Cent. Eur. J. Math. • 11(2) • 2013 • 357-367
DOI: 10.2478/s11533-012-0051-5
Central European Journal of Mathematics
Uniformly bounded composition operators
in the Banach space of bounded (p, k)-variation
in the sense of Riesz–Popoviciu
Research Article
Francy Armao1∗ , Dorota Głazowska2† , Sergio Rivas3‡ , Jessica Rojas1§
1 Facultad de Ciencias, Departamento de Matemática, Universidad Central de Venezuela, Caracas, Venezuela
2 Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra,
Poland
3 Departamento de Matemática, Universidad Nacional Abierta, Caracas, Venezuela
Received 11 October 2011; accepted 27 February 2012
Abstract: We prove that if the composition operator F generated by a function f : [a, b] × R → R maps the space of bounded
(p, k)-variation in the sense of Riesz–Popoviciu, p ≥ 1, k an integer, denoted by RV(p,k) [a, b], into itself and is
uniformly bounded then RV(p,k) [a, b] satisfies the Matkowski condition.
MSC:
47H30, 26A45, 47B38
Keywords: Nemytskij (composition, superposition) operator • Uniformly bounded mapping • Uniformly continuous mapping •
de la Vallée Poussin second-variation • Popoviciu k-th variation
© Versita Sp. z o.o.
1.
Introduction
Let a, b ∈ R, a < b, be fixed. For a given function f : [a, b] × R → R the nonlinear composition operator (also called
superposition operator, substitution operator or Nemytskij operator) F : R[a,b] → R[a,b] , generated by f, is defined by
F x(t) = f(t, x(t)),
∗
†
‡
§
E-mail:
E-mail:
E-mail:
E-mail:
x ∈ R[a,b] ,
t ∈ [a, b].
(1)
[email protected]
[email protected]
[email protected]
[email protected]
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Uniformly bounded composition operators in the Banach space of bounded (p, k)-variation in the sense of Riesz–Popoviciu
Here R[a,b] denotes the family of all functions x : [a, b] → R. In spite of its simple form, the behaviour of this locally
defined operator, cf. [19, 28–30, 43–46], exhibits many surprising and even pathological features in various function
spaces. For instance, applying the Banach contraction principle (or one of its generalizations) to find the solutions x in
a function Banach space (X , k · k), X ⊂ R[a,b] , of the nonlinear iterative functional equation
x(t) = f(t, x(φ(t))),
cf. for instance [17], it is necessary to guarantee that F is Lipschitzian, i.e. that
kF x − F yk ≤ K kx − yk,
x, y ∈ X .
(2)
It turns out, however, that sometimes this leads to a strong degeneracy: in some classical function spaces X , the operator
(1) satisfies (2) if and only if the corresponding function f has the form
f(t, u) = α(t) + β(t)u,
α, β ∈ X .
(3)
This means, roughly speaking, that one may apply classical fixed point principles for contraction type maps only if the
underlying problem is actually linear.
To the best of our knowledge, the first who observed the kind of degeneracy phenomenon for composition operators
described above was Janusz Matkowski. More specifically, Matkowski (in part with coauthors) proved that Lipschitz
continuous operators (1) in X are generated only by affine functions (3), if X is the space Cm [a, b] of m times continuously
differentiable functions [22], the Sobolev space W1p [a, b] of functions with distributional first derivative in Lp [a, b] [26], or the
space BVp2 [a, b] of functions of bounded (p, 2)-variation [25]. Likewise, an analogous result was proved by Matkowska [21]
for the space Cα [a, b] of Hölder continuous functions of order α < 1, by Lupa [20] for the space Cn,α [a, b] of functions
with Hölder continuous n-th derivative, by Sieczko [39] for the space ACn [a, b] of functions with absolutely continuous
n-th derivative, by Knop [15] for the space Lipn [a, b] of functions with Lipschitz continuous n-th derivative, by Merentes
and Rivas [33] for the space RVp [a, b] of functions of bounded generalized p-variation in the Riesz sense, and by
Merentes [31, 32] for the space RVφ [a, b] of functions of bounded generalized φ-variation in the Riesz sense.
The Nemytskij operator (1) appears frequently in connection with integral equations and iterative functional equations.
Moreover, Nemytskij operators satisfying condition (2) are considered also in the study of systems with hysteresis [16]
and difference equations [40].
Like in [2], we shall say that the composition operator (1) has Matkowski’s property if, whenever this operator maps the
space X into the space Y and satisfies some additional condition like global Lipschitz continuity, uniform continuity or
other, the generator function f has the form (3).
In the case when X = Y = BV[a, b], where BV[a, b] is the Banach space of functions of bounded variation, the
degeneracy one encounters when a Lipschitz condition is imposed is somewhat different. Recall that, given a function
f : [a, b] × R → R, if for any fixed y, f(·, y) has the left-hand limit lims→x− f(s, y) at each point x ∈ (a, b], then the left
regularization f − of f is defined by
f − (x, y) =

h(a, y)
for x = a,
 lim h(s, y)
for a < x ≤ b.
s→x−
(4)
Similarly, we define the right regularization f + of f. These regularizations are different from f only if f(·, y) is discontinuous from the left or right, respectively.
Matkowski and Miś in [27] showed that if the composition operator F maps the space BV[a, b] into itself and satisfies
condition (2), then there exist two functions α, β ∈ BV[a, b] continuous from the left on (a, b], such that
f − (x, y) = α(x)y + β(x),
(x, y) ∈ (a, b] × R.
(5)
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Clearly, an analogous result is true for the right regularization.
From this fact, Appell, Guanda and Väth observed in [2] that this is a weaker form of the Matkowski property. We shall
say that the composition operator (1) has the weak Matkowski property if, whenever this operator maps the space X
into the space Y and satisfies some additional condition like globally Lipschitz, uniformly continuous or other, the left
regularization (4) (or right regularization) of the generator function f has the form (5).
The above result has been further extended to several spaces of functions of generalized bounded variation in one variable
[7–9, 12] and two variables [10, 11]. Below we give an example from [2] which shows that the Matkowski condition and
the weak Matkowski condition are not equivalent.
Example 1.1.
Let {r0 , r1 , . . .} be an enumeration of all rational numbers in [0, 1], r0 = 0, and let g : R → R be any function satisfying
g(0) = 0 and |g(u) − g(v)| ≤ L|u − v|. We define f : [0, 1] × R → R by


 g(u)
2k
f(t, u) =

0
if t = rk ,
otherwise.
Denote by P[0, 1] the family of all partitions of the interval [0, 1]. For any partition P = {t0 , t1 , . . . , tm } ∈ P[0, 1] and
x ∈ BV[0, 1] we then have
m
X
|F x(tj ) − F x(tj−1 )| ≤ 2
j=1
∞
X
|f(rk , x(rk ))| = 2
k=0
∞
X
|g(x(rk ))|
≤ 2L,
2k
k=0
which shows that F maps the space BV[0, 1] into itself. Furthermore, for any x, y ∈ BV[0, 1] and P ∈ P[0, 1], as above
we obtain the following estimation:
var(F x − F y, P, [0, 1]) =
m
m
X
X
F x(tj ) − F y(tj ) − F x(tj−1 ) + F y(tj−1 ) ≤ 2
f(tj , x(tj )) − f(tj , y(tj ))
j=1
j=1
∞
∞
X
X
|g(x(rk )) − g(y(rk ))|
f(rk , x(rk )) − f(rk , y(rk )) ≤ 2
≤
2k
k=0
k=0
≤ 2L
∞
X
|x(rk ) − y(rk )|
≤ 2Lkx − ykBV .
2k
k=0
This, together with the trivial estimate |F x(0) − F y(0)| ≤ L|x(0) − y(0)|, shows that F satisfies the global Lipschitz
condition (2) with K = 2L, although f is not of the form (3). We can see that f − = f + = 0 for the function f in
Example 1.1, in accordance with [2, Theorem 2].
Notice that there are important function spaces having neither the Matkowski property nor the weak Matkowski property.
For example, Appell, Guanda and Väth proved in [2] that the condition (2) in the space C[a, b] equipped with the norm
kukC = max |u(t)|
a≤t≤b
is equivalent to the Lipschitz condition
|f(t, u) − f(t, v)| ≤ K |u − v|,
a ≤ t ≤ b,
u, v ∈ R,
for the function f(x, ·) (with the same Lipschitz constant as in (2)); this is of course what one should expect in “reasonable”
function spaces. A similar result holds for the Lebesgue space Lp [a, b], 1 ≤ p < ∞, equipped with the norm
b
Z
kukLp =
|u(t)|p dt
1/p
.
a
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Uniformly bounded composition operators in the Banach space of bounded (p, k)-variation in the sense of Riesz–Popoviciu
The strong degeneracy described above, occuring also in many familiar functions spaces, emphasizes a need of proving
Lipschitz condition (2) or weaker uniform continuity, Matkowski [23], in order to get the Matkowski property.
Matkowski [23] proved that the uniformly continuous Nemytskij operator, acting between the Banach spaces of Hölder
functions, has the Matkowski property. Uniform continuity of composition operators has been considered in other
functional Banach spaces [1, 3–6, 13, 14, 23].
It turns out [24], Lipschitz continuity and uniform continuity of the composition operator can be replaced by a rather weak
uniform boundedness of the Nemytskij operator, cf. Definition 4.2. This notion has been recently applied by Wróbel [47]
to the space of functions of bounded k-th variation in the sense of Popoviciu, cf. also [42]. In this paper, following an
idea of Wróbel [47], we show that for a fixed p > 1, any uniformly bounded composition operator, mapping RV(p,k) [a, b]
into itself, has the Matkowski property.
2. Riesz–Popoviciu space of functions of bounded (p, k)-variation and some
related Banach spaces
Throughout we will consider several normed spaces endowed with the following norms.
I.
Let Lip[a, b] denote the Banach space of all Lipschitz continuous functions u : [a, b] → R, equipped with the natural
norm
kukLip = |u(a)| + Lba (u),
where
u(s) − u(t) : a≤s<t≤b
Lba (u) = sup s−t is the smallest Lipschitz constant of u on [a, b].
II.
Let k ≥ 1 be an integer, u ∈ R[a,b] and t1 , . . . , tk+1 be distinct points, not necessarily in linear order, of [a, b]. Define
the k-th divided difference of u as
u[t1 ] = u(t1 ),
u[t1 , t2 ] =
u(t2 ) − u(t1 )
,
t2 − t1
t2 6= t1 ,
u[t1 , . . . , tk , tk+1 ] =
u[t2 , . . . , tk+1 ] − u[t1 , . . . , tk ]
,
tk − t1
tk 6= t1 ,
see [37]. For a partition P : a = t1 < . . . < tn = b of the interval [a, b], we define
σk (u, P) =
n−k
n−k
X
X
u[tj+1 , . . . , tj+k ] − u[tj , . . . , tj+k−1 ] =
(tj+k − tj ) u[tj , . . . , tj+k ].
j=1
j=1
In 1933–34, Popoviciu [35] defined the concept of function of bounded k-variation of u as
Vk (u, [a, b]) = Vk (u) = sup σk (u, P).
P
If Vk (u) < ∞ we say that u has finite k-variation and we denote by BVk [a, b] the vector space of such functions. For
k = 1, BV1 [a, b] is the classic space of functions of bounded variation. For k = 2, BV2 [a, b] is the space of functions of
second bounded variation given by de la Vallée Poussin in 1908 [41]. In the general case, it is known [38] that BVk [a, b]
has a structure of Banach space with respect to the norm
kukk = |u(a)| + · · · + |u(k−1) (a)| + Vk (u, [a, b]),
u ∈ BVk [a, b].
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There is another way to get this space using partitions of the interval [a, b] of the type
P : a = t1 < . . . < tk ≤ tk+1 < . . . < t2k ≤ t2k+1 < . . . < tkn = b,
(6)
with at least 2k − 1 points considered and the functionals σbk defined as
σbk (u, P) = σbk (u) =
n−1
X
u[tjk+1 , . . . , t(j+1)k ] − u[t(j−1)k+1 , . . . , tjk ].
j=1
Put
bk (u, [a, b]) = V
bk (u) = sup σbk (u, P).
V
P
bk (u) < ∞ denote by BV
c k [a, b]. In Lemma 3.1 we prove that BVk [a, b] = BV
c k [a, b],
The space of the functions u that satisfy V
k ∈ N.
III.
In [34] Merentes, Sánchez and Rivas generalized the concept of functions of bounded variation introduced by Riesz in
[36] as follows. Given a number p > 1, an integer k ≥ 1 and a function u ∈ R[a,b] , consider a partition of [a, b] by blocks
of the type (6) and functions
R
σb(p,k)
(u, P)
n−1
X
=
j=1
!p
u[tjk+1 , . . . , t(j+1)k ] − u[t(j−1)k+1 , . . . , tjk ]
t(j+1)k − t(j−1)k+1 .
t(j+1)k − t(j−1)k+1 Put
b R (u, [a, b]) = V
b R (u) = sup σbR (u, P).
V
(p,k)
(p,k)
(p,k)
P
b R (u) < ∞. The vector space of such
It is said that u has bounded (p, k)-variation in the sense of Riesz–Popoviciu if V
(p,k)
functions is denoted by RV(p,k) [a, b]. For a function u ∈ RV(p,k) [a, b] we define the norm
b R (u)
kukR(p,k) = |u(a)| + |u0 (a)| + · · · + |u(k−1) (a)| + V
(p,k)
1/p
.
One can show that RV(p,k) [a, b] with the norm k · kR(p,k) is a Banach space and it is called the space of functions of bounded
(p, k)-variation in the sense of Riesz–Popoviciu. In [34] the following has been shown.
Theorem 2.1 (generalization of the Riesz lemma).
Let p > 1, k ∈ N. Then u ∈ RV(p,k) [a, b] if and only if u(k−1) ∈ AC[a, b] and u(k) ∈ Lp [a, b]. Moreover,
Z
b(p,k) (u, [a, b]) =
V
1
((k − 1)!)p
b
|u(t)|p dt,
a
u ∈ RV(p,k) [a, b].
From this result we immediately obtain that Ck [a, b] ⊂ RV(p,k) [a, b].
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3.
Auxiliary results
We begin this section with some lemmas.
Lemma 3.1.
If k ∈ N, then
bk (u) ≤ Vk (u) ≤ 3k V
bk (u),
V
u ∈ RV(p,k) [a, b],
and
c k [a, b].
BVk [a, b] = BV
Proof.
c k [a, b] and a ≤ t1 < . . . < tk+1 ≤ b. Choose arbitrarily b1 , . . . bk , c1 , . . . , ck so that
Take u ∈ BV
t1 < b1 < . . . < bk = t2 ,
tk < c1 < . . . < ck = tk+1 .
Then
u[t2 , . . . , tk+1 ] − u[t1 , . . . , tk ] ≤ u[t2 , . . . , tk+1 ] − u[b1 , . . . , bk ] + u[b1 , . . . , bk ] − u[c1 , . . . , ck ]
bk (u, [t1 , tk+1 ]).
+ u[c1 , . . . , ck ] − u[t1 , . . . , tk ] ≤ 3 V
Take arbitrary n ∈ N, n ≥ k + 1. Then for any partition P : a = t1 < . . . < tn = b of the interval [a, b] we have
n−k
n−k
X
X
bk (u, [tj , tj+k ]) =
u[tj+1 , . . . , tj+k ] − u[tj , . . . , tj+k−1 ] ≤
3V
j=1
j=1
bk (u, [t1 , t1+k ]) + · · · + V
bk (u, [t1+k , t1+2k ]) + V
bk (u, [t2+k , t2+2k ]) + · · ·
=3 V
bk (u, [tn−2k , tn−k ]) + · · · + V
bk (u, [tn−k , tn ])
bk (u, [t2+2k , t2+3k ]) + · · · + V
+V
bk (u, [t1 , t1+k ]) + V
bk (u, [t2+k , t2+2k ]) + · · · + V
bk (u, [tn−2k , tn−k ]) + · · ·
=3 V
bk (u, [t1+k , t1+2k ]) + V
bk (u, [t2+2k , t2+3k ]) + · · · + V
bk (u, [tn−k , tn ]) ,
+ V
bk (u, [a, b]) and, consequently,
whence σk (u, P) ≤ 3k V
bk (u, [a, b]).
Vk (u, [a, b]) ≤ 3k V
c k [a, b] ⊂ BVk [a, b].
It follows that u ∈ BVk [a, b] which shows that BV
Now assume that u ∈ BVk [a, b] and take n ∈ N and a partition
P : a = t1 < . . . < tk ≤ tk+1 < . . . < t2k ≤ t2k+1 < . . . < tnk = b
of the interval [a, b]. From the triangle inequality we get
u[tjk+1 , . . . , t(j+1)k − u[t(j−1)k+1 , . . . , tjk ] ≤ u[tjk+1 , . . . , t(j+1)k ] − u[tjk , tjk+1 , . . . , tjk+k−1 ]
+ u[tjk , tjk+1 . . . , tjk+k−1 ] − u[tjk−1 , tjk , tjk+1 , . . . , tjk+k−2 ] + · · ·
+ u[tjk−k+2 , tjk−k+3 , . . . , tjk, tjk+1 ] − u[t(j−1)k+1 , . . . , tjk ]
=
k−1
X
u[tjk−i+1 , . . . , t(j+1)k−i ] − u[tjk−i , . . . , t(j+1)k−i−1 ],
i=0
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for j = 1, . . . , n. Summing up these inequalities we obtain
n−1
n−1 X
k−1
X
X
u[tjk , . . . , t(j+1)k−1 ] − u[t(j−1)k+1 , . . . , tjk ]
u[t(j−1)k+i+1 , . . . , t(j+1)k−i ] − u[tjk−i , . . . , t(j+1)k−ii−1 ] ≤ Vk (u, [a, b]),
j=1
j=1 i=0
whence
bk (u, [a, b]) ≤ Vk (u, [a, b]),
V
which completes the proof.
c k [a, b], k ∈ N.
This lemma guarantees that BVk [a, b] = BV
Lemma 3.2 ([34, Proposition 2.1]).
If p > 1 and k ∈ N, then
1/p
bk (u) ≤ (b − a)1−1/p V
b R (u) ,
V
(p,k)
u ∈ RV(p,k) [a, b].
Lemma 3.3.
Let k ≥ 2 be a positive integer and 1 < p < ∞. Then there exists a positive constant s(k, p) > 0 such that
kukLip ≤ s(k, p)kukR(p,k) ,
u ∈ RV(p,k) [a, b].
(7)
Apply [47, Lemma 3] and the continuous embedding given in [34, Proposition 2.1] to get (7).
4.
Main results
We begin this section with the following
Theorem 4.1.
Let [a, b] ∈ R, a < b, be an interval, p > 1 a real number, k ≥ 2 a positive integer and let a function f : [a, b] × R → R
be continuous with respect to the second variable. Suppose that the composition operator F generated by f acts from
the space RV(p,k) [a, b] into itself and satisfies the following inequality:
kF (u) − F (v)kR(p,k) ≤ γ ku − vkR(p,k) ,
u, v ∈ RV(p,k) [a, b],
for some function γ : [0, ∞) → [0, ∞). Then there exist functions α, β ∈ RV(p,k) [a, b] such that
f(t, x) = α(t)x + β(t),
t ∈ [a, b],
x ∈ R.
(8)
Proof.
By hypothesis, for x ∈ R fixed, the constant function u(t) = x, t ∈ [a, b], is in RV(p,k) [a, b] and therefore
F (u) = f(·, x) ∈ RV(p,k) [a, b], so f(·, x) is continuous for every x ∈ R. Let s, s ∈ [a, b], s < s, x1 , x2 , x 1 , x 2 ∈ R and
consider the functions
xi − x i
(t − s) + xi ,
i = 1, 2.
ui (t) =
s−s
These functions are straight lines that pass through the points (s, x1 ) and (s, x 1 ) in the case of u1 and points (s, x2 ) and
(s, x 2 ) in the case of u2 . Thus, it follows that both functions have (p, k)-bounded variation. In addition,
R
x1 − x 1 − x2 + x 2
ku1 − u2 kR(p,k) = (t
−
s)
+
x
−
x
1
2
s−s
= |x1 − x2 |.
(p,k)
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Uniformly bounded composition operators in the Banach space of bounded (p, k)-variation in the sense of Riesz–Popoviciu
On the other hand, as F (ui ) ∈ RV(p,k) [a, b], i = 1, 2, from Lemma 3.1 and Lemma 3.3, we get
kF (u1 ) − F (u2 )kLip =
|f(s, x1 ) − f(s, x2 ) − f(s, x 1 ) + f(s, x 2 )|
≤ K kF (u1 ) − F (u2 )kR(p,k) ,
|s − s|
where K = 3s(p, k) max 1, (b − a)1−1/p .
In this way, by hypothesis, it follows that
|f(s, x1 ) − f(s, x2 ) − f(s, x 1 ) + f(s, x 2 )|
≤ K γ(|x1 − x2 |).
|s − s|
We set constants p, q ∈ R and let x1 = x 2 = (p + q)/2, x 1 = p, x2 = q in the above inequality. Then
p + q
p + q − f(s, q) − f(s, p) + f s,
≤ K γ(|x1 − x2 |)|s − s|.
f s,
2
2
By continuity of f in the first variable, taking limit as s → s, we get
p + q
= f(s, p) + f(s, q),
2f s,
2
s ∈ [a, b],
p, q ∈ R.
Since the function f(t, ·) is continuous and satisfies the Jensen equation, see [18, p. 315], there are functions
α, β : [a, b] → R satisfying (8), whence, for every x ∈ R the function f(·, x) ∈ RV(p,k) [a, b], we have
β(t) = f(t, 0),
α(t) = f(t, 1) − β(t),
t ∈ [a, b],
which implies that α, β ∈ RV(p,k) [a, b].
Matkowski [24] introduced the notion of a uniformly bounded operator and proved that any uniformly bounded composition
operator acting between general Lipschitz function normed spaces must be of the form (3).
Definition 4.2.
Let Y and Z be two metric (or normed) spaces. We say that a mapping F : Y → Z is uniformly bounded if, for any
t > 0, there exists a nonnegative real number γ(t) such that for any nonempty set B ⊂ Y we have
diam B ≤ t
=⇒
diam H(B) ≤ γ(t).
Remark 4.3.
Every uniformly continuous operator or Lipschitzian operator is uniformly bounded. Note that, under the assumptions of
this definition, every bounded operator is uniformly bounded.
Recently Wróbel [47] has shown that if the composition operator F , generated by f : [a, b] × R → R, maps the space
BVk [a, b], k ≥ 2, into itself and is uniformly bounded, then the space BVk [a, b] satisfies the Matkowski condition; i.e.,
the function f has the form (3). Applying Theorem 4.1 we show the relevant result for the space RV(p,k) [a, b], which reads
as follows:
Theorem 4.4.
Let a, b ∈ R, a < b, p > 1 a real number and k ≥ 2 a positive integer. Suppose that a function f : [a, b] × R → R
is continuous with respect to the second variable. If the composition operator F generated by f acts from the space
RV(p,k) [a, b] into itself and is uniformly bounded then there exist functions α, β ∈ RV(p,k) [a, b] with (8).
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Proof.
Take any t ≥ 0 and u, v ∈ RV(p,k) [a, b] such that ku − vkR(p,k) ≤ t. Since diam {u, v} ≤ t, by uniform
boundedness of F , we have diam F ({u, v}) ≤ γ(t), i.e.,
kF (u) − F (v)kR(p,k) = diam F ({u, v}) ≤ γ ku − vkR(p,k) ,
and the result follows from Lemma 4.1.
Following the ideas developed in [47, Remark 1] we get the following corollary of this theorem, changing the condition
that the function f is continuous in the second variable to the hypothesis of continuity from the right of the function γ
and γ(0) = 0.
Corollary 4.5.
Let a, b ∈ R, a < b, p > 1 a real number and k ≥ 2 a positive integer. If the composition operator F , generated by f,
acts from the space RV(p,k) [a, b] into itself and satisfies the following inequality:
kF (u) − F (v)kR(p,k) ≤ γ ku − vkR(p,k) ,
u, v ∈ RV(p,k) [a, b],
for some right continuous function γ : [0, ∞) → [0, ∞) such that γ(0) = 0, then there exist functions α, β ∈ RV(p,k) [a, b]
satisfying (8).
In a similar way as [23, Theorem 2] we get the following
Corollary 4.6.
Let a, b ∈ R, a < b, p > 1 a real number and k ≥ 2 a positive integer. If the composition operator F , generated by f,
acts from the space RV(p,k) [a, b] into itself and is uniformly continuous with respect to the norm k · kR(p,k) , then there exist
functions α, β ∈ RV(p,k) [a, b] satisfying (8).
In addition, if we take γ(t) = ct, for some c > 0, we get [33, Theorem 1].
Acknowledgements
This research has been partly supported by the Central Bank of Venezuela. We also want to give thanks to the library
staff of B.C.V. for compiling the references.
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