Shared value in a finite-dimensional Banach manifold

Acta Mathematica Scientia 2010,30B(5):1687–1695 http://actams.wipm.ac.cn

SHARED VALUE IN A FINITE-DIMENSIONAL BANACH MANIFOLD∗ J.M. Soriano Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Sevilla, Aptdo. 1160, Sevilla 41080, Spain E-mail: [email protected]

Abstract Sufficient conditions are given to assert that two C 1 -mappings share only one value in a connected compact Banach manifold modelled over Rn . The proof of the result, which is based upon continuation methods, is constructive. Key words regular value; continuation methods; atlas; chart; Banach manifold; compactness 2000 MR Subject Classification

1

58C30; 65H20

Introduction

Perhaps the most important concept in mathematical physics is the concept of Banach manifold. The reason for this importance is that different parameter spaces or coordinate spaces can be used in a description of a scientific phenomenon in terms of a local description. It is important to have a transformation rule for these coordinates used in the process of measurement since this rule exists in a Banach manifold. Let X, Y be two Banach Spaces. Let u : U ⊂ X → Y be a continuous mapping. One way of solving the equation u(x) = y (1) for any fixed y ∈ Y , is to embed (1) in a continuum of problem H(x, t) = y (0 ≤ t ≤ 1),

(2)

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 [1–21]. A continuation method is introduced to solve (1) when u : M → Rn , where M is a connected compact C 1 -Banach manifold modelled on Rn , and H(·, ·) is a C 1 -mapping. Sufficient conditions are given to prove that two C 1 -mappings share only one value on a Banach manifold by using continuation methods on charts. Other conditions, sufficient to ∗ Received

December 5, 2007; revised December 14, 2008. This work is partially supported by D.G.E.S. Pb 96-1338-CO 2-01 and the Junta de Andalucia.

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guarantee the existence of zero points, were given by the author in several other papers [6–21]. This shared value can be estimated by following a curve. The proof supplies the existence of a curve which leads to the point whose image is the shared value. The key is the use of the chart spaces [24], the compactness of M , together with the use of the continuous dependece theorem [23] and a consequence of the properties of Banach algebra of a linear continuous mapping from a Banach space into itself [22]. We briefly recall some theorems and notations to be used. Definitions and Notations [24] Let M be a topological space. A chart (U, ϕ) in M is a pair where the set U is open in M and ϕ : U → Uϕ is a homeomorphism onto an open subset Uϕ of a Banach-space Xϕ . We call ϕ a chart map, Xϕ is called a chart space, and Uϕ the chart image. For x ∈ U , xϕ = ϕ(x) is called the representative of x in the chart (U, ϕ) or the local coordinate of x in the local coordinate system ϕ. The point x ∈ M may 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 C k -compatible if and only if U ∩ V = ∅ or ϕ ◦ ψ −1 : ψ(U ∩ V ) → ϕ(U ∩ V ) and ψ ◦ ϕ−1 : ϕ(U ∩ V ) → ψ(U ∩ V ) are C k -mappings, k ≥ 0. A C k -atlas for M, 0 ≤ k ≤ ∞, is a collection of charts (Ui , ϕi ) (i ranging in some index set) which satisfies the following conditions: (i) Ui cover M, (ii) any two charts are C k -compatible, (iii) all chart spaces Xi are Banach spaces over K. If there is a C k -atlas for M, then M is said to be a C k -Banach manifold. If all chart spaces are of the same dimension d, 0 < d ≤ ∞, then d is called the dimension of M. If all chart spaces are equal to a fixed Banach space X, M is called a C k -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 M and N be C k -Banach manifolds with chart spaces over K, k ≥ 1. Then f : M → N is called a C r -mapping, r ≤ k, if and only if f is C r at each point x ∈ M in fixed admissible charts. This means the following: If (U, ϕ) and (V, ψ) are charts in M and N , respectively, with x ∈ U and f (x) ∈ V , then the mapping f = ψ ◦ f ◦ ϕ−1 , which is well defined in a sufficiently small neighbourhood of xϕ , is C r in the usual sense. f is called a representative of f. Let f : M → N be a C k -mapping, k ≥ 1, where M and N are C k -Banach manifolds with chart space over K, f is called a submersion at x if and only if f  (x) is surjective and the null space N (f  (x)) splits the tangent space of M at point x ([24], p.538–540), which is automatic when M and N are finite-dimensional. A point x ∈ M is called a regular point of f if and only if f is a submersion at x. A point y ∈ N is called a regular value of f if and only if the set f −1 (y) is empty or consists only of regular points. If f is a submersion at every point x ∈ M , f is called a submersion. 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 isomorphisms L : X → Y. Let B(x0 , ρ) be the open ball with centre x0 and of radius ρ. If u : X → Y is a linear continuous bijective operator, then the inverse linear continuous operator will be denoted by u−1 . If U is a set in X, let ∂U denote the boundary of the set U , and U its closure.

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Hx (x, t) denotes the partial F-derivative of H with respect to X at the point (x, t), where H : X × [0, 1] → Y. Theorem 1 Continuous Dependence Theorem ([23], p.18–19) Let the following conditions be satisfied: (i) P is a metric space, called the parameter space. (ii) For each parameter p ∈ P , mapping Tp satisfies the following hypotheses: (1) Tp : M → M , i.e., M is mapped into itself by Tp . (2) M is a closed non-empty set in a complete metric space (X, d). (3) Tp is k-contractive for any fixed k ∈ [0, 1). (iii) For each p0 ∈ P , and any x ∈ M, lim Tp (x) = Tp0 (x). p→p0

Thus for each p ∈ P , the equation xp = Tp xp has exactly one solution xp , where xp ∈ M and lim xp = xp0 . p→p0

Theorem 2 ([22], p.23–24) Let X, Y be Banach spaces over K. Hence, the following statements hold: a) Isom(X, Y ) is open in L(X, Y ) . b) The mapping β: Isom(X, Y ) → L(Y, X), β(u) := u−1 is continuous.

2

Shared Value in a Finite-Dimensional Banach Manifold

Theorem 3 Let f, g : M → Rn be two C 1 -mappings,where M is a compact connected C -Banach manifold modelled on Rn . (Ui , ϕi )i∈I , I = 1, 2, · · · , N , is a C 1 -atlas for M, where Uϕi , i ∈ I are bounded. Suppose that the following conditions hold: (i) Mapping f has only one zero x in M with x∗ ∈ Uj . (ii) Zero is a regular value of each mapping H(·, t) : M × {t} → Rn , H(·, t) := f (·) − tg(·), for any fixed t ∈ [0, 1]. (iii) Any chart map ϕi , i ∈ I, can be extended homeomorphically on U i , and the representative of H over Uϕi , i ∈ I can be extended continuously over U i , where H = H(·, ·) : M × [0, 1] → Rn , H(x, t) = f (x) − tg(x). (iv) The representative H x : Uϕi × [0, 1] → L(Rn , Rn ) has a continuous extension over U ϕi , i ∈ I. Hence the following statement holds: (a) The mappings f and g share only one value x∗∗ on M. (b) There is C-mapping x(·) : [0, 1] → M with 1

H(x(t), t) = 0, ∀t ∈ [0, 1], x(0) = x∗ , x(1) = x∗∗ . Proof L(Rn , Rn ) is provided by the topology given by its operator norm and Rn × R is provided by a product topology. x∗ belongs to Uj , j ∈ I, and x∗ϕj = ϕj (x∗ ) is its representative in the chart (Uj , ϕj ). The representative mapping of H : M × [0, 1] → Rn , H(x, t) = f (x) − tg(x) in this chart is the C 1 -mapping H : Ujϕj × [0, 1] ⊂ Rn+1 → Rn , H(xϕj , t) := (H ◦ (ϕ−1 j , Id))(xϕj , t)

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with Id(t) := t. This verifies that H(x∗ϕj , 0) = 0. In the same way the representative of mapping H is defined in any chart, which will also be denoted by H for simplicity. We also observe that any extended mapping will be denoted as the original mapping. (a) Let us fix any chart of the atlas (Uj , ϕj ), j ∈ I, which will be called (U, ϕ), and let us suppose that (xa , ta ) ∈ H −1 (0) with xa ∈ U. Therefore, the representative of H, H : Uϕ × [0, 1] ⊂ Rn+1 → Rn has the zero (xaϕ , ta ) with xaϕ , = ϕ(xa ). We will prove there is a neighbourhood Ia of ta and a continuous mapping α(·) : Ia ⊂ [0, 1]→Uϕ ⊂ Rn with H(α(t), t) = 0, ∀t ∈ Ia : (a1) By hypothesis (ii), H x (xϕ , t)(·) ∈ L(Rn , Rn ) maps onto Rn for any (xϕ , t) ∈ (Uϕ × [0, 1]) ∩ H −1 (0), therefore rank(H x (xϕ , t)) = n, and hence H x (xϕ , t) belongs to Isom(Rn , Rn ). −1 We will prove here that there is a real number C > 0, such that if (xϕ , t) belongs to (H (0)) ∩ (Uϕ × [0, 1]), then

H x (xϕ , t)−1 ≤ C. Since H is a C 1 -mapping, the mapping H x : Uϕ × [0, 1] ⊂ Rn × [0, 1] → L(Rn , Rn ) (xϕ , t) → H x (xϕ , t) is continuous. From Theorem 2, the inverse formation β : Isom(Rn , Rn ) → L(Rn , Rn ), β(u) = u−1 is a continuous mapping. Since

· ◦β ◦ H x : (H

−1

(0)) ∩ (Uϕ × [0, 1]) → R, (xϕ , t) → Hx (xϕ , t)−1

is continuous due to it being a composition of continuous mappings, and gives that (H (Uϕ × [0, 1]) is a compact set, the Weierstrass theorem implies that

H x (xϕ , t)−1 ≤ C, ∀(xϕ , t) ∈ (H

−1

−1

(0)) ∩

(0)) ∩ (Uϕ × [0, 1]).

The number C hence has been defined for a particular chart, and since the atlas (Ui , ϕi )i∈I has N charts, then there are N positive numbers Ci , i ∈ I. We select C = min{Ci : i ∈ I}. (a2) Two positive numbers r, r0 must be found to be used later. Hypothesis (iii) lets us define the mapping h : B := ((H

−1

(0)) ∩ (Uϕ × [0, 1])) × U ϕ × [0, 1] ⊂ Rn × R × Rn × R → Rn ,

h(xaϕ , ta ; (xaϕ + xϕ ), t) := H x (xaϕ , ta )(xϕ ) − H((xaϕ + xϕ ), t). Since H is a C 1 -mapping, h is a composition of continuous mappings, and since B is a compact set of the topological product space Rn × R × Rn × R, therefore, for any r > 0 and for C given in Section (a1), there is a  r  δ > 0, (3) 2C

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such that if (xaϕ , ta ; (xaϕ + xϕ ), t), (xaϕ , taϕ ; (xaϕ + xϕ ), t ) ∈ B with

(xaϕ , ta ; (xϕ + xaϕ ), t) − (xaϕ , ta ; (xϕ + xaϕ ), t ) < δ

 r  , 2C

then

r . (4) 2C On the other hand, from hypothesis (iv) the mapping hx : B → L(Rn , Rn ) can be defined as

h(xaϕ , ta ; (xaϕ + xϕ ), t) − h(xaϕ , ta ; (xaϕ + xϕ ), t ) <

hx (xaϕ , ta ; (xaϕ + xϕ ), t) := H x (xaϕ , ta ) − H x ((xaϕ + xϕ ), t), and is also uniformly continuous in the compact set B, and therefore there is an r := δ

 1  > 0, 2C

(5)

such that if (xaϕ , ta ; (xaϕ + xϕ ), t), (xaϕ , taϕ ; (xaϕ + x ), t ) ∈ B with

(xaϕ , ta ; (xϕ + xaϕ ), t) − (xaϕ , ta ; (xaϕ + xϕ ), t ) < δ

 1  , 2C

then

1 . (6) 2C r We take r given by (5) and fix r0 := δ( 2C ) given by (3), and we define the number r0 :=  min{r, r0 }. Hence the numbers r and r0 have been defined in a particular chart, and since the atlas (Ui , ϕi )i∈I has N charts there are positive numbers ri , r0i , i ∈ I. We select

hx (xaϕ , ta ; (xaϕ + xϕ ), t) − hx (xaϕ , taϕ ; (xϕ + xaϕ ), t ) <

r = min{ri : i ∈ I} and r0 = min{r0i : i ∈ I}. (a3) Let us suppose that the representative (xaϕ , ta ) of (xa , ta ) in the chart (Uϕ × −1 [0, 1], (ϕ, Id)) verifies (xaϕ , ta ) ∈ H (0) ∩ (Uϕ × [0, 1]). Such a point (xaϕ , ta ) will be called a ”starting point” in Uϕ × [0, 1], and there is a positive number R such that the closed ball B(xaϕ , R) ⊂ Uϕ . We select ra := min {R, r} and r0a = min {R, r0 } with r0 , r found in Section (a2). ker(A) will mean ker(H x (xaϕ , ta )), and (ker(A))⊥ its topological complement The sets Ia := {t ∈ [0, 1] :| t − ta |≤ r0a }, Ma := {xϕ ∈ (ker(A))⊥ : xϕ ≤ ra } will be associated to the “representative starting point” (xaϕ , ta ). Since

xϕ ≤ R, ∀xϕ ∈ Ma , therefore (xϕ + xaϕ ) ∈ Uϕ ⊂ Xϕ . Given a “starting point” (xa , ta ) and its “representative starting point” (xaϕ , ta ), we will prove here the existence of two continuous mappings α(·) : Ia ⊂ R→Ma +xaϕ ⊂ Uϕ ⊂ Xϕ , such that H(α(t), t) = 0, ∀t ∈ Ia

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and α(·) : Ia ⊂ R →U ⊂ M, such that H(α(t), t) = 0, ∀t ∈ Ia . Let us solve the equation H((xaϕ + xϕ ), t) = 0

(7)

for fixed t ∈ Ia when xϕ is in Ma . Obviously H(xaϕ , ta ) = 0. Equation (7) is equivalent to the following equation H x (xaϕ , ta )−1 [H x (xaϕ , ta )(xϕ ) − H((xaϕ + xϕ ), t)] = xϕ ,

(8)

which leads us to define the mappings ht : Ma × {t} → Rn , for fixed t ∈ Ia , ht (xϕ ) := H x (xaϕ , ta )(xϕ ) − H((xaϕ + xϕ ), t) = h(xϕ , t), and Tt : Ma → Rn , Tt (xϕ ) := H x (xaϕ , ta )−1 ht (xϕ ). Observe that ht (xϕ ) is h(xϕ , t) defined in Section (a2) when t is fixed and belongs to Ia . Let us also observe that t is in the definitions of ht and Tt is an index and not a partial derivative as usually written. Evidently hta (0) = 0, (9) and

d ht (0) = 0. dx Equation (10) is equivalent to the following key fixed point equation Tt (xϕ ) = xϕ ,

(10)

(11)

which will be studied later in the article. Let xϕ , xϕ ∈ Ma , t ∈ Ia , and hence the Taylor theorem together with equations (6) and (10) implies that      d   

ht (xϕ ) − ht (xϕ ) ≤ sup  ht (xϕ + θ(xϕ − xϕ )) : θ ∈ [0, 1] xϕ − xϕ dx 1 1

xϕ − xϕ ≤ r. (12) ≤ 2C 2C Equations (9) and (12) imply that

ht (xϕ ) ≤ ht (xϕ ) − hta (0) + hta (0) ≤

r , 2C

hence

Tt (xϕ ) ≤ H x (xaϕ , ta )−1

h(xϕ , t) ≤ r.

(13)

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We will apply Theorem 1 to the sets and mappigs which have been just defined. The metric space (Ia , | · |) is the parameter space of hypothesis (i) needed in Theorem 1. Ma is considered as the closed and non-empty set and (ker(A))⊥ as the complete metric space of hypothesis (ii), which is verified below: From equation (13), for any fixed t ∈ Ia , and for all xϕ ∈ Ma , we have Tt xϕ ≤ r, therefore Tt xϕ ∈ Ma , and hence Tt : Ma → Ma , i.e., Tt maps the closed and non-empty set Ma of the Banach space (ker(A))⊥ into itself. From equation (12), and the Taylor theorem, for any xϕ , xϕ ∈ Ma , and for any t ∈ Ia the following holds:

Tt (xϕ ) − Tt (xϕ ) ≤ H x (xaϕ , ta )−1

ht (xϕ ) − ht (xϕ ) 1 ≤ xϕ − xϕ . 2 Therefore Tt is half-contractive for any fixed t ∈ Ia . Hence, hypothesis (ii) of Theorem 1 is verified. For any fixed t0 ∈ Ia and for all xϕ ∈ Ma , the following holds: Tt (xϕ ) = H x (xaϕ , ta )−1 (H x (xaϕ , ta )(xϕ ) − H(xaϕ + xϕ , t)) → H x (xaϕ , ta )−1 (H x (xaϕ , ta )(xϕ ) − H(xaϕ + xϕ , t0 )) = Tt0 (xϕ ) as t → t0 , t ∈ Ia , therefore hypothesis (iii) of Theorem 1 is also verified. Hence, Theorem 1 implies, for any t ∈ Ia , that Tt has a unique fixed point xϕ ∈ Ma , Tt (xϕ ) = xϕ := xϕ (t), and xϕ (t) → xϕ (t0 ) as t → t0 ; t, t0 ∈ Ia , i.e., xϕ (·) is a continuous mapping. Thus for any t ∈ Ia there is only one xϕ ∈ Ma , i.e., Tt (xϕ ) = xϕ := xϕ (t) and that the mapping xϕ (·) just defined verifies xϕ (t) → xϕ (t0 ) as t → t0 , ∀t, t0 ∈ Ia , which implies that xϕ (·) is a continuous mapping. Thus, for any t ∈ Ia , there is an xϕ (t) such that H(xaϕ + xϕ (t), t) = 0, (14) and furthermore, H(xaϕ + xϕ (t), t) → H(xaϕ + xϕ (t0 ), t0 ) = 0 as t → t0 , t, t0 ∈ Ia . Let us observe that Tta (0) = 0, xϕ (ta ) = 0. Equation (14) can be written as H(α(t), t) = 0, which is verified for all t ∈ Ia , where α is the continuous mapping (a curve in a chart space) α : Ia → Uϕ ⊂ (ker(A))⊥ , α(t) := xaϕ + xϕ (t), where α(ta ) = xaϕ .

(15)

We construct the mapping αa (a curve in the manifold M ), which we will write as α in a shorter way α : Ia → U ⊂ M, α := (ϕ−1 ◦ α),

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which is continuous as composition of continuous mappings. This definition implies that α(ta ) = xa .

(16)

H(α(t), t) = (H ◦ (ϕ−1 , Id))(α(t), t) = H(α(t), t) = 0, ∀t ∈ Ia.

(17)

Eq.(14) implies that

(b) Conclusions (a) and (b) will be proved here. Since from hypothesis (i) , f −1 (0) = x∗ , there is Uj such that H −1 (0) ∩ (Uj × [0, 1]) = ∅, therefore there is a point (x∗ , 0) ∈ H −1 (0), x∗ ∈ Uj . −1 Since (x∗ϕ , 0) ∈ H (0) ∩ (Uϕ × [0, 1]), i.e., (x∗ϕ , 0) is a “starting point”, therefore from Section (a3) there are Ma , Ia , α such that α : Ia → Ma + x∗ϕ , H(α(t), t) = 0, ∀t ∈ Ia , and hence H(α(t), t) = (H ◦ (ϕ−1 , Id))(α(t), t) = H ◦ ((ϕ−1 ◦ α)(t), t) = 0, ∀t ∈ Ia .

(18)

We construct the continuous mapping α∗ : Ia → M, α∗ (t) := (ϕ−1 ◦ α)(t), and hence equation (18) can be writen as H(α∗ (t), t) = 0, ∀t ∈ Ia .

(19)

Let us suppose that α(t) ∈ Uϕ , ∀t ∈ [0, b], then mapping α is extended to the right of b by −1 taking (α(b), b), which belongs to H (0) ∩ (Uϕ × [0, 1]), as the following “starting point”. We also call α and α∗ the continuous prolonged mappings. Mapping α∗ is 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 [0, b) = Iα∗ , such that H(α∗ (t), t) = 0 has a solution, ∀t ∈ Iα∗ . This solution is unique with maximal interval of existence equal to  J= Iα∗ . α∗

Since M is compact if J = [0, b), b < 1, it follows from the compactness of M that α∗ (t) → x ∈ M , as t → b− , with x ∈ Uj, j ∈ I. From the continuity of H, we know that H(x, b) = 0, therefore the point (xϕj, , b) is a starting point in Uϕj × [0, 1] and the solution of H(α∗ (t), t) = 0 can be continuated beyond b, which is a contradiction. Hence we can continuate α∗ until t = 1. Hence H(α∗ (1), 1) = f (α∗ (1)) − g(α∗ (1)) = 0. That is, an x∗∗ with f (x∗∗ ) = g(x∗∗ ) is reached by the continuous mapping α∗ : [0, 1] → M , with H(α∗ (t), t) = 0, ∀t ∈ [0, 1]. If f and g share more than one value, then the previous construction starts at two different zeros of f − g, which would lead us to conclude that there are two zeros for f , which contradicts one of the hypotheses, and therefore there is only one shared value.

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