Convergence of iterates of pre-mean-type mappings

Abstract

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.

Keywords

• mean
• pre-mean
• pre-mean-type mapping
• invariant mean
• invariant pre-mean
• iterate
• convergence
• functional equation