Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 2, 1–24

On the Autonomous Nemytskij Operator in Hölder spaces

M. Goebel and F. Sachweh

Dedicated in gratitude to Lothar von Wolfersdorf on his 65th birthday

Abstract. The paper is devoted to the autonomous Nemytskij operator (superposition operator) in Hölder spaces H^{k+α}[a, b], (k, α) ∈ ℤ_{+} × (0, 1]. We study acting, continuity, Lipschitz continuity, and Fréchet differentiability conditions. For k = 0, α ∈ (0, 1] and k ∈ ℕ, α = 1 the respective conditions are both necessary and sufficient. For k ∈ ℕ, α ∈ (0, 1) only the acting condition is both necessary and sufficient; the other investigated properties are characterized by necessary and sufficient conditions different from each other.

Keywords: Hölder spaces, Lipschitz spaces, Nemytskij operator, superposition operator, acting conditions, boundedness, continuity and Lipschitz continuity, Fréchet differentiability

AMS subject classification: 47 H 30

1. Introduction

To describe the contents of this paper properly we start with introducing the function spaces needed in the following.

Let [a, b] ⊂ ℝ be a given bounded interval, α ∈ (0, 1] and h_{α} : C[a, b] → \overline{ℝ} defined by

h_{α}(u) = sup \{\frac{|u(t) − u(s)|}{|t − s|^{α}} | t ≠ s, t, s ∈ [a, b]\} .

Then u ∈ C[a, b] is said to be α-Hölder continuous on [a, b] if h_{α}(u) < ∞. The set of all such functions is called (classical) Hölder space and denoted by H^{α}[a, b]. Introducing the norm

∥u∥_{H^{α}[a,b]} = ∥u∥_{C[a,b]} + h_{α}(u)  (1.1)

the set H^{α}[a, b] becomes a Banach space. In the case of α = 1 we write Lip[a, b] instead of H^{1}[a, b]; the elements of Lip[a, b] are Lipschitz continuous functions on [a, b]. If further a natural number k ∈ ℕ is given, then a function u ∈ C^{k}[a, b] is said to be (k, α)-Hölder continuous on [a, b] if its k-th derivative u^{(k)} is an α-Hölder continuous

M. Goebel: Martin-Luther-Universität Halle-Wittenberg, FB Math. und Inform., D-06099 Halle (Saale); goebel@mathematik.uni-halle.de F. Sachweh: Bispinckpl. 1-3, D-48683 Ahaus; f sachweh@lauder.de

ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

function. The set of all such functions is denoted by H^{k+α}[a, b]. When endowed with the norm

∥u∥_{H^{k+α}[a,b]} = ∑_{i=0}^{k} ∥u^{(i)}∥_{C[a,b]} + h_{α}(u^{(k)})  (1.2)

it is also a Banach space. Again, in the ase of α = 1 we use the notation Lip^{k}[a, b] instead of H^{k+1}[a, b]. By convention, H^{0+α}[a, b] = H^{α}[a, b] and Lip^{0}[a, b] = Lip[a, b] so that the space H^{k+α}[a, b] is defined for any pair (k, α) ∈ ℤ_{+}×(0, 1], where ℤ_{+} = {0}∪ℕ.

Now we turn to functions f = f(x) acting in ℝ. For given α ∈ (0, 1] a function f ∈ C(ℝ) is said to be locally α-Hölder continuous on ℝ if for each x_{0} ∈ ℝ there is a positive number δ such that f ∈ H^{α}[x_{0} − δ, x_{0} + δ]. This is equivalent to the condition f ∈ H^{α}[−r, r] for all positive numbers r. The symbol H_{loc}^{α}(ℝ) denotes the set of all locally α-Hölder continuous functions defined on ℝ. We write f ∈ H_{loc}^{k+α}(ℝ), where again k ≥ 1 is a natural number and α ∈ (0, 1], provided f ∈ C^{k}(ℝ) and f^{(k)} ∈ H_{loc}^{α}(ℝ). In the case of α = 1 we write Lip_{loc}(ℝ) and Lip_{loc}^{k}(ℝ) instead of H_{loc}^{1}(ℝ) and H_{loc}^{k+1}(ℝ), respectively.

The subject of our paper is the autonomous Nemytskij operator F u = f ∘ u, generated by a given function f : ℝ → ℝ, as a map of the Hölder space H^{k+α}[a, b], (k, α) ∈ ℤ_{+} × (0, 1], into itself. There are given conditions to f ensuring F acting in H^{k+α}[a, b], boundedness of F, its continuity, local Lipschitz continuity and Fréchet differentiability, respectively. While for H^{α}[a, b], α ∈ (0, 1], and Lip^{k}[a, b], k ∈ ℕ, all respective conditions are both necessary and sufficient, for H^{k+α}[a, b] with (k, α) ∈ ℕ × (0, 1) this is only the case for the acting condition. Concerning the other properties for these parameters we have always separate necessary and sufficient conditions with a certain distinction between both.

The plan of the paper is as follows. In Section 2 we consider the case k = 0, α ∈ (0, 1], in Section 4 the case k ∈ ℕ, α = 1, and eventually in Section 5 the case (k, α) ∈ ℕ × (0, 1). Section 3 is devoted to some preliminaries for the subsequent sections. Most of the results in Section 2 are known; the results in Sections 4 and 5 seem to be new. One of the purposes of this paper is to present a self-contained unified and not too sophisticated theory of the autonomous Nemytskij operator in H^{k+α}[a, b]. Unfortunately, in Section 5 some questions remain open.

Dealing with nonlinear analysis means at first to study Nemytskij operators. We were led to the problems considered in this paper in studying an optimal control problem, in which the state equation is a nonlinear singular integral equation of Cauchy type. Concerning existence and uniqueness of solutions to the state equation we could refer to L. v. Wolfersdorf [26]. However, we needed also a simple and convenient condition ensuring e.g. differentiability of Nemytsjkij operators in Hölder spaces. First results suitable for our purposes can be found in M. Goebel [8]. An application of these results to the control problem just mentioned is given in M. Goebel and D. Oestreich [10]. The material to be presented now is mainly based on F. Sachweh [21], which was written under the auspices of the other author. We have announced our results in M. Goebel and F. Sachweh [11], where the reader can find a summary in form of a table (with an obvious misprint in the last row).

From the papers dealing explicitly with autonomous Nemytskij operators in Hölder spaces we quote at the moment only M. Z. Berkolajko [4], P. Drábek [7] and M. Goebel [8]. In case of a non-autonomous Nemytskij operator the generating function f depends on t ∈ [a, b] and x ∈ ℝ. For such operators considered in a Hölder space H^{α}[a, b] the reader is referred to the papers R. Nugari [19, 20], R. Chiappinelli and R. Nugari [6], and M. Goebel [9], and in H^{k+α}[a, b] to T. Valent [24: Chapter II], E. Wegert [26: Section 1.3], and M. Lanza de Cristoforis [12]. Most of the quoted papers deal in spaces of functions depending on several variables. A profound account of all relevant literature for both autonomous and non-autonomous Nemytskij operators in different function spaces until about 1988 can be found in the monograph J. Appell and P. P. Zabrejko [2].

We finish this introductory section with some notations used throughout the whole paper. If X is a Banach space, the set of all continuous operators from X into itself is denoted by C(X) and by C^{1}(X) the subset of C(X), whose elements are continuously Fréchet differentiable on X. Lip_{loc}(X) denotes the set of all locally Lipschitz continuous operators acting in X and L(X) the set of all linear bounded operators acting in X. The symbol B(x_{0}, δ) ⊂ X stands for an open ball in X centered at x_{0} ∈ X with radius δ > 0. For any function f we denote its i-th derivative by D^{i}f or f^{(i)}. Finally, c denotes a generic constant.

2. The Nemytskij operator in H^{α}[a, b]

Before starting to consider autonomous Nemytskij operators F generated by f : ℝ → ℝ in the Hölder space H^{α}[a, b] with α ∈ (0, 1], we give two well-known properties of this space in our first lemma.

Lemma 2.1.

(a) For any u, v ∈ H^{α}[a, b], uv ∈ H^{α}[a, b] and ∥uv∥_{H^{α}[a,b]} ≤ ∥u∥_{H^{α}[a,b]}∥v∥_{H^{α}[a,b]}.

(b) For any β ∈ (0, α), H^{α}[a, b] ⊂ H^{β}[a, b] with h_{β}(u) ≤ (b − a)^{α−β}h_{α}(u) for all u ∈ H^{α}[a, b].

Our first theorem presents a necessary and sufficient acting condition proved by M. Z. Berkolajko [4] for generalized Hölder spaces. The proof of the necessity part given below is an simplified adaption of P. Drábek [7]. It is an easy task to derive a necessary and sufficient condition if acting of F is combined with its boundedness as it was done e.g. by A. A. Babajev [3]. The subsequent corollary is an immediate consequence of Theorem 2.1.

Theorem 2.1. The autonomous Nemytskij operator F generated by f maps H^{α}[a, b] into itself if and only if f ∈ Lip_{loc}(ℝ).

Proof. It suffices to show the necessity of f ∈ Lip_{loc}(ℝ) provided F is acting in H^{α}[a, b].

1. Let x_{0} ∈ ℝ be arbitrarily fixed. The linear function u_{0} defined by u_{0}(t) = t − a + x_{0} − \frac{b−a}{2} (t ∈ [a, b]) belongs to H^{α}[a, b] and hence, by our assumption, there is a constant L_{0} > 0 with

|f(u_{0}(t)) − f(u_{0}(s))| ≤ L_{0}|t − s|^{α} (t, s ∈ [a, b])

which implies

|f(x) − f(y)| ≤ L_{0}|x − y|^{α} (x, y ∈ B(x_{0}, \frac{b−a}{2}) ⊂ ℝ).

Consequently, f ∈ H_{loc}^{α}(ℝ) and the statement is already proved for α = 1.

2. Let α ∈ (0, 1). Assuming f ∉ Lip_{loc}(ℝ) we can find a number r_{0} > 0 and two convergent sequences (x_{n}), (y_{n}) ⊂ [−r_{0}, r_{0}], x_{n} ≠ y_{n} for all n ∈ ℕ, such that

|f(x_{n}) − f(y_{n})| > n|x_{n} − y_{n}|  (n ∈ ℕ).  (2.1)

Because of the first point of the proof, the left-hand side of (2.1) is bounded and so (x_{n}) and (y_{n}) have the same limit, say x_{0}. We put

B_{n} = B(x_{0}, γ_{n+1}) ⊂ ℝ,  where γ_{n} = \frac{b − a}{2^{n_{0}+n+2}}  (n ∈ ℕ),

with n_{0} ∈ ℕ so large that γ_{n} < 1 for all n ∈ ℕ. Then for every n ∈ ℕ there is a k_{n} ∈ ℕ with x_{k_{n}}, y_{k_{n}} ∈ B_{n} and hence |x_{k_{n}} − y_{k_{n}}| < 2γ_{n+1} = γ_{n} for all n ∈ ℕ. We may assume k_{n} > k_{m} for n > m. By setting δ_{n} = |x_{k_{n}} − y_{k_{n}}|^{1/α} (n ∈ ℕ) we get another sequence δ_{n} ⊂ (0, 1) with δ_{n} < γ_{n}. Now we choose a number t_{1} ∈ (a, a + \frac{b−a}{4}) and define

t_{n} = t_{1} + ∑_{i=1}^{n−1}(γ_{i} + δ_{i}) and s_{n} = t_{n} + δ_{n} (n ∈ ℕ).

Obviously, (t_{n}), (s_{n}) ⊂ [a, b] and γ_{m} < t_{n} − t_{m} for arbitrary m, n ∈ ℕ with n > m. Since 0 < t ≤ 1 implies t ≤ t^{α} and t > 1 implies t^{α} > 1, we find γ_{m} < (t_{n} − t_{m})^{α} and in the same way also γ_{m} < (t_{n} − s_{m})^{α} and γ_{m} < (s_{n} − s_{m})^{α} for all m, n ∈ ℕ with n > m. We introduce on M = {t_{n}}_{n∈ℕ} ∪ {s_{n}}_{n∈ℕ} a function u = u(t) by setting

u(t) = \{ \begin{matrix}x_{k_{n}} & for t = t_{n} \\ y_{k_{n}} & for t = s_{n} \end{matrix}

and claim

|u(t) − u(s)| ≤ |t − s|^{α} (t, s ∈ M). (2.2)

Indeed, let m, n ∈ ℕ with n > m. Then we have

|u(t_{n}) − u(t_{m})| = |x_{k_{n}} − x_{k_{m}}| ≤ γ_{n+1} + γ_{m+1} ≤ γ_{m} ≤ (t_{n} − t_{m})^{α}

and, analogously,

|u(t_{n}) − u(s_{m})| ≤ (t_{n} − s_{m})^{α}  and  |u(s_{n}) − u(s_{m})| ≤ (s_{n} − s_{m})^{α}.

If m = n, then

|u(t_{n}) − u(s_{n})| = |x_{k_{n}} − y_{k_{n}}| = δ_{n}^{α} = (s_{n} − t_{n})^{α}.

Thereby (2.2) is proved. Due to E. J. McShane [18] (compare also J.-E. Björk [5] and W. Walter [24: p. 65]) u can be extented to [a, b] preserving the Hölder condition (2.2). This extension is again denoted by u. Then because of our assumption there exists a positive constant L_{u} such that

|f(u(t)) − f(u(s))| ≤ L_{u}|t − s|^{α} (t, s ∈ [a, b]).

For t = t_{n} and s = s_{n} this results in

|f(x_{k_{n}}) − f(y_{k_{n}})| ≤ L_{u}|t_{n} − s_{n}|^{α} = L_{u}δ_{n}^{α} = L_{u}|x_{k_{n}} − y_{k_{n}}|.

Taking advantage of (2.1) we come to L_{u} ≥ k_{n} for all n ∈ ℕ. Since this is impossible the assertion is proved Corollary 2.1. Each autonomous Nemytskij operator acting in H^{α}[a, b] is bounded.

Unfortunately, not each Nemytskij operator acting in H^{α}[a, b] is continuous. As a simple example one may take F generated by the function f, which is defined by f(x) = |x| (x ∈ ℝ). Obviously, F maps H^{α}[0, 1] into itself. But this map is not continuous. To see this, take u_{n} defined by u_{n}(t) = t^{α} − ε_{n} (t ∈ [0, 1]), with (ε_{n}) ⊂ (0, 2) and ε_{n} → 0. Then u_{n} ∈ H^{α}[0, 1] and u_{n} → u_{0} in H^{α}[0, 1] as n → ∞, where u_{0}(t) = t^{α} (t ∈ [0, 1]). However, h_{α}(f ∘ u_{n} − f ∘ u_{0}) ≥ 2 for all n ∈ ℕ, such that F u_{n} → F u_{0} in H^{α}[0, 1].

In the literature we can find also f defined by f(x) = min{x, 1} (x ∈ ℝ) as another example for a function generating a non-continuous Nemytskij operator in H^{α}[0, 1]. But the given arguments are correct only for α = 1, i.e. in the space Lip[0, 1], and false for α ∈ (0, 1) (see F. Sachweh [21: pp. 16 and 52 – 54]).

The continuity and Lipschitz continuity of an autonomous Nemytskij operator are completely characterized in the next two theorems. Thereby, Theorem 2.3 shows that there is no sense to assume F satisfies a (global) Lipschitz condition on H^{α}[a, b]. Only local Lipschitz continuity makes sense.

Theorem 2.2. The autonomous Nemytskij operator F generated by f maps the space H^{α}[a, b] continuously into itself if and only if f ∈ C^{1}(ℝ).

Proof. 1. We begin with the sufficiency of the condition f ∈ C^{1}(ℝ). Due to Theorem 2.1 F u = f ∘ u ∈ H^{α}[a, b] holds for all u ∈ H^{α}[a, b]. To show the continuity of F : H^{α}[a, b] → H^{α}[a, b] let there be given a sequence (u_{n}) ⊂ H^{α}[a, b] converging to u_{0} ∈ H^{α}[a, b]. Since obviously f ∘ u_{n} → f ∘ u_{0} in C[a, b] it remains to show that

h_{α}(f ∘ u_{n} − f ∘ u_{0}) ⟶ 0.  (2.3)

For any n ∈ ℕ we define

Δ_{n}(t, s) = \frac{|f(u_{0}(t)) − f(u_{0}(s)) − f(u_{n}(t)) + f(u_{n}(s))|}{|t − s|^{α}} (t, s ∈ [a, b], t ≠ s).

Using the Lagrange formula

f(x) − f(y) = (x − y) ∫_{0}^{1} f′(y + τ(x − y)) dτ  (x, y ∈ ℝ)  (2.4)

we may verify that

Δ_{n}(t, s) ≤ h_{α}(u_{0}) ∫_{0}^{1} |f′(u_{0}(s) + τ(u_{0}(t) − u_{0}(s)))

− f′(u_{n}(s) + τ(u_{n}(t) − u_{n}(s)))|dτ

+ h_{α}(u_{0} − u_{n}) ∫_{0}^{1} |f′(u_{n}(s) + τ(u_{n}(t) − u_{n}(s)))| dτ

for all t, s ∈ [a, b] and all n ∈ ℕ. Since if n → ∞ the first integrand tends to zero uniformly on [a, b] × [a, b] × [0, 1] and since the second integrand is uniformly bounded on the same set, we get (2.3).

2. Now we prove the necessity of f ∈ C^{1}(ℝ). Again due to Theorem 2.1 we know f ∈ Lip_{loc}(ℝ). Hence, both

L_{1}(x) = lim inf_{ξ→0} \frac{f(x + ξ) − f(x)}{ξ} and L_{2}(x) = lim sup_{ξ→0} \frac{f(x + ξ) − f(x)}{ξ}

are finite for all x ∈ ℝ and, in virtue of the Rademacher Theorem (see, e.g., [28: p. 50]), L_{1}(x) = L_{2}(x) = f′(x) for almost all x ∈ ℝ. Suppose there exists a point x_{0} ∈ ℝ at which f is not differentiable, i.e. L_{1}(x_{0}) < L_{2}(x_{0}). We denote by (ξ_{n}), (η_{n}) ⊂ ℝ two zero sequences with

L_{1}(x_{0}) = lim_{n→∞} \frac{f(x_{0} + ξ_{n}) − f(x_{0})}{ξ_{n}} and L_{2}(x_{0}) = lim_{n→∞} \frac{f(x_{0} + η_{n}) − f(x_{0})}{η_{n}}

and by (δ_{n}) ⊂ ℝ a further zero sequence such that f has in all points x_{0} + δ_{n} a finite derivative f′(x_{0} + δ_{n}). Without loss of generality, for all n ∈ ℕ we may assume ξ_{n} > 0 or ξ_{n} < 0 and η_{n} > 0 or η_{n} < 0. First assuming ξ_{n} > 0 for all n we consider u_{n} defined by

u_{n}(t) = (t − a)^{α} + x_{0} + δ_{n} (t ∈ [a, b], n ∈ ℕ).

Obviously, u_{n} ∈ H^{α}[a, b], and if n → ∞, then, in H^{α}[a, b], u_{n} → u_{0} ∈ H^{α}[a, b] where u_{0} is defined by u_{0}(t) = (t − a)^{α} + x_{0} (t ∈ [a, b]). So, by our assumption, f ∘ u_{n} → f ∘ u_{0} in H^{α}[a, b]. Taking ε = \frac{L_{2}(x_{0})−L_{1}(x_{0})}{4} we find a natural number n_{0} = n(ε) such that h_{α}(f ∘ u_{n} − f ∘ u_{0}) ≤ ε for all n ≥ n_{0} which implies

\begin{matrix}|f((t − a)^{α} + x_{0} + δ_{n}) − f((s − a)^{α} + x_{0} + δ_{n}) \\ −f((t − a)^{α} + x_{0}) + f((s − a)^{α} + x_{0})| & ≤ ε |t − s|^{α} \end{matrix}

for all t, s ∈ [a, b] and all n ≥ n_{0}. Substituting here t = a + ξ_{m}^{1/α} with m ∈ ℕ so large that t ∈ [a, b], and s = a, we get

|\frac{f(x_{0} + δ_{n} + ξ_{m}) − f(x_{0} + δ_{n})}{ξ_{m}} − \frac{f(x_{0} + ξ_{m}) − f(x_{0})}{ξ_{m}} | ≤ ε

for all n ≥ n_{0} and m sufficiently large. If in this inequality m → ∞ we arrive at

|f′(x_{0} + δ_{n}) − L_{1}(x_{0})| ≤ ε  (n ≥ n_{0}).  (2.5)

If ξ_{n} < 0 for all n, the same arguments but with u_{n} defined by u_{n}(t) = −(b−t)^{α}+x_{0}+δ_{n} and later t = b − (−ξ_{m})^{1/α}, s = b lead to the same estimate (2.5). Completely in the same way one can show that also |f′(x_{0} + δ_{n}) − L_{2}(x_{0})| ≤ ε for all n ≥ n_{0}. So we have

L_{2}(x_{0}) − L_{1}(x_{0}) ≤ |L_{2}(x_{0}) − f′(x_{0} + δ_{n})| + |f′(x_{0} + δ_{n}) − L_{1}(x_{0})|
≤ 2ε
< L_{2}(x_{0}) − L_{1}(x_{0})

for all n ≥ n_{0} which is a contradiction. Consequently, the generating function f has at each point x ∈ ℝ a finite derivative. Because of (2.5) can now be verified for any x_{0} ∈ ℝ, any zero sequence (δ_{n}) ⊂ ℝ, and any ε > 0, the continuity of f′ at x_{0} is also proved, i.e. f ∈ C^{1}(ℝ) as claimed

In the proof of the next two theorems we make use of the following lemma. Its proof is quite elementary and hence omitted (see [8]).

Lemma 2.2. Let h ∈ C([a, b] × [0, 1]) and α ∈ (0, 1] be given. If there exists a positive constant L such that

|h(t, τ) − h(s, τ)| ≤ L |t − s|^{α} (t, s ∈ [a, b], τ ∈ [0, 1]),

then for the parameter integral g(t) = ∫_{0}^{1} h(t, τ) dτ (t ∈ [a, b])

g ∈ H^{α}[a, b] with h_{α}(g) ≤ L  and  ∥g∥_{H^{α}[a,b]} ≤ ∫_{0}^{1} ∥h(·, τ)∥_{H^{α}[a,b]} dτ.

Theorem 2.3. If the autonomous Nemytskij operator F generated by f maps the space H^{α}[a, b] into itself, then:

(a) F is locally Lipschitz continuous if and only if f ∈ Lip_{loc}^{1}(ℝ).

(b) F is (globally) Lipschitz continuous if and only if there are two constants c_{1}, c_{2} ∈ ℝ such that f(x) = c_{1}x + c_{2} for all x ∈ ℝ.

Proof. (a)/1. Let f ∈ Lip_{loc}^{1}(ℝ) and let F_{1} denote the Nemytskij operator generated by the derivative f′. In virtue of Theorem 2.1 and Corollary 2.1 F_{1} is a bounded operator acting in H^{α}[a, b]. Because of the Lagrange formula (2.4)

F u − F v = (u − v) ∫_{0}^{1} F_{1}(v + τ(u − v)) dτ (u, v ∈ H^{α}[a, b])

holds and by means of Lemma 2.2 and Lemma 2.1/(a) we get the estimation

∥F u − F v∥_{H^{α}[a,b]} ≤ ∥u − v∥_{H^{α}[a,b]} ∫_{0}^{1} ∥F_{1}(v + τ(u − v))∥_{H^{α}[a,b]} dτ  (2.6)

for all u, v ∈ H^{α}[a, b] which gives the local Lipschitz continuity of F.

(a)/2. Assuming now F ∈ Lip_{loc}(H^{α}[a, b]), by Theorem 2.2 we have f ∈ C^{1}(ℝ). So it remains to show the local Lipschitz continuity of f′. Let x_{0} ∈ ℝ be fixed and u_{0} ∈ H^{α}[a, b] defined by u_{0}(t) = (t − a)^{α} + x_{0} (t ∈ [a, b]). By our assumption, we find two constants L_{0} > 0 and δ_{0} > 0 such that

∥F u − F v∥_{H^{α}[a,b]} ≤ L_{0}∥u − v∥_{H^{α}[a,b]} (u, v ∈ B(u_{0}, δ_{0}) ⊂ H^{α}[a, b]).

Clearly, for any x ∈ B(x_{0}, δ_{0}) ⊂ ℝ the function u_{x} defined by u_{x}(t) = (t − a)^{α} + x (t ∈ [a, b]) belongs to B(u_{0}, δ_{0}) ⊂ H^{α}[a, b]. So we get

∥F u_{x} − F u_{y}∥_{H^{α}[a,b]} ≤ L_{0}|x − y| (x, y ∈ B(x_{0}, δ_{0}) ⊂ ℝ)

and, consequently,

\frac{|f((t − a)^{α} + x) − f((t − a)^{α} + y) − f((s − a)^{α} + x) + f((s − a)^{α} + y)|}{|t − s|^{α}} ≤ L_{0} |x − y|

for all t, s ∈ [a, b] and all x, y ∈ B(x_{0}, δ_{0}) ⊂ ℝ. For s = a we obtain eventually

|\frac{f(x + (t − a)^{α}) − f(x)}{(t − a)^{α}} − \frac{f(y + (t − a)^{α}) − f(y)}{(t − a)^{α}} | ≤ L_{0}|x − y|

for all t ∈ [a, b] and all x, y ∈ B(x_{0}, δ_{0}) ⊂ ℝ. Letting t → a we come to

|f′(x) − f′(y)| ≤ L_{0}|x − y| (x, y ∈ B(x_{0}, δ_{0}) ⊂ ℝ)

which proves the desired property.

(b) Here we have to prove only that the (global) Lipschitz continuity of F in H^{α}[a, b] implies the linearity of the generating function f. In H^{α}[a, b] let us take the norm ∥u∥_{H^{α}[a,b]} = |u(a)| + h_{α}(u) equivalent to the norm given in (1.1). Let τ, σ ∈ [a, b] with τ > σ and x, y, ξ, η ∈ ℝ be fixed. Consider two functions u, v ∈ H^{α}[a, b] defined by

u(t) = \{ \begin{matrix}x & if a ≤ t ≤ σ \\ \frac{ξ − x}{τ − σ} (t − σ) + x & if σ < t < τ \\ ξ & if τ ≤ t ≤ b \end{matrix}

and

v(t) = \{ \begin{matrix}y & if a ≤ t ≤ σ \\ \frac{η − y}{τ − σ} (t − σ) + y & if σ < t < τ \\ η & if τ ≤ t ≤ b . \end{matrix}

Then

∥u − v∥_{H^{α}[a,b]} = |x − y| + \frac{|ξ − η − x + y|}{|τ − σ|^{α}}.

Since F is Lipschitz continuous in H^{α}[a, b], there exists a constant L > 0 such that

h_{α}(f ∘ u − f ∘ v) ≤ L∥u − v∥_{H^{α}[a,b]} (u, v ∈ H^{α}[a, b]).

Substituting here the functions u, v defined above we get

|f(x) − f(y) − f(ξ) + f(η)| ≤ L(|x − y| |τ − σ|^{α} + |ξ − η − x + y|)

which, for y = x + h and η = ξ + h with h ∈ ℝ different from zero, results in

|\frac{f(x + h) − f(x)}{h} − \frac{f(ξ + h) − f(ξ)}{h}| ≤ L |τ − σ|^{α}.

Since by Theorem 2.2 we know that f ∈ C^{1}(ℝ), we obtain |f′(x) − f′(ξ)| ≤ L |τ − σ|^{α}. Therefore f′(x) = f′(ξ) for all x, ξ ∈ ℝ, which implies our assertion Finally, we give a necessary and sufficient differentiability criterion.

Theorem 2.4. The autonomous Nemytskij operator F generated by f is continuously Fréchet differentiable in H^{α}[a, b] if and only if f ∈ C^{2}(ℝ), where its Fréchet derivative F′(u) at u ∈ H^{α}[a, b] is given by

F′(u)v = (f′ ∘ u)v  (2.7)

for all v ∈ H^{α}[a, b].

Proof. 1. First assume f ∈ C^{2}(ℝ). Again let F_{1} denote the Nemytskij operator generated by f′. Due to Theorem 2.2, we know that F_{1} ∈ C(H^{α}[a, b]). Like (2.6) one can verify that

∥F (u + v) − F u − F_{1}(u)v∥_{H^{α}[a,b]} ≤ ∥v∥_{H^{α}[a,b]} ∫_{0}^{1} ∥F_{1}(u + τv) − F_{1}u∥_{H^{α}[a,b]}dτ (2.8)

for all u, v ∈ H^{α}[a, b]. Let now u ∈ H^{α}[a, b] be fixed. Then for any ε > 0 there is δ > 0 such that

∥F_{1}(u + τv) − F_{1}u∥_{H^{α}[a,b]} < ε

for all v ∈ B(0, δ) ⊂ H^{α}[a, b] and all τ ∈ [0, 1]). Hence, from (2.8)

∥F (u + v) − F u − F_{1}(u)v∥_{H^{α}[a,b]} ≤ ε∥v∥_{H^{α}[a,b]}

for all v ∈ B(0, δ) ⊂ H^{α}[a, b]) follows. Since evidently the map v ↦ F_{1}(u)v is linear and bounded in H^{α}[a, b], we see that the operator F is Fréchet differentiable at u ∈ H^{α}[a, b], and its Fréchet derivative at u is given by F_{1}u, i.e. relation (2.7) holds. Furthermore, we have

∥F′(u) − F′(v)∥_{L(H^{α}[a,b])} ≤ ∥F_{1}u − F_{1}v∥_{H^{α}[a,b]} (v ∈ H^{α}[a, b]),

showing the continuity of F′ : H^{α}[a, b] → L(H^{α}[a, b]) at u ∈ H^{α}[a, b]. So we have proved that F ∈ C^{1}(H^{α}[a, b]).

2. Now we show that F ∈ C^{1}(H^{α}[a, b]) implies f ∈ C^{2}(ℝ) and its Fréchet derivative has the form (2.7). First, because of Theorem 2.2, we know that f ∈ C^{1}(ℝ). To verify (2.7) we fix an arbitrary u ∈ H^{α}[a, b]. Then for any ε > 0 there is δ > 0 such that

∥F (u + τv) − F u − τF′(u)v∥_{H^{α}[a,b]} ≤ ε|τ| ∥v∥_{H^{α}[a,b]}

and, in particular,

|\frac{f(u(t) + τv(t)) − f(u(t))}{τ} − (F′(u)v) (t)| ≤ ε∥v∥_{H^{α}[a,b]}

for all v ∈ H^{α}[a, b], 0 ≠ τ ∈ ℝ with |τ|∥v∥_{H^{α}[a,b]} ≤ δ and all t ∈ [a, b]. If in the last inequality τ → 0, we obtain

|f′(u(t))v(t) − (F′(u)v)(t)| ≤ ε ∥v∥_{H^{α}[a,b]}

for all v ∈ H^{α}[a, b] and all t ∈ [a, b] which implies the validity of (2.7). Putting there v = 1 we get f′ ∘ u ∈ H^{α}[a, b] for all u ∈ H^{α}[a, b]. Furthermore,

\begin{matrix}∥f′ ∘ u − f′ ∘ v∥_{H^{α}[a,b]} & = ∥F′(u)1 − F′(v)1∥_{H^{α}[a,b]} \\ & ≤ ∥F′(u) − F′(v)∥_{L(H^{α}[a,b])} \end{matrix} (u, v ∈ H^{α}[a, b]).

This means the Nemytskij operator generated by the function f′ : ℝ → ℝ maps H^{α}[a, b] continuously into itself. Again in virtue of Theorem 2.2 f′ ∈ C^{1}(ℝ) which completes the proof

We want to mention that particularly for Theorem 2.2 and Theorem 2.4 (but also for Theorem 2.3/(a)) there are well-known criteria even for generalized Hölder spaces (see J. Appell and P. P. Zabrejko [2: Chapter 7]). Our proofs are independent of these somehow sophisticated criteria. The sufficient parts of the theorems just mentioned were proved in M. Goebel [8], their necessary parts seem to be new. However, compare P. Drábek [7] concerning Theorem 2.2, and R. Chiappinelli and R. Nugari [6] for the case of non-autonomous Nemytskij operators. Theorem 2.3/(b) was first proved in J. Matkowski [15] for α = 1, in A. Matkowska [13] for α ∈ (0, 1) and rediscovered in J. Appell, E. De Pascale and P. P. Zabrejko [1]. Further results concerning globally Lipschitzian Nemytskij operators in different function spaces can be found in [14, 16, 17]. Locally Lipschitz continuous Nemytskij operators in spaces of bounded α-Hölder continuous functions defined on ℝ were investigated by E. P. Sobolevskij [23, 24].

3. Some auxiliary results

The study of F u = f ∘u in H^{k+α}[a, b], (k, α) ∈ ℕ×(0, 1], first raises the question whether or not f ∘ u belongs to H^{k+α}[a, b] if u is an element of the same space. In other words, we need for the k-th derivative of the composed function f ∘ u a convenient formula, by means of which we may easily answer this and other questions. Such a formula is given below in Lemma 3.3 showing that we have to study also certain nonlinear differential operators. At first, however, we give in Lemma 3.1 some further properties of Hölder and Lipschitz continuous functions required in the subsequent sections.

Lemma 3.1.

(a) The inclusion C^{1}[a, b] ⊂ Lip[a, b] holds, where h_{1}(u) = ∥u′∥_{C[a,b]} for all u ∈ C^{1}[a, b].

(b) For any pairs (k, α), (l, β) ∈ ℤ_{+} × (0, 1] with l + β < k + α the space H^{k+α}[a, b] is continuously embedded into H^{l+β}[a, b].

(c) There exists a constant c > 0 such that ∥D^{i}u∥_{H^{α}[a,b]} ≤ c∥u∥_{H^{k+α}[a,b]} for each u ∈ H^{k+α}[a, b] and i ∈ {0, 1, …, k}.

(d) For any u, v ∈ H^{k+α}[a, b] we have uv ∈ H^{k+α}[a, b], and there exists a constant c > 0 such that ∥uv∥_{H^{k+α}[a,b]} ≤ c∥u∥_{H^{k+α}[a,b]}∥v∥_{H^{k+α}[a,b]} for all u, v ∈ H^{k+α}[a, b].

Proof. (a) Due to the mean value theorem, for any u ∈ C^{1}[a, b] and any s, t ∈ [a, b] with s ≠ t there is some τ ∈ [a, b] such that |u(t) − u(s)| = |u′(τ)| |t − s|. Hence we have u ∈ Lip[a, b] with h_{1}(u) ≤ ∥u′∥_{C[a,b]}. To show the converse inequality let t_{0} ∈ [a, b] be such that |u′(t_{0})| = ∥u′∥_{C[a,b]}. Then

h_{1}(u) ≥ \frac{|u(t_{0}) − u(t)|}{|t_{0} − t|} (t ∈ [a, b], t ≠ t_{0}),

and if t → t_{0}, we get h_{1}(u) ≥ |u′(t_{0})| which proves the wanted equality.

(b) If l = k, then β < α and the statement follows via Lemma 2.1/(b). If l + 1 ≤ k, the inclusion H^{k+α}[a, b] ⊂ H^{l+β}[a, b] is obvious, and because of Lemma 2.1/(b) and property (a) for any u ∈ H^{k+α}[a, b] we have

∥u∥_{H^{l+β}[a,b]} = ∑_{i=0}^{l} ∥D^{i}u∥_{C[a,b]} + h_{β}(D^{l}u) ≤ c ∑_{i=0}^{l+1} ∥D^{i}u∥_{C[a,b]} ≤ c∥u∥_{H^{k+α}[a,b]}

showing the continuity of this embedding.

(c) Let u ∈ H^{k+α}[a, b] be fixed. Because of (1.2) the statement is evident if i = k. In the case of i < k we use Lemma 2.1/(b) and property (a) proved above to get

\begin{aligned}∥D^{i}u∥_{H^{α}[a,b]} & ≤ ∥D^{i}u∥_{C[a,b]} + (b − a)^{1−α}h_{1}(D^{i}u) \\ & = ∥D^{i}u∥_{C[a,b]} + (b − a)^{1−α}∥D^{i+1}u∥_{C[a,b]} \end{aligned}

from which the statement follows.

(d) If u, v ∈ H^{k+α}[a, b], then D^{k−j}u, D^{j}v ∈ H^{α}[a, b] (j = 0, …, k) and due to the Leibniz formula

D^{k}(uv) = ∑_{j=0}^{k} \binom{k}{j}D^{k−j}uD^{j}v  (3.1)

and Lemma 2.1/(a) we get D^{k}(uv) ∈ H^{α}[a, b], which means uv ∈ H^{k+α}[a, b]. Furthermore, we have

∥uv∥_{H^{k+α}[a,b]} ≤ ∥uv∥_{H^{α}[a,b]} + ∑_{i=1}^{k} ∥D^{i}(uv)∥_{H^{α}[a,b]}.

Taking advantage of (3.1), Lemma 2.1/(a) and statement (c) just proved we obtain

\begin{aligned}∥uv∥_{H^{k+α}[a,b]} & ≤ ∥uv∥_{H^{α}[a,b]} + ∑_{i=1}^{k} ∑_{j=0}^{i} \binom{i}{j}∥D^{i−j}uD^{j}v)∥_{H^{α}[a,b]} \\ & ≤ c(1 + ∑_{i=1}^{k} ∑_{j=0}^{i} \binom{i}{j})∥u∥_{H^{k+α}[a,b]}∥v∥_{H^{k+α}[a,b]} \end{aligned}

where the constant c is the same as in statement (c)

12 M. Goebel and F. Sachweh

Let P_{k} (k ∈ ℕ) be the set of all differential operators P defined by

P u = p(u′, …, u^{(k)}) (u ∈ C^{k}[a, b])

where p ∈ C(ℝ^{k}) is a polynomial of the form

p(x_{1}, …, x_{k}) = ∑_{i=1}^{m} c_{i}( ∏_{j=1}^{k} x_{j}^{a_{ij}})  ((x_{1}, …, x_{k}) ∈ ℝ^{k})  (3.2)

with m, c_{i} ∈ ℕ, a_{ij} ∈ ℤ_{+} and a_{i_{0}k} > 0 for at least one index i_{0}.

We consider P ∈ P_{k} as a map defined on H^{k+α}[a, b]. To prepare the most important Lemmas 3.3 - 3.5 we prove firstly the following one.

Lemma 3.2. For any P ∈ P_{k} there exists Q ∈ P_{k+1} such that DP u = Qu for all u ∈ C^{k+1}[a, b].

Proof. Firstly, for any k ∈ ℕ let a_{1}, …, a_{k−1} ∈ ℤ_{+} and a_{k} ∈ ℕ be given. Using induction it can be shown that there exists Q ∈ P_{k+1} such that

D( ∏_{j=1}^{k}(D^{j}u)^{a_{j}}) = Qu (u ∈ C^{k+1}[a, b]).

Then for an arbitrary P ∈ P_{k} generated by p ∈ C(ℝ^{k}) as given in (3.2) we have

DP u = ∑_{i=1}^{m} c_{i}D( ∏_{j=1}^{k}(D^{j}u)^{a_{ij}})  (u ∈ C^{k+1}[a, b])  (3.3)

where a_{i_{0}k} > 0 for at least one index i_{0}. Since there exists Q_{i_{0}} ∈ P_{k+1} with

D( ∏_{j=1}^{k}(D^{j}u)^{a_{i_{0}j}}) = Q_{i_{0}}u (u ∈ C^{k+1}[a, b])

the desired statement follows

Lemma 3.3. For each f ∈ C^{k}(ℝ) there are k operators P_{i} ∈ P_{k−i+1} (i = 1, …, k) such that

D^{k}(f ∘ u) = ∑_{i=1}^{k}(f^{(i)} ∘ u) P_{i}u  (3.4)

for all u ∈ C^{k}[a, b].

Proof. The assertion will be proved by induction. If f ∈ C^{1}(ℝ) and u ∈ C^{1}[a, b], formula (3.4) is true because of D(f ∘ u) = (f′ ∘ u)P_{1}u with P_{1} ∈ P_{1} generated by p_{1}(x_{1}) = x_{1}. Assume for k ∈ ℕ the statement to be true with certain P̃_{i} ∈ P_{k−i+1} (i = 1, …, k). Then in virtue of Lemma 3.2 there are Q̃_{i} ∈ P_{k−i+2} with DP̃_{i}u = Q̃_{i}u for all u ∈ C^{k−i+2}[a, b] (i = 1, …, k). If now f ∈ C^{k+1}(ℝ) and u ∈ C^{k+1}[a, b], then

D^{k+1}(f ∘ u) = D(D^{k}(f ∘ u))

= ∑_{i=1}^{k} D[(f^{(i)} ∘ u)P̃_{i}u]

= ∑_{i=1}^{k}(f^{(i+1)} ∘ u) u′P̃_{i}u + ∑_{i=1}^{k}(f^{(i)} ∘ u)Q̃_{i}u

= ∑_{i=1}^{k+1}(f^{(i)} ∘ u)P_{i}u

where in the last equation we have put

\begin{matrix}P_{1}u & = Q̃_{1}u \\ P_{i}u & = u′P̃_{i−1}u + Q̃_{i}u (i = 2, …, k) \\ P_{k+1}u & = u′P̃_{k}u. \end{matrix} \}

Since P_{i} ∈ P_{k−i+2} (i = 1, …, k + 1), the statement is proved for k + 1

Lemma 3.4. Each differential operator P ∈ P_{k} maps H^{k+α}[a, b] into H^{α}[a, b] and is locally Lipschitz continuous.

Proof. By the definition of P ∈ P_{k} it is evident that P u ∈ H^{α}[a, b] for each u ∈ H^{k+α}[a, b]. So if u_{0} ∈ H^{k+α}[a, b] and δ > 0 are fixed, the statement follows after showing that there is another constant c = c(u_{0}, δ) > 0 such that

∥P u − P v∥_{H^{α}[a,b]} ≤ c∥u − v∥_{H^{k+α}[a,b]}  (3.5)

for all u, v ∈ B(u_{0}, δ) ⊂ H^{k+α}[a, b]. Let P be generated by p ∈ C(ℝ^{k}) as given in (3.2). Then using the identity

∏_{j=1}^{k} x_{j} − ∏_{j=1}^{k} y_{j} = ∑_{j=1}^{k} [( ∏_{l=1}^{j−1} x_{l})(x_{j} − y_{j})( ∏_{l=j+1}^{k} y_{l})] (x_{j}, y_{j} ∈ ℝ, k ∈ ℕ)

and Lemma 2.1/(a), for arbitrary u, v ∈ H^{k+α}[a, b] we get

\begin{aligned}∥P u − P v∥_{H^{α}[a,b]} & = ∥ ∑_{i=1}^{m} c_{i}[ ∏_{j=1}^{k}(D^{j}u)^{a_{ij}} − ∏_{j=1}^{k}(D^{j}v)^{a_{ij}}]∥_{H^{α}[a,b]} \\ & ≤ c ∑_{i=1}^{m} ∑_{j=1}^{k} [( ∏_{l=1}^{j−1} ∥D^{l}u∥_{H^{α}[a,b]}^{a_{il}})∥(D^{j}u)^{a_{ij}} \\ & − (D^{j}v)^{a_{ij}} ∥_{H^{α}[a,b]}( ∏_{l=j+1}^{k} ∥D^{l}v∥_{H^{α}[a,b]}^{a_{il}})]. \end{aligned}  (3.6)

Because of Lemma 3.1/(c) we have for all u ∈ B(u_{0}, δ) ⊂ H^{k+α}[a, b]

∥D^{l}u∥_{H^{α}[a,b]} ≤ c and ∥(D^{j}u)^{a_{ij}} − (D^{j}v)^{a_{ij}} ∥_{H^{α}[a,b]} ≤ c∥u − v∥_{H^{k+α}[a,b]}

where in proving the second estimate we may take advantage of the elementary identity

x^{k} − y^{k} = (x − y) ∑_{i=0}^{k−1} x^{k−i−1}y^{i} (x, y ∈ ℝ, k ∈ ℕ).

Combining the last two estimates with (3.6) we obtain the desired property (3.5)

Lemma 3.5. If i ∈ {1, …, k − 1} is fixed, then each differential operator P ∈ P_{k−i} maps H^{k+α}[a, b] into H^{1+α}[a, b] and is locally Lipschitz continuous.

Proof. Let i ∈ {1, …, k − 1} and P ∈ P_{k−i} be fixed. By the definition of the class P_{k−i}, P u ∈ H^{1+α}[a, b] for each u ∈ H^{k+α}[a, b]. In virtue of Lemma 3.2 there exists another differential operator Q ∈ P_{k−i+1} with DP u = Qu for all u ∈ C^{k−i+1}[a, b]. Thus we have

\begin{aligned}∥P u − P v∥_{H^{1+α}[a,b]} & ≤ ∥P u − P v∥_{H^{α}[a,b]} + ∥D(P u − P v)∥_{H^{α}[a,b]} \\ & = ∥P u − P v∥_{H^{α}[a,b]} + ∥Qu − Qv∥_{H^{α}[a,b]} \end{aligned}

for all u, v ∈ H^{k+α}[a, b]. Applying Lemma 3.4 to P and Q for any u_{0} ∈ H^{k+α}[a, b] and any δ > 0 we find some constants c > 0 and δ > 0 such that, for all u, v ∈ B(u_{0}, δ) ⊂ H^{k+α}[a, b],

\begin{aligned}∥P u − P v∥_{H^{α}[a,b]} & ≤ c∥u − v∥_{H^{k−i+α}[a,b]} \\ ∥Qu − Qv∥_{H^{α}[a,b]} & ≤ c∥u − v∥_{H^{k−i+1+α}[a,b]}. \end{aligned}

Observe that M = B(u_{0}, δ) ⊂ H^{k+α}[a, b] implies M ⊂ B(u_{0}, δ) ⊂ H^{k−j+α}[a, b] for any j = 1, …, k − 1. From the given estimates the desired local Lipschitz continuity of P : H^{k+α}[a, b] → H^{1+α}[a, b] immediately follows

Lemma 3.3 shows that D^{k}(f ∘ u) can be written as a sum of products, where in the products the term f^{(i)} ∘ u is one factor and in the other u^{(k−i+1)} is the highest derivative of u which occurs. Taking into account these auxiliary results we are now in a position to prove new properties of the Nemytskij operator in Lip^{k}[a, b] and H^{k+α}[a, b], respectively.

4. The Nemytskij operator in Lip^{k}[a, b]

In this section we consider the autonomous Nemytskij operator F in Lip^{k}[a, b] with k ∈ ℕ. The theorems below are analogous to those given in Section 2 for the space H^{α}[a, b] with α ∈ (0, 1]. Since H^{1}[a, b] = Lip[a, b] = Lip^{0}[a, b], the statements are also true for k = 0.

Theorem 4.1. The autonomous Nemytskij operator F generated by f maps the space Lip^{k}[a, b] into itself if and only if f ∈ Lip_{loc}^{k}(ℝ).

Proof. 1. We show the sufficiency of f ∈ Lip_{loc}^{k}(ℝ). Let u ∈ Lip^{k}[a, b] be given. Obviously, f ∘ u ∈ C^{k}[a, b] and relation (3.4) holds with certain P_{i} ∈ P_{k−i+1}. Since for i = 1, …, k we have

f^{(i)} ∈ Lip_{loc}(ℝ)  and  Lip^{k}[a, b] ⊂ Lip^{k−i+1}[a, b] ⊂ Lip[a, b]

from Theorem 2.1 we get f^{(i)} ∘ u ∈ Lip[a, b] and from Lemma 3.4 also P_{i}u ∈ Lip[a, b]. Hence by formula (3.4) D^{k}(f ∘ u) ∈ Lip[a, b] follows which means f ∘ u ∈ Lip^{k}[a, b].

2. Let now f : ℝ → ℝ be such that f ∘ u ∈ Lip^{k}[a, b] for all u ∈ Lip^{k}[a, b]. For arbitrary x_{0} ∈ ℝ and δ = x_{0} − a − \frac{b−a}{2} the linear function u_{0} defined by u_{0}(t) = t + δ (t ∈ [a, b]) is an element of Lip^{k}[a, b] and therefore also (f ∘ u_{0})(t) = f(t + δ). Hence f(·+δ) ∈ C^{k}[a, b], that means f ∈ C^{k}[x_{0}− \frac{b−a}{2}, x_{0}+ \frac{b−a}{2}]. Moreover, D^{k}(f ∘u_{0}) = f^{(k)} ∘ u_{0} ∈ Lip[a, b], i.e. there exists a constant L_{0} > 0 such that

|f^{(k)}(u_{0}(t)) − f^{(k)}(u_{0}(s))| ≤ L_{0}|t − s| = L_{0}|u_{0}(t) − u_{0}(s)| (t, s ∈ [a, b])

which implies |f^{(k)}(x) − f^{(k)}(y)| ≤ L_{0}|x − y| for all x, y ∈ B(x_{0}, \frac{b−a}{2}) ⊂ ℝ. So we have f ∈ Lip^{k}[x_{0} − \frac{b−a}{2}, x_{0} + \frac{b−a}{2}] and consequently f ∈ Lip_{loc}^{k}(ℝ)

Corollary 4.1. Each autonomous Nemytskij operator F acting in Lip^{k}[a, b] is bounded.

Theorem 4.2. The autonomous Nemytskij operator F generated by f maps the space Lip^{k}[a, b] continuously into itself if and only if f ∈ C^{k+1}(ℝ).

Proof. 1. Firstly we assume f ∈ C^{k+1}(ℝ). Because of C^{k+1}(ℝ) ⊂ Lip_{loc}^{k}(ℝ) and Theorem 4.1, the Nemytskij operator F generated by f maps Lip^{k}[a, b] into itself. To show its continuity let the sequence (u_{n}) ⊂ Lip^{k}[a, b] converge to u ∈ Lip^{k}[a, b]. By Lemma 3.1/(b) this implies

u_{n} ⟶ u  in Lip^{i}[a, b] for all i = 0, …, k − 1.  (4.1)

Applying Lemma 3.3 we find

\begin{aligned}∥F u_{n} − F u∥_{Lip^{k}[a,b]} & ≤ ∥f ∘ u_{n} − f ∘ u∥_{Lip[a,b]} \\ & + ∑_{i=1}^{k} ∥D^{i}(f ∘ u_{n}) − D^{i}(f ∘ u)∥_{Lip[a,b]} \\ & ≤ ∥f ∘ u_{n} − f ∘ u∥_{Lip[a,b]} \\ & + ∑_{i=1}^{k} ∑_{j=1}^{i} ∥(f^{(j)} ∘ u_{n})P_{ij}u_{n} − (f^{(j)} ∘ u)P_{ij}u∥_{Lip[a,b]} \end{aligned} (4.2)

with certain P_{ij} ∈ P_{i−j+1}. Since by assumption f^{(j)} ∈ C^{1}(ℝ) (j = 0, …, k), Theorem 2.2 implies f^{(j)} ∘ u_{n} → f^{(j)} ∘ u in Lip[a, b], and since in virtue of (4.1) and Lemma 3.4 P_{ij}u_{n} → P_{ij}u in Lip[a, b], we obtain F u_{n} → F u in Lip^{k}[a, b] which proves the continuity of F at the point u ∈ Lip^{k}[a, b].

2. Let now F ∈ C(Lip^{k}[a, b]). By Theorem 4.1 we have f ∈ Lip_{loc}^{k}(ℝ) which implies D^{k}f ∈ Lip_{loc}(ℝ), and it remains to show that D^{k}f ∈ C^{1}(ℝ). Like in the second part of the proof to Theorem 2.2, for each x_{0} ∈ ℝ there exists a sequence (x_{n}) ⊂ ℝ converging to x_{0} and f^{(k)} differentiable at each x_{n}. Consider the functions u_{n} ∈ Lip^{k}[a, b] for each n ∈ ℤ_{+} defined by u_{n}(t) = t − ξ + x_{n} (t ∈ [a, b], ξ ∈ (a, b)). Evidently, D^{k}(f ∘ u_{n}) = f^{(k)} ∘ u_{n} and u_{n} → u_{0} in Lip^{k}[a, b] if n → ∞. Since F is continuous at u_{0}, for arbitrary ε > 0 there is an n_{0} = n_{0}(ε) ∈ ℕ with ∥F u_{n} − F u_{0}∥_{Lip^{k}[a,b]} ≤ ε for all n ≥ n_{0}, and therefore

\frac{|f^{(k)}(u_{n}(t)) − f^{(k)}(u_{0}(t)) − f^{(k)}(u_{n}(s)) + f^{(k)}(u_{0}(s))|}{|t − s|} ≤ ε

for all t, s ∈ [a, b] and all n ≥ n_{0}. In particular, for s = ξ we obtain

|\frac{f^{(k)}(t − ξ + x_{n}) − f^{(k)}(x_{n})}{t − ξ} − \frac{f^{(k)}(t − ξ + x_{0}) − f^{(k)}(x_{0})}{t − ξ}| ≤ ε  (4.3)

for all t ∈ [a, b] and all n ≥ n_{0}. Supposing f^{(k)} to be not differentiable at x_{0} we can find two zero sequences (τ_{n}), (θ_{n}) ⊂ ℝ and two reell numbers L_{1}, L_{2} such that

L_{1} = lim_{m→∞} \frac{f^{(k)}(x_{0} + τ_{m}) − f^{(k)}(x_{0})}{τ_{m}} < lim_{m→∞} \frac{f^{(k)}(x_{0} + θ_{m}) − f^{(k)}(x_{0})}{θ_{m}} = L_{2}.

If in estimate (4.3) we put t = ξ + τ_{m}, m sufficiently large, and if afterwards m → ∞, we find

|f^{(k+1)}(x_{n}) − L_{1}| ≤ ε  (n ≥ n_{0})  (4.4)

and in the same way |f^{(k+1)}(x_{n})−L_{2}| ≤ ε for all n ≥ n_{0}. Thus we come to |L_{1}−L_{2}| ≤ ε, which contradicts L_{1} < L_{2}. Hence f^{(k)} is differentiable at any point x_{0} ∈ ℝ. The continuity of f^{(k+1)} follows from (4.4) (see the proof of Theorem 2.2)

Theorem 4.3. The autonomous Nemytskij operator F generated by f is a locally Lipschitz continuous map from Lip^{k}[a, b] into itself if and only if f ∈ Lip_{loc}^{k+1}(ℝ).

Proof. 1. Firstly let us assume f ∈ Lip_{loc}^{k+1}(ℝ). By formula (4.2), with u_{n} replaced by v, for all u, v ∈ Lip^{k}[a, b]

∥F u − F v∥_{Lip^{k}[a,b]} ≤ ∥f ∘ u − f ∘ v∥_{Lip[a,b]}

+ ∑_{i=1}^{k} ∑_{j=1}^{i} [∥P_{ij}v∥_{Lip[a,b]}∥f^{(j)} ∘ u − f^{(j)} ∘ v∥_{Lip[a,b]}

+ ∥f^{(j)} ∘ u∥_{Lip[a,b]}∥P_{ij}u − P_{ij}v∥_{Lip[a,b]}]

holds where P_{ij} ∈ P_{i−j+1}. The desired local Lipschitz continuity of F : Lip^{k}[a, b] → Lip^{k}[a, b] follows now from Theorem 2.3/(a) applied to the Nemytskij operators F, F_{1}, …, F_{k} generated by f, f′, …, f^{(k)}, respectively, Lemma 3.4 and finally Lemma 3.1/(b).

2. If, on the other hand, we assume F : Lip^{k}[a, b] → Lip^{k}[a, b] to be locally Lipschitz continuous, then due to Theorem 4.2 the generating function f belongs to C^{k+1}(ℝ). So we have to show only that f^{(k+1)} is locally Lipschitz continuous on ℝ. To prove this for any x ∈ ℝ we define u_{x} by u_{x}(t) = t − a + x (t ∈ [a, b]). In particular, for fixed x_{0} ∈ ℝ we set u_{0} = u_{x_{0}}. Then ∥u_{x} − u_{y}∥_{Lip^{k}[a,b]} = |x − y| for all x, y ∈ ℝ and hence, for any δ > 0, u_{x} ∈ B(u_{0}, δ) ⊂ Lip^{k}[a, b] provided |x − x_{0}| < δ. So by our assumption there are two constants L_{0} > 0 and δ_{0} > 0 such that ∥F u_{x} − F u_{y}∥_{Lip^{k}[a,b]} ≤ L_{0}|x − y| for all x, y ∈ B(x_{0}, δ_{0}) ⊂ ℝ which because of D^{k}(f ∘ u_{x}) = f^{(k)} ∘ u_{x} implies

|\frac{f^{(k)}(t − a + x) − f^{(k)}(t − a + y) − f^{(k)}(s − a + x) + f^{(k)}(s − a + y)}{t − s}| ≤ L_{0}|x − y|

for all t, s ∈ [a, b] and all x, y ∈ B(x_{0}, δ_{0}) ⊂ ℝ. For s = a this estimate gives

|\frac{f^{(k)}(x + t − a) − f^{(k)}(x)}{t − a} − \frac{f^{(k)}(y + t − a) − f^{(k)}(y)}{t − a}| ≤ L_{0}|x − y|

for all t ∈ [a, b] and all x, y ∈ B(x_{0}, δ_{0}) ⊂ ℝ from which, if t → a, we get |f^{(k+1)}(x) − f^{(k+1)}(y)| ≤ L_{0}|x − y| for all x, y ∈ B(x_{0}, δ_{0}) ⊂ ℝ. This shows the claimed local Lipschitz continuity of f^{(k+1)} on ℝ

Theorem 4.4. The autonomous Nemytskij operator F generated by f is continuously Fréchet differentiable in Lip^{k}[a, b] if and only if f ∈ C^{k+2}(ℝ). Its Fréchet derivative F′(u) at u ∈ Lip^{k}[a, b] is given by

F′(u)v = (f′ ∘ u)v  (4.5)

for all v ∈ Lip^{k}[a, b]

Proof. That F ∈ C^{1}(Lip^{k}[a, b]) implies f ∈ C^{k+2}(ℝ) and the validity of (4.5) can be shown in the same way as in the proof of Theorem 2.4 for k = 0. So assuming conversely f ∈ C^{k+2}(ℝ) we are going to show F ∈ C^{1}(Lip^{k}[a, b]) and formula (4.5). Due to Theorem 4.2 the Nemytskij operator F_{1} defined by F_{1}u = f′ ∘ u (u ∈ Lip^{k}[a, b]) belongs to C(Lip^{k}[a, b]). Hence for any u, v ∈ Lip^{k}[a, b] the function φ(τ) = F_{1}(u + τv) maps continuously the interval [0, 1] into Lip^{k}[a, b] and thus is also integrable. Let u ∈ Lip^{k}[a, b] and ε > 0 be fixed. Then there exists some δ > 0 such that

∥∫_{0}^{1} (F_{1}(u + τv) − F_{1}u)dτ∥_{Lip^{k}[a,b]} ≤ ∫_{0}^{1} ∥F_{1}(u + τv) − F_{1}u∥_{Lip^{k}[a,b]}dτ < ε (4.6)

for all v ∈ B(0, δ) ⊂ Lip^{k}[a, b]. The classical Lagrange formula (2.4) yields

f(u(t) + v(t)) − f(u(t)) − f′(u(t))v(t) = v(t) ∫_{0}^{1} (f′(u(t) + τv(t)) − f′(u(t)))dτ

for all t ∈ [a, b] and all u, v ∈ Lip^{k}[a, b]. Since here the parameter integral is the same as the integral on the left-hand side of estimate (4.6), we can apply Lemma 3.1/(d). So taking advantage of (4.6) we get

\begin{aligned}∥F (u + v) − F u − F_{1}(u)v∥_{Lip^{k}[a,b]} & ≤ c∥v∥_{Lip^{k}[a,b]} ∥∫_{0}^{1} (F_{1}(u + τv) − F_{1}u) dτ∥_{Lip^{k}[a,b]} \\ & ≤ cε∥v∥_{Lip^{k}[a,b]} \end{aligned}

for all v ∈ B(0, δ) ⊂ Lip^{k}[a, b] which means the Fréchet derivative of F at u is given by F_{1}u = f′ ∘ u ∈ L(Lip^{k}[a, b]). Since its continuity is obvious the theorem is proved

These results are quite satisfactory and in a certain sense final. By no means the same can be said concerning those of the next section.

5. The Nemytskij operator in H^{k+α}[a, b]

Now we come to the most interesting case, namely to H^{k+α}[a, b] with (k, α) ∈ ℕ×(0, 1). Not very much papers are dealing with the Nemytskij operator in this general Hölder space (see J. Appell and P. P. Zabrejko [2: Section 8.3]). However, let us mention the introductory Section 1.3 of the monograph E. Wegert [26], which is devoted to nonautonomous Nemytskij operators acting in H^{k+α}(Ω) (Ω ⊂ ℝ^{n}), but also in C^{k}(Ω) and Sobolev spaces. Sufficient acting conditions combined with boundedness, continuity and differentiability conditions are given. Naturally, our sufficient conditions given below are sharper than those which one gets by specifying Wegert’s results.

Again we begin with an acting condition, which in this section will be the only necessary and sufficient condition. Note that Theorem 5.1 is not true for k = 0, however for α = 1 (compare with Theorem 2.1 and Theorem 4.1).

Theorem 5.1. The autonomous Nemytskij operator F generated by f maps the space H^{k+α}[a, b], (k, α) ∈ ℕ × (0, 1), into itself if and only if f ∈ H_{loc}^{k+α}(ℝ).

Proof. 1. Let first f ∈ H_{loc}^{k+α}(ℝ). Since for any i = 1, …, k and any u ∈ H^{k+α}[a, b] the derivative D^{i}(f ∘ u) belongs to C[a, b], it remains to show D^{k}(f ∘ u) ∈ H^{α}[a, b]. By Lemma 3.3

D^{k}(f ∘ u) = ∑_{i=1}^{k} (f^{(i)} ∘ u) P_{i}u  (u ∈ H^{k+α}[a, b])  (5.1)

holds with certain differential operators P_{i} ∈ P_{k−i+1}. Because of Lemma 3.1/(b) and Lemma 3.4 we have P_{i}u ∈ H^{α}[a, b] for i = 1, …, k, and because of Theorem 2.1 we have f^{(i)} ∘u ∈ H^{α}[a, b] for i = 1, …, k−1. Consider now f^{(k)} ∘u for any fixed u ∈ H^{k+α}[a, b]. Choosing r > 0 so large that |u(t)| < r for each t ∈ [a, b], we find a constant L_{r} > 0 such that

|f^{(k)}(x) − f^{(k)}(y)| ≤ L_{r}|x − y|^{α} (x, y ∈ [−r, r]),

hence, by the mean value theorem,

|f^{(k)}(u(t)) − f^{(k)}(u(s))| ≤ L_{r}|u(t) − u(s)|^{α} ≤ L_{r}∥u′∥_{C[a,b]}^{α}|t − s|^{α}

for all t, s ∈ [a, b]. Thus we also have f^{(k)} ∘ u ∈ H^{α}[a, b] and formula (5.1) yields D^{k}(f ∘ u) ∈ H^{α}[a, b] for each u ∈ H^{k+α}[a, b], which shows that F generated by f ∈ H_{loc}^{k+α}(ℝ) maps H^{k+α}[a, b] into itself.

2. The proof of the necessity of f ∈ H_{loc}^{k+α}(ℝ) is almost the same as in Theorem 4.1 (for α = 1) and therefore omitted

Corollary 5.1. Each autonomous Nemytskij operator F acting in H^{k+α}[a, b], (k, α) ∈ ℕ × (0, 1), is bounded.

We change over to continuity conditions for F in H^{k+α}[a, b]. Like before, continuity of F is not already implied by its acting in H^{k+α}[a, b]. As a simple illustrating example in H^{1+α}[0, 1], with α ∈ (0, 1), one may take the Nemytskij operator F generated by the function f defined by

f(x) = \{ \begin{matrix}\frac{x^{1+α}}{1+α} & if x > 0 \\ 0 & if x ≤ 0. \end{matrix}

Looking to what we have found concerning continuity in the foregoing sections, we could expect that the autonomous Nemytskij operator F generated by f is continuous in H^{k+α}[a, b] if and only if f ∈ C^{k+1}(ℝ). But in the following Theorem 5.2, which is an improvement of the corresponding theorem announced in [11], we have proved a sufficient condition weaker than the inclusion f ∈ V^{k+1}(ℝ). Note, on the other hand, that this sufficient continuity criterion seems to be only a very little touch more restrictive than the necessary condition from Theorem 5.1.

Theorem 5.2. The autonomous Nemytskij operator F generated by f maps the space H^{k+α}[a, b], (k, α) ∈ ℕ × (0, 1), continuously into itself if, for any r > 0, there exists β ∈ (α, 1] such that f ∈ H^{k+β}[−r, r].

Proof. Since the assumption implies f ∈ H_{loc}^{k+α}(ℝ), in virtue of Theorem 5.1 the related Nemytskij operator F maps H^{k+α}[a, b] into itself. At first we prove the continuity of this map in the case of k = 1. So let (u_{n}) be an arbitrary sequence in H^{1+α}[a, b] converging to u_{0} ∈ H^{1+α}[a, b]. We choose r ≥ 1 and n_{0} ∈ ℕ such that

∥u_{n}∥_{H^{1+α}[a,b]} ≤ r  and  ∥u_{n} − u_{0}∥_{H^{1+α}[a,b]} ≤ 1

for each n ≥ n_{0}. By assumption f ∈ Lip[−r, r] and f′ ∈ H^{β}[−r, r]. Let L and L_{β} denote the Lipschitz and Hölder constant, respectively. Then we have the estimate

\begin{matrix}∥F u_{n} − F u_{0}∥_{H^{1+α}[a,b]} \\ = ∥f ∘ u_{n} − f ∘ u_{0}∥_{C[a,b]} + ∥(f′ ∘ u_{n})u′_{n} − (f′ ∘ u_{0})u′_{0}∥_{H^{α}[a,b]} \\ ≤ L∥u_{n} − u_{0}∥_{C[a,b]} + ∥f′ ∘ u_{0}∥_{H^{α}[a,b]}∥u′_{n} − u′_{0}∥_{H^{α}[a,b]} \\ + r∥f′ ∘ u_{n} − f′ ∘ u_{0}∥_{H^{α}[a,b]} \\ ≤ c∥u_{n} − u_{0}∥_{H^{1+α}[a,b]} + rL_{β}∥u_{n} − u_{0}∥_{C[a,b]}^{β} + rh_{α}(f′ ∘ u_{n} − f′ ∘ u_{0}) \\ ≤ c∥u_{n} − u_{0}∥_{H^{1+α}[a,b]}^{β} + rh_{α}(f′ ∘ u_{n} − f′ ∘ u_{0}) \end{matrix}  (5.2)

for all n ≥ n_{0}. To show that the second term tends to zero provided n → ∞, for each n ≥ n_{0} we introduce Δ_{n}(t, s) defined by

Δ_{n}(t, s) = |f′(u_{n}(t)) − f′(u_{0}(t)) − f′(u_{n}(s)) + f′(u_{0}(s))| (t, s ∈ [a, b]).

Because of f′ ∈ H^{β}[−r, r] we have

\begin{matrix}Δ_{n}(t, s) & ≤ L_{β} (|u_{n}(t) − u_{0}(t)|^{β} + |u_{n}(s) − u_{0}(s)|^{β}) \\ & ≤ 2L_{β}∥u_{n} − u_{0}∥_{H^{1+α}[a,b]}^{β} \\ & ≤ 2r^{β}L_{β}∥u_{n} − u_{0}∥_{H^{1+α}[a,b]}^{β} \end{matrix}  (t, s ∈ [a, b]).  (5.3)

On the other hand, again using that f′ ∈ H^{β}[−r, r] and additionally Lemma 3.1/(a), we find

\begin{matrix}Δ_{n}(t, s) & ≤ L_{β}(|u_{n}(t) − u_{n}(s)|^{β} + |u_{0}(t) − u_{0}(s)|^{β}) \\ & ≤ L_{β}(∥u′_{n}∥_{C[a,b]}^{β} + ∥u′_{0}∥_{C[a,b]}^{β})|t − s|^{β} \\ & ≤ 2r^{β}L_{β}|t − s|^{β} \end{matrix}  (t, s ∈ [a, b]).  (5.4)

We put γ = αβ^{−1} which implies γ ∈ (0, 1), βγ = α, and β(1 − γ) = β − α > 0. By (5.3) and (5.4) we obtain

\begin{matrix}Δ_{n}(t, s) & = (Δ_{n}(t, s))^{1−γ}(Δ_{n}(t, s))^{γ} \\ & ≤ 2r^{β}L_{β}∥u_{n} − u_{0}∥_{H^{1+α}[a,b]}^{β−α}|t − s|^{α} \end{matrix} (t, s ∈ [a, b]).

Hence for all n ≥ n_{0} we have the estimate

h_{α}(f′ ∘ u_{n} − f′ ∘ u_{0}) ≤ 2r^{β}L_{β}∥u_{n} − u_{0}∥_{H^{1+α}[a,b]}^{β−α},

showing together with (5.2) that F u_{n} → F u_{0} in H^{1+α}[a, b] as claimed.

Consider now the general case k ≥ 2. Again let (u_{n}) be a sequence in H^{k+α}[a, b] converging to u_{0} ∈ H^{k+α}[a, b]. Observe that this implies u_{n} → u_{0} in Lip^{k−2}[a, b] and u_{n} → u_{0} in H^{1+α}[a, b]. Since by assumption f ∈ C^{k−1}(ℝ), Theorem 4.2 yields

f ∘ u_{n} ⟶ f ∘ u_{0}  in Lip^{k−2}[a, b].  (5.5)

Furthermore, by assumption, for any i = 1, …, k − 1 and r > 0 we can find β ∈ (α, 1] with f^{(i)} ∈ H^{1+β}[−r, r]. Thus, by what we have just proved for k = 1,

f^{(i)} ∘ u_{n} ⟶ f^{(i)} ∘ u_{0}  in H^{1+α}[a, b] for i = 1, …, k − 1.  (5.6)

Finally, because of Lemma 3.5, for any differential operator P_{i} ∈ P_{k−i} (i = 1, …, k −1) we have

P_{i}u_{n} ⟶ P_{i}u_{0}  in H^{1+α}[a, b].  (5.7)

We estimate for any n ∈ ℕ now the norm of F u_{n} − F u_{0} in H^{k+α}[a, b]:

\begin{matrix}∥F u_{n} − F u_{0}∥_{H^{k+α}[a,b]} \\ ≤ ∥f ∘ u_{n} − f ∘ u_{0}∥_{Lip^{k−2}[a,b]} + ∥D^{k−1}(f ∘ u_{n} − f ∘ u_{0})∥_{H^{1+α}[a,b]} \\ ≤ ∥f ∘ u_{n} − f ∘ u_{0}∥_{Lip^{k−2}[a,b]} \\ + ∑_{i=1}^{k−1} ∥(f^{(i)} ∘ u_{n})P_{i}u_{n} − (f^{(i)} ∘ u_{0})P_{i}u_{0}∥_{H^{1+α}[a,b]} \end{matrix}  (5.8)

where P_{i} ∈ P_{k−i} (see Lemma 3.3). Because of (5.5) - (5.8) the proof is complete

The situation is similar in the case of local Lipschitz continuity and continuous Fréchet differentiability of F. Taking into account the results from the previous sections one could expect F to be locally Lipschitz continuous or continuously Fréchet differentiable in H^{k+α}[a, b] if and only if the generating function f belongs to Lip_{loc}^{k+1}(ℝ) or C^{k+2}(ℝ), respectively. As it can be seen in the subsequent two theorems these conditions are indeed sufficient. However, we could not verify their necessity. So, there are still differences between the sufficient and the necessary conditions.

Theorem 5.3.

(a) The autonomous Nemytskij operator F generated by f is a locally Lipschitz continuous map in H^{k+α}[a, b], (k, α) ∈ ℕ × (0, 1), if f ∈ Lip_{loc}^{k+1}(ℝ).

(b) If conversely F is a locally Lipschitz continuous map in H^{k+α}[a, b], (k, α) ∈ ℕ × (0, 1), then f ∈ H_{loc}^{(k+1)+α}(ℝ).

Proof. (a) First let k = 1. So we suppose f ∈ Lip_{loc}^{2}(ℝ). We fix an arbitrary element u_{0} ∈ H^{1+α}[a, b] and a number δ > 0. Let r > 0 be so large that ∥u∥_{H^{1+α}[a,b]} ≤ r for all u ∈ B(u_{0}, δ) ⊂ H^{1+α}[a, b]. Let L_{r} denote the Lipschitz constant of f with respect to the interval [−r, r]. Because of f′ ∈ Lip_{loc}^{1}(ℝ), Theorem 2.3 gives a constant L > 0 such that

∥f′ ∘ u − f′ ∘ v∥_{H^{α}[a,b]} ≤ L∥u − v∥_{H^{α}[a,b]} (u, v ∈ B(u_{0}, δ) ⊂ H^{1+α}[a, b]).

Then we have

∥F u − F v∥_{H^{1+α}[a,b]}
= ∥f ∘ u − f ∘ v∥_{C[a,b]} + ∥(f′ ∘ u)u′ − (f′ ∘ v)v′∥_{H^{α}[a,b]}
≤ L_{r}∥u − v∥_{C[a,b]} + ∥(f′ ∘ u)∥_{H^{α}[a,b]}∥u′ − v′∥_{H^{α}[a,b]}
+ ∥v′∥_{H^{α}[a,b]}∥f′ ∘ u − f′ ∘ v∥_{H^{α}[a,b]}
≤ c∥u − v∥_{H^{1+α}[a,b]}

for all u, v ∈ B(u_{0}, δ) ⊂ H^{1+α}[a, b] and the statement is proved.

Let now k ≥ 2. Again we take an arbitrary u_{0} ∈ H^{k+α}[a, b] and an arbitrary number δ > 0. Since the assumption f ∈ Lip_{loc}^{k+1}(ℝ) implies f ∈ Lip_{loc}^{k−1}(ℝ) and f′, …, f^{(k−1)} ∈ Lip_{loc}^{2}(ℝ), by Theorem 4.3 and the assertion already proved for k = 1 we find a constant c > 0 such that

∥f ∘ u − f ∘ v∥_{Lip^{k−2}[a,b]} ≤ c∥u − v∥_{Lip^{k−2}[a,b]}

and

∥f^{(i)} ∘ u − f^{(i)} ∘ v∥_{H^{1+α}[a,b]} ≤ c∥u − v∥_{H^{1+α}[a,b]} (i = 1, …, k − 1)

for each u, v ∈ B(u_{0}, δ) ⊂ H^{k+α}[a, b]. By Lemma 3.5, without loss of generality with the same constant c, for any P_{i} ∈ P_{k−i} (i = 1, …, k − 1) we have

∥P_{i}u − P_{i}v∥_{H^{1+α}[a,b]} ≤ c∥u − v∥_{H^{k+α}[a,b]} (u, v ∈ B(u_{0}, δ) ⊂ H^{k+α}[a, b]).

Therefore from estimate (5.8) with u instead of u_{n} and v instead of u_{0} the assertion follows also in this case.

(b) If we suppose now F ∈ Lip_{loc}(H^{k+α}[a, b]), then by Theorem 5.1 we have f ∈ C^{k}(ℝ). So it is to show that f^{(k)} ∈ H_{loc}^{1+α}(ℝ). By assumption, for any u_{0} ∈ H^{k+α}[a, b] there exist two constants L_{0} > 0 and δ_{0} > 0 such that

∥F u − F v∥_{H^{k+α}[a,b]} ≤ L_{0}∥u − v∥_{H^{k+α}[a,b]}  (5.9)

for all u, v ∈ B(u_{0}, δ_{0}) ⊂ H^{k+α}[a, b]. Firstly as u_{0} we take u_{0}(t) = t − a + x_{0} (t ∈ [a, b]) where x_{0} ∈ ℝ is arbitrarily fixed. Since the function u_{x} defined by u_{x}(t) = t−a+x (t ∈ [a, b]) belongs to B(u_{0}, δ_{0}) ⊂ H^{k+α}[a, b] if x ∈ B(x_{0}, δ_{0}) ⊂ ℝ, from (5.9) we obtain

|f^{(k)}(t − a + x) − f^{(k)}(t − a + y)| ≤ ∥D^{k}(f ∘ u_{x} − f ∘ u_{y})∥_{C[a,b]} ≤ L_{0}|x − y|

for all t ∈ [a, b] and all x, y ∈ B(x_{0}, δ_{0}) ⊂ ℝ which implies

|f^{(k)}(x) − f^{(k)}(y)| ≤ L_{0}|x − y| (x, y ∈ B(x_{0}, δ_{0}) ⊂ ℝ),

i.e. f^{(k)} ∈ Lip_{loc}(ℝ). Suppose f^{(k)} is not differentiable at x_{0}. Then there are two zero sequences (τ_{n}), (θ_{n}) ⊂ ℝ and two constants L_{1} > 0 and L_{2} > 0 with L_{1} < L_{2} such that

L_{1} = lim inf_{h→0} \frac{f^{(k)}(x_{0} + h) − f^{(k)}(x_{0})}{h} = lim_{n→∞} \frac{f^{(k)}(x_{0} + τ_{n}) − f^{(k)}(x_{0})}{τ_{n}},
L_{2} = lim sup_{h→0} \frac{f^{(k)}(x_{0} + h) − f^{(k)}(x_{0})}{h} = lim_{n→∞} \frac{f^{(k)}(x_{0} + θ_{n}) − f^{(k)}(x_{0})}{θ_{n}}.

In (5.9) we take now u_{0} defined by u_{0}(t) = t − ξ + x_{0} (t ∈ [a, b]) with fixed ξ ∈ (a, b). Furthermore, we define u_{n} by u_{n}(t) = t − ξ + x_{0} + τ_{n} (t ∈ [a, b]). Let n_{0} = n_{0}(δ_{0}) ∈ ℕ so large that |τ_{n}| < δ_{0} for all n ≥ n_{0} and hence u_{n} ∈ B(u_{0}, δ_{0}) ⊂ H^{k+α}[a, b]. Estimate (5.9) yields

∥F u_{n} − F u_{0}∥_{H^{k+α}[a,b]} ≤ L_{0}|τ_{n}| (n ≥ n_{0}).

In particular, we have

h_{α}(D^{k}(f ∘ u_{n} − f ∘ u_{0})) ≤ L_{0}|τ_{n}| (n ≥ n_{0})

and hence

\frac{|f^{(k)}(t − ξ + x_{0}) − f^{(k)}(t − ξ + x_{0} + τ_{n}) − f^{(k)}(x_{0}) + f^{(k)}(x_{0} + τ_{n})|}{|t − ξ|^{α}} ≤ L_{0}|τ_{n}| (5.10)

valid for all t ∈ [a, b]. Because of f^{(k)} ∈ Lip_{loc}(ℝ) and the Rademacher Theorem (see, e.g., [28: p. 50]) there exists a further zero sequence (h_{m}) ⊂ ℝ such that f^{(k)} is differentiable at all points x_{0} + h_{m}. Without loss of generality we may assume ξ + h_{m} ∈ [a, b] for all m ≥ n_{0}. Hence substituting t = ξ + h_{m} into (5.10) for all n, m ≥ n_{0} we get

|\frac{f^{(k)}(x_{0} + h_{m} + τ_{n}) − f^{(k)}(x_{0} + h_{m})}{τ_{n}} − \frac{f^{(k)}(x_{0} + τ_{n}) − f^{(k)}(x_{0})}{τ_{n}} | ≤ L_{0}|h_{m}|^{α},

On the Autonomous Nemytskij Operator in Hölder Spaces 23

from which

|f^{(k+1)}(x_{0} + h_{m}) − L_{1}| ≤ L_{0}|h_{m}|^{α} (m ≥ n_{0})

follows. Consequently, for any ε > 0 there exists an m_{0} = m_{0}(ε) ≥ n_{0} with

|f^{(k+1)}(x_{0} + h_{m}) − L_{1}| ≤ ε (m ≥ m_{0}).

Since the same estimate is true with L_{2} instead of L_{1}, L_{1} = L_{2} follows and hence f^{(k)} is differentiable at x_{0}.

It remains to show that f^{(k+1)} ∈ H_{loc}^{α}(ℝ). Again let x_{0} ∈ ℝ be arbitrary but fixed, u_{0} ∈ H^{k+α}[a, b] be given by u_{0}(t) = t − a + x_{0} (t ∈ [a, b]) and let L_{0} > 0 and δ_{0} > 0 be two constants such that (5.9) holds. Let δ_{1} = min\{\frac{δ_{0}}{2} , b − a\}. Then u definded by u(t) = t−a+x+Δ (t ∈ [a, b]) is an element of the ball B(u_{0}, δ_{0}) ⊂ H^{k+α}[a, b] provided x ∈ B(x_{0}, δ_{1}) ⊂ ℝ and Δ ∈ B(0, δ_{1}) ⊂ ℝ. Since also v defined by v(t) = t − a + x (t ∈ [a, b]) belongs to B(u_{0}, δ_{0}) ⊂ H^{k+α}[a, b] from (5.9) we obtain

\frac{|f^{(k)}(t − a + x + Δ) − f^{(k)}(t − a + x) − f^{(k)}(x + Δ) + f^{(k)}(x)|}{|t − a|^{α}} ≤ L_{0}|Δ|

valid for all t ∈ [a, b]. Let now y ∈ B(x_{0}, δ_{1}) ⊂ ℝ be another arbitrary element. Without loss of generality we may assume x < y and hence a < a + y − x < b. Substituting t = a + y − x into the last estimate we find

|\frac{f^{(k)}(y + Δ) − f^{(k)}(y)}{Δ} − \frac{f^{(k)}(x + Δ) − f^{(k)}(x)}{Δ}| ≤ L_{0}|y − x|^{α}

for all Δ ∈ B(0, δ_{1}) ⊂ ℝ. If here Δ → 0, we obtain

|f^{(k+1)}(y) − f^{(k+1)}(x)| ≤ L_{0}|y − x| (x, y ∈ B(x_{0}, δ_{1}) ⊂ ℝ)

showing that f^{(k+1)} belongs to H_{loc}^{α}(ℝ). This completes the proof

Theorem 5.4.

(a) The autonomous Nemytskij operator F generated by f is a continuously Fréchet differentiable map in the space H^{k+α}[a, b], (k, α) ∈ ℕ × (0, 1), if f ∈ C^{k+2}(ℝ).

(b) If conversely F is a continuously Fréchet differentiable map in H^{k+α}[a, b], (k, α) ∈ ℕ × (0, 1), then f ∈ H_{loc}^{(k+1)+α}(ℝ).

In both cases the Fréchet derivative at any point u ∈ H^{k+α}[a, b] is given by

F′(u)v = (f′ ∘ u) v

for all v ∈ H^{k+α}[a, b].

The proof of this theorem is nearly the same as the proofs of the analogous statements in the previous sections and hence omitted.

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Received 27.05.1998