Ordinary differential equations with measurable right-hand side and parameters in metric spaces

Friedemann Schuricht^{∗}

Max-Planck-Institut für Mathematik in den Naturwissenschaften Inselstraße 22–26 D-04103 Leipzig, Germany

Heiko von der Mosel^{∗∗}

Mathematisches Institut Universität Bonn Beringstraße 1 D-53115 Bonn, Germany

Abstract

We consider initial value problems for ordinary differential equations with a measurable right-hand side containing parameters in general metric spaces. We prove existence and uniqueness of solutions and the continuous dependence of the solution on the initial data and the parameters under weak assumptions on the right-hand side. In the case where the parameters belong to a normed linear space we show that the solution depends differentiably on the parameters and initial values. These generalizations of classical results in the theory of ordinary differential equations have applications, e.g., in control theory and variational problems with nonholonomic constraints.

AMS Classification: 34A12, 34A34

key words: ordinary differential equations, initial value problems, existence, uniqueness, dependence on parameters, nonlinear equations

∗schur@mis.mpg.de ∗∗heiko@math.uni-bonn.de

Introduction

In nature, engineering and economics many processes are governed by systems of ordinary differential equations of the form

ẋ = f(t, x(t), λ) on I ⊂ IR, x(τ) = ξ ,  (1)

where the right-hand side f depends on certain parameters λ. In addition to the fundamental questions concerning existence and uniqueness of solutions of the initial value problem (1), the continuous dependence of the solution x(t; τ, ξ, λ) on the initial data (τ, ξ) and the parameters λ is a central issue in applications. Due to inaccuracies of practically all measurements, the investigation of how a solution of a differential system depends on the data and parameters is of great importance.

Usually one is interested in the solution of the initial value problem (1) in its own and the knowledge of the continuous dependence of the solution on the data is sufficient for many applications. However, an ordinary differential equation such as (1) may also occur as part of a more comprehensive problem, e.g., as a nonholonomic constraint in a variational problem, which is a typical situation in optimal control theory. In that type of problems it can be convenient to remove the differential equation (1) as constraint by substituting the solution x(t; τ, ξ, λ) for x(t) everywhere. As consequence the derivatives of x(t; ·) may enter explicitly the Euler-Lagrange equation of the modified problem (cf. the application described at the end of this introduction). Hence for a variety of applications it turns out to be necessary to study the differentiability properties of the solution x(t; τ, ξ, λ) with respect to initial values and parameters.

The dependence of solutions of ordinary differential systems (1) on parameters is extensively treated in the literature in the case where f is a smooth function in all of its variables, and where the parameter λ belongs to some subset Λ of the Euclidean space IR^{n}, see e.g. the monograph by Walter, [6]. In many applications, however, the function t ↦ f(t, x, λ) is merely measurable and the parameter space may be some infinite dimensional function space. For this more general case only results concerning continuous dependence on parameters can be found in the literature, whereas the differentiable dependence on parameters is investigated only for parameter sets Λ ⊂ IR^{n} and smooth right-hand sides, (cf. the appendix in the treatise of Hestenes [3]). With the present paper we want to close this gap.

In Section 1 we treat the continuous dependence of solutions x(t; τ, ξ, λ) on the initial data and parameters (τ, ξ, λ). Section 2 is concerned with the differentiable dependence on initial values and parameters (ξ, λ). The main results are stated as Theorem 1.1 and Theorem 2.1 in Sections 1.1 and 2.1, respectively. The proofs can be found in Sections 1.2 and 2.2. In both cases we proceed basically as in the smooth case, i.e. we replace the ordinary differential system (1) with an equivalent integral equation and look for a solution in a suitable Banach space by means of Banach’s fixed point theorem. However, the proofs are much more technical than in the smooth case because of our weak assumptions on f.

Lemma 1.2 in Section 1.2 presents a result which can basically be found in Hestenes [3], where it is, however, not completely correctly stated. Hestenes considers compact subsets S ⊂ IR × IR^{n} with elements (t, x), where each slice of S with t = const. is convex. One argument in the proof is based on the erroneous conclusion that for each sufficiently small δ > 0 also the δ-neighbourhood B_{δ}(S) of S has only convex slices t = const. Our counterexample in the appendix demonstrates that this is not true in general.

Let us finally mention that we were led to questions concerning the parameter dependence of solutions of ordinary differential equations in the course of investigating mechanical problems. In particular, we are interested in the mechanical behaviour of DNA-molecules which can be modeled as elastic rods. Using a general rod theory we are able to describe self-contact phenomena of such long slender molecules by a rigorous variational approach. Here, ordinary differential equations of the form (1) occur naturally as side conditions in the variational problem, and the differentiable dependence of the solution on parameters is needed to derive the Euler–Lagrange equation, [2], [5]. Notice that the problem of self-contact in elasticity could not be treated rigorously before.

Notation: We use the standard notation B_{R}(x) for open balls with radius R > 0 and center x, where x is a point in a metric space X. If for an open, bounded set Ω contained in Ω_{0} ⊂ X also its closure \overline{Ω} is contained in Ω_{0}, then we simply write Ω ⊂⊂ Ω_{0}. The space of continuous functions defined on X with values in a metric space Y will be denoted by C^{0}(X, Y). In addition, if X and Y are normed linear spaces, we define C^{k}(X, Y) to be the space of functions from X to Y whose k-th derivative exists and is continuous on X. L^{1}(I, Y) is the space of integrable Y -valued functions on some interval I ⊂ IR with respect to the one-dimensional Lebesgue-measure, and we write L^{1}(I, IR) = L^{1}(I).

Acknowledgements. We would like to express our gratitude both to the Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig and to the Sonderforschungsbereich 256 in Bonn for generously supporting our research.

1 Local existence, uniqueness and continuous dependence of the solution

1.1 Formulation of the result

Consider the initial value problem

ẋ(t) = f(t, x(t), λ), x(τ) = ξ ∈ IR^{n},  (2)

where λ ∈ Λ is a parameter in a metric space Λ. Here, the function f maps Ω_{0} × Λ into IR^{n}, where Ω_{0} is a region in IR × IR^{n}.

Our hypotheses on the right-hand side of (2) are: For any ω = (τ, ξ) ∈ Ω_{0} there is δ = δ(ω) > 0, and there are measurable functions M_{ω}, K_{ω}, such that the cylinder C_{δ}(ω) := B_{δ}(τ) × B_{δ}(ξ) is contained in Ω_{0}, and M_{ω}, K_{ω} ∈ L^{1}(B_{δ}(τ)) with the following properties:

(f1) For every x ∈ B_{δ}(ξ) ⊂ IR^{n} and λ ∈ Λ the function f(., x, λ) is measurable on B_{δ}(τ) ⊂ IR with

|f(t, x, λ)| ≤ M_{ω}(t) for a.e. t ∈ B_{δ}(τ).  (3)

Note that this implies that f(., x, λ) ∈ L^{1}(B_{δ}(τ), IR^{n}) for all x ∈ B_{δ}(ξ), λ ∈ Λ.

(f2) For every x, y ∈ B_{δ}(ξ) ⊂ IR^{n} and for each λ ∈ Λ one has

|f(t, x, λ) − f(t, y, λ)| ≤ K_{ω}(t)|x − y| for a.e. t ∈ B_{δ}(τ).  (4)

(f3) For each x ∈ B_{δ}(ξ), λ_{0} ∈ Λ

lim_{λ→λ_{0}} ∫_{B_{δ}(τ)} |f(s, x, λ) − f(s, x, λ_{0})| ds = 0.  (5)

Using standard arguments one can easily check that the conditions (f2) and (f3) follow from the following stronger condition, where we assume that Λ ⊂ V is an open subset of some normed linear space V, compare with slightly stronger condition (f5) in Section 2.1:

(f4) f(t, ., λ) is differentiable on B_{δ}(ξ) for a.e. t ∈ B_{δ}(τ) and for each λ ∈ Λ. Moreover f(t, x, .) is Fréchet–differentiable on Λ for a.e. t ∈ B_{δ}(τ) and for each x ∈ B_{δ}(ξ). There is a function P_{ω} ∈ L^{1}(B_{δ}(τ)), such that

|f_{x}(t, x, λ)| + ∥f_{λ}(t, x, λ)∥_{L(V,IR^{n})} ≤ P_{ω}(t)  (6)

for all λ ∈ Λ, x ∈ B_{δ}(ξ), and for a.e. t ∈ B_{δ}(τ).

We consider open bounded subsets Ω ⊂⊂ Ω_{0} with the property that there is some δ_{Ω} > 0, such that

Ω and B_{δ}(Ω) are x-convex for all 0 < δ < δ_{Ω}.  (7)

A set S ⊂ IR × IR^{n} is convex in x or x-convex, if for all pairs (t, x), (t, y) ∈ S the straight line segments [(t, x), (t, y)] lie in S, compare Hestenes [3, p.377]. Moreover we choose Ω for simplicity in such a way, that

J := {t ∈ IR : (t, x) ∈ Ω for some x ∈ IR^{n}} is an open interval.  (8)

Remark. In the appendix we present an example of a bounded open set Ω, such that Ω and even \overline{Ω} are x-convex, but such that there is no δ_{Ω}, such that (7) holds. Hence the reasoning in Hestenes [3, p.377] under the weaker assumption that only Ω is x-convex, contains a gap, see our proof of Lemma 1.2. Note that if the set Ω is convex, then B_{δ}(Ω) is convex (hence also x-convex) for all δ > 0.

Now we are in the position to formulate a local existence and uniqueness theorem for an initial value problem of the form (2) including the continuous dependence on parameters λ ∈ Λ and initial data (τ, ξ) ∈ Ω.

Theorem 1.1. Let f satisfy (f1)-(f3) (or (f1) and (f4)), and let Ω ⊂⊂ Ω_{0} be an open, bounded set, such that (7) and (8) hold. Then there exists a constant ρ > 0, such that for each w = (τ, ξ, λ) ∈ Ω × Λ there is a unique solution x = x(t; τ, ξ, λ) on B_{ρ}(τ) of the initial value problem

ẋ(t) = f(t, x(t), λ), x(τ) = ξ.  (9)

Moreover, x depends continuously on the initial data and the parameter, i.e., for any fixed τ^{∗} ∈ J, one has

x ∈ C^{0}(B_{ρ}(τ^{∗}) × Ω^{∗} × Λ, IR^{n})

where Ω^{∗} := {(τ, ξ) ∈ Ω : τ ∈ B_{ρ}(τ^{∗})}.

Note that for each (τ, ξ) ∈ Ω_{0} one can choose an open, bounded convex set Ω ⊂⊂ Ω_{0} containing (τ, ξ). Hence also (7) holds for all δ > 0, in particular for all δ > 0, such that B_{δ}(Ω) ⊂ Ω_{0}. Thus, Theorem 1.1 gives a local statement for each (τ, ξ) ∈ Ω_{0}. Note also that the radius ρ in Theorem 1.1 does not depend on (τ, ξ, λ), but merely on Ω and f.

1.2 Proofs

First we will see that, since Ω ⊂⊂ Ω_{0} satisfies (7) and is bounded, we can find integrable functions M, K, and a constant δ > 0 all independent of the point ω ∈ Ω, such that (f1)–(f3) hold uniformly with M, K in place of M_{ω}, K_{ω}. This result can be found in Hestenes [3, p.377], but for completeness we will give a proof here. As pointed out above we actually need the assumption (7) on Ω, which is stronger than just x-convexity of Ω or \overline{Ω}, see our example in the appendix.

Lemma 1.2. In addition to the assumptions (f1)–(f3), let Ω ⊂⊂ Ω_{0} be an open, bounded set, such that (7) and (8) hold. Then there is a number δ > 0, such that B_{δ}(Ω) ⊂ Ω_{0}, and there are functions M, K ∈ L^{1}(IR), such that

|f(t, x, λ)| ≤ M(t) for a.e. t ∈ B_{δ}(J),  (10)
|f(t, x, λ) − f(t, y, λ)| ≤ K(t)|x − y| for a.e. t ∈ B_{δ}(J),  (11)

for each λ ∈ Λ and all x, y ∈ IR^{n} with (t, x), (t, y) ∈ B_{δ}(Ω). Furthermore, for (τ, x) ∈ \overline{Ω} one has

lim_{λ→λ_{0}} ∫_{B_{δ}(τ)} |f(s, y, λ) − f(s, y, λ_{0})| ds = 0  (12)

for any y ∈ B_{δ}(x), λ_{0} ∈ Λ.

Note that {t ∈ IR : (t, x) ∈ B_{δ}(Ω) for some x ∈ IR^{n}} = B_{δ}(J).

Proof. For given ω = (τ, ξ) ∈ \overline{Ω} consider the concentric cylinders C_{δ(ω)}(ω), C_{δ(ω)/2}(ω), C_{δ(ω)/4}(ω) ⊂ Ω_{0}, such that (f1)-(f3) hold on C_{δ(ω)}(ω). Then

\overline{Ω} ⊂ ⋃_{ω∈\overline{Ω}} B_{δ(ω)/4}(ω),  (13)

and according to the theorem of Heine-Borel there is a finite number of points ω_{i} = (τ_{i}, ξ_{i}) ∈ \overline{Ω}, i = 1, …, N, such that for δ_{i} := δ(ω_{i})

\overline{Ω} ⊂ ⋃_{i=1}^{N} B_{δ_{i}/4}(ω_{i}).  (14)

Take 0 < 4δ < min{δ_{1}, …, δ_{N}, 4δ_{Ω}}, where δ_{Ω} is the constant in condition (7). Then

B_{δ}(Ω) ⊂ ⋃_{i=1}^{N} B_{δ_{i}/2}(ω_{i}),  (15)

since for ω ∈ B_{δ}(Ω) there is ω^{∗} ∈ Ω, such that |ω − ω^{∗}| < δ. Then by (14) we find i^{∗} ∈ {1, …, N}, such that |ω^{∗} − ω_{i^{∗}}| < δ_{i^{∗}}/4, which implies

|ω − ω_{i^{∗}}| ≤ |ω − ω^{∗}| + |ω^{∗} − ω_{i^{∗}}| < δ + δ_{i^{∗}}/4 ≤ δ_{i^{∗}}/2.

Let M_{i} := M_{ω_{i}}, K_{i} := K_{ω_{i}} ∈ L^{1}(B_{δ_{i}}(τ_{i})) for i ∈ {1, …, N} be the functions of our hypotheses. Extending these finitely many functions to all of IR by 0 and denoting these extensions by M̃_{i}, K̃_{i}, i = 1, …, N, we can define

M := ∑_{i=1}^{N} M̃_{i}, K := ∑_{i=1}^{N} K̃_{i} on IR.  (16)

Since 0 ≤ M̃_{i}(t) ≤ M(t), we immediately verify (10). Indeed, for (t, x) ∈ B_{δ}(Ω) we find by (15) some i ∈ {1, …, N} with (t, x) ∈ B_{δ_{i}/2}(ω_{i}) ⊂ B_{δ_{i}}(ω_{i}), hence by assumption (f1) |f(t, x, λ)| ≤ M_{i}(t), unless t belongs to the null-set excluded in (f1). In that case we ignore that point. To be more precise, let J_{0} := ⋃_{i=1}^{N}{t ∈ B_{δ}(τ_{i}) :

(3) or (4) does not hold }, then we obtain (10) for all t ∈ B_{δ}(J) − J_{0}. Note that J_{0} has measure zero as the finite union of null-sets.

To show (11) take (t, x), (t, y) ∈ B_{δ}(Ω) with t ∈ B_{δ}(J) − J_{0}. If |x − y| < δ, we are done, since then by (15) we find i ∈ {1, …, N}, such that (t, x) ∈ B_{δ_{i}/2}(ω_{i}), in particular x ∈ B_{δ_{i}/2}(ξ_{i}) ⊂ IR^{n}. But then y ∈ B_{δ_{i}}(ξ_{i}), since δ ≤ δ_{i}/4, and (f2) on B_{δ_{i}}(ξ_{i}) implies the desired estimate (11) for K as defined in (16). If |x − y| ≥ δ, however, take k ∈ IN , such that |x − y| < kδ. Setting y_{j} := y + (j/k)(x − y) for j = 0, …, k, we obtain

y_{0} = y, y_{k} = x, |y_{j} − y_{j−1}| = |x − y|/k < δ.  (17)

The set B_{δ}(Ω) is x-convex, since δ < δ_{Ω}, hence all the points (t, y_{j}), j = 0, …, k, are contained in B_{δ}(Ω). For each j ∈ {1, …, k} one can find i_{j} ∈ {1, …, N} by (15), such that (t, y_{j}) ∈ B_{δ_{i_{j}} /2}(ω_{i_{j}}), hence by (17) (t, y_{j−1}) ∈ B_{3δ_{i_{j}} /4}(ω_{i_{j}}), since δ < δ_{i_{j}}/4.

Now (f2) implies for all λ ∈ Λ

|f(t, x, λ) − f(t, y, λ)| ≤ ∑_{j=1}^{k} |f(t, y_{j}, λ) − f(t, y_{j−1}, λ)|

≤ ∑_{j=1}^{k} K_{i_{j}}(t)|y_{j} − y_{j−1}|

≤ K(t) ∑_{j=1}^{k} |y_{j} − y_{j−1}|

=_{(17)} K(t) ∑_{j=1}^{k} |x − y|/k = K(t)|x − y|,

which proves (11).

Now, (14) implies that for (τ, x) ∈ Ω we can find i ∈ {1, …, N} such that (τ, x) ∈ B_{δ_{i}/4}(ω_{i}), where ω_{i} = (τ_{i}, ξ_{i}) ∈ \overline{Ω}. Since δ < δ_{i}/4 one obtains B_{δ}((τ, x)) ⊂ B_{δ_{i}/2}(ω_{i}), i.e., B_{δ}(x) ⊂ B_{δ_{i}/2}(ξ_{i}) and B_{δ}(τ) ⊂ B_{δ_{i}/2}(τ_{i}). Thus we can apply (5) to get

lim_{λ→λ_{0}} ∫_{B_{δ}(τ)} |f(s, y, λ) − f(s, y, λ_{0})| ds

≤ lim_{λ→λ_{0}} ∫_{B_{δ_{i}/2}(τ_{i})} |f(s, y, λ) − f(s, y, λ_{0})| ds = 0

for all y ∈ B_{δ}(x) ⊂ B_{δ_{i}/2}(ξ_{i}). □

For a subinterval J^{∗} ⊂ B_{δ}(J) let D^{0}(J^{∗}, IR^{n}) be the space of piecewise continuous functions x on J^{∗} with

graph x := {(t, x(t)) : t ∈ J^{∗}} ⊂ B_{δ}(Ω)  (18)

equipped with the norm given by sup_{J^{∗}} |x(.)|. Here δ > 0 is the constant appearing in Lemma 1.2. We introduce the integral expression

E(x, y, λ, μ) := ∫_{J^{∗}} |f(s, x(s), λ) − f(s, y(s), μ)| ds  (19)

for x, y ∈ D^{0}(J^{∗}, IR^{n}), λ, μ ∈ Λ. By (11), for all x, y ∈ D^{0}(J^{∗}, IR^{n}), λ ∈ Λ,

0 ≤ E(x, y, λ, λ) ≤ sup_{J^{∗}} |x(.) − y(.)| ∫_{J^{∗}} K(s) ds.  (20)

Lemma 1.3. (i) Let x ∈ C^{0}(J^{∗}, IR^{n})∩D^{0}(J^{∗}, IR^{n}), then f(., x(.), λ) ∈ L^{1}(J^{∗}, IR^{n}) for each λ ∈ Λ. Moreover, if λ_{0} ∈ Λ, then

lim_{λ→λ_{0}} ∫_{J^{∗}} |f(s, x(s), λ) − f(s, x(s), λ_{0})| ds = 0.  (21)

(ii) For given x ∈ D^{0}(J^{∗}, IR^{n}), λ_{0} ∈ Λ, ϵ > 0, there exists σ = σ(x, λ_{0}, ϵ) > 0, such that

B_{σ}(x) ⊂ D^{0}(J^{∗}, IR^{n}), B_{σ}(λ_{0}) ⊂ Λ,  (22)

and such that for all y, z ∈ B_{σ}(x), λ ∈ B_{σ}(λ_{0}) one has the inequality

E(y, z, λ, μ) < ϵ.  (23)

Proof. Approximate x with a sequence of step functions {x_{k}} ⊂ D^{0}(J^{∗}, IR^{n}) in the L^{∞}-sense. Apply (11) now for λ ∈ Λ for each k ∈ IN to get

|f(t, x_{k}(t), λ) − f(t, x(t), λ)| ≤ K(t)|x_{k}(t) − x(t)| for a.e. t ∈ J^{∗}.  (24)

Since for a.e. t ∈ J^{∗} we have K(t) < ∞, we get the pointwise limit

lim_{k↗∞} f(t, x_{k}(t), λ) = f(t, x(t), λ) for a.e. t ∈ J^{∗}.

Since x_{k} is a step function, we have that f(., x_{k}(.), λ) is measurable for every k ∈ IN . Indeed, the image x_{k}(J^{∗}) consists of finitely many points in IR^{n}. Look at the set X_{B} := {x ∈ IR^{n} : f(t, x, λ) ∈ B} for some Borel set B ∈ B(IR^{n}), where B(IR^{n}) denotes the Borel algebra of IR^{n}. We find that X_{B}∩x_{k}(J^{∗}) is finite, hence measurable. Since x_{k} is measurable, we know that

{t ∈ J^{∗} : x_{k}(t) ∈ X_{B}} = {t ∈ J^{∗} : x_{k}(t) ∈ X_{B} ∩ x_{k}(J^{∗})} ∈ B(IR)

Hence also the set {t ∈ J^{∗} : f(t, x_{k}(t), λ) ∈ B} is in the Borel algebra B(IR), which is what we needed to prove that f(., x_{k}(.), λ) is measurable.

As a pointwise limit of measurable functions we obtain f(., x(.), λ) as a measurable function for each λ ∈ Λ. Since by (10) f(., x(.), λ) is bounded by M ∈ L^{1}(IR), we get that f(., x(.), λ) ∈ L^{1}(J^{∗}, IR^{n}).

Statement (21) follows from (23) for y = z = x, since ϵ > 0 can be chosen arbitrarily small.

To prove (23), let ϵ > 0 be given. Select a step function x^{∗} ∈ D^{0}(J^{∗}, IR^{n}) such that sup_{J^{∗}} |x(.) − x^{∗}(.)| < ϵ/(8γ) for some positive constant γ > ∫_{J^{∗}} K(s) ds. Then apply (20) to get

E(x, x^{∗}, μ, μ) < ϵ/8 for all μ ∈ Λ.  (25)

Since x^{∗} is a step function with graph x^{∗} ⊂ B_{δ}(Ω), we may apply (12) to find σ = σ(x, λ_{0}, ϵ) > 0, such that

E(x^{∗}, x^{∗}, λ, λ_{0}) < ϵ/4 for all λ ∈ B_{σ}(λ_{0}) ⊂ Λ  (26)

and also B_{σ}(x) ⊂ D_{0}(J^{∗}, IR^{n}), B_{σ}(λ_{0}) ⊂ Λ, i.e., such that (22) holds.

Now we use

E(x, x, λ, λ_{0})  ≤  E(x, x^{∗}, λ, λ) + E(x^{∗}, x^{∗}, λ, λ_{0}) + E(x^{∗}, x, λ_{0}, λ_{0})
<_{(25),(26)} ϵ/8 + ϵ/4 + ϵ/8 = ϵ/2 for all λ ∈ B_{σ}(λ_{0}).  (27)

Diminish σ if necessary to have also σ < ϵ/(4γ). For y, z ∈ B_{σ}(x), λ ∈ B_{σ}(λ_{0}) we conclude with (20),(22) and (27)

E(y, z, λ, λ_{0}) ≤ E(y, x, λ, λ) + E(x, x, λ, λ_{0}) + E(x, z, λ_{0}, λ_{0})
≤ ϵ/4 + ϵ/2 + ϵ/4 = ϵ.
□

For J^{∗} ⊂ B_{δ}(J) let Ω^{∗} := {(τ, ξ) ∈ Ω : τ ∈ J^{∗}}, where Ω is as before. We define the function space

C_{J^{∗}}^{0} := {x ∈ C^{0}(J^{∗} × Ω^{∗} × Λ, IR^{n}) : ∥x∥_{∞} < ∞},  (28)

where

∥x∥_{∞} := sup_{(t,w)∈J^{∗}×Ω^{∗}×Λ} |x(t, w)| for x ∈ C^{0}(J^{∗} × Ω^{∗} × Λ, IR^{n}).  (29)

It is well known that (C_{J^{∗}}^{0}, ∥.∥_{∞}) is in fact a Banach space, see e.g. Dunford-Schwartz [1, Lemma I.4.18, Corollary I.7.7].

From now on we focus on functions x ∈ C_{J^{∗}}^{0}, such that

graph x(., w) = {(t, x(t, w)) : t ∈ J^{∗}} ⊂ B_{δ}(Ω) for all w ∈ Ω^{∗} × Λ,  (30)

and we define the operator A on such functions x as

Ax(t, w) := ξ + ∫_{τ}^{t} f(s, x(s, w), λ) ds for (t, w) = (t, τ, ξ, λ) ∈ J^{∗} × Ω^{∗} × Λ. (31)

Lemma 1.4. (i) Ax ∈ C_{J^{∗}}^{0} for all x ∈ C_{J^{∗}}^{0} satisfying (30).

(ii) If |J^{∗}| is so small that

∫_{J^{∗}} K(s) ds ≤ 1/2,  (32)

then

∥Ax_{1} − Ax_{2}∥_{∞} ≤ \frac{1}{2}∥x_{1} − x_{2}∥_{∞}  (33)

for all x_{1}, x_{2} ∈ C_{J^{∗}}^{0} satisfying (30).

Proof. Let x ∈ C_{J^{∗}}^{0}, such that (30) holds. For (t, w) = (t, τ, ξ, λ) and (t′, w′) = (t′, τ′, ξ′, λ′) in J^{∗} × Ω^{∗} × Λ, we estimate

|Ax(t, w) − Ax(t′, w′)|
≤ |ξ − ξ′| + | ∫_{τ}^{t} f(s, x(s, w), λ) ds − ∫_{τ′}^{t′} f(s, x(s, w′), λ′) ds|
≤ |ξ − ξ′| + | ∫_{τ}^{t} f(s, x(s, w), λ) ds − f(s, x(s, w′), λ′) ds|
+| ∫_{τ}^{τ′} f(s, x(s, w′), λ′) ds| + | ∫_{t′}^{t} f(s, x(s, w′), λ′) ds|

≤ |ξ − ξ′| + ∫_{J^{∗}} |f(s, x(s, w), λ) − f(s, x(s, w), λ′)| ds
+ ∫_{J^{∗}} |f(s, x(s, w), λ′) − f(s, x(s, w′), λ′)| ds
+| ∫_{τ}^{τ′} f(s, x(s, w′), λ′) ds| + | ∫_{t′}^{t} f(s, x(s, w′), λ′) ds|
≤ |ξ − ξ′| + E(x(., w), x(., w), λ, λ′) + E(x(., w), x(., w′), λ′, λ′)  (34)
+| ∫_{τ}^{τ′} M(s) ds| + | ∫_{t′}^{t} M(s) ds|.

For any given ϵ > 0 we find some η = η(ϵ, t, w) > 0, such that by (23) the right-hand side of (34) is less than 5ϵ for |ξ − ξ′| < η, λ′ ∈ B_{η}(λ) ⊂ Λ, τ′ ∈ B_{η}(τ) ∩ J^{∗} and t′ ∈ B_{η}(t) ∩ J^{∗}. Note that we also use the continuity of x to get x(., w′) sufficiently close to x(., w) in D^{0}(J^{∗}, IR^{n}). This concludes the proof of (i).

To prove (33) choose x_{1}, x_{2} ∈ C_{J^{∗}}^{0} satisfying (30). In particular, x_{1}(., w), x_{2}(., w) ∈ D^{0}(J^{∗}, IR^{n}). Now we conclude

|Ax_{1}(t, w) − Ax_{2}(t, w)| ≤ E(x_{1}(., w), x_{2}(., w), λ, λ)

≤ sup_{s∈J^{∗}} |x_{1}(., w) − x_{2}(., w)| ∫_{J^{∗}} K(s) ds by (20)

≤ \frac{1}{2}∥x_{1} − x_{2}∥_{∞} by (32).

Taking the supremum over all (t, w) ∈ J^{∗} × Ω^{∗} × Λ on the left gives the desired estimate (33). □

Proof of Theorem 1.1 . Let δ > 0, M, K ∈ L^{1}(IR) as in Lemma 1.2. Fix any τ^{∗} ∈ J. We choose ρ ∈ (0, δ) so small that

∫_{B_{ρ}(τ^{∗})} K(s) ds < \frac{1}{2} ,  (35)
∫_{B_{ρ}(τ^{∗})} M(s) ds < \frac{δ}{2} .  (36)

Set

J^{∗} := B_{ρ}(τ^{∗})  (37)
Ω^{∗} := {(τ, ξ) ∈ Ω : τ ∈ J^{∗}}  (38)
C^{∗} := \{y ∈ C_{J^{∗}}^{0} : graph y(., w) ⊂ \overline{B_{δ/2}(Ω)} for all w ∈ Ω^{∗} × Λ\}.  (39)

Obviously, C^{∗} is closed in C_{J^{∗}}^{0}.

By (10),(11), (35) and (36) we obtain for all y, z ∈ C^{∗} and all (t, w) = (t, τ, ξ, λ) ∈ J^{∗} × Ω^{∗} × Λ

| ∫_{τ}^{t} f(s, y(s, w), λ) ds − ∫_{τ}^{t} f(s, z(s, w), λ) ds| < \frac{∥y − z∥_{∞}}{2} ,  (40)
| ∫_{τ}^{t} f(s, y(s, w), λ) ds| < \frac{δ}{2} .  (41)

We now fix any x_{1} ∈ C^{∗} and define a sequence {x_{k}} ⊂ C^{∗} ⊂ C_{J^{∗}}^{0} by

x_{1}(t, w) := x_{1},  (42)
x_{k}(t, w) := Ax_{k−1}(t, w) for all (t, w) ∈ J^{∗} × Ω^{∗} × Λ.  (43)

Clearly x_{1} ∈ C^{∗}. If x_{k−1} ∈ C^{∗} for k ≥ 2, then x_{k} ∈ C_{J^{∗}}^{0} by Lemma 1.4 (i). Furthermore, by (31) and (41) one has

|x_{k}(t, w) − ξ| = | ∫_{τ}^{t} f(s, x_{k−1}(s, w), λ) ds| < \frac{δ}{2}

for all (t, w) = (t, τ, ξ, λ) ∈ J^{∗} × Ω^{∗} × Λ. Since (τ, ξ) ∈ Ω^{∗}, we get x_{k} ∈ C^{∗} ⊂ C_{J^{∗}}^{0} for all k ∈ IN . By (35) and because C^{∗} is a closed subset of C_{J^{∗}}^{0}, the contraction property (33) together with the Banach Fixed Point Theorem leads to a fixed point x ∈ C_{J^{∗}}^{0} of the equation x = Ax, i.e.,

x(t, w) = x(t; τ, ξ, λ) = ξ + ∫_{τ}^{t} f(s, x(s, w), λ) ds  (44)

for all (t, w) ∈ J^{∗} × Ω^{∗} × Λ.

Obviously t ↦ x(t; τ, ξ, λ) solves (9) on B_{ρ}(τ^{∗}) for arbitrary (τ, ξ, λ) ∈ Ω^{∗} ×Λ. In particular, t ↦ x(t; τ^{∗}, ξ, λ) provides a solution of (9) on B_{ρ}(τ^{∗}) for any (τ^{∗}, ξ, λ) ∈ Ω × Λ. On the other hand, (t, τ, ξ, λ) ↦ x(t; τ, ξ, λ) is continuous on J^{∗} × Ω^{∗} × Λ, where J^{∗} = B_{ρ}(τ^{∗}), i.e.,

x ∈ C^{0}(B_{ρ}(τ^{∗}) × Ω^{∗} × Λ, IR^{n}).

It remains to show uniqueness. Let t ↦ y(t) also be a solution of (9) for given w^{∗} = (τ^{∗}, ξ, λ) ∈ Ω^{∗} × Λ in a small open neighbourhood J̃ ⊂ J^{∗} of τ^{∗}. By continuity one finds that graph y ⊂ B_{δ/2}(Ω). for sufficiently small J̃, because (τ^{∗}, ξ) ∈ Ω^{∗}. Since y satisfies the integral equation as (44) for τ = τ^{∗} on J̃, we get by (11) and (35)

|x(t; τ^{∗}, ξ, λ) − y(t)| ≤ ∫_{τ^{∗}}^{t} |f(s, x(s, w^{∗}), λ) − f(s, y(s), λ)| ds

≤ \frac{1}{2} sup_{s∈J̃} |x(s, w^{∗}) − y(s)| for all t ∈ J̃.

Taking the supremum also on the left-hand side leads to

x(t, w^{∗}) = y(t) for all t ∈ J̃,

and we get uniqueness on J̃. But this is in fact sufficient for uniqueness on all of B_{ρ}(τ^{∗}), since we have constructed a solution for each initial value (τ, ξ), which is locally unique. □

2 Differentiability properties of the solution

2.1 Formulation of the result

We again consider the system of differential equations

ẋ(t) = f(t, x(t), λ), x(τ) = ξ ∈ IR^{n},  (1)

but now we are interested in the differentiable dependence of the solution on ξ ∈ IR^{n} and λ ∈ Λ. While in Section 1 Λ was a metric space, we assume here that Λ is a open subset of a normed linear space V. In addition, we extend our set of assumptions (f1)–(f3) on the right-hand side f of (2): For any ω = (τ, ξ) ∈ Ω_{0} there is a constant δ = δ(ω) > 0 such that C_{δ}(ω) := B_{δ}(τ) × B_{δ}(ξ) ⊂ Ω_{0}, and such that the following properties hold:

(f5) f(t, ., .) is continuously Fréchet–differentiable on B_{δ}(ξ) × Λ for a.e. t ∈ B_{δ}(τ). There is a function P_{ω} ∈ L^{1}(B_{δ}(τ)), such that

|f_{x}(t, x, λ)| + ∥f_{λ}(t, x, λ)∥_{L(V,IR^{n})} ≤ P_{ω}(t)  (45)

for all λ ∈ Λ, x ∈ B_{δ}(ξ), and for a.e. t ∈ B_{δ}(τ).

(f6) There is a function Q_{ω} : B_{δ}(τ) × IR → IR, such that

(Q1) Q_{ω}(., s) ∈ L^{1}(B_{δ}(τ)) for all s ∈ IR, (Q2) Q_{ω}(t, .) is monotone for a.e. t ∈ B_{δ}(τ), (Q3) Q_{ω}(t, s) → 0 for s → 0 for a.e. t ∈ B_{δ}(τ),

and such that

|f_{x}(t, x, λ) − f_{x}(t, y, λ)| + ∥f_{λ}(t, x, λ) − f_{λ}(t, y, λ)∥_{L(V,IR^{n})}  (46)
≤ Q_{ω}(t, |x − y|)

for all λ ∈ Λ, x, y ∈ B_{δ}(ξ), and for a.e. t ∈ B_{δ}(τ). Here ∥.∥_{L(V,IR^{n})} denotes the usual operator norm on the space of linear mappings from V to IR^{n}.

As in Section 1.1 one can easily check that (f6) follows from the following stronger condition

(f6^{∗}) f(t, ., .) is twice differentiable on B_{δ}(ξ) × Λ for a.e. t ∈ B_{δ}(τ). There is a function R_{ω} ∈ L^{1}(B_{δ}(τ)), such that

∥D^{2}f(t, x, λ)∥ ≤ R_{ω}(t)  (47)

for all λ ∈ Λ, x ∈ B_{δ}(ξ), and for a.e. t ∈ B_{δ}(τ). Here D^{2}f denotes the operator consisting of all second (Fréchet–) derivatives of f.

In the next theorem we formulate the differentiability properties of the solution x = x(t; τ, ξ, λ) of (2) with respect to the initial value ξ ∈ IR^{n} and the parameter λ ∈ Λ for any fixed τ ∈ J, where, as in Section 1,

J := {t ∈ IR : (t, x) ∈ Ω for some x ∈ IR^{n}}

is an open interval by choice of an open, bounded set Ω ⊂⊂ Ω_{0} with the properties (7) and (8). We use the notation

Ξ(τ) := {ξ ∈ IR^{n} : (τ, ξ) ∈ Ω}.

Theorem 2.1. Let Ω ⊂⊂ Ω_{0} be open and bounded, such that (7) and (8) hold, and let f satisfy (f1),(f5),(f6). Then there exists a constant ρ > 0, such that for each τ ∈ J, ξ ∈ Ξ(τ) and λ ∈ Λ there is a unique solution x = x(t; ξ, λ) on B_{ρ}(τ) of the initial value problem

ẋ(t) = f(t, x(t), λ), x(τ) = ξ.  (9)

Moreover, x is continuously differentiable with respect to the initial value and the parameter, more precisely

x ∈ C^{1}(B_{ρ}(τ) × Ξ(τ) × Λ, IR^{n}).

For notational convenience we write w = (ξ, λ) ∈ Ξ(τ) × Λ. The Fréchet–derivative of x with respect to w will be denoted by D_{w}x.

Corollary 2.2. Under the assumptions of Theorem 2.1 one has

D_{w}x(t, w)v = ζ + ∫_{τ}^{t}[f_{x}(s, x(s, w), λ)D_{w}x(s, w)v + f_{λ}(s, x(s, w), λ)μ] ds (48)

for all v = (ζ, μ) ∈ IR^{n} × V. Moreover, ẋ is Fréchet–differentiable with respect to w ∈ Ξ×Λ and D_{w}x is differentiable with respect to t a.e. on B_{ρ}(τ), and one obtains

\frac{d}{dt}D_{w}x(t, w)v = D_{w}(\frac{d}{dt}x(t, w))v  (49)

for all v ∈ IR^{n} × V, w ∈ Ξ × Λ and for a.e. t ∈ B_{ρ}(τ).

Remarks. 1. The careful reader will ask why the theorem does not treat the dependence of x on the initial time τ. The classical case where f is continuously differentiable in all variables tells us that the partial derivative x_{τ} (t; τ, ξ, λ) is a sum containing the term f(τ, ξ, λ), see e.g., Walter [6, p.154]. The function f, however, is not continuous but merely measurable in τ in our situation. For simplicity we have omitted the formulation of our differentiability results in that generality, instead we refer the interested reader to Hestenes [3, pp. 390–397] for results in that direction.

2. If the set of parameters Λ ⊂ V is compact (or locally compact), then one can omit (f6) in Theorem 2.1, since a continuous function is uniformly continuous on a compact set.

2.2 Proofs

As in Section 1.2 we can prove a global version of (45) for open, bounded subsets Ω ⊂⊂ Ω_{0} satisfying (8). Since we will not need a global version of (46), condition (7) is not necessary for proving the following Lemma.

Lemma 2.3. In addition to (f5),(f6) assume that Ω ⊂⊂ Ω_{0} is an open and bounded set, such that (8) holds. Then there is a number δ_{0} > 0 with B_{δ_{0}}(Ω) ⊂ Ω_{0}, such that f(t, ., .) is continuously Fréchet–differentiable on {x ∈ IR^{n} : (t, x) ∈ B_{δ_{0}}(Ω)} × Λ for a.e. t ∈ B_{δ_{0}}(J). Moreover, there is a function P ∈ L^{1}(IR), such that

|f_{x}(t, x, λ)| + ∥f_{λ}(t, x, λ)∥_{L(V,IR^{n})} ≤ P (t)  (50)

for all λ ∈ Λ, for a.e. t ∈ IR, such that (t, x) ∈ B_{δ_{0}}(Ω).

Remark. Note that (50) together with the continuity of f_{x}, f_{λ} imply that f_{x}(., x(.), λ) ∈ L^{1}(J^{∗}, IR^{n}) and f_{λ}(., x(.), λ) ∈ L^{1}(J^{∗}, L(V, IR^{n})) for any subinterval J^{∗} ⊂ J, x ∈ D^{0}(J^{∗}, IR^{n}) and any λ ∈ Λ, where D^{0}(J^{∗}, IR^{n}) is the space of piecewise continuous functions x on J^{∗} with

graph x := {(t, x(t)) : t ∈ J^{∗}} ⊂ B_{δ_{0}}(Ω).  (51)

In fact, one may argue exactly as in the proof of Lemma 1.3 (i), using (50) and the continuity of f_{x}, f_{λ} instead of (10) and (11).

Proof of Lemma 2.3 . We proceed analogously to the proof of Lemma 1.2 using a finite covering as in (14),(15). For given ω = (τ, ξ) ∈ \overline{Ω} consider the concentric cylinders C_{δ(ω)}(ω), C_{δ(ω)/2}(ω), C_{δ(ω)/4}(ω) ⊂ Ω_{0}, such that (f5),(f6) hold on C_{δ(ω)}(ω). Then

\overline{Ω} ⊂ ⋃_{ω∈\overline{Ω}} B_{δ(ω)/4}(ω),  (52)

and according to the theorem of Heine-Borel there is a finite number of points ω_{i} = (τ_{i}, ξ_{i}) ∈ \overline{Ω}, i = 1, …, Ñ, such that for δ_{i} := δ(ω_{i})

\overline{Ω} ⊂ ⋃_{i=1}^{Ñ} B_{δ_{i}/4}(ω_{i}).  (53)

Then one obtains

B_{δ_{0}}(Ω) ⊂ ⋃_{i=1}^{Ñ} B_{δ_{i}/2}(ω_{i})  (54)

for δ_{0} := min{δ_{1}, …, δÑ}/4. In order to show (50) take the function P ∈ L^{1}(IR) defined as

P (t) := ∑_{i=1}^{Ñ} P̃_{i}(t),

where P̃_{i} := P_{ω_{i}} on B_{δ(ω_{i})}(τ_{i}) and P̃_{i} := 0 on IR − B_{δ(ω_{i})}(τ_{i}). The continuity of f_{x}(t, ., .) and f_{λ}(t, ., .) being a pointwise property follows from (46). □

As in Section 1.2 we consider a subinterval J^{∗} ⊂ B_{δ^{∗}}(J) for δ^{∗} := min{δ, δ_{0}}, where δ and δ_{0} are the positive constants in Lemma 1.2 and Lemma 2.3, respectively. We fix a parameter τ^{∗} ∈ J^{∗} and consider the slice Ξ^{∗} := Ξ(τ∗). Let us define the function spaces

C_{J^{∗}}^{0} := {x ∈ C^{0}(J^{∗} × Ξ^{∗} × Λ, IR^{n}) : ∥x∥_{∞} < ∞},  (55)
C_{J^{∗},L}^{0} := {X ∈ C^{0}(J^{∗} × Ξ^{∗} × Λ, L(IR^{n} × V, IR^{n})) : ∥X∥_{∞} < ∞},  (56)
C_{J^{∗}}^{1} := {x ∈ C_{J^{∗}}^{0} : D_{w}x ∈ C_{J^{∗},L}^{0}},  (57)

where we used the norms

∥x∥_{∞} := sup_{(t,w)∈J^{∗}×Ξ^{∗}×Λ} |x(t, w)| for x ∈ C^{0}(J^{∗} × Ξ^{∗} × Λ, IR^{n}),  (58)
∥X∥_{∞} := sup_{(t,w)∈J^{∗}×Ξ^{∗}×Λ} ∥X(t, w)∥_{L(IR^{n}×V,IR^{n})}  (59)
for X ∈ C^{0}(J^{∗} × Ξ^{∗} × Λ, L(IR^{n} × V, IR^{n})).

Recall that for w = (ξ, λ) ∈ Ξ^{∗} × Λ

∥X(t, w)∥_{L(IR^{n}×V,IR^{n})} := sup_{∥v∥_{I R^{n}×V} ≤1} |X(t, w)v|

for v = (ζ, μ) ∈ IR^{n} × V with

∥v∥_{IR^{n}×V} := |ζ| + ∥μ∥_{V} .

We introduce the norm

∥x∥_{1,∞} := ∥x∥_{∞} + ∥D_{w}x∥_{∞} on C_{J^{∗}}^{1}.  (60)

Lemma 2.4. The spaces (C_{J^{∗}}^{0}, ∥.∥_{∞}), (C_{J^{∗},L}^{0}, ∥.∥_{∞}) and (C_{J^{∗}}^{1}, ∥.∥_{1,∞}) are Banach spaces.

Proof. For (C_{J^{∗}}^{0}, ∥.∥_{∞}) and (C_{J^{∗},L}^{0}, ∥.∥_{∞}) we refer to [1] as before. For C_{J^{∗}}^{1} take a Cauchy sequence {x_{n}} ⊂ C_{J^{∗}}^{1}, which means that x_{n} → x in C^{0}(J^{∗} × Ξ^{∗} × Λ, IR^{n}) and D_{w}x_{n} → X in C^{0}(J^{∗} × Ξ^{∗} × Λ, L(IR^{n} × V, IR^{n})). We have to show that the Fréchet–derivative D_{w}x exists and satisfies

D_{w}x = X.  (61)

In order to do that, observe that for v ∈ IR^{n} × V with ∥v∥_{IR^{n}×V} sufficiently small,

x_{n}(t, w + v) − x_{n}(t, w) = ∫_{0}^{1} \frac{d}{ds}x_{n}(t, w + sv) ds
= ∫_{0}^{1} D_{w}x_{n}(t, w + sv) ds v
= ∫_{0}^{1}[D_{w}x_{n}(t, w + sv) − X(t, w + sv)] ds v
+ ∫_{0}^{1} X(t, w + sv) ds v
= ∫_{0}^{1}[D_{w}x_{n}(t, w + sv) − X(t, w + sv)] ds v
+ ∫_{0}^{1}[X(t, w + sv) − X(t, w)] ds v
+ ∫_{0}^{1} X(t, w) ds v.

Now we estimate

|x_{n}(t, w + v) − x_{n}(t, w) − ∫_{0}^{1}[X(t, w + sv) − X(t, w)] ds v − ∫_{0}^{1} X(t, w) ds v|

≤ ∥D_{w}x_{n} − X∥_{∞}∥v∥_{IR^{n}×V} for all n ∈ IN .

Going to the limit n → ∞ on both sides of this inequality we obtain

x(t, w + v) − x(t, w) = X(t, w)v + o(∥v∥_{IR^{n}×V}) for ∥v∥_{IR^{n}×V} → 0,

where we used the uniform convergence x_{n} → x in C^{0}(J^{∗} ×Ξ^{∗} ×Λ, IR^{n}), D_{w}x_{n} → X in C^{0}(J^{∗} × Ξ^{∗} × Λ, L(IR^{n} × V, IR^{n})) and the continuity of X. This implies (61). □

We consider x ∈ C_{J^{∗}}^{1}, such that

graph x(., w) = {(t, x(t, w)) : t ∈ J^{∗}} ⊂ B_{δ^{∗}}(Ω)  (62)

for all w = (ξ, λ) ∈ Ξ^{∗} ×Λ. Similarly as in (31) we define the operator A on functions x ∈ C_{J^{∗}}^{1} which satisfy (62).

Ax(t, w) := ξ + ∫_{τ^{∗}}^{t} f(s, x(s, w), λ) ds  (63)

for all (t, w) = (t, (ξ, λ)) ∈ J^{∗} × Ξ^{∗} × Λ. Note that, in contrast to (31), τ^{∗} is fixed here.

Lemma 2.5. For all x ∈ C_{J^{∗}}^{1} with (62) we have that Ax ∈ C_{J^{∗}}^{1}, and

D_{w}Ax(t, w)v = ζ + ∫_{τ^{∗}}^{t} f_{x}(s, x(s, w), λ)D_{w}x(s, w)v ds
+ ∫_{τ^{∗}}^{t} f_{λ}(s, x(s, w), λ)μ ds  (64)

for all (t, w) ∈ J^{∗} × Ξ^{∗} × Λ and v = (ζ, μ) ∈ IR^{n} × V.

Proof. One can show that Ax ∈ C_{J^{∗}}^{0} for x ∈ C_{J^{∗}}^{1} ⊂ C_{J^{∗}}^{0} satisfying (62) by a reasoning analogous to the proof of Lemma 1.4 (i), by simply fixing τ := τ^{∗}. Let us now show that Ax ∈ C_{J^{∗}}^{1} and that (64) holds. For w = (ξ, λ) ∈ Ξ^{∗} × Λ and v = (ζ, μ) ∈ IR^{n} × V we consider the function

α(σ) := Ax(t, w + σv) = ξ + σζ + ∫_{τ^{∗}}^{t} f(s, x(s, w + σv), λ + σμ) ds

for all |σ| so small, that w + σv ∈ Ξ^{∗} × Λ. The assumption (f5) on f implies that the integrand on the right-hand side is differentiable with respect to σ for a.e. s ∈ J^{∗}, and we can estimate

|\frac{d}{dσ} f(s, x(s, w + σv), λ + σμ)|

= |f_{x}(s, x(s, w + σv), λ + σμ)D_{w}x(s, w + σv)v + f_{λ}(s, x(s, w + σv), λ + σμ)μ|
≤ P (s)(∥D_{w}x∥_{∞}∥v∥_{IR^{n}×V} + ∥μ∥_{V}) for a.e. s ∈ J^{∗}, by (50).

Hence we may apply Zeidler [8, p.1018] to obtain

α′(0) = ζ + ∫_{τ^{∗}}^{t} [f_{x}(s, x(s, w), λ)D_{w}x(s, w)v + f_{λ}(s, x(s, w), λ)μ] ds
=: D_{w}Ax(t, w)v.  (65)

Analogously to the previous estimate we get

|D_{w}Ax(t, w)v| ≤ |ζ| + ∫_{τ^{∗}}^{t} P (s)(∥D_{w}x∥_{∞}∥v∥_{IR^{n}×V} + ∥μ∥_{V}) ds
≤ ∥v∥_{IR^{n}×V} [1 + (∥D_{w}x∥_{∞} + 1) ∫_{J^{∗}} P (s) ds],

which implies that D_{w}Ax(t, w) ∈ L(IR^{n} × V, IR^{n}), i.e., that D_{w}Ax(t, w) is the Gâteaux–derivative of Ax(t, .). Below we show the continuity of D_{w}Ax(., .) on J^{∗} × Ξ^{∗}×Λ. Hence D_{w}Ax(t, w) is the Fréchet–derivative of Ax(t, .) (cf. Zeidler [7, p.137]), and D_{w}Ax ∈ C_{J^{∗},L}^{0}. Since Ax ∈ C_{J^{∗}}^{0} we conclude that Ax ∈ C_{J^{∗}}^{1}.

Let us finally show that D_{w}Ax(., .) is continuous on J^{∗} × Ξ^{∗} × Λ. Indeed, taking sequences (t_{n}, w_{n}) → (t, w) ∈ J^{∗} × Ξ^{∗} × Λ one estimates for v = (ζ, μ) ∈ IR^{n} × V:

|D_{w}Ax(t_{n}, w_{n})v − D_{w}Ax(t, w)v|

≤ |D_{w}Ax(t_{n}, w_{n})v − D_{w}Ax(t_{n}, w)v| + |D_{w}Ax(t_{n}, w)v − D_{w}Ax(t, w)v|
≤ | ∫_{τ^{∗}}^{t_{n}} |f_{x}(s, x(s, w_{n}), λ_{n})D_{w}x(s, w_{n})v − f_{x}(s, x(s, w), λ)D_{w}x(s, w)v| ds|
+| ∫_{τ^{∗}}^{t_{n}} |f_{λ}(s, x(s, w_{n}), λ_{n})μ − f_{λ}(s, x(s, w), λ)μ| ds|
+| ∫_{t}^{t_{n}} |f_{x}(s, x(s, w), λ)D_{w}x(s, w)v + f_{λ}(s, x(s, w), λ)μ| ds|
≤ ∫_{J^{∗}} |f_{x}(s, x(s, w_{n}), λ_{n}) − f_{x}(s, x(s, w), λ)|∥D_{w}x(s, w_{n})∥_{L(IR^{n}×V,IR^{n})}∥v∥_{IR^{n}×V} ds
+ ∫_{J^{∗}} |f_{x}(s, x(s, w), λ)|∥D_{w}x(s, w_{n}) − D_{w}x(s, w)∥_{L(IR^{n}×V,IR^{n})}∥v∥_{IR^{n}×V} ds
+ ∫_{J^{∗}} |f_{λ}(s, x(s, w_{n}), λ_{n}) − f_{λ}(s, x(s, w), λ)|∥μ∥_{V} ds
+| ∫_{t}^{t_{n}} [|f_{x}(s, x(s, w), λ)|∥D_{w}x(s, w)∥_{L(IR^{n}×V,IR^{n})}∥v∥_{IR^{n}×V}
+|f_{λ}(s, x(s, w), λ)|∥μ∥_{V} ] ds|

≤ ∥D_{w}x∥_{∞}∥v∥_{IR^{n}×V} ∫_{J^{∗}} |f_{x}(s, x(s, w_{n}), λ_{n}) − f_{x}(s, x(s, w), λ)| ds

+∥v∥_{IR^{n}×V} sup_{J^{∗}} ∥D_{w}x(., w_{n}) − D_{w}x(., w)∥_{L(IR^{n}×V,IR^{n})} ∫_{J^{∗}} P (s) ds

+∥μ∥_{V} ∫_{J^{∗}} |f_{λ}(s, x(s, w_{n}), λ_{n}) − f_{λ}(s, x(s, w), λ)| ds  (66)

+(∥D_{w}x∥_{∞}∥v∥_{IR^{n}×V} + ∥μ∥_{V}) · | ∫_{t}^{t_{n}} P (s) ds| .

Note that by (50) the integrands on the right-hand side are all dominated by 2P ∈ L^{1}(J^{∗}), and since f_{x}(s, ., .) and f_{λ}(s, ., .) are both continuous on {x ∈ IR^{n} : (s, x) ∈ B_{δ_{0}}(Ω)} × Λ, we obtain pointwise for a.e. s ∈ J^{∗} :

lim_{n→∞} |f_{x}(s, x(s, w_{n}), λ_{n}) − f_{x}(s, x(s, w), λ)| = 0,
lim_{n→∞} |f_{λ}(s, x(s, w_{n}), λ_{n}) − f_{λ}(s, x(s, w), λ)| = 0,

where we have also used that x(., w_{n}) → x(., w) as n → ∞. After taking the supremum in (66) over all v ∈ IR^{n} × V with ∥v∥_{IR^{n}×V} ≤ 1 we can apply Lebegue’s Dominated Convergence Theorem to get

∥D_{w}Ax(t_{n}, w_{n}) − D_{w}Ax(t, w)∥_{L(IR^{n}×V,IR^{n})}
≤ ∥D_{w}x∥_{∞} ∫_{J^{∗}} |f_{x}(s, x(s, w_{n}), λ_{n}) − f_{x}(s, x(s, w), λ)| ds

+ sup_{J^{∗}} ∥D_{w}x(., w_{n}) − D_{w}x(., w)∥_{L(IR^{n}×V,IR^{n})} ∫_{J^{∗}} P (s) ds

+ ∫_{J^{∗}} |f_{λ}(s, x(s, w_{n}), λ_{n}) − f_{λ}(s, x(s, w), λ)| ds  (67)

+ (∥D_{w}x∥_{∞} + 1) · | ∫_{t}^{t_{n}} P (s) ds| ⟶ 0 as n → ∞,

by the integrability of P, and since for x ∈ C_{J^{∗}}^{1} one has

sup_{J^{∗}} ∥D_{w}x(., w_{n}) − D_{w}x(., w)∥_{L(IR^{n}×V,IR^{n})} → 0 as n → ∞.

□

We define an operator B on C_{J^{∗}}^{0} × C_{J^{∗},L}^{0}, which assigns to each (x, X) ∈ C_{J^{∗}}^{0} × C_{J^{∗},L}^{0} a mapping

B(x, X) : J^{∗} × Ξ^{∗} × Λ ⟶ L(IR^{n} × V, IR^{n})

given by

B(x, X)(t, ξ, λ)v := ζ + ∫_{τ^{∗}}^{t} f_{x}(s, x(s, ξ, λ), λ)X(s, ξ, λ)v ds
+ ∫_{τ^{∗}}^{t} f_{λ}(s, x(s, ξ, λ), λ)μ ds  (68)

for v = (ζ, μ) ∈ IR^{n} × V. Note that in fact B(x, X)(t, ξ, λ) ∈ L(IR^{n} × V, IR^{n}) for all (t, ξ, λ) ∈ J^{∗} × Ξ^{∗} × Λ.

Lemma 2.6. (i) B(x, X) ∈ C_{J^{∗},L}^{0} for all x ∈ C_{J^{∗}}^{0} satisfying (62) and all X ∈ C_{J^{∗},L}^{0}, i.e., the mapping (t, ξ, λ) ↦ B(x, X)(t, ξ, λ) is continuous on J^{∗}×Ξ^{∗}×Λ for each (x, X) ∈ C_{J^{∗}}^{0} × C_{J^{∗},L}^{0}, where x satisfies (62).

(ii) If, in addition, |J^{∗}| is so small that

∫_{J^{∗}} P (s) ds < \frac{1}{2},  (69)

then

∥B(x, X_{1}) − B(x, X_{2})∥_{∞} ≤ \frac{1}{2}∥X_{1} − X_{2}∥_{∞}  (70)

for each x ∈ C_{J^{∗}}^{0} with (62), and all X_{1}, X_{2} ∈ C_{J^{∗},L}^{0}. Here, ∥.∥_{∞} is the sup-norm as defined in (59).

(iii) Let {x_{k}} ⊂ C_{J^{∗}}^{0} be a sequence satisfying (62), x_{k} → x ∈ C_{J^{∗}}^{0}, such that x satisfies (62) and X ∈ C_{J^{∗},L}^{0}. Then ∥B(x_{k}, X) − B(x, X)∥_{∞} → 0 as k → ∞.

Remark. By Lemma 2.5 we readily see that for all x ∈ C_{J^{∗}}^{1} satisfying (62) the identity

D_{w}Ax(t, w)v = B(x, D_{w}x)(t, w)v

holds for all (t, w) ∈ J^{∗} × Ξ^{∗} × Λ, v ∈ IR^{n} × V, or, equivalently,

D_{w}Ax = B(x, D_{w}x).  (71)

Proof of Lemma 2.6 . We already realized that B(x, X)(t, ξ, λ) ∈ L(IR^{n} × V, IR^{n}) for all (t, ξ, λ) ∈ J^{∗} × Ξ^{∗} × Λ, (x, X) ∈ C_{J^{∗}}^{0} × C_{J^{∗},L}^{0}. The continuity of the mapping (t, ξ, λ) ↦ B(x, X)(t, ξ, λ) on J^{∗} × Ξ^{∗} × Λ can be shown as that of (t, w) ↦ D_{w}Ax(t, w) in the previous proof as long as (x, X) ∈ C_{J^{∗}}^{0} × C_{J^{∗},L}^{0} with x satisfying (62).

For the contraction property (70) we estimate with (50)

|B(x, X_{1})(t, w)v − B(x, X_{2})(t, w)v|
≤ | ∫_{τ^{∗}}^{t} |f_{x}(s, x(s, w), λ)(X_{1}(s, w) − X_{2}(s, w))v| ds|
≤ ∥X_{1} − X_{2}∥_{∞}∥v∥_{IR^{n}×V} ∫_{J^{∗}} P (s) ds
≤ \frac{1}{2}∥X_{1} − X_{2}∥_{∞}∥v∥_{IR^{n}×V}  by (69),

for all (t, w) := (t, ξ, λ) ∈ J^{∗}×Ξ^{∗}×Λ and v ∈ IR^{n}×V. Since B(x, X_{j})(t, w) ∈ L(IR^{n}× V, IR^{n}), j = 1, 2, we can take the supremum over v ∈ IR^{n} × V with ∥v∥_{IR^{n}×V} ≤ 1, and get

∥B(x, X_{1})(t, w) − B(x, X_{2})(t, w)∥_{L(IR^{n}×V,IR^{n})} ≤ \frac{1}{2}∥X_{1} − X_{2}∥_{∞}.

Note that B(x, X_{j}) ∈ C_{J^{∗},L}^{0} for j = 1, 2, hence we can take the supremum over J^{∗} × Ξ^{∗} × Λ on the left-hand side to conclude that (70) holds.

In order to show (iii) we estimate similarly as before

|B(x_{k}, X)(t, w)v − B(x, X)(t, w)v|

≤ | ∫_{τ^{∗}}^{t} |(f_{x}(s, x_{k}(s, w), λ) − f_{x}(s, x(s, w), λ))X(s, w)v| ds|
+| ∫_{τ^{∗}}^{t} |(f_{λ}(s, x_{k}(s, w), λ) − f_{λ}(s, x(s, w), λ))μ| ds|
≤ ∥X∥_{∞}∥v∥_{IR^{n}×V} ∫_{J^{∗}} |f_{x}(s, x_{k}(s, w), λ) − f_{x}(s, x(s, w), λ)| ds
+∥μ∥_{V} ∫_{J^{∗}} |f_{λ}(s, x_{k}(s, w), λ) − f_{λ}(s, x(s, w), λ)| ds.

First we take the supremum over all v ∈ IR^{n} × V with ∥v∥_{IR^{n}×V} to obtain

∥B(x_{k}, X)(t, w) − B(x, X)(t, w)∥_{L(IR^{n}×V,IR^{n})}
≤ ∥X∥_{∞} ∫_{J^{∗}} |f_{x}(s, x_{k}(s, w), λ) − f_{x}(s, x(s, w), λ)| ds
+ ∫_{J^{∗}} |f_{λ}(s, x_{k}(s, w), λ) − f_{λ}(s, x(s, w), λ)| ds.  (72)

Now we use the finite covering (54) to define the sets

J_{l} := {s ∈ J^{∗} : (s, x(s, w)) ∈ B_{δ_{l}}(ω_{l})} for l = 1, …, Ñ.

Note that since x satisfies (62) we have

J^{∗} ⊂ ⋃_{l=1}^{Ñ} J_{l}.

On each J_{l} for l = 1, …, Ñ, we can apply (46) on the integrands on the right-hand side of (72) to obtain

∥B(x_{k}, X)(t, w) − B(x, X)(t, w)∥_{L(IR^{n}×V,IR^{n})}
≤ ∥X∥_{∞} ∑_{l=1}^{Ñ} ∫_{J_{l}} |f_{x}(s, x_{k}(s, w), λ) − f_{x}(s, x(s, w), λ)| ds

+ ∑_{l=1}^{Ñ} ∫_{J_{l}} |f_{λ}(s, x_{k}(s, w), λ) − f_{λ}(s, x(s, w), λ)| ds

≤ (∥X∥_{∞} + 1) ∑_{l=1}^{Ñ} ∫_{J_{l}} Q_{ω_{l}}(s, r_{k}) ds,  (73)

where r_{k} := ∥x_{k} − x∥_{∞}. Note that we used the monotonicity property (Q2) for each Q_{ω_{i}}, i = 1, …, Ñ. Finally, taking the supremum over J^{∗} × Ξ^{∗} × Λ on the left-hand side we arrive at

∥B(x_{k}, X) − B(x, X)∥_{∞} ⟶ 0 as k → ∞,

where we have applied Lebegue’s Dominated Convergence Theorem using Q_{ω_{l}}(., 2∥x∥_{∞}), as integrable majorizing functions for the integrands Q_{ω_{l}}(s, r_{k}), l = 1, …, Ñ on the right-hand side of (73) for sufficiently large k ∈ IN . □

Proof of Theorem 2.1 . Let δ^{∗} = min{δ, δ_{0}}, where δ_{0} > 0, P ∈ L^{1}(IR) as in Lemma 2.3 and δ > 0 and the functions M, K ∈ L^{1}(IR) as in Lemma 1.2. Fix any τ^{∗} := τ ∈ J and choose ρ ∈ (0, δ^{∗}), such that

∫_{B_{ρ}(τ^{∗})} K(s) ds < \frac{1}{2},  (74)
∫_{B_{ρ}(τ^{∗})} M(s) ds < \frac{δ}{2},  (75)
∫_{B_{ρ}(τ^{∗})} P (s) ds < \frac{1}{2}.  (76)

Set

J^{∗} := B_{ρ}(τ^{∗}),  (77)
Ξ^{∗} := {ξ ∈ IR^{n} : (τ^{∗}, ξ) ∈ Ω} = Ξ(τ^{∗}).  (78)

Define C_{J^{∗}}^{0}, C_{J^{∗},L}^{0} and C_{J^{∗}}^{1} according to (55),(56) and (57), and set

C^{1∗} := \{y ∈ C_{J^{∗}}^{1} : graph y ⊂ \overline{B_{δ/2}(Ω)}\},  (79)

(compare also with (37),(38) and (39).) Fix any x_{1} ∈ C^{1∗}, set x_{k+1} := Ax_{k}, which implies that x_{k} ∈ C_{J^{∗}}^{1} for all k ∈ IN by Lemma 2.5. Moreover, by the same reasoning as in the proof of Theorem 1.1 one shows that x_{k} ∈ C^{1∗} for all k ∈ IN . By the contraction property (33), which therefore holds on the sequence {x_{k}} by the definition of C^{1∗}, and the Banach Fixed Point Theorem we obtain

x_{k} → x ∈ C_{J^{∗}}^{0} with Ax = x.

For this fixed x the operator B(x, .) on C_{J^{∗},L}^{0} is a contraction mapping according to (70) of Lemma 2.6, hence has also a fixed point X ∈ C_{J^{∗},L}^{0}, i.e., B(x, X) = X. Now set X_{1} := D_{w}x_{1}, and consider the sequence X_{k+1} := B(x_{k}, X_{k}) with the x_{k} from above. Observe that X_{k} = D_{w}x_{k} for all k ∈ IN by (71). We claim that

X_{k} → X in C^{0}(J^{∗} × Ξ^{∗} × Λ, L(IR^{n} × V, IR^{n})).  (80)

In fact, we can use (70) and Lemma 2.6 (iii) to get

∥X_{k+1} − X∥_{∞} = ∥B(x_{k}, X_{k}) − B(x, X)∥_{∞}
≤ ∥B(x_{k}, X_{k}) − B(x_{k}, X)∥_{∞} + ∥B(x_{k}, X) − B(x, X)∥_{∞}
≤ \frac{1}{2}∥X_{k} − X∥_{∞} + β_{k} with β_{k} → 0 for k → ∞.  (81)

For a given ϵ > 0 there is a k_{0} ∈ IN , such that β_{k} ≤ ϵ/2 for all k ≥ k_{0}. Thus from (81) we infer for k > k_{0} + 1

∥X_{k+1} − X∥_{∞} ≤ \frac{1}{2}(∥X_{k} − X∥_{∞} + ϵ)

≤ \frac{1}{2}(\frac{1}{2}(∥X_{k−1} − X∥_{∞} + ϵ) + ϵ)

≤ \frac{1}{2^{k−k_{0}+1}} ∥X_{k_{0}} − X∥_{∞} + ϵ ∑_{i=1}^{k−k_{0}+1} \frac{1}{2^{i}}

≤ \frac{1}{2^{k−k_{0}+1}} ∥X_{k_{0}} − X∥_{∞} + ϵ.

Consequently, ∥X_{k+1} − X∥_{∞} ≤ 2ϵ for k sufficiently large, since the first term on the right-hand side can be made arbitrarily small choosing k large enough. This finishes the proof of (80).

Finally we observe that D_{w}x = X, since C_{J^{∗}}^{1} is a Banach space according to Lemma 2.4. This implies that x is in fact in C_{J^{∗}}^{1}. Theorem 1.1 shows that x is the unique solution of (9) on B_{ρ}(τ^{∗}), which finishes the proof of Theorem 2.1. □

Proof of Corollary 2.2 . (68),(71) and the fact that Ax = x from the previous proof imply

D_{w}x(t, w)v = D_{w}Ax(t, w)v = B(x, D_{w}x)(t, w)v
= ζ + ∫_{τ}^{t}[f_{x}(s, x(s, w), λ)D_{w}x(s, w)v + f_{λ}(s, x(s, w), λ)μ] ds

for v = (ζ, μ) ∈ IR^{n} × V, which proves (i). Hence, we can differentiate D_{w}x with respect to t (a.e. on B_{ρ}(τ)) and obtain

\frac{d}{dt}D_{w}x(t, w)v = f_{x}(t, x(t, w), λ)D_{w}x(t, w)v + f_{λ}(t, x(t, w), λ)μ  (82)

for a.e. t ∈ B_{ρ}(τ) and for v = (ζ, μ) ∈ IR^{n} × V. On the other hand, looking at the differential system (9) for x = x(t, w), we see that under the conditions (f1)-(f6) on f together with Lemma 2.3, that ẋ is Fréchet–differentiable with respect to w ∈ Ξ×Λ, since the right-hand side of (9) is. Computing this Fréchet derivative using the chain rule one obtains exactly the expression in (82), hence

\frac{d}{dt}D_{w}x(t, w)v = D_{w}(\frac{d}{dt}x(t, w))v,

i.e., (49) as claimed. □

Appendix: A counterexample

We are going to construct an open, bounded x-convex set Ω ⊂ IR^{3}, such that also its closure \overline{Ω} is x-convex, with the property that for every δ > 0 sufficiently small the corresponding δ-neighbourhood B_{δ}(Ω) fails to be x-convex. This set Ω clearly violates condition (7). In the following we write x = (ξ, η) ∈ IR^{2}.

Consider the curve γ : (0, 1) → IR^{3} defined as

γ(t) := ( \begin{matrix}t \\ ξ(t) \\ η(t) \end{matrix} ) := ( \begin{matrix}t \\ t cos ϕ(t) \\ t sin ϕ(t) \end{matrix} )

where ϕ(t) := −2π log_{3/2} t for t ∈ (0, 1). Then we have

η(t) = 0 and ξ(t) = t for t = 1, \frac{2}{3}, \frac{4}{9}, …, \frac{2^{k}}{3^{k}} , ….

For the planar disk D_{ρ} := {(0, ξ, η) ∈ IR^{3} : ξ^{2} + η^{2} < ρ} and for ϵ > 0 set

Ω(ϵ) := {(t, ξ, η) ∈ IR^{3} : t ∈ (0, 1), (t, ξ, η) ∈ γ(t) + D_{tϵ}}.

Obviously, Ω(ϵ) and \overline{Ω(ϵ)} are bounded and (ξ, η)-convex. For 0 < ϵ < 1 the set Ω(ϵ) is a spiral, which does not intersect the t-axis, and the closure \overline{Ω(ϵ)} intersects the t-axis only in the origin. It will turn out that for ϵ > 0 sufficiently small Ω := Ω(ϵ) is the desired set.

Given any 0 < δ < \sqrt{2}/3 , set t_{0} := \sqrt{2}δ, and t_{1} := 3t_{0}/2, hence [t_{0}, t_{1}] ⊂ (0, 1). We claim that the circle

Ct̃:= \{(t̃, ξ, η) : ξ^{2} + η^{2} = t̃^{2}\}, t̃:= \frac{t_{0} + t_{1}}{2} ,

is contained in the δ-neighbourhood of Γ := γ((0, 1)), i.e.,

Ct̃ ⊂ B_{δ}(Γ) ⊂ B_{δ}(Ω(ϵ)) for all ϵ > 0.  (83)

Indeed, if one parametrizes Ct̃ by the mapping c : [t_{0}, t_{1}] → IR^{3} defined by

σ ↦ c(σ) := t̃(1, cos ϕ(σ), sin ϕ(σ)),

one easily checks that

dist (Γ, c(σ)) ≤ |γ(σ) − c(σ)| ≤ δ/2 for all σ ∈ [t_{0}, t_{1}].

Now observe that Γ lies on the boundary of the cone K, whose axis is equal to the t-axis and which is centered at the origin with opening angle π/4, since |γ(t) − (t, 0, 0)| = t for all t ∈ (0, 1). Consequently, we have that

Ω(ϵ) ⊂ Ω_{K}(ϵ), where Ω_{K}(ϵ) := ⋃_{(t,ξ,η)∈∂K}(t, ξ, η) + D_{tϵ}.  (84)

By elementary geometric arguments one can easily deduce (see Figure 1) that

dist ((t, 0, 0), Ω_{K}(ϵ)) = t sin(arctan(1 − ϵ)).

Fig. 1.

Since sin(arctan(1 − ϵ)) → \frac{1}{\sqrt{2}} as ϵ → 0, there is some ϵ_{0} > 0 such that

sin(arctan(1 − ϵ_{0})) > \frac{4}{5\sqrt{2}}.

Thus for t = t̃= 5t_{0}/4 = 5\sqrt{2}δ/4

dist ((t̃, 0, 0), Ω_{K}(ϵ_{0})) = \frac{5}{4}δ\sqrt{2} sin(arctan(1 − ϵ_{0})) > δ.

That implies that the point (t̃, 0, 0), which is the center of the circle Ct̃, is not contained in B_{δ}(Ω_{K}(ϵ_{0})). By (84) we get (t̃, 0, 0) ∉ B_{δ}(Ω(ϵ_{0})). This together with (83) shows that B_{δ}(Ω(ϵ_{0}) is not (ξ, η)-convex for all δ ∈ (0, \sqrt{2}/3).

References

[1] N. Dunford, J.T. Schwartz, Linear Operators I, Interscience, New York, 1958.

[2] O. Gonzalez, J.H. Maddocks, F. Schuricht, H. von der Mosel, Global Curvature and Self-Contact of Nonlinearly Elastic Rods, Manuscript.

[3] M.R. Hestenes, Calculus of Variations and Optimal Control Theory, J. Wiley & Sons, New York, 1966.

[4] E. Kamke, Das Lebesgue–Stieltjes Integral, Teubner, Leipzig, 1960.

[5] F. Schuricht, H. von der Mosel, Euler–Lagrange Equations for Nonlinearly Elastic Rods with Self-Contact, in preparation.

[6] W. Walter, Ordinary differential equations, (Graduate texts in math. 182, Readings in mathematics), Springer, New York Berlin Heidelberg, 1998.

[7] E. Zeidler, Nonlinear functional analysis and its applications, vol.I: Fixed–Point Theorems, Springer, New York Berlin Heidelberg, 1985.

[8] E. Zeidler, Nonlinear functional analysis and its applications, vol.IIB: Nonlinear Monotone Operators, Springer, New York Berlin Heidelberg, 1990.