EXTENDED BANACH G -FLOW SPACES ON DIFFER

Electronic Journal of Mathematical Analysis and Applications,Vol. 3(2) July 2015, pp. 59-91.ISSN: 2090-729(online)http://fcag-egypt.com/Journals/EJMAA/

EXTENDED BANACH G -FLOW SPACES ON DIFFERENTIALEQUATIONS WITH APPLICATIONSLINFAN MAO

Abstract. Let V be a Banach space over a field F . A G -flow is a graphG embedded in a topological space S associated with an injectivemappingsL : uv L(uv ) V such that L(uv ) = L(vu ) for (u, v) X G holdingwith conservation lawsXL (vu ) = 0 for v V G ,uNG (v)

wheredenotes the semi-arc of (u, v) X G , which is a mathematicalobject 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.uv

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 graphG is an embedding of an oriented graph G in a topological space C . All elementsin 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-arcvuu and end semi-arc v , such as those shown in Fig.1 following.

1991 Mathematics Subject Classification. 05C78,34A26,35A08,46B25,46E22,51D20.Key words and phrases. Banach space, topological graph, conservation flow, topological graph,differential flow, multi-space solution of equation, system control.Submitted Dec. 15, 2014.59

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LINFAN MAO

vu u

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-

L(uv )

vu

v

Fig.1All these semi-arcs of a topological graph G are denoted by X 21 G .A vector labeling G L on Gis a 1 1 mapping L : G V such that L : uv

L(uv ) V for uv X 12 G , such as those shown in Fig.1. For all labelings G Lon G , defineL1 G + G L2 = G L1 +L2 and G L = G L .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 21 G hold withL (uv ) = L (v u ) and conservation lawsXL (v u ) = 0uNG (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(v

u4

) u4

-

- L(v

u5

) u5

u1

L(v u1 )

u2

L(v u2 )

u3

-

L(v u3 )

v

-F (v

u6

) u6

Fig.2Clearly, if V = Z and O = {1}, then the G -flow G L is nothing else but the networkflow X( G) Z on G . Let G L , G L1 , G L2 be G -flows on a topological graph G and F a scalar.L1 L2LIt is clear that G + Gand G are also G -flows, which implies that allconservation G -flows on G also form a linear space over F with unit G 0 underV0operations + and, denotedby G , where G is such a G -flow with vector 0 onuv 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 appliedto differential operators in [6], which has shown its important role both in mathematics and applied sciences. It should be noted that a conservation law naturallydetermines an autonomous systems in the world. We can also find G -flows bysolving conservation equationsXL (v u ) = 0,vV G .uNG (v)

EXTENDED BANACH G -FLOW SPACES

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61

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 systems (h1 ; R1 ),ie Re(2 ; R2 ), , (m ; Rm ), different two by two. A topological structure GL ;

e Re is inherited byon ;

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 6= j m, i.e., a topological vertex-edge labeled graph. Clearly,i 1 TL e eG ; R is a 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 withoperator 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 industrialsystems. All of these discussions provide new viewpoint for mathematical elements,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.conservative if

Let V be a Banach space. A family V of vectors v V isX

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 thatv=

dimVX

xvi i .

i=1

Consequently,X dimVX

vV

i=1

xvi i =

dimVXi=1

X

vV

xvi

!

i = 0

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LINFAN MAO

implies that

X

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xvi = 0

vV

for integers 1 i dimV .Conversely, if

X

xvi = 0, 1 i dimV ,

vV

define

vi =

dimVX

xvi i

i=1

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, a vector 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 following resultestablishes its inverse.Theorem 2.3. A G -flow G L 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 ) and\L(v) (L(u)) = L(v u ) or .Proof Notice that

for v V G implies

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 }


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63

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

with an orientation v u on its each arcs. Then, it is clear that G L is a G -flowby definition.Conversely, if G L is a G -flow, letL(v) = {L(v u ) V for (v, u) X G }

G . Then, it is also clear that L(v), v V G are conservationfamilies 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 -flowsona topologicalgraphG with

weights v for v V , particularly, e i on e X G if X G V G +1.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 equationsX(v,u) = 0, v V GProof

uV

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 G

kL(uv )k

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

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

(2) G L = G L for any scalar .

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(3) G L1 + G L2 G L1 + G L2 because of

X

L1

kL1 (uv ) + L2 (uv )k

G + G L2 =

(u,v)X G

X

(u,v)X G

kL1 (uv )k +

X

(u,v)X G

kL2 (uv )k = G L1 + G L2 .

Whence, k k is a norm on linear space G V .Furthermore, if V is an inner space with inner product h, i, defineDXL1 EG , G L2 =hL1 (uv ), L2 (uv )i .Then we know thatD E(4) G L , G L =

(u,v)X G

P

hL(uv ), L(uv )i 0 and

(u,v)X G

DL EG , G L = 0 if and only

if L(uv ) = 0 for (u, v) X( G), i.e., G L = O.DEDE (5) G L1 , G L2 = G L2 , G L1 for G L1 , G L2 G V because ofDXXL1 EG , G L2=hL1 (uv ), L2 (uv )i =hL2 (uv ), L1 (uv )i(u,v)X G

=

X

(u,v)X G

(u,v)X G

hL2 (uv ), L1 (uv )i =

DL2 EG , G L1

(6) For G L , G L1 , G L2 G V , there isD E G L1 + G L2 , G LDD E E= G L1 , G L + G L2 , G L

because ofD G L1

+=

E D E G L2 , G L = G L1 + G L2 , G LXhL1 (uv ) + L2 (uv ), L(uv )i

(u,v)X( G )

=

X

(u,v)X( G )

==

hL1 (uv ), L(uv )i +

X

(u,v)X( G )

DL1 E D EG, G L + G L2 , G LDD E E G L1 , G L + G L2 , G L .

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

rDL E

L G , GL

G =

hL2 (uv ), L(uv )i


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65

for G L G V . Then it is a normed space. Furthermore, we know the followingresult. Theorem 2.5. For any topological graph G , G V is a Banach space, andfurthermore, if V is a Hilbert space, G V is a Hilbert space also.Proof As shown in the previous, G V is a linear normed space or inner spaceif V is an inner space. We show thatit is also complete, i.e., any Cauchy sequencen oin G V is converges. In fact, let G Ln be a Cauchy sequence in G V . Thus forany number > 0, there always exists an integer N () such that

Ln

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

kLn (uv ) Lm (uv )k G Ln G Lm < i.e., {Ln (uv )} is also a Cauchy sequence for (u, v) X G , which is convergeson in V by definition.Let L(uv ) = lim Ln (uv ) for (u, v) X G . Then it is clear thatn

lim G Ln = G L .

n

However, we are needed to show G L G V . By definition,XLn (uv ) = 0vNG (u)

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

vNG (u)

vNG (u)

=

X

n

L(uv ) = 0.

vNG (u)

Thus, G L G V .

L2L1Similarly,Etwo conservation G -flows G and G are said to be orthogonal ifDL1 G , G L2 = 0. The following result characterizes those of orthogonal pairs ofconservation G -flows. L1L2Theorem 2.6. Let G L1 , G L2 G V . Then G is orthogonal to G if andonly 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,DXL1 EG , G L2 =hL1 (uv ), L2 (uv )i = 0,(u,v)X G

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LINFAN MAO

EJMAA-2015/3(2)

i.e., G L1 is orthogonal to G L2 .Conversely, if G L1 is indeed orthogonal to G L2 , thenDL1 EG , G L2 =

X

(u,v)X G

hL1 (uv ), L2 (uv )i = 0

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

Theorem 2.7. Let V be a Hilbert space with an orthogonal decompositionV = V V for a closed subspace V V . Then there is a decompositionVe Ve ,G =V

where,

no e = VG L1 G V L1 : X G Vno e = VG L2 G V L2 : X G V ,

i.e., for G L G V , there is a uniquely decompositionL G = G L1 + G L2with L1 : X G V and L2 : X G V .

Proof By definition, L(uv ) V for (u, v) X G . Thus, there is a decompositionL(uv ) = L1 (uv ) + L2 (uv )

with uniquelydeterminedL1 (uv ) V but L2 (uv ) V .hhL1 iL2 iLet Gand Gbe two labeled graphs on G with L1 : X 12 G V andh i h i D EL2 : X 21 G V . We need to show that G L1 , G L2 G , V . In fact,the conservation laws show thatX

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

vNG (u)

for u V

X

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

vNG (u)

G . Consequently,X

vNG (u)

L1 (uv ) +

X

vNG (u)

L2 (uv ) = 0.


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67

Whence,0 =

=

+

=

*

*

*

*

+

X

v

L1 (u ) +

vNG (u)

X

X

L1 (u ),

vNG (u)

v

X

L1 (u ),

v

X

X

vNG (u)

+

L1 (uv )

vNG (u)v

v

hL1 (u ), L2 (u )i +

vNG (u)

=

*

+

L2 (u )

vNG (u)

L1 (uv ),

+

L1 (u )

vNG (u)

X

L2 (u ),

vNG (u)v

X

v

vNG (u)v

X

X

X

+

+

+

X

v

L1 (u ),

vNG (u)

Notice that*XL1 (uv ),vNG (u)

v

X

v

+

L1 (u )

vNG (u)

0,

*

X

L2 (u )

L2 (u ),

vNG (u)

X

X

v

+

v

L2 (u )

vNG (u)

X

v

L2 (u ),

vNG (u)

vNG (u)

X

X

L2 (uv ),

vNG (u)

+

v

+

L1 (u )

+

L2 (uv )

vNG (u)v

hL2 (u ), L1 (u )i

+

L1 (u )

vNG (u)

*

v

vNG (u)

v

vNG (u)

X

L1 (u ) +

vNG (u)

*

X

v

+

*

*

X

v

L2 (u ),

vNG (u)

X

X

v

L2 (u ) .

vNG (u)

X

v

L2 (u ),

v

+

0.

v

+

= 0,

L2 (u )

vNG (u)

vNG (u)

+

We therefore get that*

X

vNG (u)

v

L1 (u ),

+

*

X

L1 (u )

X

L1 (uv ) = 0 and

v

vNG (u)

= 0,

X

X

L2 (u ),

X

L2 (uv ) = 0.

v

vNG (u)

vNG (u)

L2 (u )

i.e.,

vNG (u)

vNG (u)

h i h i Thus, G L1 , G L2 G V . This completes the proof.

2.3 Solvable G -Flow Spaces

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

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

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

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LINFAN MAO

-

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(0, 1)

v1

v2

(1, 0)

(0, 1)

? 6(1, 0)

(1, 0)

-

? 6(0, 1)

(1, 0)v4

v3

(0, 1)

Fig.3

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

+dv; 1 i n L(v u ); L u , u NG(v) = 0Lvdxior

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

+dunless L v u0 for v V G , where, Lv; 1 i n , Lv;1 i ndxixidenote an ordinary or partial differential operators, respectively.Notice that for a strong-connected graph G , there must be a decomposition!!m1m[ [ [2 G=CiTi ,i=1

i=1

where C i , T j are respectively directed circuit or path in G with m1 1, m2 0.The following result depends on the structure of G .Theorem 2.8. For a strong-connected topological graph G with decomposition!!mm[1 [ [2 G=CiT i , m1 1, m2 0,i=1

i=1

there always exist linear G -flows G L , not all flows being zero on G .

kk kkk ui+1Proof For an integer 1 k m1 , let C k = u1 u2 usk and L ui= vk ,where i + 1 (mods). Similarly, for integers 1 j m2 , if T j = w1j w2j wtj , let


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69

wtL wjt j+1 = 0. Clearly, the conservation law hold at v V G by definition,

kk ui+1and each flow L ui, i + 1 (modsk ) is linear determined by

kk ui+1L ui==

kXk uiL ui1+0

X

j6=i vN (uk )iC

vk + 0 +

m2X

X

j

m2 k XL v ui +

X

j=1 vN (uk )iTj

kL v ui

0 = vk .

j=1 vN (uk )iTj

Thus, G L 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). Furthermore, if m1 2, there is at least two circuits C , C in the decomposition of G . Let flows on C and C be respectively x andf(x, t). We then know the conservation laws hold for vertices in G , and similarly,there are indeed flows on G determined by ordinary differential equations.Theorem 2.9. For a strong-connected topological graph G with decomposition!!mm[1 [ [2 T i , m1 2, m2 0,CiG=i=1

i=1

there always exist ordinary differential G -flows G L , 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)

? 6xv4

x

x-

? 6f(x, t)v3

f(x, t)Fig.4Similarly, we know the existence of non-trivial partial differential G -flows. Letx = (x1 , x2 , , xn ). If xi = xi (t, s1 , s2 , , sn1 )u = u(t, s1 , s2 , , sn1 )pi = pi (t, s1 , s2 , , sn1 ),i = 1, 2, , n

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is a solution of systemdx1Fp1

dx2dxndu= == PnFp2Fpnpi Fpi

=

i=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 ), i = 1, 2, , n

such that F (x10 , xn20 , , xn0 , u, p10 , p20 , , pn0 ) = 0u0 Xxi0pi0= 0, j = 1, 2, , n 1, sj sji=0

then it is the solution of Cauchy problem

F (x1 , x2 , , xn , u, p1 , p2 , , pn ) = 0xi = xi0 (s1 , s2 , , sn1 ), u0 = u0 (s1 , s2 , , sn1 ) , 0pi0 = pi0 (s1 , s2 , , sn1 ), i = 1, 2, , n

uFand Fpi =for integers 1 i n.xipiFor partial differential equations of second order, the solutions of Cauchy problemon heat or wave equationsn22n u = a2 X u2X u u = a2t2x2ii=1

tx2i ,i=1u = (x)u|t=0 = (x) u|t=0 = (x) ,t where pi =

t=0

are respectively knownu (x, t) =

1n(4t) 2

Z

+

e

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

(y1 , , yn )dy1 dyn

for heat equation andu (x1 , x2 , x3 , t) =

1t 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. Then, the result following on partial G -flows is similarly known tothat of Theorem 2.9.


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71

Theorem 2.10. For a strong-connected topological graph G with decomposition!!mm[1 [ [2 G=CiT i , m1 2, m2 0,i=1

i=1

there always exist partial differential G -flows G L , not all flows being zero on G .3. Operators on G -Flow Spaces3.1 Linear Continuous Operators

Definition 3.1. Let T O be an operator on Banach space V over a field F .An operator T : G V G V is bounded if

T G L G L for G L G V with a constant [0, ) and furthermore, is a contractor if

T G L1 T G L2 G L1 G L2 ) for G L1 , G L1 G V with [0, 1).

Theorem 3.2. Let T : G V G V be a contractor. Then there is a uniquelyconservation G-flow G L G V such that T G L = G L.n oProof Let G L0 G V be a G-flow. Define a sequence G Ln by L1G = T G L0 , L2G = T G L1 = T 2 G L0 ,. . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . .. ,LnG = T G Ln1 = Tn G L0 ,..................................

n oWe prove G Ln is a Cauchy sequence in G V . Notice that T is a contractor.For any integer m 1, we know that

Lm+1

G Lm = T G Lm T G Lm1

G

G Lm G Lm1 = T G Lm1 T G Lm2

2 G Lm1 G Lm2 m G L1 G L0 .

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

Lm

G Ln

G

G Lm G Lm1 + + + G Ln1 G Ln

m + m1 + + n1 G L1 G L0 n1 m

= G L1 G L0 1 n1

G L1 G L0 for 0 < < 1.1

n o

Consequently, G Lm G Ln 0 if m , n . So the sequence G Lnis a Cauchy sequence and converges to G L . Similar to the proof of Theorem 2.5,we know it is a G-flow, i.e., G L G V . Notice that

L

G L G Lm + G Lm T G L

G T GL

G L G Lm + G Lm1 G L .

Let m . For 0 < < 1, we therefore get that G L T G L = 0, i.e.,

T G L = G L. For theuniqueness,if there is an another conservation G-flow G L G V holding

with T G L = G L , by

L

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

Definition 3.3. An operator T : G V G V is linear if T G L1 + G L2 = T G L1 + T G L2

for G L1 , G L2 G V and , F , and is continuous at a G -flow G L0 if therealways exist a number () for > 0 such that

T G L T G L0 < if G L G L0 < ().The following result reveals the relation between conceptions of linear continuouswith that of linear bounded.Theorem 3.4. An operator T : G V G V is linear continuous if and only ifit is bounded.Proof If T is bounded, then

T G L T G L0 = T G L G L0 G L G L0

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73

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

L

6= 0,

G G L0 < () with () = ,then there must be

T G L G L0 < ,i.e., T is linear continuous on G V . However, this is obvious for = 0. n oNow if T is linear continuous but unbounded, there exists a sequence G Ln inVG such that

Ln

G n G Ln .Let

LnG =

Ln1

Ln G .n G

1

Then G Ln = 0, i.e., T G Ln 0 if n . However, by definitionn

Ln

G

T

T G Ln =

Ln

n G

n G Ln

T G Ln

= 1,=

n G Ln n G Ln a contradiction. Thus, T must be bounded.

The following result is a generalization of the representation theorem of Frechetand Riesz on linear continuous functionals, i.e., T : G V C on G -flow space G V ,where C is the complex field.Theorem 3.5. Let T : G V C be a linear continuous functional. Then thereLb Vis a unique G G such that D b ET G L = G L, G Lfor G L G V .

Proof Define a closed subset of G V byno N (T) = G L G V T G L = 0

for the linear continuous functional T. If N (T) = G V , i.e., T G L = 0 forb G L G V , choose G L = O. We then easily obtain the identity D bET G L = G L, G L .

Whence, we assume that N (T) 6= G V . In this case, there is an orthogonal decompositionVG = N (T) N (T)with N (T) 6= {O} and N (T) 6= {O}.

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Choose a G -flow G L0 N (T) with G L0 6= O and defineL L0 LG = T GLG T G L0G

for G L G V . Calculation shows that T G L = T G L T G L0 T G L0 T G L = 0, i.e., G L N (T). We therefore get thatDL E0 =G , G L0D L0 L E=T GLG T G L0G , G L0 DL0 E DL E= T GLG , G L0 T G L0G , G L0 .Notice that

DL0 E

2G , G L0 = G L0 6= 0. We find that

T GL=Let

L0 *+T G L 0 DL L0 EL T GL0G ,G= G , G.

L0 2

2

G

G L0 L0 TGLbL0 G = G = G L0 ,L0

2

G

T G L0L DL Lb Ewhere = .WeconsequentlygetthatTG=G,G .

2

G L0 b D bENow if there is another G L G V such that T G L = G L , G L for G L DVb bE G , there must be G L , G L G L = 0 by definition. Particularly, let G L =bLb G G L . We know that

Dbb b b bE

Lb

G G L = G L G L , G L G L = 0,b bb b which implies that G L G L = O, i.e., G L = G L .

3.2 Differential and Integral OperatorsLet V be Hilbert space consisting of measurable functions f (x1 , x2 , , xn ) ona 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 []


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75

and G V its G -extension on a topological graph G . The differential operator andintegral operatorsZZnXand,D=aixii=1

on G V are respectively defined by

vD G L = G DL(u )

andZ

Z

LG

=

LG

=

Z

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

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

Z

aifor (u, v) X G , where ai , C0 () for integers 1 i, j n andxjK(x, y) : C L2 ( , C) withZ

K(x, y)dxdy < .

Such integral operators are usually called adjoint for K (x, y). Clearly, for G L1 , G L2 G V and , F ,

Z

=

Z

by K (x, y) =

vvvvvvD G L1 (u ) + G L2 (u ) = D G L1 (u )+L2 (u ) = G D(L1 (u )+L2 (u ))vvvv= G D(L1 (u ))+D(L2 (u )) = G D(L1 (u )) + G D(L2 (u ))

vvvv= D G (L1 (u )) + G (L2 (u )) = D G L1 (u ) + D G L2 (u )

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

D G L1 + G L2 = D G L1 + D G L2 .

Similarly, we know also thatZ ZZ L1L2 G L1 + G L2 = G +G ,

Z

Thus, operators D,

Z

Z G L1 + G L2 =

and

Z

L1G

are al linear on G V .

Z+

L2G .

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--ete ?=t et}6-t

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

t

t

D

t

t

et

t

-e-tRt6e }= e ?t-t

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

t

t

[0,1]

t

t

b(t)

,

R

[0,1]

et

Fig.5For example, let f (t) = t, g(t) = et , K(t, ) = t2 + 2 for = [0, 1] and let G Lbe the G -flow shown on the left in Fig.6. Then, we know that Df = 1, Dg = et ,Z

1

K(t, )f ( )d =

0

Z

1

K(t, )g( )d =

0

Z

Z

1

K(t, )f ( )d =0

Z

1

0

1

K(t, )g( )d =0

Z

1

0

t21t2 + 2 d =+ = a(t),24

t2 + 2 e d

= (e 1)t2 + e 2 = b(t)

and the actions D G L ,

Z

LG and

[0,1]

Z

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

Furthermore, we know that both of them are injections on G V .ZTheorem 3.6.D : G V G V and: GV GV .

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

i.e., the conservation lawsX

v

DL(u ) = 0 and

vNG (u)

hold with v V

X

vNG (u)

Z

L(uv ) = 0

G .

vHowever, because of G L(u ) G V , there must beX

vNG (u)

Z

L(uv ) = 0 for v V

G ,

L G GV ,


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we immediately know that0 = D

and

0=

Z

for v V

G .

X

vNG (u)

X

vNG (u)

L(uv ) =

L(uv ) =

X

77

DL(uv )

vNG (u)

X

vNG (u)

Z

L(uv )

4. G -Flow Solutions of EquationsAs we mentioned, all G-solutions of non-solvable systems on algebraic, ordinaryor partial differential equations determined in [13]-[19] are in fact G -flows. Weshow there are also G -flow solutions for solvable equations, which implies that theG -flow solutions are fundamental for equations.4.1 Linear EquationsLet V be a field (F ; +, ). We can further defineL1 G G L2 = G L1 L2with L1 L2 (uv ) = L1 (uv ) L2 (uv ) for (u, v) X G . Then it can be verified

c isomorphic to F ifeasily that G F is also a field G F ; +, with a subfield Fthe conservation laws is not emphasized, wherenoc= .G L G F |L (uv ) is constant in F for (u, v) X GF

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

EClearly, G F F

aX = G L

1is uniquely solvable for X = G a L in G F if 0 6= a F . Particularly,if one viewsLvan element b F as b = G if L(u ) = b for (u, v) X G and 0 6= a F , thenan algebraic equationax = b 1in F also is an equation in G F with a solution x = G a L such as those shown in Fig.6 for G = C 4 , a = 3, b = 5 following.

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-

53

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 linearsystems is known following.nTheorem 4.1. A linear system (LESm) of equationsa11 X1 + a12 X2 + + a1n Xn = G L1a21 X1 + a22 X2 + + a2n Xn = G L2n(LESm).......................................am1 X1 + am2 X2 + + amn Xn = G Lmwith aij C and G Li G V for integers 1 i n and 1 j m is solvable forXi G V , 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 a21= ...am1

a12a22...am2

...

a1na2n....amn

L1L2 .... Lm

Let Xi = G Lxi with Lxi (uv ) V on (u, v) X G for integersn1 i n. For (u, v) X G , the system (LESm) appears as a common linearsystema11 Lx1 (uv ) + a12 Lx2 (uv ) + + a1n Lxn (uv ) = L1 (uv )a21 Lx1 (uv ) + a22 Lx2 (uv ) + + a2n Lxn (uv ) = L2 (uv )..........................................................am1 Lx1 (uv ) + am2 Lx2 (uv ) + + amn Lxn (uv ) = Lm (uv )Proof

By linear algebra, such a system is solvable if and only if ([4])+

rank [aij ]mn = rank [aij ]m(n+1) (uv )for (u, v) G .Labeling the semi-arcuv respectively by solutions Lx1 (uv ), Lx2 (uv ), , Lxn (uv )for (u, v) X G , we get labeled graphs G Lx1 , G Lx2 , , G Lxn . We prove thatLx G 1 , G L x2 , , G L xn G V .


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79

Let rank [aij ]mn = r. Similar to that of linear algebra, we are easily know thatmPXj1 =c1i G Li + c1,r+1 Xjr+1 + c1n Xjni=1m X = Pc2i G Li + c2,r+1 Xjr+1 + c2n Xjnj2,i=1...........................................mP Xjr =cri G Li + cr,r+1 Xjr+1 + crn Xjni=1

Lxwhere {j1 , , jn } = {1, , n}. Whence, if G jr+1 , , G Lxjn G V , thenX

Lxk (uv )

X

=

mX

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

X

Lxjn (uv ) = 0

vNG (u)

vNG (u)

nWhence, the system (LESm) is solvable in G V .

The following result is an immediate conclusion of Theorem 4.1.Corollary 4.2. A linear system of equationsa11 x1 + a12 x2 + + a1n xn = b1a21 x1 + a22 x2 + + a2n xn = b2 ...................................am1 x1 + am2 x2 + + amn xn = bm

with aij , bj F for integers 1 i n, 1 j m holding with+

rank [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 considerdifferential equations in G V following.Theorem 4.3. For GL G V , the Cauchy problem on differential equationDGX = GLX|is uniquely solvable prescribed with G xn =x0n = G L0 .

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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 equationnXF (uv )ai= L (uv )xii=1

on the semi-arc uv . By assumption, ai C0 () and L (uv ) L2 [], whichimplies 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, leti (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 equations

1 x1 , x2 , , xn1 , x0n , F = 12 x1 , x2 , , xn1 , x0n , F = 2. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .n x1 , x2 , , xn1 , x0n , F = nClearly,

D

Notice that

X

vNG (u)

F (uv ) =

X

vNG (u)

v F (u )

vNG (u)X

We therefore know that

DF (uv ) =

=

xn =x0n

X

X

L(uv ) = 0.

vNG (u)

X

L0 (uv ) = 0.

vNG (u)

F (uv ) = 0.

vNG (u)

Thus, we get a uniquely solution G X = G F G V for the equationDGX = GLX|xn =x0Ln = G 0.prescribed with initial data GWe know that the Cauchy problem on heat equationnXu2u= c2tx2ii=1

is solvable in Rn R if u(x, t0 ) = (x) is continuous and bounded in Rn , c anon-zero constant in R. For G L G V in Subsection 3.2, if we define L L GL GLt=Gand= G xi , 1 i n,txithen we can also consider the Cauchy problem in G V , i.e.,nXX2X= c2tx2ii=1


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81

with initial values X|t=t0 , and know the result following. Theorem 4.4. For G L G V and a non-zero constant c in R, the Cauchyproblems on differential equationsnXX2X= c2tx2ii=1

with initial value X|t=t0 = G L G Vis solvable in G V if L (uv ) is continuousand 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=1with initial value u|t=0 = L (uv ) (x) if X = G F . 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 Labeling the semi-arc uv by F (uv ) (x, t) for (u, v) X G , we get a labeledgraph G F on G . We prove G F G V . By assumption, G L G V , i.e., for u V G ,XL (uv ) (x) = 0,vNG (u)

we know thatXF (uv ) (x, t)vNG (u)

=

X

vNG (u)

1=n(4t) 2

1n(4t) 2Z

+

e

Z

+

e

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

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

Z

+

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

X

vNG (u)

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

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

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

with initial value X|t=t0 = G L

nXX2X= c2tx2ii=1 G V is solvable in G V .

Similarly, we can also get a result on Cauchy problem on 3-dimensional waveequation in G V following.

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Theorem 4.5. For G L G V and a non-zero constant c in R, the Cauchyproblems on differential equations 2

X2X2X2X2=c++t2x21x22x23LVwith initial value X|t=t0 = G G is solvable in G V if L (uv ) is continuousand bounded in Rn for (u, v) X G .

For an integral kernel K(x, y), the two subspaces N , N L2 [] are determined by

Z(x) L2 []|K (x, y) (y)dy = (x) ,N =

ZN =(x) L2 []|K (x, y)(y)dy = (x) .

Then we know the result following.

Theorem 4.6. For GL G V , if dimN = 0, then the integral equationZXXG G = GL

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

DL EG , G L = 0,

GL 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 L (uv ) [x]L (uv ) [x]dx = 0 for G L N .

Applying Hilbert and Schmidts theorem ([20]) on integral equation, we knowthe integral equationZF (x) K (x, y) F (y) dy = L (uv ) [x]

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

for G L N . Thus, there are functions F (x) L2 [] hold for the integralequationZF (x) K (x, y) F (y) dy = L (uv ) [x]


EJMAA-2015/3(2)

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

ZX vvF (u ) [x] K(x, y)F (u ) [x] =

vNG (u)

which implies that,ZXK(x, y)

vNG (u)

Thus,

X

F (uv ) [x] =

X

83

L (uv ) [x] = 0,

vNG (u)

X

F (uv ) [x].

vNG (u)

F (uv ) [x] N .

vNG (u)

However, if dimN = 0, there must beXF (uv ) [x] = 0vNG (u)

for u V G , i.e., G F G V . Whence, if dimN = 0, the integral equationZXXG G = GL

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

Theorem 4.7. Let the integral kernel K(x, y) : C L2 ( ) begiven withZ|K(x, y)|2 dxdy > 0,dimN = 0 and K(x, y) = K(x, y)

for almostall (x, y) . Then there is a finite or countably infinite systemnLi oG -flows G L2 (, C) with associate real numbers {i }i=1,2, Ri=1,2,

such that the integral equationsZK(x, y) G Li [y] dy = i G Li [x]

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

and

lim i = 0.

i

Proof Notice that the integral equationsZK(x, y) G Li [y] dy = i G Li [x]

is appeared as

Z

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

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on (u, v) X G . By the spectral theorem of Hilbert and Schmidt ([20]), there isindeed a finite or countably system of functions {Li (uv ) [x]}i=1,2, hold with thisintegral equation, and furthermore,|1 | |2 | 0

with

lim i = 0.

i

Similar to the proof of Theorem 4.5, if dimN = 0, we know thatXLi (uv ) [x] = 0vNG (u)

for u V G , i.e., G Li G V 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.decomposition

If the topological graph G is strong-connected with circuitl [CiG=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 , ) = 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 , ) = 0X|x0 = G Lsuch that L (uv ) = Li (x) for (u, v) X C i is solvable for X G V .

Proof Let X = G Lu(x) with Lu(x) (uv ) = u(x) for (u, v) X G . Notice thatthe Cauchy problem

Fi (x, X, Xx1 , , Xxn , Xx1 x2 , ) = 0X|x0 = GLthen appears as

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


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85

Let G Lu(x) be a labeling on G with u (uv ) (x) on uv for (u, v) X G . Weshow that G Lu(x) G V . 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 knowthatXLx0 (uv ) = 0,uV G .vNG (u)

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

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 onevariable 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 decompositionl [G=Cii=1

and Li (uv ) = ai C for (u, v) X C i and integers 1 i l, then thepolynomialF (X) = G L1 X n + G L2 X n1 + + G Ln X + G Ln+1always has roots, i.e., X0 G C such that F (X0 ) = O if G L1 6= O and n 1.Particularly, an algebraic equationa1 xn + a2 xn1 + + an x + an+1 = 0with a1 6= 0 has infinite many G -flow solutions in G C 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 linearand non-linear equations in physics. For example, we know the spherical solution

rg 2122222ds2 = f (t) 1 dt r dr r (d + sin d )r1 rgfor the Einsteins gravitational equations ([9])

1R Rg = 8GT 2with R = R= g R , R = g R , G = 6.673 108 cm3 /gs2 , =8G/c4 = 2.08 1048 cm1 g 1 s2 . By Theorem 4.8, we get their G -flowsolutions following.

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Corollary 4.10. The Einsteins gravitational equations1R Rg = 8GT ,2has infinite many G -flow solutions in G C , 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 Einsteinsgravitational equations,such as those shown in Fig.7.v1

S4

-S

v2

1

?S

6v4

y

S3

2

v3

Fig.7where, each Si is a spherical solution

1rs 2dt dr2 r2 (d2 + sin2 d2 )ds2 = f (t) 1 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 ( ) begiven withZ|K(x, y)|2 dxdy > 0,K(x, y) = K(x, y)

for almost all (x, y) , andlL [ G =Cii=1

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

is solvable in G V with V = L2 [] if and only ifDL E G , G L = 0, G L N .


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87

Corollary 4.12. Let the integral kernel K(x, y) : C L2 ( ) begiven withZ|K(x, y)|2 dxdy > 0,K(x, y) = K(x, y)

for almost all (x, y) , and

lL [ G =Cii=1

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

i=1,2,

with associate real numbers {i }i=1,2, R such that the integral equationsZK(x, y) G Li [y] dy = i G Li [x]

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 SolutionsLet X = G Lu(x) and X2 = G Lu1 (x) be respectively solutions ofF (x, Xx1 , , Xxn , Xx1 x2 , ) = 0on the initial values X|x0 = G L or X|x0 = G L1 in G V with V = L2 [], theHilbert space. The G -flow solution X is said to be stable if there exists a number() for any number > 0 such that

kX1 X2 k = G Lu1 (x) G Lu(x) <

if G L1 G L (). By definition,

X

L1

kL1 (uv ) L (uv )k

G G L =and

Lu1 (x)

G Lu(x) =

G

(u,v)X G

X

(u,v)X G

ku1 (uv ) (x) u (uv ) (x)k .

Clearly, if these G -flow solutions X are stable, thenXku1 (uv ) (x) u (uv ) (x)k ku1 (uv ) (x) u (uv ) (x)k < (u,v)X G

ifkL1 (uv ) L (uv )k

X

(u,v)X G

kL1 (uv ) L (uv )k (),

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

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Conversely, if u (uv ) (x) is stable on uv for (u, v) X G , i.e., for any number/ G > 0 there always is a number () (uv ) such thatku1 (uv ) (x) u (uv ) (x)k

Documents

EXTENDED BANACH G -FLOW SPACES ON DIFFER