Convergence of iterates of pre-mean-type mappings
Pre-mean in an interval I, being defined as a function M:I2 → I such that M(x,x) = x for x ∈ I,is an essential generalization of the mean. If M and N are pre-means, a map (M,N):I2 → I2 is called pre-mean-type mapping. The problem of converg...
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doaj-fb96bbf07d2649d987a69ee71653cc8e2021-07-15T14:10:15ZengEDP SciencesESAIM: Proceedings and Surveys2267-30592014-11-014619621210.1051/proc/201446016proc144616Convergence of iterates of pre-mean-type mappingsMatkowski Janusz0Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4aPre-mean in an interval I, being defined as a function M:I2 → I such that M(x,x) = x for x ∈ I,is an essential generalization of the mean. If M and N are pre-means, a map (M,N):I2 → I2 is called pre-mean-type mapping. The problem of convergence of iterates of pre-mean type mappings of the form \hbox{$% \left( B_{s,t}^{[p,q]},B_{1-s,1-t}^{[-p,-q]}\right) $} B s,t [ p,q ] ( , B 1 − s, 1 − t [ − p, − q ] ) with s,t ∈ (0,1);p,q ∈ R, p ≠ q, where \hbox{$B_{s,t}^{[p,q]}:\left( 0,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) ,$} B s,t [ p,q ] : ( 0 , ∞ ) 2 → ( 0 , ∞ ) , \begin{equation*} B_{s,t}^{[p,q]}=\left( \frac{sx^{p}+\left( 1-s\right) y^{p}}{tx^{q}+\left( 1-t\right) y^{q}}\right) ^{1/\left( p-q\right) },\text{ \ \ \ \ \ }x,y>0, \end{equation*} B s,t [ p,q ] = s x p + ( 1 − s ) y p t x q + ( 1 − t ) y q 1 / p − q , x,y > 0 , is considered. It is proved, in particular, that for p = 2r, q = r and s ≤ t< 2s, the sequence of iterates at the point (x,y) converges to \hbox{$\left( \sqrt{xy},\sqrt{xy}\right) $} ( xy , xy ) . For some s and t the iterates behave in ”chaotic” way. An application in solving a functional equation is presented.http://dx.doi.org/10.1051/proc/201446016meanpre-meanpre-mean-type mappinginvariant meaninvariant pre-meaniterateconvergencefunctional equation |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Matkowski Janusz |
spellingShingle |
Matkowski Janusz Convergence of iterates of pre-mean-type mappings ESAIM: Proceedings and Surveys mean pre-mean pre-mean-type mapping invariant mean invariant pre-mean iterate convergence functional equation |
author_facet |
Matkowski Janusz |
author_sort |
Matkowski Janusz |
title |
Convergence of iterates of pre-mean-type
mappings |
title_short |
Convergence of iterates of pre-mean-type
mappings |
title_full |
Convergence of iterates of pre-mean-type
mappings |
title_fullStr |
Convergence of iterates of pre-mean-type
mappings |
title_full_unstemmed |
Convergence of iterates of pre-mean-type
mappings |
title_sort |
convergence of iterates of pre-mean-type
mappings |
publisher |
EDP Sciences |
series |
ESAIM: Proceedings and Surveys |
issn |
2267-3059 |
publishDate |
2014-11-01 |
topic |
mean pre-mean pre-mean-type mapping invariant mean invariant pre-mean iterate convergence functional equation |
url |
http://dx.doi.org/10.1051/proc/201446016 |
work_keys_str_mv |
AT matkowskijanusz convergenceofiteratesofpremeantypemappings |
_version_ |
1721300308992196608 |
description |
Pre-mean in an interval I, being defined as a function M:I2 →
I such that M(x,x) =
x for x ∈ I,is an essential
generalization of the mean. If M and N are pre-means, a map (M,N):I2
→ I2 is called pre-mean-type mapping. The
problem of convergence of iterates of pre-mean type mappings of the form
\hbox{$% \left( B_{s,t}^{[p,q]},B_{1-s,1-t}^{[-p,-q]}\right) $}
B
s,t
[
p,q
]
(
,
B
1
−
s,
1
−
t
[
−
p,
−
q
]
)
with s,t ∈ (0,1);p,q ∈ R, p ≠ q,
where
\hbox{$B_{s,t}^{[p,q]}:\left( 0,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) ,$}
B
s,t
[
p,q
]
:
(
0
,
∞
)
2
→
(
0
,
∞
)
,
\begin{equation*} B_{s,t}^{[p,q]}=\left( \frac{sx^{p}+\left( 1-s\right) y^{p}}{tx^{q}+\left( 1-t\right) y^{q}}\right) ^{1/\left( p-q\right) },\text{ \ \ \ \ \ }x,y>0, \end{equation*}
B
s,t
[
p,q
]
=
s
x
p
+
(
1
−
s
)
y
p
t
x
q
+
(
1
−
t
)
y
q
1
/
p
−
q
,
x,y
>
0
,
is considered. It is proved, in particular, that for
p =
2r, q = r and s ≤ t<
2s, the sequence of iterates at the point
(x,y)
converges to
\hbox{$\left( \sqrt{xy},\sqrt{xy}\right) $}
(
xy
,
xy
)
. For some s and t the iterates behave in ”chaotic” way. An
application in solving a functional equation is presented. |