About amenability of subgroups of the group of diffeomorphisms of the interval

Averaging linear functional on the space continuous functions of the group of diffeomorphisms of interval is found. Amenability of several discrete subgroups of the group of diffeomorphisms ${\text{Diff}}^3([0,1])$ of interval is prove. In particular, a solution of the problem of amenability of the Thompson’s group $F$ is given.