J. Ko ean Ma h. Soc. 46 (2009), No. 3, pp. 463–473
DOI 10.4134/JKMS.2009.46.3.463
FREDHOLM MAPPINGS AND BANACH MANIFOLDS
Jos´
e Ma ´
ıa So iano A bizu
Abs ac . Two C1-mappings, whose domain is a connec ed compac
C1-Banach mani old modelled o e a Banach space Xo e K=Ro C
and whose ange is a Banach space Yo e K,a e in oduced. Su icien
condi ions a e gi en o asse hey sha e only a alue. The p oo o he
esul , which is based upon con inua ion me hods, is cons uc i e.
1. P elimina ies
Scien i ic phenomena a e locally desc ibed by pa ame e s, whose choice
is some imes a bi a y. This implies he impo ance o he a ailabili y o a
me hodology o he compa ison o esul s o measu emen s. Locally, a Banach
mani old looks like a Banach space. Fo a local desc ip ion, di e en Banach
(o coo dina e o pa ame e ) spaces a e allowed and ans o ma ion ules exis
o hese coo dina es.
Le X, Y be wo Banach spaces. Le u:U⊂X→Ybe a con inuous
mapping. One way o sol ing he equa ion
(1) u(x) = y
o any ixed y∈Y, is o embed (1) in a con inuum o p oblems
(2) H(x, ) = y, (0 ≤ ≤1),
which is sol ed when = 0.When = 1,p oblem (2) becomes (1). I i is
possible o con inue he solu ion o all ∈[0,1], hen (1) is sol ed. This is
he con inua ion me hod wi h espec o a pa ame e [1-25]. A con inua ion
me hod was in oduced o sol e (1) when u:M→Rn,whe e Mis a connec ed
compac C1-Banach mani old modelled on Rn,and H(·,·) is a C1-mapping [25].
He e Mis a Banach mani old modelled on an in ini e-dimensional Banach
space Xo e K=Ro Cand u anges o e an in ini e-dimensional Banach
space Yo e K.
Recei ed July 11, 2007.
2000 Ma hema ics Subjec Classi ica ion. P ima y 58C30, Seconda y 65H20.
Key wo ds and ph ases. egula alue, con inua ion me hods, a las, cha , Banach mani-
old, compac ness.
This wo k is pa ially suppo ed by D.G.E.S. Pb 96-1338-CO 2-01 and he Jun a de
Andalucia.
c
°2009 The Ko ean Ma hema ical Socie y
463
464 JOS´
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IA SORIANO ARBIZU
Su icien condi ions a e gi en o p o e ha wo C1-mappings, o which
one is F edholm o index ze o, sha e only one alue on a Banach mani old by
using con inua ion me hods on cha s. O he condi ions, su icien o gua an ee
he exis ence o ze o poin s, ha e been gi en by he au ho in se e al o he
pape s [7-25].This sha ed alue can be es ima ed ollowing a cu e. The p oo
supplies he exis ence o a cu e which leads o he poin whose image is he
sha ed alue. The keys a e he use o he cha spaces [27], he compac ness
and connec ness o M, oge he wi h he use o he Con inuous Dependence
Theo em (Theo em 2) [26] and Theo em 1 [28].
We b ie ly ecall some heo ems and no a ion o be used.
De ini ions and No a ion. [26-28]. Le F:D(F) : X→Y, whe e X, Y
a e Banach spaces o e K.I D(F) is open, hen mapping Fis said o be a
F edholm mapping i and only i bo h Fis a C1-mapping and F0(x) : X→Y
is a F edholm linea mapping o all x∈D(F). Tha L:X→Yis a linea
F edholm mapping means ha Lis linea and con inuous and bo h he num-
be s dim(ke (L)) and codim(R(L)) a e ini e, whe e dim signi ies dimension,
codim codimension, ke ke nel and R(L) s ands o he ange o mapping L.
The e o e ke (L) = X1is a Banach space and has opological complemen X2,
since dim(X1) is ini e. The in ege numbe ind(L)= dim(ke (L))-codim(R(L))
is called he index o L. Le F(X, Y ) deno e he se o all linea F edholm map-
pings L:X→Y.
Le Mbe a opological space. A cha (U, ϕ) in Mis a pai whe e he se
Uis open in Mand ϕ:U→Uϕis a homeomo phism on o an open subse Uϕ
o a Banach space Xϕ. We call ϕacha map, Xϕis called cha space, and
Uϕcha image. Fo x∈U,xϕ=ϕ(x) is called he ep esen a i e o xin he
cha (U, ϕ) o he local coo dina e o xin he local coo dina e sys em ϕ. The
poin x∈Mmay ha e di e en local coo dina es xϕ=ϕ(x) and xψ=ψ(x) o
wo di e en cha s (U, ϕ) and (V, ψ), espec i ely. The ans o ma ion ules
be ween hem a e xϕ=ϕ(ψ−1(xψ)) and xψ=ψ(ϕ−1(xϕ)).
Two cha s, (U, ϕ) and (V, ψ) in M, a e called Ck-compa ible i and only i
U∩V=∅,o bo h ϕ◦ψ−1:ψ(U∩V)→ϕ(U∩V) and ψ◦ϕ−1:ϕ(U∩V)→
ψ(U∩V) a e Ck-mappings, k≥0.
ACk-a las o M, 0≤k≤ ∞ is a collec ion o cha s (Ui, ϕi),whe e i∈I,
which sa is ies he ollowing condi ions:
(i) he Uico e M,
(ii) any wo cha s a e Ck-compa ible,
(iii) all cha spaces Xia e Banach spaces o e K.
I he e is a Ck-a las o M, hen Mis said o be a Ck-Banach mani old. I
all cha spaces a e equal o a ixed Banach space X, M is called a Ck-Banach
mani old modelled on X. He e mani olds wi hou bounda ies will be conside ed,
such as he su ace o a ball in Rn, an open se in a Banach space X, e c.
Le Mand Nbe Ck-Banach mani olds wi h cha spaces o e K, k ≥1.The
mapping :M→Nis called a C -mapping,whe e ≤k, i and only i is C
FREDHOLM MAPPINGS AND BANACH MANIFOLDS 465
a each poin x∈Min ixed admissible cha s. This means he ollowing: I
(U, ϕ) and (V, ψ) a e cha s in Mand N espec i ely, wi h x∈Uand (x)∈V,
hen he mapping =ψ◦ ◦ϕ−1,which is well de ined in a su icien ly small
neighbou hood o xϕ, is C in he usual sense. is called a ep esen a i e o
.
Two C1-cu es in M, which pass h ough he poin x∈M, a e called equi -
alen he poin xi and only i he ep esen a i es ha e he same angen ec o
a xin some ixed admissible cha . A angen ec o (o he wise known as
x0( 0)) o Ma xconsis s o all C1-cu es which a e equi alen a x o a ixed
C1-cu e. The angen “abs ac ” ec o o he p e ious de ini ion o he
cu e x(·) : U( 0)⊂R→M, x =x( ) a he poin x( 0),has i s ep esen a i e
o local coo dina e ϕ=x0
ϕ( 0) in he cha (U, ϕ),whe e xϕ=xϕ( ) = ϕ(x( )).
The angen space TMx o Ma he poin xis by de ini ion he se o all
angen ec o s. I is p o en ha his is a opological ec o space which is
linea homeomo phic o each cha space Xϕa he poin x.
The map 0(x) : T Mx→T N (x)is called he angen map o :M→N
a poin x, which is clea ly he no mal F-de i a i e in local coo dina es.
A mapping :M→Nis called a F edholm ope a o a xi and only i he
linea iza ion 0(x) : Mx→N (x)is a F edholm ope a o . Fu he mo e, is a
F edholm ope a o s a xi and only i he ep esen a i es o in local cha s
a e F edholm ope a o a he co esponding poin s.
Le :M→Nbe a Ck-mapping, k≥1,whe e Mand Na e Ck-Banach
mani olds wi h cha space o e K.The mapping is called a subme sion a x
i and only i 0(x) is su jec i e and he null space ke ( 0(x)) spli s he angen
space o Ma poin x(which is au oma ic when ke ( 0(x)) = {0}). A poin
x∈Mis called a egula poin o i and only i is a subme sion a x. A
poin y∈Nis called a egula alue o i and only i he se −1(y) is emp y
o consis s only o egula poin s.
I X, Y a e Banach spaces, le L(X, Y ) deno e he se o all linea con in-
uous mappings L:X→Y. Le Isom(X, Y ) deno e he se o all he linea
homeomo phisms L:X→Y. Le B(x0, ρ) be he open ball o cen e x0and
adius ρ. I u:X→Yis a linea con inuous bijec i e ope a o , hen he
in e se linea con inuous ope a o will be deno ed by u−1.
Mapping Hx(xϕ, ) deno es he pa ial F-de i a i e o mapping Hwi h e-
spec o Xa he poin (xϕ, ), whe e H:Uϕ×[0,1] ⊂X×R→Y.
A ep esen a i e poin always has i s co esponding cha map as subindex.
Theo em 1 ([28], p. 300).I S∈ F(X, Y ), whe e X, Y a e Banach spaces
o e K, hen he e is a numbe ε > 0such ha
T∈ F(X, Y )and Ind T= Ind S
o all linea F edholm mappings T∈ L(X, Y )wi h kT−Sk< ε.
Theo em 2 (Con inuous Dependence Theo em [26], pp. 18–19).Le he ol-
lowing condi ions be sa is ied:
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(i) P is a me ic space, called he pa ame e space.
(ii) Fo each pa ame e p∈P, mapping Tpsa is ies he ollowing hypo heses:
(1) Tp:M→M, i.e. Mis mapped in o i sel by Tp.
(2) Mis a closed non-emp y se in a comple e me ic space (X, d).
(3) Tpis k-con ac i e o any ixed k∈[0,1).
(iii) Fo each p0∈P, and any x∈M, lim
p→p0
Tp(x) = Tp0(x).
Then o each p∈P, he equa ion xp=Tpxphas exac ly one solu ion xp,whe e
xp∈Mand lim
p→p0
xp=xp0.
2. F edholm mappings on compac Banach mani olds
Theo em 3. Le , g :M→Ybe wo C1-mappings, whe e Mis a compac ,
connec ed, C1-Banach mani old modelled on X, whe e X, Y a e wo Banach
spaces o e K=Ro C. Le (Ui, ϕi)i∈I, I = 1,2, . . . , N be a C1-a las o M.
Suppose ha he ollowing condi ions hold:
(i) Mapping has only one ze o x?in M, wi h x∗∈Uj, and is a F edholm
mapping o index ze o a x∗.
(ii) Fo any ixed ∈[0,1],ze o is a egula alue o any mapping
H(·, ) : M× { } → Y, H(·, ) := (·)− g(·).
Hence he ollowing s a emen holds ue:
(a) The mappings and gsha e only one alue x∗∗ on M.
(b) The e is a C1-mapping α(·) : [0,1] →M, wi h
H(α( ), ) = 0,∀ ∈[0,1], α(0) = x∗, α(1) = x∗∗,whe e
H:M×[0,1] →Y, H(x, ) = (x)− g(x).
P oo . L(X, Y ) is p o ided by he opology gi en by i s ope a o no m and
X×Ris p o ided by a p oduc opology.
The poin x∗belongs o Uj, j ∈I, and x∗
ϕj=ϕj(x∗) is i s ep esen a i e
poin in he cha (Uj, ϕj).The ep esen a i e mapping o he mapping Hin
his cha (Uj, ϕj) is he C1-mapping
H:Ujϕj ×[0,1] ⊂X×R→Y, H(xϕj, ) := (H◦(ϕ−1
j, Id))(xϕj, ),
wi h Id( ) := . This mapping e i ies ha H(x∗
ϕj,0) = 0.
The ep esen a i e mapping o mapping Hon any cha is also w i en as
H o simplici y, and he same c i e ium will be used o any mapping. Any
ex ended mapping will be deno ed as he o iginal mapping.
(a) By hypo hesis (i), is a F edholm mapping o index ze o a x∗,i.e., he
ep esen a i e o in local cha s a e F edholm mappings o index ze o a he
co esponding ep esen a i e poin s. Since is a C1mapping and since index
is an in ege , he e o e Theo em 1 implies ha Ind 0(x) is locally cons an .
FREDHOLM MAPPINGS AND BANACH MANIFOLDS 467
Hence, since Mis connec ed, is a F edholm mapping o index ze o on M.
Hence
(3) 0(x)∈ F(X, Y ),and Ind 0(x) = 0,∀x∈M.
Since His a C1mapping, he e o e
∀ε > 0,k 0(xϕi)−( 0(xϕi)− g0(xϕi)) k=| |k g0(xϕi)k< ε, when | |< δ(ε),
and since he index is an in ege and [0,1] is connec ed, Theo em 1 and Equa-
ion (3) imply o any ixed x∈Min local cha s, ha
0(xϕi)− g0(xϕi)∈ F(X, Y )
and
Ind( 0(xϕi)− g0(xϕi)) = 0,∀i∈I, ∀ ∈[0,1].
Hence o any ixed ∈[0,1], H(·, ) is a F edholm mapping and Ind(Hx(x, )) =
Ind( 0(x)) = 0,and
(4) Ind(Hx(xϕi, )) = 0, i ∈I, x ∈Ui, ∈[0,1].
By hypo hesis (ii), since ze o is a egula alue o he mapping H(·, ) o
any ixed ∈[0,1], he e o e he mapping
Hx(xϕi, )(·)∈ L(X, Y )
maps Xon o Y o any ixed (xϕi, )∈(Uϕi×[0,1])∩H−1(0), he e o e
(5) codim( ange(Hx(xϕi, ))) = 0,
and hence
Ind(Hx(xϕi, )) = dim(ke (Hx(xϕi, ))).
Equa ions (4) and (5) imply ha
Ind(Hx(xϕi, )) = 0 = dim(ke (Hx(xϕi, ))),
i.e., (Hx(xϕi, )) is injec i e. The e o e Hx(xϕi, )(·) is a linea con inuous
bijec i e mapping, and since X, Y a e Banach spaces, he linea mapping
Hx(xϕi, )−1(·) is con inuous. Hence Hx(xϕi, )(·) is a linea homeomo phism,
i.e.,
Hx(xϕi, )(·)∈Isom(X, Y ).
(b) Le us ix any cha o he a las (Uj, ϕj), j ∈I, which will be called
(U, ϕ),whose co esponding cha image is Uϕ.Le us suppose ha (xa, a)∈
H−1(0) wi h xa∈U. Such a poin (xa, a) will be call a “s a ing poin ”.
The ep esen a i e mapping o Hin he cha (U, ϕ)
H:Uϕ×[0,1] ⊂X×R→Y,
clea ly has as ze o (xaϕ, a) wi h xaϕ,=ϕ(xa).
The ep esen a i e poin o (xa, a) in he cha (U×[0,1],(ϕ, Id)) e i ies
(xaϕ, a)∈H−1(0) ∩(Uϕ×[0,1]). Such a poin (xaϕ, a) will be called a

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“ ep esen a i e s a ing poin ” in Uϕ×[0,1], and he e is a posi i e numbe R
such ha he ball B(xaϕ, R)⊂Uϕ,and u he mo e
kHx(xaϕ, a)−1k=C.
(b1) Two posi i e numbe s a, 0amus be ound o la e use.
Le us cons uc he se
B=B(xaϕ, R)×[0,1],
and de ine he mapping
h:B⊂Uϕ×R→Y,
h((xaϕ+xϕ), ) := Hx(xaϕ, a)((xaϕ+xϕ)−xaϕ)−H((xaϕ+xϕ), ).
Since his a con inuous mapping as a composi ion o con inuous mappings,
he e o e o any > 0 and o Cgi en abo e, he e is a
(6) δ³
2C´>0,
such ha , i ((xaϕ+xϕ), ),∈Bwi h
kxϕ, − ak< δ ³
2C´ hen
(7) kh((xaϕ+xϕ), )−h(xaϕ, a)k<
2C.
On he o he hand, ano he mapping can be de ined as
hx:B→ L(X, Y )
hx((xaϕ+xϕ), ) := Hx(xaϕ, a)−Hx(xaϕ+xϕ, a)
and is also con inuous in he se B, and he e o e he e is an
(8) := δµ1
2C¶>0
such ha , i ((xaϕ+xϕ), )∈Bwi h
k(xϕ, − a)k< δ µ1
2C¶, hen
(9) khx((xaϕ+xϕ), )−hx(xaϕ, a)k<1
2C.
By aking gi en by Equa ion (8) and ixing 0
0:= δ(
2C),gi en by Equa ion
(6), he numbe 0:= min{ , 0
0}can be de ined. We selec a:= min {R, }
and 0a= min {R, 0}.
(b2) The se s
Ia:= { ∈[0,1] :| − a|≤ 0a}, Aa:= {xϕ∈X:kxϕk≤ a},
FREDHOLM MAPPINGS AND BANACH MANIFOLDS 469
will be associa ed o he “ ep esen a i e s a ing poin ” (xaϕ, a). Since
kxϕk≤ R, ∀xϕ∈Aa, he e o e
(xϕ+xaϕ)∈Uϕ⊂Xϕ.
Gi en a “s a ing poin ” (xa, a) and i s “ ep esen a i e s a ing poin ” (xaϕ,
a), he exis ence o wo con inuous mappings a e p o ed:
α(·) : Ia⊂R→Aa+xaϕ⊂Uϕ⊂X, such ha H(α( ), ) = 0,∀ ∈Ia
and
α(·) : Ia⊂R→U⊂M
such ha
H(α( ), ) = 0,∀ ∈Ia.
Le us sol e he equa ion
(10) H((xaϕ+xϕ), ) = 0,
o ixed ∈Iawhen xϕis in Aa.Ob iously, H(xaϕ, a) = 0.Equa ion (10) is
equi alen o he ollowing equa ion
(11) Hx(xaϕ, a)−1[Hx(xaϕ, a)(xϕ)−H((xaϕ+xϕ), )] = xϕ,
which leads us o de ine he mappings
h :Aa× { } → Y o ixed ∈Ia,
h (xϕ) := Hx(xaϕ, a)(xϕ)−H((xaϕ+xϕ), ) = h((xaϕ+xϕ), ),
and
T :Aa→X, T (xϕ) := Hx(xaϕ, a)−1h (xϕ).
Obse e ha h (xϕ) is h((xaϕ+xϕ), ) de ined in (b1) when is ixed and
belongs o Ia.Le us also obse e ha in he de ini ions o h and T is an
index o highligh ha is ixed.
E iden ly
(12) h a(0) = 0,
and
(13) h0
a(0) = 0.
Equa ion (11) is equi alen o he ollowing key Fixed Poin Equa ion
(14) T (xϕ) = xϕ,
which is s udied below.
Le xϕ, x0
ϕ∈Aa, ∈Ia,and hence he Taylo Theo em oge he wi h Equa-
ions (9) and (13) imply ha
(15)
kh (xϕ)−h (x0
ϕ)k
≤sup nkh0
(x0
ϕ+θ(xϕ−x0
ϕ)) k:θ∈[0,1]o· k xϕ−x0
ϕk ≤ 1
2C a.
470 JOS´
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Equa ions (12) and (15) imply ha
kh (xϕ)k ≤ k h (xϕ)−h a(0) k+kh a(0) k ≤ a
2C,
hence
(16) kT (xϕ)k ≤ k Hx(xaϕ, a)−1kk h(xϕ, )k ≤ a.
We will apply Theo em 2 o he se s and mappings which ha e jus been de ined.
The me ic space (Ia,|·|) is he pa ame e space o hypo hesis (i) needed in
Theo em 2. The se Aais conside ed as he closed and non-emp y se and X
as he comple e me ic space o hypo hesis (ii), which is e i ied below:
F om Equa ion (16), o any ixed ∈Ia,and o all xϕ∈Aa,we ha e
kT xϕk≤ a, he e o e T xϕ∈Aa,and hence T :Aa→Aa,i.e., T maps he
closed and non-emp y se Aao he Banach space Xin o i sel .
F om Equa ions (9), (13), and he Taylo Theo em, o any xϕ, x0
ϕ∈Aa,
and o any ∈Ia, he ollowing holds:
kT (xϕ)−T (x0
ϕ)k ≤ k Hx(xaϕ, a)−1kk h0
(x0
ϕ+θ(xϕ−x0
ϕ)) kk xϕ−x0
ϕk
≤Ckh0
(x0
ϕ+θ(xϕ−x0
ϕ)) −h0
a(0) kk xϕ−x0
ϕk
≤1
2kxϕ−x0
ϕk,(θ∈[0,1]).
The e o e T is hal -con ac i e o any ixed ∈Ia.Hence hypo hesis (ii) o
Theo em 2 is e i ied.
Fo any ixed 0∈Iaand o all xϕ∈Aa, he ollowing holds:
T (xϕ) = Hx(xaϕ, a)−1(Hx(xaϕ, a)(xϕ)−H(xaϕ+xϕ, ))
→Hx(xaϕ, a)−1(Hx(xaϕ, a)(xϕ)−H(xaϕ+xϕ, 0))
=T 0(xϕ) as → 0, ∈Ia,
he e o e hypo hesis (iii) o Theo em 2 is also e i ied. Hence Theo em 2 im-
plies, o any ∈Ia, ha T has a unique ixed poin xϕ∈Aa, T (xϕ) = xϕ:=
xϕ( ),and xϕ( )→xϕ( 0) while → 0; , 0∈Ia,i.e., xϕ(·) is a con inuous
mapping. Thus o any ∈Ia he e is only one xϕ∈Aa, i.e.,
T (xϕ) = xϕ:= xϕ( )
and he mapping, we ha e jus de ined, xϕ(·) e i ies
xϕ( )→xϕ( 0),while → 0,∀ , 0∈Ia,
which implies ha xϕ(·) is a con inuous mapping. Thus o any ∈Ia he e is
one xϕ( ) such ha
(17) H(xaϕ+xϕ( ), )=0,
and u he mo e
H(xaϕ+xϕ( ), )→H(xaϕ+xϕ( 0), 0) = 0 while → 0, , 0∈Ia.
FREDHOLM MAPPINGS AND BANACH MANIFOLDS 471
Le us obse e ha T a(0) = 0, xϕ( a)=0.
Equa ion (17) can be w i en as H(α( ), ) = 0,which is e i ied o all
∈Ia,whe e αis he ollowing cu e on he cha space (U, ϕ),
(18) α:Ia→Uϕ⊂X, α( ) := xaϕ+xϕ( ),whe e α( a) = xaϕ,
which is one o he goals o his sec ion.
The con inui y o bo h αand ϕ−1le s us cons uc he ollowing cu e αon
he opological space M.
(19) α:Ia⊂R→U⊂M, α( ) := (ϕ−1◦α).
Equa ions (18) and (19) implies ha
(20) α( a) = xa
o be used in he nex sec ion.
Equa ion (17) implies ha
H(α( ), ) = (H◦(ϕ−1, Id))(α( ), ) = H(α( ), ) = 0,∀ ∈Ia,
which is he ano he goal o his sec ion.
(c) Conclusions (a) and (b) will be p o ed he e.
Since −1(0) = x∗ om hypo hesis (i), hen he e exis s Ujsuch ha
H−1(0) ∩(Uj×[0,1]) 6=∅, he e o e he e is a poin (x∗,0) ∈H−1(0), x∗∈Uj,
i.e., (x∗,0) is a “s a ing poin ”.
Since (x∗
ϕj,0) ∈H−1(0) ∩(Uϕj×[0,1]),i.e., (x∗
ϕj,0) is a “ ep esen a i e
s a ing poin ”, he e o e om (b2) he e exis a se I0,and wo con inuous
mappings αand αsuch ha
α:I0→Uϕj⊂X, which e i ies H(α( ), )=0,∀ ∈I0,
and
(21) α:I0→Uj⊂M, which e i ies H(α( ), ) = 0,∀ ∈I0.
We wan o ex end he con inuous mapping α:I0→M o be a con inuous
mapping α: [0,1] →M, and o ex end Equa ion (21) o become
H(α( ), ) = 0,∀ ∈[0,1].
Le us suppose ha α( )∈M, ∀ ∈[0, b], b ∈Ui.Mapping αis ex ended o
he igh o bby aking (α(b), b),which belongs o H−1(0)∩(Ui×[0,1]), i ∈I, as
he ollowing “s a ing poin ”. Equa ion (20) enables he con inuous ex ension
o he con inuous mapping α o he igh . The con inuous ex ended mapping
is also known as α.
Mapping αis successi ely ex ended o he igh in he same way by using i s
ep esen a i e in he di e en cha s o he a las. Now we conside all in e als