Proof

The Banach space F can be identified with F ₁ × F ₂, and since D i (a) is an isomorphism from E onto F ₁, E can be identified with F ₁. By translation, we may assume that a = 0. Let

φ : U × F 2 → F 1 × F 2 : x y 2 ↦ i x + 0 y 2 .

We have φ (x, 0) = i (x) and Dφ(0, 0) = D i(0) + (0, 1 _{F 2} ). Since D i (0) is an isomorphism from F ₁ onto F ₁ × 0, Dφ (0, 0) is an isomorphism from F ₁ × F ₂ onto F ₁ × F ₂. By Theorem 1.29, φ is a local diffeomorphism of class C ^p that admits an inverse diffeomorphism r of class C ^p . Hence, there exists an open neighborhood U of a in F ₁ such that, for all x ∈ U, r (i (x)) = x.

Recall that every closed subspace of a Hilbert space E splits in E ([P2], section 3.10.2(II), Theorem 3.147(2)). Corollary 1.33 shows that any immersion of class C ^p admits a local retraction r of class C ^p . Immersions are therefore local sections (see [P1], section 1.1.1 (III)).

Corollary-Definition 1.33

1. 1)  Let E, F be Banach spaces, A a non-empty open subset of E, a ∈ A and s : A → F a mapping of class C ^p such that D s (a) is surjective with a kernel E ₂ that splits in E (i.e. E ≅ E 1 × E 2 ). Then, there exist an open neighborhood U ⊂ A of a and a diffeomorphism ψ of class C ^p

   ψ : V 1 × V 2 → U ,

   

   
   where V ₁ × V ₂ ⊂ A (V _i ⊂ E _i ) is an open neighborhood of a in E, such that s ∘ ψ is the first projection V ₁ × V ₂ ↠ V ₁ .
2. 2)  The mapping s defined above is called a submersion of class C ^p .

Proof

By translation, we may assume that a = (a ₁, a ₂) = (0, 0). Thus, D ₁ s (0, 0) is an isomorphism from E ₁ onto F that allows these two Banach spaces to be identified. Let

φ : A → E 1 × E 2 : x 1 x 2 ↦ s x 1 x 2 x 2 .

Then, Dφ (0) is represented by the matrix

D 1 s 0 0 D 2 s 0 0 0 1 E 2 ,

which is an automorphism of E ₁ × E ₂. By Theorem 1.29, there exist a neighborhood U ⊂ A of 0 in E ₁ × E ₂ and a diffeomorphism φ : U → V ₁ × V ₂ of class C ^p with inverse diffeomorphism ψ : V ₁ × V ₂ → U of class C ^p . For all y ₁ ∈ V ₁, we have f (ψ (y ₁, x ₂)) = y ₁.

([P2], section 3.2.2 (IV), Theorem 3.5(3)) implies the following result:

Corollary-Definition 1.34

1. 1)  Let E, F be Banach spaces, A a non-empty open subset of E, a ∈ A, and f : A → F a mapping of class C ^p . The following conditions are equivalent:
   1. i)  There exist an open neighborhood U of a, U ⊂ A, a Banach space G , a submersion s of U into G of class C ^p and an immersion i of G into F of class C ^p such that f | _U  = i ∘ s.
   2. ii)  There exist an open neighborhood U ⊂ A of a, a diffeomorphism ξ : U → ~ U ′ of class C ^p , where U′ is an open subset of E, a mapping Φ ∈ ℒ E F and a diffeomorphism ψ : V → ~ V ′ of class C ^p , where V is an open neighborhood of f (a) in F and V′ = Φ (U′) is an open subset of F, such that

      f U ⊂ V , Φ ξ U ⊂ ψ V , f U = ψ − 1 ∘ Φ ∘ ξ ,

         [1.18]

      
      and the kernel and image of Φ split.
2. 2)  The mapping f defined above is called a subimmersion of class C ^p .

The composition of two immersions is an immersion, the composition of two submersions is a submersion and, given a subimmersion f, an immersion i, and a submersion s, the mapping i ∘ f ∘ s is a subimmersion (exercise). However, the composition of two subimmersions is not always a subimmersion ([DIE 93], Volume 3, section 16.8, Problem 1(b)).

Theorem 1.35

(rank) Let E, F be Banach spaces, A some non-empty open subset of E and f : A → F a mapping ofclass C ¹.

1. i)  If f is a subimmersion, there exists an open neighborhood U ⊂ A of a in E such that rk _x (f) = rk _a (f) for all x ∈ U.
2. ii)  Conversely, if there exists an open neighborhood U ⊂ A of a in E such that rk _x (f) = rk _a (f) for all x ∈ U and if the spaces E, F are finite-dimensional, then f is a subimmersion at the point a.
3. iii)  Write rk _x (f) = +∞ if rk _x (f) is not finite. The mapping x ↦ rk _x (f) from A into the discrete subspace ℕ ¯ ≔ ℕ ∪ + ∞ of the extended real line ℝ ¯ is lower semi-continuous ([P2], section 2.3.3 (III)) in A.

Proof

(i): By [1.18], rk _x (f)= rk (Φ) for all x ∈ U. (ii): See [DIE 93], Volume 1, (10.3). (iii): If r = rk _a (f) < +∞, we may assume that E = K r and extract a square submatrix M _x that has rank r at x = a from the Jacobian matrix of f at the point x. The determinant Δ_x of this submatrix is therefore non-zero for x = a. But the mapping x ↦ Δ _x is continuous, so there exists a neighborhood U of a in which Δ _x ≠ 0, and so rk _x (f) ≥ r. If rk _a (f) = +∞, then, for all r ∈ ℕ , there exist vectors e ₁, …, e _r ∈ E such that the vectors D f (a).e ₁, …, D f (a).e _r generate a subspace of F of dimension r. Arguing by contradiction, we deduce that there exists a neighborhood U of a in which rk _x (f) = +∞.

1.3 Other approaches to differential calculus

1.3.1 Lagrange variations and Gateaux differentials

In this section, K = ℝ .

(I) Affine spaces Given a vector space E, an affine space E attached to the space E is a homogeneous space of the additive group E ([P1], section 2.2.8(II)) such that the (transitive) action of E on E is faithful, i.e. such that the neutral element 0 is the only element of E that fixes every element of E . The action of x ∈ E on P ∈ E is written as P + x. We say that E is the space of translations of E , its elements are the translations of E and, if dim (E) < ∞, this quantity is called the dimension of E . Given some origin O chosen from E , the elements of E are all of the form O + x (x ∈ E).

Remark 1.36

The mapping O + x ↦ x is a bijection from E onto E that allows these two sets to be identified.

If Q = P + x, we write x = PQ → (the bipoint of origin P and endpoint Q) 7 . If E is a locally convex space, the sets O + U = {O + x : x ∈ U}, where the U are the open sets of E, define a topology on E . When equipped with this topology, E is called a locally convex affine space. Every point of such a space admits a fundamental system of convex neighborhoods. We can similarly define the concepts of affine topological space, affine normed space, affine pre-Hilbert space, etc.

(II) Lagrange variations Let E = O + E be a locally convex affine space, F a locally convex space, A some non-empty subset of E , a some point of A and f : A → F a mapping.

We say that f admits a Lagrange first variation δ f (a): E → F : h ↦ δ f (a) [h] at the point a if, for all h ∈ E,

lim t → 0 , t ≠ 0 f a + t . h − f a − t . δ f a h t = 0 .

If this condition is satisfied, we say that f admits a Lagrange second variation δ ² f (a): E → F if

lim t → 0 , t ≠ 0 f a + t . h − f a − t . δ f a h − t 2 . δ 2 f a h t 2 = 0 .

The Lagrange variation of order n, δ ⁿ f (a) : E ⁿ → F : h ↦ δ ⁿ f (a) [h], is defined inductively in the same way.

(III) Gateaux differentiability It is easy to show using the generalized Goursat theorem (section 1.2.5 (III)) that, if E and F are complex locally convex spaces and f admits a Lagrange first variation at the point a, then δ f (a) is linear ([HIL 57], Theorem 26.3.2). In general, we have the following result:

Lemma 1.37

If f admits a Lagrange first variation δ f (a), then the mapping δ f (a): E → F is homogeneous, i.e. δ f (a) [λ.h] = λ.δ f (a) [h] for all λ ∈ ℝ and all h ∈ E (exercise).

Definition 1.38

We say that f is Gateaux differentiable (or G-differentiable) at the point a if it admits a Lagrange first variation δ f (a) at a and δ f (a) is a bounded linear mapping from E into F ([P2], section 3.4.4 (I) , Definition 3.63). If so, δ f (a) is written as D ^G f (a) and is called the Gateaux differential of f at the point a.

The set D a G A F of G-differentiable mappings at the point a is an affine space. A G-differentiable mapping is not necessarily continuous. If E is a normed vector space and f is differentiable at the point a, then it is also G-differentiable at this point and D f (a) = D ^G f (a) (exercise). On product spaces, we may define the Gateaux partial differential D ₁ ^G f (a ₁, a ₂) in the first variable and, similarly, in the second variable, etc. In the conditions of Theorem 1.9, where f ∈ D a G A F and g ∈ D f a B G (and B denotes an open subset of F containing f (A)), we have g ∘ f ∈ D a G A G and (instead of [1.5])

D G g ∘ f = D g f a ∘ D G f a

   [1.19]

(exercise ^*: see [ALE 87] section 2.2.2).

Let E be a normed affine space, F a locally convex space, A a non-empty open subset of E , [a, a + h] a segment contained in A and f : A → F a G-differentiable mapping. Then, the mean value theorem (Theorem 1.13(i)) remains valid (exercise ^*: see [ALE 87], section 2.2.3) in the form

f a + h − f a γ ≤ h . sup t ∈ Θ D G f a + t h γ ,

   [1.20]

where |.|_γ is a continuous semi-norm on F. The claims (ii) and (iii) of this theorem also hold after making analogous adjustments. From this, we deduce the following result:

Corollary 1.39

Let E be a normed vector space, A some non-empty open subset of E, F a locally convex space and f : E → F a mapping that is G-differentiable in A. If D G f : A → ℒ E F is continuous, then f is differentiable in A.

Proof

Let ε > 0 and γ ∈ Γ, where (|.|)_γ ∈ Γis the family of semi-norms which defines the topology of F. There exists δ = δ (ε, γ) > 0 such that, ∀ x ∈ A,

x − a < δ ⇒ D G f x − D G f a γ < ε .

Let h ∈ E be such that | h | < δ. The relation [1.8] with the modifications stated above implies that | f (a + h) − f (a) − D ^G f (a).h |_γ ≤ ε | h |.

Thus, we do not need to distinguish between Fréchet and Gateaux differentiation when talking about mappings of class C ^p (p > 0).

1.3.2 Calculus of variations: elementary concepts

In this section, K = ℝ .

(I) Euler condition Let E be a locally convex affine space, A some non-empty open subset of E , a some point of A and J : A → ℝ a mapping that admits a Lagrange first variation δJ (a) at the point a.

Theorem 1.40

(Euler) For the mapping J to have a relative extremum (or local extremum) at the point a, the Euler stationarity condition δJ (a) = 0 must necessarily be satisfied.

Proof

By translating the origin, we may assume that a = 0. Similarly, we can assume that J has a minimum at 0 for clarity. We will argue by contradiction. Assume that δJ (0) ≠ 0. There exists h ∈ A such that β := δJ (0) [h] ≠ 0. By Lemma 1.37, we may assume without loss of generality that β < 0 (replacing h by − h if necessary). There exists a function φ defined in some open neighborhood of 0 in ℝ satisfying lim _t → 0 φ (t) = 0 and J (t.h) = J (0) + t.β + t.φ(t). Pick t > 0 sufficiently small that β + φ (t) < 0. Then, J (t.h) < J (0), contradiction.

Definition 1.41

We say that a is an extreme point (or an extremal point) of J if δJ (a) = 0.

Theorem 1.42

Suppose that the Euler stationarity condition is satisfied. A necessary condition for J to admit a relative minimum at the point a is given by δ ² J (a) [h] ≥ 0 for all h ≠ 0.

Proof

We have f (a + t.h) − f (a) = t ².δ ² J (a) [h] + o (t ²).

(II) Euler-Lagrange equation The Euler–Lagrange equation is essentially the Euler condition applied to calculus of variations. A full treatment would require another volume ⁸ ; we will content ourselves with briefly mentioning the simplest part, which is sufficient for our purposes. Let E be a Banach space and suppose that t 1 , t 2 ∈ ℝ , t ₁ < t ₂.

Lemma 1.43

(fundamental lemma of the calculus of variations) Let f : [t ₁, t ₂] → E ^∨ be a continuous mapping. If every mapping h :[t ₁, t ₂] → E of class C ¹ such that h (t ₁) = h (t ₂) = 0 satisfies ∫ _{t 1} ^{t ₂} 〈f(t), h(t)〉dt = 0, then f = 0.

Proof

We will argue by contradiction from the assumption that f (t ₀) ≠ 0. There must therefore exist z ∈ E such that 〈f (t ₀), z〉 > 0. Since f is continuous, there exists an interval [α, β] ⊂ [t ₁, t ₂] containing t ₀ such that α < β and 〈f (t), z〉 > 0 for all t ∈ [α, β]. There exists a function φ : t 1 t 2 → ℝ of class C ¹ that is zero in ∁ t 1 t 2 ] α , β [ and which satisfies φ (t) > 0 for t ∈ ]α, β[; for example, φ (t) = (t − α)² (β − t)² if t ∈ ]α, β[ and φ (t) = 0 if t ∈ ∁ t 1 t 2 ] α , β [ . Therefore, setting h (t) = φ (t).z, we have ∫ _{t 1} ^{t ₂} 〈f(t), h(t)〉dt > 0, contradiction.

Let Ω₁, Ω₂ be non-empty open subsets of E and let

ℒ : t 1 t 2 × Ω 1 × Ω 2 → ℝ : t x u ↦ ℒ t x u

be a mapping (known as the Lagrangian in mechanics) of class C ¹ with partial differential D 3 ℒ = ∂ℒ ∂ u . Let x ₁, x ₂ ∈ Ω₁ and write X for the set of mappings x : [t ₁, t ₂] → Ω₁ of class C ¹ satisfying x (t ₁) = x ₁, x (t ₂) = x ₂ whose derivatives x . take values in Ω₂. The set X is an open subset of the normed affine space O + X, where X is equipped with the norm h 1 ≔ sup t ∈ t 1 t 2 h t + h . t ; X is a Banach space whose elements satisfy the condition h (t ₁) = h (t ₂) = 0 (exercise). Let

J : X → ℝ : x ↦ ∫ t 1 t 2 ℒ t x t x . t . dt .

Theorem 1.44

Let x ∗ ∈ X be a mapping of class C ² 9 . The relation δJ (x ^*) = 0 holds, i.e. x ^* is an extreme point of J, if and only if x ^* is a solution of the Euler–Lagrange equation:

d dt ∂ℒ ∂ x . − ∂ℒ ∂ x = 0 .

   [1.21]

Proof

Let h ∈ X. If ε > 0 is sufficiently small, then x ∗ + ε . h ∈ X . Thus,

1 ε J x ∗ + ε . h − J x ∗ = 1 ε ∫ t 1 t 2 ℒ t x ∗ + ε . h x . ∗ + ε . h . − ℒ t x ∗ x . ∗ . dt .

Let f ε t = ℒ t x ∗ t + ε . h t x . ∗ t + ε . h . t . The mapping (ε, t) ↦ f (ε, t) has a partial derivative ∂ f ∂ ε whose absolute value is upper bounded by a function g : t ↦ g (t) that is integrable in [t ₁, t ₂]. Therefore, as ε ↦ 0, 1 ε J x ∗ + ε . h − J x ∗ converges ([P2], section 4.1.2(II), Theorem 4.11) to

δJ x ∗ h = ∫ t 1 t 2 ∂ ∂ ε ℒ t x ∗ t + ε . h t x . ∗ t + ε . h . t ε = 0 . dt = ∫ t 1 t 2 ∂ℒ ∂ x x = x ∗ . h t + ∂ℒ ∂ x . x = x ∗ . h . t . dt = ∫ t 1 t 2 ∂ℒ ∂ x − d dt ∂ℒ ∂ x . x = x ∗ . h t . dt ,

which gives the desired result by Lemma 1.43.

The Lagrangian ℒ is said to be regular at x = x ^* if ∂ 2 ℒ ∂ x . 2 t x ∗ t x . ∗ t is invertible in ℒ 2 s E ℝ for every t ∈ [t ₁, t ₂].

Corollary 1.45

(Hilbert) If x ∗ ∈ X is a solution of the Euler–Lagrange equation [1.21] and the Lagrangian is regular at x = x ^*, then x ^* is of class C ².

Proof

The Euler–Lagrange equation can be written as φ (t, x, u, q) = 0, φ t x u q = ∂ℒ ∂ u − q , q = ∫ ∂ℒ ∂ x . dt + C te ; φ is of class C ¹ and such that ∂ φ ∂ u = ∂ 2 ℒ ∂ u 2 . If the Lagrangian is regular at the point x ^*, then, for any t ∈ [t ₀, t ₁], the implicit function theorem (Theorem 1.30) implies that there exists an open neighborhood of t x ∗ t x . ∗ t in [t ₁, t ₂] × Ω₁ × Ω₂, in which the relation φ (t, x, u, q) = 0 is equivalent to u = ψ (t, x, q), where ψ is of class C ¹; thus, x . ∗ t = ψ t x ∗ t x . ∗ t is of class C ¹, and x ^* is of class C ².

(III) Legendre condition Suppose that ℒ is of class C ² and x ^* satisfies the Euler–Lagrange equation. Expanding to second order, we have

δ 2 J x ∗ h = ∫ t 1 t 2 ∂ 2 ℒ ∂ x 2 . h h + 2 ∂ 2 ℒ ∂ x ∂ x . ( h h . ) + ∂ 2 ℒ ∂ x . 2 . ( h . h . ) . dt = ∫ t 1 t 2 ∂ 2 ℒ ∂ x . 2 ⏟ P . h . h . + ∂ 2 ℒ ∂ x 2 − d dt ∂ 2 ℒ ∂ x ∂ x . ⏟ Q . ( h h ) . dt ,

where P and Q are evaluated at the point (t, x ^* (t)) and h is evaluated at the point t.

Theorem 1.46

(Legendre) A necessary condition for x ^* to minimize J over X is given by the weak Legendre condition:

P t x ∗ ≥ 0 , t 1 ≤ t ≤ t 2 .

Proof

Suppose that there exists t ₀ ∈ ]t ₁, t ₂[ such that P(t ₀, x ^∗(t ₀)) ≱ 0. This means that there is v ∈ E ^× such that P(t ₀, x ^* (t ₀)). (v, v) = − 2β with β > 0. Since t ↦ P (t, x ^* (t)). (v, v) is continuous, there exists a real number α > 0 such that t ₁ ≤ t ₀ – α < t ₀ + α ≤ t ₂ and P (t, x ^* (t)). (v, v) ≤ − β for all t ∈ [t ₀ – α, t ₀ + α]. We will construct a function ρ : t 1 t 2 → ℝ for which the above integral is < 0. Let h (t) = λ (t).v with λ t = sin 2 π t − t 0 α for t ∈ [t ₀ − α, t ₀ + α] and λ (t) = 0 for t ∈ [t ₁, t ₂] − [t ₀ − α, t ₀ + α]. It is easy to see that λ is of class C ¹ in [t ₁, t ₂] and (exercise)

∫ t 1 t 2 P . v v . λ . 2 + Q . ( v v ) . λ 2 dt ≤ − β . π 2 α + 2 M . α ,

where M = sup _{t ∈[t1, t2]} Q (t, x ^* (t)).(v, v). For α sufficiently small, the right-hand side is < 0. After replacing h by ε.h if necessary, where ε > 0 is sufficiently small, x ^* (t) + ε.h (t) ∈ Ω₁ and x ∗ t + ε . h . . t ∈ Ω 2 for all t ∈ [t ₁, t ₂]. The necessity of the weak Legendre condition then follows from Theorem 1.42.

Remark 1.47

If E is finite-dimensional, the strong Legendre condition can be stated as P (t, x ^* (t)) > 0(t ₁ ≤ t ≤ t ₂), i.e. P (t, x ^* (t)).(v, v) > 0 for all v ≠ 0. If the strong Jacobi condition ¹⁰ is also assumed, we obtain the sufficient condition for a “weak” local minimum proved by Weierstrass in 1879.

1.3.3 “Convenient” differentials

(I) c ∞ topology Let E be a real locally convex space and suppose that A is a non-empty subset of E. A smooth curve in A is defined as a mapping c : I → A of class C ^∞, where I is a non-empty open interval of ℝ , for example ]−1, 1[. Any such curve is said to be analytic if it is of class C ^ω .

If E is a complex locally convex space, suppose again that A is a non-empty subset of E. Then, A can be viewed as a subset A ₀ of the real locally convex space E ₀ obtained from E by restriction of the field of scalars ([P2], section 3.2.2 (II)), and any smooth curve in A ₀ is said to be a smooth curve in A.

Definition 1.48

[KRI 97] Let E be a locally convex space. The topology c ∞ of E is the finest topology that makes all smooth curves in E ( section 1.3.3 ) continuous. When equipped with this topology, the space E is denoted by c ∞ E .

The topology of c ∞ E is finer than the topology of E, so every open subset of E is open in c ∞ E . If the space E is bornological ([P2], section 3.4.4 (I), Definition 3.61), it has the finest locally convex topology of all locally convex topologies coarser than the topology of c ∞ E ([KRI 97], Corollary 4.6). “Convenient” differential calculus is performed with mappings defined in open sets of c ∞ E .

Recall that every Fréchet space and every Silva space is bornological and complete ([P2], sections 3.4.4 (I) and 3.8.2(II)). We have the following result ([KRI 97], Theorem 4.11):

Theorem 1.49

Let E be a metrizable locally convex space or a Silva space. Then, the topology of E coincides with the topology of c ∞ E .

Remark 1.50

If a locally convex space E is bornological but non-normable, then c ∞ E × E ∨ is not a topological vector space ( [KRI 97] , Corollary 4.21).

(II) K ℳ spaces

Definition 1.51

[KRI 97] A locally convex space F is said to be convenient if it is Mackey-complete, i.e. if the normed vector space F _B ( Definition 1.19 ) is a Banach space for every bounded, balanced and convex set B ⊂ F.

The following definition will be useful:

Definition 1.52

A locally convex space E is called a K ℳ space if it is quasi-complete and bornological, and the topologies of E and c ∞ E coincide.

Theorem 1.49 shows that Fréchet (and in particular Banach spaces) and Silva spaces are K ℳ spaces. Any quasi-complete locally convex space, and hence any K ℳ space, is convenient (exercise). Nonetheless, K ℳ spaces are sufficiently general for our purposes, so we will restrict attention to them to simplify the statements of results.

(II) Mappings of class c r r ∈ ∞ ω

Definition 1.53

Let E and F be real locally convex K ℳ spaces and let A be an open subset of E. A mapping f from A into F is said to be of class c ∞ if f ∘ c is a smooth curve in F for any smooth curve c in A. This mapping f is said to be of class c ω if it is of class c ∞ and f ∘ c is an analytic curve in F for every analytic curve c in A. ¹¹

If E is a Banach space and f is of class C ^r (r ∈ {∞, ω}), then f is of class c r . J. Boman showed the following result in 1967 ([KRI 97], Corollary 3.14):

Theorem 1.54

(Boman) Let A be a non-empty open subset of ℝ n . The mapping f : A → ℝ m is of class C ^∞ if and only if it is of class c ∞ .

Theorem 1.55

Let E and F be K ℳ spaces and suppose that A is an open subset of E.

1. 1)  Suppose that f : A → F is of class c ∞ . Then, f has a Gateaux differential D G f a ∈ ℒ E F at every point of A and the mapping D G f : A → ℒ E F is of class c ∞ .
2. 2)  Let G be a locally convex space and suppose that g : B → G is a mapping of class c ∞ , where B is an open subset of F containing f (A). Then, the following chain differentiation rule holds (compare with [1.5] and [1.19] ):

D G g ∘ f a = D G g f a ∘ D G f a .

Proof

(1) Let h ∈ E; there exists ε > 0 such that a + t h ∈ A for all t ∈ ]− ε, + ε[ and the curve t ↦ a + th is smooth. Therefore, f has a Lagrange first variation δ f (a). It can be shown that δ f (a) is bounded linear ([KRI 97], Chapter I, 3.18), so δ f (a) = D ^G f (a). Furthermore, D ^G f (a) is continuous linear, since E and F are bornological ([P2], section 3.4.4 (I), Theorem 3.62). If c is a smooth curve in A, then δ f (a) ∘ c is a smooth curve in ℒ E × F , which gives the stated result by induction (ibid.). (2): ibid.

In the real analytic case, the following result ([KRI 97], Chapter II, section 10.4) generalizes Boman’s theorem:

Theorem 1.56

Let E and F be real K ℳ spaces, U a non-empty open subset of E and f : U → F a mapping. The following conditions are equivalent: (i) f is of class c ω ; (ii) f is of class c ∞ and λ ∘ f ∘ μ is analytic from Ω into ℝ for any linear form λ ∈ F ^∨ and any affine mapping μ : Ω → U (Ω = μ ^− 1 (U)).

(III) Holomorphic mappings Assume that K = ℂ .

A holomorphic curve in a non-empty open subset A of a complex K ℳ space E is a holomorphic mapping from a disk in the complex plane, for example the open disk D of center 0 and radius 1, into A. A c -holomorphic mapping (or a mapping of class c ∞ ) from A into F, where F is a complex K ℳ space, is a mapping that transforms the holomorphic curves in A into holomorphic curves in F. With this notation, the mapping f : A → F is holomorphic if and only if λ ∘ f ∘ c is a holomorphic function for every continuous linear form λ ∈ F ^∨ and every holomorphic curve c : D → A . The following result ([KRI 97], Chapter II, section 7.9) generalizes the classical Hartogs theorem ([P2], section 4.3.2 (II), Corollary 4.80):

Theorem 1.57

(generalized Hartogs theorem) Let E ₁, E ₂, F be complex K ℳ spaces and suppose that A = A ₁ × A ₂ is a non-empty open subset of E ₁ × E ₂. The mapping f : A → F is -holomorphic if and only if it is c -holomorphic separately in each variable, i.e. f (., z ₂) and f (z ₁,.) are -holomorphic for every z ∈ A _i (i = 1, 2).

1.4 Smooth partitions of unity

In this section, K = ℝ .

1.4.1 C^∞ -paracompactness of Banach spaces

Let E be a normed vector space. The topological notions of normal space and paracompact space ([P2], sections 2.3.10 and 2.3.11) inspire the following definitions:

Definition 1.58

1. i)  E is said to be C ^r -normal (0 ≤ r ≤ ∞) if, given any two disjoint closed subspaces A and B of E, there exists a mapping f : X → [0, 1] of class C ^r taking the values 1 in A and 0 in B.
2. ii)  E is said to be C ^r -paracompact (0 ≤ r ≤ ∞) if it admits partitions of unity of class C ^r , i.e. given any open covering (U _i ) _i ∈ I of X, there exists a subordinate partition of unity (ψ _i ) _i ∈ I such that each ψ _i is of class C ^r .

Recall that (ψ _i ) _i ∈ I is a subordinate partition of unity of (U _i ) _i ∈ I if and only if supp (ψ _i ) ⊂ U _i , the family (supp (ψ _i )) _i ∈ I is locally finite and ∑ _i ∈ I ψ _i  = 1 ([P2], section 2.3.12).

Theorem 1.59

The normed vector space E is C ^r -paracompact if and only if it is C ^r -normal.

Proof

The necessary condition is clear. We will show the sufficient condition: let (U _i ) _i ∈ I be an open covering of E. If E is paracompact, there exists a locally finite open covering (V _j ) _i ∈ J finer than (U _i ) _i ∈ I such that each V _j is contained in some U _i(j). There also exist other open refinements (W _j ) _j ∈ J and (Z _j ) _j ∈ J of (V _j ) _j ∈ J such that Z j ¯ ⊂ W j ⊂ W j ¯ ⊂ V j . If E is C ^r -normal, then, for all j, there exists a function ψ _j : X → [0, 1] of class C ^r that is equal to 1 in Z j ¯ and 0 in ∁ E V j . Let ψ = ∑ _j ψ _j and θ _j = ψ _j /ψ. Thus, (θ _j ) _j ∈ J is a subordinate partition of unity of class C ^r of (V _j ) _i ∈ J and hence of (U _i(j)) _j ∈ J , which shows that E is C ^r -paracompact.

Remark 1.60

If a Banach space E is C ^r -paracompact, then every vector subspace of E and every open set of E is C ^r -paracompact ([ABR 83] , Proposition 5.5.20; [TOR 73], Corollary 1 of Theorem 1).

As a metrizable space, every Banach space is paracompact by Stone’s theorem ([P2], section 2.3.10, Theorem 2.57). The next result ([ABR 83], Propositions 5.5.18 and 5.5.19), whose proof was established by Bonic and Frampton in 1966, follows from the fact that any separable Banach space is a Lindelöf space ([P2], section 2.6.3) (exercise).

Theorem 1.61

For a separable Banach space to be C ^r -paracompact (1 ≤ r ≤ ∞), it is sufficient for its norm |.| : x ↦ | x | to be of class C ^r in E − {0}.

Corollary 1.62

Every separable Hilbert space E is C ^∞ -paracompact.

Proof

Write N for the norm of E. We have N (x)² = 〈x | x〉, so 2N (x) DN (x).h = 2 〈x, | h〉, and if x ≠ 0, DN (x).h = 〈x, | h〉/N (x). Hence, N is differentiable in E − {0} and DN (x) = 〈x |.〉/N(x) (x ≠ 0). This mapping DN is continuous from E − {0} into E ^∨ ≅ E. It is easy to see that it is differentiable from E − {0} into E ^∨, and so we can deduce by induction (exercise) that N is of class C ^∞.

It was shown in [TOR 73] that every reflexive (not necessarily separable) Banach space is C ¹-paracompact and that the separability condition is not required in Corollary 1.62:

Theorem 1.63

Every Hilbert space E is C ^∞ -paracompact.

This implies the result stated in ([P2], section 4.4.1, Theorem 4.88)

Corollary 1.64

(Whitney’s theorem) The space ℝ n is C ^∞ -paracompact.

1.4.2 c ∞ -paracompactness

We can define c ∞ -regularity, c ∞ -normality and c ∞ -paracompactness of a locally convex space in the obvious ways; the statement of Theorem 1.59 still holds if C ^∞ is replaced by c ∞ ; moreover, if c ∞ E is a c ∞ -regular Lindelöf space, then it is c ∞ -paracompact ([KRI 97], Proposition 16.2). We already know that any nuclear space E has a topology defined by a family of pre-Hilbert norms ([P2], section 3.11.3(III)). Each of these semi-norms is of class c ∞ in E − {0}. Nuclear Fréchet spaces (also known as ℱ N spaces), nuclear Silva spaces (also known as SN spaces), and countable products of ℱ N spaces and SN spaces are paracompact Lindelöf spaces (ibid.). The following result is analogous to Corollary 1.62 ([KRI 97], Theorem 16.10):

Theorem 1.65

Every ℱ N space, every strict inductive limit of ℱ N spaces and every SN space is c ∞ -paracompact.

1.5 Ordinary differential equations

1.5.1 Existence and uniqueness theorems

(I) Notion of the solution of a differential equation Let I be an interval of ℝ with non-empty interior I ∘ , Ω a non-empty open subset of E = ℝ n and f a mapping from I × Ω into E. We say that a mapping φ : I → E is a solution (or integral) of the differential equation

x . = f t x

   [1.22]

if the conditions (ODE _1,2,3) below are satisfied:

• (ODE ₁) φ (t) ∈ Ω for all t ∈ I;
• (ODE ₂) φ is locally absolutely continuous in I (i.e. each of its components with respect to the canonical basis of ℝ n is locally absolutely continuous: see [P2], section 4.1.7(III));
• (ODE ₃) φ . t = f t φ t λ-almost everywhere in I, where λ is the Lebesgue measure on ℝ ([P2], section 4.1.1(II)).

A Cauchy problem involves determining a function φ that is a solution of [1.22] and that satisfies the Cauchy condition:

φ t 0 = x 0 t 0 ∈ I ∘ x 0 ∈ Ω .

   [1.23]

If φ satisfies the conditions (ODE _1,2,3) and [1.23], then, for all t ∈ I,

φ t = x 0 + ∫ t 0 t f τ φ τ dτ .

   [1.24]

Conversely, suppose that [1.24] holds, (ODE ₁) is satisfied and t ↦ f (t, φ (t)) is locally λ-integrable. Then, [1.23] holds; furthermore, Lusin’s measurability criterion ([P2], section 4.1.6(II)) implies that the function t ↦ f (t, φ (t)) is λ-measurable for every locally absolutely continuous function φ : I → E if the conditions (Cat _1,2) below are satisfied:

• (Cat ₁) the function x ↦ f (t, x) from Ω into E is continuous for every t ∈ I;
• (Cat ₂) the function t ↦ f (t, x) from I into E is λ-measurable for every x ∈ Ω.

Suppose further that:

(Cat ₃) For all x ₀ ∈ Ω and every r > 0 such that B _r (x ₀) ⊂ Ω, where B _r (x ₀) is the open ball of center x ₀ and radius r in E, there exists a locally λ-integrable function m from I into ℝ + such that | f (t, x)| ≤ m (t) for all (t, x) ∈ I × B _r (x ₀).

Then, for every locally absolutely continuous function φ : I → E satisfying φ (I) ⊂ B _r (x ₀), the mapping t ↦ | f (t, φ (t))| is locally λ-integrable in I, so t ↦ f (t, φ (t)) is locally λ-integrable in I ([P2], section 4.1.2(I)).

Definition 1.66

The conditions (Cat _1,2,3 ) are known as the Carathéodory conditions.

(II) Existence theorem

Theorem 1.67

(Carathéodory) Suppose that the Carathéodory conditions (Cat _1,2,3 ) are satisfied. Then, for all (t ₀, x ₀) ∈ I × Ω, there exists an interval J ⊂ I with some point t ₀ in its interior and a mapping φ : J → E such that φ (J) ⊂ B _r (x _o ), φ is a solution of [1.22] and this solution satisfies the Cauchy condition [1.23] .

Proof

We will show that there exist an interval J _β = [t ₀, t ₀ + β] ⊂ I (β > 0) and an absolutely continuous mapping φ : J _β → E such that φ (J _β ) ⊂ B _r (x ₀) and φ satisfies [1.24]. The same argument works on an interval J′_α = [t ₀ − α, t ₀] (α > 0). Let M : [ t 0 , + ∞ [ ∩ I → ℝ + be the mapping:

M t = 0 t < t 0 , M t = ∫ t 0 t m τ dτ t ∈ t 0 , + ∞ ∩ I .

Since M is continuous and non-decreasing, there exists an interval J _β as specified above satisfying the property that, for all t ∈ J _β ,

0 ≤ M t < r .

   [1.25]

We can now inductively define a sequence of absolutely continuous mappings φ _i : J _β → E using the conditions:

φ i t = x 0 t 0 ≤ t ≤ β / i , φ i t = x 0 + ∫ t 0 t − β / i f τ φ i τ dτ t 0 + β / i < t ≤ t 0 + β .

By (Cat ₃), the second equation implies that, for all t ∈ J _β ,

φ i t − x 0 ≤ ∫ t 0 t − β / i f ( τ φ i τ ) dτ ≤ ∫ t 0 t 0 + β m τ dτ < r ,

so φ _i (t) ∈ B _r (x ₀) for all t ∈ J _β . If t ₁, t ₂ ∈ J _β , then

φ i t 2 − φ i t 1 ≤ M t 2 − β / i − M t 1 − β / i ,

and M is uniformly continuous in the compact set J _β by Heine’s theorem ([P2], section 2.4.5, Theorem 2.86), so | M (t ₂ − β/i) − M (t ₁ − β/i)| → 0 uniformly in i if t ₂ − t ₁ → 0, and the set H ≔ φ i : i ∈ ℕ × is equicontinuous ([P2], section 2.7.3). Since H t ≔ φ i t : i ∈ ℕ × is contained in B _r (x ₀) for all t ∈ J _β , the third Ascoli–Arzelà theorem (ibid.) implies that H is relatively compact in φ ∈ C J β E equipped with the uniform structure of uniform convergence. Hence, there exists a subsequence (φ _{i k} ) that converges uniformly to some mapping φ ∈ C J β E . Moreover, | f(t, φ _{i k} (t))| ≤ m(t)(t ₀ ≤ t ≤ t ₀ + β) and f(t, φ _{i k} (t)) → f(t, φ(t)) for i _k → ∞ by (Cat ₁); furthermore, as we saw earlier, (Cat ₂) implies that t ↦ f (t, φ (t)) is measurable. The Lebesgue dominated convergence theorem ([P2], section 4.1.2(II), Theorem 4.9) therefore implies that, for all t ∈ J _β ,

∫ t 0 t f τ φ i k τ dτ → ∫ t 0 t f τ φ τ dτ i k → ∞ .

But, for all t ∈ J _β ,

φ i k t = x 0 + ∫ t 0 t f τ φ i k τ dτ − ∫ t − β / i t f τ φ i k τ dτ ,

and the second integral tends to 0 as i _k → ∞, so the equality [1.24] is satisfied for all t ∈ J _β . Finally, φ _{i k}  → φ in the Banach space AC (J _β ; E) of absolutely continuous mappings from J _β into E ([P2], section 4.1.7(III)), so φ is absolutely continuous.

Corollary 1.68

(Peano’s theorem) Suppose that f is continuous in I × Ω. Let J be a compact interval that is a neighborhood of t ₀ in I and let m = sup t ∈ J , x ∈ B r x 0 f t x . For every compact interval [t ₀, t ₀ + β] contained in J satisfying β < r/m, there exists a solution of [1.24] that takes values in B _r (x ₀).

Proof

We have M = mβ, so the inequality [1.25] is satisfied if and only if β < r/m.

Remark 1.69

Corollary 1.68 (and hence Theorem 1.67 ) fails if E is replaced by an arbitrary Banach space ([BOU 76], Chapter 4 , section 1, Exercise 18). We can define an absolutely continuous mapping φ : J _β → E as we did in ([P2], section 4.1.7(I)), but it is not true in general that φ t − φ t 0 = ∫ t 0 t φ . τ dτ for all t ∈ J _β (however, this property does hold if E is assumed to be reflexive). Furthermore, since the ball B _r (x ₀) is not relatively compact when E is infinite-dimensional, the proof of Theorem 1.67 no longer works.

(III) Uniqueness theorem The fourth Carathéodory condition can be stated as follows (reusing some of the notation of Cat ₃):

(Cat ₄) For all x ₀ ∈ E and every real number r > 0 such that B _r (x ₀) ⊂ Ω, there exists a locally λ-integrable function k from I into ℝ + such that

f t x ′ − f ( t x ″ ) ≤ k t . x ′ − x ″ , ∀ t x ′ , t x ″ ∈ I × B r x 0 .

Theorem 1.70

(Carathéodory) Suppose that the Carathéodory conditions Cat _1,2,3,4 are satisfied. Then, for all t 0 x 0 ∈ I ∘ × Ω , there exists an interval J ⊂ I with interior point t ₀ and a unique mapping φ : J → Ω that is a solution of [1.22] and which satisfies the Cauchy condition [1.23] .

Proof

Let

K t = ∫ t 0 t 0 + t k τ dτ 0 ≤ t ≤ b

and choose the real number β > 0 in the proof of Theorem 1.67 to satisfy the additional condition K (β) < 1. Furthermore, let

X = ψ ∈ C 0 β E : ψ 0 = 0 & ψ t ∈ B r c 0 ∀ t ∈ 0 β

and, for every function ψ ∈ X , set

T . ψ t ≔ ∫ t 0 t 0 + t f τ ψ τ − t 0 + x 0 dτ 0 ≤ t ≤ β .

Clearly, T X ⊂ X . The set X is a closed subset of the space C 0 β E equipped with the uniform structure of uniform convergence; the norm of C 0 β E is ‖ψ‖_∞ ≔ sup _{t ∈ [0, β]}| ψ(t)|. The space C 0 β E is complete ([P2], section 2.7.2, Corollary 2.115), so X is complete ([P2], section 2.4.4(II), Lemma 2.77). Furthermore, ψ ∈ C 0 β E is a fixed point of T if and only if the function φ ∈ C t 0 t 0 + β E defined by

φ t ≔ ψ t − t 0 + x 0 0 ≤ t ≤ β

satisfies [1.24]; finally, ψ (t – t ₀) ∈ B _r (0) if and only if φ (t) ∈ B _r (x ₀). If ψ 1 , ψ 2 ∈ X for all t ∈ [0, β], then

T . ψ 1 t − T . ψ 2 t ≤ ∫ t 0 t 0 + t k τ . ψ 1 τ − t 0 − ψ 2 τ − t 0 dτ ≤ K b ψ 1 − ψ 2 ∞

and hence ‖T. ψ ₁ − T. ψ ₂‖_∞ ≤ K(b)‖ψ ₁ − ψ ₂‖_∞. Therefore, T has a unique fixed point by Theorem 1.27.

Definition 1.71

1. 1)  We say that f : I × Ω → E is locally Lipschitz in the second variable in I × Ω if, for every (t, x) ∈ I × Ω, there exists a neighborhood V of t, a neighborhood S of x and a constant k _V,S > 0 such that f (., x) is regulated on V ( [DIE 93] , Volume 1, section 7.6) and f satisfies the Lipschitz condition:

   f t x ′ − f ( t x ″ ) ≤ k V , S . x ′ − x ″ , ∀ t ∈ V , ∀ x ′ , x ″ ∈ S .

   

2. 2)  We say that f is Lipschitz with constant k > 0 in the second variable if the above statement is satisfied when V = I, S = Ω and k _V,S = k.

Any locally Lipschitz function in the second variable clearly satisfies the conditions Cat _1,2,3,4, and we may therefore apply Theorem 1.70 to this function. We also have the following result:

Corollary 1.72

(Cauchy–Lipschitz theorem) If f is Lipschitz with constant k in the second variable in I × Ω, let J be a compact interval contained in I with non-empty interior, t ₀ a point of I ∘ , x ₀ a point of E, r > 0 a real number such that B _r (x ₀) ⊂ Ω, m = sup t ∈ J , x ∈ B r x 0 f t x and ρ = min {r/m, 1/k}. For every compact interval K contained in J ∩ ]t ₀ − ρ, t ₀ + ρ[, there exists a unique mapping φ that is a solution of [1.22] and which satisfies the Cauchy condition [1.23] .

Remark 1.73

1. i)  Theorem 1.70 fails if E is replaced by an arbitrary Banach space. However, Corollary 1.72 remains valid, with an identical proof. Furthermore, ρ = r/m ( [BOU 76] , Chapter 4 , section 1.5 , Theorem 1) ¹² . Interested readers can find additional existence and uniqueness results for the solutions of [1.22] in infinite dimensions in [DEI 77] , section 8, and [SCH 89] . Note, however, that “infinite-dimensional systems” are not governed by a functional differential equation of the form [1.22], where E is a Banach space: see [HAL 77] .
2. ii)  By the mean value theorem ( Theorem 1.13 ), in order for f to be locally Lipschitz, it is sufficient for it to be of class C ¹ .

Theorem 1.74

Let E be the space ℝ n (respectively any Banach space), I ⊂ ℝ an interval with some interior point t ₀, Ω a non-empty open subset of E and f : I × Ω → E a mapping satisfying the conditions Cat _1,2,3,4 (respectively a locally Lipschitz function in the second variable). For all x ₀ ∈ Ω, there exists a maximal interval J (t ₀, x ₀) ⊂ I with interior point t ₀ in which [1.22] has a solution φ satisfying the Cauchy condition [1.23] and such that φ (J (t ₀, x ₀)) ⊂ Ω. This solution φ (.; t ₀, x ₀) is unique.

Proof

Let M be the set of intervals L ⊂ I with non-empty interior containing t ₀ as an interior point and such that there exists a solution φ of [1.22] in L satisfying the Cauchy condition [1.23] and φ (L) ⊂ Ω. The set M is not empty by Theorem 1.70 (respectively Corollary 1.72 and Remark 1.73). Let L , L ′ ∈ M with L ⊂ L′. If φ, φ′ are solutions of [1.22] in L, L′, respectively, satisfying the Cauchy condition [1.23], then it follows from Theorem 1.70 (respectively Corollary 1.72 and Remark 1.73) (arguing by contradiction: exercise) that φ′ is a continuation of φ. Let J t 0 x 0 = ∪ L ∈ M L ; there must therefore exist a unique solution φ of [1.22] in J (t ₀, x ₀) satisfying the Cauchy condition [1.23] and φ (J (t ₀, x ₀)) ⊂ Ω.

Definition 1.75

The solution φ (.; t ₀, x ₀) defined in J is called the maximal integral of [1.22] satisfying [1.23] .

Remark 1.76

Suppose that f is locally Lipschitz in the second variable.

1. 1)  Let t _f := sup (J (t ₀, x ₀)) ≤ +∞ (the argument below also works when t _i := inf (J (t ₀, x ₀)) ≥ −∞, mutatis mutandis). If f (t, φ (.;t ₀, x ₀)) is bounded in J (t ₀, x ₀), then ψ (t; t ₀, x ₀) admits a limit c := φ (t _f − 0; t ₀, x ₀); furthermore, c is a frontier point of Ω if J(t ₀, x ₀) ∩ [t ₀, +  ∞ [ ≠ I ∩ [t ₀, +  ∞ [. This inequality, together with the condition Ω = E, implies that lim _{t → t f}  | φ(t; t ₀, x ₀)| =  + ∞; by contrast, if t _f ∈ I and Ω = E, then J(t ₀, x ₀) ∩ [t ₀, +  ∞ [ = I ∩ [t ₀, +  ∞ [ ( [BOU 76] , Chapter 4 , section 1.5 , Theorems 2 and Corollaries 1 and 2). In addition to this remark, see Theorem 5.67 in section 5.7.1 .
2. 2)  Let (τ, ξ) be an arbitrary point of I × Ω. There exists an interval K ⊂ I, a neighborhood of τ in I and a neighborhood S of ξ in Ω, such that, for every point (t ₀, x ₀) ∈ K × S, there is a unique solution φ (., t ₀, x ₀) of [1.22] satisfying [1.23] , definedin K (i.e. J(t ₀, x ₀) ⊃ K). The mapping (t, t ₀, x ₀) ↦ φ(t; t ₀, x ₀) from K × K × S into Ω is uniformly continuous ([BOU 76] , Chapter 4 ,section 1.7, Theorem 4).

(IV) Differential equations in implicit form Let I be an open interval of ℝ , t ₀ a point of I, F a Banach space and Ω a non-empty open subset of F ^n + 1, where n is a natural integer. Let g : I × Ω → F be a mapping of class C ¹ and consider the differential equation:

g t y y . … y n − 1 y n = 0 .

   [1.26]

Let η ₀ = (η ₀ ⁰, η ₀ ¹,…, η ₀ ^n − 1) ∈ F ⁿ and ξ ₀ ∈ F such that (η ₀, ξ ₀) ∈ Ω, g (t ₀, η ₀, ξ ₀) = 0, and suppose that the following condition (Inv) is satisfied:

(Inv) ∂ g ∂ ξ t 0 η 0 ξ 0 is invertible in ℒ F .

By the implicit function theorem (Theorem 1.30), there exists an open neighborhood J × U of (t ₀, η ₀) in I × F ⁿ , an open neighborhood V of ξ ₀ in F, where U × V ⊂ Ω, and a mapping h : J × U → V of class C ¹, such that, for all (t, η, ξ) ∈ J × U × V,

g t η ξ = 0 ⇔ ξ = h t η .

Therefore, any mapping ψ : J → F such that ψ t ψ . t … ψ n − 1 t ∈ U and ψ ⁽ⁿ⁾ (t) ∈ V for all t ∈ J is a solution of [1.26] in J if and only if ψ is a solution of the following differential equation on J, said to be in explicit form:

y n = h t y y . … y n − 1 .

   [1.27]

Let x _i = y ^(i − 1) (i = 1,…, n) and x =(x ₀, …, x _n ). For (t, x) ∈ J × U × V, the differential equation [1.27] is equivalent to [1.22], where f = (f ₁,…, f _n ) and

f 1 t x = x 2 , … , f n − 1 t x = x n , f n t x = h t x .

The mapping f : J × U → F ⁿ is of class C ¹ and hence locally Lipschitz in x by the mean value theorem (Theorem 1.13(i)). We can therefore apply Theorem 1.72 according to Remark 1.73. Let x ₀ = (η ₀ ¹, …, η ₀ ^n − 1); any solution φ = (φ ₀, …, φ _n ) of [1.22] of class C ¹ in J satisfies the Cauchy condition [1.23] if and only if φ = φ ₁ is a solution of [1.26], of class C ⁿ in J and ψ (t ₀) = η ₀ ¹, …, ψ ^(n − 1) (t ₀) = η ₀ ^n − 1.

Remark 1.77

If the condition (Inv) is not satisfied, the differential equation [1.26] is singular. We already encountered this situation in the linear case ([P2], section 5.4.6), where it was necessary to introduce solutions in the form of hyperfunctions. The nonlinear case does not have an equivalent general theory.

1.5.2 Linear differential equations

(I) Let I be an interval of ℝ with non-empty interior I ∘ and let E = ℝ n . Consider the linear differential equation

x . = A t . x + b t ,

   [1.28]

where A : I → ℒ E and b : I → E are locally λ-integrable. The Carathéodory conditions Cat _1,2,3,4 are all satisfied. (In particular, setting f (t, x) = A (t).x + b (t), we have

f t x ′ − f ( t x ″ ) ≤ A t . x ′ − x ″ ) ,

which shows that Cat ₄ is satisfied.) Hence, for all t 0 ∈ I ∘ and every x ₀ ∈ E, there exists a unique solution φ (.; t ₀, x ₀) of [1.28] in I satisfying [1.23].

The linear equation [1.28] is said to be homogeneous if b = 0, in which case

x . = A t . x .

   [1.29]

(II) The set of solutions of [1.29] (where A is locally λ-integrable) is an ℝ -vector space. Let φ ₀ (.; t ₀, x ₀) be a solution of [1.29] in I satisfying [1.23]. The mapping x ₀ ↦ φ ₀ (t; t ₀, x ₀) is a bijective linear mapping Φ (t, t ₀) from E onto E, and Φ(., t ₀) is identical to the solution of the differential equation

dU dt = A t U

   [1.30]

for U (t ₀) = 1 _E . For all t ₁, t ₂, t ₃ ∈ I, we have Φ(t ₃, t ₁) = Φ (t ₃, t ₂) ∘ Φ(t ₂, t ₁) and Φ(t ₁, t ₂) = Φ(t ₂, t ₁)^− 1.

Definition 1.78

The mapping Φ is called the resolvent of the equation [1.29] . The matrix representing this resolvent with respect to the canonical basis of E is called the transition matrix.

Theorem 1.79

We have

det Φ t t 0 = exp ∫ t 0 t Tr A τ . dτ .

Proof

Set Φ(t, t ₀) = U (t), Δ(t) = det (U (t)) and write U (t + h) = U (t) + h.V + o (h), where V = U . t . Then, Δ(t + h) = Δ (t).det (I + h.V.U ^− 1 (t) + o (h)). But

det I + h . V . U − 1 t + o h = h n det h − 1 . I + V . U − 1 t + o 1 .

Furthermore, by [P1], section 2.3.11 (VII),

det h − 1 . I + V . U − 1 t + o 1 = h − n I + h − n + 1 Tr V . U − 1 t + o h − n + 1 ,

so det (I + h.V.U ^− 1 (t) + o(h)) = 1 + h.Tr (V.U ^− 1 (t)) + o (h). Hence, Δ(t + h) = Δ (t).(1 + h.Tr (V.U ^− 1 (t ₀)) + o (h)), so

Δ . t = Tr U . t . U − 1 t . Δ t .

We now simply apply the relation U . t U − 1 t = A t , which follows from [1.30].

(III) The above shows that the general solution of [1.29] is of the form t ↦ Φ(t, t ₀)ξ. The “variation of constants” method (exercise) allows us to obtain the following solution (defined in I) of [1.28] and the Cauchy condition [1.23]:

φ t t 0 x 0 = Φ t t 0 . x 0 + ∫ t 0 t Φ τ t 0 . b τ . dτ .

(IV) The integration of linear differential equations with constant coefficients is a classical problem and is performed using the Jordan normal form ([P1], section 3.4.3 (IV)): see, for example, [BOU 10], section 12.5.2.

1.5.3 Parameter dependence of solutions

Let I be an interval of ℝ with non-empty interior, E a Banach space (see Remark 1.73), Ω a non-empty open subset of E, Λ a topological space and f a mapping from I × Ω × Λ into E. Write f _λ (t, x) for the value of f at the point (t, x, λ) ∈ I × Ω × Λ. It is possible to show the following result ([BOU 76], Chapter 4, section 1.6, Theorem 3):

Theorem 1.80

(parameter dependence of solutions) Suppose that, for all λ ∈ Λ, (t, x) ↦ f _λ (t, x) is Lipschitz in the second variable x in I × Ω and that f _λ (t, x) → f _{λ 0} (t, x) uniformly in I × Ω as λ → λ ₀. Let φ _{λ 0} be a solution of x . = f λ 0 t x satisfying the Cauchy condition φ λ 0 t 0 = x 0 t 0 ∈ I ∘ x 0 ∈ Ω , defined on an interval J = [t ₀, t ₀ + β[ ⊂ I and taking values in Ω. For every compact interval [t ₀, t ₁] ⊂ J, there exists a neighborhood V of λ ₀ in Λ such that, for all λ ∈ V, the differential equation

x . = f λ t x

   [1.31]

admits a unique solution φ _λ defined in [t ₀, t ₁] satisfying the Cauchy condition φ _λ (t ₀) = x ₀ and taking values in Ω; furthermore, as λ → λ ₀, φ _λ  → φ _{λ 0} uniformly in [t ₀, t ₁].

Suppose now that Λ is an open subset of a normed vector space F and that the mapping (t, x, λ) ↦ f _λ (t, x) is continuous with continuous partial differentials 13 t x λ ↦ ∂ f λ ∂ x t x and t x λ ↦ ∂ f λ ∂ λ t x . Then, f _λ is locally Lipschitz in the second variable x (see section 1.5.1 (V)). Suppose further that the mappings x ₀ : Λ → Ω: λ → x ₀ (λ) and t ₀ : Λ → I : λ ↦ t ₀ (λ) are of class C ¹ in Λ. For all λ _ο ∈ Λ, Theorem 1.80 implies that there exist an open neighborhood V of λ ₀ in Λ and an open interval J ⊂ I such that t ₀ (λ ₀) ∈ J, where the sets V and J satisfy the following condition: for all λ ∈ V, t ₀ (λ) ∈ J and there exists a solution φ _λ = φ (.,t ₀ (λ), x ₀ (λ)) of [1.31] in J taking the value x ₀ (λ) ∈ Ω at the point t ₀ (λ). Then, we have the following result:

Theorem 1.81

For every t ∈ J, the mapping λ ↦ φ _λ (t) from V into Ω has a differential h t ∈ ℒ F E at the point λ ₀ of V ; h is of class C ¹ in J and is the unique solution of the linear equation:

y . = A t ∘ y + b t

   [1.32]

where

A t ≔ ∂ f λ ∂ x ( t φ λ t ) λ = λ 0 , b t ≔ ∂ f λ ∂ λ ( t φ λ t ) λ = λ 0 .

This differential h satisfies the Cauchy condition:

h t 0 λ 0 = Dx 0 λ 0 − f λ ( t 0 λ x 0 λ ) λ = λ 0 ∘ Dt λ 0 .

Proof

For all t ∈ J,

φ λ t = x 0 λ + ∫ t 0 λ t f λ τ φ λ τ dτ .

Hence, by [DIE 93], Volume 1, section 8.11, (8.11.2) and Problem 1,

h t = ∫ t 0 λ 0 t ∂ f λ ∂ x ( τ φ λ τ ) h τ + ∂ f λ ∂ λ ( τ φ λ τ ) λ = λ 0 dτ + Dx 0 λ 0 − f λ ( τ φ λ τ ) λ = λ 0 ∘ Dt λ 0 .

Therefore, h is of class C ¹ in J and

h . t = ∂ f λ ∂ x ( t φ λ t ) λ = λ 0 h t + ∂ f λ ∂ λ ( τ φ λ τ ) λ = λ 0 , h t 0 λ 0 = Dx 0 λ 0 − f λ ( τ φ λ τ ) λ = λ 0 ∘ Dt λ 0

Note that, in the linear equation [1.32], A t ∈ ℒ F E ; moreover, y , A(t) ∘ y , and b (t) belong to ℒ F E . In the Cauchy condition, f _λ (t ₀ λ), x 0 λ ) λ = λ 0 ∈ E ≅ ℒ ℝ E , Dx 0 λ 0 ∈ ℒ F E , Dt λ 0 ∈ ℒ F ℝ = F ∨ , and f _λ (t ₀ (λ), x 0 λ ) λ = λ 0 ∘ Dt λ 0 ∈ ℒ F E .

Corollary 1.82

(differentiability of the solution with respect to x ₀) Suppose that F = E, Λ = Ω, and λ = x ₀, and adopt the same hypotheses as Theorem 1.81 (mutatis mutandis). Write f and φ for f _λ and φ _λ respectively. Then:

∂ φ ∂ x 0 t t 0 x 0 = Φ t t 0

where Φ is the resolvent ( Definition 1.78 ) of the linear differential equation

y . = A t ∘ y , A t ≔ ∂ f ∂ x t φ t t 0 x 0 .

   [1.33]

Proof

By Theorem 1.81, h . t = A t h t and h (t ₀) = 1 _E .

Corollary 1.83

(differentiability with respect to the initial time) Suppose that F = ℝ , Λ = I, and λ = t ₀ . With the notation of Corollary 1.82 and the same hypotheses (exercise):

∂ φ ∂ t 0 t t 0 x 0 = − Φ t t 0 f t 0 x 0 .

Remark 1.84

Let E be finite-dimensional. The statement of Corollary 1.82 still holds under the weaker condition ( [ALE 87] , Chapter 2 ,section 2.5.6) that f satisfies the Carathéodory conditions ( Cat _1,2,3 ) and Lusin’s condition (L) holds:

(L): For all t ∈ I, the mapping x ↦ f (t, x) is continuously differentiable on Ω and, for every compact set K ⊂ Ω, there exists a locally λ-integrable function k : I → ℝ + such that

∂ f ∂ x t x ≤ k t , ∀ t x ∈ I × K .

It is clear that condition (L) implies (Cat ₄ ) by the mean value theorem. This condition does not require f to be continuous in t, which is very important in optimal control theory (Pontryagin maximum principle): see op. cit. and [PON 62] . We can further weaken (L) by considering generalized gradients and differential inclusions ( [CLA 90] ,Theorem 7.4.1), but this exceeds the scope of the present book.