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
60
LINFAN MAO
vu u
EJMAA-2015/3(2)
-
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
EJMAA-2015/3(2)
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
62
LINFAN MAO
implies that
X
EJMAA-2015/3(2)
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 }
EXTENDED BANACH G -FLOW SPACES
EJMAA-2015/3(2)
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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LINFAN MAO
EJMAA-2015/3(2)
(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
EXTENDED BANACH G -FLOW SPACES
EJMAA-2015/3(2)
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.
EXTENDED BANACH G -FLOW SPACES
EJMAA-2015/3(2)
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
-
EJMAA-2015/3(2)
(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
EXTENDED BANACH G -FLOW SPACES
EJMAA-2015/3(2)
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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LINFAN MAO
EJMAA-2015/3(2)
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.
EXTENDED BANACH G -FLOW SPACES
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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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LINFAN MAO
EJMAA-2015/3(2)
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
EJMAA-2015/3(2)
EXTENDED BANACH G -FLOW SPACES
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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LINFAN MAO
--ete ?=t et}6-t
EJMAA-2015/3(2)
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 ,
EXTENDED BANACH G -FLOW SPACES
EJMAA-2015/3(2)
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
EXTENDED BANACH G -FLOW SPACES
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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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LINFAN MAO
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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]
EXTENDED BANACH G -FLOW SPACES
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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LINFAN MAO
EJMAA-2015/3(2)
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)
EXTENDED BANACH G -FLOW SPACES
EJMAA-2015/3(2)
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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LINFAN MAO
EJMAA-2015/3(2)
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 .
EXTENDED BANACH G -FLOW SPACES
EJMAA-2015/3(2)
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 .
88
LINFAN MAO
EJMAA-2015/3(2)
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