Urysohn and Hammerstein operators on Hölder spaces

Christian Pötzsche

doi:10.1515/anly-2021-0052

Artikel Open Access

Urysohn and Hammerstein operators on Hölder spaces

Christian Pötzsche

Veröffentlicht/Copyright: 30. August 2022

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Analysis Band 42 Heft 4

Abstract

We present an application-oriented approach to Urysohn and Hammerstein integral operators acting between spaces of Hölder continuous functions over compact metric spaces. These nonlinear mappings are formulated by means of an abstract measure theoretical integral involving a finite measure. This flexible setting creates a common framework to tackle both such operators based on the Lebesgue integral like frequently met in applications, as well as, e.g., their spatial discretization using stable quadrature/cubature rules (Nyström methods). Under suitable Carathéodory conditions on the kernel functions, properties like well-definedness, boundedness, (complete) continuity and continuous differentiability are established. Furthermore, the special case of Hammerstein operators is understood as composition of Fredholm and Nemytskii operators. While our differentiability results for Urysohn operators appear to be new, the section on Nemytskii operators has a survey character. Finally, an appendix provides a rather comprehensive account summarizing the required preliminaries for Hölder continuous functions defined on metric spaces.

Keywords: Urysohn integral operator; Hammerstein integral operator; Nemytskii operator; nonlinear operator; Hölder continuity; Lipschitz continuity

MSC 2010: 47H30; 45P05; 45G15

1 Introduction

This treatise is devoted to Urysohn operators, a class of nonlinear integral operators arising in various contexts of nonlinear analysis [11, 30, 38, 44, 48], as right-hand sides of certain integrodifferential (Barbashin) equations [7], as well as in recent applications from control theory [28], mathematical biology [12], economic theory [20] (integral over an unbounded domain) or system identification [45] (sums as integrals). Urysohn operators are traditionally well-studied when acting between spaces of continuous functions over a compact domain (see [38, pp. 164ff, Section V.3], [48, pp. 35–37, Section 3.1] or [46, Appendix B.2]), spaces of integrable functions [32] with possibly different exponents, or in mixed form [38, pp. 175ff]. In such a setup, their well-definedness and continuity is addressed in, e.g., [38, pp. 172ff] and [44, p. 85], while conditions yielding that they are set contractions with respect to ambient measures of non-compactness can be found in [2, pp. 227ff] (for L p -spaces). Both necessary and sufficient conditions for the complete continuity of Urysohn operators between different function spaces are given in [41]. Furthermore, [52, pp. 162–298] provides an extensive analysis of such mappings between abstract ideal spaces; see also [51]. Properties of Urysohn operators over compact intervals having values in a real Banach space are discussed in [25, pp. 54–90, Section 2.1], and differentiability conditions were given in [19] (see also [30, pp. 41ff] or [32, pp. 417ff, Section 20] in L p -spaces). Finally, we would like to point out the paper [34] containing complete continuity results for Urysohn operators on the continuous functions over merely locally compact (and possibly unbounded) domains.

A highly relevant special case is given in terms of Hammerstein operators [30, 32, 38, 44]. Our given approach tackles them as composition of (linear) Fredholm integral operators determined by an integral kernel [21, 26, 33] with (nonlinear) Nemytskii operators [6, 13, 18, 24, 39, 40, 42, 44]. Classically, Hammerstein operators arise in fixed point problems related to nonlinear boundary value problems (see [38, pp. 177ff, Section V.5] or [48, p. 71, Theorem 5.5]), where the kernel is a corresponding Green’s function. A more recent application are integrodifference equations originating in theoretical ecology [31, 36], where the kernel models the spatial dispersal of species over a habitat, while the Nemytskii operator describes their growth phase. The classical L p -theory of Hammerstein operators is covered for instance in [44, p. 84] or [48, pp. 68ff, Section 5.3].

The paper at hand supplements the above contributions. We provide a comprehensive approach to Urysohn operators acting between possibly different spaces of Hölder continuous functions over compact metric spaces. We restrict to Hölder spaces with exponents α ≤ 1 , i.e. the functions under consideration are not necessarily differentiable with Hölder continuous derivatives of positive order. This endows us with a wide scale of Banach spaces whose elements range from nowhere differentiable to Lipschitz functions, being differentiable almost everywhere. On one hand, an early contribution to this area is the note [43] addressing well-definedness and complete continuity of general Urysohn operators. On the other hand, Hölder spaces are meanwhile widely used when dealing with linear integral operators having singular kernels [33, pp. 103ff, Chapter 7] or in the field of (quasilinear) elliptic boundary value problems [23]; moreover, [49] investigates nonlinear integral equations in Hölder spaces. Our motivation, nevertheless, is different. It rather comes from the numerical analysis of integral equations [8] and the numerical dynamics of integrodifference equations [31, 36, 46]. In the latter context, one aims to show that such infinite-dimensional dynamical systems given by the iterates of integral operators share the long term dynamics with their spatial discretizations. Certain problems in this area require to establish that Fréchet derivatives of Urysohn operators and of their spatial discretization converge to each other in the operator norm. Among the techniques for the numerical solution of integral equations, this can be justified for semi-discretizations of projection or degenerate kernel type, cf. [8, 26, 33, 46]. However, uniform convergence is not feasible when working with full discretizations of Nyström type on the continuous functions (see, e.g., [33, p. 225, Theorem 12.8]). In contrast, when working with Hölder spaces, appropriate estimates can be established [47].

Having applications from theoretical ecology to numerical dynamics in mind, it is advantageous to establish a rather flexible setting we are aiming to provide here: First, we consider vector-valued operators (in finite dimensions though), which arise in ecological models describing various interacting species. Second, we allow general measure theoretical integrals induced by a finite measure such that both integral operators based on the Lebesgue integral, as well as their spatial discretization using, e.g., Nyström methods, fit into a common framework (see Example 2.2). For this reason, we content ourselves to provide sufficient conditions guaranteeing that an integral operator is well-defined, bounded, (completely) continuous or differentiable. Necessary conditions for the above properties exist for operators on compact intervals, but are beyond the scope of this paper.

Our presentation is subdivided into three parts: In Section 2, we provide conditions of Carathéodory type on the kernel functions such that the associated Urysohn operators are well-defined, bounded, (completely and Hölder) continuous, resp. continuously differentiable. We successively study such operators, first having values in the continuous, and second in the Hölder functions. In particular, a subsection is devoted to convolutive Urysohn operators 𝒰 ~ , where Hölder continuity of the arguments u extends to the values 𝒰 ~ ⁢ ( u ) . The Section 3 on Hammerstein operators follows a similar scheme. These mappings are compositions of Fredholm and Nemytskii operators. Since Nemytskii operators between Hölder spaces have rather degenerate mapping and differentiability properties [6, Chapter 7], we retreat to the case that they map into the continuous functions. Hölder continuity of the images is then guaranteed by appropriate assumptions on the kernel of the subsequent Fredholm operator. Addressing well-studied objects, Section 3.2 on Nemytskii operators between Hölder spaces has a survey character. Finally, Section A provides a broad perspective over the class of Hölder continuous functions defined on a metric space and having values in a normed space.

Notation and terminology.

Let ℝ + := [ 0 , ∞ ) and let X , Y be nonempty sets. We write F ⁢ ( X , Y ) for the set of all functions f : X → Y . In the setting of metric spaces X , Y , a subset Ω ⊆ X is called bounded if it has finite diameter

diam ⁡ Ω := sup x , x ¯ ∈ Ω ⁡ d ⁢ ( x , x ¯ ) .

A function f : X → Y is called bounded if it maps bounded sets into bounded sets, i.e. f ⁢ ( Ω ) ⊆ Y is bounded for every bounded Ω ⊆ X , and globally bounded if f ⁢ ( X ) ⊆ Y is bounded. A completely continuous mapping is continuous and maps bounded sets into relatively compact images.

If X , Y are normed spaces, then L k ⁢ ( X , Y ) , k ∈ ℕ 0 , is the normed space of continuous k-linear maps from X k to Y, where L 0 ⁢ ( X , Y ) := Y and L ⁢ ( X , Y ) := L 1 ⁢ ( X , Y ) . We write B r ⁢ ( x 0 , X ) := { x ∈ X : ∥ x - x 0 ∥ < r } for the open and B ¯ r ⁢ ( x 0 , X ) := { x ∈ X : ∥ x - x 0 ∥ ≤ r } for the closed r-ball around x 0 ∈ X in ( X , ∥ ⋅ ∥ ) . Norms on finite-dimensional spaces are denoted by | ⋅ | , and B r ⁢ ( x 0 ) , B ¯ r ⁢ ( x 0 ) are the corresponding r-balls.

The remaining introduction anticipates notions from Section A on Hölder spaces: A function u : Ω → ℝ n on a metric space ( Ω , d ) is called α-Hölder (with Hölder exponent α ∈ ( 0 , 1 ] ) if it satisfies

[ u ] α := sup x , x ¯ ∈ Ω x ≠ x ¯ ⁡ | u ⁢ ( x ) - u ⁢ ( x ¯ ) | d ⁢ ( x , x ¯ ) α < ∞ ;

the finite quantity [ u ] α is denoted as Hölder constant of u. One speaks of a Hölder continuous function u if it is α-Hölder for some α ∈ ( 0 , 1 ) . In case α = 1 , one denotes u as Lipschitz continuous with Lipschitz constant [ u ] 1 , and for convenience we denote a continuous function also as 0-Hölder. For the linear space of all bounded and α-Hölder functions, we write^[1] C n α ⁢ ( Ω ) := C α ⁢ ( Ω , ℝ n ) , supplemented by C n 0 ⁢ ( Ω ) := C 0 ⁢ ( Ω , ℝ n ) for the bounded, continuous functions and C α ⁢ ( Ω ) := C α ⁢ ( Ω , ℝ ) . Note that C n α ⁢ ( Ω ) is a Banach space with respect to the norm (cf. Theorem A.15)

∥ u ∥ α := { sup x ∈ Ω ⁡ | u ⁢ ( x ) | , α = 0 , max ⁡ { sup x ∈ Ω ⁡ | u ⁢ ( x ) | , [ u ] α } , α ∈ ( 0 , 1 ] .

Throughout the remaining paper, our setup is as follows: Let Ω and Ω 1 be metric spaces. Suppose additionally that Ω is compact and can be interpreted as measure space ( Ω , 𝔄 , μ ) with μ ⁢ ( Ω ) < ∞ whose σ-algebra 𝔄 contains the Borel sets. The notions of measurability and integrability refer to this measure space from now on. In particular, ∫ Ω ⋅ d ⁢ μ stands for the abstract integral associated to the measure μ (e.g., [15]).

Moreover, Z ⊆ ℝ n denotes a nonempty subset throughout. Given a Hölder exponent α ∈ [ 0 , 1 ] , we write

U α := { u : Ω → Z ∣ u ∈ C n α ( Ω ) }

for the α-Hölder functions over Ω having values in Z. If 0 ≤ α ≤ β ≤ 1 , then Theorem A.18 guarantees the embedding U β ⊆ U α between the continuous and the Lipschitz continuous functions.

2 Urysohn integral operators

An Urysohn operator ^[2] is a nonlinear integral operator of the form

(2.1) 𝒰 : U α → F ⁢ ( Ω 1 , ℝ d ) , 𝒰 ⁢ ( u ) := ∫ Ω f ⁢ ( ⋅ , y , u ⁢ ( y ) ) ⁢ d μ ⁢ ( y ) ,

determined by a kernel function f : Ω 1 × Ω × Z → ℝ d and a measure μ as above. Its overall analysis is based on the following Carathéodory-like conditions.

Hypothesis.

Let m ∈ ℕ 0 . With 0 ≤ k ≤ m , one assumes the following:

The partial derivative D 3 k ⁢ f ⁢ ( x , y , ⋅ ) : Z → L k ⁢ ( ℝ n , ℝ d ) exists and is continuous for all x ∈ Ω 1 and almost all y ∈ Ω .
For all r > 0 there exists a function h r k : Ω 1 2 × Ω → ℝ + , measurable in the third argument and satisfying

(2.2) lim x → x 0 ⁡ ∫ Ω h r k ⁢ ( x , x 0 , y ) ⁢ d μ ⁢ ( y ) = 0 for all ⁢ x 0 ∈ Ω 1 ,

so that for almost all y ∈ Ω the following holds:

(2.3) | D 3 k ⁢ f ⁢ ( x , y , z ) - D 3 k ⁢ f ⁢ ( x 0 , y , z ) | ≤ h r k ⁢ ( x , x 0 , y ) for all ⁢ x , x 0 ∈ Ω 1 , z ∈ Z ∩ B ¯ r ⁢ ( 0 ) .
D 3 k ⁢ f ⁢ ( x , ⋅ , z ) : Ω → L k ⁢ ( ℝ n , ℝ d ) is measurable for all x ∈ Ω 1 , z ∈ Z , and suppose that for every r > 0 there exists a function b r k : Ω 1 × Ω → ℝ + measurable in the second argument and satisfying

ess ⁢ sup ξ ∈ Ω 1 ⁢ ∫ Ω b r k ⁢ ( ξ , y ) ⁢ d μ ⁢ ( y ) < ∞ ,

so that for almost all y ∈ Ω the following holds:

(2.4) | D 3 k ⁢ f ⁢ ( x , y , z ) | ≤ b r k ⁢ ( x , y ) for all ⁢ x ∈ Ω 1 , z ∈ Z ∩ B ¯ r ⁢ ( 0 ) .

Because we are working with a general (finite) measure on Ω, both spatially continuous and discrete integral operators fit into our framework.

Example 2.1 (Lebesgue measure).

In most applications, for example, [12, 20, 31, 36], μ is the κ-dimensional Lebesgue measure λ κ on compact sets Ω ⊂ ℝ κ yielding the Lebesgue integral in (2.1), and thus

(2.5) 𝒰 ⁢ ( u ) = ∫ Ω f ⁢ ( ⋅ , y , u ⁢ ( y ) ) ⁢ d λ κ ⁢ ( y ) = ∫ Ω f ⁢ ( ⋅ , y , u ⁢ ( y ) ) ⁢ d y : Ω 1 → ℝ d

is a spatially continuous integral operator. One clearly has μ ⁢ ( Ω ) < ∞ .

Example 2.2 (Nyström methods).

Suppose that Ω ⊂ ℝ κ is a countable set Ω ( l ) , and that η ∈ Ω ( l ) and w η denote nonnegative reals. Then μ ⁢ ( Ω ( l ) ) := ∑ η ∈ Ω ( l ) w η defines a measure on the family of countable subsets of ℝ κ , and precisely the empty set has measure 0. Moreover, the assumption

∑ η ∈ Ω ( l ) w η < ∞

guarantees that μ ⁢ ( Ω ( l ) ) is finite. The resulting μ-integral

∫ Ω u ⁢ d μ = ∑ η ∈ Ω ( l ) w η ⁢ u ⁢ ( η )

leads to spatially discrete Urysohn operators

𝒰 ⁢ ( u ) = ∫ Ω ( l ) f ⁢ ( ⋅ , y , u ⁢ ( y ) ) ⁢ d μ ⁢ ( y ) = ∑ η ∈ Ω ( l ) w η ⁢ f ⁢ ( ⋅ , η , u ⁢ ( η ) ) : Ω 1 → ℝ d ,

which cover Nyström methods with nodes η and weights w η as used for numerical approximations of spatially continuous integral operators (2.5); cf. [8, Section 3], [26, pp. 128ff, Section 4.7] or [33, pp. 219ff, Chapter 12] (the latter two references address linear operators only). Alternatively, such mappings arise in theoretical ecology by means of models for populations spreading between finitely many different patches (metapopulation models, see [31, Example 1]).

Example 2.3 (Evaluation map).

In case of singletons Ω = { η } and the measure from Example 2.2, one obtains that F ⁢ ( Ω , ℝ n ) ≅ ℝ n and the Urysohn operator (2.1) becomes an evaluation map 𝒰 ⁢ ( u ) = w η ⁢ f ⁢ ( ⋅ , η , u ⁢ ( η ) ) , which is simply a mapping from ℝ n into F ⁢ ( Ω 1 , ℝ d ) .

Remark 2.4 (Differentiability on Z).

We imposed no further conditions on the sets Z ⊆ ℝ n , and therefore some remarks on the existence of the partial derivative D 3 k ⁢ f for k > 0 are due:

For interior points of Z the partial derivatives are understood in the Fréchet sense.
If z 0 ∈ Z is not an interior point of Z, then we assume that there exists a neighborhood V ⊆ ℝ n of z 0 and an extension f ¯ : Ω 1 × Ω × ( Z ∪ V ) → ℝ d such that the partial derivatives D 3 ⁢ f ¯ ⁢ ( x , y , ⋅ ) exist in z 0 as assumed in ( U 0 k ) . Alternatively, there is the notion of cone differentiability [16, pp. 225–226].

Under continuity, the above hypothesis can be simplified as follows.

Proposition 2.5.

Let k ∈ N 0 , let Ω 1 be compact and let Z ⊆ R n be closed. If the partial derivative

D 3 k ⁢ f : Ω 1 × Ω × Z → L k ⁢ ( ℝ n , ℝ d )

exists as continuous function, then ( U 0 k ) , ( U 1 k ) and ( U 2 k ) are satisfied and the limit relation (2.2) holds uniformly in x 0 ∈ Ω 1 .

Proof.

Since Z is assumed to be closed, Z r := Z ∩ B ¯ r ⁢ ( 0 ) ⊆ ℝ n is compact for all r > 0 . Then the continuous function D 3 k ⁢ f is uniformly continuous and globally bounded on each compact product Ω 1 × Ω × Z r . Moreover, since the σ-algebra 𝔄 contains the Borel sets, continuous functions are measurable. Given this, the assertions hold with the continuous functions

h r k ⁢ ( x , x 0 , y ) := sup z ∈ Z r ⁡ | D 3 k ⁢ f ⁢ ( x , y , z ) - D 3 k ⁢ f ⁢ ( x 0 , y , z ) | , b r k ⁢ ( x , y ) := sup z ∈ Z r ⁡ | D 3 k ⁢ f ⁢ ( x , y , z ) | .

This concludes the proof. ∎

2.1 Well-definedness and complete continuity

We begin with basic properties of Urysohn operators (2.1) and assume α ∈ ( 0 , 1 ] .

Proposition 2.6 (Well-definedness of U ).

Assume that ( U 0 0 ) , ( U 1 0 ) and ( U 2 0 ) hold. Then an Urysohn operator

𝒰 : U α → C d 0 ⁢ ( Ω 1 )

is well-defined, bounded and continuous.

Proof.

Without loss of generality, let μ ⁢ ( Ω ) > 0 , since otherwise 𝒰 ⁢ ( u ) ≡ 0 on U α .

(I) Claim: U : U α → C d 0 ⁢ ( Ω 1 ) is well-defined and bounded. Choose u ∈ U α and r > 0 such that ∥ u ∥ 0 ≤ r . Given x , x 0 ∈ Ω 1 , the Carathéodory conditions ( U 0 0 ) and ( U 2 0 ) yield that f ⁢ ( x , ⋅ , u ⁢ ( ⋅ ) ) : Ω → ℝ d is measurable (see [48, p. 62, Lemma 5.1]). From ( U 1 0 ) we conclude

| 𝒰 ⁢ ( u ) ⁢ ( x ) - 𝒰 ⁢ ( u ) ⁢ ( x 0 ) | ≤ ∫ Ω | f ⁢ ( x , y , u ⁢ ( y ) ) - f ⁢ ( x 0 , y , u ⁢ ( y ) ) | ⁢ d μ ⁢ ( y ) ⁢ (by (2.1))

≤ ∫ Ω h r 0 ⁢ ( x , x 0 , y ) ⁢ d μ ⁢ ( y ) ⁢ (by (2.3))

→ x → x 0 0 ⁢ (by (2.2))

for each x 0 ∈ Ω 1 , which guarantees that 𝒰 ⁢ ( u ) is continuous. Furthermore, because ( U 2 0 ) yields

| 𝒰 ⁢ ( u ) ⁢ ( x ) | ⁢ ≤ ( 2.1 ) ⁢ ∫ Ω | f ⁢ ( x , y , u ⁢ ( y ) ) | ⁢ d μ ⁢ ( y ) ⁢ ≤ ( 2.4 ) ⁢ ess ⁢ sup ξ ∈ Ω 1 ⁢ ∫ Ω b r 0 ⁢ ( ξ , y ) ⁢ d μ ⁢ ( y ) for all ⁢ x ∈ Ω 1 ,

we see that 𝒰 ⁢ ( u ) is bounded, and thus 𝒰 ⁢ ( u ) ∈ C d 0 ⁢ ( Ω 1 ) . In addition, 𝒰 maps bounded subsets of U 0 into bounded subsets of C d 0 ⁢ ( Ω 1 ) .

(II) Claim: U : U α → C d 0 ⁢ ( Ω 1 ) is continuous. Suppose that u ∈ U α and that ( u l ) l ∈ ℕ is a sequence in U α with lim l → ∞ ⁡ ∥ u l - u ∥ 0 = 0 and r > 0 sufficiently large so that u , u l ∈ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) holds for all l ∈ ℕ . By using ( U 0 0 ) , this gives

lim l → ∞ ⁡ f ⁢ ( x , y , u l ⁢ ( y ) ) = f ⁢ ( x , y , u ⁢ ( y ) )

for all x ∈ Ω 1 and almost all y ∈ Ω . For each ε > 0 , there exists a δ > 0 such that subsets Ω ~ ⊆ Ω with μ ⁢ ( Ω ~ ) ≤ δ fulfill

∫ Ω ~ b r 0 ⁢ ( x , y ) ⁢ d μ ⁢ ( y ) ≤ ε 4 ,

and Egoroff’s theorem [15, p. 87, Proposition 3.1.3] guarantees that there exist a Ω ′ ⊆ Ω with μ ⁢ ( Ω ′ ) ≤ δ and an L ∈ ℕ such that

| f ⁢ ( x , y , u l ⁢ ( y ) ) - f ⁢ ( x , y , u ⁢ ( y ) ) | ≤ ε 2 ⁢ μ ⁢ ( Ω ) for all ⁢ x ∈ Ω 1 , y ∈ Ω ∖ Ω ′ ⁢ and ⁢ l ≥ L .

This implies that we have pointwise convergence due to

≤ ∫ Ω ∖ Ω ′ ε 2 ⁢ μ ⁢ ( Ω ) ⁢ d μ ⁢ ( y ) + 2 ⁢ ∫ Ω ′ b r 0 ⁢ ( x , y ) ⁢ d μ ⁢ ( y ) (by (2.4))

≤ ε 2 + 2 ⁢ ε 4

= ε for all ⁢ x ∈ Ω 1 , l ≥ L .

Passing to the supremum over x ∈ Ω 1 yields the limit relation

(2.6) lim l → ∞ ⁡ ∥ 𝒰 ⁢ ( u l ) - 𝒰 ⁢ ( u ) ∥ 0 = 0 .

This shows the continuity of 𝒰 . ∎

Corollary 2.7 (Complete continuity of U ).

An Urysohn operator U : U α → C d 0 ⁢ ( Ω 1 ) is completely continuous, provided one of the following conditions hold:

α ∈ ( 0 , 1 ] .
Ω 1 is compact, α = 0 , and the limit relation ( 2.2 ) holds uniformly in x 0 ∈ Ω 1 .

Proof.

We write 𝒰 0 : U 0 → C d 0 ⁢ ( Ω 1 ) for the operator defined in (2.1).

(I). For exponents α ∈ ( 0 , 1 ] , we observe 𝒰 = 𝒰 0 ∘ ℐ α 0 with the compact embedding operator ℐ α 0 from (A.3) (cf. Theorem A.22). Therefore, 𝒰 inherits the claimed properties from steps (I) and (II) of the proof to Proposition 2.6. In particular, as composition of the continuous 𝒰 0 with the compact ℐ α 0 it is completely continuous due to, e.g., [48, pp. 25–26, Theorem 2.1 (2)].

(II) Claim: If (ii) holds, then U 0 : U 0 → C d 0 ⁢ ( Ω 1 ) is completely continuous. In Proposition 2.6 it was shown that 𝒰 0 : U 0 → C d 0 ⁢ ( Ω 1 ) is bounded and continuous. If α = 0 and (2.2) holds uniformly in x 0 ∈ Ω 1 , then the first limit relation in the proof of Proposition 2.6) is true uniformly in x 0 as well, and each image

𝒰 0 ⁢ ( U 0 ∩ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) ) ⊂ C d 0 ⁢ ( Ω 1 )

is equicontinuous. Therefore, by the Arzelà–Ascoli theorem [38, p. 31, Theorem 3.2] we obtain its relative compactness. ∎

Corollary 2.8.

Let Ω 1 be compact and let Z ⊆ R n be closed. If f : Ω 1 × Ω × Z → R d is continuous, then

𝒰 : U α → C d 0 ⁢ ( Ω 1 )

is completely continuous and uniformly continuous on each set U α ∩ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) , r > 0 .

Proof.

Due to Proposition 2.5, the assumptions of Proposition 2.6 and Corollary 2.7 are fulfilled. Therefore, 𝒰 is completely continuous. Hence, it remains to show the uniform continuity on bounded sets. For this purpose, let ε > 0 and r > 0 . Because Z is closed, Z r := Z ∩ B ¯ r ⁢ ( 0 ) ⊆ ℝ n is compact. Since the continuous function f is uniformly continuous on the compact Ω 1 × Ω × Z r , there exists a δ > 0 such that for all x ∈ Ω 1 , y ∈ Ω and z , z ¯ ∈ Z r one has that

| z - z ¯ | < δ implies | f ⁢ ( x , y , z ) - f ⁢ ( x , y , z ¯ ) | < ε 2 ⁢ μ ⁢ ( Ω ) .

Suppose u , u ¯ ∈ U α ∩ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) with ∥ u - u ¯ ∥ 0 < δ . Then the inclusions u ⁢ ( y ) , u ¯ ⁢ ( y ) ∈ Z r and the estimate | u ⁢ ( y ) - u ¯ ⁢ ( y ) | < δ hold for all y ∈ Ω . They yield

| 𝒰 ⁢ ( u ) ⁢ ( x ) - 𝒰 ⁢ ( u 0 ) ⁢ ( x ) | ⁢ ≤ ( 2.1 ) ⁢ ∫ Ω | f ⁢ ( x , y , u ⁢ ( y ) ) - f ⁢ ( x , y , u 0 ⁢ ( y ) ) | ⁢ d μ ⁢ ( y ) ≤ ε 2 for all ⁢ x ∈ Ω 1 .

Passing to the least upper bound over x ∈ Ω 1 results in ∥ 𝒰 ⁢ ( u ) - 𝒰 ⁢ ( u 0 ) ∥ 0 < ε , i.e. 𝒰 is uniformly continuous on each set U α ∩ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) . ∎

The subsequent assumption allows us to infer Hölder continuity of 𝒰 .

Hypothesis.

Let ϑ ∈ ( 0 , 1 ] .

For every r > 0 , there exists a function l r : Ω 1 × Ω → ℝ + , measurable in the second argument and satisfying

ess ⁢ sup ξ ∈ Ω 1 ⁢ ∫ Ω l r ⁢ ( ξ , y ) ⁢ d μ ⁢ ( y ) < ∞ ,

so that for almost all y ∈ Ω the following holds:

(2.7) | f ⁢ ( x , y , z ) - f ⁢ ( x , y , z ¯ ) | ≤ l r ⁢ ( x , y ) ⁢ | z - z ¯ | ϑ for all ⁢ x ∈ Ω 1 , z , z ¯ ∈ Z ∩ B ¯ r ⁢ ( 0 ) .

Obviously, condition ( U 0 ′ ) is sufficient for ( U 0 0 ) .

Corollary 2.9.

If additionally ( U 0 ′ ) holds, then U : U α → C d 0 ⁢ ( Ω 1 ) is Hölder on bounded sets, that is,

(2.8) [ 𝒰 | U α ∩ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) ] ϑ ≤ ess ⁢ sup ξ ∈ Ω 1 ∫ Ω l r ( ξ , y ) d μ ( y ) for all r > 0 .

Proof.

Since ( U 0 ′ ) implies ( U 0 0 ) , we obtain from Proposition 2.6 that 𝒰 : U α → C d 0 ⁢ ( Ω 1 ) is well-defined. Given r > 0 , for u , u ¯ ∈ U α ∩ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) we derive from ( U 0 ′ ) that

| [ 𝒰 ⁢ ( u ) - 𝒰 ⁢ ( u ¯ ) ] ⁢ ( x ) | ≤ ∫ Ω | f ⁢ ( x , y , u ⁢ ( y ) ) - f ⁢ ( x , y , u ¯ ⁢ ( y ) ) | ⁢ d μ ⁢ ( y ) ⁢ (by (2.1))

≤ ∫ Ω l r ⁢ ( x , y ) ⁢ d μ ⁢ ( y ) ⁢ ∥ u - u ¯ ∥ 0 ϑ ⁢ (by (2.7))

≤ ess ⁢ sup ξ ∈ Ω 1 ⁢ ∫ Ω l r ⁢ ( ξ , y ) ⁢ d μ ⁢ ( y ) ⁢ ∥ u - u ¯ ∥ 0 ϑ ⁢ for all ⁢ x ∈ Ω 1 ,

and consequently

∥ 𝒰 ⁢ ( u ) - 𝒰 ⁢ ( u ¯ ) ∥ 0 ≤ ess ⁢ sup ξ ∈ Ω 1 ⁢ ∫ Ω l r ⁢ ( ξ , y ) ⁢ d μ ⁢ ( y ) ⁢ ∥ u - u ¯ ∥ 0 ϑ

after passing to the least upper bound over all x ∈ Ω 1 . ∎

Let us proceed to Urysohn operators having values in Hölder spaces with positive exponent β rather than in C d 0 ⁢ ( Ω 1 ) . This requires to sharpen our above assumptions beyond ( U 0 k ) .

Hypothesis.

Let m ∈ ℕ 0 and β ∈ ( 0 , 1 ] . With 0 ≤ k ≤ m , one assumes that for every r > 0 there exist the following functions:

An integrable function h ¯ r k : Ω → ℝ + , so that for almost all y ∈ Ω the following holds:

(2.9) | D 3 k ⁢ f ⁢ ( x , y , z ) - D 3 k ⁢ f ⁢ ( x ¯ , y , z ) | ≤ h ¯ r k ⁢ ( y ) ⁢ d ⁢ ( x , x ¯ ) β for all ⁢ x , x ¯ ∈ Ω 1 , z ∈ Z ∩ B ¯ r ⁢ ( 0 ) .

A function c r k : ℝ + × Ω → ℝ + measurable in the second argument with

lim δ ↘ 0 ⁡ ∫ Ω c r k ⁢ ( δ , y ) ⁢ d μ ⁢ ( y ) = 0 ,

so that for almost all y ∈ Ω and all δ > 0 the following holds:

(2.10) | z - z ¯ | ≤ δ implies | D 3 k ⁢ f ⁢ ( x , y , z ) - D 3 k ⁢ f ⁢ ( x , y , z ¯ ) - [ D 3 k ⁢ f ⁢ ( x ¯ , y , z ) - D 3 k ⁢ f ⁢ ( x ¯ , y , z ¯ ) ] | ≤ c r k ⁢ ( δ , y ) ⁢ d ⁢ ( x , x ¯ ) β

for all x , x ¯ ∈ Ω 1 , z , z ¯ ∈ Z ∩ B ¯ r ⁢ ( 0 ) .

Remark 2.10.

(1) Note that ( U ¯ 1 k ) implies ( U 1 k ) with the function h r k ⁢ ( x , x 0 , y ) := h ¯ r k ⁢ ( y ) ⁢ d ⁢ ( x , x 0 ) β , and in particular the limit relation (2.2) holds uniformly in x 0 ∈ Ω 1 .

(2) Since it might be tedious to verify the implication (2.10), we note some sufficient conditions:

If Z ⊆ ℝ n is convex, then ( U ¯ 1 k + 1 ) implies ( U ¯ 3 k ) with c r k ⁢ ( δ , y ) := δ ⁢ h ¯ r k + 1 ⁢ ( y ) . Indeed, the mean value theorem [35, p. 341, Theorem 4.2] yields for almost all y ∈ Ω that

| D 3 k ⁢ f ⁢ ( x , y , z ) - D 3 k ⁢ f ⁢ ( x , y , z ¯ ) - [ D 3 k ⁢ f ⁢ ( x ¯ , y , z ) - D 3 k ⁢ f ⁢ ( x ¯ , y , z ¯ ) ] |

= | ∫ 0 1 D 3 k + 1 ⁢ f ⁢ ( x , y , z ¯ + θ ⁢ ( z - z ¯ ) ) - D 3 k + 1 ⁢ f ⁢ ( x ¯ , y , z ¯ + θ ⁢ ( z - z ¯ ) ) ⁢ d ⁢ θ ⁢ [ z - z ¯ ] |

≤ h ¯ r k + 1 ⁢ ( y ) ⁢ | z - z ¯ | ⁢ d ⁢ ( x , x ¯ ) β (by (2.9)) for all ⁢ x , x ¯ ∈ Ω 1 , z , z ¯ ∈ Z ∩ B ¯ r ⁢ ( 0 ) .
Let Ω 1 ⊂ ℝ ν be bounded and convex. Assume for all r > 0 that there is a function γ r k : ℝ + × Ω → ℝ + measurable in the second argument satisfying

lim δ ↘ 0 ⁡ ∫ Ω γ r k ⁢ ( δ , y ) ⁢ d μ ⁢ ( y ) = 0 ,

so that for almost all y ∈ Ω and all δ > 0 the following holds:

D 1 ⁢ D 3 k ⁢ f ⁢ ( ⋅ , y , z ) : Ω 1 → L ⁢ ( ℝ ν , L k ⁢ ( ℝ n , ℝ d ) )

exists and is continuous for all z ∈ Z , almost all y ∈ Ω ,

| z - z ¯ | < δ implies | D 1 ⁢ D 3 k ⁢ f ⁢ ( x , y , z ) - D 1 ⁢ D 3 k ⁢ f ⁢ ( x , y , z ¯ ) | ≤ γ r k ⁢ ( δ , y ) for all ⁢ x , x ¯ ∈ Ω 1

and z , z ¯ ∈ Z ∩ B ¯ r ⁢ ( 0 ) . If | z - z ¯ | < δ , then the mean value theorem [35, p. 341, Theorem 4.2] yields

| D 3 k ⁢ f ⁢ ( x , y , z ) - D 3 k ⁢ f ⁢ ( x , y , z ¯ ) - [ D 3 k ⁢ f ⁢ ( x ¯ , y , z ) - D 3 k ⁢ f ⁢ ( x ¯ , y , z ¯ ) ] |

= | ∫ 0 1 D 1 ⁢ D 3 k ⁢ f ⁢ ( x ¯ + θ ⁢ ( x - x ¯ ) , y , z ) - D 1 ⁢ D 3 k ⁢ f ⁢ ( x ¯ + θ ⁢ ( x - x ¯ ) , y , z ¯ ) ⁢ d ⁢ θ | ⁢ | x - x ¯ |

≤ ( diam ⁡ Ω 1 ) 1 - β ⁢ γ r k ⁢ ( δ , y ) ⁢ | x - x ¯ | β for all ⁢ x , x ¯ ∈ Ω 1 ,

which allows us to choose c r k ⁢ ( δ , y ) := ( diam ⁡ Ω 1 ) 1 - β ⁢ γ r k ⁢ ( δ , y ) .

However, this requires one higher order of continuous partial differentiability for the kernel function f.

(iii) Replacing the function c r k : ℝ + × Ω → ℝ + in ( U ¯ 3 k ) with

(2.11) c ¯ r k : ℝ + × Ω → ℝ + , c ¯ r k ⁢ ( δ , y ) := sup ρ ≤ δ ⁡ c r k ⁢ ( ρ , y )

yields a nondecreasing function c r k ⁢ ( δ , y ) ≤ c ¯ r k ⁢ ( δ , y ) inheriting the other relevant properties from c r k .

Theorem 2.11 (Well-definedness of U ).

Assume that ( U 0 0 ) , ( U ¯ 1 0 ) and ( U 2 0 ) hold. Then an Urysohn operator U : U α → C d β ⁢ ( Ω 1 ) is well-defined and bounded. If additionally ( U ¯ 3 0 ) holds, then U is continuous.

Proof.

Choose u ∈ U α and r > 0 so large that ∥ u ∥ 0 ≤ r holds. Referring to Remark 2.10 (1), we can apply Proposition 2.6, which guarantees that 𝒰 : U α → C d 0 ⁢ ( Ω 1 ) is well-defined, bounded and continuous.

(I) Claim: U : U α → C d β ⁢ ( Ω 1 ) is well-defined and bounded. Given arbitrary x , x ¯ ∈ Ω 1 , by using ( U ¯ 1 0 ) , the estimate

| 𝒰 ⁢ ( u ) ⁢ ( x ) - 𝒰 ⁢ ( u ) ⁢ ( x ¯ ) | ⁢ ≤ ( 2.1 ) ⁢ ∫ Ω | f ⁢ ( x , y , u ⁢ ( y ) ) - f ⁢ ( x ¯ , y , u ⁢ ( y ) ) | ⁢ d μ ⁢ ( y ) ⁢ ≤ ( 2.9 ) ⁢ ∫ Ω h ¯ r 0 ⁢ ( y ) ⁢ d μ ⁢ ( y ) ⁢ d ⁢ ( x , x ¯ ) β

implies 𝒰 ⁢ ( u ) ∈ C d β ⁢ ( Ω 1 ) ( 𝒰 is well-defined) and sup ∥ u ∥ ≤ r ⁡ [ 𝒰 ⁢ ( u ) ] β < ∞ ( 𝒰 is bounded).

(II) Claim: If ( U ¯ 3 0 ) holds, then U 0 : U α → C d β ⁢ ( Ω 1 ) is continuous. Let ( u l ) l ∈ ℕ be a sequence in U α satisfying lim l → ∞ ⁡ ∥ u l - u ∥ 0 = 0 . If r > 0 is chosen sufficiently large that u , u l ∈ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) holds for all l ∈ ℕ , then ( U ¯ 3 0 ) yields

| [ 𝒰 ⁢ ( u l ) - 𝒰 ⁢ ( u ) ] ⁢ ( x ) - [ 𝒰 ⁢ ( u l ) - 𝒰 ⁢ ( u ) ] ⁢ ( x ¯ ) |

≤ ∫ Ω | f ⁢ ( x , y , u l ⁢ ( y ) ) - f ⁢ ( x , y , u ⁢ ( y ) ) - [ f ⁢ ( x ¯ , y , u l ⁢ ( y ) ) - f ⁢ ( x ¯ , y , u ⁢ ( y ) ) ] | ⁢ d μ ⁢ ( y ) ⁢ ⁢ (by (2.1))

≤ ∫ Ω c r 0 ⁢ ( | u l ⁢ ( y ) - u ⁢ ( y ) | , y ) ⁢ d μ ⁢ ( y ) ⁢ d ⁢ ( x , x ¯ ) β ⁢ ⁢ (by (2.10))

≤ ∫ Ω c ¯ r 0 ⁢ ( ∥ u l - u ∥ 0 , y ) ⁢ d μ ⁢ ( y ) ⁢ d ⁢ ( x , x ¯ ) β ⁢ ⁢ (by (2.11))

for all x , x ¯ ∈ Ω 1 , and hence

[ 𝒰 ⁢ ( u l ) - 𝒰 ⁢ ( u ) ] β ≤ ∫ Ω c r 0 ⁢ ( ∥ u l - u ∥ 0 , y ) ⁢ d μ ⁢ ( y ) .

This shows lim l → ∞ ⁡ [ 𝒰 ⁢ ( u l ) - 𝒰 ⁢ ( u ) ] β = 0 and, combined with (2.6), the claim results. ∎

Complete continuity of 𝒰 can be achieved by, e.g., slightly increasing the image space.

Corollary 2.12 (Complete continuity of U ).

If additionally ( U ¯ 3 0 ) holds, then U : U α → C d γ ⁢ ( Ω 1 ) is completely continuous, provided one of the following holds:

α ∈ ( 0 , 1 ] , γ = β and Z is closed.
Ω 1 is bounded, α ∈ ( 0 , 1 ] , γ ∈ [ 0 , β ) , and Z is closed.
Ω 1 is compact, γ ∈ [ 0 , β ] , and ( U 1 0 ) holds with

lim x → x 0 ⁡ ∫ Ω h r 0 ⁢ ( x , x 0 , y ) ⁢ d μ ⁢ ( y ) d ⁢ ( x , x 0 ) β = 0

uniformly in x 0 ∈ Ω 1 .
Ω 1 is compact and γ ∈ [ 0 , β ) .

Proof.

We write 𝒰 0 : U α → C d β ⁢ ( Ω 1 ) and 𝒰 α γ : U α → C d γ ⁢ ( Ω 1 ) for the operator given in (2.1).

(I) Claim: If α ∈ ( 0 , 1 ] , then U 0 : U α → C d β ⁢ ( Ω 1 ) is completely continuous.

Let ( u l ) l ∈ ℕ be a bounded sequence in U α , i.e. there exists a r > 0 such that

(2.12) sup x ∈ Ω ⁡ | u l ⁢ ( x ) | ≤ r , sup x , x ¯ ∈ Ω , x ≠ x ¯ ⁡ | u l ⁢ ( x ) - u l ⁢ ( x ¯ ) | d ⁢ ( x , x ¯ ) α ≤ r for all ⁢ l ∈ ℕ .

Thus, the subset { u l } l ∈ ℕ ⊂ C n 0 ⁢ ( Ω ) is bounded and equicontinuous. By the Arzelà–Ascoli theorem [38, p. 31, Theorem 3.2], there exist a subsequence ( u k l ) l ∈ ℕ and a u ∈ C n 0 ⁢ ( Ω ) with lim l → ∞ ⁡ ∥ u k l - u ∥ 0 = 0 . Because Z is closed, we have u ⁢ ( x ) ∈ Z for all x ∈ Ω , i.e. u ∈ U 0 . Since (2.12) also holds for each u k l , passing to the limit l → ∞ shows u ∈ U α . The continuity shown in Theorem 2.11 implies

lim l → ∞ ⁡ ∥ 𝒰 0 ⁢ ( u k l ) - 𝒰 0 ⁢ ( u ) ∥ β = 0 .

This establishes that every bounded sequence in 𝒰 0 ⁢ ( U α ) ⊂ C d β ⁢ ( Ω 1 ) has a convergent subsequence, i.e. the image

𝒰 0 ⁢ ( U α ∩ B ¯ r ⁢ ( 0 , C n α ⁢ ( Ω ) ) )

is relatively compact. Therefore, 𝒰 0 maps bounded subsets of U α into relatively compact sets. This shows that 𝒰 0 is completely continuous.

(II) Claim: If (iii) holds, then U 0 β : U 0 → C d β ⁢ ( Ω 1 ) is completely continuous. Let B ⊂ C n 0 ⁢ ( Ω ) be bounded and choose r > 0 so large that ∥ u ∥ 0 ≤ r holds for all u ∈ B . We establish that 𝒰 0 β ⁢ ( B ) ⊂ C d β ⁢ ( Ω 1 ) fulfills the assumptions of Theorem A.21. First, 𝒰 0 β ⁢ ( B ) is bounded due to Theorem 2.11. Second, given ε > 0 , by assumption (iii) there exists a δ > 0 such that d ⁢ ( x , x ¯ ) ≤ δ implies the estimate

∫ Ω h r ⁢ ( x , x ¯ , y ) ⁢ d μ ⁢ ( y ) d ⁢ ( x , x ¯ ) β ≤ ε for all ⁢ x , x ¯ ∈ Ω 1 , x ≠ x ¯ .

Consequently, we obtain that

| 𝒰 ⁢ ( u ) ⁢ ( x ) - 𝒰 ⁢ ( u ) ⁢ ( x ¯ ) | ⁢ ≤ ( 2.1 ) ⁢ ∫ Ω | f ⁢ ( x , y , u ⁢ ( y ) ) - f ⁢ ( x ¯ , y , u ⁢ ( y ) ) | ⁢ d μ ⁢ ( y ) ⁢ ≤ ( 2.3 ) ⁢ ∫ Ω h r 0 ⁢ ( x , x 0 , y ) ⁢ d μ ⁢ ( y ) ≤ ε ⁢ d ⁢ ( x , x ¯ ) β .

Thus, the bounded 𝒰 0 β ⁢ ( B ) ⊂ C d β ⁢ ( Ω 1 ) is relatively compact. Hence, 𝒰 0 β is completely continuous.

(III) Under (i), the mapping 𝒰 = 𝒰 0 is completely continuous due to step (I). Under (ii), the map 𝒰 = ℐ β γ ∘ 𝒰 0 is a composition with a continuous embedding ℐ β γ (cf. Theorem A.18) with the completely continuous 𝒰 0 . Under assumption (iii), the operator

𝒰 = ℐ β γ ⁢ 𝒰 0 β ⁢ 𝒰 α 0

is a composition of bounded embeddings (see Theorem A.18) with the due to step (II) completely continuous 𝒰 0 β . Finally, under (iv), the embedding ℐ β γ in 𝒰 = ℐ β γ ⁢ 𝒰 0 is compact thanks to Theorem A.22. In conclusion, at least one function in the above compositions is completely continuous and the claim results from [48, pp. 25–26, Theorem 2.1 (2)]. ∎

For a Lipschitz condition we have to investigate continuous differentiability of the kernel function.

Corollary 2.13.

If additionally ( U ¯ 1 1 ) and ( U 0 ′ ) with ϑ = 1 hold on a convex set Z ⊆ R n , then U : U α → C d β ⁢ ( Ω 1 ) is Lipschitz on bounded sets, that is,

[ 𝒰 | U α ∩ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) ] 1 ≤ max { ess ⁢ sup ξ ∈ Ω 1 ∫ Ω l r ( ξ , y ) d μ ( y ) , ∫ Ω h ¯ r 1 ( y ) d μ ( y ) } for all r > 0 .

Proof.

From Remark 2.10 (1) and Corollary 2.9, we obtain that 𝒰 : U α → C d 0 ⁢ ( Ω 1 ) is Lipschitz on bounded sets. Let r > 0 and choose u , u ¯ ∈ U α ∩ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) . Since Z is convex, the inclusion u ⁢ ( y ) + θ ⁢ ( u l ⁢ ( y ) - u ⁢ ( y ) ) ∈ Z holds for all y ∈ Ω and θ ∈ [ 0 , 1 ] . Hence, [35, p. 341, Theorem 4.2] applies and implies

[ 𝒰 ⁢ ( u ) - 𝒰 ⁢ ( u ¯ ) ] ⁢ ( x ) - [ 𝒰 ⁢ ( u ) - 𝒰 ⁢ ( u ¯ ) ] ⁢ ( x ¯ )

= ∫ Ω f ⁢ ( x , y , u ⁢ ( y ) ) - f ⁢ ( x , y , u ¯ ⁢ ( y ) ) - [ f ⁢ ( x ¯ , y , u ⁢ ( y ) ) - f ⁢ ( x ¯ , y , u ¯ ⁢ ( y ) ) ] ⁢ d ⁢ μ ⁢ ( y ) (by (2.1))

= ∫ Ω ∫ 0 1 D 3 ⁢ f ⁢ ( x , y , u ¯ ⁢ ( y ) + θ ⁢ ( u ⁢ ( y ) - u ¯ ⁢ ( y ) ) ) - D 3 ⁢ f ⁢ ( x ¯ , y , u ¯ ⁢ ( y ) + θ ⁢ ( u ⁢ ( y ) - u ¯ ⁢ ( y ) ) ) ⁢ d ⁢ θ ⁢ [ u ⁢ ( y ) - u ¯ ⁢ ( y ) ] ⁢ d ⁢ μ ⁢ ( y ) ,

and consequently ( U ¯ 1 1 ) leads to

| [ 𝒰 ⁢ ( u ) - 𝒰 ⁢ ( u ¯ ) ] ⁢ ( x ) - [ 𝒰 ⁢ ( u ) - 𝒰 ⁢ ( u ¯ ) ] ⁢ ( x ¯ ) |

≤ ∫ Ω ∫ 0 1 | D 3 ⁢ f ⁢ ( x , y , u ¯ ⁢ ( y ) + θ ⁢ ( u ⁢ ( y ) - u ¯ ⁢ ( y ) ) ) - D 3 ⁢ f ⁢ ( x ¯ , y , u ¯ ⁢ ( y ) + θ ⁢ ( u ⁢ ( y ) - u ¯ ⁢ ( y ) ) ) | ⁢ d θ ⁢ | u ⁢ ( y ) - u ¯ ⁢ ( y ) | ⁢ d μ ⁢ ( y )

≤ ∫ Ω h ¯ r 1 ⁢ ( y ) ⁢ d μ ⁢ ( y ) ⁢ d ⁢ ( x , x ¯ ) β ⁢ ∥ u - u ¯ ∥ 0 (by (2.9)) for all ⁢ x , x ¯ ∈ Ω 1 .

This guarantees the estimate

[ 𝒰 ⁢ ( u ) - 𝒰 ⁢ ( u ¯ ) ] β ≤ ∫ Ω h ¯ r 1 ⁢ ( y ) ⁢ d μ ⁢ ( y ) ⁢ ∥ u - u ¯ ∥ 0 .

Combined with (2.8), it results that 𝒰 is Lipschitz on bounded sets. ∎

2.2 Continuous differentiability

In the following, we investigate the smoothness of the Urysohn operators (2.1).

Lemma 2.14.

Assume that ( U 0 k ) , ( U 1 k ) and ( U 2 k ) hold for some k ∈ N . Then

𝒰 k : U α → L k ⁢ ( C n α ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) )

given by

(2.13) 𝒰 k ⁢ ( u ) ⁢ v 1 ⁢ ⋯ ⁢ v k := ∫ Ω D 3 k ⁢ f ⁢ ( ⋅ , y , u ⁢ ( y ) ) ⁢ v 1 ⁢ ( y ) ⁢ ⋯ ⁢ v k ⁢ ( y ) ⁢ d μ ⁢ ( y ) for all ⁢ v 1 , … , v k ∈ C n α ⁢ ( Ω ) ,

is well-defined and continuous.

Proof.

The well-definedness of 𝒰 k is shown verbatim to step (I) of the proof for Proposition 2.6, and we hence focus on continuity. Given u ∈ U α , let ( u l ) l ∈ ℕ be a sequence in U α with lim l → ∞ ⁡ ∥ u l - u ∥ 0 = 0 . Choose r > 0 so large that u , u l ∈ B r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) for all l ∈ ℕ . By using ( U 0 k ) , this leads to

lim l → ∞ ⁡ D 3 k ⁢ f ⁢ ( x , y , u l ⁢ ( y ) ) = D 3 k ⁢ f ⁢ ( x , y , u ⁢ ( y ) ) for all ⁢ x ∈ Ω 1 ⁢ and almost all ⁢ y ∈ Ω .

For ε > 0 , there exists a δ > 0 such that subsets Ω ~ ⊆ Ω satisfying μ ⁢ ( Ω ~ ) ≤ δ fulfill

∫ Ω ~ b r k ⁢ ( x , y ) ⁢ d μ ⁢ ( y ) ≤ ε 4 ,

and Egoroff’s theorem [15, p. 87, Proposition 3.1.3] yields a Ω ′ ⊆ Ω with μ ⁢ ( Ω ′ ) ≤ δ and an L 1 ∈ ℕ such that

| D 3 k ⁢ f ⁢ ( x , y , u l ⁢ ( y ) ) - D 3 k ⁢ f ⁢ ( x , y , u ⁢ ( y ) ) | ≤ ε 2 ⁢ μ ⁢ ( Ω ) for all ⁢ x ∈ Ω 1 , y ∈ Ω ∖ Ω ′ , l ≥ L 1 .

From this, for any x ∈ Ω 1 and integers l ≥ L 1 , we arrive at

| [ 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ] ⁢ v 1 ⁢ ⋯ ⁢ v k ⁢ ( x ) | ≤ ∫ Ω ∖ Ω ′ | D 3 k ⁢ f ⁢ ( x , y , u l ⁢ ( y ) ) - D 3 k ⁢ f ⁢ ( x , y , u ⁢ ( y ) ) | ⁢ d μ ⁢ ( y )

+ ∫ Ω ′ | D 3 k ⁢ f ⁢ ( x , y , u l ⁢ ( y ) ) - D 3 k ⁢ f ⁢ ( x , y , u ⁢ ( y ) ) | ⁢ d μ ⁢ ( y ) (by (2.13))

≤ ∫ Ω ∖ Ω ′ ε 2 ⁢ μ ⁢ ( Ω ) ⁢ d μ ⁢ ( y ) + 2 ⁢ ∫ Ω ′ b r k ⁢ ( x , y ) ⁢ d μ ⁢ ( y )

≤ ε 2 + 2 ⁢ ε 4

= ε .

Passing first to the supremum over x ∈ Ω 1 therefore implies

(2.14) ∥ [ 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ] ⁢ v 1 ⁢ ⋯ ⁢ v k ∥ 0 ≤ ε for all ⁢ l ≥ L 1 , v 1 , … , v k ∈ B ¯ 1 ⁢ ( 0 , C n α ⁢ ( Ω ) ) ,

and second over the vectors v 1 , … , v k yields

∥ 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ∥ L k ⁢ ( C n α ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) ) ≤ ε for all ⁢ l ≥ L 1 .

Hence, since u was arbitrary, 𝒰 k is continuous. ∎

Proposition 2.15 (Continuous differentiability of U ).

Let m ∈ N . Assume that ( U 0 k ) , ( U 1 k ) and ( U 2 k ) hold for all 0 ≤ k ≤ m on a convex set Z ⊆ R n . Then an Urysohn operator U : U α → C d 0 ⁢ ( Ω 1 ) is m-times continuously differentiable with D k ⁢ U = U k for every 1 ≤ k ≤ m .

Proof.

(I) Thanks to Lemma 2.14, the mappings

𝒰 k : U α → L k ⁢ ( C n α ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) )

are well-defined and continuous for 0 ≤ k ≤ m . Let u ∈ U α and h ∈ C d α ⁢ ( Ω ) such that u + h ∈ U α . Due to the convexity of Z, the inclusion u ⁢ ( y ) + θ ⁢ h ⁢ ( y ) ∈ Z holds for all y ∈ Ω and θ ∈ [ 0 , 1 ] . Then the remainder functions

r k ⁢ ( h ) := sup θ ∈ [ 0 , 1 ] ⁡ ∥ 𝒰 k + 1 ⁢ ( u + θ ⁢ h ) - 𝒰 k + 1 ⁢ ( u ) ∥ L k + 1 ⁢ ( C n α ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) )

satisfy lim h → 0 ⁡ r k ⁢ ( h ) = 0 for all 0 ≤ k < m . Now we obtain from [35, p. 341, Theorem 4.2] that

[ 𝒰 k ⁢ ( u + h ) - 𝒰 k ⁢ ( u ) - 𝒰 k + 1 ⁢ ( u ) ⁢ h ] ⁢ ( x )

= ∫ Ω D 3 k ⁢ f ⁢ ( x , y , u ⁢ ( y ) + h ⁢ ( y ) ) - D 3 k ⁢ f ⁢ ( x , y , u ⁢ ( y ) ) - D 3 k + 1 ⁢ f ⁢ ( x , y , u ⁢ ( y ) ) ⁢ h ⁢ ( y ) ⁢ d ⁢ μ ⁢ ( y ) ⁢ ⁢ (by (2.13))

= ∫ Ω ∫ 0 1 [ D 3 k + 1 ⁢ f ⁢ ( x , y , u ⁢ ( y ) + θ ⁢ h ⁢ ( y ) ) - D 3 k + 1 ⁢ f ⁢ ( x , y , u ⁢ ( y ) ) ] ⁢ h ⁢ ( y ) ⁢ d θ ⁢ d μ ⁢ ( y )

= ∫ 0 1 [ ( 𝒰 k + 1 ⁢ ( u + θ ⁢ h ) - 𝒰 k ⁢ ( u ) ) ⁢ h ] ⁢ ( x ) ⁢ d θ ⁢ ⁢ (by (2.13))

by Fubini’s theorem [17, p. 155, Theorem 14.1]. Consequently,

| [ 𝒰 k ⁢ ( u + h ) - 𝒰 k ⁢ ( u ) - 𝒰 k + 1 ⁢ ( u ) ⁢ h ] ⁢ ( x ) | ≤ ∫ 0 1 ∥ 𝒰 k + 1 ⁢ ( u + θ ⁢ h ) - 𝒰 k + 1 ⁢ ( u ) ∥ L k + 1 ⁢ ( C n α ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) ) ⁢ d θ ⁢ ∥ h ∥ 0

≤ r k + 1 ⁢ ( h ) ⁢ ∥ h ∥ α for all ⁢ x ∈ Ω 1 .

After passing to the least upper bound over x ∈ Ω 1 , it results in

(2.15) ∥ 𝒰 k ⁢ ( u + h ) - 𝒰 k ⁢ ( u ) - 𝒰 k + 1 ⁢ ( u ) ⁢ h ∥ L k ⁢ ( C n α ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) ) ≤ r k ⁢ ( h ) ⁢ ∥ h ∥ α .

This establishes that

𝒰 k : U α → L k ⁢ ( C n α ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) )

is differentiable in u with the derivative 𝒰 k + 1 ⁢ ( u ) .

(II) Applying step (I) in case k = 0 shows that 𝒰 is differentiable on U α with the derivative 𝒰 1 . Given this, mathematical induction yields that 𝒰 : U α → C d 0 ⁢ ( Ω 1 ) is actually m-times differentiable with the derivatives D k ⁢ 𝒰 = 𝒰 k for all 1 ≤ k ≤ m , which in turn are continuous due to Lemma 2.14. ∎

We proceed to Urysohn operators having values in a Hölder space.

Lemma 2.16.

Assume that ( U 0 k ) , ( U ¯ 1 k ) and ( U 2 k ) hold for some k ∈ N . Then

𝒰 k : U α → L k ⁢ ( C n α ⁢ ( Ω ) , C d β ⁢ ( Ω 1 ) )

given by (2.13) is well-defined. If additionally ( U ¯ 3 k ) holds, then U k is continuous.

Proof.

The well-definedness of 𝒰 k follows as in step (I) from the proof of Theorem 2.11. Let u ∈ U α and ( u l ) l ∈ ℕ denote a sequence in U α fulfilling the limit relation lim l → ∞ ⁡ ∥ u l - u ∥ 0 = 0 . In addition, choose r > 0 sufficiently large so that the inclusion u , u l ∈ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) for each l ∈ ℕ holds. Therefore, by (2.13),

[ ( 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ) ⁢ v 1 ⁢ ⋯ ⁢ v k ] ⁢ ( x ) - [ ( 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ) ⁢ v 1 ⁢ ⋯ ⁢ v k ] ⁢ ( x ¯ )

= ∫ Ω [ D 3 k ⁢ f ⁢ ( x , y , u l ⁢ ( y ) ) - D 3 k ⁢ f ⁢ ( x , y , u ⁢ ( y ) ) - [ D 3 k ⁢ f ⁢ ( x ¯ , y , u l ⁢ ( y ) ) - D 3 k ⁢ f ⁢ ( x ¯ , y , u ⁢ ( y ) ) ] ] ⋅ v 1 ⁢ ( y ) ⁢ ⋯ ⁢ v k ⁢ ( y ) ⁢ d μ ⁢ ( y ) .

After passing to the norm, our assumption ( U ¯ 3 k ) results in

| [ ( 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ) ⁢ v 1 ⁢ ⋯ ⁢ v k ] ⁢ ( x ) - [ ( 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ) ⁢ v 1 ⁢ ⋯ ⁢ v k ] ⁢ ( x ¯ ) |

≤ ∫ Ω c r k ⁢ ( | u l ⁢ ( y ) - u ⁢ ( y ) | , y ) ⁢ d μ ⁢ ( y ) ⁢ d ⁢ ( x , x ¯ ) β ⁢ ⁢ (by (2.10))

≤ ∫ Ω c ¯ r k ⁢ ( ∥ u l - u ∥ 0 , y ) ⁢ d μ ⁢ ( y ) ⁢ d ⁢ ( x , x ¯ ) β ⁢ ⁢ (by (2.11)) for all ⁢ x , x ¯ ∈ Ω 1 .

Consequently, for v 1 , … , v k ∈ B ¯ 1 ⁢ ( 0 , C n α ⁢ ( Ω ) ) we derive the estimate

[ ( 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ) ⁢ v 1 ⁢ ⋯ ⁢ v k ] β ≤ ∫ Ω c ¯ r k ⁢ ( ∥ u l - u ∥ 0 , y ) ⁢ d μ ⁢ ( y ) .

In particular, given ε > 0 , there exists a L 2 ∈ ℕ such that

[ ( 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ) ⁢ v 1 ⁢ ⋯ ⁢ v k ] β ≤ ε for all ⁢ l ≥ L 2 , v 1 , … , v k ∈ B ¯ 1 ⁢ ( 0 , C n α ⁢ ( Ω ) ) .

In conclusion, with (2.14) this implies

∥ ( 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ) ⁢ v 1 ⁢ ⋯ ⁢ v k ∥ β = max ⁡ { ∥ ( 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ) ⁢ v 1 ⁢ ⋯ ⁢ v k ∥ 0 , [ ( 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ) ⁢ v 1 ⁢ ⋯ ⁢ v k ] β }

≤ ε for all ⁢ v 1 , … , v k ∈ B ¯ 1 ⁢ ( 0 , C n α ⁢ ( Ω ) ) ,

and thus

∥ 𝒰 k ⁢ ( u l ) - 𝒰 k ⁢ ( u ) ∥ L k ⁢ ( C n α ⁢ ( Ω ) , C d β ⁢ ( Ω 1 ) ) ≤ ε for all ⁢ l ≥ max ⁡ { L 1 , L 2 } .

Therefore, because the function u was arbitrarily chosen, 𝒰 k is continuous on U α . ∎

In contrast to Proposition 2.15, establishing continuous differentiability now requires to invest one additional order of differentiability on the kernel function.

Theorem 2.17 (Continuous differentiability of U ).

Let m ∈ N . Assume that ( U 0 k ) , ( U ¯ 1 k ) , ( U 2 k ) , and ( U ¯ 3 k ) hold for all 0 ≤ k ≤ m on a convex set Z ⊆ R n . Then an Urysohn operator U : U α → C d β ⁢ ( Ω 1 ) is m-times continuously differentiable with D k ⁢ U = U k for every 1 ≤ k ≤ m .

Proof.

We establish the assertion for 𝒰 : U α → C d β ⁢ ( Ω 1 ) first. Let u ∈ U α and h ∈ C n α ⁢ ( Ω ) such that u + h ∈ U α . Moreover, choose r > 0 so large that u , u + h ∈ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) holds.

(I) Let 0 ≤ k < m . Above all, with the function c ¯ r k : ℝ + × Ω 1 → ℝ + in (2.11) we observe that

ρ k ⁢ ( h ) := ∫ 0 1 ∫ Ω c ¯ r k + 1 ⁢ ( θ ⁢ ∥ h ∥ 0 , y ) ⁢ d μ ⁢ ( y ) ⁢ d θ

satisfies lim h → 0 ⁡ ρ k ⁢ ( h ) = 0 . Given arbitrary x , x ¯ ∈ Ω 1 , again [35, p. 341, Theorem 4.2] yields

[ 𝒰 k ⁢ ( u + h ) - 𝒰 k ⁢ ( u ) - 𝒰 k + 1 ⁢ ( u ) ] ⁢ ( x ) - [ 𝒰 k ⁢ ( u + h ) - 𝒰 k ⁢ ( u ) - 𝒰 k + 1 ⁢ ( u ) ] ⁢ ( x ¯ )

= ∫ Ω ∫ 0 1 [ D 3 k + 1 f ( x , y , u ( y ) + θ h ( y ) ) - D 3 k + 1 f ( x , y , u ( y ) ) (by (2.13))

- ( D 3 k + 1 f ( x ¯ , y , u ( y ) + θ h ( y ) ) - D 3 k + 1 f ( x ¯ , y , u ( y ) ) ) ] d θ h ( y ) d μ ( y )

= ∫ 0 1 ∫ Ω [ D 3 k + 1 f ( x , y , u ( y ) + θ h ( y ) ) - D 3 k + 1 f ( x , y , u ( y ) )

- ( D 3 k + 1 f ( x ¯ , y , u ( y ) + θ h ( y ) ) - D 3 k + 1 f ( x ¯ , y , u ( y ) ) ) ] h ( y ) d μ ( y ) d θ

due to Fubini’s theorem [17, p. 155, Theorem 14.1]. Using assumption ( U ¯ 3 k + 1 ) , we obtain

| [ 𝒰 k ⁢ ( u + h ) - 𝒰 k ⁢ ( u ) - 𝒰 k + 1 ⁢ ( u ) ⁢ h ] ⁢ ( x ) - [ 𝒰 k ⁢ ( u + h ) - 𝒰 k ⁢ ( u ) - 𝒰 k + 1 ⁢ ( u ) ⁢ h ] ⁢ ( x ¯ ) |

≤ ∫ 0 1 ∫ Ω c r k + 1 ⁢ ( θ ⁢ | h ⁢ ( y ) | , y ) ⁢ | h ⁢ ( y ) | ⁢ d μ ⁢ ( y ) ⁢ d θ ⁢ d ⁢ ( x , x ¯ ) β ⁢ ⁢ (by (2.10))

≤ ∫ 0 1 ∫ Ω c ¯ r k + 1 ⁢ ( θ ⁢ ∥ h ∥ 0 , y ) ⁢ d μ ⁢ ( y ) ⁢ d θ ⁢ d ⁢ ( x , x ¯ ) β ⁢ ∥ h ∥ α ⁢ ⁢ (by (2.11)) ,

which in turn implies

[ 𝒰 k ⁢ ( u + h ) - 𝒰 k ⁢ ( u ) - 𝒰 k + 1 ⁢ ( u ) ⁢ h ] β ≤ ρ k ⁢ ( h ) ⁢ ∥ h ∥ α .

If we combine this with inequalities (2.15), then

∥ 𝒰 k ⁢ ( u + h ) - 𝒰 k ⁢ ( u ) - 𝒰 k + 1 ⁢ ( u ) ⁢ h ∥ β ≤ R k ⁢ ( h ) ⁢ ∥ h ∥ α

with the remainder term R k ⁢ ( h ) := max ⁡ { r k ⁢ ( h ) , ρ k ⁢ ( h ) } satisfying the desired limit relation lim h → 0 ⁡ R k ⁢ ( h ) = 0 .

(II) Applying step (I) in case k = 0 shows that 𝒰 is differentiable on U α with the derivative 𝒰 1 . Given this, mathematical induction yields that 𝒰 : U α → C d β ⁢ ( Ω 1 ) is actually m-times differentiable with the derivatives D k ⁢ 𝒰 = 𝒰 k for all 0 ≤ k ≤ m . Their continuity is guaranteed by Lemma 2.16. ∎

We close our general analysis of Urysohn operators with several remarks.

Remark 2.18 (Boundedness and continuity of U ).

(1) The boundedness of Urysohn operators 𝒰 stated in Proposition 2.6 and Theorem 2.11 actually means that the 𝒰 -images of merely ∥ ⋅ ∥ 0 -bounded subsets B ⊂ U α are bounded. This means that the functions in B need not to have uniformly bounded Hölder constants.

(2) The continuity statements for the Urysohn operator 𝒰 in Proposition 2.6 and Theorem 2.11, as well as for its derivatives D k ⁢ 𝒰 in Proposition 2.15 and Theorem 2.17 are to be understood in the following strong form: Already convergence in the domain U α with respect to the norm ∥ ⋅ ∥ 0 is sufficient for convergence of the 𝒰 -values in the norm ∥ ⋅ ∥ 0 , resp. ∥ ⋅ ∥ β . A corresponding statement applies to both Corollary 2.9 and 2.13.

Remark 2.19 (Urysohn operators U : U α → C d γ ⁢ ( Ω 1 ) ).

The above statements extend to Urysohn operators mapping into the γ-Hölder functions over bounded metric spaces Ω 1 . This is due to the corresponding representation ℐ β γ ⁢ 𝒰 : U α → C d γ ⁢ ( Ω 1 ) , where the embedding ℐ β γ from Theorem A.18 is continuous.

Remark 2.20 (Nyström methods).

Let Ω ( l ) be a discrete subset of a compact set Ω 1 ⊂ ℝ κ and suppose w η ≥ 0 are nonnegative reals, η ∈ Ω ( l ) . In Example 2.1, resp. Example 2.2, we pointed out that both Urysohn operators

𝒰 : U α → C d β ⁢ ( Ω 1 ) , ⁢ 𝒰 ⁢ ( u ) := ∫ Ω 1 f ⁢ ( ⋅ , y , u ⁢ ( y ) ) ⁢ d y ,

𝒰 l : U α ( l ) → C d β ⁢ ( Ω 1 ) , ⁢ 𝒰 l ⁢ ( u ) := ∑ η ∈ Ω ( l ) w η ⁢ f ⁢ ( ⋅ , η , u ⁢ ( η ) ) ,

fit well into our abstract setting, where we abbreviated

U α ( l ) := { u : Ω ( l ) → Z ∣ u ∈ C n α ( Ω ( l ) ) } .

However, when dealing with Nyström methods or for iterating integral operators 𝒰 l , it is desirable to work with Urysohn operators defined on U α rather than U α ( l ) . For this purpose, let us introduce the linear operator E l : C n α ⁢ ( Ω 1 ) → C n α ⁢ ( Ω ( l ) ) given by E l u := u | Ω ( l ) . It satisfies E l ⁢ U α ⊆ U α ( l ) and is easily seen to be bounded with

∥ E l ∥ C n α ⁢ ( Ω 1 ) , C n α ⁢ ( Ω ( l ) ) ≤ 1 .

Hence, rather than 𝒰 l we consider the composition

𝒰 l ′ : U α → C d β ⁢ ( Ω 1 ) , 𝒰 l ′ ⁢ ( u ) := 𝒰 l ⁢ ( E l ⁢ u ) = ∑ η ∈ Ω ( l ) w η ⁢ f ⁢ ( ⋅ , η , u ⁢ ( η ) ) ,

which, under appropriate assumptions on the kernel function f, inherits its properties from 𝒰 l .

2.3 Convolutive operators

In our above analysis, the Hölder continuity of an image 𝒰 ⁢ ( u ) : Ω 1 → ℝ d of a general Urysohn operator (2.1) was guaranteed and prescribed by the exponent of the kernel function f in its first variable from assumption ( U ¯ 1 0 ) . A higher degree of smoothness cannot be expected, as simple examples like the kernel function f ⁢ ( x , y , z ) := f 1 ⁢ ( x ) illustrate, where

𝒰 ⁢ ( u ) ⁢ ( x ) = ∫ Ω f 1 ⁢ ( x ) ⁢ d μ ⁢ ( y ) = μ ⁢ ( Ω ) ⁢ f 1 ⁢ ( x )

inherits its smoothness from f 1 . This situation changes for kernel functions of convolution type. Here the smoothness (Hölder continuity, differentiability) of the arguments u transfers to the images 𝒰 ⁢ ( u ) , i.e. such integral operators possess a smoothing property we are about to analyze over the course of this section. Our results generalize those of [26, pp. 52ff, Section 3.4.2] obtained for linear operators.

To be more precise, let us restrict to a compact interval Ω = Ω 1 = [ a , b ] , a < b , equipped with the 1-dimensional Lebesgue measure μ = λ 1 in (2.1). Moreover, the kernel function is of the form

f ⁢ ( x , y , z ) = f ~ ⁢ ( x - y , z )

with a function f ~ : [ a - b , b - a ] × Z → ℝ d . This yields a convolutive Urysohn operator

(2.16) 𝒰 ~ : U α → F ⁢ ( [ a , b ] , ℝ d ) , 𝒰 ~ ⁢ ( u ) ⁢ ( x ) := ∫ a b f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ d y for all ⁢ x ∈ [ a , b ] .

Hypothesis.

Let m ∈ ℕ 0 and Ω ~ := [ a - b , b - a ] . With 0 ≤ k ≤ m , one assumes the following:

The partial derivative D 2 k ⁢ f ~ ⁢ ( y , ⋅ ) : Z → ℝ d exists and is continuous for Lebesgue-almost all y ∈ Ω ~ .
D 2 k ⁢ f ~ ⁢ ( ⋅ , z ) : Ω ~ → ℝ d is measurable for all z ∈ Z , and for every r > 0 there exists an integrable function b ~ r k : Ω ~ → ℝ + so that for Lebesgue-almost all y ∈ Ω ~ the following holds:

(2.17) | D 2 k ⁢ f ~ ⁢ ( y , z ) | ≤ b ~ r k ⁢ ( y ) for all ⁢ z ∈ Z ∩ B ¯ r ⁢ ( 0 ) .

For all r > 0 , there exists an integrable function l ~ r : Ω ~ → ℝ + , so that for Lebesgue-almost all y ∈ Ω ~ the following holds:

(2.18) | f ~ ⁢ ( y , z ) - f ~ ⁢ ( y , z ¯ ) | ≤ l ~ r ⁢ ( y ) ⁢ | z - z ¯ | for all ⁢ z , z ¯ ∈ Z ∩ B ¯ r ⁢ ( 0 ) .

Note that the Lipschitz condition ( C 2 ) implies ( C 0 0 ) , but also (2.17) for k = 1 with b ~ r 1 = l ~ r .

Under these assumptions, the Hölder continuity of u ∈ U α carries over to the values 𝒰 ~ ⁢ ( u ) .

Theorem 2.21 (Hölder continuity of U ~ ⁢ ( u ) ).

Assume that ( C 1 0 ) and ( C 2 ) hold and α ∈ ( 0 , 1 ] . If u ∈ U α , r > ∥ u ∥ 0 and there exists a real C ≥ 0 satisfying

(2.19) ∫ x x ¯ b ~ r 0 ⁢ ( y ) ⁢ d y ≤ C ⁢ ( x ¯ - x ) α for all ⁢ a - b ≤ x ≤ x ¯ ≤ b - a ,

then the image satisfies

𝒰 ~ ⁢ ( u ) ∈ C d α ⁢ [ a , b ] .

Proof.

Let u ∈ U α and r > ∥ u ∥ 0 .

(I) Let x ∈ [ a , b ] be given. Above all, ( C 2 ) implies that f ~ ⁢ ( y , ⋅ ) : Z → ℝ d is continuous for Lebesgue-almost all y ∈ Ω ~ (i.e. assumption ( C 0 0 ) holds). Combined with the measurability assumed in ( C 1 0 ) , we conclude from [48, p. 62, Lemma 5.1] that y ↦ f ~ ⁢ ( x - y , u ⁢ ( y ) ) is measurable. Moreover, due to (2.17), one has the estimate

| f ~ ⁢ ( x - y , u ⁢ ( y ) ) | ≤ b ~ r 0 ⁢ ( x - y ) ,

where

∫ a b b ~ r 0 ⁢ ( x - y ) ⁢ d y ≤ ∫ Ω ~ b ~ r 0 ⁢ ( η ) ⁢ d η for all ⁢ x ∈ [ a , b ] .

Consequently, the function y ↦ f ~ ⁢ ( x - y , u ⁢ ( y ) ) is integrable. Hence, 𝒰 ~ ⁢ ( u ) : [ a , b ] → ℝ d is well-defined.

(II) For x , x ¯ ∈ [ a , b ] with x ¯ = x + Δ > x , this results in

𝒰 ~ ⁢ ( u ) ⁢ ( x ¯ ) = ∫ a b f ~ ⁢ ( x ¯ - y , u ⁢ ( y ) ) ⁢ d y (by (2.16))

= ∫ a b f ~ ⁢ ( x + Δ - y , u ⁢ ( y ) ) ⁢ d y

= ∫ a - Δ a f ~ ⁢ ( x - η , u ⁢ ( η + Δ ) ) ⁢ d η + ∫ a b - Δ f ~ ⁢ ( x - η , u ⁢ ( η + Δ ) ) ⁢ d η

via the substitution η := y - Δ , and analogously

𝒰 ~ ⁢ ( u ) ⁢ ( x ) ⁢ = ( 2.16 ) ⁢ ∫ a b - Δ f ~ ⁢ ( x - η , u ⁢ ( η ) ) ⁢ d η + ∫ b - Δ b f ~ ⁢ ( x - η , u ⁢ ( η ) ) ⁢ d η .

Whence, the difference

𝒰 ~ ⁢ ( u ) ⁢ ( x ¯ ) - 𝒰 ~ ⁢ ( u ) ⁢ ( x ) = I 0 + I 1 + I 2

can be written as sum of the terms

I 0 := ∫ a b - Δ f ~ ⁢ ( x - η , u ⁢ ( η + Δ ) ) - f ~ ⁢ ( x - η , u ⁢ ( η ) ) ⁢ d ⁢ η ,

I 1 := ∫ a - Δ a f ~ ⁢ ( η - Δ , u ⁢ ( η + Δ ) ) ⁢ d η ,

I 2 := - ∫ b - Δ b f ~ ⁢ ( x - η , u ⁢ ( η ) ) ⁢ d η = ∫ x - b x - b + Δ f ~ ⁢ ( y , u ⁢ ( x - y ) ) ⁢ d y ,

which can be estimated separately as

| I 0 | ≤ ∫ a b - Δ | f ~ ⁢ ( x - η , u ⁢ ( η + Δ ) ) - f ~ ⁢ ( x - η , u ⁢ ( η ) ) | ⁢ d η

≤ ∫ a b - Δ l ~ r ⁢ ( x - η ) ⁢ | u ⁢ ( η + Δ ) - u ⁢ ( η ) | ⁢ d η (by (2.18))

≤ [ u ] α ⁢ ∫ a b l ~ r ⁢ ( x - η ) ⁢ d η ⁢ Δ α

≤ [ u ] α ⁢ ∫ Ω ~ l ~ r ⁢ ( η ) ⁢ d η ⁢ Δ α

and, by (2.19),

| I 1 | ≤ ∫ a - Δ a b ~ r 0 ⁢ ( x - η ) ⁢ d η ≤ C ⁢ Δ α ,

| I 2 | ≤ ∫ x - b x - b + Δ b ~ r 0 ⁢ ( y ) ⁢ d y ≤ C ⁢ Δ α .

Hence, with I 0 , I 1 and I 2 also their sum is α-Hölder due to Theorem A.12, and thus 𝒰 ~ ⁢ ( u ) ∈ C α ⁢ [ a , b ] . ∎

Theorem 2.22 (Hölder continuity of U ~ ⁢ ( u ) ).

Assume that ( C 1 0 ) and ( C 2 ) hold, α ∈ ( 0 , 1 ] , and the kernel function f ~ and the partial derivative D 1 ⁢ f ~ exist as continuous functions on both sets [ a - b , 0 ) × Z and ( 0 , b - a ] × Z . If u ∈ U α , r > ∥ u ∥ 0 and there exists an h 0 > 0 such that

∫ - h 0 h 0 l ~ r ⁢ ( y ) ⁢ d y < ∞ ,

then U ~ ⁢ ( u ) is α-Hölder on every subinterval compact in [ a , b ] .

Proof.

Let u ∈ U α . As in step (I) of the proof to Theorem 2.21, one shows that 𝒰 ~ ⁢ ( u ) is well-defined. Now, suppose that I ⊆ ( a , b ) is a compact subinterval. For each x ∈ I , we choose h ∈ ( 0 , h 0 ] so small that a ≤ x - h and x + h ≤ b hold. This allows us to represent

(2.20) 𝒰 ~ ⁢ ( u ) ⁢ ( x ) = ∫ a b f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ d y = I 1 ⁢ ( x ) + I 2 ⁢ ( x ) + I 2 ⁢ ( x )

with the functions I 0 , I 1 , I 2 : I → ℝ d given by

I 0 ⁢ ( x ) := ∫ x - h x + h f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ d y = ∫ - h h f ~ ⁢ ( η , u ⁢ ( x - η ) ) ⁢ d η ,

I 1 ⁢ ( x ) := ∫ a x - h f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ d y ,

I 2 ⁢ ( x ) := ∫ x + h b f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ d y ,

where we applied the substitution η = x - y in order to rewrite I 0 ⁢ ( x ) . First, we investigate the parameter integral I 0 . Thereto, using ( C 2 ) , we obtain

| I 0 ⁢ ( x ) - I 0 ⁢ ( x ¯ ) | ≤ ∫ - h h | f ~ ⁢ ( η , u ⁢ ( x - η ) ) - f ~ ⁢ ( η , u ⁢ ( x ¯ - η ) ) | ⁢ d y

≤ ∫ - h h l ~ r ⁢ ( y ) ⁢ | u ⁢ ( x - η ) - u ⁢ ( x ¯ - η ) | ⁢ d y (by (2.18))

≤ [ u ] α ⁢ ∫ - h h l ~ r ⁢ ( y ) ⁢ d y ⁢ | x - x ¯ | α for all ⁢ x , x ¯ ∈ I ,

and consequently also I 0 is α-Hölder on the compact subinterval I. Second, the integration variable y in the parameter integrals I 1 ⁢ ( x ) and I 2 ⁢ ( x ) satisfies 0 < h ≤ x - y , resp. x - y ≤ - h < 0 . Thus, the functions I 1 , I 2 are differentiable with the derivatives

I 1 ′ ⁢ ( x ) = ∫ a x - h D 1 ⁢ f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ d y + f ~ ⁢ ( h , u ⁢ ( x - h ) ) , I 2 ′ ⁢ ( x ) = ∫ x + h b D 1 ⁢ f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ d y + f ~ ⁢ ( - h , u ⁢ ( x + h ) )

for all x ∈ I . Because these derivatives are bounded on the interval I, we conclude from Example A.4 that I 1 , I 2 are α-Hölder on I. In conclusion, Theorem A.12 yields that the sum (2.20) is α-Hölder. ∎

Theorem 2.23 (Continuous differentiability of U ~ ⁢ ( u ) ).

Assume that ( C 0 1 ) and ( C 1 1 ) hold on a convex set Z ⊆ R n , f ~ : Ω ~ × Z → R d and D 2 ⁢ f ~ ⁢ ( ⋅ , z ) are continuous for all z ∈ Z , and α ∈ ( 0 , 1 ] . If u : [ a , b ] → Z is continuously differentiable, then also the image U ~ ⁢ ( u ) : [ a , b ] → R d is continuously differentiable with the derivative

(2.21) 𝒰 ~ ⁢ ( u ) ′ ⁢ ( x ) = f ~ ⁢ ( x - a , u ⁢ ( a ) ) - f ~ ⁢ ( x - b , u ⁢ ( b ) ) - ∫ a b D 2 ⁢ f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ u ′ ⁢ ( y ) ⁢ d y for all ⁢ x ∈ [ a , b ] .

Since the derivative 𝒰 ~ ⁢ ( u ) ′ is bounded as a continuous function over the compact interval [ a , b ] , it results from Example A.4 that 𝒰 ~ ⁢ ( u ) is α-Hölder, α ∈ ( 0 , 1 ] .

Proof.

Let u : [ a , b ] → Z be continuously differentiable and choose r > ∥ u ∥ 0 . Since f ~ is continuous, assumptions ( C 0 0 ) and ( C 1 0 ) are satisfied, and consequently as shown in step (I) of the proof to Theorem 2.21, the expression 𝒰 ~ ⁢ ( u ) is well-defined. Now fix some x ∈ [ a , b ] .

(I) Claim: One has the limit relation

(2.22) lim h → 0 ⁡ ∫ a b 1 h ⁢ ( f ~ ⁢ ( x - y , u ⁢ ( y + h ) ) - f ~ ⁢ ( x - y , u ⁢ ( y ) ) ) ⁢ d y = ∫ a b D 2 ⁢ f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ u ′ ⁢ ( y ) ⁢ d y .

We define the function

F ⁢ ( h , y ) := { 1 h ⁢ ( f ~ ⁢ ( x - y , u ⁢ ( y + h ) ) - f ~ ⁢ ( x - y , u ⁢ ( y ) ) ) , h ≠ 0 , D 2 ⁢ f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ u ′ ⁢ ( y ) , h = 0 ,

having the following properties: First, F ⁢ ( h , ⋅ ) : [ a , b ] → ℝ d is integrable for every fixed h. In order to see this, let us distinguish two cases. If h ≠ 0 , then the continuity of f ~ and u yield that F ⁢ ( h , ⋅ ) is integrable. If h = 0 , then, due to ( C 1 1 ) , the derivative D 2 ⁢ f ~ ⁢ ( ⋅ , z ) is measurable for all z ∈ Z and the continuity of u and u ′ guarantee that F ⁢ ( 0 , ⋅ ) is measurable due to [48, p. 62, Lemma 5.1]. Moreover, for Lebesgue-almost all y ∈ [ a , b ] one has

| F ⁢ ( 0 , y ) | = | D 2 ⁢ f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ u ′ ⁢ ( y ) | ≤ b ~ r 1 ⁢ ( x - y ) ⁢ ∥ u ′ ∥ 0

from (2.17). Due to

(2.23) ∫ a b b ~ r 1 ⁢ ( x - y ) ⁢ d y ≤ ∫ Ω ~ b r 1 ⁢ ( η ) ⁢ d η ,

also the function F ⁢ ( 0 , ⋅ ) is integrable.

Second, F ⁢ ( ⋅ , y ) is continuous in 0, which readily results from the chain rule [35, p. 337]. Third, applying the mean value theorem [35, p. 341, Theorem 4.2] twice leads to

F ⁢ ( h , y ) = ∫ 0 1 D 2 ⁢ f ~ ⁢ ( x - y , u ⁢ ( y ) + θ ⁢ ( u ⁢ ( y + h ) - u ⁢ ( y ) ) ) ⁢ d θ ⁢ ∫ 0 1 u ′ ⁢ ( y + θ ⁢ h ) ⁢ d θ ,

and hence

| F ⁢ ( h , y ) | ⁢ ≤ ( 2.17 ) ⁢ ∫ 0 1 b ~ r 1 ⁢ ( x - y ) ⁢ d θ ⁢ ∥ u ′ ∥ 0 ≤ b ~ r 1 ⁢ ( x - y ) ⁢ ∥ u ′ ∥ 0

for all h. Thanks to (2.23), the right-hand side of this inequality is bounded above by an integrable function independent of h (cf. ( C 1 1 ) ). Combining these three aspects, it is a consequence of the dominated convergence theorem [17, p. 149, Theorem 10.1] that taking the limit and integration in (2.22) can be exchanged, which yields the claim.

(II) Using the substitution η = x - y , we obtain the representation

𝒰 ~ ⁢ ( u ) ⁢ ( x ) ⁢ = ( 2.16 ) ⁢ ∫ a b f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ d y = ∫ x - b x - a f ~ ⁢ ( η , u ⁢ ( x - η ) ) ⁢ d η ,

which in turn yields

𝒰 ~ ⁢ ( u ) ⁢ ( x + h ) - 𝒰 ~ ⁢ ( u ) ⁢ ( x )

= ∫ x + h - b x + h - a f ~ ⁢ ( η , u ⁢ ( x + h - η ) ) ⁢ d η - ∫ x - b x - a f ~ ⁢ ( η , u ⁢ ( x - η ) ) ⁢ d η

= - ∫ x - b x + h - b f ~ ⁢ ( η , u ⁢ ( x + h - η ) ) ⁢ d η + ∫ x - b x - a f ~ ⁢ ( η , u ⁢ ( x + h - η ) ) - f ~ ⁢ ( η , u ⁢ ( x - η ) ) ⁢ d ⁢ η + ∫ x - a x + h - a f ~ ⁢ ( η , u ⁢ ( x + h - η ) ) ⁢ d η

= - ∫ x - b x + h - b f ~ ⁢ ( η , u ⁢ ( x + h - η ) ) ⁢ d η - ∫ a b f ~ ⁢ ( x - y , u ⁢ ( y + h ) ) - f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ d ⁢ y + ∫ x - a x + h - a f ~ ⁢ ( η , u ⁢ ( x + h - η ) ) ⁢ d η

for all h ∈ [ a - x , b - x ] ; note that we re-substituted η = x - y in the center term of the above sum. The mean value theorem from the integral calculus applies to each continuous component function f ~ i , 1 ≤ i ≤ d , in the first and third term in the above sum. Hence, there exist reals

ξ 1 i ⁢ ( h ) ∈ ( min ⁡ { x - b , x + h - b } , max ⁡ { x - b , x + h - b } ) ,

ξ 2 i ⁢ ( h ) ∈ ( min ⁡ { x - a , x + h - a } , max ⁡ { x - a , x + h - a } ) ,

such that the identities

- ∫ x - b x + h - b f ~ i ⁢ ( η , u ⁢ ( x + h - η ) ) ⁢ d η = - h ⁢ f ~ i ⁢ ( ξ 1 i ⁢ ( h ) , u ⁢ ( x + h - ξ 1 i ⁢ ( h ) ) ) ,

∫ x - a x + h - a f ~ i ⁢ ( η , u ⁢ ( x + h - η ) ) ⁢ d η = h ⁢ f ~ i ⁢ ( ξ 2 i ⁢ ( h ) , u ⁢ ( x + h - ξ 2 i ⁢ ( h ) ) )

hold, which allow us to conclude

1 h ⁢ ( 𝒰 ~ ⁢ ( u ) i ⁢ ( x + h ) - 𝒰 ~ ⁢ ( u ) i ⁢ ( x ) )

= 1 h ( ∫ x - a x + h - a f ~ ( η , u ( x + h - η ) ) d η - ∫ x - b x + h - b f ~ i ( η , u ( x + h - η ) ) d η

- ∫ a b f ~ i ( x - y , u ( y + h ) ) - f ~ i ( x - y , u ( y ) ) d y )

= f ~ i ⁢ ( ξ 2 i ⁢ ( h ) , u ⁢ ( x + h - ξ 2 i ⁢ ( h ) ) ) - f ~ i ⁢ ( ξ 1 i ⁢ ( h ) , u ⁢ ( x + h - ξ 1 i ⁢ ( h ) ) )

- ∫ a b 1 h ⁢ ( f ~ i ⁢ ( x - y , u ⁢ ( y + h ) ) - f ~ i ⁢ ( x - y , u ⁢ ( y ) ) ) ⁢ d y for all ⁢ 1 ≤ i ≤ d .

Thanks to the limit relations

lim h → 0 ⁡ ξ 1 i ⁢ ( h ) = x - b , lim h → 0 ⁡ ξ 2 i ⁢ ( h ) = x - a , for all ⁢ 1 ≤ i ≤ d ,

and returning to vector notation, we consequently arrive at

lim h → 0 ⁡ 1 h ⁢ ( 𝒰 ~ ⁢ ( u ) ⁢ ( x + h ) - 𝒰 ~ ⁢ ( u ) ⁢ ( x ) )

= f ~ ⁢ ( x - a , u ⁢ ( a ) ) - f ~ ⁢ ( x - b , u ⁢ ( b ) ) - lim h → 0 ⁡ ∫ a b 1 h ⁢ ( f ~ ⁢ ( x - y , u ⁢ ( y + h ) ) - f ~ ⁢ ( x - y , u ⁢ ( y ) ) ) ⁢ d y

= f ~ ⁢ ( x - a , u ⁢ ( a ) ) - f ~ ⁢ ( x - b , u ⁢ ( b ) ) - ∫ a b D 2 ⁢ f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ u ′ ⁢ ( y ) ⁢ d y (by (2.22)).

This establishes that the image 𝒰 ~ ⁢ ( u ) is differentiable in x ∈ [ a , b ] with the derivative (2.21). Hence, in order to show that 𝒰 ⁢ ( u ) ′ is continuous, it suffices to establish the continuity of the parameter integral

x ↦ ∫ a b F ~ ⁢ ( x , y ) ⁢ d y

with F ~ ⁢ ( x , y ) := D 2 ⁢ f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ u ′ ⁢ ( y ) . By assumption, it follows that F ~ ⁢ ( ⋅ , y ) is continuous on [ a , b ] . From [48, p. 62, Lemma 5.1] we conclude that F ⁢ ( x , ⋅ ) is measurable and, because of

| F ~ ⁢ ( x , y ) | = | D 2 ⁢ f ~ ⁢ ( x - y , u ⁢ ( y ) ) ⁢ u ′ ⁢ ( y ) | ⁢ ≤ ( 2.17 ) ⁢ b ~ r 1 ⁢ ( x - y ) ⁢ ∥ u ′ ∥ 0 for all ⁢ x ∈ [ a , b ] ,

integrable (uniformly in x, cf. (2.23)). Then the dominated convergence theorem [17, p. 149, Theorem 10.1] shows that

x ↦ ∫ a b F ~ ⁢ ( x , y ) ⁢ d y

is continuous, and thus 𝒰 ~ ⁢ ( u ) is continuously differentiable. ∎

3 Hammerstein integral operators

Let α ∈ [ 0 , 1 ] . Hammerstein operators are of the form

(3.1) ℋ : U α → F ⁢ ( Ω 1 , ℝ d ) , ℋ ⁢ ( u ) := ∫ Ω k ⁢ ( ⋅ , y ) ⁢ g ⁢ ( y , u ⁢ ( y ) ) ⁢ d μ ⁢ ( y )

and represent a relevant special case of the Urysohn operators studied in Section 2 having the kernel function f ⁢ ( x , y , z ) := k ⁢ ( x , y ) ⁢ g ⁢ ( y , z ) . Nevertheless, we investigate them as composition of Fredholm and Nemytskii operators. For this reason, let us study these operator classes independently first.

3.1 Fredholm integral operators

A Fredholm operator is a linear integral operator of the form

(3.2) 𝒦 : C p α ⁢ ( Ω ) → F ⁢ ( Ω 1 , ℝ d ) , 𝒦 ⁢ u := ∫ Ω k ⁢ ( ⋅ , y ) ⁢ u ⁢ ( y ) ⁢ d μ ⁢ ( y ) ,

determined by a matrix-valued kernel k : Ω 1 × Ω → ℝ d × p .

Fredholm operators apparently fit in the setup of Section 2 with kernel function f ⁢ ( x , y , z ) = k ⁢ ( x , y ) ⁢ z . Nevertheless, we take the opportunity to formulate our assumptions in terms of integrals over the kernels, rather than over the kernels. Then the corresponding counterparts to ( U 1 0 ) and ( U 2 0 ) read as follows.

Hypothesis.

Assume

lim x → x 0 ⁡ ∫ Ω | k ⁢ ( x , y ) - k ⁢ ( x 0 , y ) | ⁢ d μ ⁢ ( y ) = 0 for all ⁢ x 0 ∈ Ω 1 .
k ⁢ ( x , ⋅ ) is measurable for all x ∈ Ω 1 , and

sup ξ ∈ Ω 1 ⁡ ∫ Ω | k ⁢ ( ξ , y ) | ⁢ d μ ⁢ ( y ) < ∞ .

Proposition 3.1 (Well-definedness of K ).

Assume that ( K 1 ) and ( K 2 ) hold. Then a Fredholm operator K satisfies K ∈ L ⁢ ( C p α ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) ) and

∥ 𝒦 ∥ L ⁢ ( C p α ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) ) ≤ max ⁡ { 1 , ( diam ⁡ Ω ) α } ⁢ sup ξ ∈ Ω 1 ⁡ ∫ Ω | k ⁢ ( ξ , y ) | ⁢ d μ ⁢ ( y ) .

Proof.

Let us abbreviate

M := sup ξ ∈ Ω 1 ⁡ ∫ Ω | k ⁢ ( ξ , y ) | ⁢ d μ ⁢ ( y )

and write 𝒦 0 : C p 0 ⁢ ( Ω ) → C d 0 ⁢ ( Ω 1 ) instead of 𝒦 . The inclusion 𝒦 0 ∈ L ⁢ ( C n 0 ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) ) with ∥ 𝒦 0 ∥ ≤ M is shown in [21, p. 244, Satz 1]. In case α ∈ ( 0 , 1 ] , we consider the composition 𝒦 = 𝒦 0 ⁢ ℐ α 0 , where the embedding operator ℐ α 0 from (A.3) satisfies the estimate ∥ ℐ α 0 ∥ ≤ max ⁡ { 1 , ( diam ⁡ Ω ) α } by Theorem A.18. This implies the remaining assertions. ∎

Corollary 3.2 (Compactness of K ).

A Fredholm operator K ∈ L ⁢ ( C p α ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) ) is compact, provided one of the following holds:

α ∈ ( 0 , 1 ] .
Ω 1 is compact, α = 0 and ( K 1 ) holds uniformly in x 0 ∈ Ω 1 .

Proof.

For compactness of 𝒦 0 ∈ L ⁢ ( C n 0 ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) ) , we refer to [21, p. 247, Satz 4]. In case α ∈ ( 0 , 1 ] , we consider the composition 𝒦 = 𝒦 0 ⁢ ℐ α 0 of the continuous 𝒦 0 and the embedding operator ℐ α 0 introduced in (A.3), which is compact due to Theorem A.22. ∎

Remark 3.3.

If Ω 1 is compact and k : Ω 1 × Ω → ℝ d × p is continuous, then ( K 1 ) and ( K 2 ) are fulfilled. Hence, Proposition 3.1 and Corollary 3.2 guarantee that 𝒦 ∈ L ⁢ ( C p α ⁢ ( Ω ) , C d 0 ⁢ ( Ω 1 ) ) is compact.

In order to handle Fredholm operators which map into the Hölder continuous functions, a refinement of assumption ( U ¯ 1 0 ) is due.

Hypothesis.

Let β ∈ ( 0 , 1 ] .

There exists a continuous function h ~ : Ω 1 2 → ℝ + such that

(3.3) ∫ Ω | k ⁢ ( x , y ) - k ⁢ ( x 0 , y ) | ⁢ d μ ⁢ ( y ) ≤ h ~ ⁢ ( x , x 0 ) ⁢ d ⁢ ( x , x ¯ ) β for all ⁢ x , x 0 ∈ Ω 1 .

Obviously, ( K ¯ 1 ) implies ( K 1 ) .

Theorem 3.4 (Well-definedness of K ).

Assume that ( K ¯ 1 ) and ( K 2 ) hold. Then a Fredholm operator K satisfies K ∈ L ⁢ ( C p α ⁢ ( Ω ) , C d β ⁢ ( Ω 1 ) ) and

∥ 𝒦 ∥ L ⁢ ( C p α ⁢ ( Ω ) , C d β ⁢ ( Ω 1 ) ) ≤ max ⁡ { max ⁡ { 1 , ( diam ⁡ Ω ) α } ⁢ sup ξ ∈ Ω 1 ⁡ ∫ Ω | k ⁢ ( ξ , y ) | ⁢ d μ ⁢ ( y ) , sup x , x 0 ∈ Ω 1 ⁡ h ~ ⁢ ( x , x 0 ) } .

Proof.

We abbreviate N := sup x , x 0 ∈ Ω 1 ⁡ h ~ ⁢ ( x , x 0 ) . First,

∥ 𝒦 ⁢ u ∥ 0 ≤ M ⁢ max ⁡ { 1 , ( diam ⁡ Ω ) α } ⁢ ∥ u ∥ α

holds due to Proposition 3.1 ( M ≥ 0 is defined in its proof). Second, the inequality

| 𝒦 ⁢ u ⁢ ( x ) - 𝒦 ⁢ u ⁢ ( x ¯ ) | ≤ ∫ Ω | k ⁢ ( x , y ) - k ⁢ ( x ¯ , y ) | ⁢ | u ⁢ ( y ) | ⁢ d μ ⁢ ( y ) ⁢ (by (3.2))

≤ N ⁢ d ⁢ ( x , x ¯ ) β ⁢ ∥ u ∥ α ⁢ (by (3.3)) for all ⁢ x , x ¯ ∈ Ω 1

consequently implies that [ 𝒦 ⁢ u ] β ≤ N ⁢ ∥ u ∥ α holds. A combination of these two estimates finally guarantees that

∥ 𝒦 ⁢ u ∥ β = max ⁡ { ∥ 𝒦 ⁢ u ∥ 0 , [ 𝒦 ⁢ u ] β } ≤ max ⁡ { M ⁢ max ⁡ { 1 , ( diam ⁡ Ω ) α } , N } ⁢ ∥ u ∥ α ,

and thus

∥ 𝒦 ∥ L ⁢ ( C n α ⁢ ( Ω ) , C d β ⁢ ( Ω 1 ) ) ≤ max ⁡ { M ⁢ max ⁡ { 1 , ( diam ⁡ Ω ) α } , N }

holds. ∎

Corollary 3.5 (Compactness of K ).

A Fredholm operator K ∈ L ⁢ ( C p α ⁢ ( Ω ) , C d γ ⁢ ( Ω 1 ) ) is compact, provided one of the following holds:

α ∈ ( 0 , 1 ] and γ = β .
Ω 1 is bounded, α ∈ ( 0 , 1 ] and γ ∈ [ 0 , β ] .
Ω 1 is compact, γ ∈ [ 0 , β ] and lim x → x 0 ⁡ h ~ ⁢ ( x , x 0 ) = 0 uniformly in x 0 ∈ Ω 1 .
Ω 1 is compact and γ ∈ [ 0 , β ) .

Proof.

We write 𝒦 0 : C p α ⁢ ( Ω ) → C d β ⁢ ( Ω 1 ) and 𝒦 α γ : C p α ⁢ ( Ω ) → C d γ ⁢ ( Ω 1 ) instead of 𝒦 .

(I) Claim: If (iii) holds, then K 0 β ∈ L ⁢ ( C n 0 ⁢ ( Ω ) , C d β ⁢ ( Ω 1 ) ) is compact. Given the unit ball B := B ¯ 1 ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) , we apply the compactness criterion from Theorem A.21 in order to show that 𝒦 0 β ⁢ B ⊆ C d β ⁢ ( Ω 1 ) is relatively compact. First, 𝒦 0 β ⁢ B is bounded due to the above Theorem 3.4. Second, given ε > 0 , we obtain by the assumption that there exists a δ > 0 such that d ⁢ ( x , x ¯ ) < δ yields h ~ ⁢ ( x , x ¯ ) < ε for all x , x ¯ ∈ Ω 1 . Hence, assumption ( K ¯ 1 ) implies that

| ( 𝒦 ⁢ u ) ⁢ ( x ) - ( 𝒦 ⁢ u ) ⁢ ( x ¯ ) | ⁢ ≤ ( 3.2 ) ⁢ ∫ Ω | k ⁢ ( x , y ) - k ⁢ ( x ¯ , y ) | ⁢ d μ ⁢ ( y ) ⁢ ≤ ( 3.3 ) ⁢ ε ⁢ d ⁢ ( x , x ¯ ) β for all ⁢ x , x ¯ ∈ Ω 1 ⁢ and all ⁢ u ∈ B ,

which guarantees the relative compactness of 𝒦 0 β ⁢ B .

(II) Under (i), the operator 𝒦 = 𝒦 0 0 ⁢ ℐ α 0 is a composition of the continuous 𝒦 0 0 (see Proposition 3.1) with the compact mapping ℐ α 0 (see Theorem A.22). Under (ii), one has 𝒦 = ℐ β γ ⁢ 𝒦 0 with the bounded embedding ℐ β γ (see Theorem A.18) and the compact 𝒦 0 (due to (i)). Under assumptions (iii), we have

𝒦 = ℐ β γ ⁢ 𝒦 0 β ⁢ ℐ α 0

with bounded embeddings and the compact 𝒦 0 β (thanks to step (I)). Finally, in case (iv), one has 𝒦 = ℐ β γ ⁢ 𝒦 0 , where 𝒦 0 is continuous and ℐ β γ is compact. Since at least one operator in the above compositions is compact, the compactness of 𝒦 results from [35, p. 417, Theorem 1.2]. ∎

3.2 Nemytskii operators

A Nemytskii operator ^[3] is a mapping of the form

(3.4) 𝒢 : U α → F ⁢ ( Ω , ℝ p ) , 𝒢 ⁢ ( u ) ⁢ ( x ) := g ⁢ ( x , u ⁢ ( x ) ) for all ⁢ x ∈ Ω ,

which is generated by a function g : Ω × Z → ℝ p . Our terminology using the letter “g” comes from growth function met in applications [31, 36].

Hypothesis.

Let m ∈ ℕ 0 . With 0 ≤ k ≤ m , one assumes the following:

The partial derivative D 2 k ⁢ g : Ω × Z → L k ⁢ ( ℝ n , ℝ p ) exists and can be continuously extended to Ω × Z ¯ .

Proposition 3.6 (Well-definedness of G ).

Assume that ( N 0 0 ) holds. Then a Nemytskii operator G : U α → C p 0 ⁢ ( Ω ) is well-defined, bounded and continuous. Moreover, G is completely continuous, provided α ∈ ( 0 , 1 ] .

Remark 3.7.

For α = 0 and a compact interval Ω ⊂ ℝ , a Nemytskii operator 𝒢 : C 0 ⁢ ( Ω ) → C p 0 ⁢ ( Ω ) is well-defined if and only if g : Ω × ℝ → ℝ p is continuous [5, Theorem 3.1]. Indeed, due to [5, Table 1], one has the following equivalences:

Proof.

We denote 𝒢 defined on U 0 as 𝒢 0 . Since every u ∈ U α is continuous, also 𝒢 ⁢ ( u ) : Ω → ℝ p is continuous and bounded. As a result, 𝒢 : U α → C p 0 ⁢ ( Ω ) is well-defined. Let u 0 ∈ U α and choose r := ∥ u 0 ∥ 0 + 1 . Now g can be extended continuously to Ω × Z ¯ by assumption ( N 0 0 ) and g is uniformly continuous on Ω × ( Z ∩ B ¯ r ⁢ ( 0 ) ) . Given ε > 0 , this means that there exists a δ > 0 such that

d ⁢ ( x , x ¯ ) < δ , | z - z ¯ | < δ imply | g ⁢ ( x , z ) - g ⁢ ( x ¯ , z ¯ ) | < ε for all ⁢ x , x ¯ ∈ Ω , z , z ¯ ∈ Z ∩ B ¯ r ⁢ ( 0 ) .

If u ∈ U α ∩ B δ ⁢ ( u 0 , C n 0 ⁢ ( Ω ) ) and δ < 1 , then

| u ⁢ ( x ) | ≤ | u 0 ⁢ ( x ) | + | u ⁢ ( x ) - u 0 ⁢ ( x ) | ≤ r ,

and consequently

| [ 𝒢 ⁢ ( u ) - 𝒢 ⁢ ( u 0 ) ] ⁢ ( x ) | = | g ⁢ ( x , u ⁢ ( x ) ) - g ⁢ ( x , u 0 ⁢ ( x ) ) | ≤ ε for all ⁢ x ∈ Ω .

Passing to the supremum over x ∈ Ω yields ∥ 𝒢 ⁢ ( u ) - 𝒢 ⁢ ( u 0 ) ∥ 0 ≤ ε , i.e. 𝒢 is continuous. The boundedness of 𝒢 results from the uniform continuity of g on Ω × Z ¯ , as well as properties of the norm ∥ ⋅ ∥ 0 . In conclusion, 𝒢 is well-defined, bounded and continuous.

Thanks to Theorem A.22, the embedding ℐ α 0 is compact, and therefore 𝒢 = 𝒢 0 ⁢ ℐ α 0 is even completely continuous for α ∈ ( 0 , 1 ] (see [48, pp. 25–26, Theorem 2.1 (2)]). ∎

Hypothesis.

Let ϑ ∈ ( 0 , 1 ] .

For every r > 0 , there exists a l r ′ ≥ 0 such that

(3.5) | g ⁢ ( x , z ) - g ⁢ ( x , z ¯ ) | ≤ l r ′ ⁢ | z - z ¯ | ϑ for all ⁢ x ∈ Ω , z , z ¯ ∈ Z ∩ B ¯ r ⁢ ( 0 ) .

Corollary 3.8.

Assume that g ⁢ ( ⋅ , z ) : Ω → R p is continuous for all z ∈ Z and ( N 0 ′ ) holds. Then a Nemytskii operator G : U α → C p 0 ⁢ ( Ω ) is well-defined and Hölder on bounded sets, that is,

[ 𝒢 | U α ∩ B ¯ r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) ] ϑ ≤ l r ′ for all r > 0 .

The same argument in the case sup r > 0 ⁡ l r ′ < ∞ yields a global Hölder condition for 𝒢 : U α → C p 0 ⁢ ( Ω ) .

Remark 3.9.

Note that, for α = 0 and a compact interval Ω ⊂ ℝ , Nemytskii operators 𝒢 : C 0 ⁢ ( Ω ) → C p 0 ⁢ ( Ω ) satisfy a local (resp. global) Lipschitz condition if and only if g : Ω × ℝ → ℝ p does in the second variable. In case p = 1 , even the Lipschitz constants (uniformly in x ∈ Ω ) are the same; see [4, Theorem 1] and [5, Theorem 3.2].

Proof.

As a consequence of Theorem A.11, g : Ω × Z → ℝ p is continuous and, using Lemma A.7, we can show that g satisfies assumption ( N 0 0 ) . Hence, Proposition 3.6 yields that 𝒢 is well-defined. Moreover, if r > 0 and u , u ¯ ∈ U α with ∥ u ∥ 0 , ∥ u ¯ ∥ 0 ≤ r , then

Passing to the supremum over x ∈ Ω implies ∥ 𝒢 ⁢ ( u ) - 𝒢 ⁢ ( u ¯ ) ∥ 0 ≤ l r ′ ⁢ ∥ u - u ¯ ∥ α ϑ . ∎

We next show that the derivative of a Nemytskii operator is a multiplication operator.

Lemma 3.10.

Assume that ( N 0 k ) holds for some k ∈ N . Then G k : U α → L k ⁢ ( C n α ⁢ ( Ω ) , C p 0 ⁢ ( Ω ) ) given by

(3.6) ( 𝒢 k ⁢ ( u ) ⁢ v 1 ⁢ ⋯ ⁢ v k ) ⁢ ( x ) := D 2 k ⁢ g ⁢ ( x , u ⁢ ( x ) ) ⁢ v 1 ⁢ ( x ) ⁢ ⋯ ⁢ v k ⁢ ( x ) for all ⁢ x ∈ Ω , v 1 , … , v k ∈ C n α ⁢ ( Ω )

is well-defined and continuous.

Proof.

Let v 1 , … , v k ∈ C n α ⁢ ( Ω ) be given. With u ∈ U α , also

x ↦ D 2 k ⁢ g ⁢ ( x , u ⁢ ( x ) ) ⁢ v 1 ⁢ ( x ) ⁢ ⋯ ⁢ v k ⁢ ( x )

is continuous, and consequently

𝒢 k ⁢ ( u ) ⁢ v 1 ⁢ ⋯ ⁢ v k ∈ C d 0 ⁢ ( Ω )

holds, i.e. 𝒢 k is well-defined. Let ( u l ) l ∈ ℕ be a sequence in U α with lim l → ∞ ⁡ ∥ u l - u ∥ 0 = 0 . Choose r > 0 so large that u , u l ∈ B r ⁢ ( 0 , C n 0 ⁢ ( Ω ) ) for all l ∈ ℕ . Then D 2 k ⁢ g is uniformly continuous on Ω × ( Z ∩ B ¯ r ⁢ ( 0 ) ) and, given ε > 0 , there exists a δ > 0 such that

| z - z ¯ | < δ implies | D 2 k ⁢ g ⁢ ( x , z ) - D 2 ⁢ g k ⁢ ( x , z ¯ ) | < ε for all ⁢ x ∈ Ω , z , z ¯ ∈ Z ∩ B ¯ r ⁢ ( 0 ) .

Moreover, there exists an L ∈ ℕ such that | u l ⁢ ( x ) - u ⁢ ( x ) | ≤ δ for all l ≥ L , and consequently

| ( ( 𝒢 k ⁢ ( u l ) - 𝒢 k ⁢ ( u ) ) ⁢ v 1 ⁢ ⋯ ⁢ v k ) ⁢ ( x ) | = | ( D 2 k ⁢ g ⁢ ( x , u l ⁢ ( x ) ) - D 2 k ⁢ g ⁢ ( x , u ⁢ ( x ) ) ) ⁢ v 1 ⁢ ( x ) ⁢ ⋯ ⁢ v k ⁢ ( x ) | ⁢ (by (3.6))

≤ | D 2 k ⁢ g ⁢ ( x , u l ⁢ ( x ) ) - D 2 k ⁢ g ⁢ ( x , u ⁢ ( x ) ) | ⁢ | v 1 ⁢ ( x ) | ⁢ ⋯ ⁢ | v k ⁢ ( x ) | ⁢ for all ⁢ x ∈ Ω .

Passing to the supremum over x ∈ Ω yields

∥ [ 𝒢 k ⁢ ( u l ) - 𝒢 k ⁢ ( u ) ] ⁢ v 1 ⁢ ⋯ ⁢ v k ∥ 0 ≤ ε for ⁢ v 1 , … , v k ∈ B ¯ 1 ⁢ ( 0 , C n α ⁢ ( Ω ) )

and, in turn,

∥ 𝒢 k ⁢ ( u ) - 𝒢 k ⁢ ( u 0 ) ∥ L k ⁢ ( C n α ⁢ ( Ω ) , C p 0 ⁢ ( Ω ) ) ≤ ε for all ⁢ l ≥ L .

Since u ∈ U α was arbitrary, 𝒢 k is continuous. ∎

Proposition 3.11 (Continuous differentiability of G ).

Let m ∈ N . Assume that ( N 0 k ) holds for all 0 ≤ k ≤ m on a convex set Z ⊆ R n . Then a Nemytskii operator G : U α → C p 0 ⁢ ( Ω ) is m-times continuously differentiable with D k ⁢ G = G k for every 1 ≤ k ≤ m .

Proof.

(I) Let 0 ≤ k < m . Thanks to Lemma 3.10, the mappings 𝒢 k : U α → L k ⁢ ( C n α ⁢ ( Ω ) , C p 0 ⁢ ( Ω ) ) are well-defined and continuous. Thus, given u ∈ U α and h ∈ C n α ⁢ ( Ω ) with u + h ∈ U α , the remainders

r k ⁢ ( h ) := sup θ ∈ [ 0 , 1 ] ⁡ ∥ 𝒢 k + 1 ⁢ ( u + θ ⁢ h ) - 𝒢 k + 1 ⁢ ( u ) ∥ L k + 1 ⁢ ( C n α ⁢ ( Ω ) , C p 0 ⁢ ( Ω ) )

satisfy lim h → 0 ⁡ r k ⁢ ( h ) = 0 . Now we obtain from [35, p. 341, Theorem 4.2] that

[ 𝒢 k ⁢ ( u + h ) - 𝒢 k ⁢ ( u ) - 𝒢 k + 1 ⁢ ( u ) ⁢ h ] ⁢ ( x )

= D 2 k ⁢ g ⁢ ( x , u ⁢ ( x ) + h ⁢ ( x ) ) - D 2 k ⁢ g ⁢ ( x , u ⁢ ( x ) ) - D 2 k + 1 ⁢ g ⁢ ( x , u ⁢ ( x ) ) ⁢ h ⁢ ( x ) (by (3.6))

= ∫ 0 1 [ D 2 k + 1 ⁢ g ⁢ ( x , u ⁢ ( x ) + θ ⁢ h ⁢ ( x ) ) - D 2 k + 1 ⁢ g ⁢ ( x , u ⁢ ( x ) ) ] ⁢ h ⁢ ( x ) ⁢ d θ .

Consequently,

| [ 𝒢 k ⁢ ( u + h ) - 𝒢 k ⁢ ( u ) - 𝒢 k + 1 ⁢ ( u ) ⁢ h ] ⁢ ( x ) |

≤ ∫ 0 1 ∥ 𝒢 k + 1 ⁢ ( u + θ ⁢ h ) - 𝒢 k + 1 ⁢ ( u ) ∥ L k + 1 ⁢ ( C n α ⁢ ( Ω ) , C p 0 ⁢ ( Ω ) ) ⁢ d θ ⁢ ∥ h ∥ 0 (by (3.6))

≤ r k ⁢ ( h ) ⁢ ∥ h ∥ α .

After passing to the least upper bound over x ∈ Ω , this results in

∥ 𝒢 k ⁢ ( u + h ) - 𝒢 k ⁢ ( u ) - 𝒢 k + 1 ⁢ ( u ) ⁢ h ∥ L k ⁢ ( C n α ⁢ ( Ω ) , C p 0 ⁢ ( Ω ) ) ≤ r k ⁢ ( h ) ⁢ ∥ h ∥ α .

This establishes that the mapping 𝒢 k is differentiable in u with the derivative 𝒢 k + 1 ⁢ ( u ) and, in turn, 𝒢 k + 1 is continuous due to Lemma 3.10.

(II) Applying step (I) in case k = 0 shows that 𝒢 is continuously differentiable on U α with the derivative 𝒢 1 . Given this, mathematical induction yields that 𝒢 : U α → C p 0 ⁢ ( Ω ) is actually m-times continuously differentiable with the derivatives D k ⁢ 𝒢 = 𝒢 k for all 0 ≤ k ≤ m . ∎

Remark 3.12 (Boundedness and continuity of G ).

(1) The boundedness of the Nemytskii operators 𝒢 guaranteed in Proposition 3.6 means that the 𝒢 -images of merely ∥ ⋅ ∥ 0 -bounded subsets B ⊂ U α are bounded. In particular, the functions in B are not required to have uniformly bounded Hölder constants.

(2) The continuity of the Nemytskii operator 𝒰 in Proposition 3.6, as well as for the derivatives D k ⁢ 𝒰 in Proposition 3.11 are to be understood in the following strong form: Already convergence in the domain U α with respect to the norm ∥ ⋅ ∥ 0 is sufficient for convergence of the 𝒢 -values in the norm ∥ ⋅ ∥ 0 . A corresponding statement holds in Corollary 3.8.

In contrast to the above situation, Nemytskii operators 𝒢 behave rather differently when mapping into the space of Hölder functions of exponent α ∈ ( 0 , 1 ] . For instance, [5, Example 3.10] constructs a discontinuous function g (hence 𝒢 fails to map C 0 ⁢ [ 0 , 1 ] into itself by Remark 3.7) such that 𝒢 maps C α ⁢ [ 0 , 1 ] into itself. Below, we survey some properties relating the mappings

g : Ω × ℝ → ℝ p , 𝒢 : C α ⁢ ( Ω ) → C p α ⁢ ( Ω ) ,

when Ω ⊂ ℝ κ is compact; one denotes g as autonomous if it does not depend on the first argument:

Well-definedness and boundedness: In [13, Theorem 1.1], it is shown that the condition

(3.7) for all ⁢ r > 0 ⁢ there exists ⁢ k ⁢ ( r ) > 0 ⁢ such that ⁢ | g ⁢ ( x , z ) - g ⁢ ( x ¯ , z ¯ ) | ≤ k ⁢ ( r ) ⁢ ( | x - x ¯ | α + | z - z ¯ | r )

for all x , x ¯ ∈ Ω , z , z ¯ ∈ B ¯ r ⁢ ( 0 ) is equivalent to 𝒢 being well-defined and bounded (see also [6, Theorem 7.3]). In comparison, a necessary and sufficient condition for 𝒢 to be merely well-defined is more clumsy, restricted to Ω = [ a , b ] , and given in terms of (see [5, Theorem 3.8] or [6, Theorem 7.1])

(3.8) { for all ⁢ ( x 0 , z 0 ) ∈ Ω × ℝ ⁢ and all ⁢ r > 0 ⁢ there exist ⁢ k ⁢ ( r ) > 0 ⁢ and ⁢ δ > 0 ⁢ such that | g ⁢ ( x , z ) - g ⁢ ( x ¯ , z ¯ ) | ≤ k ⁢ ( r ) ⁢ ( | x - x ¯ | α + | z - z ¯ | r )

for all x , x ¯ ∈ Ω , z , z ¯ ∈ ℝ with

x , x ¯ ∈ B r ⁢ ( x 0 ) , | z - z 0 | ≤ r ⁢ | x - x 0 | α , | z ¯ - z 0 | ≤ r ⁢ | x ¯ - x 0 | α .

If g is autonomous, then the Lipschitz condition (3.5) with ϑ = 1 is even necessary and sufficient for 𝒢 being well-defined; see [18, Theorem 1].

Let g be autonomous and let Ω = [ a , b ] . Now every well-defined 𝒢 is bounded (see [24, Corollary 2.1]) and g is continuous (see [6, Theorem 7.5]).
Continuity: If the partial derivative D 2 ⁢ g exists and satisfies

(3.9) { D 2 ⁢ g : Ω × ℝ → ℝ p ⁢ is α -Hölder in the first argument uniformly in the second argument from compact subsets of ⁢ ℝ ,

then (3.7) implies that 𝒢 is continuous (cf. [42, Theorem 2.2]). Conversely, the partial derivative D 2 ⁢ g exists if 𝒢 is continuous and (3.7) is valid (see [42, Theorem 2.2]), or if 𝒢 is bounded and Ω = [ a , b ] (see [42, Corollary 2.3]). A characterization of 𝒢 being uniformly continuous on bounded sets can be found in [13, Theorem 2.1]. For intervals Ω = [ a , b ] , it follows from [5, Table 5] that

More can be said whenever g is autonomous and Ω = [ a , b ] : Then, by reproducing [5, Table 4], the following implications hold:
Lipschitz condition: It is shown in [13, Theorem 3.1] that 𝒢 is Lipschitz on bounded sets if and only if both g and D 2 ⁢ g satisfy an estimate (3.7). The necessity to assume the existence of the partial derivatives also arose for Urysohn operators (see Corollary 2.13). Yet, the assumption of a global Lipschitz condition for 𝒢 leads to a degeneracy in g. Indeed, [39] shows that 𝒢 is globally Lipschitz if and only if all components g 1 , … , g p : Ω × ℝ → ℝ of g are affine linear, i.e. there exist a ¯ i , b ¯ i ∈ C α ⁢ ( Ω ) with

(3.10) g i ⁢ ( x , z ) = z ⁢ a ¯ i ⁢ ( x ) + b ¯ i ⁢ ( x ) for all ⁢ 1 ≤ i ≤ p , x ∈ Ω , z ∈ ℝ .

Yet, even uniform continuity of 𝒢 is sufficient for (3.10) to hold (see [40, Theorem 2]). Nonetheless, if ℐ α β ⁢ 𝒢 satisfies a global Lipschitz condition, then sup x ∈ Ω ⁡ [ g ⁢ ( x , ⋅ ) ] 1 < ∞ (cf. [5, Theorem 2.5]) and (3.10) holds in case β = 1 (cf. [5, Theorem 2.6]).

Let g be autonomous and let Ω = [ a , b ] : Then a well-defined Nemytskii operator 𝒢 is globally Lipschitz if and only if g : ℝ → ℝ p is affine-linear, i.e. there exist a ¯ , b ¯ ∈ ℝ p such that g ⁢ ( z ) = z ⁢ a ¯ + b ¯ (see [24, Theorem 2.3 (b)]).
Continuous differentiability: If g ⁢ ( x , ⋅ ) is twice differentiable such that g , D 2 ⁢ g satisfy (3.8), and (3.9) holds with D 2 2 ⁢ g (instead of D 2 ⁢ g ), then 𝒢 is continuously differentiable (cf. [42, Theorem 4.1]). Note that also for the continuous differentiability of Urysohn operators we needed to assume that the second order derivative of the kernel function exists (cf. Theorem 2.17). For a characterization of 𝒢 being continuously differentiable with a derivative being uniformly continuous on bounded sets, we refer to [13, Theorem 4.1]. In case Ω = [ a , b ] , differentiability of ℐ α β ⁢ 𝒢 is characterized in [6, Theorem 7.11]. Furthermore, if 𝒢 is differentiable with a globally bounded derivative, then g is affine linear, i.e. g satisfies (3.10).

For autonomous g and Ω = [ a , b ] , an elegant characterization holds: The Nemytskii operator 𝒢 is continuously differentiable if and only if g : ℝ → ℝ p is twice continuously differentiable (see [24, Theorem 2.4]).

3.3 Hammerstein operators

In the following, we understand Hammerstein operators (3.1) as composition

ℋ = 𝒦 ⁢ 𝒢 : C n α ⁢ ( Ω ) → F ⁢ ( Ω 1 , ℝ d )

of Fredholm operators 𝒦 ∈ L ⁢ ( C p 0 ⁢ ( Ω ) , C d β ⁢ ( Ω 1 ) ) and Nemytskii operators 𝒢 : C n α ⁢ ( Ω ) → C p 0 ⁢ ( Ω ) given in (3.2), resp. (3.4). Hence, our above preparations immediately yield properties of ℋ .

Theorem 3.13 (Well-definedness of H ).

Assume that ( K ¯ 1 ) , ( K 2 ) and ( N 0 0 ) hold. Then a Hammerstein operator H : U α → C d β ⁢ ( Ω 1 ) is well-defined, bounded and continuous.

Proof.

As composition of

𝒢 : C n α ⁢ ( Ω ) → C p 0 ⁢ ( Ω ) and 𝒦 ∈ L ⁢ ( C p 0 ⁢ ( Ω ) , C d β ⁢ ( Ω 1 ) ) ,

the claims for ℋ = 𝒦 ⁢ 𝒢 result directly from Theorem 3.4 (with α = 0 ) and Proposition 3.6. ∎

Corollary 3.14 (Complete continuity of H ).

A Hammerstein operator H : U α → C d γ ⁢ ( Ω 1 ) is completely continuous, provided one of the following holds:

α ∈ ( 0 , 1 ] and γ = β .
Ω 1 is bounded, α ∈ ( 0 , 1 ] and γ ∈ [ 0 , β ] .
Ω 1 is compact, γ ∈ [ 0 , β ] and lim x → x 0 ⁡ h ~ ⁢ ( x , x 0 ) = 0 uniformly in x 0 ∈ Ω 1 .
Ω 1 is compact and γ ∈ [ 0 , β ) .

Proof.

It results from Proposition 3.6 (case (i)) and Corollary 3.5 (cases (ii) and (iii)) that at least one of the functions in the composition ℋ = 𝒦 ⁢ 𝒢 is completely continuous. ∎

Theorem 3.15 (Continuous differentiability of H ).

Let m ∈ N . Assume that ( K ¯ 1 ) , ( K 2 ) and ( N 0 k ) hold for all 0 ≤ k ≤ m on a convex set Z ⊆ R n . Then a Hammerstein operator H : U α → C d β ⁢ ( Ω 1 ) is m-times continuously differentiable with D k ⁢ H = K ⁢ G k for every 1 ≤ k ≤ m .

Proof.

This results from the chain rule [35, p. 337], Theorem 3.4 (with α = 0 ) and Proposition 3.11. ∎

Remark 3.16 (Convolutive Hammerstein operators).

Assume that a growth function g : [ a , b ] × Z → ℝ p generates a Nemytskii operator 𝒢 mapping into C p α ⁢ [ a , b ] . Then the smoothing properties from Section 2.3 extend to convolutive Hammerstein operators

𝒦 ~ ⁢ u ⁢ ( v ) := ∫ a b k ~ ⁢ ( x - y ) ⁢ g ⁢ ( y , u ⁢ ( y ) ) ⁢ d y

with an ambient kernel k ~ : [ a - b , b - a ] → ℝ d × p .

A Hölder continuous functions

The definition of continuity for a function u in, e.g., a point x 0 is not quantitative in the sense that it provides no information on how fast its values u ⁢ ( x ) approach u ⁢ ( x 0 ) as x → x 0 . In consequence, the modulus of continuity ω : ℝ + → ℝ + of a continuous function u satisfying the estimate ∥ u ⁢ ( x ) - u ⁢ ( x 0 ) ∥ ≤ ω ⁢ ( d ⁢ ( x , x 0 ) ) may decrease arbitrarily slowly. For this reason, the space of continuous functions is often not suitable for a quantitative analysis in fields such as numerical analysis or partial differential equations. A straightforward and feasible way to strengthen the notion of continuity of u is to assume that its modulus of continuity is proportional to a power of d ⁢ ( x , x 0 ) α for some exponent α ∈ ( 0 , 1 ] . Such functions are denoted as Hölder continuous and in the focus of this appendix.

Our overall setting is as follows. Let ( Ω , d ) be a metric space and let ( Y , ∥ ⋅ ∥ ) be a normed space over 𝕂 , which stands for the real or complex field.

A function u : Ω → Y is said to be α-Hölder (with Hölder exponent α ∈ ( 0 , 1 ] ) if it satisfies

[ u ] α := sup x , x ¯ ∈ Ω x ≠ x ¯ ⁡ ∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ d ⁢ ( x , x ¯ ) α < ∞ ;

the finite quantity [ u ] α ≥ 0 is called Hölder constant of u. One speaks of a Hölder continuous function u if it is α-Hölder for some α ∈ ( 0 , 1 ) , and in case α = 1 one denotes u as Lipschitz continuous with Lipschitz constant [ u ] 1 ; a comprehensive approach to this class of functions is given in [14]. For the set of all such α-Hölder functions u : Ω → Y we write C α ⁢ ( Ω , Y ) , supplemented by C 0 ⁢ ( Ω , Y ) for the linear space of continuous functions. It is convenient to denote a continuous function as 0-Hölder, and, unless indicated otherwise, let us assume α ∈ [ 0 , 1 ] throughout.

Remark A.1.

(1) A function u : Ω → Y is constant if and only if its Hölder constant vanishes.

(2) For α ∈ ( 0 , 1 ] , there is an evident relation between Hölder functions and Lipschitz functions: Indeed, u : ( Ω , d ) → Y is α-Hölder if and only if u : ( Ω , d α ) → Y is Lipschitz with the metric d α : Ω × Ω → ℝ + given by d α ⁢ ( x , x ¯ ) := d ⁢ ( x , x ¯ ) α . Of course, the metrics d and d α on Ω are not equivalent.

(3) One does restrict to exponents α ∈ ( 0 , 1 ] for the following reason. Suppose that a function u : Ω → Y on an open subset Ω ⊆ ℝ κ satisfies [ u ] α < ∞ for an exponent α > 1 . Then

∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) - 0 ⁢ ( x - x ¯ ) ∥ = ∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ ≤ [ u ] α ⁢ | x - x ¯ | α - 1 ⁢ | x - x ¯ | for all ⁢ x , x ¯ ∈ Ω

yields that u is differentiable on Ω with derivative 0, and thus constant on the components of Ω.

(4) Suppose that Ω has a finite, positive diameter and that ϕ : [ 0 , 1 ] → ℝ + is a function satisfying ϕ ⁢ ( 0 ) = 0 and ϕ ⁢ ( 1 ) = 1 such that t ↦ ϕ ⁢ ( t ) and t ↦ t ϕ ⁢ ( t ) are positive and increasing on ( 0 , 1 ) . Then u : Ω → Y is called generalized Hölder if (see [6, Chapter 7] and [10])

sup x , x ¯ ∈ Ω , x ≠ x ¯ ⁡ ∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ ⁢ ϕ ⁢ ( d ⁢ ( x , x ¯ ) diam ⁡ Ω ) - 1 < ∞ .

In case ϕ ⁢ ( t ) := t α , α ∈ ( 0 , 1 ] , this reduces to the situation studied here.

(5) Differentiable functions on Ω ⊆ ℝ κ whose m-th derivative satisfies an α-Hölder condition, and associated function spaces C m , α ⁢ ( Ω , Y ) , are addressed in [22, pp. 30ff, Section 1.5] or [23, pp. 51ff, Section 4.1].

(6) The inner structure of Hölder spaces from an abstract Banach spaces perspective is studied in [29].

Example A.2.

Let Ω = Y = ℝ and α ∈ ( 0 , 1 ] . The function u α : ℝ → ℝ , u α ⁢ ( x ) := | x | α , is not differentiable in 0 (see Figure 1), but α-Hölder. Indeed, given x , x ¯ ∈ ℝ , the case 0 < | x | ≤ | x ¯ | leads to the inequality

1 - | x | α | x ¯ | α ≤ 1 - | x | | x ¯ | ≤ ( 1 - | x | | x ¯ | ) α ,

and therefore

| u α ⁢ ( x ¯ ) - u α ⁢ ( x ) | = u α ⁢ ( x ¯ ) - u α ⁢ ( x ) ≤ | x ¯ - x | α .

A similar argument in case 0 < | x ¯ | ≤ | x | yields the assertion with Hölder constant [ u α ] α ≤ 1 .

$Figure 1 Graphs of the Hölder functions u α : ℝ → ℝ {u_{\alpha}:{\mathbb{R}}\to{\mathbb{R}}} given by u α ⁢ ( x ) := | x | α {u_{\alpha}(x):=\lvert x\rvert^{\alpha}} from Example A.2 for α ∈ { 1 4 , 1 2 , 1 } {\alpha\in\{\frac{1}{4},\frac{1}{2},1\}} .Their decay to 0 as x → 0 {x\to 0} is faster for decreasing values of the Hölder exponent α.$

Figure 1

Graphs of the Hölder functions u α : ℝ → ℝ given by u α ⁢ ( x ) := | x | α from Example A.2 for α ∈ { 1 4 , 1 2 , 1 } .Their decay to 0 as x → 0 is faster for decreasing values of the Hölder exponent α.

Theorem A.3 (Local Hölder continuity).

Let Ω be compact and let α ∈ ( 0 , 1 ] . If u : Ω → Y is locally α-Hölder, i.e. every x ∈ Ω has a neighborhood U ⊆ Ω such that [ u | U ] α < ∞ holds, then u is α-Hölder.

Proof.

We proceed indirectly and suppose that for each c ≥ 0 there exist x , x ¯ ∈ Ω yielding the inequality ∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ > c ⁢ d ⁢ ( x , x ¯ ) α . In particular, there are sequences ( x l ) l ∈ ℕ , ( x ¯ l ) l ∈ ℕ in Ω so that

(A.1) ∥ u ⁢ ( x l ) - u ⁢ ( x ¯ l ) ∥ > l ⁢ d ⁢ ( x l , x ¯ l ) α for all ⁢ l ∈ ℕ ,

holds. Since Ω is compact, without loss of generality, we can assume that these sequences converge to x ∗ , x ¯ ∗ ∈ Ω , respectively. Passing to the limit l → ∞ in (A.1) therefore enforces lim l → ∞ ⁡ d ⁢ ( x l , x ¯ l ) α = 0 , i.e. one has x ∗ = x ¯ ∗ . Because the function u is assumed to be locally Hölder, there exists a neighborhood U ⊆ Ω of x ∗ and a real C ≥ 0 with

∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ ≤ C ⁢ d ⁢ ( x , x ¯ ) α for all ⁢ x , x ¯ ∈ U .

Thanks to the inclusion x l , x ¯ l ∈ U for sufficiently large l ∈ ℕ , this implies

∥ u ⁢ ( x l ) - u ⁢ ( x ¯ l ) ∥ ≤ C ⁢ d ⁢ ( x l , x ¯ l ) α ,

contradicting (A.1) for l > C . ∎

The relationship between differentiable and Hölder continuous functions is addressed in the following example.

Example A.4.

(1) Differentiable functions u : Ω → Y on open, bounded and convex sets Ω ⊂ ℝ κ having a bounded derivative D ⁢ u : Ω → L ⁢ ( ℝ κ , Y ) are α-Hölder for each α ∈ ( 0 , 1 ] . This follows from the mean value inequality [38, p. 35, Theorem 4.1], because

∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ ≤ sup ξ ∈ Ω ⁡ ∥ D ⁢ u ⁢ ( ξ ) ∥ ⁢ | x - x ¯ | ≤ ( diam ⁡ Ω ) 1 - α ⁢ sup ξ ∈ Ω ⁡ ∥ D ⁢ u ⁢ ( ξ ) ∥ ⁢ | x - x ¯ | α for all ⁢ x , x ¯ ∈ Ω ,

and thus

[ u ] α ≤ ( diam ⁡ Ω ) 1 - α ⁢ sup ξ ∈ Ω ⁡ ∥ D ⁢ u ⁢ ( ξ ) ∥ .

A version of this result on not necessarily convex sets Ω can be found in [22, p. 11, Proposition 1.1.13]. In the Lipschitz case α = 1 , boundedness of Ω is not required.

(2) Rademacher’s theorem [17, p. 414, Theorem 21.2] guarantees that Lipschitz functions u : Ω → ℝ d on open sets Ω ⊆ ℝ κ are differentiable Lebesgue-almost everywhere in Ω. This situation changes for exponents α ∈ ( 0 , 1 ) , and [27] shows that the Weierstraß function u : ℝ → ℝ given as Fourier series

u ⁢ ( x ) = ∑ k = 0 ∞ a k ⁢ cos ⁡ ( b k ⁢ π ⁢ x ) with ⁢ a ∈ ( 0 , 1 ) ⁢ and integers ⁢ b > 1 ,

satisfying a ⁢ b > 1 + 3 ⁢ π 2 , is α-Hölder with exponent α = - log b ⁡ a , but nowhere differentiable.

Theorem A.5.

Hölder continuous functions are uniformly continuous.

Proof.

Let ε > 0 and let u ∈ C α ⁢ ( Ω , Y ) (without loss of generality, u is not constant). Setting

δ := ( ε [ u ] α ) 1 / α

guarantees

d ⁢ ( x , x ¯ ) < δ implies ∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ ≤ [ u ] α ⁢ d ⁢ ( x , x ¯ ) α ≤ ε for all ⁢ x , x ¯ ∈ Ω ,

and thus u is uniformly continuous. ∎

The converse of Theorem A.5 does not hold.

$Figure 2 Graphs of the function u : ℝ → ℝ {u:{\mathbb{R}}\to{\mathbb{R}}} (blue) from Example A.6,which decays to 0 as x → 0 {x\to 0} faster than any Hölder function (orange).$

Figure 2

Graphs of the function u : ℝ → ℝ (blue) from Example A.6,which decays to 0 as x → 0 faster than any Hölder function (orange).

Example A.6.

Let Ω = [ - 1 2 , 1 2 ] and let Y = ℝ . The continuous function (see Figure 2)

u : [ - 1 2 , 1 2 ] → ℝ , u ⁢ ( x ) := { - 1 ln ⁡ | x | , x ∈ [ - 1 2 , 1 2 ] ∖ { 0 } , 0 , x = 0 ,

on the compact interval [ - 1 2 , 1 2 ] is uniformly continuous. However, it is not Hölder continuous, because otherwise there would exist reals α ∈ ( 0 , 1 ] and C ≥ 0 such that

- 1 ln ⁡ x = | u ⁢ ( x ) - u ⁢ ( 0 ) | ≤ C ⁢ | x - 0 | α = C ⁢ x α for all ⁢ x ∈ ( 0 , 1 2 ] ,

leading to the contradiction

C ≥ - 1 x α ⁢ ln ⁡ x → x ↘ 0 ∞

by the l’Hospital rule.

Lemma A.7.

Let α ∈ ( 0 , 1 ] and let Y be a Banach space. Every α-Hölder function u : U → Y on an open set U ⊆ Ω allows an α-Hölder extension to the closure U ¯ with the same Hölder constant.

Proof.

Let u : U ⊆ Ω → Y be α-Hölder. By Theorem A.5, it follows that u : U → Y is uniformly continuous. Hence, because Y is complete, there exists a continuous extension u ¯ : U ¯ ⊆ Ω → Y of u to the boundary. We next show that u ¯ is α-Hölder with [ u ¯ ] α = [ u ] α . Thereto, given arbitrary y , y ¯ ∈ U ¯ and the estimate ∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ ≤ [ u ] α ⁢ d ⁢ ( x , x ¯ ) α for all x , x ¯ ∈ U , we first pass to the limit x → y , then to x ¯ → y ¯ , and it results from the extension property that

∥ u ¯ ⁢ ( y ) - u ¯ ⁢ ( y ¯ ) ∥ ≤ [ u ] α ⁢ d ⁢ ( y , y ¯ ) α for all ⁢ y , y ¯ ∈ U ¯ .

This guarantees that u ¯ is α-Hölder with [ u ¯ ] α ≤ [ u ] α . Moreover, it evidently holds

∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ = ∥ u ¯ ⁢ ( x ) - u ¯ ⁢ ( x ¯ ) ∥ ≤ [ u ¯ ] α ⁢ d ⁢ ( x , x ¯ ) α for all ⁢ x , x ¯ ∈ U

and [ u ] α ≤ [ u ¯ ] α . Thus, the minimum Hölder coefficients of u and u ¯ are equal, yielding the claim. ∎

Theorem A.8.

Hölder continuous functions are bounded (on bounded sets).

Proof.

Let u ∈ C α ⁢ ( Ω , Y ) , let B ⊆ Ω be bounded and choose a fixed x 0 ∈ B . Then

(A.2) ∥ u ⁢ ( x ) ∥ ≤ ∥ u ⁢ ( x 0 ) ∥ + ∥ u ⁢ ( x ) - u ⁢ ( x 0 ) ∥ ≤ ∥ u ⁢ ( x 0 ) ∥ + [ u ] α ⁢ d ⁢ ( x , x 0 ) α ≤ ∥ u ⁢ ( x 0 ) ∥ + [ u ] α ⁢ ( diam ⁡ B ) α

for all x ∈ Ω holds, and therefore the image u ⁢ ( B ) ⊆ Y is bounded. ∎

For globally bounded α-Hölder functions u : Ω → Y , we define

∥ u ∥ α := { sup x ∈ Ω ⁡ ∥ u ⁢ ( x ) ∥ , α = 0 , max ⁡ { sup x ∈ Ω ⁡ ∥ u ⁢ ( x ) ∥ , [ u ] α } , α ∈ ( 0 , 1 ] .

On the product Y 1 × Y 2 of two normed spaces Y 1 , Y 2 , we use the product norm

∥ ( y 1 , y 2 ) ∥ = max ⁡ { ∥ y 1 ∥ , ∥ y 2 ∥ } for all ⁢ y 1 ∈ Y 1 , y 2 ∈ Y 2 .

Theorem A.9.

A function u = ( u 1 , u 2 ) : Ω → Y 1 × Y 2 is α-Hölder if and only if both component functions u j : Ω → Y j are α-Hölder for j = 1 , 2 . In this case and for α ∈ ( 0 , 1 ] , one has

[ u j ] α ≤ [ u ] α ≤ max ⁡ { [ u 1 ] α , [ u 2 ] α } ,

and for globally bounded functions u this results in

∥ u j ∥ α ≤ ∥ u ∥ α ≤ max ⁡ { ∥ u 1 ∥ α , ∥ u 2 ∥ α } for all ⁢ j = 1 , 2 .

Proof.

We restrict to the case α ∈ ( 0 , 1 ] .

“ ⇒ ”: If u : Ω → Y 1 × Y 2 is α-Hölder, then also the components u 1 , u 2 are α-Hölder due to

∥ u j ⁢ ( x ) - u j ⁢ ( x ¯ ) ∥ ≤ max i = 1 2 ⁡ ∥ u i ⁢ ( x ) - u i ⁢ ( x ¯ ) ∥ = ∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ ≤ [ u ] α ⁢ d ⁢ ( x , x ¯ ) α for all ⁢ x , x ¯ ∈ Ω , j = 1 , 2 .

“ ⇐ ”: Conversely, if the component functions u 1 , u 2 are α-Hölder, then also u is α-Hölder because of

∥ u ( x ) - u ( x ¯ ) ∥ = max i = 1 2 ∥ u i ( x ) - u i ( x ¯ ) ∥ ≤ max i = 1 2 [ u i ] α d ( x , x ¯ ) α for all x , x ¯ ∈ Ω .

These inequalities also imply the claimed estimates for the Hölder constants. Combining them with

∥ u j ( x ) ∥ ≤ ∥ u ( x ) ∥ = max i = 1 2 ∥ u i ( x ) ∥ ≤ max i = 1 2 ∥ u i ∥ α for all x ∈ Ω , j = 1 , 2 ,

yields the estimates for ∥ ⋅ ∥ α after passing to the supremum over x ∈ Ω . ∎

The product of two metric spaces Ω 1 , Ω 2 is equipped with the product metric

d ⁢ ( ( x 1 , x 2 ) , ( x ¯ 1 , x ¯ 2 ) ) := max ⁡ { d ⁢ ( x 1 , x ¯ 1 ) , d ⁢ ( x 2 , x ¯ 2 ) } for all ⁢ x 1 , x ¯ 1 ∈ Ω 1 , x 2 , x ¯ 2 ∈ Ω 2 .

Theorem A.10.

Let α ∈ ( 0 , 1 ] . A function u : Ω 1 × Ω 2 → Y defined on the product of metric spaces Ω 1 , Ω 2 is α-Hölder if and only if all functions u ⁢ ( ⋅ , x 2 ) : Ω 1 → Y and u ⁢ ( x 1 , ⋅ ) : Ω 2 → Y are α-Hölder uniformly in x 2 ∈ Ω 2 , resp. x 1 ∈ Ω 1 .

Proof.

Suppose that x 1 , x ¯ 1 ∈ Ω 1 and x 2 , x ¯ 2 ∈ Ω 2 are arbitrary.

“ ⇒ ”: Let u : Ω 1 × Ω 2 → Y be α-Hölder. From the estimates

∥ u ⁢ ( x 1 , x 2 ) - u ⁢ ( x ¯ 1 , x 2 ) ∥ ≤ [ u ] α ⁢ d ⁢ ( x 1 , x ¯ 1 ) α , ∥ u ⁢ ( x 1 , x 2 ) - u ⁢ ( x 1 , x ¯ 2 ) ∥ ≤ [ u ] α ⁢ d ⁢ ( x 2 , x ¯ 2 ) α

one deduces that u ⁢ ( ⋅ , x 2 ) and u ⁢ ( x 1 , ⋅ ) are α-Hölder (uniformly in x 2 , resp. x 1 ).

“ ⇐ ”: Conversely, the estimate

∥ u ⁢ ( x 1 , x 2 ) - u ⁢ ( x ¯ 1 , x ¯ 2 ) ∥ ≤ ∥ u ⁢ ( x 1 , x 2 ) - u ⁢ ( x ¯ 1 , x 2 ) ∥ + ∥ u ⁢ ( x ¯ 1 , x 2 ) - u ⁢ ( x ¯ 1 , x ¯ 2 ) ∥

≤ sup x ∈ Ω 2 ⁡ [ u ⁢ ( ⋅ , x ) ] α ⁢ d ⁢ ( x 1 , x ¯ 1 ) α + sup x ∈ Ω 1 ⁡ [ u ⁢ ( x , ⋅ ) ] α ⁢ d ⁢ ( x 2 , x ¯ 2 ) α

≤ ( sup x ∈ Ω 2 [ u ( ⋅ , x ) ] α + sup x ∈ Ω [ u ( x , ⋅ ) ] α ) max { d ( x 1 , x ¯ 1 ) , d ( x 2 , x ¯ 2 ) } α

implies a Hölder condition for u. ∎

Theorem A.11.

Let α ∈ ( 0 , 1 ] . Let a function u : Ω 1 × Ω 2 → Y satisfy the following conditions:

sup x 1 ∈ Ω 1 ⁡ [ u ⁢ ( x 1 , ⋅ ) ] α < ∞ .
u ⁢ ( ⋅ , x 2 ) : Ω 1 → Y is continuous for all x 2 ∈ Ω 2 .

Then u is continuous.

Proof.

Let ( x 1 ∗ , x 2 ∗ ) ∈ Ω 1 × Ω 2 be the limit of sequences ( x l 1 ) l ∈ ℕ , ( x l 2 ) l ∈ ℕ in the respective metric spaces Ω 1 and Ω 2 . If k 2 := sup x ∈ Ω 1 ⁡ [ u ⁢ ( x , ⋅ ) ] α , then

0 ≤ ∥ u ⁢ ( x l 1 , x l 2 ) - u ⁢ ( x 1 ∗ , x 2 ∗ ) ∥

≤ ∥ u ⁢ ( x l 1 , x l 2 ) - u ⁢ ( x l 1 , x 2 ∗ ) ∥ + ∥ u ⁢ ( x l 1 , x 2 ∗ ) - u ⁢ ( x 1 ∗ , x 2 ∗ ) ∥

≤ k 2 ⁢ d ⁢ ( x l 2 , x 2 ∗ ) α + ∥ u ⁢ ( x l 1 , x 2 ∗ ) - u ⁢ ( x 1 ∗ , x 2 ∗ ) ∥ ⁢ (by (i))

→ l → ∞ 0 ⁢ (by (ii))

establishes the continuity of u, since ( x 1 ∗ , x 2 ∗ ) was arbitrary. ∎

Let us next investigate the algebraic structure of the space of α-Hölder functions.

Theorem A.12 (Sum rule).

With functions u 1 , u 2 : Ω → Y , also λ 1 ⁢ u 1 + λ 2 ⁢ u 2 is α-Hölder for all λ 1 , λ 2 ∈ K . In case α ∈ ( 0 , 1 ] , one has

[ λ 1 ⁢ u 1 + λ 2 ⁢ u 2 ] α ≤ | λ 1 | ⁢ [ u 1 ] α + | λ 2 | ⁢ [ u 2 ] α ,

and for globally bounded u 1 , u 2 it holds

∥ λ 1 ⁢ u 1 + λ 2 ⁢ u 2 ∥ α ≤ | λ 1 | ⁢ ∥ u 1 ∥ α + | λ 2 | ⁢ ∥ u 2 ∥ α for all ⁢ λ 1 , λ 2 ∈ 𝕂 .

Proof.

The straightforward proof is left to the reader. ∎

A mapping ⋅ : Y 1 × Y 2 → Y is called a product if there exists a constant C ≥ 0 such that

y 1 ⋅ ( y 2 + y ¯ 2 ) = y 1 ⋅ y 2 + y 1 ⋅ y ¯ 2 , ( y 1 + y ¯ 1 ) ⋅ y 2 = y 1 ⋅ y 2 + y ¯ 1 ⋅ y 2 , ∥ y 1 ⋅ y 2 ∥ ≤ C ⁢ ∥ y 1 ∥ ⁢ ∥ y 2 ∥

for all y 1 , y ¯ 1 ∈ Y 1 , y 2 , y ¯ 2 ∈ Y 2 .

Theorem A.13 (Product rule).

If functions u 1 : Ω → Y 1 and u 2 : Ω → Y 2 are globally bounded and α-Hölder, then also their product u 1 ⋅ u 2 : Ω → Y is α-Hölder. In case α ∈ ( 0 , 1 ] , one has the estimates

[ u 1 ⋅ u 2 ] α ≤ C ⁢ ( ∥ u 2 ∥ 0 ⁢ [ u 1 ] α + ∥ u 1 ∥ 0 ⁢ [ u 2 ] α ) 𝑎𝑛𝑑 ∥ u 1 ⋅ u 2 ∥ α ≤ C ⁢ ( ∥ u 2 ∥ 0 ⁢ ∥ u 1 ∥ α + ∥ u 1 ∥ 0 ⁢ ∥ u 2 ∥ α ) .

Proof.

We restrict to α ∈ ( 0 , 1 ] . Using properties of a product, we obtain from the triangle inequality

∥ ( u 1 ⋅ u 2 ) ⁢ ( x ) - ( u 1 ⋅ u 2 ) ⁢ ( x ¯ ) ∥ ≤ ∥ ( u 1 ⁢ ( x ) - u 1 ⁢ ( x ¯ ) ) ⋅ u 2 ⁢ ( x ) ∥ + ∥ u 1 ⁢ ( x ¯ ) ⋅ ( u 2 ⁢ ( x ) - u 2 ⁢ ( x ¯ ) ) ∥

≤ C ⁢ ∥ u 1 ⁢ ( x ) - u 1 ⁢ ( x ¯ ) ∥ ⁢ ∥ u 2 ⁢ ( x ) ∥ + C ⁢ ∥ u 1 ⁢ ( x ¯ ) ∥ ⁢ ∥ u 2 ⁢ ( x ) - u 2 ⁢ ( x ¯ ) ∥

≤ C ⁢ ( ∥ u 2 ∥ 0 ⁢ [ u 1 ] α + ∥ u 1 ∥ 0 ⁢ [ u 2 ] α ) ⁢ d ⁢ ( x , x ¯ ) α for all ⁢ x , x ¯ ∈ Ω ,

and this implies that u 1 ⋅ u 2 is α-Hölder. Then the estimate for ∥ u 1 ⋅ u 2 ∥ α follows easily. ∎

Theorem A.14 (Chain rule).

Let α 1 , α 2 ∈ [ 0 , 1 ] . If the function u 1 : Ω → Y 1 is α 1 -Hölder and the function u 2 : u 1 ⁢ ( Ω ) → Y is α 2 -Hölder, then the composition u 2 ∘ u 1 : Ω → Y is α 1 ⁢ α 2 -Hölder. In case α 1 , α 2 ∈ ( 0 , 1 ] , one has the estimate

[ u 2 ∘ u 1 ] α 1 ⁢ α 2 ≤ [ u 1 ] α 1 α 2 ⁢ [ u 2 ] β ,

and for globally bounded u 2 this results in

∥ u 2 ∘ u 1 ∥ α 1 ⁢ α 2 ≤ max ⁡ { 1 , [ u 1 ] α 1 α 2 } ⁢ ∥ u 2 ∥ α 2 .

Proof.

We focus on the situation α 1 , α 2 ∈ ( 0 , 1 ] . The following holds:

∥ u 2 ∘ u 1 ⁢ ( x ) - u 2 ∘ u 1 ⁢ ( x ¯ ) ∥ ≤ [ u 2 ] α 2 ⁢ | u 1 ⁢ ( x ) - u 1 ⁢ ( x ¯ ) | α 2 ≤ [ u 1 ] α 1 α 2 ⁢ [ u 2 ] α 2 ⁢ d ⁢ ( x , x ¯ ) α 1 ⁢ α 2 for all ⁢ x , x ¯ ∈ Ω ,

and so the composition u 2 ∘ u 1 is α-Hölder. The remaining norm estimate is readily derived. ∎

We abbreviate by B ⁢ ( Ω , Y ) the space of globally bounded functions, and we define the space of globally bounded α-Hölder functions by

C α ⁢ ( Ω , Y ) := { u ∈ B ⁢ ( Ω , Y ) : u ⁢ is α -Hölder } .

Due to Theorem A.8, the characterization

C α ( Ω , Y ) = { u : Ω → Y ∣ u is α -Hölder }

holds on bounded spaces Ω. By Lemma A.7, it is C α ⁢ ( U , Y ) = C α ⁢ ( U ¯ , Y ) for subsets U ⊆ Ω and Banach spaces Y.

Theorem A.15.

The set C α ⁢ ( Ω , Y ) is a normed space over K with respect to the norm ∥ ⋅ ∥ α . Furthermore, with Y, also C α ⁢ ( Ω , Y ) is a Banach space.

Proof.

We merely show the completeness of C α ⁢ ( Ω , Y ) with respect to the norm ∥ ⋅ ∥ α for α ∈ ( 0 , 1 ] . Thereto, let ( u l ) l ∈ ℕ be a Cauchy sequence in C α ⁢ ( Ω , Y ) . Since C 0 ⁢ ( Ω , Y ) is complete in the sup -norm, ( u l ) l ∈ ℕ converges to a continuous function u : Ω → Y . It remains to show that lim l → ∞ ⁡ [ u l - u ] α = 0 and that u is α-Hölder. Thereto, for ε > 0 first choose L ∈ ℕ such that ∥ u l - u m ∥ α ≤ ε 3 for all l , m ≥ L , and, given x , x ¯ ∈ Ω , x ≠ x ¯ , choose a fixed l ¯ ≥ L such that

∥ u l ¯ ⁢ ( x ) - u ⁢ ( x ) ∥ ≤ ε 3 ⁢ d ⁢ ( x , x ¯ ) α and ∥ u l ¯ ⁢ ( x ¯ ) - u ⁢ ( x ¯ ) ∥ ≤ ε 3 ⁢ d ⁢ ( x , x ¯ ) α .

Now this results in

∥ ( u l - u ) ⁢ ( x ) - ( u l - u ) ⁢ ( x ¯ ) ∥ d ⁢ ( x , x ¯ ) α ≤ ∥ ( u l - u l ¯ ) ⁢ ( x ) - ( u l - u l ¯ ) ⁢ ( x ¯ ) ∥ d ⁢ ( x , x ¯ ) α + ∥ u l ¯ ⁢ ( x ) - u ⁢ ( x ) ∥ d ⁢ ( x , x ¯ ) α + ∥ u l ¯ ⁢ ( x ¯ ) - u ⁢ ( x ¯ ) ∥ d ⁢ ( x , x ¯ ) α

≤ [ u l - u n ¯ ] α + 2 ⁢ ε 3

≤ ε for all ⁢ l ≥ L ,

and therefore [ u l - u ] α ≤ ε . If we set ε = 1 in the above inequality and note that ( [ u l ] α ) l ∈ ℕ is bounded, then the generalized triangle inequality guarantees

∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ d ⁢ ( x , x ¯ ) α ≤ 1 + ∥ u l ⁢ ( x ) - u l ⁢ ( x ¯ ) ∥ d ⁢ ( x , x ¯ ) α ≤ 1 + [ u l ] α ≤ 1 + sup l ∈ ℕ ⁡ [ u l ] α for all ⁢ x , x ¯ ∈ Ω ,

and consequently u ∈ C α ⁢ ( Ω , Y ) . ∎

Remark A.16.

(1) The positive homogeneity [ λ ⁢ u ] α = | λ | ⁢ [ u ] α for λ ∈ 𝕂 and Theorem A.12 guarantee that [ ⋅ ] α defines a semi-norm on C α ⁢ ( Ω , Y ) . It is not a norm, since [ ⋅ ] α vanishes on the constant functions. However, if α ∈ ( 0 , 1 ] and x 0 ∈ Ω is fixed, then ∥ u ∥ α ′ := max ⁡ { | u ⁢ ( x 0 ) | , [ u ] α } defines an equivalent norm on C α ⁢ ( Ω , Y ) . Indeed, any globally bounded α-Hölder function u : Ω → Y satisfies (A.2) for all x ∈ Ω , which implies the inequality ∥ u ∥ 0 ≤ ∥ u ⁢ ( x 0 ) ∥ + [ u ] α ⁢ ( diam ⁡ Ω ) α , and consequently

∥ u ∥ α ′ ≤ ∥ u ∥ α

≤ max ⁡ { ∥ u ⁢ ( x 0 ) ∥ + [ u ] α ⁢ ( diam ⁡ Ω ) α , [ u ] α }

≤ max ⁡ { ∥ u ⁢ ( x 0 ) ∥ , [ u ] α } + max ⁡ { ( diam ⁡ Ω ) α , 1 } ⁢ [ u ] α

≤ ( 1 + max ⁡ { ( diam ⁡ Ω ) α , 1 } ) ⁢ ∥ u ∥ α ′

guarantees that both norms are equivalent.

(2) Let Ω be compact. Then

C α ⁢ ( Ω ) + := { u ∈ C α ⁢ ( Ω , ℝ ) : 0 ≤ u ⁢ ( x ) ⁢ for all ⁢ x ∈ Ω }

is an order cone in C α ⁢ ( Ω ) with nonempty interior. However, if Ω has at least one accumulation point, then C α ⁢ ( Ω ) + is not normal (see [3]).

The following result establishes that α-Hölder functions on bounded sets form a decreasing scale of spaces between the Lipschitz continuous and the uniformly continuous functions (cf. Theorem A.5).

Lemma A.17.

Let Ω be bounded. If 0 ≤ α ≤ β ≤ 1 , then β-Hölder functions are α-Hölder and satisfy

[ u ] α ≤ ( diam ⁡ Ω ) β - α ⁢ [ u ] β .

Proof.

Given u ∈ C β ⁢ ( Ω , Y ) , one has

∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ ≤ d ⁢ ( x , x ¯ ) β - α ⁢ [ u ] β ⁢ d ⁢ ( x , x ¯ ) α ≤ ( diam ⁡ Ω ) β - α ⁢ [ u ] β ⁢ d ⁢ ( x , x ¯ ) α for all ⁢ x , x ¯ ∈ Ω ,

yielding the assertion. ∎

Theorem A.18 (Continuous embeddings).

Let Ω be bounded. If 0 ≤ α ≤ β ≤ 1 , then C β ⁢ ( Ω , Y ) ⊆ C α ⁢ ( Ω , Y ) is a continuous embedding with

∥ u ∥ α ≤ max ⁡ { 1 , ( diam ⁡ Ω ) β - α } ⁢ ∥ u ∥ β

for all u ∈ C β ⁢ ( Ω , Y ) .

In other words, one has a bounded embedding operator

(A.3) ℐ β α : C β ⁢ ( Ω , Y ) → C α ⁢ ( Ω , Y ) for all ⁢ 0 ≤ α ≤ β ≤ 1 .

Proof.

If u ∈ C β ⁢ ( Ω , Y ) , then Lemma A.17 implies [ u ] α < ∞ , and thus

max ⁡ { ∥ u ∥ 0 , [ u ] α } ≤ max ⁡ { ∥ u ∥ 0 , ( diam ⁡ Ω ) β - α ⁢ [ u ] β } ≤ max ⁡ { 1 , ( diam ⁡ Ω ) β - α } ⁢ ∥ u ∥ β

yields that u is also α-Hölder and satisfies the claimed estimate. ∎

Remark A.19 (Differentiable and Sobolev functions).

Let Ω ⊂ ℝ κ be open and bounded.

(1) If Ω is convex and C ¯ 1 ⁢ ( Ω , Y ) denotes the (canonically normed) space of continuously differentiable functions allowing a continuous extension to Ω ¯ , then one has the continuous embedding [1, pp. 11–12, 1.34 Theorem]

(A.4) C ¯ 1 ⁢ ( Ω , Y ) ⊆ C α ⁢ ( Ω ¯ , Y ) for all ⁢ 0 ≤ α ≤ 1 .

(2) Let dim ⁡ Y < ∞ . If Ω has a Lipschitz boundary and k ∈ ℕ and p ≥ 1 satisfy ( k - α ) ⁢ p ≥ κ , then the Sobolev space W k , p ⁢ ( Ω , Y ) satisfies the following continuous embedding (see [1])

(A.5) W k , p ⁢ ( Ω , Y ) ⊆ C α ⁢ ( Ω ¯ , Y ) for all ⁢ 0 < α ≤ 1 .

For Hölder exponents 0 < α < β ≤ 1 , the inclusion from Theorem A.22 will typically be strict.

Example A.20.

Let Ω = [ 0 , 1 ] and Y = ℝ . In case α ∈ ( 0 , 1 ) , the function u : [ 0 , 1 ] → ℝ , u ⁢ ( x ) := x α , is contained in C α ⁢ [ 0 , 1 ] . Now if u would be β-Hölder with exponent β > α , then there exists a C ≥ 0 such that

x α = | u ⁢ ( x ) - u ⁢ ( 0 ) | ≤ C ⁢ | x - 0 | β = C ⁢ x β for ⁢ x ∈ ( 0 , 1 ] ,

yielding the contradiction

C ≥ x α - β → x ↘ 0 ∞ .

Concerning the case α = 0 , we refer to Example A.6 for a continuous function not being Hölder.

On the space C α ⁢ ( Ω , Y ) exist several measures of non-compactness, which even are not necessarily equivalent (cf. [37, Section 5]). Among them, and for finite-dimensional spaces Y, we use (see [9, 10] and [50])

χ ⁢ ( B ) := lim ε ↘ 0 ⁡ sup u ∈ B ⁡ { ∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ d ⁢ ( x , x ¯ ) α : 0 < d ⁢ ( x , x ¯ ) ≤ ε }

and obtain a sufficient compactness criterion.

Theorem A.21 (Compactness in C α ⁢ ( Ω , Y ) , cf. [9, Theorem 4]).

Let Ω be compact and let dim ⁡ Y < ∞ . A subset B ⊆ C α ⁢ ( Ω , Y ) is relatively compact, provided the following two conditions hold:

B is bounded.
For every ε > 0 , there exists a δ > 0 such that for all x , x ¯ ∈ Ω one has the following implication:

d ⁢ ( x , x ¯ ) ≤ δ 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 ∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ ≤ ε ⁢ d ⁢ ( x , x ¯ ) α for all ⁢ u ∈ B .

For α = 0 , this is essentially the sufficiency part of the Arzelà–Ascoli theorem [38, p. 31, Theorem 3.2], which establishes that (i) and (ii) characterize the relatively compact subsets of C 0 ⁢ ( Ω , Y ) .

Theorem A.22 (Compact embeddings).

Let Ω be compact and let dim ⁡ Y < ∞ . If 0 ≤ α < β ≤ 1 , then

C β ⁢ ( Ω , Y ) ⊆ C α ⁢ ( Ω , Y )

is a compact embedding. Moreover, the embedding C β ⁢ ( Ω , Y ) ⊆ C 0 ⁢ ( Ω , Y ) is even dense, provided one has Ω ⊂ R κ .

This means that bounded subsets of C β ⁢ ( Ω , Y ) are relatively compact in C α ⁢ ( Ω , Y ) . In case α ∈ ( 0 , 1 ) , the embedding (A.4) is compact. Similarly, for ( k - α ) ⁢ p > κ , also the embedding (A.5) is compact (see [1, pp. 11–12, 1.34 Theorem]).

Proof.

(I) Let B ⊆ C β ⁢ ( Ω , Y ) be bounded, that is, there exists a C ≥ 0 such that ∥ u ∥ β ≤ C for all u ∈ B . This implies

∥ u ⁢ ( x ) ∥ ≤ C and ∥ u ⁢ ( x ) - u ⁢ ( x ¯ ) ∥ ≤ C ⁢ d ⁢ ( x , x ¯ ) β ≤ C ⁢ d ⁢ ( x , x ¯ ) β - α ⁢ d ⁢ ( x , x ¯ ) α

for all x , x ¯ ∈ Ω and u ∈ B , which guarantees that B ⊆ C β ⁢ ( Ω , Y ) satisfies the assumptions of Theorem A.21. Consequently, B is a relatively compact subset of C α ⁢ ( Ω , Y ) .

(II) Referring to the Stone–Weierstraß theorem [17, p. 218, Theorem 16.1], the polynomials over a compact Ω ⊂ ℝ κ form a set of β-Hölder functions being dense in the continuous functions. ∎

However, note that the embedding C β ⁢ ( Ω , Y ) ⊆ C α ⁢ ( Ω , Y ) is not dense for 0 < α < β ≤ 1 .

Example A.23.

Let Ω = [ 0 , 1 ] , let Y = ℝ and let u ∈ C α ⁢ [ 0 , 1 ] be given by u ⁢ ( x ) := x α . Choose v ∈ C β ⁢ [ 0 , 1 ] and consider the difference u - v ∈ C α ⁢ [ 0 , 1 ] satisfying

| ( u - v ) ⁢ ( x ) - ( u - v ) ⁢ ( 0 ) | | x - 0 | α ≥ | u ⁢ ( x ) - u ⁢ ( 0 ) | | x - 0 | α - | v ⁢ ( x ) - v ⁢ ( 0 ) | | x - 0 | α = 1 - | x | β | x | α ⁢ | v ⁢ ( x ) - v ⁢ ( 0 ) | | x - 0 | β → x ↘ 0 1 .

This implies that any function v ∈ C β ⁢ [ 0 , 1 ] has α-norm greater than or equal to 1 from u.

The final example demonstrates that C α ⁢ ( Ω , Y ) is not separable.

Example A.24.

Let Ω = [ 0 , 1 ] , let Y = ℝ and for reals c ∈ ( 0 , 1 ) define the α-Hölder functions

u c : [ 0 , 1 ] → ℝ , u c ⁢ ( x ) := { 0 , 0 ≤ x ≤ c , ( x - c ) α , a < x ≤ 1 ,

where α ∈ ( 0 , 1 ] . This implies the inequality

∥ u a - u b ∥ α ≥ [ u a - u b ] α

= sup x , x ¯ ∈ [ 0 , 1 ] x ≠ x ¯ ⁡ ( u a - u b ) ⁢ ( x ) - ( u a - u b ) ⁢ ( x ¯ ) | x - x ¯ | α

≥ | ( u a - u b ) ⁢ ( b ) - ( u a - u b ) ⁢ ( a ) | | b - a | α

= ( b - a ) α | b - a | α

= 1 for all ⁢ a , b ∈ ( 0 , 1 )

with the uncountable family { u c } c ∈ ( 0 , 1 ) ⊆ C α ⁢ [ 0 , 1 ] .

References

[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed. Pure Appl. Math. (Amsterdam) 140, Elsevier/Academic, Amsterdam, 2002. Suche in Google Scholar

[2] R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Oper. Theory Adv. Appl. 55, Birkhäuser, Basel, 1992. 10.1007/978-3-0348-5727-7Suche in Google Scholar

[3] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. 10.1137/1018114Suche in Google Scholar

[4] J. Appell, Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl. 83 (1981), no. 1, 251–263. 10.1016/0022-247X(81)90261-4Suche in Google Scholar

[5] J. Appell, N. Guanda, N. Merentes and J. L. Sanchez, Boundedness and continuity properties of nonlinear composition operators: A survey, Commun. Appl. Anal. 15 (2011), no. 2–4, 153–182. Suche in Google Scholar

[6] J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge Tracts in Math. 95, Cambridge University, Cambridge, 1990. 10.1017/CBO9780511897450Suche in Google Scholar

[7] J. M. Appell, A. S. Kalitvin and P. P. Zabrejko, Partial Integral Operators and Integro-Differential Equations, Monogr. Textb. Pure Appl. Math. 230, Marcel Dekker, New York, 2000. 10.1201/9781482270402Suche in Google Scholar

[8] K. E. Atkinson, A survey of numerical methods for solving nonlinear integral equations, J. Integral Equations Appl. 4 (1992), no. 1, 15–46. 10.1216/jiea/1181075664Suche in Google Scholar

[9] J. Banaś and R. Nalepa, On the space of functions with growths tempered by a modulus of continuity and its applications, J. Funct. Spaces Appl. 2013 (2013), Article ID 820437. 10.1155/2013/820437Suche in Google Scholar

[10] J. Banaś and R. Nalepa, On a measure of noncompactness in the space of functions with tempered increments, J. Math. Anal. Appl. 435 (2016), no. 2, 1634–1651. 10.1016/j.jmaa.2015.11.033Suche in Google Scholar

[11] C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Ser. Nonlinear Anal. Appl. 9, Walter de Gruyter, Berlin, 2003. 10.1515/9783110199277Suche in Google Scholar

[12] A. Ben Amar, A. Jeribi and M. Mnif, Some fixed point theorems and application to biological model, Numer. Funct. Anal. Optim. 29 (2008), no. 1–2, 1–23. 10.1080/01630560701749482Suche in Google Scholar

[13] R. Chiappinelli and R. Nugari, The Nemitskiĭ operator in Hölder spaces: Some necessary and sufficient conditions, J. Lond. Math. Soc. (2) 51 (1995), no. 2, 365–372. 10.1112/jlms/51.2.365Suche in Google Scholar

[14] Ş. Cobzaş, R. Miculescu and A. Nicolae, Lipschitz Functions, Lecture Notes in Math. 2241, Springer, Cham, 2019. 10.1007/978-3-030-16489-8Suche in Google Scholar

[15] D. L. Cohn, Measure Theory, Birkhäuser, Boston, 1980. 10.1007/978-1-4899-0399-0Suche in Google Scholar

[16] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. 10.1007/978-3-662-00547-7Suche in Google Scholar

[17] E. DiBenedetto, Real Analysis, 2nd ed., Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser/Springer, New York, 2016. 10.1007/978-1-4939-4005-9Suche in Google Scholar

[18] P. Drábek, Continuity of Nĕmyckij’s operator in Hölder spaces, Comment. Math. Univ. Carolin. 16 (1975), 37–57. Suche in Google Scholar

[19] J. Durdil, On the differentiability of Urysohn and Nemyckij operators, Comment. Math. Univ. Carolin. 8 (1967), 515–553. Suche in Google Scholar

[20] C. Edmond, An integral equation representation for overlapping generations in continuous time, J. Econom. Theory 143 (2008), no. 1, 596–609. 10.1016/j.jet.2008.03.006Suche in Google Scholar

[21] S. Fenyö and H.-W. Stolle, Theorie und Praxis der linearen Integralgleichungen. 1, VEB Deutscher Verlag der Wissenschaften, Berlin, 1982. 10.1007/978-3-0348-7664-3Suche in Google Scholar

[22] R. Fiorenza, Hölder and Locally Hölder Continuous Functions, and Open Sets of Class C k , C k , λ , Front. Math., Birkhäuser/Springer, Cham, 2016. 10.1007/978-3-319-47940-8Suche in Google Scholar

[23] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer, Berlin, 2001. 10.1007/978-3-642-61798-0Suche in Google Scholar

[24] M. Goebel and F. Sachweh, On the autonomous Nemytskij operator in Hölder spaces, Z. Anal. Anwend. 18 (1999), no. 2, 205–229. 10.4171/ZAA/878Suche in Google Scholar

[25] D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Math. Appl. 373, Kluwer Academic, Dordrecht, 1996. 10.1007/978-1-4613-1281-9Suche in Google Scholar

[26] W. Hackbusch, Integral Equations, Internat. Ser. Numer. Math. 120, Birkhäuser, Basel, 1995. 10.1007/978-3-0348-9215-5Suche in Google Scholar

[27] G. H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), no. 3, 301–325. 10.1090/S0002-9947-1916-1501044-1Suche in Google Scholar

[28] N. Huseyin, A. Huseyin and K. Guseinov, Approximation of the set of trajectories of the nonlinear control system with limited control resources, Math. Model. Anal. 23 (2018), no. 1, 152–166. 10.3846/mma.2018.010Suche in Google Scholar

[29] N. J. Kalton, Spaces of Lipschitz and Hölder functions and their applications, Collect. Math. 55 (2004), no. 2, 171–217. Suche in Google Scholar

[30] V. Khatskevich and D. Shoiykhet, Differentiable Operators and Nonlinear Equations, Oper. Theory Adv. Appl. 66, Birkhäuser, Basel, 1994. 10.1007/978-3-0348-8512-6Suche in Google Scholar

[31] M. Kot and W. M. Schaffer, Discrete-time growth-dispersal models, Math. Biosci. 80 (1986), no. 1, 109–136. 10.1016/0025-5564(86)90069-6Suche in Google Scholar

[32] M. A. Krasnosel’skiĭ, P. P. Zabreĭko, E. I. Pustyl’nik and P. E. Sobolevskiĭ, Integral Operators in Spaces of Summable Functions, Monogr. Textb. Mech. Solids Fluids, Noordhoff International, Leiden, 1976. 10.1007/978-94-010-1542-4Suche in Google Scholar

[33] R. Kress, Linear Integral Equations, 3rd ed., Appl. Math. Sci. 82, Springer, Heidelberg, 2014. 10.1007/978-1-4614-9593-2Suche in Google Scholar

[34] M. Krukowski and B. Przeradzki, Compactness result and its applications in integral equations, J. Appl. Anal. 22 (2016), no. 2, 153–161. 10.1515/jaa-2016-0016Suche in Google Scholar

[35] S. Lang, Real and Functional Analysis, Grad. Texts in Math. 142, Springer, New York, 1993. 10.1007/978-1-4612-0897-6Suche in Google Scholar

[36] F. Lutscher, Integrodifference Equations in Spatial Ecology, Interdiscip. Appl. Math. 49, Springer, Cham, 2019. 10.1007/978-3-030-29294-2Suche in Google Scholar

[37] J. Mallet-Paret and R. D. Nussbaum, Inequivalent measures of noncompactness, Ann. Mat. Pura Appl. (4) 190 (2011), no. 3, 453–488. 10.1007/s10231-010-0158-xSuche in Google Scholar

[38] R. H. Martin, Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Pure Appl. Math. 11, Wiley-Interscience, New York, 1976. Suche in Google Scholar

[39] A. Matkowska, On characterization of Lipschitzian operators of substitution in the class of Hölder’s functions, Zeszyty Nauk. Politech. Łódz. Mat. 17 (1984), 81–85. Suche in Google Scholar

[40] J. Matkowski, Uniformly continuous superposition operators in the Banach space of Hölder functions, J. Math. Anal. Appl. 359 (2009), no. 1, 56–61. 10.1016/j.jmaa.2009.05.020Suche in Google Scholar

[41] I. V. Misyurkeev and Y. V. Nepomnyashchikh, A criterion for the complete continuity of the Uryson operator, Sov. Math. 35 (1991), no. 4, 31–41. Suche in Google Scholar

[42] R. Nugari, Further remarks on the Nemitskiĭ operator in Hölder spaces, Comment. Math. Univ. Carolin. 34 (1993), no. 1, 89–95. Suche in Google Scholar

[43] H. Pachale, Über den Urysohnschen Integraloperator, Arch. Math. 10 (1959), 134–136. 10.1007/BF01240775Suche in Google Scholar

[44] H. K. Pathak, An Introduction to Nonlinear Analysis and Fixed Point Theory, Springer, Singapore, 2018. 10.1007/978-981-10-8866-7Suche in Google Scholar

[45] M. Poluektov and A. Polar, Modelling non-linear control systems using the discrete Urysohn operator, J. Franklin Inst. 357 (2020), no. 6, 3865–3892. 10.1016/j.jfranklin.2020.02.030Suche in Google Scholar

[46] C. Pötzsche, Numerical dynamics of integrodifference equations: Basics and discretization errors in a C 0 -setting, Appl. Math. Comput. 354 (2019), 422–443. 10.1016/j.amc.2019.02.033Suche in Google Scholar

[47] C. Pötzsche, Uniform convergence of Nyström discretizations on Hölder spaces, J. Integral Equations Applications 34 (2022), no. 2, 247–255. 10.1216/jie.2022.34.247Suche in Google Scholar

[48] R. Precup, Methods in Nonlinear Integral Equations, Kluwer Academic, Dordrecht, 2002. 10.1007/978-94-015-9986-3Suche in Google Scholar

[49] S. Saiedinezhad, Existence and asymptotically stable solution of a Hammerstein type integral equation in a Hölder space, Bull. Belg. Math. Soc. Simon Stevin 25 (2018), no. 3, 453–465. 10.36045/bbms/1536631238Suche in Google Scholar

[50] S. Saiedinezhad, On a measure of noncompactness in the Holder space C k , γ ⁢ ( Ω ) and its application, J. Comput. Appl. Math. 346 (2019), 566–571. 10.1016/j.cam.2018.07.030Suche in Google Scholar

[51] M. Väth, Complete continuity of the Uryson operator, Nonlinear Anal. 30 (1997), 527–534. 10.1016/S0362-546X(96)00216-7Suche in Google Scholar

[52] M. Väth, Volterra and Integral Equations of Vector Functions, Marcel Dekker, New York, 2000. Suche in Google Scholar

Received: 2021-08-09

Accepted: 2021-12-14

Published Online: 2022-08-30

Published in Print: 2022-11-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/anly-2021-0052

Schlagwörter für diesen Artikel

Urysohn integral operator; Hammerstein integral operator; Nemytskii operator; nonlinear operator; Hölder continuity; Lipschitz continuity

Creative Commons

BY 4.0