Cent. Eur. J. Math. • 10(2) • 2012 • 609-618
DOI: 10.2478/s11533-012-0002-1
Central European Journal of Mathematics
Uniformly bounded set-valued Nemytskij operators
acting between generalized Hölder function spaces
Research Article
y
Janusz Matkowski1,2∗ , Małgorzata Wróbel3†
op
1 Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Podgórna 50, 65-246 Zielona Góra,
Poland
2 Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland
rc
3 Institute of Mathematics and Computer Science, Jan Długosz University, 42-200 Częstochowa, Poland
A
ut
ho
Received 27 August 2011; accepted 2 December 2011
Abstract: We show that the generator of any uniformly bounded set-valued Nemytskij composition operator acting between
generalized Hölder function metric spaces, with nonempty, bounded, closed, and convex values, is an affine
function.
MSC:
47H30
Keywords: Nemytskij composition operator • Uniformly bounded operator • Set-valued function • Generalized Hölder function
metric space
© Versita Sp. z o.o.
1.
Introduction
Given arbitrary nonempty sets X , Y , Z and a function h : X × Y → Z , the mapping H : Y X → Z X defined by
H(φ)(x) = h(x, φ(x)),
x ∈ X,
φ ∈ Y X,
is called the composition (Nemytskij or superposition) operator of a generator h. (Here Y X denotes the set of all
functions φ : X → Y .)
∗
†
E-mail: [email protected]
E-mail: [email protected]
609
Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces
Composition operators play an important role in the theory of differential equations, integral equations and functional
equations. In the theory of functional equations of iterative type,
φ(x) = h x, φ(f(x)) ,
existence and uniqueness of the solution φ strongly depends on the class (more precisely, on the function Banach
space) to which it belongs. An application of the Banach fixed point theorem leads to the following question. Given a
composition operator H which maps Banach function spaces F(X , Y ) ⊂ Y X into F(X , Z ) ⊂ Z X, under what conditions
the operator H is norm-Lipschitz continuous?
In [7] it was shown that if H, mapping the Banach space Lip([0, 1], R) of Lipschitzian functions into itself, is Lipschitz
norm-continuous, then
h(x, y) = a(x)y + b(x),
x ∈ [0, 1], y ∈ R,
op
y
for some a, b ∈ Lip([0, 1], R). Later, similar results were proved for BV([0, 1], R), the space of functions of bounded
variation, [3, 14], as well as for some other Banach function spaces with norms essentially stronger that the supremum
norm; cf. [1, 8, 9] for other references. W. Smajdor et al. extended some of these results for multivalued mappings; cf. for
instance [5, 6, 16, 17]. In [10] it was observed that these results remain true if the Lipschitz norm-continuity of H is
replaced by its uniform continuity; cf. also [2, 4, 11, 12].
rc
Recently, under very general assumptions, it has been proved that for the function spaces, including the classical Hölder
spaces as a special case, uniform continuity of the single-valued operator H can be replaced by a much weaker condition
of uniform boundedness (cf. Definition 4.1), which is weaker than norm boundedness of H [13]. In the present paper we
extend this result to the case of set-valued functions.
A
ut
ho
Let (X , ρ1 ) and (X , ρ2 ) be metric spaces, (Y , |·|Y ) and (Z , |·|Z ) real normed spaces, C ⊂ Y a convex cone and let a function
h : X × C → clb(Z ), where clb(Z ) is the set of all nonempty, bounded, closed and convex subsets of Z , be continuous
with respect to the second variable. Suppose that the composition operator H of the generator h maps Lip((X , ρ1 ), C )
into Lip (X , ρ2 ), clb(Z ) . If H is uniformly bounded, then our main result (Theorem 4.4) says in particular, that there
exist a : X × C → clb(Z ) and b : X → clb(Z ) such that a is linear with respect to the second variable (more precisely,
a(x, ·) is a linear map from C into clb(Z ) for any fixed x ∈ X ), the functions a(·, y) and b belong to Lip (X , ρ2 ), clb(Z )
for every y ∈ C , and
∗
H(φ)(x) = a(x, φ(x)) + b(x),
φ ∈ Lip((X , ρ1 ), C ), x ∈ X .
This generalizes all earlier results on Nemytskij operators acting between Hölder spaces. For instance, 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 Mainka [6].
2.
Preliminaries
Let (Z , | · |Z ) be a real normed space. By clb(Z ) we denote the class of all nonempty, bounded, closed and convex subsets
∗
of Z . If A, B ⊂ Z , then we define A + B = {a + b : a ∈ A, b ∈ B} and we introduce a binary operation + in clb(Z ) by
the formula
∗
A + B = cl(A + B),
∗
where cl A denotes the closure of A. The class clb(Z ) with the operation + is an Abelian semigroup with {0} as the zero
element, which satisfies the cancellation law. Moreover, we can multiply elements of clb(Z ) by nonnegative numbers
and, for all A, B ∈ clb(Z ) and λ, µ ≥ 0, the following conditions hold:
1 · A = A,
λ(µA) = (λµ)A,
∗
∗
λ(A + B) = λA + λB,
Let d stand for the Hausdorff metric on the space clb(Z ), defined by the formula
d(A, B) = max {e(A, B), e(B, A)},
610
∗
(λ + µ)A = λA + µA.
A, B ∈ clb(Z ),
J. Matkowski, M. Wróbel
where
e(A, B) = sup {ρ(a, B) : a ∈ A},
ρ(z, B) = inf {|z − b|Z : b ∈ B},
for all nonempty and bounded sets A and B. The Hausdorff metric has the following properties:
∗
∗
d(A + B, A + C ) = d(A + B, A + C ) = d(B, C ),
∗
(1)
∗
d(A + B, C + D) = d(A + B, C + D) ≤ d(A, C ) + d(B, D),
(2)
d(λA, λB) = λd(A, B)
(3)
for all λ ≥ 0 and A, B, C , D ∈ clb(Z ).
Let (Y , | · |Y ) and (Z , | · |Z ) be real normed spaces. A subset C ⊂ Y is said to be a convex cone if λC ⊂ C for all λ ≥ 0
and C + C ⊂ C . It is obvious that 0 ∈ C . A set-valued function F : C → clb(Z ) is said to be ∗ additive, if
2F
Lemma 2.1 ([17, Corollary 4]).
∗
= F (x) + F (y)
rc
for all x, y ∈ C . We will need the following
x+y
2
op
and ∗ Jensen if
y
∗
F (x + y) = F (x) + F (y),
Let C be a convex cone in a real linear space and let (Z , | · |Z ) be a Banach space. A set-valued function F : C → clb(Z )
is ∗ Jensen if and only if there exists a ∗ additive set-valued function A : C → clb(Z ) and a set B ∈ clb(Z ) such that
∗
A
ut
ho
F (x) = A(x) + B
for all x ∈ C .
For the normed spaces (Y , | · |Y ) and (Z , | · |Z ) by L(Y , Z ), k · kL(Y ,Z ) , L(Y , Z ) for short, we denote the normed space
of all additive and continuous mappings a ∈ Z Y . Let C be a convex cone in a real normed space (Y , | · |Y ). From now
on, let the set L(C , clb(Z )) consist of all set-valued functions a : C → clb(Z ) which are ∗ additive and continuous (thus
positively homogeneous), i.e.,
L(C , clb(Z )) = a ∈ clb(Z )C : a is ∗ additive and continuous .
The set L(C , clb(Z )) can be equipped with the metric defined by
dL(C ,clb (Z )) (a, b) = sup
y∈C \{0}
d(a(y), b(y))
.
kykY
(4)
611
Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces
3.
Composition operators satisfying a nonlinear Lipschitz condition
Let (X , ρX ) be a metric space and (Y , | · |Y ) a real normed space. By Lip (X , ρX ), (Y , | · |Y ) , Lip(X , Y ) for short, we
denote the family of all Lipschitz functions φ ∈ Y X , i.e. such that
L(φ) = sup
|φ(x) − φ(x)|Y
: x, x ∈ X , x 6= x
ρX (x, x)
< ∞.
Given x0 ∈ X , the pair Lip(X , Y ), k · kLip (X ,Y ),x0 , where k · kLip (X ,Y ),x0 : Lip(X , Y ) → [0, ∞) is defined by
kφkLip (X ,Y ),x0 = |φ(x0 )|Y + L(φ),
(5)
y
is a normed space.
Remark 3.1.
op
By [13, Remark 2], for any x0 , x1 ∈ X , the norms kφkLip (X ,Y ),x0 and kφkLip (X ,Y ),x1 are equivalent, so, to simplify the notation,
we shall write kφkLip (X ,Y ) instead of kφkLip (X ,Y ),x0 .
Remark 3.2.
For a set C ⊂ Y we put
rc
Since Lip(X , Y ) in a special case becomes the classical Hölder function space, cf. Remark 4.5 below, it can be called
a generalized Hölder function space.
Lip((X , ρ), C ) = φ ∈ Lip((X , ρ), Y ) : φ(X ) ⊂ C .
A
ut
ho
In the proof of the main result the following result plays a crucial role.
Lemma 3.3 ([13, Lemma 1]).
Let (X , ρ) be a metric space, (Y , | · |Y ) 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
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 and φy,y (x) = y;
(b) φy,y ∈ Lip((X , ρ), Y ) and
L(φy,y ) =
|y − y|Y
;
ρ(x, x)
(c) if y, y ∈ C then φy,y ∈ Lip((X , ρ), C );
(d) the set
K(x, x) = φy,y : y, y ∈ Y
is a linear subspace of Lip((X , ρ), Y ) containing all the constant functions.
612
(6)
J. Matkowski, M. Wróbel
In the sequel by Lip (X , ρ), clb(Z ) we denote the set of all set-valued functions φ : X → clb(Z ) such that
sup
x1 ,x2 ∈X x1 6=x2
d φ(x1 ), φ(x2 )
< ∞.
ρ(x1 , x2 )
In Lip (X , ρ), clb(Z ) we can introduce a metric dLip (X ,clb(Z )) by putting
dLip (X ,clb(Z )) (φ1 , φ2 ) = d φ1 (x0 ), φ2 (x0 ) +
sup
x1 ,x2 ∈X x1 6=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
y
K (x, x; C ) = K (x, x) ∩ Lip((X , ρ), C ).
Definition 3.4.
H(φ)(x) = h(x, φ(x)),
op
Let (X , ρ) be a metric space, (Y , | · |Y ) and (Z , | · |Z ) real normed spaces, C ⊂ Y a convex cone. Given a set-valued
function h : X × C → clb(Z ), the mapping H : Y X → clb(Z )X defined by
x ∈ X,
φ ∈ Y X,
Theorem 3.5.
rc
is called the composition (Nemytskij or superposition) operator of a generator h.
A
ut
ho
Let (X , ρ1 ) and (X , ρ2 ) be metric spaces, (Y , |·|Y ) and (Z , |·|Z ) real normed spaces, C ⊂ Y a convex cone and let a function
h(x, ·) : C → Z be continuous for all x ∈ X . Suppose that for all x, x ∈ X , x 6= x, the composition operator H of the
generator h maps the set K (x, x; C ) into the space Lip (X , ρ2 ), clb(Z ) . If for all x, x ∈ X , x 6= x and φ, ψ ∈ K (x, x; C )
the operator H satisfies the inequality
dLip ((X ,ρ2 ),clb(Z )) H(φ), H(ψ) ≤ γ kφ − ψkLip ((X ,ρ1 ),Y )
(7)
for some function γ : [0, ∞) → [0, ∞), then there exist a : X × C → clb(Z ) and b : X → clb(Z ) such that a(·, y), b ∈
Lip (X , ρ2 ), clb(Z ) for every y ∈ C , a(x, ·) ∈ L(C , clb(Z )) for every x ∈ X , and
∗
h(x, y) = a(x, y) + b(x),
x ∈ X,
y ∈ C.
Moreover, the function X 3 x 7→ a(x, ·) ∈ L(C , clb(Z )) fulfils the following condition:
dL(C ,clb(Z )) a(x, ·), a(x, ·) ≤ γ(1)ρ2 (x, x),
x, x ∈ X .
(8)
Proof.
By Lemma 3.3, for arbitrary y ∈ C , the constant function φ(t) = y, t ∈ X , belongs to K (x, x; C ). Since H
maps K (x, x; C ) into Lip (X , ρ2 ), clb(Z ) , the function
H(φ) = h(·, y) ∈ Lip (X , ρ2 ), clb(Z )
(9)
and, consequently, h is continuous with respect to the first variable.
Let us fix x, x ∈ X , take arbitrary y1 , y2 , y1 , y2 ∈ C and define a pair of functions φy1 ,y1 , φy2 ,y2 by (6). Taking into
account Lemma 3.3, the functions φy1 ,y1 , φy2 ,y2 belong to K (x, x; C ) and, by Remark 3.1, with x0 = x in (5),
kφy1 ,y1 − φy2 ,y2 kLip ((X ,ρ1 ),Y ) = |y1 − y2 | +
|y1 − y2 − y1 + y2 |Y
.
ρ1 (x, x)
(10)
613
Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces
Applying (7) with φ = φy1 ,y1 , ψ = φy2 ,y2 , by the definition of the metric dLip ((X ,ρ2 ),clb(Z )) , we get
d H(φy1 ,y1 )(x) + H(φy2 ,y2 )(x), H(φy1 ,y1 )(x) + H(φy2 ,y2 )(x)
≤ dLip ((X ,ρ2 ),clb(Z )) H(φy1 ,y1 ), H(φy2 ,y2 )
ρ2 (x, x)
≤ γ kφy1 ,y1 − φy2 ,y2 kLip ((X ,ρ1 ),Y )
and since
φy1 ,y1 (x) = y1 ,
we have
φy1 ,y1 (x) = y1 ,
φy2 ,y2 (x) = y2 ,
φy2 ,y2 (x) = y2 ,
(11)
d h(x, y1 ) + h(x, y2 ), h(x, y1 ) + h(x, y2 )
≤ γ kφy1 ,y1 − φy2 ,y2 kLip ((X ,ρ1 ),Y )
ρ2 (x, x)
for all x, x ∈ X , y1 , y2 , y1 , y2 ∈ C .
and
|u − v|Y
2
op
kφy1 ,y1 − φy2 ,y2 kLip ((X ,ρ1 ),Y ) =
y
Taking arbitrary u, v ∈ C and setting y1 = y2 = (u + v)/2, y1 = u, y2 = v, by (10), we obtain
u+v
|u − v|Y
u+v
+ h x,
, h(x, u) + h(x, v) ≤ γ
ρ2 (x, x)
d h x,
2
2
2
rc
for all x, x ∈ X , x 6= x, u, v ∈ C . Letting here x tend to x and making use of the continuity of h with respect to the first
variable we get
u+v
d 2h x,
, h(x, u) + h(x, v) = 0,
2
A
ut
ho
which shows that for every x ∈ X the function h(x, ·) is ∗ Jensen in C , that is
u+v
∗
= h(x, u) + h(x, v),
2h x,
2
u, v ∈ C .
By Lemma 2.1 there exist functions a : X × C → clb(Z ) and b : X → clb(Z ) such that a(x, ·) is ∗ additive for x ∈ X and
∗
h(x, y) = a(x, y) + b(x).
(12)
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 continuity of h(x, ·) implies continuity of a(x, ·) and, consequently, being ∗ additive,
a(x, ·) ∈ L(C , clb(Z ))
for every
x ∈ X.
To prove that b ∈ Lip (X , ρ2 ), clb(Z ) , let us note that the ∗ additivity of a(x, ·) implies a(x, 0) = {0}. Therefore, applying
(12), we get
∗
h(x, 0) = a(x, 0) + b(x)
(13)
and since, by (9), h(·, y) ∈ Lip (X , ρ2 ), clb(Z ) for all y ∈ C , (13) shows that b ∈ Lip (X , ρ2 ), clb(Z ) .
614
J. Matkowski, M. Wróbel
Now we shall prove that a(·, y) ∈ Lip (X , ρ2 ), clb(Z ) for all y ∈ C . To this end let us fix arbitrarily x, x ∈ X , x 6= x,
y ∈ C and take a constant function φ(t) = y, t ∈ X , belonging, by Lemma 3.3, 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 ), clb(Z ) , H(φ) belongs to Lip (X , ρ2 ), clb(Z ) . Moreover, according to what
has already been proved, b ∈ Lip (X , ρ2 ), clb(Z ) , therefore we get the required claim.
To prove that the function X 3 x →
7 a(x, ·) ∈ L(C , clb(Z )) fulfils (8), take x, x ∈ X , x 6= x, y1 , y2 , y1 , y2 ∈ C and define
φy1 ,y1 , φy2 ,y2 ∈ K (x, x; C ) by (6). By a similar reasoning as in the first part of the proof, applying (7) with φ = φy1 ,y1 ,
ψ = φy2 ,y2 , (10) and (11), we get
and, consequently, by (1) and (12),
op
y
|y1 − y2 − y1 + y2 |Y
ρ2 (x, x),
d h(x, y1 ) + h(x, y2 ), h(x, y1 ) + h(x, y2 ) ≤ γ |y1 − y2 |Y +
ρ1 (x, x)
|y1 − y2 − y1 + y2 |Y
d a(x, y1 ) + a(x, y2 ), a(x, y1 ) + a(x, y2 ) ≤ γ |y1 − y2 |Y +
ρ1 (x, x)
ρ2 (x, x)
rc
for all x, x ∈ X , x 6= 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
A
ut
ho
d a(x, y + y) + a(x, y + y), a(x, 2y + y) + a(x, y) ≤ γ(|y|Y )ρ2 (x, x)
and, by ∗ additivity of a(x, ·),
d a(x, y), a(x, y) ≤ γ(|y|Y )ρ2 (x, x)
(14)
for all x, x ∈ X and y ∈ C . Take arbitrary u ∈ C \ {0} and put y = u/|u|Y ∈ C . Hence and by (14),
u
u
d a x,
, a x,
≤ γ(1)ρ2 (x, x),
|u|Y
|u|Y
whence, by linearity of a(x, ·) for all x ∈ X and (3) we get,
d a(x, u), a(x, u)
≤ γ(1)ρ2 (x, x)
|u|Y
which, in view of (4), completes the proof of (8).
Remark 3.6.
If the function γ : [0, ∞) → [0, ∞) is right continuous at 0 and γ(0) = 0, then the assumption of continuity of h with
respect to the second variable can be omitted.
Indeed, let us observe that from the definition of the metric dLip ((X ,ρ2 ),clb(Z )) and by (7) we have
d H(φ)(x) + H(ψ)(x0 ), H(φ)(x0 ) + H(ψ)(x)
d H(φ)(x0 ), H(ψ)(x0 ) +
≤ γ kφ − ψk(Lip (X ,ρ1 ),Y )
ρ2 (x, x0 )
(15)
615
Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces
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,
t ∈ X.
φ(t) = y,
By Lemma 3.3 (d), φ, φ ∈ K (x0 , x; C ) and
kφ − φkLip ((X ,ρ1 ),Y ) = |y − y|Y .
Hence, by (15), we get
d h(x, y) + h(x0 , y), h(x0 , y) + h(x, y)
d h(x0 , y), h(x0 , y) +
≤ γ |y − y|Y .
ρ2 (x, x0 )
Consequently,
d h(x0 , y), h(x0 , y) ≤ γ |y − y|Y
y
and
(16)
d h(x, y) + h(x0 , y), h(x0 , y) + h(x, y) ≤ γ |y − y|Y ρ2 (x, x0 ).
op
Since, by (2),
(17)
d h(x, y), h(x, y) = d h(x, y) + h(x0 , y) + h(x0 , y), h(x, y) + h(x0 , y) + h(x0 , y)
≤ d h(x, y) + h(x0 , y), h(x, y) + h(x0 , y) + d h(x0 , y), h(x0 , y) ,
therefore, by (16) and (17),
rc
d h(x, y), h(x, y) ≤ γ |y − y|Y (1 + ρ2 (x, x0 ))
which, by continuity of γ at 0 and the equality γ(0) = 0, implies that h is continuous with respect to the second variable.
As an immediate corollary of Theorem 3.5 we obtain the following
A
ut
ho
Theorem 3.7.
Let (X , ρ1 ) and (X , ρ2 ) be metric spaces, (Y , | · |Y ) and (Z , | · |Z ) real normed spaces, C ⊂ Y a convex cone and let a
function h : X × C → clb(Z ) be such that for any x ∈ X the function h(x, ·) : C → clb(Z ) is continuous.
Suppose that
the composition operator H of the generator h maps Lip((X , ρ1 ), C ) into the space Lip (X , ρ2 ), clb(Z ) . If there exists a
function γ : [0, ∞) → [0, ∞) such that
dLip ((X ,ρ2 ),clb(Z )) (H(φ), H(ψ)) ≤ γ kφ − ψkLip ((X ,ρ1 ),Y ) ,
then
∗
h(x, y) = a(x, y) + b(x),
x ∈ X,
φ, ψ ∈ Lip((X , ρ1 ), C ),
(18)
y ∈ C,
for some b ∈ Lip (X , ρ2 ), clb(Z ) and a : X × C → clb(Z ) such that a(·, y) ∈ Lip (X , ρ2 ), clb(Z ) for every y ∈ C ,
A(x, ·) ∈ L(C , clb(Z )) for every x ∈ X , and the function X 3 x 7→ a(x, ·) ∈ L(C , clb(Z )) fulfils the Lipschitz condition
with the constant γ(1), i.e.,
dL(C ,clb(Z )) a(x, ·), a(x, ·) ≤ γ(1)ρ2 (x, x),
x, x ∈ X .
(19)
Corollary 3.8.
Let (X , ρ1 ) and (X , ρ2 ) be metric spaces, (Y , | · |Y ) and (Z , | · |Z ) real normed spaces, C ⊂ Y a convex cone and let
a function h : X × C → Z be such that for any x ∈ X the function h(x, ·) : C → Z is continuous. Suppose that the
composition operator H of the generator h maps Lip((X , ρ1 ), C ) into the space Lip((X , ρ2 ), Z ). If there exists a function
γ : [0, ∞) → [0, ∞) such that
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),
for some a ∈ Lip (X , ρ2 ), L(C , Z ) and b ∈ Lip((X , ρ2 ), Z ).
616
x ∈ X,
y ∈ C,
(20)
J. Matkowski, M. Wróbel
Indeed, since (18) reduces to (20) in the space Lip((X , ρ2 ), Z ), by Theorem 3.7, 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
|a(x)y − a(x)y|Z
≤ γ(1)ρ2 (x, x),
|y|Y
y∈C \{0}
sup
Since
a(x) − a(x)
∈ L(C , Z ),
ρ2 (x, x)
x, x ∈ X ,
x, x ∈ X .
x 6= x,
it follows that
a(x) − a(x) ≤ γ(1),
ρ2 (x, x) L(C ,Z )
which shows that a ∈ Lip (X , ρ2 ), L(C , Z ) .
x 6= x,
y
x, x ∈ X ,
Remark 3.9.
Uniformly bounded composition operators
From [13] we quote the following
Definition 4.1.
rc
4.
op
In [13, Theorem 1] it is assumed that C is a convex set with nonempty interior and a function γ is bounded in a right-hand
side neighborhood of 0. Assuming, as in [13, Theorem 1], that C is a convex cone, one can omit this assumption about γ.
A
ut
ho
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 is a nonnegative real number γ(t) such that for any set B ⊂ Y we have
diam B ≤ t
=⇒
diam H(B) ≤ γ(t).
Remark 4.2.
Let (Y, ρY ) and (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 4.3.
Obviously, every uniformly continuous operator or Lipschitzian operator is uniformly bounded. Note that, under the
assumptions of this definition, every bounded operator is uniformly bounded and the converse is not true.
The main result of this paper reads as follows.
Theorem 4.4.
Let (X , ρ1 ) and (X , ρ2 ) be metric spaces, (Y , | · |Y ) and (Z , | · |Z ) real normed spaces, C ⊂ Y a convex cone, and let a
function h : X × C → clb(Z ) be such that for any x ∈ X the function h(x, ·) : C → clb(Z ) is continuous.
Suppose that
the composition operator H of the generator h maps Lip((X , ρ1 ), C ) into the space Lip (X , ρ2 ), clb(Z ) . If H is uniformly
bounded, then there exist a : X × C → clb(Z ) and b : X → clb(Z ) such that a(·, y), b ∈ Lip (X , ρ2 ), clb(Z ) for every
y ∈ C , a(x, ·) ∈ L(C , clb(Z )) for every x ∈ X , the function X 3 x 7→ a(x, ·) ∈ L(C , clb(Z )) fulfils the Lipschitz condition
and
∗
H(φ)(x) = a(x, φ(x)) + b(x),
φ ∈ Lip((X , ρ1 ), C ), x ∈ X .
617
Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces
Proof.
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 diam H({φ, ψ}) ≤ γ(t), i.e.,
dLip ((X ,ρ2 ),clb(Z )) (H(φ), H(ψ)) = diam H({φ, ψ}) ≤ γ kφ − ψkLip ((X ,ρ1 ),Y ) ,
and the result follows from Theorem 3.7.
Remark 4.5.
op
References
y
Put 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]. Of
course, (X , ρ1 ) and (X , ρ2 ) are metric spaces. Applying Corollary 3.8 with γ(t) = Lt for some L ≥ 0 and a generator h
mapping X × C into the space cc(Z ) of all nonempty, convex and compact subsets of Z , we obtain the result of Ludew [5]
∗
(in this case the operation + reduces to the usual algebraic sum of two sets). Since any uniformly continuous operator
is uniformly bounded, from Theorem 4.4 we obtain the result of Mainka [6]. Theorem 4.4 improves also the main result
of our recent work [15], where we assume that h maps X × C into the space cc(Z ).
A
ut
ho
rc
[1] Appell J., Zabrejko P.P., Nonlinear Superposition Operators, Cambridge Tracts in Math., 95, Cambridge University
Press, Cambridge, 1990
[2] Azócar A., Guerrero J.A., Matkowski J., Merentes N., Uniformly continuous set-valued composition operators in the
spaces of functions of bounded variation in the sense of Wiener, Opuscula Math., 2010, 30(1), 53–60
[3] Chistyakov V.V., Lipschitzian superposition operators between spaces of functions of bounded generalized variation
with weight, J. Appl. Anal., 2000, 6(2), 173–186
[4] Guerrero J.A., Leiva H., Matkowski J., Merentes N., Uniformly continuous composition operators in the space of
bounded φ-variation functions, Nonlinear Anal., 2010, 72(6), 3119–3123
[5] Ludew J.J., On Lipschitzian operators of substitution generated by set-valued functions, Opuscula Math., 2007, 27(1),
13–24
[6] Mainka E., On uniformly continuous Nemytskij operators generated by set-valued functions, Aequationes Math.,
2010, 79(3), 293–306
[7] Matkowski J., Functional equations and Nemytskii operators, Funkcial. Ekvac., 1982, 25(2), 127–132
[8] Matkowski J., Lipschitzian composition operators in some function spaces, Nonlinear Anal., 1997, 30(2), 719–726
[9] Matkowski J., Remarks on Lipschitzian mappings and some fixed point theorems, Banach J. Math. Anal., 2007, 2(1),
237–244
[10] Matkowski J., Uniformly continuous superposition operators in the spaces of differentiable functions and absolutely
continuous functions, In: Inequalities and Applications, Noszvaj, September 9–15, 2007, Internat. Ser. Numer. Math.,
157, Birkhäuser, Basel, 2009, 155–166
[11] Matkowski J., Uniformly continuous superposition operators in the Banach space of Hölder functions, J. Math. Anal.
Appl., 2009, 359(1), 56–61
[12] Matkowski J., Uniformly continuous superposition operators in the space of bounded variation functions, Math.
Nachr., 2010, 283(7), 1060–1064
[13] Matkowski J., Uniformly bounded composition operators between general Lipschitz function normed spaces, Topol.
Methods Nonlinear Anal., 2011, 38(2), 395–406
[14] Matkowski J., Miś J., On a characterization of Lipschitzian operators of substitution in the space BVha, bi, Math.
Nachr., 1984, 117, 155–159
[15] Matkowski J., Wróbel M., Uniformly bounded Nemytskij operators generated by set-valued functions between generalized Hölder function spaces, Discuss. Math. Differ. Incl. Control Optim., 2011, 31(2), 183–198
[16] Smajdor A., Smajdor W., Jensen equation and Nemytskiı̆ operator for set-valued functions, Rad. Math., 1989, 5(2),
311–320
[17] Smajdor W., Note on Jensen and Pexider functional equations, Demonstratio Math., 1999, 32(2), 363–376
618