Even cycles in dense graphs

We will show that for $\alpha >0$ there is $n_0$ such that if $G$ is a graph on $n\ge n_0$ vertices such that $\alpha n<\delta (G)<(n-1)\mathrm{/}2$, then for every $n_1+n_2+\cdots +n_l=\delta (G)$, $G$ contains a disjoint union of $C_{2n_1},C_{2n_2},\dots ,C_{2n_l}$ unless $G$ has a very specific structure.