Convergence of iterates of pre-mean-type mappings

Janusz Matkowski¹

¹ Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, PL-65-516 Zielona Góra, Poland,

ESAIM. Proceedings, Tome 46 (2014), pp. 196-212

Résumé

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 with s,t ∈ (0,1);p,q ∈ R, p ≠ q, where \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 . For some s and t the iterates behave in ”chaotic” way. An application in solving a functional equation is presented.

Détail

DOI : 10.1051/proc/201446016