Fredholm mappings and Banach manifolds

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´ E MAR´ 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: 466 JOS´ E MAR´ IA SORIANO ARBIZU (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 468 JOS´ E MAR´ IA SORIANO ARBIZU “ 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´ E MAR´ IA SORIANO ARBIZU 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