On disjoint matchings in cubic graphs

For $i=2,3$ and a cubic graph $G$ let $\nu{i}(G)$ denote the maximum number of edges that can be covered by $i$ matchings. We show that $\nu{2}(G)\geq {4/5}| V(G)| $ and $\nu{3}(G)\geq {7/6}| V(G)| $. Moreover, it turns out that $\nu{2}(G)\leq \frac{|V(G)|+2\nu_{3}(G)}{4}$.