Restate what was said in my comments. Using a computer to view my answer will be better.

**Theorem.** Let $A>0,\alpha>0$ and $f(x)$ be the function satisfying the following condition
    	$$f(x)=x-Ax^{1+\alpha}+Bx^{1+2\alpha}+O\left(x^{1+3\alpha}\right),\ x\to 0.$$
    	If $a_{n+1}=f(a_n)$ and the sequence $(a_n)_{n\geqslant 1}$ is monotone decreasing to zero, then
    	$$a_n=\frac{1}{C^{\beta}n^{\beta}}+\left(B-\frac{(\alpha+1)A^2}{2}\right)\frac{\log n}{C^{2+\beta}n^{1+\beta}}+O\left(n^{-1-\beta}\right)$$
    	holds as $n\to \infty$, where $C=A\alpha,\beta=\frac{1}{\alpha}$.