Prove whether or not there exists a positive integer $N$ such that for all $\epsilon\gt0,$ $\vert f^{(n)}(x_k)-f^{(n)}(x_1)\vert<\epsilon$ if $n\gt N$. Function $f$ is defined as follows.
For positive integer $k$, define $\pmb f:\mathbb R^k\rightarrow\mathbb R^k ,\pmb f(x_1,x_2,...,x_k)=(f(x_1),f(x_2),...,f(x_k))$, where
$x_1\le x_2\le...\le x_k$ and $$f(x_j)=(0.2+0.1\ln x_j)x_j+\frac{1}{k}\sum_{i\ne j}(0.2+0.1\ln x_i)x_i$$
for each $j\in\{1,2,...,k\}$.
Denote $f^{(2)}(x_i)=f(f(x_i))$ and $f^{(n)}(x_i)=f(f^{(n-1)}(x_i))$ for positive integer $n$.
This problem has bothered me for a long time. Is there a way to solve the problem? Or do I need additional conditions? What ideas do I need to solve this problem?