UNIFORMLY BOUNDED NEMYTSKIJ OPERATORS GENERATED BY SET-VALUED FUNCTIONS
BETWEEN GENERALIZED HÖLDER FUNCTION SPACES
Janusz Matkowski
Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra
Prof. Z. Szafrana 4a, 65–516 Zielona Góra, Poland and
Institute of Mathematics Silesian University
Bankowa 14, 40–007 Katowice, Poland e-mail: J.Matkowski@wmie.uz.zgora.pl
and
Małgorzata Wróbel
Institute of Mathematics and Computer Science Jan Dlugosz University
42–200 Częstochowa, Poland e-mail: m.wrobel@ajd.czest.pl
Abstract
We prove that the generator of any uniformly bounded set-valued Nemytskij operator acting between generalized Hölder function metric spaces, with nonempty compact and convex values is an affine function with respect to the function variable.
Keywords and phrases: Nemytskij composition operator, uniformly bounded operator, set-valued function, generalized Hölder function metric space.
1. Introduction
Let X, Y, Z be nonempty sets and h : X × Y → Z be an arbitrary function. By YX denote the set of all functions ϕ : X → Y. The mapping H : YX → ZX defined by
H(ϕ)(x) := h(x, ϕ(x)), (x ∈ X), ϕ∈ YX,
is called a composition (Nemytskij or superposition) operator, and h is re-ferred to as a generator of H.
In [6] it was shown that if H, mapping the Banach space Lip([0, 1], R) into itself, satisfies Lipschitz condition with respect to the Lip-norm, then
h(x, y) = a(x)y + b(x), x∈ [0, 1], y ∈ R,
for some a, b ∈ Lip([0, 1], R), which means that h is affine with respect to the second variable. It follows that the Banach contraction principle is not appli-cable in solving some important nonlinear problems. Analogous results have been proved for some other function Banach spaces ([3, 13, 7], cf. also [1] for other references). Some of these results have been extended for multivalued mappings (cf. for instance [5, 15, 16]). In all these results the norm-Lipschitz continuity of the Nemytskij operator H is the basic assumption.
In this situation a natural question has appeared, if the Lipschitz conti-nuity can be replaced by a weaker condition. In [9] it was observed that the above results remain true if the Lipchitz-norm-continuity of H is replaced by its uniform continuity (cf. also [2, 4, 10, 11]). The suitable results for the class of Hölder set-valued functions were obtained by Ludew [5].
An essential progress in weakening of the basic assumption was made in a recent paper [12]. It turns out that the uniform continuity of H can be replaced by a much weaker condition of its uniform boundedness (cf. Definition 2), that is even weaker than the boundedness of H.
In the present paper we extend the main result of [12] to the class of set-valued functions.
prove that if H is uniformly bounded, then h is affine with respect to the second variable, and we give a more detailed representation of the operator H (Theorem 3).
Taking X := [0, 1], ρ1(x, y) = |x − y|α, ρ2(x, y) = |x − y|β for some fixed α, β ∈ (0, 1], we obtain the result of Ludew [5].
2. Preliminaries
Given a real normed space (Z, |·|Z), denote by cc(Z) the class of all nonempty, compact and convex subsets of Z. For A, B ⊂ Z and α ∈ R, define A + B := {a + b : a ∈ A, b ∈ B} and αA := {αa : a ∈ A}. It is well known that the cc(Z) with the addition is a commutative semigroup which satisfies the cancellation law.
Let d stand for the Hausdorff metric on the space cc(Z), that is, for all nonempty and bounded sets A, B ⊂ Z,
d(A, B) := max{e(A, B), e(B, A)}, A, B∈ cc(Z), where
e(A, B) = sup{ρ(a, B) : a ∈ A}, ρ(z, B) = inf{|z − b|Z : b ∈ B}. The Hausdorf metric has the following properties, for all λ ≥ 0 and A, B, C, D∈ cc(Z),
(1) d(A + B, A + C) = d(B, C),
(2) d(A + B, C + D) ≤ d(A, C) + d(B, D),
(3) d(λA, λB) = λd(A, B).
Let (Y, | · |Y) be a real normed space. A subset C ⊂ Y is said to be a convex cone if λC ⊂ C for all λ ≥ 0 and C + C ⊂ C. Note that 0 ∈ C.
Let (Y, | · |Y), (Z, | · |Z) be real normed spaces and C ⊂ Y a convex cone. A set-valued function F : C → cc(Z) is said to be additive, if
and Jensen if
2F x + y 2
= F (x) + F (y), x, y∈ C. Let us quote the following
Lemma 1[14, Theorem 5.6]. Let C be a convex cone in a real linear space (Y, | · |Y) and let (Z, | · |Z) be a Banach space. A set-valued function F : C → cc(Z) is Jensen if and only if there exists an additive set-valued function A: C → cc(Z) and a set B ∈ cc(Z) such that
F(x) = A(x) + B for allx∈ C.
For the normed spaces (Y, | · |Y), (Z, | · |Z), traditionally, by (L(Y, Z), k · kL(Y,Z)), briefly L(Y, Z), we denote the normed space of all additive and continuous mappings a ∈ ZY.
Let C be a convex cone in a real normed space (Y, | · |Y). By L(C, cc(Z)) we denote the family of all set-valued functions a : C → cc(Z) which are additive and continuous (so positively homogeneous) that is,
L(C, cc(Z)) = {a ∈ cc(Z)C : a is additive and continuous}. The L(C, cc(Z)) with dL(C,cc(Z)) defined by
(4) dL(C,cc(Z))(a, b) := sup y∈C\{0}
d(a(y), b(y)) kykY is a metric space.
3. Composition operators satisfying a general Lipschitz condition
Let (X, ρX) be a metric space and (Y, | · |Y) be a real normed space. By Lip((X, ρX), (Y, | · |Y)), briefly Lip(X, Y ), we denote the family of all Lips-chitz functions ϕ ∈ YX.Thus ϕ ∈ Lip(X, Y ) if
L(ϕ) := sup |ϕ(x) − ϕ(x)|Y ρX(x, x)
: x, x ∈ X, x 6= x
Let x0 ∈ X. Then the pair (Lip(X, Y ), k · kLip(X,Y ),x0) where k · kLip(X,Y ),x0 :
Lip(X, Y ) → [0, ∞) is defined by
(5) kϕkLip(X,Y ),x0 := |ϕ(x0)|Y + L(ϕ),
is a normed space.
Remark 1. Since for any x0, x1 ∈ X, the norms kϕkLip(X,Y ),x0 and
kϕkLip(X,Y ),x1 are equivalent [12, Remark 2], to simplify the notation, we
shall write kϕkLip(X,Y ) instead of kϕkLip(X,Y ),x0.
For the metric space (X, ρ) with X = [0, 1], ρ(x, y) = |x − y|α for some fixed α ∈ (0, 1], and Y := R, the space Lip(X, Y ) becomes the classical Hölder function space. This fact justifies the name a generalized Hölder function space for Lip(X, Y ).
For a set C ⊂ Y we put
Lip((X, ρ), C) := {ϕ ∈ Lip((X, ρ), Y ) : ϕ(X) ⊂ C}.
In the proof of the main result the following result plays a crucial role. Lemma 2 [12, Lemma 1]. Let (X, ρ) be a metric space, (Y, | · |Y) be a real normed space, C⊂ Y a convex set, and let x, x ∈ X, x 6= x, be fixed.
Then, for arbitrary y, y∈ Y, the function ϕy,y: X → Y defined by
(6) ϕy,y(t) := ρ(t, x)y + ρ(t, x)y
ρ(t, x) + ρ(t, x) , t∈ X; has the following properties:
(a) ϕy,y(x) = y, ϕy,y(x) = y; (b) ϕy,y∈ Lip((X, ρ), Y ) and
(d) the set
K(x, x) := {ϕy,y: y, y ∈ Y }
is a linear subspace of Lip((X, ρ), Y ) containing all constant functions. In the sequel by Lip((X, ρ), cc(Z)) we denote the set of all set-valued func-tions ϕ : X → cc(Z) such that
sup x1,x2∈X x16=x2 d(ϕ(x1), ϕ(x2)) ρ(x1, x2) <∞.
In the set Lip((X, ρ), cc(Z)) we introduce a metric dLip(X,cc(Z)) putting dLip(X,cc(Z))(ϕ1, ϕ2) := d(ϕ1(x0), ϕ2(x0)) + sup x1,x2∈X x16=x2 d(ϕ1(x1) + ϕ2(x2), ϕ1(x2) + ϕ2(x1)) ρ(x1, x2) ,
where x0 ∈ X is arbitrarily fixed.
By K(x, x; C) denote the set of all functions ϕ ∈ K(x, x) with values in C, that is
K(x, x; C) := K(x, x) ∩ Lip((X, ρ), C).
Definition 1. Let (X, ρ) be a metric space, (Y, |·|Y), (Z, |·|Z) be real normed spaces, C ⊂ Y a convex cone. Given a set-valued function h : X×C → cc(Z), the mapping H : YX → cc(Z)X defined by
H(ϕ)(x) = h(x, ϕ(x)), (x ∈ X), ϕ∈ YX,
is called the composition (Nemytskij or superposition) operator of a genera-tor h.
If for all x, x∈ X, x 6= x and ϕ, ψ ∈ K(x, x; C) the operator H satisfies the inequality
(7) dLip((X,ρ2),cc(Z))(H(ϕ), H(ψ)) ≤ γ kϕ − ψkLip((X,ρ1),Y )
for some function γ : [0, ∞) → [0, ∞), then there exist a : X × C → cc(Z) andb: X → cc(Z) such that a(·, y), b ∈ Lip((X, ρ2), cc(Z)) for every y ∈ C, a(x, ·) ∈ L(C, cc(Z)) for every x ∈ X, and
h(x, y) = a(x, y) + b(x), x∈ X, y ∈ C.
Moreover, the function X ∋ x 7→ a(x, ·) ∈ L(C, cc(Z)) fulfils the following condition
(8) dL(C,cc(Z))(a(x, ·), a(x, ·)) ≤ γ(1)ρ2(x, x), x, x∈ X.
P roof. According to Lemma 2, for arbitrary y ∈ C, the constant func-tion ϕ(t) = y, t ∈ X, belongs to K(x, x; C). Since H maps K(x, x; C) into Lip((X, ρ2), cc(Z)), the function
(9) H(ϕ) = h(·, y) ∈ Lip((X, ρ2), cc(Z))
and, consequently, h is continuous with respect to its first variable.
Let us fix x, x ∈ X, take arbitrary y1, y2, y1, y2 ∈ C and define the func-tions ϕy1,y1, ϕy2,y2 by (6). In view of Lemma 2, the functions ϕy1,y1, ϕy2,y2
belong to K(x, x; C) and, by Remark 1, with x0= x in (5), we have (10) kϕy1,y1 − ϕy2,y2kLip((X,ρ1),Y )= |y1− y2| +
|y1− y2− y1+ y2|Y ρ1(x, x)
. Applying inequality (7) with ϕ = ϕy1,y1, ψ= ϕy2,y2,by the definition of the
metric dLip((X,ρ2),cc(Z)),we get
d(H(ϕy1,y1)(x) + H(ϕy2,y2)(x), H(ϕy1,y1)(x) + H(ϕy2,y2)(x))
ρ2(x, x) ≤ dLip((X,ρ2),cc(Z))(H(ϕy1,y1), H(ϕy2,y2))
≤ γ kϕy1,y1 − ϕy2,y2kLip((X,ρ1),Y )
and, as (11) ϕy1,y1(x) = y1, ϕy1,y1(x) = y1, ϕy2,y2(x) = y2, ϕy2,y2(x) = y2, we obtain d(h(x, y1) + h(x, y2), h(x, y1) + h(x, y2)) ρ2(x, x) ≤ γ(kϕy1,y1 − ϕy2,y2kLip((X,ρ1),Y )) for all x, x ∈ X; y1, y2, y1, y2∈ C.
Taking arbitrary u, v ∈ C and setting y1 = y2 := u+v2 , y1 =: u, y2 := v in (10), we get
kϕy1,y1 − ϕy2,y2kLip((X,ρ1),Y )=
|u − v|Y 2 and d h x,u+ v 2 + h x,u+ v 2 , h(x, u) + h(x, v) ≤ γ |u − v|Y 2 ρ2(x, x) for all x, x ∈ X, x 6= x, u, v ∈ C.
Letting here x tends to x and taking into account the continuity of h with respect to its first variable we get
d 2h x,u+ v 2 , h(x, u) + h(x, v) = 0,
which shows that for every x ∈ X the function h(x, ·) satisfies the equation 2h x,u+ v 2 = h(x, u) + h(x, v), u, v ∈ C,
that is, for any x ∈ X, h(x, ·) is Jensen in C. By Lemma 1 there exist functions a : X × C → cc(Z) and b : X → cc(Z) such that a(x, ·) is additive for x ∈ X and
To show that a(x, ·) is continuous for any x ∈ X, let us fix y, y ∈ C. By (1) and (12) we have
d(a(x, y), a(x, y)) = d(a(x, y) + b(x), a(x, y) + b(x)) = d(h(x, y), h(x, y)). Therefore, the continuity of h(x, ·) with respect to the second variable implies the continuity of a(x, ·) and, consequently, being additive,
a(x, ·) ∈ L(C, cc(Z)) for every x ∈ X.
To prove that b ∈ Lip((X, ρ2), cc(Z)) let us note that the additivity of a(x, ·) implies a(x, 0) = {0}. Hence, applying (12), we get
(13) h(x, 0) = a(x, 0) + b(x)
and by (9), we have h(·, y) ∈ Lip((X, ρ2), cc(Z)) for all y ∈ C. Equality (13) implies that b ∈ Lip((X, ρ2), cc(Z)).
To prove that a(·, y) ∈ Lip((X, ρ2), cc(Z)) for all y ∈ C, let us fix arbi-trarily x, x ∈ X, x 6= x, y ∈ C and take a constant function ϕ(t) = y, (t ∈ X), belonging, by Lemma 2, to K(x, x; C). Taking into account (1), (2) and (12) we have
d(a(x, y), a(x, y)) = d(a(x, y) + b(x), a(x, y) + b(x))
≤ d(a(x, y) + b(x), a(x, y) + b(x)) + d(a(x, y) + b(x), a(x, y) + b(x)) = d(h(x, y), h(x, y)) + d(b(x), b(x))
= d(H(ϕ)(x), H(ϕ)(x)) + d(b(x), b(x)).
Since H maps K(x, x; C) into Lip((X, ρ2), cc(Z)), the function H(ϕ) belongs to Lip((X, ρ2), cc(Z)). Moreover, according to what has already been proved, b ∈ Lip((X, ρ2), cc(Z)). Now the above inequality implies that a(·, y) ∈ Lip((X, ρ2), cc(Z)).
To show that the function X ∋ x → a(x, ·) ∈ L(C, cc(Z)) fulfils (8) take x, x ∈ X, y1, y2, y1, y2∈ C and define ϕy1,y1 ϕy2,y2 ∈ K(x, x; C) by (6).
d(h(x, y1) + h(x, y2), h(x, y1) + h(x, y2)) ≤ γ
|y1− y2|Y +|y1− y2− y1+ y2|Y ρ1(x, x)
ρ2(x, x), and, consequently, by (1) and (12),
d(a(x, y1) + a(x, y2), a(x, y1) + a(x, y2)) ≤ γ |y1− y2|Y + |y1− y2− y1+ y2|Y ρ1(x, x) ρ2(x, x) for all x, x ∈ X; y1, y2, y1, y2∈ C.
Taking arbitrary y, y ∈ C and setting y1 = y2 := y + y ∈ C, y2 := y, y1 := 2y + y ∈ C we obtain
d(a(x, y + y) + a(x, y + y), a(x, 2y + y) + a(x, y)) ≤ γ(|y|Y)ρ2(x, x) and, by the additivity of a(x, ·),
(14) d(a(x, y), a(x, y) ≤ γ(|y|Y)ρ2(x, x) for all x, x ∈ X and y ∈ C.
Take arbitrary u ∈ C and put y := |u|u
Y ∈ C. Using (14) we obtain d a x, u |u|Y , a x, u |u|Y ≤ γ(1)ρ2(x, x). By the linearity of a(x, ·) for all x ∈ X and (3) we get,
d(a(x, u), a(x, u))
|u|Y ≤ γ(1)ρ2(x, x) which, in view of (4), completes the proof of (8).
P roof.By the definition of the metric dLip((X,ρ2),cc(Z)) and (7) we have
d(H(ϕ)(x0), H(ψ)(x0)) +d(H(ϕ)(x) + H(ψ)(x0), H(ϕ)(x0) + H(ψ)(x)) ρ2(x, x0)
(15) ≤ γ(kϕ − ψk(Lip(X,ρ1),Y ))
for all x ∈ X, ϕ, ψ ∈ K(x0, x; C).
Fix arbitrarily x ∈ X, y, y ∈ C and take a pair of constant functions ϕ, ϕ: X → C defined by
ϕ(t) := y, ϕ(t) := y, t∈ X. By Lemma 2(d), these functions belong to K(x0, x; C) and
kϕ − ϕkLip((X,ρ1),Y )= |y − y|Y.
Hence, by (15), we get d(h(x0, y), h(x0, y)) + d(h(x, y) + h(x0, y), h(x0, y) + h(x, y)) ρ2(x, x0) ≤ γ(|y − y|Y). It follows that
(16) d(h(x0, y), h(x0, y)) ≤ γ(|y − y|Y) and
(17) d(h(x, y) + h(x0, y), h(x0, y) + h(x, y)) ≤ γ(|y − y|Y)ρ2(x, x0). By (2) we have
d(h(x, y), h(x, y))
Hence, taking into account (16) and (17), we get
d(h(x, y), h(x, y)) ≤ γ(|y − y|)(1 + ρ2(x, x0))
which, by the continuity of γ at 0 and the equality γ(0) = 0, implies that h is continuous with respect to its second variable.
From Theorem 1 we obtain the following
Theorem 2. Let(X, ρ1), (X, ρ2) be metric spaces, (Y, | · |Y), (Z, | · |Z) be real normed spaces, C⊂ Y a convex cone and a function h : X × C → cc(Z) be such that for anyx∈ X the function h(x, ·) : C → cc(Z) is continuous with respect to second variable. Suppose that the composition operator H of the generator h maps Lip((X, ρ1), C) into the space Lip((X, ρ2), cc(Z)).
If there exists a function γ : [0, ∞) → [0, ∞) such that
(18)
dLip((X,ρ2),cc(Z))(H(ϕ), H(ψ))
≤ γ(kϕ − ψkLip((X,ρ1),Y )), ϕ, ψ∈ Lip((X, ρ1), C),
then
h(x, y) = a(x, y) + b(x), x∈ X, y ∈ C,
for some b∈ Lip((X, ρ2), cc(Z)) and a : X × C → cc(Z) such that a(·, y) ∈ Lip((X, ρ2), cc(Z)) for every y ∈ C, a(x, ·) ∈ L(C, cc(Z)) for every x ∈ X and the functionX ∋ x → a(x, ·) ∈ L(C, cc(Z)) fulfils the Lipschitz condition with the constant γ(1) i.e.,
(19) dL(C,cc(Z))(a(x, ·), a(x, ·)) ≤ γ(1)ρ2(x, x), x, x∈ X. Let us note the following
If there exists a function γ : [0, ∞) → [0, ∞) such that
(20)
kH(ϕ) − H(ψ)klip((X,ρ2),Z)≤ γ kϕ − ψkLip((X,ρ1),Y ) ,
ϕ, ψ ∈ Lip((X, ρ1), C), then
h(x, y) = a(x)y + b(x), x∈ X, y∈ C, for somea∈ Lip((X, ρ2), L(C, Z)) and b ∈ Lip((X, ρ2), Z).
P roof.In the space Lip((X, ρ2), Z) inequality (18) reduces to (20). There-fore, by Theorem 2, there exist a : X → L(C, Z) and b ∈ Lip((X, ρ2), Z) such that
h(x, y) = a(x)y + b(x), x∈ X, y ∈ C, and inequality (19) becomes
sup y∈C\{0} |a(x)y − a(x)y|Z |y|Y ≤ γ(1)ρ(x, x), x, x∈ X. Since a(x) − a(x) ρ2(x, x) ∈ L(C, Z), x, x∈ X, x6= x, we get a(x) − a(x) ρ(x, x) L(C,Z)≤ γ(1), x, x∈ X, x6= x, which shows that a ∈ Lip((X, ρ2), L(C, Z)).
Remark 3. In Theorem 1 of [12] it is assumed that C is a convex set with nonempty interior and the function γ is bounded in a right neighborhood of 0. Assuming that C is a convex cone we can omit assumption about a function γ in Theorem 1 of [12].
Definition 2 ([12]). Let Y and Z be two metric (or normed) spaces. We say that a mapping H : Y → Z is uniformly bounded if, for any t > 0 there exists a nonnegative real number γ(t) ≥ 0 such that for any set B ⊂ Y we have
diamB ≤ t ⇒ diamH(B) ≤ γ(t).
Remark 4. Let (Y, ρY), (Z, ρZ) be metric spaces. A mapping H : Y → Z is uniformly bounded if, and only if, for any t > 0 there is γ(t) ≥ 0 such that for any points y1, y2 ∈ Y,
ρY(y1, y2) ≤ t ⇒ ρZ(H(y1), H(y2)) ≤ γ(t).
Remark 5. Cleraly, every uniformly continuous operator or Lipschitz oper-ator is uniformly bounded. Moreover, under the assumptions of Definition 2, every bounded operator is uniformly bounded and the converse is not true. Now we can formulate our main result.
Theorem 3. Let(X, ρ1), (X, ρ2) be metric spaces, (Y, |·|Y), (Z, |·|Z) be real normed spaces, C⊂ Y a convex cone and a function h : X × C → cc(Z) be such that for anyx∈ X the function h(x, ·) : C → cc(Z) is continuous. Sup-pose that the composition operatorH of the generator h maps Lip((X, ρ1), C) into the space Lip((X, ρ2), cc(Z)).
If H is uniformly bounded, then there exist a : X × C → cc(Z) and b : X → cc(Z) such that a(·, y), b ∈ Lip((X, ρ2), cc(Z)) for every y ∈ C, a(x, ·) ∈ L(C, cc(Z)) for every x ∈ X, the function X ∋ x → a(x, ·) ∈ L(C, cc(Z)) fulfils the Lipschitz condition and
H(ϕ)(x) = a(x, ϕ(x)) + b(x), ϕ∈ Lip((X, ρ1), C), (x ∈ X). P roof. Take any t ≥ 0 and arbitrary ϕ, ψ ∈ Lip((X, ρ1), C) such that kϕ − ψkLip((X,ρ1),Y )≤ t. Since diam{ϕ, ψ} ≤ t, by the uniform boundedness
of H, we have diamH({ϕ, ψ}) ≤ γ(t), i.e.,
dLip((X,ρ2),cc(Z))(H(ϕ), H(ψ)) = diamH({ϕ, ψ}) ≤ γ kϕ − ψkLip((X,ρ1),Y )
Remark 7. If γ(t) := Lt for some L ≥ 0, then inequality (7) becomes the classical Lipschitz condition. Taking X = [0, 1], ρ1(x, y) = α(|x − y|), ρ2(x, y) = β(|x − y|) for some Hölder functions α, β : [0, 1] → [0, 1] ([1], p. 182) we obtain the main result of [5].
References
[1] J. Appell and P.P. Zabrejko, Nonlinear Superposition Operators, Cambridge University Press, 1990.
[2] A. Azócar, J.A. Guerrero, J. Matkowski and N. Merentes, Uniformly contin-uous set-valued composition operators in the space of contincontin-uous functions of bounded variation in the sense of Wiener, Opuscula Math. 30 (2010), 53–60. [3] V.V. Chistyakov, Lipschitzian superposition operators between spaces of
func-tions of bounded generalized variation with weight, J. Appl. Anal. 6 (2000), 173–186.
[4] J.A. Guerrero, H. Leiva, J. Matkowski and N. Merentes, Uniformly continuous composition operators in the space of boundedϕ-variation functions,Nonlinear Anal. 72 (2010), 3119–3123.
[5] J.J. Ludew, On Lipschitzian operators of substitution generated by set-valued functions, Opuscula Math. 27 (1) (2007), 13–24.
[6] J. Matkowski, Functional equations and Nemytskij operators, Funkc. Ekvacioj Ser. Int. 25 (1982), 127–132.
[7] J. Matkowski, Lipschitzian composition operators in some function spaces, Nonlin. Anal. Theory Meth. Appl. 30 (2) (1997), 719–726.
[8] J. Matkowski, Remarks on Lipschitzian mappings and some fixed point the-orems, Banach J. Math. Anal. 2 (2007), 237–244 (electronic), www.math-analysis.org.
[9] J. Matkowski, Uniformly continuous superposition operators in the spaces of differentiable functions and absolutely continuous functions, Internat. Ser. Nu-mer. Math. 157 (2008), 155–155.
[10] J. Matkowski, Uniformly continuous superposition operators in the space of Hölder functions, J. Math. Anal. Appl. 359 (2009), 56–61.
[11] J. Matkowski, Uniformly continuous superposition operators in the spaces of bounded variation functions,Math. Nachr. 283 (7) (2010), 1060–1064. [12] J. Matkowski, Uniformly bounded composition operators between general
[13] J. Matkowski and J. Miś, On a charakterization of Lipschitzian operators of substitution in the space BV[a, b], Math. Nachr. 117 (1984), 155–159.
[14] K. Nikodem, K-convex and K-concave set-valued functions, Zeszyty Naukowe Politechniki Łódzkiej, Mat. 559, Rozprawy Naukowe 114, 1989.
[15] A. Smajdor and W. Smajdor, Jensen equation and Nemytskii operator for set-valued functions, Rad. Math. 5 (1989), 311–320.
[16] W. Smajdor, Note on Jensen and Pexider functional equations, Demonstratio Math. 32 (1999), 363–376.
