Fredholm mappings and Banach manifolds

J. Korean Math. Soc. 46 (2009), No. 3, pp. 463–473 DOI 10.4134/JKMS.2009.46.3.463 FREDHOLM MAPPINGS AND BANACH MANIFOLDS Jos´ e Mar´ ıa Soriano Arbizu Abstract. Two C1-mappings, whose domain is a connected compact C1-Banach manifold modelled over a Banach space Xover K=Ror C and whose range is a Banach space Yover K,are introduced. Sufficient conditions are given to assert they share only a value. The proof of the result, which is based upon continuation methods, is constructive. 1. Preliminaries Scientific phenomena are locally described by parameters, whose choice is sometimes arbitrary. This implies the importance of the availability of a methodology for the comparison of results of measurements. Locally, a Banach manifold looks like a Banach space. For a local description, different Banach (or coordinate or parameter) spaces are allowed and transformation rules exist for these coordinates. Let X, Y be two Banach spaces. Let u:U⊂X→Ybe a continuous mapping. One way of solving the equation (1) u(x) = y for any fixed y∈Y, is to embed (1) in a continuum of problems (2) H(x, t) = y, (0 ≤t≤1), which is solved when t= 0.When t= 1,problem (2) becomes (1). If it is possible to continue the solution for all t∈[0,1], then (1) is solved. This is the continuation method with respect to a parameter [1-25]. A continuation method was introduced to solve (1) when u:M→Rn,where Mis a connected compact C1-Banach manifold modelled on Rn,and H(·,·) is a C1-mapping [25]. Here Mis a Banach manifold modelled on an infinite-dimensional Banach space Xover K=Ror Cand uranges over an infinite-dimensional Banach space Yover K. Received July 11, 2007. 2000 Mathematics Subject Classification. Primary 58C30, Secondary 65H20. Key words and phrases. regular value, continuation methods, atlas, chart, Banach manifold, compactness. This work is partially supported by D.G.E.S. Pb 96-1338-CO 2-01 and the Junta de Andalucia. c °2009 The Korean Mathematical Society 463 464 JOS´ E MAR´ IA SORIANO ARBIZU Sufficient conditions are given to prove that two C1-mappings, of which one is Fredholm of index zero, share only one value on a Banach manifold by using continuation methods on charts. Other conditions, sufficient to guarantee the existence of zero points, have been given by the author in several other papers [7-25].This shared value can be estimated following a curve. The proof supplies the existence of a curve which leads to the point whose image is the shared value. The keys are the use of the chart spaces [27], the compactness and connectness of M, together with the use of the Continuous Dependence Theorem (Theorem 2) [26] and Theorem 1 [28]. We briefly recall some theorems and notation to be used. Definitions and Notation. [26-28]. Let F:D(F) : X→Y, where X, Y are Banach spaces over K.If D(F) is open, then mapping Fis said to be a Fredholm mapping if and only if both Fis a C1-mapping and F0(x) : X→Y is a Fredholm linear mapping for all x∈D(F). That L:X→Yis a linear Fredholm mapping means that Lis linear and continuous and both the numbers dim(ker(L)) and codim(R(L)) are finite, where dim signifies dimension, codim codimension, ker kernel and R(L) stands for the range of mapping L. Therefore ker(L) = X1is a Banach space and has topological complement X2, since dim(X1) is finite. The integer number ind(L)= dim(ker(L))-codim(R(L)) is called the index of L. Let F(X, Y ) denote the set of all linear Fredholm mappings L:X→Y. Let Mbe a topological space. A chart (U, ϕ) in Mis a pair where the set Uis open in Mand ϕ:U→Uϕis a homeomorphism onto an open subset Uϕ of a Banach space Xϕ. We call ϕachart map, Xϕis called chart space, and Uϕchart image. For x∈U,xϕ=ϕ(x) is called the representative of xin the chart (U, ϕ) or the local coordinate of xin the local coordinate system ϕ. The point x∈Mmay have different local coordinates xϕ=ϕ(x) and xψ=ψ(x) for two different charts (U, ϕ) and (V, ψ),respectively. The transformation rules between them are xϕ=ϕ(ψ−1(xψ)) and xψ=ψ(ϕ−1(xϕ)). Two charts, (U, ϕ) and (V, ψ) in M, are called Ck-compatible if and only if U∩V=∅,or both ϕ◦ψ−1:ψ(U∩V)→ϕ(U∩V) and ψ◦ϕ−1:ϕ(U∩V)→ ψ(U∩V) are Ck-mappings, k≥0. ACk-atlas for M, 0≤k≤ ∞ is a collection of charts (Ui, ϕi),where i∈I, which satisfies the following conditions: (i) the Uicover M, (ii) any two charts are Ck-compatible, (iii) all chart spaces Xiare Banach spaces over K. If there is a Ck-atlas for M, then Mis said to be a Ck-Banach manifold. If all chart spaces are equal to a fixed Banach space X, M is called a Ck-Banach manifold modelled on X. Here manifolds without boundaries will be considered, such as the surface of a ball in Rn, an open set in a Banach space X, etc. Let Mand Nbe Ck-Banach manifolds with chart spaces over K, k ≥1.The mapping f:M→Nis called a Cr-mapping,where r≤k, if and only if fis Cr FREDHOLM MAPPINGS AND BANACH MANIFOLDS 465 at each point x∈Min fixed admissible charts. This means the following: If (U, ϕ) and (V, ψ) are charts in Mand Nrespectively, with x∈Uand f(x)∈V, then the mapping f=ψ◦f◦ϕ−1,which is well defined in a sufficiently small neighbourhood of xϕ, is Crin the usual sense. fis called a representative of f. Two C1-curves in M, which pass through the point x∈M, are called equivalent the point xif and only if the representatives have the same tangent vector at xin some fixed admissible chart. A tangent vector v(otherwise known as x0(t0)) to Mat xconsists of all C1-curves which are equivalent at xto a fixed C1-curve. The tangent “abstract” vector vof the previous definition to the curve x(·) : U(t0)⊂R→M, x =x(t) at the point x(t0),has its representative or local coordinate vϕ=x0 ϕ(t0) in the chart (U, ϕ),where xϕ=xϕ(t) = ϕ(x(t)). The tangent space TMxto Mat the point xis by definition the set of all tangent vectors. It is proven that this is a topological vector space which is linear homeomorphic to each chart space Xϕat the point x. The map f0(x) : T Mx→T Nf(x)is called the tangent map of f:M→N at point x, which is clearly the normal F-derivative in local coordinates. A mapping f:M→Nis called a Fredholm operator at xif and only if the linearization f0(x) : Mx→Nf(x)is a Fredholm operator. Furthermore, fis a Fredholm operators at xif and only if the representatives of fin local charts are Fredholm operator at the corresponding points. Let f:M→Nbe a Ck-mapping, k≥1,where Mand Nare Ck-Banach manifolds with chart space over K.The mapping fis called a submersion at x if and only if f0(x) is surjective and the null space ker(f0(x)) splits the tangent space of Mat point x(which is automatic when ker(f0(x)) = {0}). A point x∈Mis called a regular point of fif and only if fis a submersion at x. A point y∈Nis called a regular value of fif and only if the set f−1(y) is empty or consists only of regular points. If X, Y are Banach spaces, let L(X, Y ) denote the set of all linear continuous mappings L:X→Y. Let Isom(X, Y ) denote the set of all the linear homeomorphisms L:X→Y. Let B(x0, ρ) be the open ball of centre x0and radius ρ. If u:X→Yis a linear continuous bijective operator, then the inverse linear continuous operator will be denoted by u−1. Mapping Hx(xϕ, t) denotes the partial F-derivative of mapping Hwith respect to Xat the point (xϕ, t), where H:Uϕ×[0,1] ⊂X×R→Y. A representative point always has its corresponding chart map as subindex. Theorem 1 ([28], p. 300).If S∈ F(X, Y ), where X, Y are Banach spaces over K,then there is a number ε > 0such that T∈ F(X, Y )and Ind T= Ind S for all linear Fredholm mappings T∈ L(X, Y )with kT−Sk< ε. Theorem 2 (Continuous Dependence Theorem [26], pp. 18–19).Let the following conditions be satisfied: 466 JOS´ E MAR´ IA SORIANO ARBIZU (i) P is a metric space, called the parameter space. (ii) For each parameter p∈P, mapping Tpsatisfies the following hypotheses: (1) Tp:M→M, i.e. Mis mapped into itself by Tp. (2) Mis a closed non-empty set in a complete metric space (X, d). (3) Tpis k-contractive for any fixed k∈[0,1). (iii) For each p0∈P, and any x∈M, lim p→p0 Tp(x) = Tp0(x). Then for each p∈P, the equation xp=Tpxphas exactly one solution xp,where xp∈Mand lim p→p0 xp=xp0. 2. Fredholm mappings on compact Banach manifolds Theorem 3. Let f, g :M→Ybe two C1-mappings, where Mis a compact, connected, C1-Banach manifold modelled on X, where X, Y are two Banach spaces over K=Ror C. Let (Ui, ϕi)i∈I, I = 1,2, . . . , N be a C1-atlas for M. Suppose that the following conditions hold: (i) Mapping fhas only one zero x?in M, with x∗∈Uj, and fis a Fredholm mapping of index zero at x∗. (ii) For any fixed t∈[0,1],zero is a regular value of any mapping H(·, t) : M× {t} → Y, H(·, t) := f(·)−tg(·). Hence the following statement holds true: (a) The mappings fand gshare only one value x∗∗ on M. (b) There is a C1-mapping α(·) : [0,1] →M, with H(α(t), t) = 0,∀t∈[0,1], α(0) = x∗, α(1) = x∗∗,where H:M×[0,1] →Y, H(x, t) = f(x)−tg(x). Proof. L(X, Y ) is provided by the topology given by its operator norm and X×Ris provided by a product topology. The point x∗belongs to Uj, j ∈I, and x∗ ϕj=ϕj(x∗) is its representative point in the chart (Uj, ϕj).The representative mapping of the mapping Hin this chart (Uj, ϕj) is the C1-mapping H:Ujϕj ×[0,1] ⊂X×R→Y, H(xϕj, t) := (H◦(ϕ−1 j, Id))(xϕj, t), with Id(t) := t. This mapping verifies that H(x∗ ϕj,0) = 0. The representative mapping of mapping Hon any chart is also written as Hfor simplicity, and the same criterium will be used for any mapping. Any extended mapping will be denoted as the original mapping. (a) By hypothesis (i), fis a Fredholm mapping of index zero at x∗,i.e., the representative of fin local charts fare Fredholm mappings of index zero at the corresponding representative points. Since fis a C1mapping and since index is an integer, therefore Theorem 1 implies that Ind f0(x) is locally constant. FREDHOLM MAPPINGS AND BANACH MANIFOLDS 467 Hence, since Mis connected, fis a Fredholm mapping of index zero on M. Hence (3) f0(x)∈ F(X, Y ),and Ind f0(x) = 0,∀x∈M. Since His a C1mapping, therefore ∀ε > 0,kf0(xϕi)−(f0(xϕi)−tg0(xϕi)) k=|t|k g0(xϕi)k< ε, when |t|< δ(ε), and since the index is an integer and [0,1] is connected, Theorem 1 and Equation (3) imply for any fixed x∈Min local charts, that f0(xϕi)−tg0(xϕi)∈ F(X, Y ) and Ind(f0(xϕi)−tg0(xϕi)) = 0,∀i∈I, ∀t∈[0,1]. Hence for any fixed t∈[0,1], H(·, t) is a Fredholm mapping and Ind(Hx(x, t)) = Ind(f0(x)) = 0,and (4) Ind(Hx(xϕi, t)) = 0, i ∈I, x ∈Ui, t ∈[0,1]. By hypothesis (ii), since zero is a regular value of the mapping H(·, t) for any fixed t∈[0,1], therefore the mapping Hx(xϕi, t)(·)∈ L(X, Y ) maps Xonto Yfor any fixed (xϕi, t)∈(Uϕi×[0,1])∩H−1(0),therefore (5) codim(range(Hx(xϕi, t))) = 0, and hence Ind(Hx(xϕi, t)) = dim(ker(Hx(xϕi, t))). Equations (4) and (5) imply that Ind(Hx(xϕi, t)) = 0 = dim(ker(Hx(xϕi, t))), i.e., (Hx(xϕi, t)) is injective. Therefore Hx(xϕi, t)(·) is a linear continuous bijective mapping, and since X, Y are Banach spaces, the linear mapping Hx(xϕi, t)−1(·) is continuous. Hence Hx(xϕi, t)(·) is a linear homeomorphism, i.e., Hx(xϕi, t)(·)∈Isom(X, Y ). (b) Let us fix any chart of the atlas (Uj, ϕj), j ∈I, which will be called (U, ϕ),whose corresponding chart image is Uϕ.Let us suppose that (xa, ta)∈ H−1(0) with xa∈U. Such a point (xa, ta) will be call a “starting point”. The representative mapping of Hin the chart (U, ϕ) H:Uϕ×[0,1] ⊂X×R→Y, clearly has as zero (xaϕ,ta) with xaϕ,=ϕ(xa). The representative point of (xa, ta) in the chart (U×[0,1],(ϕ, Id)) verifies (xaϕ, ta)∈H−1(0) ∩(Uϕ×[0,1]). Such a point (xaϕ, ta) will be called a 468 JOS´ E MAR´ IA SORIANO ARBIZU “representative starting point” in Uϕ×[0,1], and there is a positive number R such that the ball B(xaϕ, R)⊂Uϕ,and furthermore kHx(xaϕ, ta)−1k=C. (b1) Two positive numbers ra, r0amust be found for later use. Let us construct the set B=B(xaϕ, R)×[0,1], and define the mapping h:B⊂Uϕ×R→Y, h((xaϕ+xϕ), t) := Hx(xaϕ, ta)((xaϕ+xϕ)−xaϕ)−H((xaϕ+xϕ), t). Since his a continuous mapping as a composition of continuous mappings, therefore for any r > 0 and for Cgiven above, there is a (6) δ³r 2C´>0, such that, if ((xaϕ+xϕ), t),∈Bwith kxϕ, t −tak< δ ³r 2C´then (7) kh((xaϕ+xϕ), t)−h(xaϕ, ta)k<r 2C. On the other hand, another mapping can be defined as hx:B→ L(X, Y ) hx((xaϕ+xϕ), t) := Hx(xaϕ, ta)−Hx(xaϕ+xϕ, ta) and is also continuous in the set B, and therefore there is an (8) r:= δµ1 2C¶>0 such that, if ((xaϕ+xϕ), t)∈Bwith k(xϕ, t −ta)k< δ µ1 2C¶,then (9) khx((xaϕ+xϕ), t)−hx(xaϕ, ta)k<1 2C. By taking rgiven by Equation (8) and fixing r0 0:= δ(r 2C),given by Equation (6), the number r0:= min{r, r0 0}can be defined. We select ra:= min {R, r} and r0a= min {R, r0}. (b2) The sets Ia:= {t∈[0,1] :|t−ta|≤ r0a}, Aa:= {xϕ∈X:kxϕk≤ ra}, FREDHOLM MAPPINGS AND BANACH MANIFOLDS 469 will be associated to the “representative starting point” (xaϕ, ta). Since kxϕk≤ R, ∀xϕ∈Aa, therefore (xϕ+xaϕ)∈Uϕ⊂Xϕ. Given a “starting point” (xa,ta) and its “representative starting point” (xaϕ, ta), the existence of two continuous mappings are proved: α(·) : Ia⊂R→Aa+xaϕ⊂Uϕ⊂X, such that H(α(t), t) = 0,∀t∈Ia and α(·) : Ia⊂R→U⊂M such that H(α(t), t) = 0,∀t∈Ia. Let us solve the equation (10) H((xaϕ+xϕ), t) = 0, for fixed t∈Iawhen xϕis in Aa.Obviously, H(xaϕ, ta) = 0.Equation (10) is equivalent to the following equation (11) Hx(xaϕ, ta)−1[Hx(xaϕ, ta)(xϕ)−H((xaϕ+xϕ), t)] = xϕ, which leads us to define the mappings ht:Aa× {t} → Yfor fixed t∈Ia, ht(xϕ) := Hx(xaϕ, ta)(xϕ)−H((xaϕ+xϕ), t) = h((xaϕ+xϕ), t), and Tt:Aa→X, Tt(xϕ) := Hx(xaϕ, ta)−1ht(xϕ). Observe that ht(xϕ) is h((xaϕ+xϕ), t) defined in (b1) when tis fixed and belongs to Ia.Let us also observe that tin the definitions of htand Ttis an index to highlight that tis fixed. Evidently (12) hta(0) = 0, and (13) h0 ta(0) = 0. Equation (11) is equivalent to the following key Fixed Point Equation (14) Tt(xϕ) = xϕ, which is studied below. Let xϕ, x0 ϕ∈Aa, t ∈Ia,and hence the Taylor Theorem together with Equations (9) and (13) imply that (15) kht(xϕ)−ht(x0 ϕ)k ≤sup nkh0 t(x0 ϕ+θ(xϕ−x0 ϕ)) k:θ∈[0,1]o· k xϕ−x0 ϕk ≤ 1 2Cra. 470 JOS´ E MAR´ IA SORIANO ARBIZU Equations (12) and (15) imply that kht(xϕ)k ≤ k ht(xϕ)−hta(0) k+khta(0) k ≤ ra 2C, hence (16) kTt(xϕ)k ≤ k Hx(xaϕ, ta)−1kk h(xϕ, t)k ≤ ra. We will apply Theorem 2 to the sets and mappings which have just been defined. The metric space (Ia,|·|) is the parameter space of hypothesis (i) needed in Theorem 2. The set Aais considered as the closed and non-empty set and X as the complete metric space of hypothesis (ii), which is verified below: From Equation (16), for any fixed t∈Ia,and for all xϕ∈Aa,we have kTtxϕk≤ ra,therefore Ttxϕ∈Aa,and hence Tt:Aa→Aa,i.e., Ttmaps the closed and non-empty set Aaof the Banach space Xinto itself. From Equations (9), (13), and the Taylor Theorem, for any xϕ, x0 ϕ∈Aa, and for any t∈Ia,the following holds: kTt(xϕ)−Tt(x0 ϕ)k ≤ k Hx(xaϕ, ta)−1kk h0 t(x0 ϕ+θ(xϕ−x0 ϕ)) kk xϕ−x0 ϕk ≤Ckh0 t(x0 ϕ+θ(xϕ−x0 ϕ)) −h0 ta(0) kk xϕ−x0 ϕk ≤1 2kxϕ−x0 ϕk,(θ∈[0,1]). Therefore Ttis half-contractive for any fixed t∈Ia.Hence hypothesis (ii) of Theorem 2 is verified. For any fixed t0∈Iaand for all xϕ∈Aa, the following holds: Tt(xϕ) = Hx(xaϕ, ta)−1(Hx(xaϕ, ta)(xϕ)−H(xaϕ+xϕ, t)) →Hx(xaϕ, ta)−1(Hx(xaϕ, ta)(xϕ)−H(xaϕ+xϕ, t0)) =Tt0(xϕ) as t→t0, t ∈Ia, therefore hypothesis (iii) of Theorem 2 is also verified. Hence Theorem 2 implies, for any t∈Ia,that Tthas a unique fixed point xϕ∈Aa, Tt(xϕ) = xϕ:= xϕ(t),and xϕ(t)→xϕ(t0) while t→t0;t, t0∈Ia,i.e., xϕ(·) is a continuous mapping. Thus for any t∈Iathere is only one xϕ∈Aa, i.e., Tt(xϕ) = xϕ:= xϕ(t) and the mapping, we have just defined, xϕ(·) verifies xϕ(t)→xϕ(t0),while t→t0,∀t, t0∈Ia, which implies that xϕ(·) is a continuous mapping. Thus for any t∈Iathere is one xϕ(t) such that (17) H(xaϕ+xϕ(t), t)=0, and furthermore H(xaϕ+xϕ(t), t)→H(xaϕ+xϕ(t0), t0) = 0 while t→t0, t, t0∈Ia. FREDHOLM MAPPINGS AND BANACH MANIFOLDS 471 Let us observe that Tta(0) = 0, xϕ(ta)=0. Equation (17) can be written as H(α(t), t) = 0,which is verified for all t∈Ia,where αis the following curve on the chart space (U, ϕ), (18) α:Ia→Uϕ⊂X, α(t) := xaϕ+xϕ(t),where α(ta) = xaϕ, which is one of the goals of this section. The continuity of both αand ϕ−1lets us construct the following curve αon the topological space M. (19) α:Ia⊂R→U⊂M, α(t) := (ϕ−1◦α). Equations (18) and (19) implies that (20) α(ta) = xa to be used in the next section. Equation (17) implies that H(α(t), t) = (H◦(ϕ−1, Id))(α(t), t) = H(α(t), t) = 0,∀t∈Ia, which is the another goal of this section. (c) Conclusions (a) and (b) will be proved here. Since f−1(0) = x∗from hypothesis (i), then there exists Ujsuch that H−1(0) ∩(Uj×[0,1]) 6=∅,therefore there is a point (x∗,0) ∈H−1(0), x∗∈Uj, i.e., (x∗,0) is a “starting point”. Since (x∗ ϕj,0) ∈H−1(0) ∩(Uϕj×[0,1]),i.e., (x∗ ϕj,0) is a “representative starting point”, therefore from (b2) there exist a set I0,and two continuous mappings αand αsuch that α:I0→Uϕj⊂X, which verifies H(α(t), t)=0,∀t∈I0, and (21) α:I0→Uj⊂M, which verifies H(α(t), t) = 0,∀t∈I0. We want to extend the continuous mapping α:I0→Mto be a continuous mapping α: [0,1] →M, and to extend Equation (21) to become H(α(t), t) = 0,∀t∈[0,1]. Let us suppose that α(t)∈M, ∀t∈[0, b], b ∈Ui.Mapping αis extended to the right of bby taking (α(b), b),which belongs to H−1(0)∩(Ui×[0,1]), i ∈I, as the following “starting point”. Equation (20) enables the continuous extension of the continuous mapping αto the right. The continuous extended mapping is also known as α. Mapping αis successively extended to the right in the same way by using its representative in the different charts of the atlas. Now we consider all intervals