Abstract
AbstractThe following MW-problem was posed independently by Janusz Matkowski and Jacek Wesołowski in different forms in 1985 and 2009, respectively: Are there increasing and continuous functions $$\varphi :[0,1]\rightarrow [0,1]$$
φ
:
[
0
,
1
]
→
[
0
,
1
]
, distinct from the identity on [0, 1], such that $$\varphi (0)=0$$
φ
(
0
)
=
0
, $$\varphi (1)=1$$
φ
(
1
)
=
1
and $$\varphi (x)=\varphi (\frac{x}{2})+\varphi (\frac{x+1}{2})-\varphi (\frac{1}{2})$$
φ
(
x
)
=
φ
(
x
2
)
+
φ
(
x
+
1
2
)
-
φ
(
1
2
)
for every $$x\in [0,1]$$
x
∈
[
0
,
1
]
? By now, it is known that each of the de Rham functions $$R_p$$
R
p
, where $$p\in (0,1)$$
p
∈
(
0
,
1
)
, is a solution of the MW-problem, and for any Borel probability measure $$\mu $$
μ
concentrated on (0, 1) the formula $$\phi _\mu (x)=\int _{(0,1)}R_p(x)\, d\mu (p)$$
ϕ
μ
(
x
)
=
∫
(
0
,
1
)
R
p
(
x
)
d
μ
(
p
)
defines a solution $$\phi _\mu :[0,1]\rightarrow [0,1]$$
ϕ
μ
:
[
0
,
1
]
→
[
0
,
1
]
of this problem as well. In this paper, we give a new family of solutions of the MW-problem consisting of Cantor-type functions. We also prove that there are strictly increasing solutions of the MW-problem that are not of the above integral form with any Borel probability measure $$\mu $$
μ
.
Publisher
Springer Science and Business Media LLC
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