two. A topological structure G Σ; R on e R e is inherited by Σ; h i e R e = {Σ1 , Σ2 , · · · , Σm }, V GL Σ; h i e R e = {(Σi , Σj ) |Σi T Σj 6= ∅, 1 ≤ i 6= j ≤ m} with labeling E GL Σ; T L : Σi → L (Σi ) = Σi and L : (Σi , Σj ) → L (Σi , Σj ) = Σi Σj for hintegers i 1 ≤ i 6= j ≤ m,Ti.e., a topological vertex-edge labeled graph. Clearly, → e R e is a − GL Σ; G -flow if Σi Σj = 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 a bijection between linear continuous functionals and elements, which enables one to solve linear functional equations in such extended space, particularly, solve algebraic or differential equations on a topological graph, i.e., find multi-space solutions for equations, such as those of algebraic equations, the Einstein gravitational equations and 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 linear spaces, [8]-[9], [21]-[22] for Smarandache multi-systems, [3], [20] and [23] for differential equations. − → §2. G -Flow Spaces 2.1 Existence Definition 2.1 Let V be a Banach space. A family V of vectors v ∈ V is conservative if

X

v = 0,

v∈V

called a conservative family.

Let V be a Banach space over a field F with a basis {α1 , α2 , · · · , αdimV }. Then, for v ∈ V there are scalars xv1 , xv2 , · · · , xvdimV ∈ F such that dimV X

v=

xvi αi .

i=1

Consequently, X dimV X

xvi αi =

v∈V i=1

implies that

dimV X

X

i=1

X

v∈V

xvi

!

αi = 0

xvi = 0

v∈V

for integers 1 ≤ i ≤ dimV . Conversely, if

X

xvi = 0, 1 ≤ i ≤ dimV ,

v∈V

define

i

v =

dimV X

xvi αi

i=1

i

and V = {v , 1 ≤ i ≤ dimV }. Clearly,

P

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

v∈V

vectors. Whence, if denoted by xvi = (v, αi ) for ∀v ∈ V , we therefore get a condition on families of conservation in V following. 4

Theorem 2.2 Let V be a Banach space with a basis {α1 , α2 , · · · , αdimV }. Then, a vector family V ⊂ V is conservation if and only if X (v, αi ) = 0 v∈V

for integers 1 ≤ i ≤ dimV .

For example, let V = {v1 , v2 , v3 , v4 } ⊂ R3 with v1 = (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 following result establishes its inverse. − → − →L − → Theorem 2.3 A G -flow G exists on G if and only if there are conservation families L(v) in a Banach space V associated an index set V with L(v) = {L(v u ) ∈ V f or some u ∈ V } such that L(v u ) = −L(uv ) and L(v) Proof Notice that

for ∀v ∈ V

− → G implies

\

(−L(u)) = L(v u ) or ∅.

X

L(v u ) = 0

u∈NG (v)

L(v u ) = −

X

L(v w ).

w∈NG (v)\{u}

Whence, if there is an index set V associated conservation families L(v) with L(v) = {L(v u ) ∈ V for some u ∈ V } T for ∀v ∈ V such that L(v u ) = −L(uv ) and L(v) (−L(u)) = L(v u ) or ∅, define a − → topological graph G by [ − → − → V G = V and X G = {(v, u)|L(v u ) ∈ L(v)} v∈V

5

− →L − → with an orientation v → u on its each arcs. Then, it is clear that G is a G -flow by definition.

− →L − → Conversely, 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) and 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

G with

graph − →

− →

− →

weights λv for v ∈ V , particularly, λe αi 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 integer − → 1 ≤ i ≤ dimV , such a G -flow exists if and only if the system of linear equations X − → λ(v,u) = 0, v ∈ V G − → u∈V G

− →

− →

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

− → 2.2 G -Flow Spaces Define

L → − G =

X

(u,v)∈X

kL(uv )k − → G

− →L − →V for ∀ 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. ξL L → → − − (2) G = ξ G for any scalar ξ. 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)∈X v

− →

− →

G

kL1 (u )k +

G

kL1 (uv ) + L2 (uv )k X

(u,v)∈X

L L → 1 − → 2 − kL2 (u )k = G + G . v

− →

G

− →V Whence, k · k is a norm on linear space G . Furthermore, if V is an inner space with inner product h·, ·i, define D L E X − → 1 → − L2 G ,G = hL1 (uv ), L2 (uv )i . − → (u,v)∈X G Then we know that D L LE − → → − (4) G , G =

D L LE − → − → v v hL(u ), L(u )i ≥ 0 and G ,G = 0 if and only − → (u,v)∈X G − → − →L if L(uv ) = 0 for ∀(u, v) ∈ X( G ), i.e., G = O. D L E D L E − → 1 − →L2 − → 2 − →L1 − →L1 → − L2 − →V (5) G , G = G ,G for ∀ G , G ∈ G because of D

− →L1 → − L2 G ,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 is D E − →L1 − →L2 − →L λG + µG , G D L E D L E − → 1 → −L − → 2 → −L = λ G ,G +µ G ,G

because of D − →L1 λG

E D λL E − →L2 − →L − → 1 − →µL2 − →L + µG , G = G +G ,G X = hλL1 (uv ) + µL2 (uv ), L(uv )i − → (u,v)∈X( G ) 7

hL2 (uv ), L1 (uv )i E

=

X

− →

hλL1 (uv ), L(uv )i +

(u,v)∈X( G )

D

X

− →

hµL2 (uv ), L(uv )i

(u,v)∈X( G )

E D µL2 E − →λL1 − →L − → − →L = G ,G + G ,G D L E D L E − → 1 − →L − → 2 − →L = λ G ,G +µ G ,G .

− →V Thus, G is an inner space also and as the usual, let L rD L L E → − → − → − G ,G G =

− →L − →V for G ∈ G . Then it is a normed space. Furthermore, we know the following result. − → − →V Theorem 2.5 For any topological graph G , G is a Banach space, and furthermore, − →V if V is a Hilbert space, G is a Hilbert space also. − →V Proof As shown in the previous, G is a linear normed space or inner space if V is an inner space. We show nthat ito is also complete, i.e., any Cauchy sequence in − →V − →Ln − →V G is converges. In fact, let G be a Cauchy sequence in G . Thus for any number ε > 0, there always exists an integer N(ε) such that L → n − →Lm − G − G < ε if n, m ≥ N(ε). By definition,

L → n − →Lm − kLn (uv ) − Lm (uv )k ≤ G − G < ε

− → 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 that n→∞

− →Ln − →L lim G = G .

n→∞

− →L − →V However, we are needed to show G ∈ G . By definition, X

Ln (uv ) = 0

v∈NG (u)

8

− → for ∀u ∈ V G and integers n ≥ 1. Let n → ∞ on its both sides. Then   X X lim  Ln (uv ) = lim Ln (uv ) n→∞

v∈NG (u)

v∈NG (u)

X

=

n→∞

L(uv ) = 0.

v∈NG (u)

− →L − →V Thus, G ∈ G .

− → − →L1 − →L2 Similarly, two conservation G -flows G and G are said to be orthogonal if E − →L1 → − L2 G ,G = 0. The following result characterizes those of orthogonal pairs of − → conservation G -flows. D

− →L1 → − L2 − →V − →L1 − →L2 Theorem 2.6 let G , G ∈ G . Then G is orthogonal to G if and only if − → hL1 (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 L E X − → 1 − →L2 G ,G = hL1 (uv ), L2 (uv )i = 0, − → (u,v)∈X G − →L1 − →L2 i.e., G is orthogonal to G . − →L1 − →L2 Conversely, if G is indeed orthogonal to G , then D L1 E X − → → − L2 hL1 (uv ), L2 (uv )i = 0 G ,G = − → (u,v)∈X G

− → by definition. We therefore know that hL1 (u ), L2 (u )i = 0 for ∀(u, v) ∈ X G v

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,

− →V e ⊕V e ⊥, G =V

n L o → 1 − →V

− → e = − V G ∈ G L1 : X G → V

n L2 o − → − →V

− → ⊥ ⊥ e V = G ∈ G L2 : X G → V , 9

− →L − →V i.e., for ∀ G ∈ G , there is a uniquely decomposition − →L − →L1 − →L2 G =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 ) v v ⊥ with uniquely h L idetermined h L iL1 (u ) ∈ V but L2 (u ) ∈ V . − → 1 − → 2 − → − → Let G and G be two labeled graphs on G with L1 : X 1 G → V 2 h L1 i h L2 i D E − → − → − → − → and L2 : X 1 G → V⊥ . We need to show that G , G ∈ G , V . In fact, 2

the conservation laws show that X

v∈NG (u)

for ∀u ∈ V

X

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

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

v∈NG (u)

− → G . Consequently, X

X

L1 (uv ) +

v∈NG (u)

L2 (uv ) = 0.

v∈NG (u)

Whence, 0 =

=

+

= +

*

*

*

*

X

L1 (uv ) +

v∈NG (u)

X

X

L1 (uv ),

v∈NG (u)

v∈NG (u)

X

X

L1 (uv ),

v∈NG (u)

v∈NG (u)

X

X

L1 (uv ),

v∈NG (u)

X

=

X

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

v∈NG (u) v

v

hL1 (u ), L2 (u )i +

v∈NG (u)

*

L2 (uv ),

v∈NG (u)

X

v∈NG (u)

X

+

+

+

X

v∈NG (u)

+

+

+

L1 (uv ),

v∈NG (u)

L1 (uv )

+

10

*

X

L2 (uv )

v∈NG (u)

L2 (uv ),

X

v∈NG (u)

v∈NG (u)

X

X

*

L2 (uv ),

v∈NG (u)

v∈NG (u)

X

X

*

L2 (uv ),

v∈NG (u) v

v∈NG (u) v

+

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

+

+

+

hL2 (u ), L1 (u )i

v∈NG (u)

X

L1 (uv ) +

X

+

*

X

v∈NG (u)

L2 (uv ),

X

v∈NG (u)

+

L2 (uv ) .

Notice that * + * + X X X X v v v v L1 (u ), L1 (u ) ≼ 0, L2 (u ), L2 (u ) ≼ 0. v∈NG (u)

v∈NG (u)

v∈NG (u)

v∈NG (u)

We therefore get that * + * + X X X X L1 (uv ), L1 (uv ) = 0, L2 (uv ), L2 (uv ) = 0, v∈NG (u)

i.e.,

v∈NG (u)

X

v∈NG (u)

X

L1 (uv ) = 0 and

v∈NG (u)

L2 (uv ) = 0.

v∈NG (u)

v∈NG (u)

h Li h Li − → 1 − → 2 − →V Thus, G , G ∈ G . This completes the proof.

− → 2.3 Solvable G -Flow Spaces

− →L − → − → Let G be a G -flow. If for ∀v ∈ V G , all flows L (v u ) , u ∈ NG+ (v) \ u+ are 0 determined 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 − d u+ −v − Lv ;1 ≤ i ≤ n L(v ); L u , u ∈ NG (v) = 0 dxi

or

∂ + v Lv ; 1 ≤ i ≤ n L(v u ); L u− , u− ∈ NG− (v) = 0 ∂xi + d ∂ − → u0 unless L v for ∀v ∈ V G , where, Lv ; 1 ≤ i ≤ n , Lv ;1 ≤ i ≤ n dxi ∂xi denote an ordinary or partial differential operators, respectively. − → − → − →L Theorem 2.8 For a strong-connected graph G , there exist linear G -flows G , not − → all flows being zero on G . − → Proof Notice that G is strong-connected. There must be a decomposition ! ! m1 m2 [ − → − → [ [− → G= Ci Ti , i=1

i=1

− → − → − → where C i , T j are respectively directed circuit or path in G with 1 ≥ 1, m2 ≥ 0. m k − → u For an integer 1 ≤ k ≤ m1 , let C k = uk1 uk2 · · · uksk and L uki i+1 = vk , where − → i+ Similarly, for integers 1 ≤ j ≤ m2 , if T j = w1j w2j · · · wtj , let 1 ≡t (mods). − → w L wjt j+1 = 0. Clearly, the conservation law hold at ∀v ∈ V G by definition, k u and each flow L uki i+1 , i + 1 ≡ (modsk ) is linear determined by L

k

u uki i+1

X uk = L uki−1 i + 0 ×

X

j6=i v∈N − (uk ) i C

= vk + 0 +

m2 X

X

j

m2 k X ui L v +

X

j=1 v∈N − (uk ) i T j

k L v ui

0 = vk .

j=1 v∈N − (uk ) i T j

− →L − → − → Thus, 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 depen− → dent on the time t, i.e., v(t). Thus, if there is a vertex v ∈ V G with ρ− (v) ≥ 2 − → − →′ − → − →′ and ρ+ (v) ≥ 2, i.e., v is in at least 2 directed circuits C , C , let flows on C and C be respectively x and f(x, t). We then know the conservation laws hold for vertices 12

− → − → in G , and there are indeed flows on G determined by ordinary differential equations similar to Theorem 2.8. − → Theorem 2.9 Let G be a strong-connected graph. If there exists a vertex v ∈ − → − → V G with Ď âˆ’ (v) ≼ 2 and Ď + (v) ≼ 2 then there exist ordinary differential G -flows − →L − → G , not all flows being zero on G . − → − → For example, the G -flow shown in Fig.4 is an ordinary differential G -flow in a vector space if xË™ = 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.4 Similarly, let x = (x1 , x2 , ¡ ¡ ¡ , xn ). If     xi = xi (t, s1 , s2 , ¡ ¡ ¡ , sn−1) u = u(t, s1 , s2 , ¡ ¡ ¡ , sn−1 )    p = p (t, s , s , ¡ ¡ ¡ , s ), i

i

1

2

n−1

i = 1, 2, ¡ ¡ ¡ , n

is a solution of system dx1 dx2 dxn du = = ¡¡¡ = = P n Fp1 Fp2 Fpn pi Fpi i=1

dp1 dpn = − = ¡¡¡ = − = dt Fx1 + p1 Fu Fxn + pn Fu

with initial values ( xi0 = xi0 (s1 , s2 , ¡ ¡ ¡ , sn−1), u0 = u0 (s1 , s2 , ¡ ¡ ¡ , sn−1) pi0 = pi0 (s1 , s2 , ¡ ¡ ¡ , sn−1),

such that

i = 1, 2, ¡ ¡ ¡ , n

   F (x10 , x20 , ¡ ¡ ¡ , xn0 , u, p10 , p20 , ¡ ¡ ¡ , pn0 ) = 0 n ∂u0 X ∂xi0  pi0 = 0, j = 1, 2, ¡ ¡ ¡ , n − 1,  ∂s − ∂sj j i=0 13

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

xi0 = xi0 (s1 , s2 , · · · , sn−1 ), u0 = u0 (s1 , s2 , · · · , sn−1 ) ,    p = p (s , s , · · · , s ), i = 1, 2, · · · , n i0 i0 1 2 n−1

∂u ∂F and Fpi = for integers 1 ≤ i ≤ n. ∂xi ∂pi Similarly, for partial differential equations of second order, the solutions of

where pi =

Cauchy problem on heat or wave equations  n  X n  ∂2u ∂2u 2 X  2 ∂ u ∂u   = a 2   =a ∂t2 ∂x2i 2 ∂t ∂x i=1

, i i=1   ∂u

   = ψ (x) u|t=0 = ϕ (x)  u|t=0 = ϕ (x) , ∂t t=0 are respectively known

1 u (x, t) = n (4πt) 2

for heat equation and

Z

+∞

e−

(x1 −y1 )2 +···+(xn −yn )2 4t

ϕ(y1 , · · · , yn )dy1 · · · dyn

−∞

∂  1 u (x1 , x2 , x3 , t) =  ∂t 4πa2 t

Z

M Sat

 ϕdS  +

1 4πa2 t

Z

ψdS

M Sat

M for wave equations in n = 3, where Sat denotes 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. − → Theorem 2.10 Let G be a strong-connected graph. If there exists a vertex v ∈ − → − → V G with ρ− (v) ≥ 2 and ρ+ (v) ≥ 2 then there exist partial differential G -flows − →L − → G , not all flows being zero on G . − → §3. Operators on G -Flow Spaces 3.1 Linear Continuous Operators Definition 3.1 Let T ∈ O be an operator on Banach space V over a field F . An − →V − →V operator T : G → G is bounded if L L → − → − T G ≤ ξ G 14

− →L − →V for ∀ G ∈ G with a constant ξ ∈ [0, ∞) and furthermore, is a contractor if L L L − → 1 − → 2 → 1 − →L2 − T G − T G ≤ ξ G − G )

− →L1 → − L1 − →V for ∀ G , G ∈ G with ξ ∈ [0, 1).

− →V − →V Theorem 3.2 Let T : G → G be a contractor. Then there is a uniquely − →L − →V conservation G-flow G ∈ G such that L − → − →L T G =G . − →L0 − →V Proof Let G ∈ G be a G-flow. Define a sequence L − →L1 − → 0 G =T G , L0 − →L2 − →L1 → 2 − G =T G =T G ,

n Ln o − → G by

................................ , L L − →Ln − → n−1 → 0 n − G =T G =T G , ................................ .

We prove

n Ln o − → − →V G is a Cauchy sequence in G . Notice that T is a contractor.

For any integer m ≥ 1, we know that L L Lm−1 − → m − → → m+1 − →Lm − −T G − G = T G G L − →Lm−1 − →Lm−2 → m − →Lm−1 − −T G ≤ ξ G − G = T G L →L1 − →L0 → m−1 − →Lm−2 − − −G ≤ ξ2 G ≤ · · · ≤ ξm G − G .

Applying the triangle inequality, for integers m ≥ n we therefore get that L → m − →Ln − G − G L L → m − →Lm−1 → n−1 − →Ln − − ≤ G − G −G + + · · · + G →L1 − →L0 − m m−1 n−1 ≤ ξ +ξ +···+ξ × G − G ξ n−1 − ξ m − →L1 − →L0 = × G − G 1−ξ L ξ n−1 → 1 − →L0 − ≤ × G − G for 0 < ξ < 1. 1−ξ 15

L n Ln o → m − →Ln − → − Consequently, G − G → 0 if m → ∞, n → ∞. So the sequence G − →L is a Cauchy sequence and converges to G . Similar to the proof of Theorem 2.5, we − →L − →V know 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 → m−1 − →L − − ≤ G − G + ξ G − G . L L → − → − Let m → ∞. For 0 < ξ < 1, we therefore get that G − T G = 0, i.e., L − →L − → T G =G . − →L′ − →V For the uniqueness, if there is an another conservation G-flow G ∈ G L′ − → − →L holding with T G = G , by L L L L′ → − →L′ − → − → → − →L′ − − G − G = T G − T G ≤ ξ G − G ,

− →L − →L′ it can be only happened in the case of G = G for 0 < ξ < 1.

− →V − →V Definition 3.3 An operator T : G → G is linear if L L − →L1 − →L2 − → 1 − → 2 T λG + µG = λT G + µT G

− →L1 − →L2 − →V − → − →L0 for ∀ G , G ∈ G and λ, µ ∈ F , and is continuous at a G -flow G if there always exist a number δ(ε) for ∀ǫ > 0 such that L L L → − →L0 − → − → 0 − < ε if G − G < δ(ε). T G − T G

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

ous with that of linear bounded. − →V − →V Theorem 3.4 An operator T : G → G is linear continuous if and only if it is bounded. Proof If T is bounded, then L L L0 L − → − → − → − →L0 − → − →L0 T G − T G = T G − G ≤ξ G −G

− →L − →L0 − →V for an constant ξ ∈ [0, ∞) and ∀ G , G ∈ G . Whence, if L ε → − →L0 − 6 0, G − G < δ(ε) with δ(ε) = , ξ = ξ 16

then there must be

L − → − →L0 T G − G < ε,

− →V i.e., T is linear continuous on G . However, this is obvious for ξ = 0. Now if T is linear continuous but unbounded, there exists a sequence − →V in G such that L L → n → n − − G ≥ n G . Let

n Ln o − → G

1 →Ln L ×− → n G . − n G L∗ 1 L∗ → n − → n − Then G = → 0, i.e., T G → 0 if n → ∞. However, by definition n   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 − →L∗n G =

a contradiction. Thus, T must be bounded.

The following result is a generalization of the representation theorem of Fr´echet − →V − → − →V and Riesz on linear continuous functionals, i.e., T : G → C on G -flow space G , where C is the complex field. − →V Theorem 3.5 Let T : G → C be a linear continuous functional. Then there is a − →Lb − →V unique G ∈ G such that L L Lb − → − → − → T G = G ,G − →L − →V for ∀ G ∈ G .

− →V Proof Define a closed subset of G by

L n L o − → − →V

− → N (T) = G ∈ G T G =0 17

L − →V − → for the linear continuous functional T. If N (T) = G , i.e., T G = 0 for b − →L − →V − →L ∀ G ∈ G , choose G = O. We then easily obtain the identity L L Lb − → − → → − T G = G ,G . − →V Whence, we assume that N (T) 6= G . In this case, there is an orthogonal decomposition

− →V G = N (T) ⊕ N ⊥ (T)

with N (T) 6= {O} and N ⊥ (T) 6= {O}. − → − →L0 − →L0 Choose a G -flow G ∈ N ⊥ (T) with G 6= O and define L L L L − →L∗ − → − → 0 − → 0 − → G = T G G − T G G

− →L − →V for ∀ G ∈ G . Calculation shows that L∗ L L L L − → − → − → 0 − → 0 − → T G = T G T G − T G T G = 0, − →L∗ i.e., G ∈ N (T). We therefore get that D L∗ E − → − →L0 0 = G ,G D L L0 L0 L L0 E − → − → − → − → → − = T G G − T G G ,G L D L E L D L L E − → − → 0 → − L0 − → 0 − → → − 0 = T G G ,G −T G G ,G . E L0 2 → − →L0 − →L0 − Notice that G , G = G 6= 0. We find that D

Let

L0 L0 * + − → − → D L LE L T G T G − → − → 0 − →L − →L0 − → G ,G = G , . T G = G →L0 2 →L0 2 − − G G L0 − → T G − →Lb − →L0 − →λL0 G = G = G , →L0 2 − G

L − → 0 L L Lb T G − → − → − → where λ = . We consequently get that T G = G ,G . 2 →L0 − G 18

L − →Lb ′ − →V − → − →L − →Lb ′ Now if there is another G ∈ G such that T G = G ,G for − →L − →Lb − →Lb ′ − →L − →V ∀ G ∈ G , there must be G , G − G = 0 by definition. Particularly, let − →L − →Lb − →Lb ′ G = G − G . We know that b b′ b b′ − b b′ − L L L L L L → − → − → − → → − → 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 Operators Let 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 product Z hf (x) , g (x)i = f (x)g(x)dx for f (x), g(x) ∈ L2 [∆] ∆

− →V − → − → and G its G -extension on a topological graph G . The differential operator and integral operators

n X

D=

i=1

∂ ai ∂xi

and

Z

Z

, ∆

∆

− →V on G are respectively defined by − →L − →DL(uv ) DG = G and Z

Z

− →L G =

∆

− →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

∆

− →L G = ∆

Z

Z

∆

,

∂ai − → for ∀(u, v) ∈ X G , where ai , ∈ C0 (∆) for integers 1 ≤ i, j ≤ n and K(x, y) : ∂xj ∆ × ∆ → C ∈ L2 (∆ × ∆, C) with Z K(x, y)dxdy < ∞. ∆×∆

19

Such integral operators are usually called adjoint for

Z

= ∆

Z

by K (x, y) =

∆

− →L1 − →L2 − →V K (x, y). Clearly, for G , G ∈ G and Îť, Âľ ∈ F , ÎťL (uv )+ÂľL (uv ) 2 − →L1 (uv ) − →L2 (uv ) − → 1 − →D(ÎťL1 (uv )+ÂľL2 (uv )) D ÎťG + ÂľG =D G =G − →D(ÎťL1 (uv ))+D(ÂľL2 (uv )) − →D(ÎťL1 (uv )) − →D(ÂľL2 (uv )) =G =G +G (ÎťL (uv )) L (uv ) − → 1 − →(ÂľL2 (uv )) − → 1 − →L2 (uv ) =D G = ÎťD G + D ÂľG +G − → for ∀(u, v) ∈ X G , i.e., − →L1 − →L2 − →L1 − →L2 D ÎťG + ÂľG = ÎťD G + ÂľD G . Similarly, we know also that Z Z Z − →L1 − →L2 − →L1 − →L2 ÎťG + ÂľG =Îť G +Âľ G , ∆ ∆ ∆ Z Z Z − →L1 − →L2 − →L1 − →L2 ÎťG + ÂľG =Îť G +Âľ G . ∆

Thus, operators D,

Z

and ∆

∆

Z

∆

− →V are al linear on G . ∆

et --t et ? t = t e 6 } t -t

et -1 t = et ?1 1 e } 6 6 -1

D

et

et

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

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

b(t)

Fig.5 − →L For example, let f (t) = t, g(t) = et , K(t, Ď„ ) = t2 + Ď„ 2 for ∆ = [0, 1] and let G − → be the G -flow shown on the left in Fig.6. Then, we know that Df = 1, Dg = et , Z 1 Z 1 Z 1 t2 1 K(t, Ď„ )f (Ď„ )dĎ„ = K(t, Ď„ )f (Ď„ )dĎ„ = t2 + Ď„ 2 Ď„ dĎ„ = + = a(t), 2 4 0 0 0 20

Z

Z

1

K(t, τ )g(τ )dτ =

0

1

K(t, τ )g(τ )dτ = 0 2

Z

1 0

= (e − 1)t + e − 2 = b(t)

− →L and the actions D G ,

Z

− →L G and [0,1]

Z

t2 + τ 2 eτ dτ

− →L G are shown on the right in Fig.5. [0,1]

− →V Furthermore, we know that both of them are injections on G . Z − →V − →V − →V − →V Theorem 3.6 D : G → G and : G →G . ∆

− →L − →V − →L Proof For ∀ G ∈ G , we are needed to show that D G and

v

DL(u ) = 0 and

v∈NG (u)

hold with ∀v ∈ V

− →L − →V G ∈ G ,

∆

i.e., the conservation laws X

Z

X Z

v∈NG (u)

L(uv ) = 0

∆

− → − →L(uv ) − →V G . However, because of G ∈ G , there must be X

L(uv ) = 0 for ∀v ∈ V

v∈NG (u)

− → G ,

we immediately know that 

0 = D

X

v∈NG (u)

and

0=

Z

∆

− → for ∀v ∈ V G .

 

X

v∈NG (u)

L(uv ) = 

L(u ) = v

X

DL(uv )

v∈NG (u)

X Z

v∈NG (u)

L(uv ) ∆

− → §4. G -Flow Solutions of Equations As we mentioned, all G-solutions of non-solvable systems on algebraic, ordinary or − → partial differential equations determined in [13]-[19] are in fact G -flows. We show − → − → there are also G -flow solutions for solvable equations, which implies that the G -flow solutions are fundamental for equations. 21

4.1 Linear Equations Let V be a field (F ; +, ¡). We can further define − →L1 − →L2 − →L1 ¡L2 G â—ŚG =G − → with L1 ¡ L2 (uv ) = L1 (uv ) ¡ L2 (uv ) for ∀(u, v) ∈ X G . Then it can be verified F − →F − → c isomorphic to F if the easily that G is also a field G ; +, â—Ś with a subfield F conservation laws is not emphasized, where n L o → − →F − → c= − F G ∈ G |L (uv ) is constant in F for ∀(u, v) ∈ X G .

− →

− →

F

E G

n

E G

− →F →

−

Clearly, G ≃ F . Thus G = p if |F | = pn , where p is a − → prime number. For this F -extension on G , the linear equation

− →L aX = G

− →a−1 L − →F is uniquely solvable for X = G in G if 0 6= a ∈ F . Particularly, if one views − →L − → an element b ∈ F as b = G if L(uv ) = b for (u, v) ∈ X G and 0 6= a ∈ F , then an algebraic equation

ax = b − →F − →a−1 L in F also is an equation in G with a solution x = G such as those shown in − → − → Fig.6 for G = C 4 , a = 3, b = 5 following.

-3 5

5 3

6

?53

5 3

Fig.6 Let [Lij ]mĂ—n be a matrix with entries Lij : uv → V . Denoted by [Lij ]mĂ—n (uv ) − → the matrix [Lij (uv )]mĂ—n . Then, a general result on G -flow solutions of linear systems is known following.

22

n Theorem 4.1 A linear system (LESm ) of equations  − →L1   a11 X1 + a12 X2 + · · · + a1n Xn = G    →L2  a X +a X +···+a X = − G 21 1 22 2 2n n  ......................................     →Lm  a X +a X +···+a X = − G m1 1 m2 2 mn n

n (LESm )

− →Li − →V with aij ∈ C and G ∈ G for integers 1 ≤ i ≤ n and 1 ≤ j ≤ m is solvable for − →V Xi ∈ G , 1 ≤ i ≤ m if and only if v rank [aij ]m×n = 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

  .  

− →Lx − → Proof Let Xi = G i with Lxi (uv ) ∈ V on (u, v) ∈ X G for integers − → n 1 ≤ i ≤ n. For ∀(u, v) ∈ X G , the system (LESm ) appears as a common linear system

  a11 Lx1 (uv ) + a12 Lx2 (uv ) + · · · + a1n Lxn (uv ) = L1 (uv )     a L (uv ) + a L (uv ) + · · · + a L (uv ) = L (uv ) 21 x1 22 x2 2n xn 2  ........................................................     am1 Lx1 (uv ) + am2 Lx2 (uv ) + · · · + amn Lxn (uv ) = Lm (uv )

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

− → for ∀(u, v) ∈ G . Labeling the uv respectively by solutions Lx1 (uv ), Lx2 (uv ), · · · , Lxn (uv ) semi-arc − → − →Lx − →Lx − →Lxn for ∀(u, v) ∈ X G , we get labeled graphs G 1 , G 2 , · · · , G . We prove that − →Lx1 → − Lx − →Lxn − →V G , G 2, · · ·, G ∈G . 23

Let rank [aij ]m×n = r. Similar to that of linear algebra, we are easily know that  m P − →Li   X c + c1,r+1 Xjr+1 + · · · c1n Xjn j1 = 1i G    i=1  m  P  − →Li  Xj 2 = c2i G + c2,r+1 Xjr+1 + · · · c2n Xjn , i=1   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    m  P − →Li   cri G + cr,r+1 Xjr+1 + · · · crn Xjn  Xj r = i=1

− →Lx − →Lx − →V where {j1 , · · · , jn } = {1, · · · , n}. Whence, if G jr+1 , · · · , G jn ∈ G , then X

v

Lxk (u ) =

m X X

cki Li (uv )

v∈NG (u) i=1

v∈NG (u)

X

+

c2,r+1 Lxjr+1 (uv ) + · · · +

v∈NG (u)

=

m X i=1

cki 

+c2,r+1

X

v∈NG (u)

X

X

c2n Lxjn (uv )

v∈NG (u)

Li (uv )

Lxjr+1 (uv ) + · · · + c2n

v∈NG (u)

X

Lxjn (uv ) = 0

v∈NG (u)

− →V n Whence, 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 equations   a11 x1 + a12 x2 + · · · + a1n xn = b1     a x +a x +···+a x = b 21 1 22 2 2n n 2  ..................................     am1 x1 + am2 x2 + · · · + amn xn = bm

with aij , bj ∈ F for integers 1 ≤ i ≤ n, 1 ≤ j ≤ m holding with rank [aij ]m×n = 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 consider − →V differential equations in G following. 24

− →V Theorem 4.3 For ∀GL ∈ G , the Cauchy problem on differential equation DGX = GL 0 − →X| − →L0 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

n X

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 the characteristic theory of partial differential equation of first order.

In fact, let

φi (x1 , x2 , · · · , xn , F ) , 1 ≤ i ≤ n be the n independent first integrals of its characteristic equations. Then F (uv ) = F ′ (uv ) − L0 x′1 , x′2 · · · , x′n−1 ∈ L2 [∆],

where, x′1 , x′2 , · · · , x′n−1 and F ′ are determined by system of equations   φ1 (x1 , x2 , · · · , xn−1 , x0n , F ) = φ1     φ (x , x , · · · , x , x0 , F ) = φ 2 1 2 n−1 2 n  ...............................     φn (x1 , x2 , · · · , xn−1 , x0n , F ) = φn

Clearly, 

D

X

v∈NG (u)

Notice that

F (uv ) =

X

v∈NG (u)

v

F (u )

v∈NG (u) X

We therefore know that

DF (uv ) =

=

X

X

L0 (uv ) = 0.

F (uv ) = 0.

v∈NG (u)

25

L(uv ) = 0.

v∈NG (u)

v∈NG (u)

xn =x0n

X

− →X − →F − →V Thus, we get a uniquely solution G = G ∈ G for the equation DGX = GL 0 − →X| − →L0 prescribed with initial data G xn =xn = G .

We know that the Cauchy problem on heat equation n

X ∂2u ∂u = c2 ∂t ∂x2i i=1 is solvable in Rn ×R if u(x, t0 ) = ϕ(x) is continuous and bounded in Rn , c a non-zero − →L − →V constant in R. For G ∈ G in Subsection 3.2, if we define − →L ∂G − → ∂L = G ∂t ∂t

and

− →L ∂G − → ∂L = G ∂xi , 1 ≤ i ≤ n, ∂xi

− →V then we can also consider the Cauchy problem in G , i.e., n

X ∂2X ∂X = c2 ∂t ∂x2i i=1 with initial values X|t=t0 , and know the result following. − →L′ − →V Theorem 4.4 For ∀ G ∈ G and a non-zero constant c in R, the Cauchy problems on differential equations

n

X ∂2X ∂X = c2 ∂t ∂x2i i=1

− →L′ − →V − →V with initial value X|t=t0 = G ∈ G is solvable in G 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 as n

X ∂2u ∂u = c2 ∂t ∂x2i i=1

− →F with initial value u|t=0 = L′ (uv ) (x) if X = G . According to the theory of partial differential equations, we know that Z +∞ (x1 −y1 )2 +···+(xn −yn )2 1 v 4t F (u ) (x, t) = e− L′ (uv ) (y1 , · · · , yn )dy1 · · · dyn . n (4πt) 2 −∞ 26

− → Labeling the semi-arc u by F (u ) (x, t) for ∀(u, v) ∈ X G , we get a labeled − →F − → − →F − →V graph G on G . We prove G ∈ G . − →L′ − →V − → By assumption, G ∈ G , i.e., for ∀u ∈ V G , X L′ (uv ) (x) = 0, v

v

v∈NG (u)

we know that X F (uv ) (x, t) v∈NG (u)

=

X

v∈NG

=

Z

1 n (4Ď€t) 2 (u)

1 n (4Ď€t) 2

Z

+∞

+∞

e−

(x1 −y1 )2 +¡¡¡+(xn −yn )2 4t

−∞

(x −y )2 +¡¡¡+(xn −yn )2 − 1 1 4t

e

−∞

Z

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

 

X

v∈NG (u)

+∞



L′ (uv ) (y1 , ¡ ¡ ¡ , yn ) dy1 ¡ ¡ ¡ dyn

1 e (0) dy1 ¡ ¡ ¡ dyn = 0 n (4Ď€t) 2 −∞ − → − →F − →V for ∀u ∈ V G . Therefore, G ∈ G and =

(x −y )2 +¡¡¡+(xn −yn )2 − 1 1 4t

n

X ∂2X ∂X = c2 ∂t ∂x2i i=1

− →L′ − →V − →V with initial value X|t=t0 = G ∈ G is solvable in G .

Similarly, we can also get a result on Cauchy problem on 3-dimensional wave − →V equation in G following. − →L′ − →V Theorem 4.5 For ∀ G ∈ G and a non-zero constant c in R, the Cauchy problems on differential equations ∂2X = c2 ∂t2

∂2X ∂2X ∂2X + + ∂x21 ∂x22 ∂x23

− →L′ − →V − →V ′ v with initial value X|t=t0 = G ∈ G is solvable in G if L (u ) is continuous and − → bounded in Rn for ∀(u, v) ∈ X G . For an integral kernel K(x, y), the two subspaces N , N

mined by N

=

2

φ(x) ∈ L [∆]|

Z

∆

27

∗

⊂ L2 [∆] are deter

K (x, y) φ(y)dy = φ(x) ,

N

∗

=

2

Ď•(x) ∈ L [∆]|

Z

K (x, y)Ď•(y)dy = Ď•(x) .

∆

Then we know the result following. − →V Theorem 4.6 For ∀GL ∈ G , if dimN = 0, then the integral equation Z − →X − →X G − G = GL ∆

− →V is solvable in G with V = L2 [∆] if and only if D L L′ E − → → − − →L′ G ,G = 0, ∀ G ∈ N ∗ .

− → Proof For ∀(u, v) ∈ X G − →X G −

Z

− →X G = GL

and

∆

D

E − →L → − L′ G ,G = 0,

− →L′ ∀G ∈ N

∗

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

if X (uv ) = F (x) and Z − → L′ L (uv ) [x]L′ (uv ) [x]dx = 0 for ∀ G ∈ N ∗ . ∆

Applying Hilbert and Schmidt’s theorem ([20]) on integral equation, we know the integral equation F (x) −

Z

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

∆

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

− →L′ for ∀ G ∈ N ∗ . Thus, there are functions F (x) ∈ L2 [∆] hold for the integral equation F (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 that X

v

F (u ) [x] −

Z

∆

v∈NG (u)

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

v∈NG (u)

which implies that, Z

∆

Thus,

K(x, y) 

X

v∈NG (u)

F (uv ) [x] =

X

X

F (uv ) [x].

v∈NG (u)

F (uv ) [x] ∈ N .

v∈NG (u)

However, if dimN = 0, there must be X

F (uv ) [x] = 0

v∈NG (u)

for ∀u ∈ V

− → − →F − →V G , i.e., G ∈ G . Whence, if dimN = 0, the integral equation − →X G −

Z

− →X G = GL

∆

− →V is solvable in G with V = L2 [∆] if and only if D L L′ E − → → − − →L′ G ,G = 0, ∀ G ∈ N ∗ . This completes the proof.

Theorem 4.7 Let the integral kernel K(x, y) : ∆ × ∆ → C ∈ L2 (∆ × ∆) be given with

Z

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

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

∆×∆

for almost n allL (x, o y) ∈ ∆ × ∆. Then there is a finite or countably infinite system − → − → i G -flows G ⊂ L2 (∆, C) with associate real numbers {λi }i=1,2,··· ⊂ R such i=1,2,···

that the integral equations

Z

− →Li [y] − →Li [x] K(x, y) G dy = λi G

∆

29

hold with integers i = 1, 2, · · ·, and furthermore, |λ1 | ≥ |λ2 | ≥ · · · ≥ 0

and

lim λi = 0.

i→∞

Proof Notice that the integral equations Z − →Li [y] − →Li [x] K(x, y) G dy = λi G ∆

is appeared as

Z

K(x, y)Li (uv ) [y]dy = λi 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, |λ1 | ≥ |λ2 | ≥ · · · ≥ 0

with

lim λi = 0.

i→∞

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

Li (uv ) [x] = 0

v∈NG (u)

− → − →Li − →V for ∀u ∈ V G , i.e., G ∈ G for integers i = 1, 2, · · ·.

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

l

− → [− → G= Ci i=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 , · · ·) = 0 u|x0 = Li (x)

30

is solvable in a Hilbert space V on domain ∆ ⊂ Rn for integers 1 ≤ i ≤ l, then the Cauchy problem

(

Fi (x, X, Xx1 , · · · , Xxn , Xx1 x2 , · · ·) = 0 − →L X|x0 = G − → − →V v such that L (u ) = Li (x) for ∀(u, v) ∈ X C i is solvable for X ∈ G . − →Lu(x) − → Proof Let X = G with Lu(x) (uv ) = u(x) for (u, v) ∈ X G . Notice that the Cauchy problem

( then appears as

Fi (x, X, Xx1 , · · · , Xxn , Xx1 x2 , · · ·) = 0 X|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 , · · ·) = 0 u|x0 = Li (x)

− →Lu(x) − → − → Let G be a labeling on G with u (uv ) (x) on uv for ∀(u, v) ∈ X G . We − →Lu(x) − →V show that G ∈ G . Notice that l

− → [− → G= Ci i=1

− → and all flows on C i is the same, i.e., the solution u (uv ) (x). Clearly, it is holden − → with conservation on each vertex in C i for integers 1 ≤ i ≤ l. We therefore know that

X

Lx0 (uv ) = 0,

u∈V

v∈NG (u)

− →Lu(x) − →V Thus, G ∈ G . This completes the proof.

− → G .

− → There are many interesting conclusions on G -flow solutions of equations by Theorem 4.8. For example, if Fi is nothing else but polynomials of degree n in one 31

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

− → [− → G= Ci i=1

− → and Li (u ) = ai ∈ C for ∀(u, v) ∈ X C i and integers 1 ≤ i ≤ l, then the v

polynomial

− →L1 − →L2 − →Ln − →Ln+1 F (X) = G ◦ X n + G ◦ X n−1 + · · · + G ◦ X + G − →C − →L1 always has roots, i.e., X0 ∈ G such that F (X0 ) = O if G 6= O and n ≥ 1. Particularly, an algebraic equation a1 xn + a2 xn−1 + · · · + an x + an+1 = 0 − → − →C with a1 6= 0 has infinite many G -flow solutions in G on those topological graphs l − → − → [− → 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 spherical solution

rg 2 1 ds2 = f (t) 1 − dt − dr 2 − r 2 (dθ2 + sin2 θdφ2 ) r 1 − rrg

for the Einstein’s gravitational equations ([9])

1 Rµν − Rg µν = −8πGT µν 2 with Rµν = Rαµαν = gαβ Rαµβν , R = gµν Rµν , G = 6.673×10−8cm3 /gs2, κ = 8πG/c4 = − → 2.08 × 10−48 cm−1 · g −1 · s2. By Theorem 4.8, we get their G -flow solutions following. Corollary 4.10 The Einstein’s gravitational equations 1 Rµν − Rg µν = −8πGT µν , 2 32

− → − →C has infinite many G -flow solutions in G , particularly on those topological graphs l − → [− → 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 Einstein’s

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

-S1

v1

S4

v2

?S2

6 y

v4

v3

S3

Fig.7 where, each Si is a spherical solution 1 rs 2 dr 2 − r 2 (dθ2 + sin2 θdφ2 ) ds2 = f (t) 1 − dt − r 1 − rrs of Einstein’s gravitational equations for integers 1 ≤ i ≤ 4. As a by-product, Theorems 4.5-4.6 can be also generalized on those topological graphs with circuit-decomposition following. Corollary 4.11 Let the integral kernel K(x, y) : ∆ × ∆ → C ∈ L2 (∆ × ∆) be given with

Z

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

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

∆×∆

for almost all (x, y) ∈ ∆ × ∆, and l

− →L [ − → G = Ci i=1

− → such that L(uv ) = L[i] (x) for ∀(u, v) ∈ X C i and integers 1 ≤ i ≤ l. Then, the integral equation

− →V

is solvable in G

− →X G −

Z

− →X G = GL

∆

with V = L2 [∆] if and only if D L L′ E − → → − − →L′ G ,G = 0, ∀ G ∈ N ∗ . 33

Corollary 4.12 Let the integral kernel K(x, y) : ∆ × ∆ → C ∈ L2 (∆ × ∆) be given with

Z

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

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

∆×∆

for almost all (x, y) ∈ ∆ × ∆, and l

− →L [ − → G = Ci i=1

− → such that L(u ) = L[i] (x) for ∀(u, v) ∈ X C i and integers 1 ≤ i ≤ l. Then, n Lo − → − → i there is a finite or countably infinite system G -flows G ⊂ L2 (∆, C) with v

i=1,2,···

associate real numbers {λi }i=1,2,··· ⊂ R such that the integral equations Z − →Li [y] − →Li [x] K(x, y) G dy = λi G ∆

hold with integers i = 1, 2, · · ·, and furthermore, |λ1 | ≥ |λ2 | ≥ · · · ≥ 0

and

lim λi = 0.

i→∞

§5. Applications to System Control − → 5.1 Stability of G -Flow Solutions − →Lu(x) − →Lu (x) Let X = G and X2 = G 1 be respectively solutions of F (x, Xx1 , · · · , Xxn , Xx1 x2 , · · ·) = 0 − →L − →L1 − →V on the initial values X|x0 = G or X|x0 = G in G with V = L2 [∆], the Hilbert − → space. The G -flow solution X is said to be stable if there exists a number δ(ε) for any number ε > 0 such that

if

By definition,

L → u (x) − →Lu(x) − kX1 − X2 k = G 1 − G <ε L → 1 − →L − G − G ≤ δ(ε).

L → 1 − →L − G − G =

X

(u,v)∈X

− →

34

G

kL1 (uv ) − L (uv )k

and

L → u1 (x) − →Lu(x) − −G G =

X

(u,v)∈X

− →

ku1 (uv ) (x) − u (uv ) (x)k .

G

− → Clearly, if these G -flow solutions X are stable, then ku1 (uv ) (x) − u (uv ) (x)k ≤

X

(u,v)∈X

if kL1 (uv ) − L (uv )k ≤

X

(u,v)∈X

− → G

− →

ku1 (uv ) (x) − u (uv ) (x)k < ε

kL1 (uv ) − L (uv )k ≤ δ(ε),

G

− → i.e., u (uv ) (x) is stable on uv for (u, v) ∈ X G .

− → Conversely, if u (uv ) (x) is stable on uv for (u, v) ∈ X G , i.e., for any number − → ε/ε G > 0 there always is a number δ(ε) (uv ) such that ku1 (uv ) (x) − u (uv ) (x)k < if

ε − → ε G

kL1 (uv ) − L (uv )k ≤ δ(ε) (uv ) , then there must be ε − → ku1 (uv ) (x) − u (uv ) (x)k < ε G × − → =ε ε G − → (u,v)∈X G X

if kL1 (uv ) − L (uv )k ≤ − → − → where ε G is the number of arcs of G and

δ(ε) , − → ε G

n o − → δ(ε) = min δ(ε) (uv ) |(u, v) ∈ X G .

Whence, we get the result following.

35

− → Theorem 5.1 Let V be the Hilbert space L2 [∆]. The G -flow solution X of equation ( F (x, X, Xx1 , ¡ ¡ ¡ , Xxn , Xx1 x2 , ¡ ¡ ¡) = 0 − →L X|x0 = G − →V in G is stable if and only if the solution u(x) (uv ) of equation ( F (x, u, ux1 , ¡ ¡ ¡ , uxn , ux1 x2 , ¡ ¡ ¡) = 0 − →L u|x0 = G − → is stable on the semi-arc u for ∀(u, v) ∈ X G . v

− → This conclusion enables one to find stable G -flow solutions of equations. For ex-

ample, we know that the stability of trivial solution y = 0 of an ordinary differential equation

dy = [A]y dx with constant coefficients, is dependent on the number Îł = max{ReÎť : Îť ∈ Ďƒ[A]} ([23]), i.e., it is stable if and only if Îł < 0, or Îł = 0 but m′ (Îť) = m(Îť) for all eigenvalues Îť with ReÎť = 0, where Ďƒ[A] is the set of eigenvalue of the matrix [A], m(Îť) the multiplicity and m′ (Îť) the dimension of corresponding eigenspace of Îť. Corollary 5.2 Let [A] be a matrix with all eigenvalues Îť < 0, or Îł = 0 but m′ (Îť) = m(Îť) for all eigenvalues Îť with ReÎť = 0. Then the solution X = O of differential equation

dX = [A]X dx

− →V − → − → is stable in G , where G is such a topological graph that there are G -flows hold with the equation. − → For example, the G -flow shown in Fig.8 following v1 v2 g(x)

-

f (x)

6 6

g(x)

v4

f (x) f (x)

j

g(x) Fig.8 36

g(x)

? ?f (x)

v3

− → is a G -flow solution of the differential equation d2 X dX +5 + 6X = 0 2 dx dx with f (x) = C1 e−2x + C1′ e−3x and g(x) = C2 e−2x + C2′ e−3x , where C1 , C1′ and C2 , C2′ are constants. Similarly, applying the stability of solutions of wave equations, heat equations and elliptic equations, the conclusion following is known by Theorem 5.1. − → Corollary 5.3 Let V be the Hilbert space L2 [∆]. Then, the G -flow solutions X of equations following 2  2 2 ∂ X ∂ X ∂ X − →L  2  − c + = G  2 2 ∂t2 ∂x1 ∂x2

, − →Lϕ(x1 ,x2 ) − →Lµ(t,x1 ,x2 ) − →Lφ(x1 ,x2 ) ∂X

  =G , X|∂∆ = G ,  X|t0 = G ∂t t0   2 ∂2X ∂2X ∂2X  →L  ∂ X − c2 ∂X = −  + + =O G ∂x21 ∂x22 ∂x23 ∂t2 ∂x1 and − →Lφ(x1 ,x2 )   →Lg(x1 ,x2 ,x3 )  X|∂∆ = − X|t0 = G G

− →V − → − → are stable in G , where G is such a topological graph that there are G -flows hold with these equations. 5.2 Industrial System Control An industrial system with raw materials M1 , M2 , · · · , Mn , products (including byproducts) P1 , P2 , · · · , Pm but w1 , w2 , · · · , ws wastes after a produce process, such as those shown in Fig.9 following, M1 M2

Mn

x1 x2

- ?F (x) 6

xn

?

w1

?

w2

Fig.9 37

y1

- P1

y2

- P2

ym

- Pm

?

ws

i.e., an input-output system, where, (y1 , y2, · · · , ym ) = F (x1 , x2 , · · · , xn ) determined by differential equations, called the production function and constrained with the conservation law of matter, i.e., m X i=1

yi +

s X

wi =

i=1

n X

xi .

i=1

Notice that such an industrial system is an opened system in general, which can be transferred into a closed one by letting the nature as an additional cell, i.e., all materials comes from and all wastes resolves by the nature, a classical one on human beings with the nature. However, the resolvability of nature is very limited. Such a classical system finally resulted in the environmental pollution accompanied with the developed production of human beings. Different from those of classical industrial systems, an ecologically industrial system is a recycling system ([24]), i.e., all outputs of one of its subsystem, including products, by-products provide the inputs of other subsystems and all wastes are − → disposed harmless to the nature. Clearly, such a system is nothing else but a G -flow − → because it is holding with conservation laws on each vertex in a topological graph G , − → where G is determined by the technological process for products, wastes disposal − →V and recycle, and can be characterized by differential equations in Banach space G . − →Lu v v Whence, we can determine such a system by G with Lu : u → u (u ) (t, x) for − → (u, v) ∈ X G , or ordinary differential equations   − →L0 dk X − →L1 dk−1X − →Lu(t,x)   G ◦ k + G ◦ k−1 + · · · + G =O dt

dt

k−1

Lh0 (x) dX

Lh1 (x) dX − → − → − →Lh (x) 

 , =G , · · · , k−1

= G k−1  X|t=t0 = G

dt t=t0 dt t=t0

for an integer k ≥ 1, or a partial differential equation  →L0 ∂X − →L1 ∂X − →Ln ∂X − →Lu(t,x)  − +G ◦ +···+ G ◦ =G G ◦ ∂t ∂x1 ∂xn − →Lu(x)  X|t=t0 = G

− →Li and characterize its stability by Theorem 5.1, where, the coefficients G , i ≥ 0 are determined by the technological process of production. 38

References [1] John B.Conway, A Course in Functional Analysis, Springer-Verlag New York,Inc., 1990. [2] J.L.Gross and T.W.Tucker, Topological Graph Theory, John Wiley & Sons, 1987. [3] F.John, Partial Differential Equations (4th edition), Springer-Verlag New York,Inc., 1982. [4] Kenneth Hoffman and Ray Kunze, Linear Algebra(Second Edition), PrenticeHall, Inc. Englewood Cliffs, New Jersey, 1971. [5] Linfan Mao, The simple flow representation for graph and its application, Reported at The First International Symposium on Combinatorial Optimization of China, Tianjing, 1988. [6] Linfan Mao, Analytic graph and its applications, Reported at The 6th Conference on Graph Theory and Its Applications of China, Qingdao, 1989. [7] Linfan Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries, First edition published by American Research Press in 2005, Second edition is as a Graduate Textbook in Mathematics, Published by The Education Publisher Inc., USA, 2011. [8] Linfan Mao, Smarandache Multi-Space Theory(Second edition), First edition published by Hexis, Phoenix in 2006, Second edition is as a Graduate Textbook in Mathematics, Published by The Education Publisher Inc., USA, 2011. [9] Linfan Mao, Combinatorial Geometry with Applications to Field Theory, First edition published by InfoQuest in 2005, Second edition is as a Graduate Textbook in Mathematics, Published by The Education Publisher Inc., USA, 2011. [10] Linfan Mao, Non-solvable spaces of linear equation systems. International J. Math. Combin., Vol.2 (2012), 9-23. [11] Linfan Mao, Graph structure of manifolds listing, International J.Contemp. Math. Sciences, Vol.5, 2011, No.2,71-85. [12] Linfan Mao, A generalization of Seifert-Van Kampen theorem for fundamental groups, Far East Journal of Math.Sciences, Vol.61 No.2 (2012), 141-160. [13] Linfan Mao, Global stability of non-solvable ordinary differential equations with applications, International J.Math. Combin., Vol.1 (2013), 1-37. [14] Linfan Mao, Non-solvable equation systems with graphs embedded in Rn , In39

ternational J.Math. Combin., Vol.2 (2013), 8-23. [15] Linfan Mao, Geometry on GL -system of homogenous polynomials, International J.Contemp. Math. Sciences (Accepted, 2014). [16] Linfan Mao, A topological model for ecologically industrial systems, International J.Math. Combin., Vol.1(2014), 109-117. [17] Linfan Mao, Cauchy problem on non-solvable system of first order partial differential equations with applications, Methods and Applications of Analysis (Accepted). [18] Linfan Mao, Mathematics on non-mathematics – a combinatorial contribution, International J.Math.Combin., Vol.3, 2014, 1-34. [19] Linfan Mao, Geometry on non-solvable equations – A review on contradictory systems, Reported at the International Conference on Geometry and Its Applications, Jardpour University, October 16-18, 2014, Kolkata, India. [20] F.Sauvigny, Partial Differential Equations (Vol.1 and 2), Springer-Verlag Berlin Heidelberg, 2006. [21] F.Smarandache, Paradoxist Geometry, State Archives from Valcea, Rm. Valcea, Romania, 1969, and in Paradoxist Mathematics, Collected Papers (Vol. II), Kishinev University Press, Kishinev, 5-28, 1997. [22] F.Smarandache, Multi-space and multi-structure, in Neutrosophy. Neutrosophic Logic, Set, Probability and Statistics, American Research Press, 1998. [23] Wolfgang Walter, Ordinary Differential Equations, Springer-Verlag New York,Inc., 1998. [234] Zengjun Yuan and Jun Bi, Industrial Ecology (in Chinese), Science Press, Beijing, 2010.

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