Fredholm quasi-linear manifolds and degree of Fredholm quasi-linear mapping between them
In this article a new class of Banach manifolds and a new class of mappings between them are presented and also the theory of degree of such mappings is given.
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© A. ABBASOV, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5 579
UDC 517.5
A. Abbasov (Canakkale Onsekiz Mart Univ., Turkey)
FREDHOLM QUASI-LINEAR MANIFOLDS AND DEGREE
OF FREDHOLM QUASI-LINEAR MAPPING BETWEEN THEM
КВАЗІЛІНІЙНІ МНОГОВИДИ ФРЕДГОЛЬМА
ТА СТЕПІНЬ КВАЗІЛІНІЙНИХ ВІДОБРАЖЕНЬ
ФРЕДГОЛЬМА МІЖ НИМИ
In this article a new class of Banach manifolds and a new class of mappings between them are presented and
also the theory of degree of such mappings is given.
Представлено новий клас многовидів Банаха та новий клас відображень між ними, а також наведено
теорію степеня таких відображень.
0. Introduction. As it is known, the degree theory for infinite-dimensional mappings
(of the kind “identical+compact”) for the first time was given by Leray and Schauder.
Afterwards, this theory was expanded up to various classes of mappings (for example,
up to class Fredholm proper mappings)1. However, these theories were not appropriate
for solution of non-linear Hilbert problem. For solution of this problem the class of
Fredholm Quasi-Linear (FQL) mappings, determined on Banach space, was introduced
by A. I. Shnirelman, and was determined the degree of such a mapping, which has all
the main properties of classical (finite-dimensional) degree (see [8]). Later,
M. A. Efendiyev expanded this theory up to FQL-mappings, determined on quasicylin-
drical domains (see [6]). In the given article, this theory is expanded up to FQL-
mappings, determined between FQL-manifolds. In more details:
In first part of this article an example of FQL-manifold, given in [2], is extended up
to example of Banach manifold from a wide class, namely up to space Hs (M, N) ,
where M and N are compact smooth manifolds of dimensions m , respectively n
and N doesn’t have boundary. First such structure is given in Hs (M,N) at m < n ,
and later, at m ! n . In the last case m ! n( ) the FQL-manifold Hs (M,N) is ap-
peared as a submanifold of the FQL-manifold Hs (M,Nk ) , where k ! n " 1( ) ≤ m <
< n ! k .
In second part of this article the degree of FQL-mapping is expanded up to FQL-
mappings between FQL-manifolds and its basic properties are proved. However, in this
part another form of FQL-mapping is used, as it is better adapted for definition of de-
gree. We named it as Fredholm Special Quasi-Linear (FSQL) mapping. The proof of
identity of FQL and FSQL-mappings is given in [1].
1 See review article [4] and later works on this subject, for example [7].
580 A. ABBASOV
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
As an example of an FQL-mapping, this mapping is given: Ff : Hs (M,N)!
! Hs (M,N) , Ff : u ! f (u) , where f : N! N is a smooth mapping with a gra-
dient distinct from zero in all points2.
Various types of Nonlinear Hilbert Problem have been solved by means of the the-
ory of degree of FQL-mapping (see [6, 8]).
The purpose of the given article is preparation of theoretical base for solution of
practical problems.
In the end we noted that some definitions and theorems from [8], which will be used
later, are given in Section 1.
1. Let X , Y be the real Banach spaces, ! be a bounded domain in X and Xn
be an n -dimensional Euclidian space. In addition, let !n : X ! Xn be a linear map-
ping and X!
n = "n#1(!) , ! "Xn .
Definition 1.1. A continuous mapping f n : !" Y is called a Fredholm Linear
(FL), if
a) on each plane X!
n , ! "Xn , which crosses with ! , f!n " f n
X!
n is an affine
invertible mapping between X!
n and its image Y!n = f (X!
n ) , which is closed in Y
and its co-dimension in Y is equal to n ;
b) f!n continuously depends on ! .
Definition 1.2. Let a sequence of FL-mappings f nk f nk : !" Y{ } uniformly
approximate to the mapping f on ! and
f!nk < C(") , ( f!nk )"1 < C(#) , ! "#nk ($) at k > k0 (!) , (1.1)
where C(!) does not depend on k . Then continuous mapping f : !" Y is called
a Fredholm Quasi-Linear (FQL).
Theorem 1.3. Any finite combination of linear (pseudo) differential operators and
operators of superposition with smooth function of finite number of arguments with a
gradient which is distinct from zero in all points, defines an FQL-mapping between
Hs and Hs!" at some ! and all sufficiently greater s .
2. Quasi-Linear manifolds. Let X be a real infinite-dimensional Banach mani-
fold, X j{ } , X j!1 " X j , j = 1, 2, ... , be a system of open sets covering to X (i.e.,
X = !X j ), ! j = Yj , Pj , Bj( ) be an affine bundle with the total space Yj , with the
base space Bj , which is a finite-dimensional manifold and with the continuous epi-
morphism Pj : Yj ! Bj . Let Dj be a bounded domain in Yj and ! j : X j ! Dj
be a homeomorphism. In this case we shall call ! j , X j( ) an L -chart on X j and we
shall say that, on X j an L -structure is introduced. If an L -structure is determined
on X j+1 , then obviously, it is determined also on X j (as an induced structure). Let
2 Proof of quasilinearity of similar mapping is given in [2].
FREDHOLM QUASI-LINEAR MANIFOLDS AND DEGREE OF FREDHOLM QUASI-LINEAR … 581
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! "j : X !j " D !j , ! ""j : X !!j " D !!j , !j , !!j " j , be an L -structures on X j . Then
the transition functions ! ""j !! "j
#1 : D !j " D !!j and ! "j !! ""j
#1 : D !!j " D !j will arise.
Let’s suppose that each of them is an FQL-mapping between affine bundles ! "j =
= Y !j , P !j , B !j( ) and ! ""j = Y ""j , P ""j , B ""j( ) , i.e., an FQL-mapping in charts of ! "j
and ! ""j in sense of Definition 1.2. In this case we shall say that two L -structures on
X j are equal.
Definition 2.1. A class of equivalent L -structures on X j is called an FQL-
structure on X j .
Obviously, the FQL-structure on X j+1 induces an FQL-structure on X j , too.
The FQL-structures on X j and X j+1 are called coordinated, if the FQL-structure on
X j coincides with the induced structure.
Definition 2.2. A collection of FQL-structures on X j , j = 1, 2, 3, ... , which are
coordinated between each other is called an FQL-structure on X .
The Banach manifold X with FQL-structure is called FQL-manifold.
Now, let us define an FQL and FSQL-mappings between FQL-manifolds.
Let X , !X be FQL-manifolds,
!j : X j ! X j+1 , X = ! X j and !i : !Xi " !Xi+1 , !X = ! !Xi .
In addition, let ! j , X j( ) , !"i , !Xi( ) be L -charts on X , !X and ! j (X j ) =
= Dj , !"i !Xi( ) = !Di be the bounded domains of ! j = Yj , Pj , Bj( ) and !"i =
= !Yi , !Pi , !Bi( ) , respectively.
Definition 2.3. A continuous mapping f : X ! "X between FQL-manifolds X
and !X is called a Fredholm Quasi-Linear (FQL), if
a) !j !i : f (X j ) ! "Xi ;
b) !"i ! f !" j
#1 : Dj ! "Di are FQL-mappings in charts of affine bundles ! j and
!"i (in sense of Definition 1.2).
Definition 2.4. A continuous mapping f ji = Yj ! "Yi is called a Fredholm Spe-
cial Linear (FSL) mapping between affine bundles ! j and !"i , if there exist sub-
bundles ! j,r = Yj , Pj,r , Bj,r( ) of ! j and !"i,r = !Yi , !Pi,r , !Bi,r( ) of !"i (respec-
tively), with identical dimension r of base spaces, such that f ji is a bimorphism
between ! j,r and !"i,r .
In this case we will denote f ji by f ji,r . The restriction of FSL-mapping onto any
domain Dj , Dj ! Yj shall be named an FSL-mapping, too.
Definition 2.5. A continuous mapping f ji = Yj ! "Yi is called a Fredholm Spe-
cial Quasi-Linear (FSQL) mapping between affine bundles ! j and !"i , if there exists
582 A. ABBASOV
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
a sequence of FSL-mappings f ji,r = Yj ! "Yi , r = 1, 2, 3, ... , which uniformly con-
verges to f ji in each bounded domain Dj ! Yj and estimates (1.1) are satisfied.
Definition 2.6. A continuous mapping f : X ! "X between FQL-manifolds X
and !X is called a Fredholm Special Quasi-Linear (FSQL), if
a) !j !i , f (X j ) ! "Xi ;
b) !"i ! f !" j
#1 : Dj ! "Di is an FSQL-mapping between ! j and !"i .
As it was mentioned in introduction, the proof of identity of FQL and FSQL-
mappings is given in [1].
3. Example of FQL-manifold (in case of m < n ). Let be X = Hs (M,N) ,
where M , N are the compact smooth manifolds of the dimensions m , n
m < n( ) respectively and N has no boundary. Besides, let N be embedded in
R2n+1 3. Obviously, on X one can introduce the smooth structure [5]; the Hilbert real
space Hs (M, Rn ) will be its tangential space.
Now let’s introduce an FQL-structure on X . Suppose that X is naturally embed-
ded in Hs (M, R2n+1) , X j = u !X u s < j{ } , where j and s are the natural
numbers, and s is a norm in Hs (M, R2n+1) ). In order to solve this problem we
shall construct an affine bundle Yj , Pj , Bj( ) with finite-dimensional base space Bj ,
shall pick out a bounded domain Dj in Yj , shall construct homeomorphisms ! j :
Dj ! X j ( L -charts), j = 1, 2, 3,…, and also shall prove that homeomorphisms
!i
"1 !! j : Dj ! Di are FQL-mappings.
Lemma 3.1. If m < n , then
!" j, s( ) > 0 !u "X j !y u( )"N : ! y, u( ) " # ,
where ! y, u( ) = min
x
! y, u(x)( ){ } , and ! y, u(x)( ) is a distance between y and
u(x) , x !M , on N .
Proof. Let’s suppose the contrary:
!" > 0 !u" #X j !y "N : ! y, u"( ) < " ,
so, u! (M) is a ! -network of N . For simplicity, let’s suppose that n = m + 1 . Let
! be an (m + 1) -dimensional unit cube, homeomorphic to a (closed) domain of N .
Besides, let k be a cube, belonging to ! with the same dimension, its sides are par-
allel to the relevant sides of ! and the distance between them is ! .
Remark 3.1. On the contrary assumption, a part of surface u! (M) , which is the
! -network of k , will belong to ! .
Let’s take m -dimensional sections of k in form of m -dimensional planes,
which are parallel to a m -dimensional side of k and are on a distance of 2! from
3 For simplicity, the embedding mappings are not written in the text.
FREDHOLM QUASI-LINEAR MANIFOLDS AND DEGREE OF FREDHOLM QUASI-LINEAR … 583
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each other. On the opposite assumption, between two (such) next planes has to be part
of surface u! (M) . The m -dimensional volume of each similar part will be more or
equal to 1! 2"( )m . A number of such parts is not less than 1
2!
"
#$
%
&'
( [!] shows the
whole part of the number). Therefore, the total volume of all similar parts will be more
or equal to 1
2!
"
#$
%
&'
(
)*
+
,-
. 1/ 2!( )m . Obviously, 1
2!
"
#$
%
&'
(
)*
+
,-
. 1/ 2!( )m 0 1 at ! " 0 ,
so, the volume of surface u! (M) , u! "X j , will increase infinitely at ! " 0 . On
the other hand, as
!u "X j : u !C " c # u s < c # j .
Then
!u "X j : Vm (u) ! c " j "Vm (M) ,
where Vm (u) , Vm (M) are m -dimensional volumes of u(M) and M respectively,
and c there is a constant which is not dependent from u (u !X j ) . In other words,
all the numbers Vm (u) , u !X j , are bounded from above (by c ! j !Vm (M) ). This
paradox proves the contention of lemma.
Now we shall start construction of FQL-structure on Hs (M,N) . Let x1,…, xN{ }
be a ! -network of M . Let’s assign pN (u) = u(x1),…, u(xN )( )![N]N to each
mapping u !X j
4. Let
Bj = y = y1,…, yN( )![N]N "u !X j{ :
u(x1) = y1, u(x2 ) = y2 ,…, u(xN ) = yN } .
Obviously, Bj is a domain in [N]N , therefore it will also be a manifold of di-
mension n ! N .
Now for every point y !Bj we shall construct mapping Hs (M,N) , Uy (xi ) =
= yi , i = 1, N , as follows: Let Uy : M! R2n+1 be such a mapping that, Uy (xi ) =
= yi , i = 1, N , and in addition, Uy s
has a minimum among all such mappings.
Such a mapping Uy (x) exists, is unique and continuously depends on y ; it results
from convexity of function u ! u s
2 . In this case, Uy s
< j , because according to
the construction, there exists such a mapping u !X j that pN (u) = y , and Uy s
≤
≤ u s for each similar u(x) .
4 In the given article (because of technical problem) the same number is designated by symbols N and
N , namely, number of elements in ! -network of M .
584 A. ABBASOV
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
As known, N has a tubular neighborhood in R2n+1 . Let’s denote its radius by !
! > 0( ) . There exists a nearest point !(y) "N for each point y from this neighbor-
hood. Moreover, the mapping y! !(y) is smooth, surjective and non-degenerative.
Let
u !Hs (M, R2n+1) , u s < j .
As u C1 ! K " u s at sufficiently greater s , then u 1C ! K " j . Therefore,
!x "M : #u (x) R2n+1 < K $ j .
Then
! "x , ""x #M : !u "Hs (M, R2n+1) :
u s < j u( !x ) " u( !!x ) R2n+1 < K # j # d( !x , !!x ) ,
where d is the distance on M . Therefore,
! "x , ""x #M : !u "Hs (M, R2n+1) :
u s < j u( !x ) " u( !!x ) R2n+1 < #
when d !x , !!x( ) < " ! = " K # j( )( ) . Let x !M . Obviously,
!i : d(x, xi ) < ! .
Therefore,
!u "Hs (M, R2n+1) : u s < j u(x) ! u(xi ) R2n+1 < " .
As a result of that u(x) , u !Hs (M, R2n+1) , belongs to the ! -tubular neighbor-
hood of N (in R2n+1 ) when u s < j and u(xi ) !N , i = 1, N . Therefore it is
possible to project it smoothly on N (by help of ! ). As Uy < j , then all of this
is true also for Uy . Let Uy (x) = ! !Uy (x) . According to the construction, this map-
ping belongs to pN!1 y( ) , so, Uy (xi ) = yi , i = 1, N .
Remark 3.2. Due to the smoothness of ! , Uy s
! C " Uy s
< C " j . Thus,
Uy !X j , but Uy !XC" j .
Let expy : Ty N! N be the exponential mapping. Obviously, expy is diffeo-
morphism between some neighborhoods of zero (in TyN ) and of point y !N . Let’s
denote these neighborhoods by !1(y) and !1(y) , relatively. We can suppose that
!1(y) and !1(y) are independent from y !N , as expy is smooth and N is com-
pact.
Analogously to proved above, one can show that the !1 -neighborhood of Uy (x) ,
y !Bj includes all u(x) from pN
!1 pN (Uy )( )! XC" j when ! is small enough.
FREDHOLM QUASI-LINEAR MANIFOLDS AND DEGREE OF FREDHOLM QUASI-LINEAR … 585
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Let y0 !Bj . Let’s take n of vector fields in neighborhood of Uy0 (x) , which
are tangential to N , orthogonal to each other and have the unit length. Let’s denote
them by
!g1(y),…, !gn (y) . According to the Lemma 3.1 this vector fields will be de-
fined lengthways of each Uy (x) , where y !"y0 , and !y0 is ! -neighborhood of
point y0 in Bj . Bj can be covered by help of finite-number of similar ! -
neighborhoods, because it is relatively compact and finite-dimensional. Let’s denote
them by !
1y
,…, !
ly
, where y1,…, yl are some points from Bj . Let
FN =
!
! "M# Rn !v "Hs ,
!v(x1) =…= !v(xN ) = 0{ } ;
it is a linear subspace of Hs (M, Rn ) of finite co-dimension nN , where
!e1,…, !en{ }
is an orthonormal basis in Rn . Obviously, any function
!v !FN has the following
form in this basis:
!v(x) = v1(x) !
!e1 +…+ vn (x) !
!en ,
where vk (k) , k = 1, n , is a scalar function, vk !Hs (M, R1) , vk (xi ) = 0 , k = 1, n ,
i = 1, N .
Let’s consider the mapping
! p : "yp # F
N $ pN%1 "yp( ) ,
! p y, !v( ) (x) = expUy x( )
!g(x) , p = 1, l ,
where
!g(x) = v1(x) !
"g1 Uy (x)( ) +…+ vn (x) !
!gn Uy (x)( ) .
Obviously,
1) ! p "y , !v( ) # ! p ""y , !w( ) !
!v, !w "FN at !y " !!y , !y , !!y "#yp , as (accord-
ing to construction) ! p "y , !v( )# pN$1 "y( ) , and ! p ""y , !w( )# pN$1 ""y( ) ;
2) ! p y, !v( ) " ! p y, !w( ) !y "#
py
!p = 1, l at
!v C < !1 ,
!w C < !1 and
!v ! !w , as expy is diffeomorphism in !1 -neighborhood of 0y !TyN .
It follows from here that ! p , p = 1, l , is a diffeomorphism between !yp !
!
!v !FN !v C < "1{ } and neighborhood u(x) Uy (x) ! u(x) C
< "1{ } , where
y !"yp , pN u(xi )( ) = pN Uy (xi )( ) , i = 1, N . According to the construction, this
neighborhood contains the set
pN!1 "yp( )! X j . Obviously, Dp =
=
! p
"1 pN"1(#yp )! X j( ) is a bounded domain in !yp " F
N . Let’s paste Dp , D !p ,
p, !p = 1, l , by the help of ! "p
#1 !! p ; as a result we shall receive some set Dj .
Let’s construct an affine bundle, in which Dj will be a bounded domain. Let
!g1, p (y),…, !gn, p (y) and
!g1, !p (y),…, !gn, !p (y) be the two vector fields, defined (as
586 A. ABBASOV
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above) in neighborhoods of Uyp (x) and Uy !p (x) respectively and
y !"yp ! "y #p .
Besides, let ! p, "p ,y (x) be an orthogonal matrix, which transforms the first basis into
the second in the point y =Uy (x) . The diffeomorphism ! "p
#1 !! p will transform
y, !v( )!"yp # F
N into
y, !w( )!"y #p $ FN , where
!w(x) = ! p, "p ,y (x) #
!v(x) . (3.1)
The function (3.1) is a linear isomorphism, which smoothly depends on y !
!
!yp ! !y "p . Pasting all !yp " F
N , p = 1, l , by the help of ! "p
#1 !! p , we shall
receive an affine bundle, which we will denote by Yj , Pj , Bj( ) . According to the con-
struction, Dj will be the bounded domain in Yj . Now let’s paste !1,…,!l by the
help of transition functions; as a consequence we shall receive one diffeomorphism be-
tween Dj and X j , which we shall denote by ! j . Thus, construction of the L-chart
! j
"1, X j( ) on X j is finished.
Now we shall show that L -structures on X j and Xi are coordinated for different
j and i . For this purpose it is enough to prove that transition function !i
"1 !! j is
a FQL-mapping between affine bundles Yj , Pj , Bj( ) and Yi , Pi , Bi( ) . Let (x1,…
…, xN ) , ( !x1,…, !xL ) be points from M , which have been used at definition of L -
structures on X j , Xi and y = y1,…, yN( ) , !y = !y1,…, !yL( ) be points from Bj ,
Bi respectively. Moreover, let Uy (x) , U !y (x) be mappings, constructed by the help
of above mentioned method,
!g1(y),…, !gn (y) and
!!g1(y),…, !!gn (y) be the vector
fields, defined (as above) in the neighborhoods of Uy (x) , U !y (x) , respectively.
Let
FN =
!
! "Hs (M, Rn )
!
!(x1) =…=
!
!(xN ) = 0{ } ,
FL =
!
! "Hs (M, Rn )
!
!( #x1) =…=
!
!( #xL ) = 0{ } ,
be vector subspaces of Hs (M, Rn , which are isomorphic to layers of affine bundles
Yj , Pj , Bj( ) , (Yi , Pi , Bi ) respectively. Without loss of generality, we can suppose
that xm ! "xr , m = 1, N , r = 1, L . Let
[N]N FN+L =
!
! "Hs (M, Rn )
!
!(xm ) =
!
!( #xr ) = 0, m = 1, N , r = 1, L{ } .
Obviously, FN = FN+L + FL , where FL is orthogonal complement to FN+L in
FN and !
py
" FN = !
py
" FL( ) " FN+L . Pasting !
py
" FL( ) " FN+L , p = 1, l ,
FREDHOLM QUASI-LINEAR MANIFOLDS AND DEGREE OF FREDHOLM QUASI-LINEAR … 587
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by the help of diffeomorphisms (3.1), we shall get a new affine bundle. Let’s denote it
by Yj , Pji , Bji( ) .
Let
y, !z( )!"yp # FL . Let’s look at the function
u(x) = expUy x( ) zk (x) + vk (x)( )
k=1
n
! " !gk Uy (x)( )#
$%
&
'(
,
where vk (xm ) = vk ( !xr ) = 0 , that is
!v = v1,…, vn( )!FN+L . For each such u(x) ,
u(xm ) = ym , u( !xr ) = !yr , m = 1, N , r = 1, L . Therefore,
expU !y (x)
"1 u(x) = !y , !w(x)( ) ,
where !y = !y1,…, !yL( ) ,
!w(x) = w1(x),…,wn (x)( ) . Thus !i
"1 !! j will transform
the layer Pji
!1 y, !z( ) above y,
!z( ) into the layer Pi!1 "y( ) above !y , where !y =
= u( !x1),…, u( !xL )( ) . Then it will transform Pji
!1 "y ,!z( ) in Pi!1 " #yq( ) , where !y "
!" #yq , ! "yq is a chart of a fixed atlas on Bi , and !y ,
!z is a neighborhood of y,
!z( )
in Bji . This transition function has the following form:
y,
!z, !v( ) " !y , !w( ) = !y , w1,…,wn( )( ) ,
where
!y = u( !x1),…, u( !xL )( ) , u = ! j y, !z + !v( ) ,
and
wk (x) = !!gk U !y (x)( ), !h(x)( ) , k = 1, n ,
is the scalar product of vectors, tangential to N in point U !y (x) ,
!
h(x) = expU !y (x)
"1 u(x)
!
h(x) !TU "y (x)N( ) .
It is obvious from the foresaid formulas that in charts of the mentioned affine bundles,
the transition function !i
"1 !! j is given by an operator of superposition with smooth
functions. As all used functions have gradients different from zero in all points, then
according to the Theorem 6.3, such a function is an FQL-mapping. Therefore, it is an
FQL-mapping between L -charts on X j and Xi . It follows from here that the struc-
ture included in X is Fredholm Quasi-Linear.
4. Example of FQL-manifold (the case m ! n ). For simplicity, let’s suppose
that m < 2n . Let R4n+2 = R2n+1 ! R2n+1 , and N2 = N ! N is embedded in
R4n+2 such that N is embedded in. Let X = Hs (M,N) . Obviously, X 2 =
= Hs (M,N2 ) . Let X j
2 = X j ! X j , X j,0 = X j !O " X j
2 , where X j = u !X{
u s < j } , O :M! 0 and N0 = N ! 0 , where 0 is the origin of R4n+2 . In ad-
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dition, let x1,…, xN{ } be a ! -network of M . Without loss of generality, also let’s
suppose that the origin of R4n+2 coincides with a point of N0 and consequently, of
N2 . To each mapping u = (u1, u2 ) !X j
2 we shall assign the point
pN 2 (u1, u2 ) = u1(x1), u2 (x1)( ) ,…, u1(xN ), u2 (xN )( )( ) ![N2 ]N ,
and to each mapping u1,O( )!X j,0 , the point
!pN (u1,O) = u1(x1),O( ) ,…, u1(xN ),O( )( ) ![N0 ]N .
Let
!Bj = !y = y1, y2( ) = y11, y21( ) , ..., y1N , y2N( )( ) ![N2 ]N " u1, u2( )!X j
2{ :
u1(x1), u2 (x1)( ) = y11, y21( ) ,…, u1(xN ), u2 (xN )( ) = y1N , y2N( )} ,
Bj,0 = y1, 0( ) = y11, 0( ) ,…, y1N , 0( )( ) ![N0 ]N " u1,O( )!X j,0{ :
u1(x1), 0( ) = y11, 0( ) ,…, u1(xN ), 0( ) = y1N , 0( )} .
Obviously,
!Bj ( Bj,0 ) is a domain in [N2 ]N [N0 ]N( ) , therefore it is a 2nN
( nN )-dimensional manifold. Moreover, Bj,0 will be submanifold of
!Bj and
! y1, 0( )"Bj,0 : !pN#1 y1, 0( )$ p
N 2
#1 y1, 0( ) .
Let’s denote a mapping by ! , which transforms each point y1, y2( ) of the ! -
tubular neighborhood of N2 (in R4n+2 ) into the nearest point of N2 . By the help
of the aforecited method (see, the case m < n ) to each point !y !
!Bj at first, we shall
assign such a mapping
U !y = U1,"""y ,U2, !y( ) !Hs M, R2n+1 " R2n+1( ) that U !y (xi ) =
= y1i , y2i( ) , i = 1, N , and later, we shall assign a mapping U !y = U1,"""y ,U2, !y( ) =
= ! U1,!!!y ,U2, "y( ) "Hs (M,N2 ) , which also will satisfy the condition U !y (xi ) =
= y1i , y2i( ) , i = 1, N .
We need the following in advance.
Lemma 4.1. Let U y1,0( ) = U1, y1,0( ),U2, y1,0( )( ) !Hs M, R2n+1 " R2n+1( ) be a
mapping such that U(y1,0)(xi ) = U1,(y1,0)(xi ),U2,(y1,0)(xi )( ) = (y1i , 0) , i = 1, N , and
in addition, U y1,0( )
!
s
has a minimum among all such mappings. Then
U2, y1,0( ) ! O , in other words, such U y1,0( ) will belong to Hs (M, R2n+1 ! R2n+1) .
Here u1, u2( ) !s = u1 s + u2 s and s are the norms in Hs (M, R4n+1)
and Hs (M, R2n+1) respectively.
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Proof. As y1i , 0( )!R2n+1 , i = 1, N , then obviously, such U y1,0( ) will belong
to Hs (M, R2n+1 ! 0) .
According to the Lemma 4.1, the mapping
U y1, 0( ) = ! !U y1, 0( ) will belong to
Hs M,N0( ) .
Let !y0 !
!Bj . Let’s take 2n vector fields in neighborhood of U !y0 (x) , which are
tangential to N2 , orthogonal to each other and have the unit length. Let’s denote
them by
!g1(y),…, !g2n (y) . According to the Lemma 3.1, these vector fields will be
defined lengthways of each U !y (x) , where !y !
!" !y0 and
!! !y0 is ! -neighborhood of
point !y0 in
!Bj . One can cover
!Bj by the help of finite number of similar ! -
neighborhoods
!!
1!y
,…, !!
"l!y
( !y1,…, are points from
!Bj ), because
!Bj is relatively
compact and finite-dimensional. Let
! p,0 = !! !yp " Bj,0 , p = 1, !l . Obviously, the
collection { !1,0 ,…, ! "l ,0{ } will cover Bj,0 . Let
F2N =
!
! "Hs (M, R2n )
!
!(x1) =…=
!
!(xN ) = 0{ } ,
it is a linear subspace of Hs (M, R2n ) of finite co-dimension 2nN . Let
!e1,…, !e2n{ } be an orthonormal basis in R2n . Obviously, each function
!v !F2N
will have in this basis the following form:
!v(x) = v1(x) !
!e1 +…+ v2n (x) !
!e2n .
Here vk (x) , k = 1, 2n , is a scalar function, vk !Hs (M, R1) , vk (xi ) = 0 , k = 1, 2n ,
i = 1, N . Let’s consider a mapping
!! p : !" !yp # F
2N $ p
N 2
%1 !" !yp( ) ,
!! p !y,
"v( ) (x) = expU !y (x)
"g(x) , p = 1, !l ,
where
!g(x) = v1(x) !
"g1 U #y (x)( ) +…+ v2n (x) !
!g2n U #y (x)( ) , (4.1)
!y ! !"yp , p = 1, !l .
As in the case m < n , one can show that the
!! p , p = 1, !l , is a diffeomorphism
between
!! !yp "
"v #F2N "v C < $1{{ } and neighborhood u(x){
U !y (x) ! u(x) C
<
< !1 } , where
!y ! !" !yp ,
pN 2 u(xi )( ) = pN 2 U !y (xi )( ) , i = 1, N . According to the
construction, this neighborhood contains the set
p
N 2
!1 !" !yp( ) " X j
2 .
Now let’s construct such subbundle of
!! !yp " F
2N , which
!! p would transform
onto !pN
!1(" p,0 ) . Let
!T !y ,
!y ! !"yp , be a space of mappings
!g :M! T ! T (N2 ) of
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form (4.1). Obviously, it is linear and isomorphic to F2N . In addition,
!T !!y "
!
"T "!!y = " at !!y " !!!y and
!T !y continuously depends on
!y ! !" !yp . Therefore, the
family
!T !y !y ! !"yp{ } , p = 1, !l will induce an affine bundle with the total space
!T !yp = !T !y" ,
!y ! !" !yp , with the layer F2N , with the projection
!P!yp (
"g) = !y and the
base space
!!yp . Let’s denote it by
!T !yp , !P!yp , !! !yp( ) . Obviously, the mapping
!Gp : !! !yp " F
2N # !T !yp ,
!Gp !y,
"v( ) = "g ,
!y ! !" !yp ,
!v !F2N , p = 1, !l ,
will be an isomorphism between Cartesian product
!! !yp " F
2N and
!T !yp , !P!yp , !! !yp( ) .
Let T y1,0( ) , y1, 0( )!" p,0 be a space of mappings
!g :M! T ! T (N0 ) , where
!g(x) = vk (x) !
!gk U y1,0( )(x)( )
k=1
2n
" , y1, 0( )!" p,0 ,
!v !F2N .
For each y1, 0( )!" p,0 , T y1,0( ) will be linear subspace of
!T !y ,
!y ! !"yp , where !y =
= !y = y1, 0( ) . In addition, T y1,0( ) continuously depends on y1, 0( )!" p,0 and
T !y1,0( ) ! T !!y1,0( ) = " , when !y1, 0( ) " !!y1, 0( ) . Therefore, the family T y1,0( ){
y1, 0( )!" p,0 } = # , p = 1, !l , will induce an affine bundle with the total space
Tp,0 = T y1,0( )! , y1, 0( )!" p,0 , with the projection Pp,0
!g( ) = y1, 0( ) and the
base space ! p,0 . Let’s denote it by Tp,0 , Pp,0 , ! p,0( ) . According to the construction,
it will be a subbundle of
!T !yp , !P!yp , !! !yp( ) . As
!Gp is the isomorphism, then
!Gp
!1(Tp,0 ) will be an affine subbundle of ! p,0 " F2N . According to the construction,
the mapping
!! p will transform
!Gp
!1(Tp,0 ) onto !pN
!1(" p,0 ) . Obviously,
!Dp =
=
!! p
"1 p
N 2
"1 !# !yp( )" X j
2( ) and( also
Dp,0 = !! p
"1 !p
N
"1(# p,0 )" (X j,0 )( ) is a bounded
domain in
!! !yp " F
2N (accordingly, in
!Gp
!1(Tp,0 ) ). Let’s paste
!Dp and
!D !p (and
also Dp,0 and D !p ,0 ), p, !p = 1, !l , by the help of mappings
!! "p
#1 " !! p ; as a result
we shall receive a set
!Dj (accordingly, Dj,0 ).
Now let’s construct such two affine bundles that in one of them each
!Dj and in
the other each Dj,0 will be a bounded domain. Let
!g1, p (y),…, !g2n, p (y) and
!g1, !p (y),…, !g2n, !p are vector fields, defined (as above) in neighborhoods of
U !yp (x)
and Uy !p (x) , respectively. Let
!y ! !" !yp " !" !y #p and µ p, !p , !y (x) is an orthogonal ma-
trix, which transforms the first basis onto second basis in point y =U !y (x) . The dif-
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feomorphism
!! "p
#1 " !! p will transform the element
!y, "v( )! !" !yp # F2N into
!y, "w( )! !" !y #p $ F2N , where
!w(x) = µ p, !p , "y (x) "
!v(x) . (4.2)
The function (4.2) is the linear isomorphism, smoothly depending on !y !
! !" !yp " !" !y #p . Pasting together all
!! !yp " F
2N (and also all
!Gp
!1(Tp,0 ) ), p = 1, !l , by
the help of mappings
!! "p
#1 " !! p , we shall receive an affine bundle (accordingly, a sub-
bundle). Let’s denote it by
!Yj , !Pj , !Bj( ) (accordingly, by (Yj,0 , Pj,0 , Bj,0 ) ). Accord-
ing to the construction,
!Dj (and also Dj,0 ) will be the bounded domain in Yj (ac-
cordingly, in Yj,0 ).
Now we shall paste together !!1,…, !!l by the help of transition functions; as a
consequence we shall receive one diffeomorphism between
!Dj (and also Dj,0 ) and
X j
2 (accordingly, X j,0 ), which we will denote by
!! j . Thus, the construction of the
L -chart ! j
"1, X j
2( ) (and also (
!! j
"1, X j,0 ) ) on X j
2 (accordingly, on X j,0 ), is
completed.
Similarly to the case m < n , the transition function between affine bundles
!Yj , !Pj , !Bj( ) and
!Yi , !Pi , !Bi( ) will be an FQL-mapping. Therefore, the L -structures
on X j
2 and Xi2 , j ! i , will be coordinated. It follows from here that
!!i
"1 " !! j
will be an FQL-mapping between subbundles Yj,0 , Pj,0 , Bj,0( ) and Yi,0 , Pi,0 , Bi,0( ) ,
too. In other words, the L -structures on X j,0 and Xi,0 , j ! i , also will be coordi-
nated. Thus, the structure, introduced in X , will also be Fredholm Quasi-Linear.
Remark. It is obvious from all of the above-established facts that at (n ! 1) " k ≤
≤ m < n ! k , k ! 3 , all constructions and proofs will be similar to the case m < 2n .
5. A degree of FSQL-mapping. At the definition of the degree of FSQL-mapping
between FQL-manifolds we shall consider a more simple case, namely when the fol-
lowing conditions are satisfied:
(1) FQL-manifolds X and !X are embedded in Banach spaces E1 and E2 ,
respectively.
(2) The open sets X j and !Xi (see, the definition of FQL-manifold) have forms
X j = X ! B1(Rj ) and !Xi = !X ! B2 (ri ) , where B1(Rj ) and B2 (ri ) are the open
balls in E1 and E2 with centers at zero and of the radiuses Rj and ri , respec-
tively, Rj , ri ! " , when j, i! " .
(3) For each j and i , the L -charts ! j , ! j
"1 , !"i , !"i( )#1 are uniformly
continuous.
(4) FSQL-mapping f : X ! "X satisfies the following a priori estimate
592 A. ABBASOV
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x 1 ! " f (x) 2( ) , (5.1)
where ! is a positive monotone function, and ! 1 , ! 2 are norms in E1 and E2 ,
respectively.
Now we shall start defining the degree of FSQL-mapping f : X ! "X . For sim-
plicity let’s suppose that ! is the identical mapping. Let’s consider the equation
f (x) = !x0 , !x0 " !X . (5.2)
When condition (5.1) is satisfied, all the solutions of the equation (5.2) will belong to
XR0 = X ! B1(R0 ) , R0 = !x0 2 . According to the assumption,
!j0 !j " j0 : X j ! XR0 ,
and according to the Definition 2.6,
!i0 !i " i0 : f (X j ) ! "Xi .
Let j and i be numbers, for which all above mentioned conditions are satisfied.
Then, while defining deg f( ) in the point !x0 " !X we may consider the restriction
of the mapping f onto X j . As ! j and !"i are the homeomorphisms, then to
solve equation (5.2) in XR0 will be equivalent to solve equation
f ji (y) = !y0 , !y0 = !"i ( !x0 )
in ! j (XR0 ) , where f ji ! "#i ! f !# j
$1 . According to Definition 2.6, f ji is an FSQL-
mapping between affine bundles ! j and !"i . Let f ji,r{ } be a sequence of FSL-
mappings, which uniformly converges to f ji on Dj . Let’s consider the equation
f ji,r (y) = !y0 , !y0 = !"i ( !x0 ) ; (5.3)
we will search its solutions in ! j (X "R0 ) , where X !R0 = X ! B1 !R0( ) , !R0 =
= !x0 2 + 2" , ! > 0 .
Remark 5.1. Obviously, X !R0 " X j when j is big enough, therefore
! j (X "R0 ) # Dj .
This problem can be transformed to finite-dimension problem. Indeed, as f ji,r is
a bimorphism, then it will induce the finite-dimensional continuous mapping
g ji,r : Bj,r ! "Bi,r
between base spaces of affine bundles ! j and !"i . Let’s consider this finite-
dimensional equation
g ji,r !( ) = "!0 , !"0 = !Pi,r ( !y0 ) , (5.4)
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where !Pi,r is the projection of subbundle !"i,r = !Yi , !Pi,r , !Bi,r( ) . Let’s prove that
when r is big enough; finding the solutions of equation (5.3) is equivalent to finding
the solutions of equation (5.4).
Indeed, let y !" j X #R0( ) and it is a solution of equation (5.3). Obviously, there
exists unique ! "Pj,r # j X $R0( )( ) , such that y !Pj,r"1 #( ) and f ji,r,! (y) = "y0 , where
Pj,r is the projection of subbundle ! j,r = Yj , Pj,r , Bj,r( ) , and f ji,r,! is the restric-
tion of f ji,r onto layer Yj,! = Pj,r"1 !( ) . Therefore, f ji,r,! Yj,!( ) = "Yi, "!0 , where
!Yi, !"0
is the layer of !"i,r = !Yi , !Pi,r , !Bi,r( ) , which contains point !y0 . Therefore, !
will be solution of the equation (5.4).
Conversely, let ! be a solution of (5.4). This means that
f ji,r,! Yj,!( ) = "Yi, "!0 .
As f ji,r,! is an isomorphism, then there exists unique point y !Pj,r"1 #( ) such that
f ji,r,! (y) = "y0 , (5.5)
i.e., the equation (5.3) is solved. Let’s show that y !" j X #R0( ) is not possible. Obvi-
ously,
f (x) ! ( "#i )!1 ! f ji,r !# j (x) 2 < $ , x !Dj ,
when r is big enough. If y !" j (X #R0 ) , then x = ! j
"1(y) #X $R0 , i.e., x 1 > !R0 .
Then, it follows from estimate (5.1) that
!"i( )#1 ! f ji,r !" j (x) 2
≥
f (x) 2 ! f (x) ! "#i( )!1 ! f ji,r !# j (x) 2
≥
≥ !x0 2 + 2"( ) # " > !x0 2 ,
i.e., !"i( )#1 ! f ji,r !" j (x) $ !x0 , hence f ji,r (y) ! "y0 . This contradicts to equal-
ity (5.5). So, y !" j (X #R0 ) .
Thus, the equation (5.3) is transformed to finite-dimension equation (5.4). Now we
can define the degree of FSL-mapping f ji,r .
Definition 5.1. deg f ji,r( ) = deg g ji,r( ) .
The sign of this degree depends on orientations in Bj,r and !Bi,r , but its absolute
value is invariable. The last circumstance is not important for proof of the existence of
a solution of (5.2) (see Theorem 5.1 and Definition 5.2).
Theorem 5.1. Let f ji,r1, f ji,r2 : Yj ! "Yi be FSL-mappings, which are close
enough to FQL-mapping fi, j : Yj ! "Yi in Dj . Then
deg f ji,r1( ) = deg f ji,r2( ) .
594 A. ABBASOV
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The proof of this theorem is similar to proof of the Theorem 2.3 from [1].
By the Theorem 5.1 the sequence deg f ji,r1( ){ } will be stable when r is big
enough. Therefore, we can give the next definition.
Definition 5.2. deg f ji( ) = lim
r!"
deg f ji,r( ) .
As ! j and !"i are homeomorphisms, then we can give the next definition.
Definition 5.3. deg f( ) = deg f ji( ) .
Obviously, deg f( ) does not depend on L -charts on X and Y .
Theorem 5.2. Let ft{ } be a family of FSQL-mappings, continuously (uniformly
in each X j ) depending on parameter t ! 0,1[ ] . Let’s suppose also that the condi-
tions (1)–(4) are satisfied for each t . Then
deg f0 , !x( ) = deg f1, !x( ) , !x " !X .
Here the function ! does not depend on t .
Proof. Using compactness of 0,1[ ] , uniform continuity (according to t ) of the
family ft{ } and also of the mappings ! j and !"i( )#1 , we can approximate the
family of FSQL-mappings ft , ji : Yj ! "Yi by the help of the family ft ,i, j,r{ } of FSL-
mappings. According to the Theorem 5.1, the absolute value of degree of FSL-mapping
will be locally stable. Therefore,
deg f0, ji,r , !y( ) = deg f1, ji,r , !y( ) , !y = !"i ( !x ) ,
when r is big enough. Hence
deg f0, ji , !y( ) = deg f1, ji , !y( ) .
From here,
deg f0 , !x( ) = deg f1, !x( ) .
Theorem 5.2 is proved.
Theorem 5.3. At the conditions (1)–(4)
deg f , !x1( ) = deg f , !x2( ) , !x1, !x2 " !X .
Proof. Let X j ! X ! B1 R( ) , R ! " max #x1 2, #x2 2 + 2${ }( ) and !Xi is
such that f (X j ) ! "Xi . Let f ji,r{ } be a sequence of FSL-mappings, which con-
verges to the f ji in Dj . As
deg f ji,r , !yl( ) = deg f ji , !yl( ) , !yl = !"i !xl( ) , l = 1, 2 ,
when r is big enough, then it is enough to prove that
deg f ji,r , !y1( ) = deg f ji,r , !y2( ) .
FREDHOLM QUASI-LINEAR MANIFOLDS AND DEGREE OF FREDHOLM QUASI-LINEAR … 595
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For this purpose it is enough to prove that
deg g ji,r , !"11( ) = deg g ji,r , !"22( ) , !"l = !Pi,r ( !yl ) , l = 1, 2 .
The last equality is known from the classical (finite-dimensional) analysis.
Theorem 5.3 is proved.
Theorem 5.4. Let the conditions (1)–(4) be satisfied and deg( f ) ! 0 . Then the
equation (5.2) has a solution for each !x0 " !X .
The similar theorem has been proved in [3] (see also [8]).
Remark 5.2. Specific examples of FQL-mappings with calculated degrees (in rela-
tively simple cases) are given in [6] and [8].
1. Abbasov A. A special quasi-linear mapping and its degree // Turkish J. Math. – 2000. – 24, № 1. –
P. 1 – 14.
2. Abbasov A. A quasi-linear manifolds and quasi-linear mapping between them // Turkish J. Math. –
2004. – 28, № 1. – P. 205 – 215.
3. Abbasov A. L-homology theory of FSQL-manifolds and the degree of FSQL-mappings // Ann. Pol.
Math. – 2010. – 98, № 2. – P. 129 – 145.
4. Borysovich Y. G., Zvyagin V. G., Sapronov Y. I. Nonlinear Fredholm maps and Leray – Schauder theory
// Uspechi Mat. Nauk. – 1977. – 22, № 4. – S. 3 – 54 (in Russian).
5. Eells J. Fredholm structures // Proc. Symp. Pure Math. – Providence, R. I.: Amer. Math. Soc., Pr., 1970.
– 18. – P. 62 – 85.
6. Efendiev M. A. The degree of FQL-mapping in quasi-linear domains and Hilbert nonlinear problem in
ring // Izv. AN Azerb. SSR. – 1979. – 5 (in Russian).
7. Zviyagin V. G. About oriented degree of one class disturbances of Fredholm mappings and bifurcations
of solutions of Nonlinear boundary problem with noncompact disturbances // Mat. Sb. – 1991. – 182,
№ 12 (in Russian).
8. Shnirelman A. I. The degree of quasi-linear mapping and Hilbert nonlinear problem // Mat. Sb. – 1972.
– 89(131), № 3. – S. 366 – 389 (in Russian).
Received 17.06.10,
after revision — 24.02.11
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| id | umjimathkievua-article-2744 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:29:32Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/2a/f777b4f4b29153ee72ed05fa565f782a.pdf |
| spelling | umjimathkievua-article-27442020-03-18T19:35:13Z Fredholm quasi-linear manifolds and degree of Fredholm quasi-linear mapping between them Квазілінійні многовиди фредгольма та степінь квазілінійних відображень фредгольма між ними Abbasоv, A. Аббасов, А. In this article a new class of Banach manifolds and a new class of mappings between them are presented and also the theory of degree of such mappings is given. Представлено новий клас многовидів Банаха та новий клас відображень між ними, а також наведено теорію степеня таких відображень. Institute of Mathematics, NAS of Ukraine 2011-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2744 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 5 (2011); 579-595 Український математичний журнал; Том 63 № 5 (2011); 579-595 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2744/2244 https://umj.imath.kiev.ua/index.php/umj/article/view/2744/2245 Copyright (c) 2011 Abbasоv A. |
| spellingShingle | Abbasоv, A. Аббасов, А. Fredholm quasi-linear manifolds and degree of Fredholm quasi-linear mapping between them |
| title | Fredholm quasi-linear manifolds and degree of Fredholm quasi-linear mapping between them |
| title_alt | Квазілінійні многовиди фредгольма та степінь квазілінійних відображень фредгольма між ними |
| title_full | Fredholm quasi-linear manifolds and degree of Fredholm quasi-linear mapping between them |
| title_fullStr | Fredholm quasi-linear manifolds and degree of Fredholm quasi-linear mapping between them |
| title_full_unstemmed | Fredholm quasi-linear manifolds and degree of Fredholm quasi-linear mapping between them |
| title_short | Fredholm quasi-linear manifolds and degree of Fredholm quasi-linear mapping between them |
| title_sort | fredholm quasi-linear manifolds and degree of fredholm quasi-linear mapping between them |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2744 |
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